Карнап Рудольф. Подготовка. Черновик. 30

Черновик. Приложение.
К анализу. Корреляции.
_______
_______
Полный текст " Логический синтаксис языка "

Intemahonal Библиотека психологии
Философия и сценический метод


ЛОГИЧЕСКИЙ СИНТАКС
ЯЗЫКА



Международная библиотека психологии
Философия и научный метод


ОБЩИЙ РЕДАКТОР. CK OGDEN.
PHIlOilOrHlCU. Стити

Мисси сс Разума
КОНТУКТ И СОН *

Tractatus LociPO-PKoosormcn.
ПСИХОЛОГИЧЕСКИЕ TVFfS *.

SctiNTinc THoiwr *. ...

MLwnvo Meandjo
INDIVIDDAI. PSTCHOLOet
SrxcvLATIONS {fycftt ty Jae «t Fptlnui
RsTCHOiocT Ria-aoning *.

Пкилософия "Как будто"

NATVRI из lNITllJCD.Ct
TnXTATHT И CljdR \ -OTANCE
Рост Млнд
МИНТАииТФ АРИС
Певхолог Религиозной МиссныМ
PHiiosorirr или Mwtc
PssaiOLOov или Мимический Prooict.
ПРИНАРЛЕЙНАЯ КРИТИЧНОСТЬ

Mefafhvsical Фонд Сакос
Мысль а. Нет.

Физические данные и CK ^ AACTER *

PsvoioLOcy OF Emotion
ПБОРЛЕМЫ ПРОБЛЕМЫ
История Кутесиализма
ЛИЧНЫЙ м *

Учебный Пиитмоиокт

Ланклаги и Младенец

Секс и размышления в Sa \ acs SoatTr *

COUFAILSTtVB PKILQMFHT

Социальный Лм в Животном \ Сорло

Как Aniauie найти через Wat AMUT

Социальные насекомые

Теоретический Biolocv

PossifliLfTy


Фундаменты или геометрия и индукция
Закон Фекунга ....
Ментальное DEVTxoFScefT или Ребенок

Eidetic Imacrt

Концентрический Siethoo. ,

Основы MATHEUATits
ФИЛОСОФТ OPTKE Unconsoou *.

РЕЗУЛЬТАТ GRIXK pHLUJSOFHT. ,

Paytholocy или Drawtncs Ходрека
Изобретения и Unco.nsctoue.

Theoey Законодательства.

ОБЩАЯ ЖИЗНЬ МОНКЕТОВ
Dhelofment ofthi шестилетних Имфулей
КОНСИ ЛЬВОН ТЛТЕС в Дюнкескте.
Науки о человеке в марине.
Этический Роттти ....


АОК. {.Vsristnf CoC <tf. CeminSet)

, ^ GE Мур, Lar.D.

Hr Карин Штфен
FY WH IL Riters. FJLS.

, , {y L. \\ nrCCsYTETN

Fv a G JL-VC.

Fv CD Broad, LittJ).

Fy CK Огден грустно. А. Ричардс
F y Альфред Адур
> y TE HuLMt
Фю Эухенио Риньяно
Г. Вайкивцер
F y U L. Thurstone
, , , ч Р. Тишнер

, h 'К. KCETRA
, \\ KOKUR
Фу 1. Х. Лефра
, Ft WP ^ FRS
К. Ретисл
Ф. И. Ричардс
F y ;. A. BlrTYjniXl.


»B ^ MORTO.N HuNCS

Fv F. A Lange
F y RC Gciwon, MJ}
от OUELS Fox
, , FyJ. PUGTT

0 0 MAUNDNA-An. O Se.
Fy P MAUQN-OlRin.
, ALSTSBES
, Fv ;. ветвь

FyW MORrO.N WmxuR
ly} TCMVOxCU.
, Fv Sforr BUQIANAN
Fyll. Б. ДОООСЛДВСКИЙ
, fyj. F.MaRSTT
> yK.C.fiaA0
Fv Liang Chi> Чао
, У. М. Марстон
F y BlRTRA.NO RlFKCLL, FJLS
FvR.C.LnDcs
, Fv C. Murfht
- JiT.! Э. Донахт
F y Chrlstlne Laod-Fra-nrun

Fyj. HW CODGER

, Фай Отто Ранр
, Fv PS FLORENCE
, Fv ;, P- Гамильтон

, Fv FRA.NK LORIAIER

Fy сэр Ричард Пейджет
. Жан Никод

, , Fy F. PavlhaN

, , ч К. BCMLER

, FY ER Jaensch

r FL LAIG-N'RL'LAAASTTNt
, Ly Fp Ramsey
F y E. AON Hartmann
, , Ф. Л. 2нин

, , Fy Helca Eng


(y k. U MONFT.KYRLR
, M \ VA WiLLEMSt
Fy LA KtRCFATRICR
Fy EA WeSTIRUarck


vliKruFi inou iFsl eVut взял на себя iFf i


ИИТААР itKbiaJ la tFr Stntt.



ЛОГИЧЕСКИЙ СИНТАКС
ЯЗЫКА


По

РУДОЛЬФ КАРНАП

PtQFsssoK OP Философия в ТБ
Uniyi litTY Of Chicago


ЛОНДОН

МАРШРУТ & KEGAN PAUL LTD

BROADWAY HOUSE 6S-74 CARTER LANE, EC4



Pint pubhthei m England igyj
Sftvnd imprraion 1949
Третий день 1951
Четвертый iwi preta cm ipSd
Пятый день рождения 1959


Переведено

AMETHE SMEATON (Couktos \ '0N ZcprtttN)


UruNTEO BV UTUOCEAFIFY IK GREAT BEITAtK
ЯЮТОЛБ И ЭОКС. tlVITEO, NOEVTCM



СОДЕРЖАНИЕ


ЛИЦО


Предисловие к английскому изданию XI

Предисловие Сюй

iNTRODUCnON

§ I Что такое л / гический синтаксис? я

§ 2 Языки как исчисление 4

ЧАСТЬ I ОПРЕДЕЛЕННЫЙ ЯЗЫК I
Правила формирования языка I
§3 Предикаты и функторы n

§4 Синтаксические готические символы 15

§ 5 Соединительные символы 18

§ 6 Универсальные и экзистенциальные предложения 20

§ 7 Оператор К 22

§ 8 Определения 23

§9 Предложения и числовые выражения 25

B Правила трансформации для языка I
§ 10 Общие замечания относительно трансформации

Правила 27

§N Примитивные предложения языка I 29

§12 Правила вывода языка I 32

§ 13 Понятия и доказательства в языке I 33

§ 14 Правила следствия для языка I 37

C Замечания об определенной форме письма
§ 15 Определенный и неопределенный 44

^ 16 На интуиционизме 46

§I6a личность 49

§17 Основа толерантности m Синтаксис 51

ЧАСТЬ II ФОРМАЛЬНОЕ СТРОИТЕЛЬСТВО
СИНТАКСИС ЯЗЫКА I
§ Синтаксис I может быть Formulateo мл 53

§ 19 Антитизация синтаксиса 54

§ 20. Роды] Термины 38



VI CONLLN'L'S

§21. Правила Fonnation: я. Числовые выражения и


Предложения 62

§ 22. Правила формирования: 2. Определения ... 66

§23. Правила трансформации 73

§ 24. Описательный синтаксис ...... 76

§ 25. Арифметический, аксиоматический и физический синтаксис 78


ЧАСТЬ III. НЕОПРЕДЕЛЕННЫЙ ЯЗЫК II


А. Правила FoitMATiott для языка II
§ 26. Символический аппарат языка II. , 83

§27. Классификация типов 84

§ 28. Правила формирования числовых выражений и

Предложения . 87

§ 29. Правила формирования определений .... 88

D. Правила трансформации языка U
§ 30. Примитивные предложения языка I II. 90

§31. Правила вывода языка II. 94

§32. Выводы и доказательства в языке II. 95

Статья 33. Сравнение примитивных предложений и правил

II тех же систем ... 96

C. Правила следствия для языка II
§340. Неполные и полные критерии действительности, 98

§346. Сокращение 102

§34c. Оценка 106

1 34 д. Определение «Аналитика во II» и «Противоречивость»

нет

§34e. Об аналитических и противоречивых приговорах

Язык II 115

I34 /. Следствие в языке II .... Нью-Йорк

§34g. Логический контент 120

§34 ^. Принципы индукции и отбора

Аналитика 121

§341. Язык II не противоречит. , , 124



СОДЕРЖАНИЕ


Vll

§35 Синтаксические предложения, которые ссылаются на себя 129


§36. Неразрешимые предложения 131

D Дальнейшее развитие языка II
§37 Предикаты как символы класса 134

§ 38. Устранение касс 136.

§ 38 fl О предположениях о существовании m Logic 140

§38 ^ Кардинальные числа 142

§38c Описания 144

§ 39 Действительные числа 147

§ 40. Язык физики 149


ЧАСТЬ IV ОБЩИЙ СИНТАКС
Объект I ^ Guace и Синтаксис-Язык


§41 О синтаксических обозначениях 153

§ 42. О необходимости различения

Выражение и его обозначение 156

§ 43 О допустимости неопределенных сроков 160

§44. О допустимости нецензурных выражений 262

§45 Неопределенные термины в синтаксисе z6 ^

B Синтаксис op ant Language

(а) Общие соображения

§46 Правила формирования 167

§47 Правила трансформации, d Условия 170

§48 c Условия 172

§49 Содержание 175

§50. Логические и описательные выражения, Sub Lan-
Guage 177

§51 Логико-физические правила 180

§52 L-термины, «аналитические» и «противоречивые» 182

(б) переменные

§ S 3 Системы уровней, предикатов и функторов r86

§54. Подстановочные переменные и константы 189

§55. Универсальные и экзистенциальные операторы 196

§ 56 Диапазон 199

§ 57 Приговор Судебных. 200



СОДЕРЖАНИЕ


TAGS


VIII


(«) Арифметика; Непротиворечивость; Антиномии

§ 58. Арифметика 205

§ 59. Непротиворечивость и полнота

язык . 207

§600. Антиномии 211

§6oA. Концепции «True * и« False ». . 214
§6oc. Синтаксические антиномии. , , 0,217
§6od. Каждая арифметика неполноценна ... 220

(г) письменный и устный перевод

§ 61. Перевод с одного языка на другой. 222

§ 62. Толкование языка , 227

(<) Экстенсиональность

§63. Квазисинтаксические предложения. • • 233

J | b4. Две интерпретации квазисинтаксического

Предложения 237

§ 65. Расширение в отношении частичных приговоров 240

§ 66. Расширение в отношении частичных выражений 243

§ 67. Тезис экстенсиональности .... 245

Статья 68. Интенсивные предложения автономного режима

речи 247

§ 69. Интенсивные предложения логики модальностей 250

§ 70. Квази-синтаксические и синтаксические методы

в логике модальностей 256

§71. Нужна ли интенсиональная логика? , . 257

(/) Реляционная теория и аксиоматика

§710. Реляционная теория 260

§716. Синтаксические термины реляционной теории. , 262

§7if. Изоморфизм 264

§7rd. Неперечислимые кардинальные числа. 267

§7if. Аксиоматический метод 271



СОДЕРЖАНИЕ

PACS

ЧАСТЬ V ФИЛОСОФИЯ И СИНТАКС

А. О форме приговоров, принадлежащих
Логика науки


§ 72. Философия заменена логикой науки 277

§73 TheLogic of Science, Синтаксис Langiisg *

Saence 2S1

§ 74. Приговоры псевдообъекта * 84

§ 75 Приговоры о значении * 88

§ 76 универсальных слов * 9 *

§ 77 Универсальные слова в материальной моде Spf * cb 297

§ 78 Путаница в философии, вызванной матфналом

Режим речи * 98

§ 79 Философские предложения в материале и т джи *

Формальный режим речи IPS

§ 80 Опасности материальной формы речи ^ 308

§ 81. Допустимость материальной моды Sp ^ ch 312

B Логика операций - это Stntax
§82 Физический язык 3 ^ 5

§ 83. Так называемые основы наук 3 * 2

§ 84. Проблема основания математики

§ 85 Синтаксические предложения в литературе

Speaal Sciences 328

§86. Логика безопасности - это синтаксис 331

Бибуография и указатель авторов 334

Указатель предметов 347



ПРЕДИСЛОВИЕ К АНГЛИЙСКОМУ ИЗДАНИЮ


Настоящее английское издание содержит несколько разделов, которые не являются
нашел м немецкую онгина! Это §§ i6iz, 34a- ^, 380- ^,
6o «-rf, 7i <je. Эти двадцать две секции были включены в состав
рукопись немецкого оригинала, когда он был отправлен для публикации
(в декабре 1933 г.), но из-за отсутствия
Пространство Tbe содержание § 340-1 было, в несколько другой формуле
, опубликованный в немецком издании Em Gulttgskntmim fur
die Satse der kleusuchen Mathemabk, и содержание §§ 6 oa-d
и 71 год до н.э.
matik § 60 ongmal здесь опущен, smce это было только
сокращенный заменитель § 6оа- <f

В Bibhography некоторые менее важные публикации были
удалены, а другие, в основном, за последние несколько лет, были добавлены

Несколько небольших дополнений и исправлений
более важные из них происходят в следующих пунктах § 8,
регрессивный дефимуон, § la, RI 2 (см. сноску), § 14, добавлены доказательства
к теоремам 3 и 7, § 21, D 29, § aa, два предложения в D 64 (см.
сноска), D 83, 29 $, сноска! 30, PSII 4 (см. Сноску к § 12),
PSII 19, условие добавлено, § 48 (сноска в разделе), § 51, определение
* L-следствие ', § 56 (см. Сноску), TKcorems 8 и 9 изъяты,
§57, исправлены теоремы 2 и 3, добавлен последний абзац,
§62, объяснение §§65 и 66, определения

'extensional' restnctcdto замкнутые частичные выражения и теорема
65 8a добавлен, §67, конец второго абзаца Величество
эти исправления и ряд других были предложены
предоставлен доктором Тарским, другими - JCC McKmseyandW V
Куайн, всем, кому я очень благодарен за их
полезная критика

Проблема рендеринга немецкой терминологии была естественной
самый сложный, в некоторых случаях нет английского слова м
существование, которое точно соответствовало онгмалу
очевидный эквивалент «быть tmavailAle из-за его
В некоторых других системах иногда необходимо было
для наших целей норды, которые ранее не несли
техническая значимость, иногда даже совершенно новые



XII ПРЕДИСЛОВИЕ К ИЗДАНИЮ ENCLISII

с первого взгляда некоторые из них кажутся непринужденными или странными, я могу только *
попросите читателя иметь в виду особые трудности,
и заверить его, что ни один термин не был выбран без самого тщательного
рассмотрение и убеждение, что ч оправдал бы себя в использовании.

Для облегчения обсуждения и ссылки, немецкий символический
аббревиатуры были сохранены во всех строго формализованных
части книги. Английские эквиваленты были заменены
только там, где они встречаются в неформальном тексте
аббревиатуры, которые не являются символическими (например, «TN» для
«термин-номер» вместо немецкого «GZ»), или как случайный
символы введены просто в целях иллюстрации (например, «фа»
для «отца» вместо немецкого «Va»). ^ Vhe ^ eve ^ немецкий
аббревиатура была использована впервые, полное немецкое слово
был вставлен в скобки; и в случае терминов
в соответствии с формальными определениями, полный ключ к символизации
приводится в сноске в начале соответствующих разделов.

Я хочу выразить свою благодарность графине фон Цеппелин
для выполнения трудной задачи перевода этой книги,
в дополнение к д-ру В. В. Куайну, есть несколько ценных предложений относительно
терминологии, и д-р EC Грэм, д-р 0. Хелмер и доктор Э.
Нагель за помощь в проверке доказательств.

RC

Кембридж, Мэн, май 1936



ПРЕДИСЛОВИЕ


В течение почти столетия математики и логики были
Трудно сделать логику точной наукой. В определенной степени
их усилия увенчались успехом, поскольку
наука о логистике научила людей манипулировать
прецизионные символы и формулы, которые похожи по своей природе
для тех, кто использовал UI математику Но книга по HGIC должна содержать,
в дополнение к формулам, пояснительный контекст, который, с
помощь слов обычного языка, объясняет формулы
и отношения между ними. и этот контекст часто оставляет много
желать лучшего в отношении ясности и точности
годами логики представляли совершенно разные тенденции мышления
разработали все более и более точку зрения, что в этом контексте
текст содержит существенную часть логики, и что важный
вещь 13, чтобы разработать точный метод для построения этих
предложения о предложениях Целью настоящей работы является
дать систематическое изложение такого метода, а именно, из
метод «логического синтаксиса». (Для получения более подробной информации см. Введение
Стр. X и 2)

В нашем «Венском круге», а также в родственных группах (м. Польша,
Франция, Англия, США и, в частности, даже
многие) убежденность выросла и неуклонно растет, что
метафизика не может заставить себя обладать
Эта часть работы философов, которая может быть
быть научным по своей природе - исключая эмпирические вопросы, которые
можно отнести к empinca! наука - состоит из логического анализа
Целью логического синтаксиса является создание системы понятий,
язык, с помощью которого результаты логического анализа
быть точно сформулированным Pktbsopfy u к bt заменен логикой
наука - то есть логический анализ концепций и
предложения науки, лиса логика науки не что иное, как
логический синтаксис языка науки Это вывод
что мы руководствуемся соображениями в последней главе этой книги

Сама книга делает попытку продвинуться в форме
точный синтаксический метод, необходимые инструменты для разработки
проблемы логики науки Это делается в первую очередь
формулировка синтаксиса двух особенно важных типов
языка, который мы будем называть, соответственно, «Язык I» и



rORCVVORD


XIV

«Язык II». Язык I прост по форме и охватывает узкий
область понятий. Язык II - это nchcr в способах выражения; в
Это все предложения как классической математики, так и классической
физика может быть сформулирована. На обоих языках расследование
не будет ограничиваться математико-логической частью языка -
как это обычно происходит в логистике - но будет существенно обеспокоен
также с синтетическими, empmeal предложениями. Последний, так называемый
«настоящие» предложения составляют ядро ;;науки; математика
логические предложения являются аналитическими, без реального содержания, и просто
формальные вспомогательные средства.

На языке I в качестве примера будет показано, в каком
следует, как синтаксис языка может быть сформулирован в
этот язык сам (часть II). Обычный страх, который тем самым
традиции - так называемые «эпистемологические» или «языковые» анти
номинации - должны возникать, это не оправдано.

Трактовка синтаксиса языков I и II будет следующей
ниже общего синтаксиса, применимого к любому языку
что бы то ни было (часть IV); и, хотя попытка очень далека от
достижение желаемой цели, но задача является одной из фундаментальных
portance. Диапазон возможных языковых форм и, следовательно,
В сущности, из всех возможных логических систем несопоставимо
больше, чем очень узкий круг, к которому более ранние исследования
в модеме логика была ограничена. До настоящего времени
было лишь очень небольшое отклонение, несколько пунктов здесь и там,
от формы языка, разработанной Расселом, который уже
стать классическим. Например, некоторые формы предложения (такие как
неограниченные экзистенциальные предложения) и правила вывода (такие как
как Закон исключенного среднего), были устранены
определенные авторы. С другой стороны, ряд расширений имеют
было предпринято несколько интересных, многозначных исчислений
логичные к двухзначному исчислению предложений,
и, наконец, привело к логике вероятности. Точно так же
так называемые интенциональные предложения были введены и, с их
помощь логика модальности разработана. Дело в том, что никаких попыток нет
было сделано, чтобы рискнуть еще дальше от классических форм
случается из-за широко распространенного мнения, что любые такие отклонения должны
быть оправданным - то есть, что новая языковая форма должна быть доказана
быть «правильным» и представлять собой верный перевод «
истинная логика ».

Чтобы устранить эту точку зрения, вместе с псевдо-проблемы



FOREnOSD XV

и утомительные споры, которые возникают в результате этого, является одним из
Главные задачи этой книги В ней, vteiv будет утверждаться, что
у нас есть полное уважение к формам
языка, что и формы построения предложений и
правила трансформации (лучше обычно обозначать как
«Постулаты» и «Правила inrentt») могут быть выбраны довольно
произвольно до сих пор, в constnictmg языке, процедура
обычно был первым, чтобы назначить встречу фундаментальному
математически-логические символы, а затем рассмотреть, какие предложения
и логические выводы считаются логически правильными в соответствии с
это среднее значение Smce выражено назначение среднего значения
м слов, и, как следствие, мексакт, никакого заключения
Таким образом, я могу плакать, а не быть точным и неоднозначным.
Соединение станет ясно только при приближении с
противоположное направление, пусть любые постулаты и любые правила вывода будут
выбранный произвольно, тогда этот выбор, каким бы он ни был,
Термин, что означает, что вы должны быть приписаны к основному логическому
Bj'mbols По этому методу также конфликт между расходящимися
взгляды на проблему основ математики
Исчезает для языка, его математический фон, может быть
построен в соответствии с предпочтениями любого из помптов зрения
представлен, так что вопрос об оправдании вообще не возникает, а только
вопрос о синтаксических последствиях для которых одно или другое
о выборе ведет, в том числе вопрос о непротиворечивости
Точка зрения, которую мы предложили - назовем ее
Принцип толерантности (см. Стр. 51) - относится не только к
Математика, но на все вопросы логики С точки зрения Tlus Pomt,
задача создания общего синтаксиса - иными словами,
определение тех синтаксических понятий, которые доступны
языки любой формы - очень важный
область общего синтаксиса, например, можно выбрать
определенная форма для языка науки в целом, а также для
что любой ветви saence, и точно указать характеристику
различия между ним и другими возможными языковыми формами
Первые попытки отогнать корабль логики с терра
Jrrma классических форм, безусловно, были смелыми, считаются
с исторической точки зрения Но они были затруднены
stnvmg после «правильности» Теперь, однако, это препятствие имеет
преодолен, и перед нами лежит безграничный океан
ограниченные возможности.



FORHWORO


XVI

В ряде мест в тексте делается ссылка на наиболее
важная литература по этому вопросу. Полный список не имеет, как
когда-либо пытались. Дальнейшая библиографическая информация может
легко получить из указанных произведений. Самый импортный
Ссылки на муравьи приведены на следующих страницах: стр. 96 и далее, ком.
сопоставление нашего языка 11 с другими логическими системами; с. 136!?.,
символика классов; с. 158 фт., о синтаксическом
ЦИИ; с. 253 ф., о логике модальностей; С. 280 ф. и 320 ф.
по логике науки.

Для развития идей в этой книге я многим обязан
стимулирование я получил от тщеславных писаний, писем и
стихи по логическим задачам. Здесь следует упомянуть
самые важные имена. Прежде всего, я обязан
сочинения и лекции Фреге. Благодаря ему мое внимание было
обращается к стандартной работе по логистике, а именно, Принципы
Малхматика Уайтхеда и Рассела. Точка \ * новизна
формальная теория языка (в нашей терминологии это называется sj'ntax)
Твас впервые разработал для математики Гильберта м его «мета *
математика », к которой обращаются польские логики, особенно А.Юдуки
wicz, Lesniewskt, Lukasiewicz и Tarski добавили «мета-
логика». Для этой теории Гддел создал свой плодотворный метод
«Антметизация». С точки зрения и метода синтакс я
в частности, получил ценные предложения от разговоров
Монтаны Тарского и Гсдела. У меня есть много, за что поблагодарить
Витгенштейн в моих размышлениях об отношениях между
синтаксис и логика науки; для расхождений в наших точках
зрения см. стр. 282 и след. (Кстати, dpropos из сделанных замечаний
- особенно в §17 и §67 - в отличие от Витгенштейна
прежняя догматическая точка зрения. Профессор Шлик теперь сообщает мне
что в течение некоторого времени, в> vritmgs, которые еще не опубликованы, Wittgen-
Штейн согласился, что правила языка могут быть выбраны с
полная свобода.) Опять же, я многому научился из писаний
авторов, с которыми я не совсем согласен; эти,
во-первых, Вейль, Брауэр и Льюис. Наконец, я хочу
выразить благодарность профессору Беманну и доктору Геделю за
прочитав рукопись этой книги в более раннем варианте (1932).
и за то, что внесли многочисленные ценные предложения в
улучшение. _ п


Прага, май 1934 г.



ВСТУПЛЕНИЕ

§ 1. Что такое логический синтаксис?

Под логическим синтаксисом языка мы подразумеваем формальное
теория об игуислических формах этого языка - систематическая
изложение формальных правил, которые регулируют это вместе с
развитие последствий, вытекающих из этих правил
Теорию, правило, определение или тому подобное следует называть формальным
когда в нем не упоминается ни о происхождении
символы (например, слова) или в смысле выражений
(например, предложения), но просто и исключительно для видов и порядка
из символов, из которых построены eicpresstons
Распространено мнение, что синтаксис и логика, несмотря на некоторые
точки соприкосновения между ними, по сути, являются
совсем другой тип Синтаксис языка должен лежать
вниз правила, в соответствии с которыми hnguistic структуры (например,
предложения) должны быть построены из элементов (таких как слова или
части слов) Основная задача логики, с другой стороны, хз
должен формулировать правила, в соответствии с которыми
может быть выведено из других суждений, другими словами, согласно
на какие выводы можно сделать из предпосылок

Но развитие логики за последние десять лет показало,
ясно, что это может быть изучено только с любой степенью точности
когда оно основано, а не на суждениях (мыслях или содержании
мысли) а скорее о языковых выражениях, из которых предложения
являются наиболее важными, потому что только для них можно заложить
вниз четко определенные правила И на самом деле, на практике, каждый
С тех пор как Anstotle, устанавливающий правила, имел дело в основном
с предложениями Но даже те современные логики, которые согласны с
нам по нашему мнению, что логика искажается с предложениями, еще для
большая часть убеждена, что логика в равной степени связана с
отношения значений между предложениями Они считают, что м
в отличие от правил синтаксиса, правила логики неформальные
На следующих страницах, которые я противостоит этой точке зрения, мнение
эта логика также касается формального обращения с
будут представлены и развиты. Мы увидим, что



ВСТУПЛЕНИЕ


логические характеристики предложений (например,% vheiher предложение
является аналитическим, нетрадиционным или противоречивым; является ли это экзистенциальным
предложение или нет; и так далее) и логические отношения между ними
(например, противоречат ли предложения таро друг другу или являются
совместимы друг с другом; является ли человек логически выводимым
от другого или нет; и т. д.) зависят исключительно от
синтаксическая структура предложений. Таким образом, логика будет
прийти часть синтаксиса, подтолкнул, что последний задуман в
достаточно широкий смысл и сформулированный с точностью. Дифференциал
разница между sj'ntactical правилами в более узком смысле и
Логические правила дедукции это только разница между образованием
правила и правила преобразования, оба из которых полностью
Формулируется в sjTitactical условиях. Таким образом, мы оправданы в
обозначая как «логический синтаксис» sj-стебель, который составляет правила
формирование и преобразование.

Вследствие несущественного и логически несовершенного
структура рутовальных слов-языков (таких как немецкий или
Латыни), утверждение их формальных правил формирования и транс
формирование будет $ 0 сложным, что вряд ли он
выполнимо на практике. И такая же трудность возникнет в
совокупность искусственных языков слов (таких как эсперанто); даже для
хотя они избегают определенных логических недостатков, которые характеризуют
естественные языковые языки, они должны, по необходимости, быть еще очень
сложный с логической точки зрения в связи с тем, что
они являются разговорными языками и, следовательно, все еще зависят от
естественные языки.

На данный момент мы оставим в стороне вопрос формального
недостатки словесных языков, и, учитывая
примеры, продолжайте убеждать себя, что правила формирования
и трансформация имеют одинаковую природу, и что оба позволяют быть
формально задержан. Например, учитывая соответствующее правило, это
Можно доказать, что слово-серия «Питотс ларулизует клатически» isa
предложение, при условии, что "Пироис *", как известно, является существенным
(во множественном числе), «karuli2e» глагол (в третьем лице множественного числа), и
"Эластично". Наречие; AU из которых. QfoQUcse.inawtU-constcucwd
язык - как, например. На эсперанто - можно было собрать
форма слов одна. Смысл слов довольно
несущественно для цели и не должно быть известно. Далее, учитывая
По соответствующему правилу предложение «А карулирес эбетически» можно



3


§I ^ VHAT IS ЛОГИЧЕСКИЙ синтаксис '

выводится из первоначального предложения и предложения «А является
Пирот »- агам при условии, что тот тип, к которому человек
слова принадлежат известно здесь также, ни значение
ни слова, ни смысл этих трех предложений не должны быть известны

К недостаткам слов-языков, логический
синтаксис языка такого рода не будет разработан, но, в
Вместо этого мы рассмотрим синтаксис двух искусственно построенных
Symbohc языки (то есть такие языки, которые используют
формальные символы вместо слов)
все модемные логические исследования, этот метод используется, только для
Ма Symbohc язык доказал, что можно достичь точного
формулировка и доказательство
символический язык такого рода можно будет сложить
система правил одновременно простая и жесткая - которая сама по себе позволит
нам ясно показать характеристики и ассортимент оборудования
логического синтаксиса

Предложения, определения и правила синтаксиса языка
связаны с формами этого языка, но теперь, как
Являются ли эти предложения, правила и правила, чтобы они соответствовали
прямо выражен? Является ли своего рода супер-язык, необходимый для
цель? И, агауи, третий язык, чтобы объяснить symtax ;;этого
супер-язык и так до бесконечности? Или можно сформулировать
синтаксис языка внутри самого этого языка? Очевидное
в последнем случае возникнет страх в связи с определенным рефлексивным
определения, противоречия природы, похоже, похожи на те,
которая знакома обеим теориям Кантора о трансфиции агрегатов
и м до-расселловская логика может появиться
Но мы увидим позже, что без какой-либо опасности противоречий или
antinomies emergmg можно выразить синтаксис языка
поддерживать этот язык в той степени, которая обусловлена
богатство средств выражения языка м вопрос

Однако сначала мы не будем довольствоваться этим
Блеф, важно, хотя это и есть. Вместо этого мы будем строить
синтаксические понятия, относящиеся к языкам, которые мы выбрали, и
отложим на время вопрос о том, можем ли мы
та-ва. SenJtiuMae & Hiaei нет. это. тот

сам язык На первых этапах теории такой наивный подход
Кажется, всегда оказывался наиболее плодотворным.
метрия, антметика и дифференциальное исчисление все появились первыми,



4 INTRODOCTJON

и только намного позже (в некоторых случаях, сотни лет спустя)
гносеологические и логические обсуждения уже разработанных
теории следуют. Следовательно, мы начнем с построения синтаксиса,
а затем, позже, приступить к формализации своих концепций и тем самым
определить его логический характер.

Следуя этой процедуре, мы имеем дело с двумя
Guages: в первую очередь с языком, который является объектом
наше исследование - мы будем называть это объектом-языком - и,
во-вторых, с языком, на котором мы говорим о синтаксическом
формы объектного языка - ^ мы будем называть это синтаксис-
язык. Как мы уже говорили, мы возьмем в качестве наших объектных языков
определенные символические языки; в качестве нашего синтаксического языка мы будем в
сначала просто используйте английский язык с помощью некоторых
дополнительные готические символы.

§ 2. Языки как исчисление

Под исчислением понимается система соглашений или правил
следующий вид. Эти правила касаются элементов -
так называемые символы - о природе и отношениях которых
нет ничего более предполагаемого, чем то, что они распространены в различных
классы. Любая конечная серия этих символов называется выражением
исчисления в вопросе.

Правила исчисления определяют, во-первых,
определения, при которых можно сказать, что выражение принадлежит определенному
таин категория выражений; и, во-вторых, под чем
условия преобразования одного или нескольких выражений в
другой или другие могут быть разрешены. Таким образом, система языка,
когда только формальная структура в смысле, описанном выше
считается, это исчисление. Два разных вида правил таковы
которые мы ранее называли правилами формирования и транс
формация - а именно, синтаксические правила в более узком смысле
(например, «Выражение этого языка называется предложением, когда оно
таким же образом символы таких и таких
вид, происходящий в таком-то порядке "), и так называемый
логические законы дедукции (например, «если предложение состоит из
символы объединяются таким-то и таким образом, и если другой
состоит из символов, скомбинированных таким-то и другим способом,
тогда второе может быть выведено из первого »). Далее каждый



5


§ 2 ЯЗЫКИ КАК КАЛЬКУЛИ

хорошо определенная математическая дисциплина является исчислением м в этом смысле
Но система правил шахмат - это тоже исчисление. Шахматные фигуры
являются символами (здесь, в отличие от тех из слов-языков,
они не имеют значения), правила формирования определяют
шахматные фигуры (особенно имитальные позиции в игре),
и правила трансформации определяют движения, которые
смягченный - то есть допустимые преобразования одного
положение другого

В самом широком смысле логический синтаксис такой же, как
построение и манипулирование исчисления, и это только потому, что
языки являются наиболее важными примерами расчета, который, как
В большинстве случаев только языки синтаксически исследуются.
исчислений (даже в тех, которые не являются языками м собственно
смысл слова), элементы - это wntten символы
«Символ», который следует, будет иметь то же значение, что и слово
«характер» Не будет предполагаться, что такой символ обладает
смысл, или что он обозначает что-либо

Когда мы поддерживаем этот логический синтаксис, язык рассматривается как
Разумеется, мы не подразумеваем под этим утверждением, что язык - это ничто
больше, чем исчисление Мы только имеем в виду, что синтаксис касается
та часть языка, которая имеет атрибуты исчисления - это
Это ограничено формальным аспектом языка. Кроме того, любой
определенный язык имеет, помимо этого аспекта, другие, которые могут
например, его слова имеют
Это означает, что это объект исследования и изучения
В то же время слова и выражения языка имеют
тесная связь с действиями и восприятием, и в этой связи
они являются объектами психологического изучения агама, языковой кон
устанавливает гистонкально данный метод кумуляции, и, таким образом,
взаимное влияние, с определенной группой людей, и
как таковой является объектом социологии. В самом широком смысле
языка оценивает языки от каждого из этих
Помпы из синтаксического (в нашем смысле формального), из
семасиологический, от психологического и от социологического
c hai * e уже сказал, что синтаксис касается исключительно
Формальные свойства выражений. Теперь сделаем это утверждение.
Предположим, что два языка {Spraehen), Sj и Sj,
использовать разные символы, но так, чтобы один-один
между символами Sj и теми могут быть установлены



6 INTBODDCTION

Sj, так что любое синтаксическое правило о Sj становится синтаксическим
правило о Sj, если вместо того, чтобы связать его с символами Sj, мы
опоздать на соответствующие символы S,; и наоборот. Потом,
хотя два языка сгнили, у них одинаковые
формальная структура (мы называем их изоморфными языками) и синтаксис
касается исключительно структуры языков в этом смысле.
С синтаксической точки зрения не имеет значения, является ли один из
два символических языка, скажем так, используют знак * & ',
где другой использует * • '(в словах ^ ^: использует ли один
«и» и другие «и») до тех пор, пока правила формирования и
дуга преобразования аналогична. Например, это полностью зависит от
формальная структура языка и предложений,
является ли определенное предложение аналитическим или нет; или один сен *
Вывод выводится из другого или нет. В таких случаях дизайн
(визуальная форма, гештальт) отдельных символов является вопросом *
разница. В точном синтаксическом определении намека не будет
сделано для этого дизайна. Кроме того, это в равной степени неважно от
Синтаксическая точка зрения, что, например, символ «и»
должно быть конкретно вещь, состоящая из чернил принтеров, если мы
согласился всегда ставить спичку на бумагу вместо этого
конкретный символ, формальная структура языка будет
оставаться без изменений.

Теперь должно быть ясно, что любая серия любых вещей будет одинаково
хорошо служить в качестве терминов или выражений в исчислении, или, более
на языке, Нужно только раздавать вещи
в конкретных классах, и тогда мы можем построить
прессы, имеющие вид серии вещей, составленных в соответствии
правилам образования. На обычных языках
Символы (выражение) - это либо временной ряд звуков, либо
пространственные серии материальных тел, изготовленные на бумаге. Пример
язык, который использует подвижные предметы для своих символов, является карточным
индексная система; карты служат именами объектов для книг
библиотека и райдеры как предикаты, обозначающие свойства (для
например, «одолжил», «на переплет» и тому подобное); карта с
всадник делает предложение.

Синтаксис o (язык или любого другого исчисления,
в общем, со структурами возможных серийных порядков (определенного
вид) из любых элементов тчтсевер. Теперь мы будем различать
между чистым и описательным синтаксисом. Чистый синтаксис обеспокоен



7


§ 2 ЯЗЫКИ КАЛЬКУЛИ

С возможными договоренностями, без ссылки на
природа вещей, составляющих пустые элементы,
или на вопрос о том, какие из возможных договоренностей
из этих элементов где-то на самом деле реализованы (то есть,
с возможными формами предложений, безотносительно к
образцы слов, из которых состоят предложения, или
существуют ли какие-либо предложения на бумаге где-то м
мир) В чистом синтаксисе формулируются только определения и
Последствия таких определений разработаны таким образом, что чистый синтаксис
полностью аналитический, и не более чем аналитический анализ, или,
другими словами, геометрия конечных, дискретных, сенальных структур
Описательный синтаксис определенного вида связан с чистым синтаксисом как
физическая геометрия к чистой математической геометрии, это касается
с синтаксическими свойствами и отношениями даны эмпирически
выражения (например, с предложениями конкретной книги)
Для этого - ^ так же, как м применение геометрии - это
необходимо выработать так называемые коррелятивные определения посредством
из которых виды объектов, соответствующих различным видам
синтаксических элементов определяются (например, «материал
тела, состоящие из чернил принтеров формы 'V', должны служить
символы джункуона «) Предложения дескриптивного синтаксиса могут, для
позиция, заявить, что четвертое и седьмое предложения партии
Тезисы противоречат друг другу, или что второе предложение
в трактате не синтаксически правильно

Когда мы говорим, что чистый синтаксис U согласовывает формы
предложения, это "связано с" предназначено в образном
смысл Аналитическое предложение на самом деле не "связано с"
что-либо, в том смысле, в котором это является основным предложением, для аналитического
предложение без содержания.
здесь в том же смысле, в котором говорят, что арифметика
касается чисел или чистой геометрии
геометрические конструкции

Таким образом, мы видим, что когда мы расследуем или судим
частная научная теория с логической точки зрения, результаты
из этого логического aralyns должны быть сформулированы как тактические предложения,
либо чистого, либо описательного синтаксиса Логика науки (логическая
методология) не что иное, как синтаксис языка
Наука Этот факт будет ясно изложен в заключительной главе
из книги бис Синтаксические проблемы приобретают все большее значение



INTRODDCTION


в силу метафизической позиции, представленной
Венский круг. Согласно этой точке зрения, предложения мета-
физика - это псевдо-предложения, которые на логическом анализе доказаны
быть либо пустые фразы или фразы, которые нарушают правила
синтаксис. Из так называемых философских проблем единственным
что-либо, что имеет какое-либо значение, является локомотивом науки.
Поделиться этим мнением является замена логического синтаксиса для философии.
вышеупомянутая антиметафизическая высота не будет, однако,
появляются в этой книге как предположение или как тезис. В
последующие запросы носят формальный характер и не зависят от
любым способом на то, что обычно называют философской доктриной.

Метод синтаксиса, который будет разработан в следующем
страницы будут полезны не только логическим аналитикам
теории - это также поможет {n логическому анализу
языки. Хотя здесь, по причинам, указанным выше, мы будем
иметь дело с символическими языками, синтаксическими понятиями и
правила - не в деталях, но в их общем характере - также могут быть
применяется к анализу невероятно сложного
языки. Прямой анализ этих, который был
распространенный до сих пор, неизбежно должен потерпеть неудачу, как физик
был бы разочарован, когда он с самого начала пытается связать
его законы для естественных вещей - деревьев, атонов и так далее. Во-первых
место, физик связывает свои законы с простейшим из построенных
формы; тонкому прямому рычагу, простому маятнику, точечному
фермерские массы и т. д. Затем с помощью личинки, относящейся к
эти построенные формы, он позже в состоянии проанализировать в
подходящие элементы сложное поведение реальных тел, и
таким образом, чтобы контролировать их. Еще одно сравнение: сложный
конфигурации горных цепей, рек, границ и тому подобного
наиболее легко представлен и исследован географическим
координаты - или, другими словами, построенными линиями, не приведенными в
природа. Таким же образом, синтаксическое свойство конкретного
слово-язык, такой как Englbh, или определенных классов
языков или конкретного подъязыка слова-языка,
лучше всего представлен и исследован сравнением% с построенным
язык, который служит системой отсчета. Такая задача, как
всегда, выходит за рамки этой книги.



§ 2 ЯЗЫКИ КАЛЬКУЛЬ


9


Терминологические замечания

Приводится причина выбора термина «(логический) синтаксис».
во введении прилагательное может быть опущено где

нет опасности путаницы с лингвистическим синтаксисом (который не
чисто в своем методе, и не удается установить точное
система правил), например, м текст этой книги и м логический
трактаты м общие

Как предполагает само слово, самые ранние исчисления в смысле
описанные выше были разработаны в математике Гильберт был
сначала относиться к математике как исчисление в строгом смысле - т.е.
установить систему правил, имеющих математические формулы для
их объекты Эту теорию он назвал метаматемами, а его
Первоначальный объект должен был получить доказательство свободного
дом из противоречия классической математики
matics IS - если рассматривать в самом широком смысле, а не только из
точка зрения только что упомянутой задачи - синтаксис
Математический язык По аналогии с обозначением Гильбертана,
Варшавские логианцы (Лукасевич и другие) говорили о
«метапропостциональное исчисление», «металогте» и т. д.
слово «metalogic» является подходящим обозначением для субдомена
синтаксиса, который имеет дело с логическими предложениями в более узком смысле
(то есть, исключая математические)

Термин семантика используется Chwjstek для обозначения теории
который он построил с тем же объектом, что и наш синтаксис, но
который использует совершенно другой метод (этого мы будем
скажу позже) Но так как м наука о языке, это слово
13 обычно принимается как синоним «семасиологии» (или «теории
Это означает, что, возможно, не совсем желательно передавать его
синтаксис - то есть к формальной теории, которая не принимает во внимание
Meanmgs (Сравните Brial, Эссе де S ^ Manttque Наука Des


язык, таким образом, является подразделом семанологии, но это должно быть
отличается от семасиологии, которая, как часть науки
язык, исследует значение выражений
исторически заданные языки



ЧАСТЬ I

ОПРЕДЕЛЕННЫЙ ЯЗЫК I
ПРАВИЛА ФОРМИРОВАНИЯ ЯЗЫКА I
§ 3. Предикаты и функторы

Синтаксический метод здесь будет разработан в связи с
с двумя конкретными символическими языками, взятыми в качестве объектных языков
Первый из этих языков - ^ мы будем называть его Язык I, или,
вкратце, я с математической точки зрения
сборник натуральных чисел в определенной степени,
примерно соответствует тем теориям, которые обозначены как
Конструктивист, финитист или интуиуомст. Ограничение состоит из
В том, что встречаются только определенные числовые свойства,
что 13 сказать, те из которых владение или не обладание
любое число может быть определено конечным числом
шаги в соответствии с фиксированным методом Именно из-за этого ограничения
я называю это дефиютным языком, хотя он не является определенным
язык в более узком смысле слова определен, то есть
скажем, разрешимые (то есть либо доказуемые, либо опровержимые) предложения
Позже мы будем иметь дело с языком II, который включает в себя
Язык, который я представляю себе в качестве подъязыка Язык II содержит
m сложение mdefimte концепций и охватывает как арифметику
действительных чисел и математического анализа до степени
который разработан м классической математики, а в дальнейшем
теория агрегатов Языки I и 11 не только включают
Математика, однако, прежде всего, они предоставляют возможность
Построение эмпирических предложений, охватывающих любую область объектов
Например, во II классической и релятивистской физике
Мы придаем особое значение синтаксическому
Синтетические (не чисто логико-математические) предложения,
wbicb - это обычно.% игнорируется модемной логикой
предложения, рассматриваемые с точки зрения языка как
в целом, являются только вспомогательными средствами для работы с empincal, то есть
нематематические, предложения



ЧАСТЬ I. TTTE DETDOTE l-KXCCACE


ЯВЛЯЕТСЯ

В первой части будет сформулирован сьютакс Языка I. Вот,
английский язык, дополненный несколькими готическими символами,
быть использован в качестве syntas-языка. В Пан II синтаксис Лан-
Guage I Trill будет оформлен, то есть он будет выражен в виде
исчисления языка; и это будет сделано на языке I isclf.
В Пан III синтаксис более богатого языка II будет detTloped,
но только более простым методом слова-языка. В Пан я
Tre должен отказаться от объектных языков 1 и Il.and создать общий
S5'ntax, который будет применим ко всем языкам ctcry.

Для понимания следующих глав, предыдущий
знание элементов логистики (логика SJM ^ HC) является желательным,
хотя не абсолютел} - обязательно. Punher подробности дополнения ^
краткие сообщения, приведенные здесь, можно найти в регулярном
изложения исчисления предложений и так называемого функционала
исчисление. Смотрите: Хильбен Карнап Льюис [LcsiO-

Язык, который связан с объектами любого домена
может обозначать эти объекты либо rrv.ts, либо systcnitic
postiianal coordotatrr, то есть символами, которые показывают место
объекты в системе, и, следовательно, их ходатайства в отношении
для другого. Примерами позиционных символов являются, например,
номера домов, в отличие от отдельных имен
как «Красный лев», которые были cusiomsiy в более ранние дни;
Osnvald'a обозначение цветов с помощью букв и цифр, а
против их дифференциации с помощью цветовых названий синего цвета,
и т.д.); обозначение гогтафических мест по их широте и
долгота, а не по именам менна *, «мыс добра»
Надежда '); и обычное обозначение космических известковых
четыре координаты (координаты пространства и времени - ^ четыре действительные числа-
берс) в тел) 'и т. Метод обозначения собственными именами является
prirruliveone; это позиционное обозначение соответствует более
продвинутая стадия обучения и имеет значительную методологическую
Преимущества о> -кр первого. Мы будем называть язык (или суб-
язык) whidi обозначает объекты, принадлежащие донууну с
что это касается позиционных обозначений, эо или ^ Mtr-
в отличие от neTrr-Lo - ^^ rs.

До сих пор было обычной символической логикой использовать имя-
языки, объекты для большинства панорамы, обозначенные
имена 'a *,' b * и т. д. (соответствующие обозначениям 'moon *,
'Menna *,' Napoleon *, из слов-языков). Здесь мы будем



* 3


§ 3 ПРЕДИКАТИТЕ АНУ ФУНКТОРОВ

принять согласованные языки для наших объектных языков, и,
В частности, m Langiuge I, мы будем использовать натуральные числа в качестве
ординаты. Рассмотрим в качестве области позиций одно-
sions с определенным направлением, если «а» обозначает позицию
В этом случае следующая позиция будет обозначена как «а».
начальная позиция обозначается буквой «О», затем
Очевидно, что обозначения будут обозначаться как 'O' * 0 ", и так далее.
такие выражения бывают с
представление более высоких должностей, они влекут за собой определенную сумму
mconvenience.we должны, с целью сокращения, miroduce
обычные числовые символы по определению Таким образом, «l * для« 0 »,« 2 »для
«0» и т. Д. Если мы хотим изменить позиции, сделанные двумя,
в трехмерной или n-мерной области мы используем упорядоченные пары или триады
или n объявлений цифр-символов

Для того, чтобы выразить свойство объекта, или позиции, или
связь между несколькими объектами или позибо, используются предикаты
Примеры (i) I> t «BlueCj)« имеют значение », позиция 3
синий ", ma name-language 'Blue (a) * - это" объект a синий "
(2) Пусть «Wr (3, 5}) означает« позиция 3 теплее, чем позиция
5 ", ma name-language * Wr {a, b) '" тело a теплее
чем тело б "," Фа (а, б) * "
b »и т. д. (3) Пусть« T (0, 8, 4,3) »означает« температура при
положение 0 намного выше, чем в положении 8, как температура
температура в положении 4 выше, чем в положении 3 "В
В приведенных выше примерах «синий» - это 3 однолистных предиката, «Wr» -
названный предикат, 'T' четырехзначный предикат In * Wr (3, 5) * 3 '
13 называется первым, а «вторым аргументом» Wr
Два класса предикатов предикаты м примеры
процитированные выше выражают (как мы обычно говорим) эмпинальные свойства или
Отношения Мы называем эти предикаты desenpitve и различаем их
из логических предикатов, которые являются те, которые (как мы говорим xisual)
выражать логико-математические свойства или отношения
примеры логических предикатов * Piim (5) * mcan3 "5 - простое число
число «,« Gr (7,5) »« 7 больше 5 »или« позиция 7
более высокое положение, чем положение 5 "Точное определение
будут даны синтаксические понятия * описательный 'и' ^ rogicaT ''
позже, без ссылки на значение как т
объяснение [Обозначение «предикат», которое ранее было
Приложено только m случаев mvolving один термин, здесь, для



14 P.U5T I. ДАФТ ЛАКСКАСА 1

пример Гильберта, применимый также в делах, подстрекающих больше ибанов
один термин; у общего слова для обозначения обоих случаев
Сам prot-cd гораздо практичнее.]

Предикаты, так сказать, собственные имена свойств
позиции. \ Мы определили должности с помощью систематического
order-sj-mbols, а именно число * s \ Tnbols. Подобным образом мы можем
также обозначьте их свойства номерами символов. Вместо того
color-names, color-numbere (или триады таких чисел) могут быть
используемый; вместо неточных обозначений «тепло», «круто», «холодно»,
и так далее, теперь я могу использовать номера temperanire. Это не имеет
только преимущество в том, что гораздо больше информации может быть
учитывая, но, кроме того, еще одно преимущество, которое имеет решающее значение
значение для науки, а именно, что только благодаря этому
«Антметизация» - это формулировка законов универсалии (например,
Достаточно: что из fdaiion между температурой и расширением,
или между icmperanire и давлением). Чтобы
c-xpress свойства или отношения положения с помощью чисел,
мы будем использовать функторы. Например: пусть 'te' будет чайной пастой
функтор; 'te (3) = 5' тогда означает; «Темперар на позиции
3 - это 5 "; если мы возьмем функтор 'tdiff' для представления температуры
разница, то «tdiff (3,4) = 2» означает: «разница
температура в позициях 3 и 4 равна 2 ". Кроме того,
ftre функторы, мы также используем lopeaJ ftmcicB-s. Например:
«сумма (3,4)» имеет значение: «3 плюс 4»; • fak (3) 'равнозначен
«3!». «Sum» - логический функтор с двумя терминами, «fat * {Fahiltit) a
однозначный логический функтор. Hcrealsointhccxpr «sion'sum (3,4)»,
* 3 'и' 4 'называются ctpwwuU; в * te (3) = 5 ',' 3 'называется
аргумент для 'le', а '5' называется значением 'te' для
аргумент * 3 '.

Выражение в любом случае обозначает число
добыто или не определено), мы называем выражение saenericol (точное
определение на с. 26). Примеры: «O», * 0 «», «3», «te (3)»,
«сумма (3,4)». Выражение whidi соответствует пропозициональному
Приговор лорд-языка мы называем происхождением (определение на стр. 26).
Примеры: 'Blue (3)', * Pnni (4) \ Выражение называется <; f-
сценарий (определение на стр. 25), когда либо описательный предикат
или в нем встречается описательный функтор; в противном случае это называется hpicel
(определение на стр. 25).



§ 4 СИНТАКТИЧЕСКИЕ ГОТИЧЕСКИЕ СИМВОЛЫ


ЯВЛЯЕТСЯ


§ 4 Синтаксические готические символы

Два символа «а» и «а» встречаются в разных местах на этом
Таким образом, они являются разными символами (не ручными),
но они eqaal (не tme ^ tud) Синтаксические правила языка
должны не только определять, какие вещи должны быть использованы в качестве символов,
но и при каких обстоятельствах эти символы должны быть
присваивается как синтаксически равный Очень часто символы, которые не
Ul> em внешний вид считается синтаксически равным, например,
m обычный язык, * 2 'и' j '[Такое заявление о равенстве
не всегда означает, что два символа должны быть
могут быть различия в использовании, которые
зависит от несинтаксических факторов. Например, это обычно
не использовать * 2 * и '3 * в одном и том же контексте, который пишут почти всегда,
либо «wbca», либо «jehta», а не «jebra»].
книга, 'z' и '3 * синтаксически неравны С другой стороны, мы
будет рассматривать '(' 1 '' как равные символы, а также

соответствующие закрывающие скобки
большие, круглые и квадратные скобки в выражениях нашего объекта-
язык, следовательно, не имеет никакого отношения к синтае.
Это сделано исключительно для удобства читателя. Далее, в
наша система (в отличие от Рассела) их символы «2» и «a *»
Мы можем написать «*» повсюду, но агам для
удобство чтения, когда между предложениями стоит «=» (и
не между числовыми выражениями) мы обычно вместо wnte '='
Мы будем называть два выражения равными выражениями, когда их
соответствующие символы являются равными символами, если два символа или два
выражения, равны (syntacticaUy), то мы говорим также, что они
имеют тот же синтаксический логово ^ Но это не m никак
предотвратить их различные визуальные формы, как, например,
регистр '(' и или * = 'и * =', или другой цвет интерфейса, или любой
другие характеристики, которые синтаксически не имеют значения
Почти все работы, заработанные в этой книге,
с чистым (не описательным) синтаксисом, и, следовательно, должны сделать,
не с выражениями как пространственно разрозненными вещами, а только с
их синтаксическое равенство или неравенство, и, следовательно, с их
Синтаксический дизайн ^ Что бы ни говорилось ни о каком явном объяснении
одновременно с любым другим выражением глаз и, возможно,



16 ПАРТИ. ОПРЕДЕЛЕННЫЙ ЯЗЫК I

сердечно, будьте основаны на экспрессивном дизайне. Поэтому для
ради краткости, часто будем говорить просто о «e.Tpression» или
'symbol *, вместо' expressional design * или 'sjTnbolic design *.
[Например, вместо саджинга: «в с-выражении« Q (3, 5) »(и
следовательно в e \ -очень равном выражении) такой символ как sj-mbol '3'
происходит », мы говорим более кратко:« в каждом выражении дизайна
'Q (3 .S)' ® sjTnbol конструкции "3" встречается "или, еще проще,
«В выражении« Q (3,5) »» встречается символ «3».]
область чистого синтаксиса, эта упрощенная форма речи не может привести
до двусмысленности.

Символы пяти видов, перечисленных ниже, встречаются в I.
позже последуют планы)

1. Одиннадцать указаний tymboU (символ-дизайн):

«V *. '. ; = *. '3'. «K *.

Следующие четыре категории, к каждой из которых неограничен
число s ^ mibols может принадлежать:

2. (Числовые) vanabUs {'u \' v \ ... 's', в определениях

§§ 20-24, '/'.— V').

3. Постоянный нюнкраб (например, «О», «1», «2» и т. Д.); sj ^ nbols
принадлежность к группам (а) и (3) называются цифрами.

4. Предикаты (группы букв с начальными заглавными буквами, например

'Prim *, также * P * Q' R ').

5. Функторы (группы букв с маленькими начальными буквами, например,
«сумма»).

Символ, который не является переменной, называется константой. Бывший
Пресса I состоит из упорядоченной серии sjTnbols I, из которых
число конечно (но оно также может быть 0 или i; то есть
скажем, выражение может быть пустым или состоять из одного символа
только).

Под sjTitactical / orm (или, коротко: формой) мы понимаем любой
вид или категория выражений, которые синтаксически определяются
(тон, так сказать, определяется только со ссылкой на серийный заказ
и sj-ntactical категория символов; а не любым
синтаксические условия, такие как место, цвет и т. д.). Форма
определенное выражение может быть указано более или менее точно: наиболее
Точная спецификация - это то, что определяет дизайн
прижимной; самое неточное, то, что просто утверждает, что это
выражение.



§4 СИНТАКТИКА. ГОТИЧЕСКИЙ СИМВОЛ ^ I7

Введем сокращенный метод записи
утверждения о форме Например, на языке слов мы
Caifmake следующее утверждение о форме выражения
'Pnm {*)' “Это выражение состоит из предиката, опе
кронштейн, вантуз и дозирующий кронштейн в следующем порядке:
Вместо этого мы будем больше говорить «это выражение имеет
форма pr {3) ”Этот метод использования готических символов состоит
m mtroducuig синтаксические имена для представления символьных категорий,
Синтаксическое описание формы затем осуществляется путем помещения
эти синтаксические имена одно за другим Мы обозначим
символы (всех конструкций) на 'o', переменные (числовые) на
«3» (ZahlvanabU), символ (обозначение символа) «0» по «im» {nuU),
цифры m в общем случае «33» (Zahheuhen), предикаты в «пи»
(и, в частности, однозначный, двухзначный, называемый
цитирует * pd * pt * 'pt * * соответственно), функция - * fu' (и,
в частности, и т. д.) Для синтаксического обозначения
одиннадцать отдельных символов, мы будем использовать сами символы,
и, кроме того, для двухзначного: imcftcm- ^ ioZr (Ferhiup-
/ ufigf 3 eui <n) - 'V', 0 обозначение 'Derfn' Таким образом,

например, в * Pnm («)», * («ts aaymbol объектного языка,
с другой стороны, в * pr (3)% * ('это символ синтаксиса *
язык, который служит синтаксическим именем для этого символа м
объектный язык, и, соответственно, не что иное, как
видение английских слов 'скобка openmg' когда символ
Таким образом, используется как имя для себя (или, более
название для его собственного символа-дизайна), мы называем его auioTtymous символом
(см. §42) В результате двойного использования не может возникнуть двусмысленности
символов "(" и т. д., потому что эти символы встречаются только автономно
mious m связь с готическими буквами всякий раз, когда мы хотим
различать разные символы одного и того же вида по их синтаксическому
обозначения, которые мы используем mdices Например, * P (x, y, *) 'имеет
форма pt (3,3,3) или, точнее, форма pt * (3i, 3j, 3i) для
наиболее важные кмдсноны мы также будем использовать синтаксический
символ (с большой буквы) выражения (любой формы) мы
обозначить * SI '{AusdnieK), числовые выражения * 3'
(ZcAi ^ ifrucA), предложения * S * {Satz) Другие обозначения будут
позже мы также используем mdices для того, чтобы
смешивать выражения в предложении вида
(SVS) oG, три составляющих предложения могут быть равны или



PARTI. ОПРЕДЕЛЕННЫЙ ЯЗЫК I


неравные; в предложении фонн (SiVSj) 3Sj, первый и
треть составляющих предложений - это равные предложения.

Посредством индексов «b *» и «I» можно указать, что
символ описательный или логический соответственно. Например, «Фуй»
обозначает логические функторы, * 3 # * описательный числовой
pressions. Вместо того, чтобы тврить символ «fl» (или e.xpression)
сформироваться. , мы часто пишем для краткости: «а .. например, вместо
«логического функтора Dvo-termcd» мы кратко напишем: * a fuj * ';
аналогично «3 *» и др.

В дальнейшем готические символы будут использоваться в связи
с английским текстом; в более поздней конструкции синтаксиса I,
который дается не на словесном языке, а посредством дальнейшего
символы, эти символы не встречаются.

T * он главный объект метода готических символов состоит в том, чтобы защитить
нас от неправильного способа выражения, очень часто в обоих
логические и математические труды, которые не делают различий между
анимация символов и то, что ^ mbolized. Например, мы
найти «в том или ином месте, где происходит x ^ y», где правильная форма
было бы ** . , происходит "или происходит". Если экс

обсуждение предметного языка обсуждается, либо
выражение должно быть написано в кавычках, или его синтаксический
Обозначение (без кавычек). Но если
Синтаксическое обозначение - это то, о чем мы говорим, а затем, в
очередь, должны быть заключены в кавычки. Позже мы покажем
как легко пренебречь этим правилом и неспособность
разграничение между символами и объектами, которые они опустошают,
приводит к ошибке и мраку (§§ 41, 42).

§ 5. Соединительные символы

Символы с одним или двумя терминами используются для
построить новое предложение из одного или двух предложений относительно.
В строго формально сконструированной системе значение этих
символы - как будет более подробно обсуждаться позже - возникает из
правила трансформации. Для того, чтобы облегчить понимание
их, мы предварительно объясним их значение (и аналогично
других символов) менее точными методами; во-первых, примерно
матовый перевод на слова английского языка, а во-вторых,
с большей точностью, с помощью так называемых таблиц истинности.



§ 5 СОЕДИНИТЕЛЬНЫЕ СИМВОЛЫ I 9

~ {Si) IS называется отрицанием (Si), (Si) V (S2), (Si) »(®j)>
называется, соответственно, дизъюнкция,
конъюнкция, импликация, эквивалентность (или уравнение) Sj и Sj,
м, который и (Sg называются теимами

Переводы этих ^ mbolsare следующие "не *," или * (м
неисключительный смысл), 'и', 'не или' (иногда также
переводится как «если тогда»), «или, или нет, и
не «Мы обычно пишем дизайн qnrabol * =» в форме
's', где это происходит между предложениями (не между числовыми
pressions), '= * и' r 'считаются равными символами, т.е.
символы одного и того же синтаксического дизайна

В большинстве принятых систем специальный символ
Используется lence, m дополнение к символу идентичности или equaLty '='
(Например, Рассел использует * = ', Гильберт') Мы, с другой

стороны, как в Языке I, так и в Языке II используется только один
Болик дизайн (но для более легкого понимания читателя, мы используем
два вида фигур) Как мы увидим позже (стр. 244 е) этот метод
Является ли допустимым и полезным для eztensional языков, таких как
Я и II

Далее для краткости я записываю любые
символические выражения, которые встречаются либо в объекте, либо в
Синтаксис-языки, мы (как обычно) опустить скобки
объемное частичное выражение IQj (которое может быть либо предложением
или синтаксическое обозначение предложения) в следующих случаях

I Когда SIi состоит только из одной буквы

z В отношении - ^ (^ Ii), или Derfn (? Ii), или (3Ij) 0Ctfn, когда 2 Ii
Прошу или с 'или с PR, или с оператором
(сек ниже)

3 ^ Vhen - это дизъюнктивный термин и сам по себе дизъюнкт

4 Когда 13 соединительный термин, и сам является соединением

5 ^ Vhen IS операнд и сам просит с оператором
(об этом позже)

Таким образом, вместо '(' ^ (^)) V (<3 ^ '[но не вместо
* ~ ((Si) V (Sj) y] пишем для краткости аналогично

'SiVS, vS 3'. 'Sx.S, .S,'

Это упрощение, однако, будет использовано здесь только для
практическая цель записать выражения - формула
синтаксические определения и правила будут переданы
прессы без скобок опущены

Есть, очевидно, четыре возможности связи с



20


ЧАСТЬ I. ТИИКСКИЙ ДВИЖЕННЫЙ ЯЗЫК I


правда и ложь двух предложений, Sj и Sj. Это будет
представлен четырьмя строками tntili-значения tahk gi ^ 'cn ниже.
В таблице показано, в каком из этих четырех случаев предложено соединение
верно и в котором оно ложно; например, дизъюнкция ложна
только в четвертом случае, иначе это правда.


SG

ВСГ,

Si.e,

Si3S |

SjSS,

(0

TT

T

T

T

T

(2)

TF

T

F

F

F

(3)

FT

T

F

T

F

(4)

FF

F

F

T

T


Таблица nvo-hne ниже является таблицей для отрицания.


е.


(1)

T

F

(А)

F

T


С помощью приведенных выше таблиц истинное значение кратно
сложное предложение может быть легко установлено для различных
случаи, прежде всего, определяя s'alues ;;для компонента
пристрастия, а затем пошагово продвигаться ко всему
приговор. Так, например, можно определить, что для
Sj V © j, справедливо то же распределение истинности T, F, T, T, что и
на случай импликации; и из этого мы получаем перевод
в слова: «нет. , .или. , , для обмана Далее это также может быть
Установлено, что истина -% распределения

T, T, T, T, и, таким образом, безусловно верно ли Sj и © j
будь правдой или ложью. Позже мы будем называть такие предложения аналитическими.
предложения.


§ 6. Универсальные и экзистенциальные предложения


Здесь мы снова приведем значения выражений с помощью
сначала о переводе, а затем о формулировке условий.
Например, пусть «Red» будет pr ^; «Красный (3)» будет означать; «


; "и

; ^ стороны

Эти обычные формы предложения, мы введем следующее.



21


§6 НЕОБХОДИМЫЕ И НАСТОЯЩИЕ ПРЕДЛОЖЕНИЯ

(*) 3 (красный (x}) 'будет означать то же, что и * красный (0) »красный (i)» красный (2) «
Красный (3) *, то есть «Каждая позиция до 3 - красная», «Qx) 3
(Красный (x)) 'будет означать то же, что и * Красный (0) VRed (i) VRed (2) V
Red (3) ', или "Есть одна позиция до 3, которая красная"

Выражения, которые встречаются в начале предложений
выше, а именно '(x)' Q x) "*, * (*) 3 '(3 *) 3 дуга, называемая tm-
bmited универсальный оператор, неограниченный экзистенциальный оператор,
универсальный оператор Ixmted и экзистенциальный оператор bmtted
соответственно в двух ограниченных операторах «3» называется
ограничивающее выражение оператора и во всех четырех операторах
x 'IS называется ^ et ^ erafor-canable' Red (x) 'называется операндом
(принадлежат оператору) В языке I только ограниченные операторы
произойдет, мы не будем использовать неограниченные операторы до тех пор, пока
м язык II

Если и 51 - операторы, то вместо WSII (2 Is (S)),
мы будем солнечно (сравните с 19, условие 5)

Vanable (или symboUdesign переменной), Jt, называется
привязан в определенной позиции m tDi (является ли символ дизайна
происходит в этой позиции или нет), когда есть (собственно или им *
правильно) частичное предложение, которое соответствует этой позиции и
имеет вид (; 5), где ^ - оператор, имеющий оператор-
переменная 3i

Vanable 3, который происходит в определенном положении в называется
свободен в этой позиции, когда Jj не связан в этой позиции
m Пример. Пусть Sj имеет вид SjVSjVSi и задает
назовите дизайн 'Pi (*) V (x) 5 (P, (x, ^)) v?, (*) * во всех позициях
из Sj, * x 'связано m Sj и, следовательно, также в Sx, m Si первый
и четвертый * x 'и * y' свободны, если
free m iHi встречается в ISi, тогда Hi называется открытым, иначе
называется закрытым

Чтобы выразить неограниченное количество волн, свободные переменные будут
Например, пусть © j будет 'sum (x, y) = sum (>, x)' This
будет означать «Для любых двух чисел, сумма первого и
секунда всегда равна сумме второго и первого »Если © 5
верно, то и каждое предложение, вытекающее из © j, является результатом
подстановки любых произвольных числовых выражений для 'x' и
', например,' sum (3, 7) = sum (7, 3) '(© J [Таким образом, m наше
система, так называемые предложения функций также ранжируются как
Наше разделение на дозированные и открытые предложения



22 ЧАСТЬ р. ОПРЕДЕЛЕННЫЙ ЯЗЫК I

проводит обычную классификацию на предложения и предложения

функции.]

В использовании free t'oriablet для cxprcssinR неограниченная универсальность,
наш язык совпадает с языком Рассела. Но когда Рассел, в его
Пояснительный текст (Prtne. ^ Fath. H говорит, что свободная (реальная) переменная
двусмысленный или имеет неопределенное значение, мы не согласны с
ему. «Красный (х)» - это правильное предложение с совершенно однозначным
смысл; это в точности эквивалентно по смыслу предложению
(происходит на нашем языке 11 и на языке Рассела *)
• (X) (красный (*)) *.


Выражение, которое возникает из данного выражения
суждение 3 i из 3 i будет обозначено синтаксически

'? t, j Это может быть точно определено следующим образом.
Позиции в Sti, в которых ji свободно встречается, называются
tubuituHon-posilions для 3) in? ti; * выражение, которое


возникает из Dli, когда 3, rebeed 3-й все замещения *
позиции в Привет ', здесь 3 я должен быть построен таким образом, что нет переменной
который связан в любом из положений замещения ^ для 31 в
91, встречается как свободная переменная в 3i * If. тн 91 ,, 3, не встречается


в качестве свободной переменной, то 91,


(3)


обозначает неизменный экс-


Прессион 91.


Пример: Пусть S ,, € ,, © 4 представляет ранее упомянутый
десять, и пусть 2 будет переменной, а 3, то S, (^ 1)
представляет значение: «Pj {0) v (*) 5 0i)) vP, (0)».

(*? ') ** *** ;' (3 *) (* = y *) 'означает: «Для каждого числа y существует
это следующее большее число * »Здесь a 3, в котором '** происходит ns a
Переменная fw, например, 'x' \ не должна заменяться на 'y';

QxlCxssx * ')', очевидно, ложно


§ 7. K-оператор

Выражение вида {K3) 3 (S) не является точным, как
соответствующие выражения, которые имеют универсальный или
Оператор exjstential - это числовое выражение. К-оператор,
(3) 3. я * не оператор предложения, а оператор записи;



23


§ 7 k-оператор

или, более конкретно, числовой оператор (K 3 i) 3 i (Si) означает
«Наименьшее число до (и включая) 3i, для которого Si
true, и, когда такого числа не существует, 0 ”Примеры
* Gr (a, b) «означает« a больше, чем b », * (Kx) 9 (Gr (3f, 7))»
эквивалентное значение мм до '8', * (K *) 9 (Gr (ср. 7) .Pnm (*)) 'равно
эквивалент мм * * * *

В целом, из указанного значения следует, что два
обозначения форм (i) и (a) ниже означают одно и то же

[~ (33i) 3i jJ3i [pr! (Si) "

(Jj3i (~ (3, = 3i) 3 ~ pr, (30) .i; t, (3i)] (2)

Упомянутые ранее обозначения оператор-переменная *,
Переменные 'Iimitmg-expression', 'operand', 'bound' и 'free',
также применяются к выражениям, имеющим K-операторы [В отличие от
с обычным (Russellian) descnptioo, описание посредством
К-оператор никогда не бывает пустым или двусмысленным, это всегда
univQcal, следовательно, использование оператора K> не требует
предостерегающие правила необходимы в нашей системе]

§ 8 Определения

Символы, для которых не определены определения, называются
defisied или pTvntbve символы Логические символы pnmitive
Язык 1 состоит из одиннадцати отдельных упомянутых символов
уже (см. p x6), вместе с im и всеми 3 как описательными
примитивные символы любой pr »или fii» могут быть установлены все остальные 33,
px, и fu, который желательно использовать, должен быть представлен
A 33 или PR всегда явно определены, фу либо
в восторге или постепенно

Определение exphat состоит из одного предложения, регрессивного
определение двух предложений Каждое из этих предложений будет иметь
form 3 i = 3 s »Of Si = St. Выражение 3 i (или Sj) называется
и contzms символ, который должен быть определен

3 a (или (S |) IS называется defimeru

В точном определении, символ, который должен быть определен
встречается только в течение определенного срока, в том числе и регрессивного
С другой стороны, это происходит также в определениях второго предложения



PJIRT - это DEUMTE LANCUAGB т

В остальном, определение может содержать только другие примитивные символы.
или такие, которые были определены ранее. Таким образом, порядок в wbidi
defimtioas установлены не могут быть изменены произвольно. К эади
определенный символ принадлежит cferin of dffjatiorj, под whidi подразумевается
самая короткая серия о! предложения, которые определены в терминах oS
этот символ вместе с определениями всех определенных символов
происходит в цепи. Двенадцать определений символа
всегда конечны и (кроме порядка наследования) уникальны
определяется.


Для явного dffadiiont, в более широком смысле, в котором слово
здесь используются оба явных определения в более узком смысле
То есть те, в которых определение состоит только из
новый символ (например, defijuoon JJ в I) - и
Определения ao-eillcd в универе - те, где содержится де ^ 'enduxn
другие sytabola, кроме нового символа (например, определение
РТ или «н фу в я).


Определение naneral, имеет вид: Sjj
Определение предиката ^ prj *, имеет вид;

Явное определение »fwutor fUj * имеет вид;

3 «) * 3 - \ Sxe7rple. '' Dl {x) ^ 3 ^ \ l> d. 1, стр. $ 9-}

Rtyreurre определение фу "имеет вид: (о) фу *
(mi.Sj. ... 3,) - 3 i; (я) фу, '' (ч ', 3 ^ ... 5,) = 3 ~ В всегда

затем следует * rguiBent- € xpt®sion v-atvshles

которые не связаны. [Fxampfr, 'Def. 3 для «прод», стр. 59;
Первое уравнение служит для преобразования fUj (mi, 3) l в
Второе выражение относится к K (3j'.3 ») обратно к fu, (3» .30, так что для
Например, в «Prod (6,3-)», используя второе уравнение шесть rimes
и первое уравнение один раз, «prod» может быть исключено.} Далее,
Это определение должно соответствовать следующим двум условиям:
ЦИИ; (r) в значениях не может быть свободной переменной wbidi docs
еще не встречаются в определении; и (2) два равных T * riabl «
не должно встречаться в dcfiniaidutn.


Если ro ^ fiCTi (i) вы не сделали, то возможно, чтобы определения были
мною ^ из ххх «мтраВакция может быть выведена. Эта
^ y U показано на примере »n (Lesniewsld дает аналогичный пример
^^, * "'десятичное исчисление [Xeuet Sj-itm]. стр. 79 f.). Мы определяем



25


§ 8 ОПРЕДЕЛЕНИЯ



P (*) s (Gr (*, y) .Gr (y, 5))

(0

(0

(Гр (7.6) .gr (6.5)) Op (7)

(2)


Гр (7.6) .gr (6.s)

(3)

(2) (3)

Р (7)

(4)

( «)

P (7) 3 (Gr (7, 4). Gr (4.s))

(Ы)

(5)

Р (7) 0Gr (4, с)

(6)

(6)

~ Гр (4.S) 0 ~ Р (7)

(7)


~ Гр (4, S),

(8)

(7) (8)

~ Р (7)

(9)


(4) и (9) противоречат друг другу

С другой стороны, нет необходимости делать обратное
условие, переменная отсутствует в определенных значениях
Дениендум. (Сравните, например, Def 3 i. P 59)

Условие (2) выполнено, а не с целью избежать
дикноны, но с целью обеспечения пенсионного обеспечения
позиция, если один определил 'P (x, *) sQ (*) *, то' P 'm' P {0, 1) 'мог бы
не быть устраненным

Если у нас есть предложение вида 3i = 3 * », то

как мы увидим позже, 3i ^ заменяется на 3s> ®i ® * »
и coarerseiy, m каждое другое предложение (p 3 6) Это означает, что
каждый определенный символ, где бы он ни находился, может быть
благодаря его определению, но в случае
определяемый символом не всегда возможно
[Пример Если 'prod (*, y)' встречается в предложении m, которое ** 'свободно
(например, 'prod (*, y) = prod (y, a :)'), lhen'prod'caiin не может быть удален]

Теперь мы можем более точно определить условия
«описательный» и «hguaV», хотя до настоящего времени имеют только
было грубо объяснено

Если символ не определен, то Q | называется дескриптивным (ot,)
когда Oi - это pr или fu Если Oi определено, то Q | называется,
когда происходит неопределенное в dcfiniUon-cbain Ци Ан
Выражение itti называется descnpDve, когда происходит Ob
Qi называется логическим (q]), когда Ox не является tUx, называется логическим
(плохо), когда SIi не является SIj

§ 9. Предложения и числовые выражения

Теперь мы назовем несколько видов выражений.
Важными из них являются предложения (S) и многочисленные
pressiom (3) До сих пор у нас были только точные объяснения
из них со ссылкой на pieanmgs Теперь, однако, эти виды
должны быть определены формально и точно мы уже обследовали



26 P- «T I. TITE ОПРЕДЕЛЕНИЯ LWJCUACE 1


все возможные способы сужения предложений и числовые
exprcssiors в I-angu ^ I, так что теперь у нас есть только перечислять
^ • различные формы, возникающие из них.

S может содержать другие © и 3 * ® pat® », аналогично 3 ttiay
содержат и другие 3 * «d также (с помощью K-оператора) S.
Отсюда и определения, которые мы выдвигаем из терминов
«предложение» и «числовое выражение», к которому wc должен добавить
вспомогательный термин «аргумент-выражение», ссылаются друг на друга, но
или просто так, чтобы определить, является ли конкретное выражение
представленный, «Ij, это S или 3» относится к вопросу
является ли определенное правильное частичное выражение SIj символом © или 3 -
Таким образом, отсюда следует, что этот процесс ссылки заканчивается
после конечного числа шагов; определения однозначны
и не содержат в себе богатый круг. [Определения строго
точная форма будет предоставлена ;;лучше в рамках
символически сформулированный синтаксис.]

Sj-мбол I, который является либо ini, либо определенной цифрой, либо 3,
называется омеил (33). Выражение I называется тц sccedted
выражение (St (.Striehautdrvek)), когда оно имеет одно из следующих
формы: 1. ню; 2 St». [Поэтому St либо либо * 0 '(неправильное
Выражение с ударением) или '0' с одним или несколькими акцентами * • '.]

Выражение I называется числовым выражением (3), если оно
имеет одну из следующих форм: 1.35; а. 3'J 3 '4 * (^^ i)

3i (S), где 3i не встречается как свободная переменная в 3i- Regres-
определение для n-термина nrcmncpt-cx pre slon * (SIrg *) в


I: 9 Irg 'представляет собой 3; имеет форму 3,

Выражение I называется предложением (S), когда оно имеет
из последующей формы: 1. 3 = 3 CEquatitm '); а.

3. '^ (S); 4. (S) perfn (S); 5. 15, (S), где? F, имеет вид
(Ji) 3 или 0 3 i) 3 n, а где 3, документы не являются свободными s-ariable
в 3 я. [ft не обязателен, чтобы переменная оператора появлялась в
операнд как свободная переменная; если нет, то?!, (©,) эквивалентно в
значение для S ,.]


Наиболее важной классификацией выражений является
в предложение ^ и не-предложения. Частое отправление
exp ^ ions wmch не являются предложениями в выражениях с 'Inde-
weaTvmB ("pt ^ r naro« '* в широком смысле) и остальное
^ '«F'f« lfill «l», «* yn $ emantic» выражения) может быть



§ 10, КАСАЮЩИЕСЯ ПРАВИЛ ТРАНСФОРМАЦИИ


27


B ПРАВИЛА ТРАНСФОРМАЦИИ ДЛЯ
ЯЗЫК I

Статья 10. Общие замечания относительно
Правила трансформации

Для построения исчисления утверждение транс
правила foimatiOQ, а также правила формирования, приведенные для
Язык I, существенный С помощью первого мы определяем
при каких условиях предложение является следствием другого
предложение или предложения (предлоги) Но тот факт, что € 2 является
Следствие не означает, что мысль о Си будет
совершенный мыслью о Св. Это не вопрос
психологическая, но логическая связь между предложениями
утверждение Sn утверждение St> s уже объективно в *
Решено. Мы увидим, что отношения, указанные здесь
Материальный путь может быть чисто формально задуман. \Пример
Пусть Si будет '{*) 5 (Red (*))', а Sj * Red (3) ', учитывая, что все
позиции до 5 красного цвета, то также дается (неявно), что
позиция 3 красная В этом особом случае, возможно, St будет иметь
«думали» одновременно с Си, но в других случаях, когда
преобразование является более сложным, следствие не будет
Обязательно нужно думать вместе с предпосылками

Это невозможно с помощью простых методов, чтобы создать
фимуон для термина «следствие» в его полном понимании
Такое определение еще не достигнуто в современной логике
конечно, м старая логика) Но вернемся к этой теме
Позже в настоящее время мы определим для языка I, вместо
термин «следствие» несколько более узкий термин «денвабельный»
[При построении систем логики обычно принято использовать
только последнее более узкое понятие, и это обычно не ясно
Понятно, что понятие денвацильности не является общим
концепция последствий] Для этой цели термин «непосредственно

будут сформулированы различные выводы [3 т.
можно извлечь из St или из Si и Sj, когда с помощью одного
из правил вывода S3 может быть получен из Si или из Si
и Sf]



28 ЧАСТЬ I. ОПРЕДЕЛЕННЫЙ ЯЗЫК I

Под влиянием д-р ^ ой прмиает S ^, S ,, ... (из которых
число является конечным и может также быть 0), мы понимаем
ряд предложений любой конечной длины, такой, что предложение Кэрри
из сенсов является либо одной из предпосылок, либо определением-предложением,
или непосредственно выводится из одного или нескольких (в наших объектных языках я
и если не более двух) предложений, предшествующих ему в
Senes. Если S, это последнее предложение деривации с предварительным
скучает по Si, ... S *. тогда Sp называется производным от Si, ... S * ,.

Если предложение в материальном толковании логически универсально
верно (и, следовательно, следствие любого предложения, что
так или иначе), тогда мы назовем это ана / юлиВ (или тавтологическим) предложением.
[Exem / Джи *; * Красный (3) v ».- Красный {3) *; это предложение верно в каждом
случай, независимо от ivaturc позиции 3.] Но это
Это еще одна концепция, которая не поддается формальному анализу
средства простых методов, и это будет обсуждаться позже. Первый,
мы предлагаем дать определение несколько более узкого
термин «доказуемый». [Это обычная процедура; Гддел был
первый, чтобы показать, что не все аналитические предложения являются наглядными.]
Si называется доказуемым, когда S] выводится из
нулевая серия предпосылок, а следовательно, из любого предложения
Когда-либо.

Если предложение, если оно интерпретируется материально, является логически недействительным,
Мы назовем это противоречивым. \ ^ ampt: ; красный (3). ~ -Red (3) ';
это предложение является ложным в каждом случае, независимо от характера
Положения 3.] Мы вернемся позже к рассмотрению
эта концепция. На данный момент мы расскажем, вместо этого, некоторые
какой более узкий термин * опровергаемый '. <3 ib называется опровержимым, когда в
хотя бы одно предложение доказуемо, G * получается из
Gi путем подстановки любых акцентированных выражений для всех 3
которые встречаются как свободные переменные, [Пример: • Prim (x) 'является опровержимым
потому что '~ Prim (0 "' i) 'доказуемо.] Закрытое предложение Gi
Таким образом, опровергается, когда и только когда. ~ Gx наглядно.

Предложение называется ^ TirtrfiV, когда оно не является ни
противоречат друг другу. Предложение называется неразрешимым, когда оно не является ни
доказуемо или опровержимо. Этот последний срок несколько больше
больше, чем термин «синтетический». Позже мы увидим, что
Логическое предложение O'crj является либо analj-Uc, либо противоречивым; и это
поэтому sj-нтктические предложения ели только среди
Описательные предложения. С другой стороны, в I - и аналогично в



§ 10, КАСАЮЩИЕСЯ ПРАВИЛ ТРАНСФОРМАЦИИ 29

каждый суффиксально нч язык - есть логические предложения, которые
неразрешимы (сравните § 36)

По причинам технической простоты принято не
сформулировать всю систему правил вывода, но лишь несколько
из них, и m место отдыха, чтобы установить определенные предложения, которые
наглядно (на основе общей системы правил)
так ^ alled prlmldre предложений Выбор правил и примитивов
предложения - например, когда определенная материальная интерпретация
исчисление исходит заранее - в значительной степени произвольно
Часто система может быть изменена (без изменения содержимого)
пропустить примитивное предложение и вместо него положить
правило mfcrence - и наоборот

Мы также установим правила вывода (то есть,
определение «непосредственно dcnvable») и установить примитивные предложения
для языковых объектов Inthismethod, aderivatlottwithcertam
предпосылки должны быть определены как предложения предложений, каждый из которых
один - это одна из предпосылок, или предложение, или
defimtiOQ предложение, или непосредственно выводится из предложений, которые
предшествовать ему. Денвация без посылок называется
доказательство Доказательство и, таким образом, смысл предложений wbch каждый
либо примитивное предложение, либо определение предложения, либо непосредственно
извлекаемые из предложений, которые предшествуют ему в смыслах
предложение доказательства называется доказуемым предложением

§ II Первоначальные предложения языка I

Мы дадим здесь не отдельные примитивные предложения, а
элементы схем из pnmiUve предложений Каждая схема будет
определить количество предложений, к которым неограниченное количество
предложения принадлежат, например, посредством схемы PSI 1
определяется, что каждое предложение вида SiD (~ 0, oSj)
13 называться примитивным предложением первого рода, где
и €, могут быть предложения, которые построены в любом случае
что бы ни было [принято ставить умные предложения
вместо схем и м Язык II мы также будем использовать, что
metacL Но для этого 9UC90сев переменных для Sv и. ^ JL »»
необходимо например pnmitiie предложение PSIIi (стр. 91)
соответствует схеме PSIi, но, потому что м язык я
у нас нет необходимых переменных в нашем распоряжении, мы не можем



ЧАСТЬ ЧАСТЬ I. ШИНА ОПРЕДЕЛЕННАЯ I ^ GUACE I

построить примитивные предложения сами, но только схемы.
Предложения, которые здесь называются примитивными предложениями
дуга первого рода, во II, косвенно демонстрируемые предложения. Oни
следовать из PSIIi путем замены.]

Схемы Примильца Приговоры танкажа 1

(а) Примитивные предложения так называемого предложения celculuf.

PSII.

PSIA. (--GpSO ^ Sj

PSI 3. {SiOS0D ((StO6,) D (S, Os3)]

(б) Примитивные предложения операторов предложений (ограничено).

PS14. (Жо ,, U (s,) эс, (^;)

rsi5. (3,) !, '(S.) s [(s.) S, (S.). S. (^,)]

PSK. a3i) 8 (S,) S ~ (3.) С. (~ ®.)

(e) Примитивные предложения iitntity.

PSI 7. 3 я = 3,

PSI8. (3v = S.) О [®iOS. (® ^)]

(d) Примитивные предложения арифметики.

PSI9. ~ (Nu = 3' )

PSIio. (3 i '«= 3,') 3 (3, = Ji)

(«) Примитивные предложения оператора.

®. (^)] Va 3.) 3. [(S ,.

(3.) 3. [~ (3 = 3.) О ~ G. (^)]). S.])

Теперь мы рассмотрим эти примитивные предложения, когда материально
истолкованная дуга и (в случае PSI5-11)
замена любого jj на свободный g, истинные предложения следуют из
их.

Для PSI 1-3: это легко показать с помощью значения истинности
Ubles (на стр. 20). Для PSI4: две стороны эквивалентности


3 *


§ II. ПРИМИТИВНЫЕ ПРЕДЛОЖЕНИЯ ЯЗЫКА I
равны по значению в соответствии с уже заданным значением
для ограниченного универсального оператора - поэтому оба верны или
оба ложны Для PSI5, когда somethmg верно для каждого
число до n + 1, то это также верно для каждого числа до n
и для n + 1, и наоборот, для 1 ^ 16 «есть число m
Считается, что такое и такое свойство ", эквивалентно
в среднем к предложению «Это не верно для каждого числа до
Это не обладает свойством м вопрос "PSI 4 а
5 представляют, так сказать, регрессивное определение ограниченного
универсальный оператор PSI 6 представляет явное определение
ограниченный экзистенциальный оператор, хотя явно определенные символы могут
всегда быть элюминированным, не всегда возможно устранить
определенные символы (сравните стр. 25)
ограниченный универсальный оператор не может быть устранен, когда
Выражение limitmg содержит свободную переменную (как, например, m
PSI 5) Ограниченные универсальные операторы и регрессивно определенные фу
это не просто сокращения, и если мы откажемся от них,
выразительность языка была бы очень важна *
С другой стороны, чтобы отказаться от ограниченного
экзистенциальный оператор, оператор K * и символы con *
соединение и смысл вместе со всеми четко определенными 33,
pt и fu только преуспеют в том, чтобы сделать язык более
неуклюжий без м
pressible

Символ идентичности или равенства 's =' между 3 “здесь m-
склонен (как м арифметика) в том смысле, что 31 = 3: верно, если и
только если 3i и 3 * обозначают одно и то же число, чтобы использовать общий
Фраза Из этого следует, что PSI 7 и 8 являются действительными PSI 9
означает, что ноль не является преемником любого другого числа, а
следовательно, первоначальный термин PSI 10 означает, что разные
номера имеют одинаковые преемники PSI 9 и 10 соответствуют
к четвертой и третьей аксиоме соответственно м системы Пеано
аксиомы для антемики. Материальная действительность PSIii следует
из значения ранее заданного K-оператора (§7)



3 =


ЧАСТЬ I. DITOIRRE UAKCDACE I


Статья 12. Правила вывода языка I


S, IS называется непосредственным переводом в I от Sx (^ * * *) •

Si и Sj (Rl3, 4), когда одно из следующих условий RI 1-4
выполняется;


RLX. (Замена).


Sj имеет вид Sj


(!) •


RIA. (Соединения.) (A) Sj получается из Sj заменой
частичное предложение (правильное или неправильное) вида S ^ VSj
; ^ SiDSj, или наоборот; * (6) также с формами и

~ (~ S4V < «- <е ^); (в) также с формами и

(S40S4). (S »0S4).

RI3. (Последствие.)


Ri4. (Завершите mduedon.) S, имеет форму Sj
имеет вид SjD S, (^ '1 j.


и 6}


То, что мы здесь сформулируем в виде определения
«непосредственно получаемый» обычно формулируется в форме правил
умозаключения. Таким образом, condidoos только что поселились бы соответствовать
следующие четыре правила:

1. Правило ruirh'run'on. Любая замена разрешена.

2. Правило / I / Neft'onr. (а) Частичное предложение StV S, всегда может

быть подчиненным и, соответственно. Соответственно с

(4) и (с).

3. Правило невосприимчивости от Sj и SjDSj, S, может быть
выводится.

4. Правило полной индукции. Пример; От при (тТу) и
Pri (3i) 3prj (jj '), pri (30).

Эти правила сформулированы таким образом, что, когда
время материально истолковано, оно всегда ведет от истинного
склонности к дальнейшим истинным предложениям. В случае RI я, это foUowa


Немецкий оригинал, RI a (a) относится к 640 © *

~ ^ заменяет определение значения символа

внимание на то, что вместо этого
RI 3 (а). что означает определение дис-
^ принято, потому что в PSI 1-3 подразумевается
P ^ lla K * 1 * J “” используется sj-mbol. По простой причине

PSII 4 также был изменен (см. 530).



33


§12 РУС НЕЗАВИСИМОСТИ ЯЗЫКА I

от толкования свободных Vambles, данного ранее м
книга, а в случае RIz и 3, из значения истины
Таблицы (стр. 20) RI 2 представляет собой, так сказать, решение ezphcit
символов дизъюнкции, конъюнкции и eqmvalence,
которые просто служат аббревиатурами RI4 соответствует
обычная арифметическая функция полного вывода, если
perty принадлежит числу 0, и если это свойство является наследственным
один (то есть тот, который, если он принадлежит числу n, также принадлежит
к п + я), то он также принадлежит каждому числу (пятый Пеано
аксиома)

§ 13 Выводы и доказательства в языке I

Что предложение ceitam является очевидным или выводится из
некоторые другие предложения, будут показаны путем предоставления доказательства или
Мы найдем более плодотворный метод
доказывая универсальные синтаксические предложения, которые означают, что все
времена такого и такого рода являются очевидными или выводятся из
другие предложения, такие и такие формы Иногда доказательство
универсальное синтаксическое предложение этого kmd может быть осуществлено
построение схемы для доказательства или опровержения
Состояния схемы приводят к доказательству, или денонсацию можно получить
Отдельные случаи Еще один плодотворный метод, который во многих случаях
устраняет необходимость построения специальных схем, основывается на том факте
что универсальные синтаксические предложения о демонстрации или
denvabihty может быть выведен из других предложений того же рода.
То есть, если € 3 13 подлежит извлечению из (3 „и © 3 из, то © j
также выводится из © 3, для них можно получить
первые два денвонации один за другим, если © 3
доказуемо, и © 3 отклоняется от © j, тогда © 3 также
Кроме того, если © jO <^ доказуемо, то © j
nvable от © j (согласно RI 3) Обратное не всегда верно,
но только следующее, если ^ замкнуто и © j обратимо из Sj,
© jO ©} Демонстрируемо [Контр-пример для открытого
следующим образом пусть © J будет 'x = 2', а будет '(*) 3 (* = 2)', © j - это
wnWrt. * 03 "si / i © pj ast 3 -m 'liflb taije -a

не доказуемо и даже ложно, потому что © 3 результаты этого
с помощью замены «2» на ** и применения

RI3]

SL 4



34


ЧАСТЬ 1. T !! E ОПРЕДЕЛЕННЫЙ ЯЗЫК 1


Мы приводим простые примеры схемы доказательства и
схема-схема вместе с несколькими универсальными символическими предложениями
о наглядности и выводимости. [Ссылки на
в левой части страницы примитивные предложения и правила
только там, чтобы облегчить undeistandmg - они не относятся к
схемы. С другой стороны, особые условия, изложенные в
слова, которым подвергается конкретное выражение (например,
в схеме деривации ниже), имеют важное значение для схемы.]


Пример схемы проверки


фунтов на квадратный дюйм 1 SiD (~ SiDe,) (0

PSl2 (-wS, oG,) dS, (2)

PSI3, в котором предложение '-SiDGi будет принято за Sj
и Si принимается за <3 с-

(S. 0 (~ S. = G,)) 3 (((~ S, 3 S.) 3 SJ 3 (S. = SJ) (3)
(;) (3) RI3 ((~ 6, 3 S,) se.) = (S. 3 S,) (4)

W (4) RI3 S, SSi (5)

«Ниторем 13,1. ®iDS, всегда (то есть для любого предложения
дизайн @ 1) наглядный.

Обозначим синтаксическую теорему № п § m
«Теорема m« n ». Синтаксические теоремы относятся к этому

часть языка 1, которая соответствует так называемой
caJcului. Эта часть включает PSI 1-3 и RI 1-3.

Теорема ijA GiV ^ Sj b всегда доказуема. Это
так называемый принцип исключенного среднего.

Теорема 13.3. Gj и взаимно выводимы.

Теорема 13.4. Если Gi опровержим, то каждое предложение Gi
rs выводится из G, - Поскольку Gi опровергаем, доказуемо
предложение »> -Gj существует так, что Gj получается из Gi с помощью
замещения. Таким образом, в дополнение к Gj, мы можем использовать ^ S3 как
предпосылка в образовании ичемо.


(1) RIt
PSI я

(3) (4) RI3
(») (5) RI3


S,

-S,

S,

S,


(1)

(2)

(3)

(4)

(5)
( ' »)



§13. Выводы и доказательства в Лос-Анджелесе I 35

Теоремы S} nt2cucal, которые следуют, относятся к этой части
язык, который выходит за рамки исчисления предложений, а именно,
к исчислению предикатов [Это обычно называется
исчисление симметрии По большей части, до настоящего времени, термин
'предикат' был добавлен только к одному, названному pt] В этом
Язык домена я отклоняюсь дальше от обычной формы
формальный язык (Рассел и Гильберт) Поскольку язык I является
язык координат, метод полного вывода
(RI4) будет часто использоваться в доказательствах и выводах

A- STVTAcncAt. Теоремы об универсальных предложениях

Теорема 13.5. Каждое предложение одной из следующих форм
доказуемый

Вт (3JA (е.) = Ы. (^).

CS) (ч) 3. (s0oS, (^ J,

(e) (3 i) 3i (®i) = 'Si * при условии, что он не является свободным
переменный в

Теорема X3.6 <

(a) (S,) 13 всегда выводимо & om (3i) 3i (Si).

(5) (3) 3 всегда выводится из

(c) Sj всегда выводится из (ji) 3i (®i) »при условии
не встречаются как свободная переменная m € 1;

(<0 (3 i) 3 i ('Si) 3 (3 i) 3 i (' = s)> 3 всегда из

(3i) 3 (<5pS0.

(e) (3 i) 3i (Si) = (3i) 3i (S,) »всегда извлекаемые из Si = ®j
(это следует из теоремы 6 б, г)

B. STNTAcncAi, Теоремы о существующих предложениях

Терем 13.7. Следующие предложения всегда являются

w (33 дня (S0sS. (^):

(S) Q3ja '(SOs [Q303. (SOvs. (^,) J,



36


PARTI. ОПРЕДЕЛЕНИЕ ФУТБОЛА I


w S, (| J = (3 Дж.) 3. (Si).

Теорема 13.8. (33 i) 3 i (®i) w, получаемых из G; а если 3 ^
документы не встречаются как свободная переменная в Gj, тогда обратное также
правда. - Дальнейшие соотношения выведения аналогичны теореме 6
может быть заявлено.


C. Синтаксические ткеобемы об уравнениях

Теорема I3.9. Следующие предложения всегда являются
стрейбл:


Вт (3 = 3) = [3. (| J = 3 (FJ.];


W (3. = 3.) = (3, = 3.);

W [(3> = 3.). (3, = 3.) L3 (3. = 3.) -

3 * 3 ** 0 - Следующие ученые получены из

D. SirrrAcricAL Теофемы о замене

Теорема 13.31. 51, 3, 01, я, darireble от 3. = 3 и% 3,%.
при условии последнего la ecntcncc. Другими словами, А: если
13, затем в любом предложении левую часть
уравнение может быть заменено правым слагаемым (и, аналогично,
термин nght-hand от левого tcmi).

n, ^ A ^ rT? '®t' '' i “'<" tv »l> lerrotnS, E®, nndSr, S, 5r„

Под ред последнее предложение. Другими словами: если
Таким образом, в любом предложении, в котором secoiid (или
поджег огонь

заминировано). Доказательство

6e) (& rnr “; -“ (“” tpatv, для вдохновения. Теорема

«Мне ; 7 ^ 0? ° P- '; 1 условие, что

наш fo ™ St и S. я, нет. необходимо в

н, боже, ие, я, легкость



37


§ 13 ПРОИЗВОДНЫЕ И ДОКАЗАТЕЛЬСТВА В ЯЗЫКЕ I

замещения, все выражения одного вида (а именно, свободный
меценаты), которые встречаются в предложении
одновременно, с другой стороны, в случае замены,
не нужно обращать внимания на остальные части предложения
Возможность представления определений в форме
уравнения зависят от теорем ii и 12 (сравните § 8)
основа явного определения, определение может в каждом
дело должно быть заменено на dtfimens, и наоборот


§ 14 Правил Последствия для Языка I

Может возникнуть ситуация, когда для определенного pn, скажем pti, каждый
предложение вида pTi (St) доказуемо, но не
версальное предложение pti (3i) Мы встретимся с pr такого рода позже
на (§ 36) Хотя каждый индивидуальный случай является переносимым, нет
возможность вывести предложение pri (3i), чтобы создать
возможно, мы введем термин «следствие», который
шире, чем термин «денвабель», и, аналогично, термин
«аналитический», который шире, чем «демонстрируемый» *, и
'противоречивый *, который шире опровержимого * Определение
будет оформлено так, что универсальное предложение m вопрос, pri (3i),
хотя и не наглядно, но будет аналитическим.

Для этого необходимо иметь дело также с классами
предложения До сих пор мы говорили только о первых смыслах
десятки или другие выражения, но класс может быть такого
природой, что она не может быть исчерпана с помощью первого чувства
затем можно назвать классом iT ^ mte, более точное определение tbis
термин не нужен для нашей цели) класс выражений
дается посредством синтаксического определения (либо defimte, либо
mdefimte) формы выражений для каждого
схема примитивных предложений бесконечный класс предложений
Говорить о массах выражений только
более удобный способ говорить о синтаксических формах
pressions [Позже мы увидим, что «класс» и «свойство»
два слова для одного и того же слова

Мы будем применять следующие обозначения (синтаксиса
язык) к классам выражений (предложения по большей части)
'ft' {Klasse) будет общим термином. * {2Ii} * будет доставлен в
представляют класс whidi, единственный элемент? Ii,



JS P.UITI. TIETE DENULTS LANCOACE I

клисс, состоящий из элементов СТСУ.
Сумма сессий С. конец S ,. Исчезает дип эйпрессии *
класс dfScriphT'i, когда хотя бы одно из выражений ^ e в нем является
дескриптивное выражение; в противном случае это называется hpicaJ dass. (В
этот раздел, «Jii» и т. д. всегда определяют классы o {предложения-)
St IS называется прямым (в I), когда один из
Следующие условия, DCi, 3 »выполнены:

DCI. Конечна, и в wludiRI 4 существует деривация
(полная индукция) не используется и из каких предпосылок
являются предложения 5ij и последнее предложение Sjl

Контроллеры домена. Существует такое джи, что FLJ - это дас

все предложения вида €,

51% называется <irert eonse ^ etue ^ lan (m I) 51j, когда каждый
предложение в 51% является прямым следствием под * dass из 51] - конечное
смыслы (не обязательно конечных) групп предложений, таких, что
esery dass (кроме первого) - прямой посвященный
Dass, который непосредственно предшествует этому в серии, является eotisequrtuf-
серия (в 1). Sj называется следствием (в I) 51j, когда есть
существует серия следствий, из которых 51, является первым и
(Sj) последний. (5 «называется следствием Ш.орофа
когда S является следствием {S,}. или {Sj, S «}, повторно
соответственно.

Jn правило DC t wc airc не обязано отменять правило RI 4 (завершено
индукция). Но его дополнительное применение было бы излишним
поскольку на основании приведенных определений можно показать, что
e, всегда является следствием js, i 5, d S, j | , Пусть это
класс будет 51%. Тогда, как легко увидеть, есть предложение в форме
от 51, и, следовательно, согласно DC I,
прямое следствие 5 ^; таким образом, дас из этих предложений, 51;
Это прямой класс последствий 51; и в соответствии с ДК. ©, есть
прямое следствие 51 ,, и, следовательно, следствие 51 ,.

Теорема 14. 1 K предложение выводится из других предложений,
тогда это тоже следствие их.



39


§ 14 ПРАВИЛ ПОСЛЕДСТВИЯ ДЛЯ L.WCU \ CE X

Последовательность »reUtjon имеет более широкое расширение, чем
Денвабихты-отношение Правило DCa может, как указано выше,
Положение shovrs, быть частично заменено на TLL4 A
эквивалент DCa невозможно получить ни с помощью
RI 4 или любые другие правила вывода бывшего kmd, то есть
скажем, любые правила, связанные с понятием «непосредственно денубируемые»
Например, деривация должна состоять из конечного числа
В терминах эти правила всегда относятся к конечному числу предпосылок.
Но DCa в целом относится к бесконечным классам предложений [Com-
сравните пример, приведенный в начале раздела pti (3i)
Не является следствием какого-либо надлежащего подкласса класса
предложения pT | (€ t) еще меньше следствие конечного подкласса]

Таким образом, теперь у нас есть различные методы вычета
ограниченный метод демации и более широкий метод
Concequenee-senes Вывод является конечным смыслом предложений,
следствие-смысл является конечным смыслом не обязательно конечным
классы В случае денвации, каждый шаг mdmdual (то есть
отношение «непосредственно денабируемое») является определенным, но не отношение
«denvable», который определяется всей цепочкой денваунов
В случае следствия, шаг шага (т.е.
отношение «прямое следствие») уже определено, и поэтому
тем более отношение «следствие»
более узкий, чем термин «последствие»
только тот, который точно соответствует тому, что мы имеем в виду, когда мы говорим
«Это предложение« против »(логически) из этого» или «Если это
предложение верно, тогда (по логическим причинам) это тоже верно »
В обычных системах символической логики вместо понятия
«следствие», более узкое, но гораздо более простое понятие
nvable 'применяется, используя определенные правила вывода
И, на самом деле, метод денервации остается фундаментальным
ментальный метод, каждая демонстрация применения) любого
Термин в конечном итоге основан на деноминации.
Существование отношения следствия - то есть
построение следствия смысла м объектно-языкового -
может быть достигнуто только е) б) средство денвации (доказательство) в
язык sjTitax

Предложение Sj называется аналитическим (m I), когда оно является следствием
нулевого класса предложений (и, следовательно, следствие
предложение), оно называется противопоставление 7, когда каждое предложение является



^ 0 ПАКТл. Отклонения UINGOAOE I

CHnsAquancE of S .; это OEM

»Alj4 или cootrodtory; это он сделал .ynthetlc wbon, t .. север
анальное или противоречивое. р,(

KuMUd eto fl, B colled cal, tic при каждом предложении o [
fl - аналитическое предложение; taatradialory, когда каждое предложение является
следствие Si, l и lynrte / fe svhen это не является ни аналитическим

противоречат друг другу. -

Два или более предложения называются несовместимыми
другое) когда класс, состоящий из них, является противоречивым классом;
в противном случае они называются совместимыми.

Теорема Т4.2. Каждое доказуемое предложение является аналитическим;
каждое опровержимое предложение противоречиво. Обратное есть. как-
никогда, не всегда верно.

Теорема 14.3. Каждый S, (и «i) является либо анальным, либо противоположным
dictory. Только Sb (или выдумка) могут быть синтетическими.

Доказательство: я. Пусть (3, замкнутый Sj. Применение правил ол
сокращение, которое будет дано позже (§ 340), приводит к конечному
количество шагов, чтобы перейти к • nu »nu *. или к его отрицанию.
Здесь каждый шаг сокращения согласуется с DCi. Там-
fore 01 является L-детерминированным.

2. Предположим, что каждое <5 \, в котором n отличается свободным
переменные происходят L-детерминировать; мы покажем, что в этом случае
То же самое верно для всех S | tvith «+ 1 свободных переменных. Пусть си он
Si с n + 1 свободными переменными. Рассмотрим


класс 5I1. предложений вида Si j • Каждый из них

предложения содержит n свободных переменных, и, следовательно, согласно
Наше предположение, это L-детерминированный. Тогда согласно DCa,
01 является прямым следствием fi |. Теперь приведите все предложения
fi arc aiulytic или, по крайней мере, один из них противоречив - скажем, Sj.
В первом случае Si также является аналитическим; во втором случае Si **
противоречив, потому что Sj является прямым следствием Si. Там-
fore 01 является L-детерминированным и, следовательно, каждое © j с n + 1
свободными переменными является L-детрминат.

3. По принципу индукции это следует из (i) и (2)
что каждый © j является L-детерминированным.

Пример: Теорема Permat'a:


• (Gr ( », 0) .GR {>». 0) .GR ( », 0) .GR («. 0" )) 3

~ (sum [pot (*, u). pot {^ ;, u)] • = pot (ff, u)) '



§ t4 ПРАВИЛА ПОГОВОРНОСТИ ДЛЯ ЯЗЫКА I 4 I

(defmitiODs 'Gr', 'sum *,' pot * будут даны в § 20)
логическое предложение и, следовательно, согласно теореме 143.
конечно, либо аналитический, либо противоречивый
не известно, какой из этих двух случай

Теорема Z4 * 4 <AA [противоречива тогда и только тогда, когда хотя бы
одно принадлежащее ему предложение противоречиво, но может
быть противоречивым без каких-либо предложений, принадлежащих
это противоречиво (по этой причине важно, чтобы не
должны быть только предложения, но также и классы предложений.
классифицируются как аналитические, противоречивые или синтетические.)

Пример Пусть prt будет неопределенным pr ^, тогда предложения pTi (nu)
и ~ pit (nu) являются синтетическими, но класс этих двух предложений
(как и их соединение) противоречиво

С помощью понятия «аналитический»,
что 13 обычно обозначается как «логически действительный» или «истинно по логическому
основания * вы достигли до сих пор это по большей части было
думал, что логическая действительность представима термином
«демонстрируемый», то есть в процессе деривации, но
хотя, для большинства практических случаев, термин «демон *
Стрейбл »представляет собой адекватное приближение, он не исчерпывает
Концепция логической справедливости То же самое верно для пар
«демонстрируемый» - ^ «атлитический» и «рефибируемый» - «противоречивый», как
для пары «производный» - ^ «следствие»

В материале mterpretatioo аналитическое предложение абсолютно
правда, какими бы ни были эмпирические факты. Следовательно, в нем не говорится
Что-нибудь о фактах На другой группе, противоречивое предложение
заявляет слишком много, чтобы быть способным быть правдой, из-за
диктаторский приговор и каждый фаа и его противоположность могут быть выведены
Синтетическое предложение иногда верно, а именно, когда
факты существуют - и иногда ложные, следовательно, он говорит что-то вроде
какие факты существуют. Синтетический senlettcet являются подлинными утверждениями
о реальности

Если мы хотим определить, что предложение (м материал
режим речи) означает, не выходя из основного формального
go xjver mvo 'ikcfi 05 tiit' iicnvrG & nfteiprOcfaon 05 плата за обслуживание,
мы должны выяснить, какие предложения являются последствиями этого
предложение Среди этих предложений мы можем игнорировать те, которые
последствия каждого предложения - то есть аналитические
предложения Неаналитические последствия Si составляют



42


ЧАСТЬ I. TIIE ОПРЕДЕЛЕННЫЙ ЯЗЫК I


вся область, из которой * вырастает ”Си
определить следующим образом; по локальному содержанию Sj или (в I) мы
понять класс неаналитических предложений (I), которые
последствия Si или rcspecAcly (в I). «Контент» или
«Смысл» предложения часто говорят о svithout dctermininf;
именно то, что следует понимать под выражением. Определенный
нам кажется, что термин «контент» представляет собой именно то, что подразумевается под
«смысл» или «смысл» - пока ничего психологического или экстра
это логично.

Мы называем предложения или классы предложений, основанные на том же
содержание равноправное. Два предложения явно равносильны
когда и только когда каждый из них является следствием
Другой.


В дискуссиях о том, имеют ли определенные предложения одинаковые
sfwe (или подлость), возражения следующего характера очень часто
сделал логуану: «Но это предложение и это не может иметь
тот же смысл (или смысл), потому что они связаны с довольно
разные мысли, образы и т. д. "На это возражение может
следует ответить, что вопрос о соответствии
р? согласование концепций и

нравиться. Ланер - это вопрос психологического характера и должен
следовательно, будет решено эмпирическим и психологическим mvesugaQon.
Это не имеет ничего общего с логикой. (Кроме того,
вопрос о том, какие идеи связаны с конкретными предложениями
неопределенный и неоднозначный; ответ будет отличаться в зависимости от
человек, который является предметом eipcnment и для конкретного
Cirrorwtances.) Вопрос, имеют ли два senfencet одинаковые
/ Ogirof смысла касается только согласия двух
tences m всех их последствий. Концепция «иметь
логически тот же смысл *, таким образом, адекватно выражен
определенный синтаксический термин «равноправный». Концепция НВО Имнр
имея то же самое, что будет понято син

Тактический термин «синонимичный» * является аналогом ^ е.

Теорема 14. 5 Взаимно выводимые предложения равносильны,
Я говорю, что это не всегда так.

Два r> ptcs, 31%, руда, смазанная нефтью, когда eech

вход S, в котором происходит 31, является равносильным (не, например,
равное m tntth-value), которое определяет S, которое возникает из

• sv-iTn "означает это понятие

Ssmonjmous отношение, которое обозначено в материале

comprehoS "" '"" "S"



43


§ 14 ПРАВИЛ ПОСЛЕДСТВИЯ ДЛЯ ЯЗЫКА I

Примеры '2', 'O' * ',' i * ',' suin (i, i) 'являются синонимами Let' te '
будьте неопределенным fua, тогда ctmi, когда 'te (3) = s' эмпирически
tme предложение, * te (3) 'не является синонимом' s ', и, более
как правило, не с 331 или 1 евро. [Но 'te (3)' является синонимом
'5' m rilanon to 'te (3) = s', об этом pomt см. § 65] В Enghsh
язык «Одиссей» и «пена Телемаха * не являются
синонимы, хотя оба они обозначают одного и того же человека

Теорема 14-6. Если 3 i == 3 »аналитическое, то 3 i 3? ^

синоним, и наоборот

Теорема X4 * 7. (а) Если аналитически, то это

следствие (i) если ЕСТЬ следствие и закрыто,
тогда SjD Sj является аналитическим.

Proofof'jh (яфо вначале из DCi и Rl3) LetSj
быть закрытым Мы сформулируем доказательство для общих случаев, для специального
В случае, когда <aj логично, доказательство значительно проще
TV e назовет S, аналитическое число (5 j, если S, является аналитическим

(A) Каждое предварительное предложение I является аналитическим подтекстом <5 i

(B) Если €) и 04 оба являются аналитическими нефитами 0 i, а если 0 $
непосредственно дербле от 0, и 04, то 0 ^ также является аналитическим
подразумевает 0 ^

(C) Из A и B следует, что если 0 $ - аналитическая импликация
из 04, и 04 получается из 0, без применения
полная индукция, то 04 также является аналитическим подтекстом 04
Поэтому, если 03 является аналитическим разогревом 04 и если, согласно
для DC I, 04 является прямым следствием 0 j, то 04 также является
аналитический подтекст 04

(D) Если, согласно DCz, 04 является прямым следствием
и у каждого предложения 5I4 есть аналитический подтекст 04, затем 04
IS также аналитический импеат 04

(E) Из C и D следует, что если каждое предложение из flj является
аналитическое значение 04, и если 04 является следствием, то 04
13 также аналитическое значение 0 |

(F) Поскольку 04 является аналитическим множителем самого себя, поэтому
верно следующее: если 03 является следствием 0i, то 03 является
аналитический импеат 04

Теорема 14.8. Два предложения являются синонимами, когда и только
когда они равносильны. [Это вахд для языков I и II
и для некоторых других языков также сравните теорему 6546]

Теорема 14.9. Если 04 = 02 является аналитическим, то 04 и 02
равносильный, и наоборот



PARTI. ДИТИНИТНЫЙ ЯЗЫК I


44

Из теорем 6. 8 и 9 следует, что определение
и определения определения-предложения являются sj-неимущими.

Рмаркф на тфрминолосы. Вместо выражения «аналитический»,
Витгенштейн [TVaefafui] - и, вслед за ним, литература
Венский круг до настоящего времени - использует выражение «тауто-
логическое "или" тавтология "(tshich, howes'cr, i * определено только для
исчисление предложений). С другой стороны, принято применять
термин «тавтологический» для преобразования предложений, а именно, к
те, которые не увеличивают содержание. \ Vc скажем, например; «
Логические выводы тавтологичны ». Это вопрос опыта,
однако использование слова «тавтологический» в этих двух
разные чувства, тем более что первое не соответствует
обычный способ речи, легко приводит к недопониманию и
фьюжн. Поэтому представляется более практичным сохранить
Пресс-м только во втором случае («тавтологическое заключение») и
принять выражение «аналитический», чтобы применить к первому случаю («аналитический
предложения'). Этот термин, который использовался в первую очередь Кантом,
был более резко определен Фреге ([Crundhsen] с. 4). Он звонит
предложение аналитическое, когда, для его ptwf, только "универсальный логический
законы »вместе с определениями необходимы. Добислав [Ана *
lytiiche] отметил, что понятие является относительным; это должно
всегда относиться к определенной системе предположений и методов
приукрашивания (pnmittve предложений и правил вывода), то есть
скажем, в нашей терминологии, к определенному языку.

Выражение «противоречивый * (или« противоречие ») было также
введенный Витгенштейном (в исчислении предложений).
В дополнение к выражениям «аналитический» и «синтетический» Кант сделал
Не используйте третье выражение для отрицания аналитических предложений.
Возможно, стоит подумать, является ли выражение «аналитическим»
следует понимать как общий термин (согласно предложению
Дубислава [AnalytUchij, в отличие от обычного использования), а затем
«аналитически верно» и «аналитически неверно» или «положительно аналитически»
и «отрицательно аналитический», используемый вместо «аналитический» и «против
dictory.


C. ЗАМЕЧАНИЯ ПО ОПРЕДЕЛЕННОЙ ФОРМЕ
ЯЗЫКА

§ 15. Определенный и неопределенный

Форма языка, наиболее часто используемая в модемной логике
Это то, что Уайтхед и Рассел [Prine. ^ Falfl.] есть
построен на фундаменте, заложенном работой Фреге. Пеано,
chrBder и другие. Гильберт использует другую символику,



§ 15 ОПРЕДЕЛЕННОЕ И НЕПРЕРЫВНОЕ 45

но его форма языка осталась прежней
При выборе символов для наших объектных языков I и II мы
приняли символику Рассела, потому что это самый
широко известный в форме языка мы следуем за мамой
выход из системы Гильберта и Рассела, но мы отклоняемся от
Это некоторые важные помпы, особенно мой язык.
Наиболее важные отклонения связаны с использованием символов
положение, название названий объектов (язык координат),
ограниченные операторы (определенный язык), и два разных кмд
umversality

Мы уже говорили (§ 3) о природе нашего языка
рассматривается как язык эоординат (символы положения как
аргументы) В этой форме языка есть существенное
Тактическая разница между Ntuation-Urmt для позиций, и
другой детирмнауона, с помощью которого любые свойства
позиции изложены Последние мы будем называть quahiaitve термины A
отношение ситуации м простейший случай будет выражен с помощью
аналитического (или противоречивого) предложения (например, «Позиции 7 и
6 соседних позиций ") С другой стороны, количественный
Соотношение, в самом простом случае, будет выражено через
синтетическое описание (например, «позиция 7 добавить позицию 6»
одного цвета "). Первое предложение определяется
логическая операция, а именно доказательство, последнее, с другой стороны,
может быть изучен только на основании эмпиничных наблюдений, то есть
сказать, путем вывода из наблюдательных предложений
существенная разница, которая затрудняется, когда язык
так сконструировано - как приемами, принятыми до сих пор - ^ что
Ситуация детализирована и выражены количественные отношения
сделано синтаксически идентичным образом

Мы будем называть символ Языки I и II определенным, когда
Это либо неопределенная константа, либо определенная константа
Цепь которого не имеет неограниченного оператора, в противном случае неопределенный

Выражение будет называться d ^ nsle, когда все константы, которые
происходят m, это определенно, и когда все переменные m это ограниченно
связанный, иначе неопределенный

Все определенные выражения закрыты. В случае
Применительно к языку I понятия «определенный» и «закрытый»
идентичные, одинаковые, «неопределенные» и «открытые». Мы называем меня определенным
язык, потому что, я все константы и все замкнутые выражения



rUJTI. TlIEDmXIISlJO.'CDXCEI


; 4 ^ '

ДФ и ИИТО. В самом строгом смысле, всего 8 ланттайре во всех
ejrj'ressinns являются определенными, может быть, он назвал определенный язык.] [На
допустимость неопределенных понятий, сравните §§ 43 - 45.]

«Вычислить» числовое выражение, скажем, 3i «означает: перевести *
форма 3i ttito ; 1; или, более cxaetlr, чтобы pioTC предложение
форма 3i = ; t. «Resoh'c ** предложение, скажем 5j» означает: либо
питии-е или опровергнуть это. Теперь можно показать, что rrvn- dffjdtf 3i
ht eati / Zarei; CNI, что Rvery Itfoatt ; | Эон он RW / Рей. Более того,
существует определенный метод, с помощью которого этот расчет
и разрешение соответственно может быть достигнуто. Это субед
сокращение, которое будет объяснено лучше. Если pri является определенным pTj * и
Фуй это определенный Фудж *. тогда ptj (; t „... St,) всегда разрешима, и
фу, (; tj, ... ; t,) является вычислимым alwaj-s.


§ 16. О турне 5 .М

Некоторые из тенденций, которые обычно обозначаются как
«финитисты» или «конструктивисты» находят, в определенном смысле, их реализацию
на нашем определенном языке I. «В определенном смысле», пусть это будет
отметил; поскольку эти тенденции, как правило, только
смутно сформулировано, точное утверждение невозможно. Они есть
chiefiy, представленный / iifiaft'omm (IV> incaji; и в современном
бои, прежде всего BrouTver; также Вейль, Хейтинг и Бетикр) и
союзнические мнения (например, Ф. Кауфман и Витгенштейн).
Точки соприкосновения в настоящее время будут указаны точно, но наши
собственная точка зрения отличается от рассматриваемой tCDdcncies одним существенным
уважать. Мы считаем, что проблемы, с которыми сталкивается интуиционизм, могут
быть четко сформулированы только посредством построения
Calwlus, и что больные неформальные люди должны быть
опоясанный лишь как более или менее расплывчатые предварительные
минусы ^ ttion. Большинство интуиционистов, однако, из


позже скажу что-нибудь о его методе,
tuiftn * понял, что все плюсы и минусы

В частности, речь идет о ccitli ihe fonns of »caf cufus,
«° mt i» галстук от; ;; \\ - это может быть

. ложь! Hot Mteste Tec DDL RF ::



47


§I6 ON INTurriosiSM

тот или иной язык, который будет построен "" или, из
Теоретическая точка зрения '^ Какие последствия наступят, если мы
структурировать язык тем или иным способом

С этой точки зрения догматическое отношение, которое делает так много
бесполезные обсуждения исчезают, когда мы здесь строим наши
Язык Я так, что это определенный язык, и, следовательно,
выполняет определенные условия, установленные интуиционизмом, мы не делаем
тем самым предположить, что это единственно возможное или оправданное
форма языка. Напротив, мы исключим
Язык I как вспомогательный язык м, тем более полный
Язык II и форма обоих языков будут рассмотрены
условно

На языке я все pti и fU | безусловно, вопрос
Дефицит ПТ | можно отнести к определенному числу или нет, или
имеет ли определенный fuj определенное значение для определенного числа или
нет, 13 всегда разрешимый Этот факт соответствует интуиционисту
Требование, чтобы не было допущено никакой концепции, для которой метод
резолюция не указана. Кроме того,
операторы м у меня есть результат, который умверсахты неограничен, хотя
Это может быть положительно выражено (а именно, с помощью свободных переменных},
нельзя отрицать, мы можем только сказать, либо «P {*)», что означает
«Все числа имеют свойство P» или «что означает

«Все числа имеют свойство nol-P», «Ни один номер не имеет
свойство П »С другой стороны. «Не все номера имеют
свойство P »не выражается m I, во II оно будет выражаться как
Это предложение будет относиться к II (как к
Гильберта и Рассела) в качестве эквивалента мне
* (3 ») (~ P (*)) ', что означает« Существует (как минимум) одно число
который имеет свойство не P "В I нет таких неограниченных
экзистенциальные предложения, и этот факт также соответствует условию
заложенный интуиционизмом, а именно, что экзистенциальное предложение
может быть указано, только если можно привести конкретный пример,
или, по крайней мере, метод, с помощью которого пример может быть
построено конечное, ограниченное число шагов для
Ученики, считающие, что не может быть решено
быть «недопустимым» или «бессмысленным» («бессмысленным»)
вполне ясно, однако, является ли (и не совсем точно, в каких пределах)
в соответствии с их мнением, экзистенциальными предложениями и, возможно,
даже отрицание универсальных предложений также должно быть исключено



т. «Я. TIIE DEFINITE L.WCUACE 1


4S

люди с формальными правилами формирования, или только определенные
возможности трансформации должны быть в каду. Вопрос в
t-ol \ Td IS, aboA-c all, вопрос косвенного доказательства посредством
опровержение нелатисального приговора.

Давайте возьмем пример: (пусть «P» будет

W (PW) (S.), ~ W (PW) (SJ, Bx) (~ PW) (S,).

В математике дасдеала (и, следовательно, также в л
Рассел и Гильберт, а также в нашем II), когда сокращается объявление
ohnadum, 6-й Sj выводится, а затем из него экзистенциальный
предложение Sj. Для того, чтобы сделать вывод, приводящий к
неограниченное, необусловленное экзистенциальное предложение, которое Брауэр вновь
существительные так называемые Lars of Exeludfd Mi'dJlf. Язык-форма
из I, однако, показывает, что тот же результат может быть достигнут другим
методы, а именно, посредством объединения неограниченного
операторы. В I Sx можно перевести в 'P (x) *, но Sj и
не переводятся в 1. Здесь Закон Exduded Middle
остается в 1 (Теорема 13 ^ ^ исключение этого закона, как
хорошо известно, приносит с собой серьезные осложнения, которые не
происходят в I. Таким образом, язык I удовлетворяет основным условиям
Интуиционизм проще, чем предложенная форма языка
Брауэром (и частично осуществленным Хейтингом).

В I универсальность выражается двумя разными способами: свободным
переменные, и необъяснимые opverators. Потому что последняя дуга
всегда ограничен в I, два метода выражения не равны
значение. Мы можем использовать эти две возможности выражения в
Для того чтобы выразить разные виды вмрррдлитв.

Давайте рассмотрим несколько примеров: «Все железяки на
этот стол круглый ». 2 а. «Все железяки - это кусочки металла».
ZB. «Все железные дуги намагничиваются».
время зависит от эмпирической проверки ряда начальных
случаи; предложение такого рода может быть определено только в ограниченном
домен. Следовательно, ограниченный универсальный оператор лучше всего приспособлен к
^ сформулируйте это. В случаях 20 и zb происходит неограниченная универсальность.
Действительность этих предложений не может быть определена тестированием
случайных случаев. Предложение 2 с является аналитическим и следует из
определение «железа». Приговор афхас (как и все так называемые законы
природы) характер гипотезы. Такое предложение
Кулон при принятии конвенции, которая в свою очередь



49


§16 ОБ ИНТУМОНСМЕ

зависит от тестирования отдельных экземпляров
свободных переменных адаптируется к формулировке неограниченного
Умверсахты из примеров аа и ай
Ф. Кауфманн справедливо подчеркнул разницу между
два вида uiuversahty (он обозначает их, общие с
Гуссерль, как индивидуальный (я) и специфический (универсальный) универсальность [WTicther
его хитрость, основанная на этой дифференциации, логики
настоящее время, особенно Рассела и Теории
совокупностей, полностью оправдано, здесь не рассматривается]
Возможно, форма языка 1 представляет собой реализацию
часть идей Кауфминна, но это невозможно решить
Точно, Смее Кауфманн, как Брауэр, не сделал ставку вниз
основы построения формальной системы A отклонение
из языковой формы языка я состоит из того факта, что
Kauftnann, как и Wittgenstem, рассматривает предложения типа zb
быть недопустимым, поскольку они не являются ни аналитическими, ни ограниченными, и
М следствие не может быть полностью проверено каким-либо образом В
В отличие от этого взгляда, языковая форма I также допускает
синтетические неограниченно универсальные предложения

§ 1 6a личность

Следующие объяснения касаются символа
* =: * рассматривается как символ идентичности в более узком смысле
(то есть, используется между 3 bctiveen objea-обозначениями)
а не как символ эквивалентности (то есть, как используется до
твин г) символ идентичности происходит на языках I и II
(как и в языках Фреге, Бемана, Гильберта) как
определенный символ После Лейбница Рассел определяет 'x = ^' m
следуйте по пути «* и вы согласны со всеми своими (элементарными)
perties "Витгенштейн полностью отвергает символ и предлагает
новый метод использования vambles, с помощью которого можно избежать

Философские дискуссии обосновывают обоснованность этих
различные методы кажутся нам неправильными
вопрос о создании коувенуона, чьи технические
эффективность может быть обсуждена
второй или третий из этих методов не должны использоваться вместо
из первых м ;;Языки I и II Как это происходит, Leibntz-
Метод Рассела применим только к языку II, здесь



50


T.SST 1. TflE DmNnX UOCCTUCE X


конечное окончание (3, = js) = (pi) (Pi (5 i) DPi (5 :)). Ag ^
Это определение иногда объективно (для инстинктов на
Пир Витгенштейна, Римси, Бемснн)
возможно, чтобы два разных объекта совпадали во всех
свойства. Но этот ohjecbon отклоняется, как только «все про *
perties »понимаются как побуждающие к положению. То есть
уже верно e \ 'en для имени * linguagcs, и, безусловно, верно для
языки координат: 3 и 33 обозначают то же место, когда
каждое свойство позиции, которая выполняется для 3, я удерживаю jdso для 3 r
В любом случае это было бы достаточно в определении вместо * aH pro *
преследует «сказать * все свойства позиции» (например, для
вместо pa можно использовать вид vanible, ограниченный ptj, скажем, saypj).

Критика Штетгенштейна идет еще дальше: он не смешивается
отвергнуть определение Рассела, но отказывается использовать символ
идентичности в АС. Но нам кажется, что все, что вытекает из его
Замечания по поводу этого символа заключаются в том, что предложения вида 3 ®S
по крайней мере, в самых простых случаях - не syntheuc, но аналитика: это
Нам не кажется, что такие предложения все равно
недопустимо. Во избежание использования символа идентичности,
Витгенштейн предлагает использовать правило замещения, которое отличается
(Например, обычно используется как в математике, так и в логике.
HU правило, что для разных Tiriables разные константы должны быть
замещенный. Более короткая форма, • P (i, y) ',

Guage соответствует обычной форме сеотенции «» *. (zaj) 3P (x, y) ».

с другой стороны, «P (*,> •) VP (x, x)» соответствует предложению
Поскольку документы Витгенштейна не формулируют никаких новых правил
замена, но только приводит ряд примеров, это не дорого
Мы собираемся выяснить, что такое метод hb. Дозирование
Метод переменных переменных приводит к определенным осложнениям.

** для нас было бы лучше сохранить обычное использование
1 ° * ^ * ° ^ * y • Тити с ним в то же время обычный

правило замещения.


instsnee. не можешь

полученный из 'Pfr step. «PfO.y)»

выведено sentPn ... «n» '** «Можно понять, почему в

^ заменить на 'у *. Отсюда в

из этого 8 ^: вызвано написанием seznethmg

должен быть установлен. • • «Для этого подойдут новые правила



§L6a. iDornTT 51

Использование Расселом символа идентичности для определения
конечные классы по перечислению их элементов одинаково
по нашему мнению, однако, в этом нет необходимости
отклонить эти классы, но только наблюдать разницу (конечно
важный) поддерживать между ними и теми классами, которые
определяются посредством свойства м в более узком смысле этого
13 осуществляется с помощью подходящих синтабкальных дифференцировок,
Существенным моментом является разница между pri (и m parbcular
Fimte Defimte PN) и Pt ^

§ 17. Принцип толерантности в синтаксисе

В foregomg мы обсудили несколько примеров
Неблагоприятные требования (особенно Брауэра, Кауфнанна,
и Wittgenstem), с помощью которого определенные формы языка
—Методы выражения и умозаключения - ^ будут исключены
Наше отношение к требованиям такого рода является общим
Это не наши заслуги
установить profubtbons, но для amxe в монастырях.

Некоторые из запретов wbtdi до сих пор предлагались
были исторически полезны м, что они служили, чтобы подчеркнуть
важные отличия и их можно заметить, но такие
запреты могут быть заменены определенным дифференциалом В
во многих случаях это вызвано одновременным
бон (аналог евклидова и неевхдийского гео-
метри) языковых форм разных кмдс - например,
defimte и mdefimte язык, или язык, допускающий и
кто-то не признает закон исключенного среднего Иногда
Можно заменить запрет, принимая во внимание
как правило, в рамках одной из форм языка
подходящей классификации выражений и мвесбгабона
Таким образом, например, в то время как Витгенштейн и
Кауфман отвергает как логические, так и антимебские свойства, я
scnpbve и логические предикаты были выделены в 11
определенные предикаты и mdefimte будут распределены, а их
дифференцированная собственность detensmed. И хзихер, м IJ, гнев шейл
различать ограниченно универсальные предложения, анализировать
ограниченно универсальные предложения, а syathebc неограниченно умверсал
предложения, в то время как> Vttgenstetxi, Kauffnann и Schhck все экс-



PARTI. TKE PEHNITE LANCDACE I


5 =

Чувак предложения третьего рода 0iws природы) из языка
в целом, как не поддающиеся полной аксификации.

В мотеле есть мотели. Evcrj'one собирается создать
собственная логика, т. е. lus собственная форма языка, как он пожелает. Ах что
От него требуется то, что если он хочет это обсудить, он должен заявить
его методы дорого, и дают синтетические правила вместо философии
Софические аргументы.

Толерантное отношение здесь, насколько это возможно,
Мейкаль исчисление дуги обеспокоен, отношение, которое Tacady разделял
большинством математиков. В конфликте над логичным
Основы матеминков, это отношение было представлено
особый акцент (и, очевидно, перед кем-либо еще) Макнгер
([/ fifintiafaimuj] стр. 324 ф.). Макнгер отмечает, что концепция
constructidty, который innubonism absolubres, можно интерпретировать
оба стали гораздо уже и в гораздо более широком смысле. Им-
Персонализация псевдопроблем философии
применения множества милосердия к форме языка как
все станет ясно позже (см. J 78).



ЧАСТЬ n


ФОРМАЛЬНОЕ СТРОИТЕЛЬСТВО
СИНТАКСИС ЯЗЫКА I

§ i8. Синтаксис I может быть сформулирован в I

До настоящего времени мы различали объект-
язык и синтаксис-язык м, который sjmtax из
объект-язык сформулирован. Это обязательно два отдельных
языки? Если на этот вопрос дается положительный ответ (как это
Хербранд м связи с метараатематикой), то третий
язык будет необходим для формулировки синтаксиса
синтаксис языка, и так до бесконечности Accordmg к другому
мнение (что Витгенштейна), существует только один язык, и
то, что мы называем синтаксисом, вообще не может быть выражено - оно может только
показано "В отличие от этих взглядов, мы намерены показать, что
на самом деле, обойтись можно только одним языком; не,
однако путем повторного вызова синтаксиса, но посредством demonstiatmg, который
возникновение каких-либо противоречий синтаксис этого
язык может быть сформулирован в рамках самого этого языка в каждом
язык S, синтаксис любого языка, будь то
совершенно другой тип языка, или подъязыка, или
даже самого S - может быть сформулирован в степени, которая ограничена
только богатством м средствами выражения языка S
Таким образом, с помощью средств выражения нашего дефинированного языка I,
определенная часть синтаксиса любого языка, что более
Например, языка Рассела или Языка II, или даже
Сам язык я - можно сформулировать на следующих страницах,
последнее обязательство будет выполнено, то есть мы
сформулировать синтаксис I - насколько он определен - в / себя В этом
Процесс может случиться так, что предложение <3i от I, когда материально
интерпретируется как синтаксическое предложение, скажет что-то о Си
само по себе и без каких-либо противоречий

Мы различаем описательный и чистый синтаксис (см. Стр. 6f)
Предложение m в описательном синтаксисе любого языка может содержать
Мстрее, что это означает, что я делаю это на своем сайте.
определенный смысл позиции [символ занимает позицию,



54 пирог н. ФОРМАЛЬНАЯ КОНСТРУКЦИЯ ТИАРСКОГО СИНТА «OF 1
выражение занимает ряд позиций.] Пример: «На странице 33
В строке 32 этой книги встречается выражение вида «3 = 33i»
(а именно, 'ar = 2'). ”Поскольку язык I имеет достаточные средства ex *
пресса по его приказу с целью описания
Perty области дискретных позиций, описательно-sjmtactical
предложение такого рода может быть сформулировано в I независимо от т \ hcthcr
он описывает выражение другого языка или выражение
Я сама. Например, можно было бы продолжить
Троединство в неопределенном пр »для различных видов символов
выражения, которые будут описаны (позже мы вместо этого создадим
один неопределенный fiib, а именно 'ici', например, thepib

«Var» для фургонов, яма «LogZz *» для логических чисел,
ptb «Id» для символа идентичности и так далее. Давайте сейчас
обозначьте позицию на стр. 33, с которой начинается «= 2», с помощью * a
Тогда вышеупомянутое описательно-синтаксическое предложение может быть
сформулировал я следующим образом:

'Var (a) «Id (a') • LogZz (a")

Это синтетическое описательное предложение. Затем мы можем, далее,
определить ptb «LogSats» так, чтобы «LogSat2 (», u) »означало:« В
ряд позиций, простирающихся от * до x + u, происходит 6i ". Затем
предложение: «Каждое выражение формы 3 * 331 является Sj»
быть оказанным в я

'(Var (x), Id (xi), LogZi (xn)) D LogSatz (x, 2)';
это аналитическое предложение, которое следует из определения
«LogSatz


Статья 19. Арииметизация синтаксиса

Как мы уже упоминали, всегда можно заменить
любой ptb y fub. Несколько difTercnt ptb можно назвать однородными
если не более одного из них может занимать любую должность. Тогда это
способы, позволяющие заменить класс однородного pfb одним fUb,
Вы пишете одно значение из систематического или произвольного
к сожалению, каждому из них. [Пример: пусть класс
цветов, которые должны быть выражены, конечно. Мы можем
prm tvtry color by pi „Blue-. «Красный» и так далее. Эти пр,
тогда однородны и U.ercfore мы можем заменить их на
Спойте, скажем, со, нумеруя цвета, и



сс


§ 19 АРИФМБТИЗАЦИЯ ОП СИНТАКС

Подчеркивая, что «coI (a) = b * должно означать« Позиция a имеет
цвет нет б '] Аналогично, в формулировке синтаксиса I
m I, мы не будем обозначать различные километры символов
разные ptb (как, например, в примере, данном m § 18
«Id» и т. Д.), Но по одному значению, а именно «Зей» Мы будем соотносить
значения * zei 'для различных символов (символы-конструкции), частично
произвольно и частично m в соответствии с определенными правилами
Значения называются числами символов символов. Например,
мы будем согласовывать термин номер 15 с символом идентичности
Это означает, что (вместо 'Id (a) *) мы будем wnte * zei (a) = i5'
когда мы хотим выразить тот факт, что символ идентичности
происходит на позиции а не только экономика м примитивна
синтаксические понятия, но другие причины, которые будут обсуждаться
Позже обосновать выбор этого метода
синтаксис. (IiithisanthmetizattoQ, wemaheuseofthemethodwhich
Гддел [VnenUchetdbart} достиг такого успеха в
математика или синтаксис математики)

В общем, установление номеров терминов для разных
символы могут быть выполнены произвольно все, что должно быть предоставлено
для того факта, что для переменных, число которых не является
ограниченное количество неограниченных номеров терминов
- аналогично для 33, pr и fu Теперь мы будем указывать бесконечное
классы чисел для этих чисел этих кмдс символов
м следующим образом Пусть р пробегает все номера чернослива
g больше, чем 2 положения, число-член 3 должно быть куча
(что 13, простое число больше 2), термин число
определенный 33 должен быть ap * (то есть вторая степень некоторого простого
число больше 2), термин число неопределенного pt
должен быть р ', то из определенного pr, ap *, то из неопределенного фу,
ap ^ (а именно, термин число * zei 'должен быть 3', который
243), а это определенное fu, ap * Но не все числа
классы, определенные таким образом, будут использоваться как
номера их выбор будет определен позже
оставшиеся символы - а именно, неопределенные логические константы - ^ мы
назначить (произвольно) другие номера, а именно
на символ 0 (),

срок № 4 6 10 12 14 15 18 20 21 22 24 26 30 33 34.
[Последние три символа являются вспомогательными символами, которые не встречаются



56 ЧАСТЬ JI. ФОРМАЛЬНОЕ СТРОИТЕЛЬСТВО СИОТАКСА 1
в выражениях самого языка; см. стр. 68 относительно
их,]

Если любая эмпирическая теория сформулирована в I, то
Писательные примитивные символы этой теории добавляются к логическому
примитивные символы языка I. Аналогично в формулировке
описательный синтаксис; здесь «зей» является единственным дополнительным примитивом
символ. В следующем построении системы синтаксического
Определения, однако, «Зей *» не будет сначала использоваться. Ибо, в ТБ
этап, мы не имеем дело с описательным, но с чистым синтаксисом,
и в этом есть дуг бо аддиональный примитивный тимбо, так как чистый
синтаксис не более чем арифметика. Так же, как термин числа
соответствуют символам, поэтому серии терминов-номеров соответствуют
выражения. Например, серия 3, 15, 4 соответствует
выражение «х = 0». Понятия и предложения чистого синтаксиса относятся
Теперь не к серии символов, но к соответствующим крикам
Термин-номер. Таким образом, они являются арифметическими понятиями или предложениями.

Формулировка s) 'ntax становится технически проще, если
мы идем на один шаг дальше с методом корреляции
номера. Мы установим правило, согласно которому для каждой серии
term-numbers, одно число - мы называем его серией «число
серия - будет однозначно исправлена. Таким образом, мы не будем
больше приходится иметь дело с серией чисел, но только с одним
номера. Правило выражается следующим образом:

принять за номер серии для scries, который состоит из n
tcrm-imrabcrs, Aj.kp, где pi (i = i to n) - простое число ilh
число в порядке величины. [Пример: серия 3, 15,
4 ^ и вместе с ним выражение * x = 0 ', имеет номер серии
2 * 3 * * 5 **] Поскольку разложение числа в его простое число
Факторы уникальны, ряд из десятинных чисел в исходном порядке
может быть восстановлен из серийного номера, и, следовательно, также
^^ возрастное выражение, с которым связан номер серии.
[c ru es, заявленные ранее относительно числа терминов, являются дополнительными
но не обязательно так устроен, что ни один термин-номер не находится на
в то же время серийный номер любой серии.]

Метод построения серийных номеров может быть
неоднократно применяется. Например, к доказательству как серии

Вначале соответствует серия серий номеров.

В соответствии с описанным методом мы можем затем
tenet-senet.numher к этому scry серийных номеров.



§19 АРИФМЕТИЗАЦИЯ СИНТАКСА 57

С помощью этих условий о числах tenn и senes,
все определения чистого синтаксиса становятся арифметическими определениями,
а именно, определения свойств или отношений между числами
Например, словесное определение «предложение * больше не будет иметь
форма "Выражение называется предложением, когда оно состоит из
символы объединены таким-то и таким-то образом », а мстед -«
выражение называется предложением, когда его номер серии выполняет такие
и такие условия », или, точнее,« число называется
senes-номер предложения, когда оно выполняет то-то и то-то
эти условия касаются только видов и
порядок символов выражения, то есть с
kmds и порядок показателей показателей чернослива множителей
Таким образом, мы сможем выразить их чисто
метрически AH предложения чистого синтаксиса следуют из этих
антметические определения и, таким образом, являются аналитическими предложениями
элементарная арифметика Определения и предложения синтаксиса
мантексид м таким образом не отличаются принципиально от
другие определения и предложения, но только в той степени, в которой
мы даем им определенную интерпретацию (а именно синтаксический
mterpretatioo) с определенной системой
Если этот метод антметизации не применяется, определенные
Точности возникают в точной формулировке синтаксиса.
давайте рассмотрим синтаксическое предложение «^ не доказуемо»,
что означает "Нет предложения Senses havmg в качестве последнего предложения
Это доказательство ». Если синтаксис не антметизирован, а вместо этого был
предложенный ранее, построен с помощью pr ^ ('Var и т. д.), мы
может быть истолковано как теория, связанная с определенными физическими
объекты, а именно, символы wntten символов в синтаксисе этого
КМД. Конечно, можно выразить "Там не существует фактического
Доказательства Gj », но предложение не
демонстрируемость Gj означает гораздо больше, а именно:
Gj - это возможно », чтобы можно было выразить такое предложение
о возможности м не антемптизированный синтаксис (неважно
будь то физически или нет), синтаксис будет
должны быть дополнены теорией (не empincal, но
аналитическое) рассматривает возможные расположения любых элементов -
То есть, с помощью чистого комбинаторного анализа доказывает, как
когда-либо, чтобы быть намного проще, вместо того, чтобы создать новый
бинаторный анализ этой кмм ма не антметической формы, чтобы использовать



§ 20 CCNZRAl. УСЛОВИЯ


59


Di. Н.Ф. (*) = * «

D2, I. sum (0.> -) => -

2. sum {x ',>) = nf (sum (x, jf))

D3. X. prod (0,3 /) = 0

2. prod (* ', y) = s \ im (pfod (x, 3'),> ')

D4. I. po (0, ^) = 0>

2. po (A ', y) = prod (po (fe, 3 ^), y)

Ds. Горшок (х, к) = ро (к, *)

D6. I. fak (0) = 0 '

2. fak (* «) = prod (fak (x), * ')

Пояснение, Di-6 Явный (Dt.s) или regressiTc (Dz, 3, 4, 6)
Определения здесь даны для иска Фу, имеющего значения Преемника
(в x), сумма (из X и y), произведение, мощность ('pot {x, y)' 'x *' m
обычные символы), фактонал (сравните стр. 14) 'po * является только
auTiIiaiy концепция «горшок», это необходимо, потому что мы предусмотрели
что первый аргумент должен быть принят за то, к которому относится
Gression Reisis

С помощью регрессивных определений, указанных для «суммы» и
* продиктовать обычные фундаментальные законы арифметики (
строгие, объединяющие и распределительные законы) и, кроме того, все известные
теоремы элементарной арифметики можно доказать с помощью
RI4 (полная информация)

Dy. X. IsO ', 2-2, 10, 10 с 9 i, 34 ai 33>

Пояснение Есть столько определенных 33, сколько нам потребуется.
Здесь деомал в нескольких местах принимается за одно неделимое

D8. Grgl (х, у) ~ (3u) х (х = 5um (у, тт))

D9. Гр (х, у) с (GrgI (х, у). ~ (Х = у))

ДХО. Tlb (x, y) = (3u) * (* = prod (y, a))


Ключ к символам, определенным вами
fak factonal {Факултай)

Grg! больше или равно {грБитер
Одер гутех)

Gr больше (grffuer)

Tib делится (я <J / 6 <это)

Pnm, pr, pnm простое число
(FTT ^ AAL)

gl term number {GltedsaJtTl
Ing длина {Lange)
letzt l23X {Uizte)
Рейхе Сенес {Рейхе)


этот раздел (продолжение)
zus coxapposedizusammengezetxt)
эпоха заменена (erretxt)

INA ЕСТЬ выражение (мт
AiadmcK)

Инар м экспрессивные смыслы
(m der Auidruchzreihe)

AlnA выражение в выражении
sion (Auidnuk m Ausdruek)
AlnAR ex pre s s ion in eipres-
Stonal Senes {Ausdruek tn der
AmdrucksTethe)



6o ЧАСТЬ U. FOKMAL CONSTKUCT.ON OF T..F SYNTAX OF.

D ... PrimMs (~ (x = 0). ~ (R =.). Wx (("= OV
(u = * W ~ Tlb (jf, «i)))


ДРА. X. pr (0. *) = ® / v

2 pr (n'.x) = (Kj) »(PfimC).). ™ (l, p).
GrO-, пр ( «, аг)))

Е, FTA «из; pr («. A) .. Uff»! • <; ”P" ™

содержит ос о фактор в х.


2, prim (Fi) = {Km) nfIf2k (prim (n)) l [PomH.

Gr (m, prim {»))]„

& pte2.io «: pnmW .. .he mh pnme ou.hher (.ceoAFS
величина).

D.4. , ., К

Exfl <MATL <м! el (".») n * «" * teno-numbe. МЧП «П« »"
Энея> онемение <г х.

DJ lng (») = (K2) 2 (стр. (N'.x) = 0)

EAPtoohoo; Лос (х) я. Uie lens * (.ht i .os «y» «« P »

слагаемые) ряда с числом сенсе х-
DX6. Iem (*) = gl (lng (ar). »)

Объяснение: одолжено (*) - это последний член-число
номер серии *.


Dx7 »x. rcihe (0 = банк (* x *)

2. rcihe2 (r, ») ss prod (rcihe (t), pot (3, t) ')

3. reihe3 (r, l, u) = prod (reihc2 (r, 0.pot (5 »»))

Объяснение: reihe (t) »s серийный номер (2 *) из»

Я - единственный термин-число; reihe2 (f, 0 - номер серии (•

из серии 0, номерами термов которых являются r и X;
(В «reiheJ * * 2» не JJ, а составная часть м «ви
символ 'rcihe2'.)

\ Vc теперь введет следующие сокращения для
exphnations. Вместо того, чтобы писать «срок-число .-- ^

напишите * TN ../ (например, 'fWnegatioti sjTubol *, который равен 2t) -
Вместо «серийный номер ...» мы напишем

позиция 'st ^ Cp'rator * и т. д.). Вместо серий

номер серии ... * wc будет wnte (fof



§20 ОБЩИЕ УСЛОВИЯ 6l

«SSNproof») Если мы читаем словесную транскрипцию определения
пренебрегая индексами, мы получим синтаксическую интерпретацию
определение (например, m объяснение D i8 «zus (*,> ')»
Это смысл, который состоит из двух частичных значений x и y ")
С другой стороны, если мы прочитаем транскрипцию, включая
индексы, мы получим (обычно в форме, не буквально точной)
антематическое толкование определения (например, в
случай D i8 «zus (*,>) - число старших, которое
состоит из двух частичных значений, имеющих числа x
и ^ ») В дальнейшем мы всегда будем работать с
индексы, но позже мы будем использовать их только тогда, когда это будет необходимо
сделать это ради блага

Di8. т. zus (*, yi) = (K2) pot ^ nm (сумма [Ing (x), Ing (y ')]),
sum (*, y)] [(«) Ing (*) (gl (n, x) = gl (n, *)). («) BG (у)
(~ («A 0) 3 [gl (sum [bg {x).«], Z) agl («,;»)])]

2. zus3 (*, y ', *) = zu3 (zus (*, y'), x)

3. 2ua4 (ar, ^, 2, u) = zus (zus3 (x, y, 2), u)
и так далее

Объяснение zus (x, ^) - это ^ senes wbch - это eotnpostd из двух
s * ^ sub-senea * andj '(не Wtertns.asdifferentfrorn'reiheS (*, »)')
Соответственно * zus 3 * и т. Д., В случае композиции из
или более ® ^ к югу от 8enes

Дх 9. ers (x, n, ^) = (Kx) pot [pnm (сумма [bg (x), bg (> -))),
сумма (x, ^)] ((3 «) x (3 t») x [(x = zus 3 («, rcihe [gl (n, x)],
. С)) (х = 2us3 (В, J -, «)), (п = IIF [BG («)]) я)

Объяснение tn (x, n, y) является выражением SNc, которое следует из
выражение x, когда n-й S''term mx заменяется на
S '* выражение у

D 20. InA (r, x) s (3 n) bg (x) (^ («= 0). [Gl (n, x) = t])

D21. InAR (r, r) = (3 k) bg (r) (~ (A = 0) .InA [r, gl (A, r)))
D22. AInA (x, y) = (3 «) y '(3f) y (y = * us 3 {B, x, o))

D 23. AlnAR (x, r) = {3 A) bg (r) (A = 0) • AlnA [x, gl (A, r)])
Пояснение, D 20 TNsymboI t встречается в выражении sn x
Dzr t происходит в выражении ssNexpression senses r
77 nf ej cpf ess i on x'qcepyy ^ fafOAieray gpnipwtif latp n iptfrparf /
m выражение y D23 выражение x встречается m выражение
выражения * senes r



62 р.л. ФОРМАЛЬНОЕ СТРОИТЕЛЬСТВО СИНТАКСА I


§21. Правила формирования: (I) Числовой
Выражения и предложения *


• D24. cinkl (») s7us3 (rcihe (6), младший, rhhe (io))

Пояснение: tlicSSbrjckcringofar,

что есть, выражение

D35. Вар (к) Н (PriTn (J) .gr (я, 2))

Объяснение; * V4r (i) * означает, что * простое число больше
3 (таким образом, в качестве числа термина это "^^ variabJe).

Д 2 $. DeftZzl (j) 5 (3 м) * ^ 'ar (м) «[* = банк (рт, 2)])

D27. DcflPradl (*) и DaS. D ; fiFul (0. IMy
Сформулировано в логике.

; .vp / (», D26> 2S: * является определенным (или подходящим * подходящим)
когда t - вторая (или четвертая или шестая соответственно) степень простого числа
число больше 2. (Coneenung the addiuon ^ '1 * см. Isier.)


Подчеркните, что обозначение термина ^ numbfr of dtfned fymhoU
Мы назначили термин-мюнхен определенным символам
различные виды чисел из трех dss $ e », а именно второго,
четвертая и шестая степени простых чисел больше 2. Мы


• Ключ к символам:
einkl: брекетинг (EinJUawme
ступенька)

Var: v-arisble [Veriahlf)

Dcftl ^ a, DeftPmd, DeftF \ i; де
оштрафованная цифра, предикат, fune
tor (dffittiertfj Zahlteichm
PrSJtkat, Funklor}

UndPrad. UndFu: не определено ...
«(Tmdr / im'rrfa.,.)

Zi: цифра {ZahUeithen)
Prfld; предикат (fVdiifcaf).
AOp, EOp. КОП, СОП: уни
versal, eiistenfia], описание
. приговор оператои
(Все; Exuieiu; K; Satt '
Oprrator)

Op; Оператор (Оператор)

ZA: числовое выражение (Zaht-
astsdrwk)

нег: отрицание (Непация)


dis: дизъюнкция (дизъюнкция)
Lon: соединение (ATonjiiwAri'iw)
имп; Impb'eation (Импликация)
#q: эквивалентность (rlgimw / ms)
Веркн: Джанкшен (ЙеркнОпфикс)
концерт: уравнение (Gleithmg)

Сата: предложение

VR; > 'ariable-scrics (Variableti')
Райя)

DKstr; непосредственно cinstiucted
(tranittelbar konstruiert)
Констр; строительство (Kwutn / **
ribft)

KonstrA; построенный экспресс-
Сион (Jionifn / irr / ff Atuiintck)
Geb: связанный (gehwtden)

Fiei, Fr: бесплатно
Оффен; открытый

Ceschl; dosed ^ ercWojjen)



63


§21. ЧИСЛЕННЫЕ ВЫРАЖЕНИЯ И ПРЕДЛОЖЕНИЯ

однако позже установит метод определения символов m
таким образом, что не все Numbeia из трех упомянутых классов
использоваться в качестве номеров терминов для определенных символов, но вместо этого только
эти числа, которые выполняют определенные условия Мы называем символом * ^
основанный, когда он либо выполняет эти условия или является примитивным
symboL Эти условия будут сформулированы таким образом, чтобы
любой символ, который их выполняет, обратится через свою цепочку
определений в pmmtive syinbob Мы называем выражением
основанный, когда каждый из его условий является основанным термином.

Те термины, которые будут определены, и из которых
обозначения (а именно, слово-обозначение, готический символ и
предикат m формальная система) связывает дополнительный '1 * или
«2» (от «определено 33I», D26, до «построено», D78} также
mclude символы и выражения, которые не основаны. Эти
только вспомогательные термины для определений, которые последуют позже

D 29. nadPrad (r, n) s (3 k) s [\ ^ ar (i] (*)

( »Страдалицы (PNM [горшок (я, п)], 3))]

Аналогично D 30: UndFu (r, 1?)

Explarustum s является неопределенным pr * (или fu *), когда pnme nisnber
k больше chan 2 существует, так что r является третьей (или пятой) степенью
номер телефона. (Ты делаешь так, чтобы
номер n, который необходим для синтаксических правил, может следовать
безоговорочно от числа-числа пи * или фу *)

D 3X-Zzl (j) s (DeftZzI (s) V Var (*) V (1 = 4))

Объяснение s - это t '^ 331, когда sis является либо определенным' ^ 3} 1, либо
3 или Ttu (см. стр. 6)

D 32. Pradl (j) = [DeftPradl (r) V (^ n) s ^ ndPiad (x, b))]

Объяснение xuai ^ pil, когда x является либо определенным pri, либо
неопределенный шт.

Аналогично D 33; Фул (х).

(Что 13, фул)

D34. AOpl (a, J, r) = [Var (x) • (a = zus (einkl [reihe (x)], o))
• ~ -InA (x, t7)]

Аналогично D 35: EOpl (7, xv), D36: KOpl (z, x, p)

D 37. SOpl (s, x, c) = (AOpl (7, x, o) V EOpl {s, x, o))

D 38. Opl (*, X, v) = (SOpl lx, t, v) V KOpl (s, X, 0))

Пояснение, 034 * auSMinnrmo / operatorl с TNopeia-
tor-vaiiable t и sv iffni t ti, то есть sr имеет вид (3 ^ Sj,



64 TART II. ФОРМАЛЬНОЕ СТРОИТЕЛЬСТВО СИНТАКСА 1

где Ji не встречается m? lj. - D 35-D 38: экзистенциальный операторl,
K-операторl, оператор предложения I (то есть универсальный или
экзистенциальный операторl), операторl (то есть, предложения или
K * opcratorl).

D 39- ZAl (s) 5 (3 i) 2 (3 t>) = (3 «) r (3 y) z ([Zzl (r). [Z = reihe»)
(r))] V [s = zus [r, reihe (i4)]] V [Ful (/). (s = zus [rcihe (r),
cinkl («)])] V [K0pl (j *, i, o)« (»= 2us [_j ', cinkl {u!)])])
Пояснение: zii anSN3i ^^ имеет одну из следующих форм:
3JL. ? Ii. FUL (2l,). (K3) 9r, (2I,) (см. С. 26). Здесь 91. 91 ,. и 91, дуга
любые выражения вообще; с другой стороны, в случае
32 (D 53) 9 (я 3 ^ 1 91, это серия, состоящая из нескольких 3 ^ ^ TL
запятые, а 91, это G2. В отличие от 32, 3 («ZA»,
D 87) основано. Аналогично в случае 6l (D 47), S2 (D 54),
и 6 («Сатц», D 83).

D40. neg (x) = zus (reihe (2i), einkl (a :))

D 41. dia (*, y) = rua3 (cinki (x), reihe (22), einkl (> •))

D42t кон (х, у); D43J imp (x, ^) и D44t aq (», y), являются
аналогично.

Пояснение: Если x и y - выражения SN, 91 ,. 91 ,, то нег (х)
aN „^^ jn <m '- (9I,), dis (x, y) даюнелион (Sj) v (21,); случаи
eonjunetton (kon), implieation (imp) и эквивалентность (Bq) являются
аналогично.


D 45. V erkn (x, y, z) s [(x = dls (jr, s)) v (x = kon (> -, z)) V (x * imp
(y, s)) v (x = aq (3 ', 2))].

Fi ^ / onotion: xis s: то есть х имеет

форма (9I) ottln (9I,), где у равен 9t, а х равен 91 ,.

D46. glg (*, y) = 2 us 3 (x, rcihe (i 5), ^)

E ^ lonatim; Если * и y - выражения дуги 9I „21 ,,, то glg (*, y) •»


D 47. Satzl (*) e (3,) X (3 p) X (3, (3 года) - ([., = Гиг (против)] V

[Pradl (1). (x = zus [reihe (i), einkl (w)])] V [x = neg (r)] V

Verkn (х, t-, тт) В [SOPI (у, т, р). (Х, = XUS [у, cinkl (тт)])])

или f ^, когда x имеет одну из следующих форм:

t® *) ® *** "^. (5) 9 Ii (9 U или (33) 91, (91,)
см. стр. 20 ;. Разница между СИ. © 2. и S аналогичен

между 31, 32 и 3.

”" * ; rft ('' 'I ”" "(* ;")] • w> ”5 Вт (3”) *

[{k ^ O) V ([A = prod (2, m) i]. Var [gl (ft, x)] ") V ([ft ^ prod
(2. «)]. [GI (ft.X) =, 2))]) ^ ^



§21 ЧИСЛЕННЫЕ ВЫРАЖЕНИЯ И ПРЕДЛОЖЕНИЯ 65

Пояснение Выражение * называется n-tenned lanable-smes
когда он состоит из n переменных и запятых

D49. UKstrl (jr, tc) = ([ZAl (to). (S = 2us [tc, reihe (14)])] V

fO '"1 - • _ r /,„.


Пояснение Выражение s вызывается непосредственно из
одно выражение te, скажем UIi, когда оно имеет одну из следующих форм
I ^ 1 ', где Hi - это 31 »a' - (Si), где 2! I - это SI, 3 ful {2Ii)
или пил (2lj, где 2ti является переменной senes, 4 2i, 3, где 9li является
vanable-Senes


D 50. UKstr 2 (r, c, w) 5 [(3 *) * (3 года) ® (ZAl (o) • Satzl (к) •
Opl (3 ', J, o) «(3 = zus [>, eudd (to)])) V (ZAl (o) * ZAl (to).
[* = glg (o, te)]) v (Satzl (t>). Satzl (to) »Verkn {z, v, to)) v
(3 n) Ing (r) (V ar [gl («,»)]. ZAl (to). (Видно (r, n, to)])]
Пояснение Выражение z вызывается непосредственно из tcmstructed из
другие выражения v, скажем, когда оно имеет одно из следующих

понижающие формы I (3)? I | (1lli) или (33)? Ii (2 *) или (K3) aj (? I *), где
^ Я 31 и% Si, 2 Sxsg 4> vvhere и 3 л>
3 (lli) t) tcfR (S |)> где и ^ - 31, или когда 4, x результаты & om
9 i если a 3] $ заменить на где ^ 1 $ a 3l


Dfi. Konstrl (r) s


Объяснение r это когда r это SSNsenes

^^ выражения, каждое из которых является 33I или построено напрямую
& om одно или два из предыдущих выражений, встречающихся в выражениях
(А в этом кмдс потребляет 3л и 3т, или, точнее, м
в соответствии со следующими определениями из 32 и 32)


D 52- KonstrAl (*) n (3 r) пот (pnm png (*)], прод [*, Ing (*)])
pionstrl (r) • Qetzt (r) = x)]

Пояснение SNespression x называется созданным! когда он является
последнее выражение в SSNconstnictioal [Предел для результатов г
из следующего соображения Пусть r будет кратчайшим
из которых последнее s '^ предложение a * Тогда lng {r) ; lng {*),
каждый простой множитель r равен Spnm (lng (*)), число этих
факторы »Slng (*), их показатели <*, поэтому
rgpnm (lng (x)) ** '“* ^ * ^]

D53-ZA2 (x) 5 (KonstrAl (*). ZAl (*) ')



66 ЧАСТЬ 11. ФОРМ.У. СТРОИТЕЛЬСТВО СИНТАКСА I


D54. Satz 2 (*) s (KonstrAl (*). Satzl (*))

Пояснение: выражение x является 3 ^ (из S 2), когда оно
constnictedl и 31 (или SI, соответственно); см. объяснение
из D 39.

D 55. Геб (с, X. n) s (3 0 * (3 *) * (3 «) ^ (3 с (3 р) (3 o) с»
[(x = zus 3 (l, s, u)) »(x = ztis [j *, einU (ta)])« Opl (^ 'jf, c).
ZA 2 (c) 4 S 3 tz 2 (ts) 4 Gr (ti, Ing (f)) 4 Grgl (сумма flng (<),

tasW]. ")]

Пояснение: ™ vanab! Esi * называется связанным с SNeipression
X в n-м месте (где в этом месте необязательная необходимость)
если выполняются следующие условия; В х выражение х из
форма 3 t (9 |) происходит, где? I, Opentorl ненавидит 3 ^ как предел
и s как оператор -t-ariable; является S 2; n-е место х принадлежит
до 2 (см. стр. в).

D 56. Frei (», x, n) s [Var (r) 4 (gl (n, x) = j), - Geb (r, x, u)]
fxpfaKaltoR. Свободно видимый t происходит в n-м месте m x.

D 57. Fr (r, x) E (3 n) Ing (x) (F rei (tx it))

Пояснение: t происходит как свободный t'OriaMe m x.

D58. Offcn (x) s (3 т) x (Fr (r, x))

D59. Geschl (х) S ^ Offcn (х)

Пояснение: х открыт; x - это cfored ($ ee p. 21).


§ 22. Правила формирования: (2) Определения *

Если исчисление должно содержать определения, то при определенных условиях *
В его формулировке возникает трудность, которая
редко учитывается. Если все это требуется от

• Ключ к символам:

VRDef: переменная-серия для Deft: определенные (de / imfTl)
определено Z: символ (ZricAen)

DefZz, DefPrBd: определение UndDeskr: неопределенное описание

цифра, предикат tive [dttkriptiv) символ

DefexpFu, DefteipFu: явный Undeft; неопределенный символ

определение функтора DefKette, DeftKctte: определение:

DefrekFu, DeftrekFu: сеть магазинов (DeJinihonenibeHe)
sive (refan-riti) определение основанного {hasiert)

функтор Deslcr; описательный

Dcf, Df: определение предложения Log; логический (/ ogircA)

(Defirdtiontsata)



§22 ПРАВИЛА ФОРМИРОВАНИЯ ОПРЕДЕЛЕНИЯ 67

Тиос признал в исчислении, что / удовлетворяют определенным правилам
образование, исчисление, как правило, будет противоречивым

Пример Например, D i SO) удовлетворяет правилам формирования
для определения m I (§ 8) с помощью D i, предложение
'nf (0) = 0' 'ДЕЙСТВИТЕЛЬНО, но предложение' nf (a:) = x '"'
аналогично определению принятого решения и с его помощью
Предложение '' «- (nf (0) = 0 ')' доказуемо Таким образом, m I, предложения
которые взаимно противоречивы, являются очевидными

Чтобы избежать противоречия, мы обычно добавляем
национальное требование ", чтобы определяемый символ не имел
произошло определение, которое уже было подставлено "Но
Требование такого рода является отходом от домена
исчисление и формальный метод В строго формальной процедуре,
обман относительно того, является ли данное предложение допустимым
определение может конкретное исчисление или нет зависит исключительно от
форма предложения и правила формирования
исчисление Но в силу вышеупомянутого неформального требования это
решение станет зависеть от исторического утверждения
относительно того, были ли определенные предложения плохо сформулированы ранее или
не. И то же самое верно для решения,
Монстрабильность данного предложения (как показывает наш пример). Теперь
как преодолеть эту трудность?

Для начала, очевидно, что трудности исчезают, если
в формировании языка S m вопрос, один из foUowmg
процедуры приняты

(а) Определения вообще не допускаются в S

(б) добавлено только конечное число частных определений
м S, и они ранжируются среди примитивных предложений S

(f) Любое количество определений, для которых правила формирования
даны, могут быть сформулированы в S Но определения не
допустил m доказательств, они принимаются только как предпосылки
[Таким образом, приведенный выше пример 'nf (0) = 0' * не является
monstrable, но только denvable из 'nf (j:) = x' '] Если предложение Sj
содержит определенные символы (le symbob, основанный на определенных определениях)
тогда, хотя это не само по себе доказуемо, это предложение, которое
следует в результате устранения определенных символов
Доказуемо

Регрессивно определенные символы не всегда могут быть устранены. В



6S ЧАСТЬ II. ЗА. \! Al. КОКСТБУКТИО ST-XTAX I
определенный язык, многие предложения элементарной арифметики
(например: 'prod (2,3) = 6') должны быть
ограниченное количество регрессивных определений, которые могут быть использованы
Уметь в доказательствах, необходимо. Таким образом, для такого языка -
например, язык I - вышеупомянутые вадж-ы из
трудно) бесполезны. Мы должны будем обнаружить некоторые другие
решение:

2. На языке I мы разрешаем неограниченное количество
окончания, в том числе регрессивные; но с помощью подходящих правил
позаботимся о том, чтобы из каждого определенного sjTObol узнаваемо
как это определяется. Это возможно в антропогенном sj'ntax. Мы
ранее установили класс чисел на срок
номера определенных символов каждого из трех видов, 53, pr,
и фу; но, несмотря на этот класс, мы оставили выбор открытым.
Теперь, однако, правила, которые будут установлены, определят этот выбор
таким образом, что из числа-члена определенного символа не
только его определение, а также, косвенно, вся его цепочка определений
будет следовать однозначно. Таким образом, каждое так называемое логическое свойство
любого предложения - из-за безумия, его demenstrability - становится
синтаксическое или формальное свойство; это зависит исключительно от формального
структура предложения, то есть по арифметическим свойствам
из числа терминов, которые составляют предложение.

RuU для выбора числа-члена определенного символа Oj в
Язык I: В определении Oj пусть Oj заменяется на
вспомогательный символ manent выглядит следующим образом: a 33 by C * с термином
число 30, подходящее 'tt' с термином число 33, фу '^' с
термин-число 34. Схема определения, которая возникает в результате
тогда этот процесс содержит только старые символы; таким образом, его серийный номер
r - или в случае схемы регрессивного определения, так как она
состоит из двух предложений, его номер серии serics r - может быть
определяется. Возьмем в качестве числа-числа для Oj, когда
0i - это 33 (или пи, или фу), второе (или четвертое, или шестое,
соответственно) степень первого простого числа. Применяя тлус
Правило число-термин для Oj b определяется однозначно; и кон
а именно, из этого числа-члена r и, следовательно, из схемы определения,
и, наконец, определение 0] однозначно определимо.

С помощью этого правила мы можем теперь установить разницу
основанные на твине и не основанные символы ^. Например, четвертый
сила простого числа р (больше 2) основана (см. стр. 63)



§ 22 ФОРМИРОВАНИЕ BTTLES РАЗРЕШЕНИЯ 69

когда /) получено в виде схемы определения цвета
со вспомогательным символом 'w' - при условии, что аналогичный
Действие выполняется для каждого определенного символа ^, встречающегося в определении
схема Чтобы сформулировать это условие, мы позже определим
концепция цепочки определений (D81) До этого, однако.
Необходимо определить список вспомогательных терминов
D 60. VRDef {x, y. б) ч [VR (х, б). (k) Ing (x) (0 Ing (x) ([Var
(гл (*, х)). (gl (ft, x) = gl (/, x))] 3 (ft = /)). (x) y (Fr (x, y) 0
Ина (х, х))]

Пояснение x - это n-термины SNyanable-senes, которые вам подходят
(как аргумент-выражение дефиниендуры) для ^^ определений у
когда верно следующее x - это n vanable-senes, нет двух
равные vanablea встречаются в x, каждая переменная, которая встречается как свободная
vanable в y также происходит mx (* VRDef 'является вспомогательным термином для
цель аббревиатуры)

D <1. DefZzl (x) s (3 x) x [(x = gIg (reihe (3o), s]) * Geschl (x)]
Объяснение x называется ^ определением 7N33 (то есть,
выражение, подобное схеме определения 33), когда х имеет
форма ^ я где Си замкнут

ДВР. DefFrldl (х, В) с (3TE) х (3t>) о (3s) х [(vszus [Reihe (33),
einkl (e)]) 4 (xs 3 q (a, «)) • VRDef (p, z, b)]

Аналогично D 63 'DefezpFul (x, B)

Объяснение x называется дефьютированием pf (или явным
finiUonl of fu "соответственно), когда x имеет вид (U]) = ^, (или
(? Ii) = 9 li соответственно), где% является n-термином vanable-senes
который 13 подходит для

D 64. DefrekFul (r, b) = (3 x,) r Q xj) r (3 Uj) Xj (3 Oj) Xj (3 k ^)

(3 c,) X, (3 x) K, (3 x) (3 m) n [(r = reihc2 (x „Xj)). (х ^ =
концерт {%. ti)) • (»s = glg («%. * k)) «Var (x). (f) r, (Fr \ t, cj 0
InA (r, «,)) 4 (футы) Ing {t ^ ([gl (футы, t ^) = 34] D (0 Ing (s) (^ (/ = 0)»)


^ 2), ж])]). (x = emkl (zus (rcihe 2 {j, ra), tr])) # VRDef (tr.
Pi, jn) 4 '-' InA (x, te)])]

Объяснение r называется определением ss 'regressiTC! фу "когда
r - смысл двух выражений X |, X | следующего кмд Xj имеет



70 ЧАСТЬ №. ФОРСТАЛ КОНОТЛТЦТРО.В ТИЕИ СИТВТАИ I

форма 9I, = 9J „x, имеет форму каждого t-ariablc ', который

', я.>, ^ <) г, Whrrrr

. ';' ; срок

•• •. ; ; -J. Сейчас

Есть два случая, которые следует отличать друг от друга. Первый
case: weO; тогда? I имеет форму ^ (n «), SI 4 - форму ^ (Si '). и
9b Uie форма (3 я). Второй случай: »n> 0; тогда Sj имеет форму 4. (nu, 91,),

91 * форма (J * ', 9IJ. И 91 * форма (s „9I,); здесь 91, является
m * tcrmcd vanable-senes адаптирован к 91, а j, не встречается в 91, -
[Это UI: RZ91,: t ',: 91 ;; у ,; 91 *; е, 91; 1: czj ;; эи 9б: 9л * л *

Д «5. DcfiZz2 (/,> •) 2 [DcfZzl O '). (<= pot [prim (y). 2])]
Аналогично Difis DeftPr2d2 (/, », y); D 67 S DcftexpFii2 (бн, у);

D 68: DcftrckFu2 («, n, r).

Пояснение: »является jj (или pr *, или fu *, соответственно), что
«определены 2» с помощью defirutionl> '(или явного defiiutionl
у, или регрессивное определение! г соответственно).

D 69. DefZz2 (*, 0 ^ (3 года) * (DeftZzZ (by) • (* »эра [y, i.
reih5 (л)])]

Аналогично D 70 s DefPfad2 (*, f », 05 D 7 * t DefexpFu2 (x,»,!).

Объяснение, D 65 - 68; «, $ Называется определением 133 1 (или пр.»

(или явное definttiott 2 для fu * t, соответственно), когда t является de *
оштрафован! с помощью у, и х получается из у, когда первый (или второй
Из первых соответственно) фНтерм. а именно '(или' '' или 'соответственно)

Заменен на Тсытнбол т.

Д 72 J DcfrckFu2 (x, n,!) S (3 r) x (3 года) r [DeflrekFu2 (f, n, r) •
(gl (i • »;) =:> ') • (* = эпоха [y, I, reihe (f)))]

Аналогично D 73 1 2 DefrekFu 2 (x, n, r).

Пояснение: х называется первым (или вторым) из

s ^^ регрессивные определения fn "! при следующих условиях
Удовлетворены: т регрессивны определены! с помощью (регрессивного
определение!) r; у является первой (или второй) частью f, а х получается из
у, когда в у '^' заменяется на первое место (или во всех местах, в которых
это произошло).

D74- Def2 (r,!) 3 [DefZz2 (x, <) V (3 n) Ing (x) (DcfPr5d2
(x, n, /) V DcfcxpFu2 (x, n, t) V iDefr ^ Fu2 (x, n, f) V
zDefrckFu! (х, н, я))]

• (Mote, т. 93 с.) Условие, что предметы из 9b свободны, -
и соответствующий член D 64 '* ~ Geb [gl [sum (ib, /), r,], t- ,,
8 мкм (A, 0 l ", очевидно необходимы, но они опущены в
Немецкий оригинал (также в § 8). Мое внимание было обращено на это
контроль со стороны доктора Тарского.



§32 ПРАВИЛА ФОРМИРОВАНИЯ ДЕФУМАЦИЙ 7 I

Объяснение X называется дифимлюмом-предложением t, когда x либо
решение2 33 т или prt, или явное defioitioD2 фу (,
или первая или вторая часть регрессивного определения 2
D 75. Deft 2 (/, n) = (3 года) t ^ DeftPrad 2 (f, n, y) V DeftexpFu 2
(t, n, y) V DeftrekFu 2 (t, n, y))

Объяснение t является символом K ^ teimed (pr * или fu *), который
fined2

D 76. Z2 (f, n) 3 [UndPrad (f, n) VUndFu (/, B) VDcft 2 (/, B)]
Объяснение t называется именованным символом, когда t является
pr * или fu * и либо не определено, либо определено ^.

D 77. Koiistr 2 (r) = (Konstrl (r) • (x) r (f) x (^) x (m) t (b) Ing (y)
[(AlnAR (x, r), (je = zus (rcihe {r), ei ^ Cy)]) »{t, Bt) •

VR (>, n)) 3 («= B) D

D TSt KonstrA 2 (x), аналогично D 52
Пояснение: T 3 An ^ comfr * trtK> n 2 r является условием, которое
выполняет fouowmg coodittoo. Неэкспрессия a> (9 |) occumog
г-н, где Qj является 01 символом! и переменная n ^ Tenned-
senes, m равно n Таким образом, ma конструкция !, каждый pr и каждый
fuhasthee корректное число аргументов - D78 ;;Последнее выражение
кодекса! называется построен!

D 79. UndDeakr (0 с (3 b) / (DndPrid (r, b) V UndFu {t, n))
Объяснение tu aix undefined desanptwe tytniol (n2miljpt или fv)
D 80. Undcft (t) s [(f = 4) V (r = 6) V (t ss 10) V (f = 12) V (fa 14) v
(t = 15) V 18) V {r ;;= 2o) V («= 2i) V (r = 22) v (<= 24) v
(/ = 26) VVar (е) VUndDesir (/)]

Объяснение t »a vndefiTied '^ tymbol, когда t является одним из
двенадцать неопределенных логических констант (см. стр. 55}, или vanable, или
неопределенный описательный символ

Д 81. DefKette (Os (n) Ing (r) (x) gl (b, r) (r) x [(- (n = 0).
[gl (b, r) = x]. InA {t, x)) o (KonstrA2 (x). (3 r) x (Def2)
(х, 0). [Undcft (0 V (3 m) «(Def2 [gl (m. R). T])]. (0 Ing (x)
[(lDefrekFu2 (x, /, /) 3 2DcfrekFu2 [gl (b ', r), /, <]).
(2DefrekFu2 (x, /, * '*) ”([” => ”'] • lDefrekFu2

[GL ( «, г) ./. Г]))])]

Объяснение r называется цепочкой определения ^^, когда foUowmg
ИСТИННО - Каждое выражение, возникающее как член цепочки
построен! и это определение определение !. Если т в ^ символ м
выражение, которое является членом r, то либо t не определено,
или Si или stxne предыдущее выражение r является определением-предложением! т.



72 ПУТЬ 11. ПРАВИЛЬНАЯ КОНСТРУКЦИЯ ШИНЫ S ^ ; NTAX OF I

Если часть этой части является рефракционной2, то
выражение nhich сразу следует за второй частью этого
definiuon; если по выражению это вторая часть регрессивного де »
finiuonJ, то непосредственно предшествующее выражение является первой частью
этого определения.

DSA. DcfiKeite (t, f) = (3 i) r [DefKette (r). [letrt (r) = x].

Dcf 2 (x, l)]

D S3. Deft (банк пота OsQ 0 (3, банк (:, банк [2 банка (2, /)])])
[DefiKettc (/, г)]

Explanatior. D $ 2: (определяется с помощью цепочки dcfinioon r.
- D 83: символ называется определенным, когда есть цепочка определений
Т, с помощью которого определяется т.

DS4. Нижнее {л) с (Undcfi (т) VR> EFT ( »))

Erfilatwtion • sjTnbol 1 называется bntrd cither, когда он не определен
или когда u определен (с помощью dcfinition-ehain).

D8S. Konstr (r) s (Konstf 2 (f) * (0 ^ (InAR (l, r) 3B3a (/))]
D86: KonstrA (v), аналогично D 52.

; .rpf <zna {> oR, D 8 $: комбинация выражений является конструкцией 2
из которых все символы являются символами. - D 86: выражение
IS называется cofurruered, когда это последнее выражение сужения.

D87. З (х) (ЗМ (х) .KonstrA (х))

DS8. Затц (*) с (Sat2l {х) .KonstrA (х))
i? xp / i2nafto «: x называется 3 ^ ®. соответственно) когда х оба


ориентироваться только на основанные выражения и, следовательно, на 3 ^ S,

соответственно) в собственном смысле.

D 89. Def (i, 0 H (Def 2 (x, r). Констра (x))

D90. Df (х) 5Q <) х (Защита (Xo)

Пояснение: D S9: x - значение по умолчанию. (Это определение
аналогично D 87 и D SS.) -— D 90: * является defimtion-sentmee.

D91. DcskrZ (03 (UndDcskr (r) V [Deft (t). (R) - (Deft *
Kctte (ff) 3 (3 ») r [InAR (x, r), UndDeskr {i)])])

D 92. DeskrA (x) = (3 f) x (InA {/, x) .DesUZ (0)

Пояснение, D 91: (является символом «оджи»), когда t
неопределенный Oj или когда определен т, и каждая цепочка определений
t содержит неопределенное значение в (предел, как в D 83) .— D 92: x - это
Священное выражение Qa, когда x содержит Oj.



73


§22 ОПРЕДЕЛЕНИЯ ФОРДЖИ \ ТИОВА

D 93. LogZ (0 = (Bas (0 • ~ DeskrZ (/))

D94. LogA (*) r (0 a: (iiA (*, *) 3 LogZ (0)

Пояснение, D 93 Языковой символ Ci основан и не
scnptive - D 94 a; является логическим выражением Si, когда все символы х
являются логическими символами

D95. DeftZz {я) = (DeftZzl (*). Осн (к))

D 96. Zz (r), D 97. Prad (r) и D 98. Fu (s) являются аналогичными
Пояснение, D 95-98 Определяется 33, 33, pr, fu- В отличие от-
к вспомогательным терминам, которые были определены на более ранней стадии
Определенные здесь термины относятся только к символам ^


§ 23. Правила трансформации *

Следующие определения составляют
ранее установленные правила преобразования языка I (§ это и
§ X2) Для этого сначала необходимо определить subntutioo (D 102),
D 99-101 ввести вспомогательные термины для определения суб
stitubon.

D99. X stfrei (0, r, x} s (Kff) Ing (x) [Fret (r, x »n)> '*«' (3 n) lng («)
(Gf (m, n) «Ffti (t, x, m))]

2 stfrei (A ', r, *) ss (Kii) stfrei (A, #, x) [' ^ (nssstfrei (ife, r, *)) «
Frei (<, X, n) ; <- (3 м) stfrei (fe, s, x) (~ [msstfrei
(A./,x)].Gr(m,n).Fr«(i,x,m))]

D 100. an ^ rei (r, x) ss (Kn) lng (x) (stfrei (n, r, x) = 0)

Dioi. 1 сБ (0, х, г, у) = х

2 sb (A ', x, r, y) = eis (sb (/ fc, x, r,>'), stfrei (ft, i, x),> ')
Пояснение, J 3 gg-iot Letsbe ^ Vj ^ stfrei (l ^, r, x) istheposition-
номер x) th 3i (отсчитывается от конца выражения
х) какой OCCUR свободно mx (0 в случае, когда нет k + i
free 3i mx) anzfrei (rx) - число свободно встречающихся


• Ключ к символу
номер позиции бесплатно 3
{Stell tiuiuiitin er dafretm 3)
количество свободных 3 (An ~
SahJ Freier 3)

sb, замена замещения (замена-
Тион)

Примитивное предложение GrS (Giwaf-
гниль)

AErs ex pression-замена
(Ausdrucksersttzun ^


KV нет свободной переменной (Jteine ;;frete
X ^ anable)

UAbIb напрямую выводится
мтулбар санититбар)

Ab! деривация (Ableitung)

Производное от Abib (ableitBar)

Bew доказательство (fleiceir)

Bewb демонстрируемый (beKnsbar)



74 ЧАСТЬ II. FORXUL СТРОИТЕЛЬСТВО СИНТАКСА I

в X. 8b (II, V, J,> ') это то выражение, которое является результатом выражения
Сион Х, когда, начиная с последнего свободного 8 "К последний последний свободный 3 я из *" ре
последовательно заменяется выражением у.

D 102. subst {x, x,> ') = sb (worfrei (j, *), jr,

Объяснение: Если * это s> * cxpress! On SIj; J.Jil тогда
subst (*, J, y) - подстановка SNcxpression, см. с. 22.)

D 103. GrSl (x) s (3 года) * (3 2) * [Satz (*) • (* = »mp O ', имп

[NefiO'i.z]))]

Соответственно D 104-1131 GrS 2 (x) - GrSll (x); дать один
дальнейший пример:

D106. GrS 4 (*) s (3 * (3> ;) * [Satz (x). (X = aci (nis lrcihe 4)

(6, j, io, 4), cinkl {3 ')], subst [>', i, reihc (4)]))]

D114. GrS (x) s (GrSI (x) vGrS 2 (x) V ... VGrSll (x))
Пояснение: Д 103-113: х - это примитивное предложение первого
Добрый; второй вид; ... одиннадцатый вид (PSI i-ii) .— D114: x 1 $ a
Примитивное предложение.

D 115. AEr 8 (xj.x, .tei.j>,) s (3 ii) x, (3 t ») Xi [(xiBrus 5 («, если ^, t))).
(j ;, = lus 3 («, K„ ti))]

Объяснение! Ezpression * замена: Xt результаты X | когда
частичное выражение Wj заменяется на w ,. (В терминах термина
'эра' символ, тогда как здесь в выражении, заменяется.)

D 116. КВ {у. X, r) s ~ (3 «) Ing (x) (3 i) y (Fr {t, y). GCB
(/,x,n).Frei(r, x, k))

Пояснение: «KV (y, x, s)» означает, что нет переменной, которая связана
в X в месте ot замена s происходит как свободная переменная в y.
(См. Стр. 22.)

D117. UAblbl (x, x) = (3jy) x (3s) x [ZA (3 -). (X = subst (x, r,> -)).

KVCy.xr)]

D118. UAhiS 7 р-

; ; .'I ; ;. ...

в, 0; j. i, u'j = Kon limp («, w), imp (c, u)])) J • [AErs (x, z,
I * 0 AErs (x, z, fOf, Wi)])

Diip. UAblb3 (2' , х, ^) с (х = IInp (J », у '))

D 120. CAblb 4 (z. X, y) 5 (3 1) s [(x = subst [z, s, reihe {4)]). •
(ysimp (x, subst [s, r, reihe2 (1,14)]))]

D 121. U Ablb (r, x,>) s (UAblbl (x, x) V UAblb 2 {z, x) v UAblb 3
(2.х,> ') VUAblb4 (х, х, 3.))



§ 23 ПРАВИЛА ТРАНСФОРМАЦИИ 75

Пояснение, D 117 с называется прямо-денвабельным от х твен
X IS и z имеет вид к RI 1, см. § 12) -

D 118-120 '<прямая таблица 2 (или 3, 4 соответственно)
с RI 2, 3, 4 - D 121 z IS непосредственно выводится из x или
от * и у

D 122. Abi (r. P) = (2q) r («) ing (r) (*> r ([r = zus (/., 9)]. - [Ing
(r) = 0]. [(- («= 0) • [gl (b, r) = x]) D (Satz (*). [Gr [n, bg
(/>)] 3 (GrS (*) VDf (*) V (3 *) «{30 n [~ (A = B). ~ (/ = N).
UAblb [*. Gl (*, 0 .gU / .r)] l)]) l)

Объяснение r - это ssNdenvation с предварительным SSNsgfjgg.
пропускает p, если следующие условия, выделенные жирным шрифтом r, состоят из p и q,
каждое выражение, являющееся членом r, является предложением, каждое выражение
Pression, который является членом 4 TS либо примитивное предложение или
определение-предложение, или может быть непосредственно лишен одного или двух
Vious предложения г-н (см. стр. 29)

D 123 AbIS4tz (r, x, _p) s (Abl (r, ^) »netzt (r) ss *])

Объяснение г является обозначением предложения х от смыслов
предпосылки р

DX24. Bew (r) sAbl (r, 0)

D 125. BewSatz (r, ») s (B« w (f) * pem (r) = *])

Пояснение, D 124 r является доказательством, когда r является денваноном без
предпосылки - D 125 р является доказательством предложения х

Пусть «Ablb (*, p)» означает, что x является способным к изменению со стороны
пропускает p, и 'Bewb (ar)' x демонстрируется. Эти синтаксические
понятия, которые относятся к языку J, не могут быть определены.
определения следующие

Ablb {x, p) r (9 f) (AblSatz (r, x, p))

Bewb (x) - (3 r) (BcwSatz (r, x))

Для формирования этих определений, неограниченная опера
Tore, wbch не встречаются на языке I, требуются
слова "денвебл" и "демонстрируемый" являются только в I
Определенные понятия денвабильности и демонстративности могут быть
оштрафованный, например, такой АС относится к самому денервации, или к
Само доказательство, соответственно (сек D 123, D 125) или такие понятия, как
«денвеблируемым из р посредством денватации, состоящей из не более
n sjTnbols ', или' демонстрируется с помощью доказательства, состоящего из
не более n символов "Если mdefimte синтаксические понятия должны быть



76 ЧАСТЬ II. rOR. \ tAL КОКСТРУКНО СИНТАКСА I

определяется, то неопределенный язык должен быть взят в качестве
синтаксис-язык - такой как, например, наш язык II.

Для определенных неопределенных понятий, хотя они не могут быть определены
в I, универсальное предложение, которое утверждает, что они дуги predlcablc
для каждого отдельного случая, однако, может быть сформулировано в I.
окончание таких понятий, как «не демонстрируемый» и «не выводимый» в
неопределенный язык, отрицательный неограниченный экзистенциальный оператор,
который может быть заменен универсальным оператором; и
ограниченная универсальность может быть выражена в 1 с помощью свободного
переменная. «~ IlewSat2 (r, a) * означает:« Каждый r не является доказательством a »,
другими словами: «а не доказуемо»; AblSatz [r, b, rcihe
(a)] 'означает: «Каждый r не является производным от b от a», в других
слова: «б не выводится из а».

Статья 24. Описательный синтаксис

Теперь мы завершили наше изложение чистого синтаксиса
Язык я; этот пример проясняет, что чистый синтаксис U ничего
кроме части арифметики. Описательный s} -ntax, с другой
рука, также использует описательные символы, и тем самым
Вне границ арифметики. Например, предложение
Синтаксический синтаксис может утверждать, что в определенном месте лингвистический
Выражение такой и такой формы происходит. Было указано
ранее (стр. 54), что возможный метод состоит в том, чтобы ввести серию
неопределенных символов в качестве дополнительных примитивных символов (например:
'Var', 'Id *,' Prad * и т. Д.). Дут, как мы уже
объявил (р, 54), мы будем действовать по-другому. Мы возьмем
undefined fijj 'zei' как единственный дополнительный примитивный символ. (Если
предложения, в которых встречается этот символ, в свою очередь
тактически лечим, будем согласовывать с ним число-число
243 (= 3 *) 0 'Построение описательного синтаксиса занимает точно
в той же форме, что и конструкция любого другого описательного oAr / offia / jc
tyrtfin A. Сначала синтаксис языка S, на котором должен быть A
сформулировано должно быть установлено. Таким образом, метод
поздние предложения и вывод их из A определяется. За
некоторые A •. , ihatS

;. -i 111 o: I.

десерипативные примитивные символы A, которые добавляются к примитиву



77


§ 24 ОПИСАТЕЛЬНЫЙ СИНТАКС

символы S, из них, согласно синтаксическим правилам S,
дополнительные символы могут быть определены, 2 аксиома как дополнительные
предложения S из них, с помощью преобразования-
правила S, последствия могут быть получены (так называемые теоремы
А), 3 дополнительных правила вывода, в большинстве случаев, однако, эти
не вводятся, если мы используем unhned pr ^ в качестве примитивных символов
дескриптивного синтаксиса, то необходимо большое количество аксиом,
посредством этого утверждается, например, что в отличие от символов могут
не происходит в одном и том же месте, и так далее. Ряд
аксиомы m требуется форма неограниченных экзистенциальных предложений,
Чтобы сделать возможным вывод даже простых предложений о
денвабильность и демонстративность Если, с другой стороны, мы берем
фу 'зей' как примитивный символ, тогда никакие аксиомы
То, что в другом случае исключается
здесь уже исключены родные аксиомы с помощью синтаксического
правила, касающиеся функторов (конкретный фу может иметь только одно значение
для определенного места), необходимые экзистенциальные предложения следуют
из антметики

С помощью примитивного символа 'zti' мы здесь дадим
определение - «3 регрессивный - только одного дополнительного символа» *
Относительно описательного синтаксиса Это fu * b 'ausdr' {Ausdruek} ^
самый важный термин описательного синтаксиса
D 126, 1 ausdr (0, x) пятно [2, zei (x)]

2 ausdr (A ', x) = prod [ausdr {k, x), горшок (pnm (it "), zet
[сумма (», * ;)])]

Пояснение ausdr (k, x) 1 $ выражение ^ (с символами A + i)
который происходит в положениях х к х + к, так как TNjymbol на
позиция y IS 2ei (y), следовательно, ausdr (ft, *) = a ** 'f * ^ »3 *** f *

P 56)

С помощью функторов 'zei * и' ausdr 'вместе с
что из ранее определенных символов чистого синтаксиса (D 1-125),
мы сейчас находимся в позиции, чтобы сформулировать предложения описательного
Синтаксис 0/1 в самом себе

Примеры предложений об отдельных символах (с помощью
из "Зей")

1 «Асимволопротивление отрицания в предложении» 'zet (a) s = 2i'

2 «Равные символы встречаются в позициях a и b» 'zei (a) s
Zei (б)»



7 $ ЧАСТЬ II. RORM \ L COKSTOUCNON OP STNT.CC OF I

B. ExampUi smtencet oboat expressiom (с помощью
'Ausdr'):

1. «В ряду позиций от a до a + b встречается 3» -
'З.А. {ausdr (Ь, а)).

2. очевидного предложения не существует »: '' x / BcwSatz
(r, ausdt (b, a) y (со свободной переменной 'r', см. стр. 76).


§ 25. Арифметическое, аксиоматическое и
Физический Синтаксис


В области описательного синтаксиса мы можем выделить два
различные теории: аксиоматический sj-ntax, который мы только что
dbcussing (с аксиомами или без них) и phj-sical s} 'ntax.
последнее относится к первому, как физическая геометрия - к аксиоматическому гео-
метрия. Phj'sical геометрия является результатом аксиоматической геометрии
средства создания так называемого юридического определения x
(См. Райхенбах [HatomaftA], (PAi / o «^ A»>]).
термин, к которому из phjVica) понятия (физики или
повседневный язык) ариоматические примитивные символы должны быть
эквивалентный по смыслу. Это только с помощью этих определений
что аксиоматический sj-ствол применим к эмпирическим предложениям.

Следующее схематическое исследование предназначено, чтобы показать больше
дорогой характер трех видов sjmtax, с помощью
аналогия с тремя видами геометрии. Кроме того, это означало
чтобы показать связь, которая обычно существует между арифметикой,
аксиоматическая система и эмпирическое применение последней.


Три вида феометрии.

I. Антметическая гфометрия.

Parria / dommt of aritfmftie, который
(в обычном методе
Немы посредством координат)
касается упорядоченных тридов реального
числа, линейные уравнения встречаются
кольцо между го и тому подобным.


Три вида синтаксиса *

I. Arithnetieal (или чистый) синтаксис.

Домен эритнетит который

(в методе антемеоза
viottily объяснил) касается
определенные продукты определенных полномочий
простые числа, отношения между
au ^ продукты и так далее.


Частичный домен TJus выбирается с помощью определенного чисто антнедальского
dehrutions. Дается практическая причина создания именно этих определений.
по некоторой модели, а именно, система структур pbj для теоретического
Обработка, к которой относятся эти определения. Это ситтем
физико-пространственных отношений], которые | физических лингвистических структур, например.
предмет физической геометрии, IIB. Я предложения, встречающиеся на листе
1 pap * r ~, который является предметом физического
1 синтаксис, II Б.



§25 Арифметический, аксиоматический и физический синтаксис 79


II Начертательная геометрия
(Это обозначение здесь не
как правило, в обычном смысле, но в
смысл синтаксического термина * de-
scnpave ')


II Описательный синтаксис


IIA Аксиоматическая геометрия


II A Аксиоматический синтаксис
Твто разных лепрецентрических форм
(a) Правильный оксиоматиз - {(6) Anthmetixation

(сравните с $ iS) (сравните §§ 19, 24)

(«Аксиоматизированное описание - I (« Арифметизированное описание -
синтаксис tive *) синтаксис tive *)


Язык с установленными логическими примитивными символами, примитивными предложениями,
и правила вывода предполагается для аксиоматической системы
Основы аксиоматической системы

1 Аксиоматические примитивные символы (описательные примитивные символы, которые добавляются
к примитивным символам языка)

«Точка», «прямо я» Вар »,« Ну *, * Pr4d », я« xei'astheonlyprimi-
Ime * ', “между ** и I * Gr (poaiCions с одинаковым живым символом)
и т. д. символы) и т. д. |


3 Аксиомы (описательные примитивные предложения, которые искусство добавило к примитиву
предложения языка)

Forezaznple, I Числовые символы Килбена, для аксиом IA / b
экземпляр аксиомы «3 не является

pt "," Cl (*, y) DGl (y,

Я а) '* и скоро


Допустимые описательные предложения аксиоматической системы:

I Аналитические предложения Для доказательства этих определений, принадлежащих
Аксиоматическая система может быть использована, но не сами аксиомы


Примеры "Каждый
точка есть точка ",« Если
каждый из трех
линии пересекаются с другой
два в разных точках,
затем сегменты
между точками
сечение образуют клубок * '
(это следует из
окончание «треугольник *»)


Примеры 'Var (x) ^ I Примеры' zei (x) »
Var (*) ',' Nu (x) DZz (x) '* ei (x)'. • [сеи (х) W4] 3
Я За (ш (х) л *

(то есть «Ини 33» *, это следует и
определение * ^ *),

'tNu (x) .Str (x1) IO I' ([sei (x) = 43. [zei ;;(xl)

ZA (x, r) 'I> = X4}) DZA [ausdr (t, x)]'

(То есть, «ню является 3». Это следует из
определение * ZA *, здесь 'ZA * есть
; pti,) 1 a pn).


3 Синтетические предложения Это сами аксиомы вместе с
синтетические предложения, которые доказаны с их помощью

Пример «Сумма I Пример * Nu (x) 3 <~ Нет, так как есть
oftbe углы tnangle Ex (x) '(то есть здесь нет аксиом, все vaLd
isequaltoaR "|“ ttu не является '3 ”*) предложения аналитичны.


IIB Физическая геометрия \ IIB Синтаксис Pfytieal
Посредством соответствующих определений определяется, какие символы
физический язык должен соответствовать примитивным символам (или определенным
определенные символы) аксиоматической системы



Итак, ПУТИЙ. RORM-U, COXSTWCNOX STNT.tt из I


EwniplfS.

I. "Физический сегмент
{для instsDCf,
«тело»)
есть длина, которую я trhen
Это так и так
количество раз длиннее
тонкая длина трафа
такой и такой спектральный
линия каднуума ".

«Физическая сек-
Говорят, что хит
длина 1 т
В этом сегменте
между знаками ттро
на стандартном счетчике
мера в Париже ».

3. «Физические объекты
Су ^ и такого рода (для
Например, светлая
Тикум или растянутый
строки) должны быть
в сторону, как прямой
вмятины «.


Я примеры. Я примеры.

1. «NmC *) * - это я». • tei (x) = 4 * - это
принимается за истину {приниматься за истину

убен и только самбен
фигура упругого эллипса CO ") можно найти на
Petsinon X. "

а. «Nufx) * должен быть j 2,« x »(i) = 4 'ii до
принято как правда | быть принятым за истину

и только о характере, который его суф *
древний тезис в дизайне персонажу
в том или ином месте (например,
эту книгу) можно найти в позитивине х. * '


[Примеры (1) представляют собой фуафиутит d! I'.Aaneni: перед тем, как термин определен государством
должен из свойств, которые объект должен иметь, чтобы он понял
по очереди. Примеры: «tUfiav * di ^ haiMeu; здесь тена определяется
условие, что объекты, понимаемые термином, должны иметь ct-rtasa
отношение (сходство *, конгруэнтность или сходство) к указанному объекту; в
Igguistic Fonnulstion остеоз принимает форму сателлита tpitio-
временная позиция. ltistobee9Tedthat.accorclingtothit, BnosTe & siTedefimtioa
^ В противном случае определяет линсбол с помощью других символов (а не с помощью
лингвистические вещи).] '


Действительный deimprfre tmtenert

J. Snimm. Это либо аналитические предложения axicsnitic

система, из которой аномитические термины приобрели физический смысл посредством
коррелятивные определения (примеры (о); см. примеры аналитических
в соответствии с IIA) или, с другой стороны (примеры (6)),
перевести такие предложения с помощью коррелирующих определений в
^ -iMomitic Tenmaology (это сказать. в терминологию, которая не
* T аксиоматическая система, но на языке gcneril).




Примеры.

(о) «Если каждый из трех
(физические) прямые
пересекаются умереть два других
в разных точках, то
(физические) сегменты
между точками
tersecDon сформировать
ол) треугольник.

_ (fc) «Если каждый из трех
световые лучи в вакууме
пересекает два других


Enmpks.

(0) * '* символ эро
физический характер в
тк) это цифра **


Примеры.

(о) "А (физиад)
Ject, который обладает
семестр 4 (то есть
определенный
perty) является нтмкралом ».


(^) "(Лирический) персонаж, имеющий дизайн
вертикальный эллипс - это цифра ».



§25 АРИФМФИЧЕСКИЙ, АКСИОМАТИЧЕСКИЙ И ФТСИЧЕСКИЙ СИНТАКСИС 8l

в разных точках, то 1
отрезки лучей
между точками зрения
сечение образует треугольник »[

2 Действительность Это либо неопределенные синтетические предложения аксиоматического
система, которая в этом случае имеет физическое значение (примеры i (o), 2 (n)) или
переводы такой неаксиоматической темоологии (примеры х (6), 2 (5)}

Примеры Примеры Нет, потому что

I (a) «Два (физических) i (a)» Если a (физические) не являются аксиомами
прямые Imes пересекаются Аэро Symbmbol происходит на
друг друга в одном месте, то нет экзистенциальной
самое большее '* символ встречается там' *

I (i) «Два луча света i (6) * Если персонаж

Вакуум пересекает один в мк, имея фигуру
другой в один час в один день восходящего эллипса
большинство шуток на месте, то нет

2 (0) «Сумма характера
углы (физического) три один вертикальный и три
угол IS 2II. ”Горизонтальные шаги происходят

a (i) «ne сумма в этом месте»
углы между тремя
световые лучи создают вакуум
которые пересекают одно
другой ты. '

Вопрос о действительности конкретной аксиоматической системы, имеющей определенные
корреляционные определения - это вопрос о законности законов, результатом которых является
из траектории аксиом до научного языка (физики)
(Например, х (я))

Здесь возникает вопрос, потому что здесь нет вопроса.
позиция, важная действительность является критической частью действительности вообще
Вопрос о действительности в отношении существующего (о необходимости
отношение к евклидовой или талиальной аксиомам, и, в частности, аксиома несправедливости для
одной частной не только аксиомам арифметики, см. с. 97)
Евклидова геометрия (например,

«Есть бесконечно
много переменных »)

3 Empiruai tenienett Настоящим следует понимать определенный синт


uiiiiiiiuiogy ^ i_iaiupies 1 \, U), i \ 0))


Exampl-s

X (а) «Этот объект А
Это световой луч вакуум

X (4) «А образует
прямой сегмент. »

2 (а) «эти три
объекты А, В, С светло-
лучи ма сакуйим друг
один из них


Примеры | Примеры

X (a) «Символ, состоящий из двух горизонтальных
инсульт происходит на месте см этой книги «

X (4) «Символ идентичности встречается в месте с
м эта книга », м символы нашей системы
'Id (c) * I' 2ei (c) = есть '

2 (и) «Чувство фигур такой-то формы
происходит в местах Rangmg {Lom A до BM это
книга **



8а ЧАСТЬ II. ФОРМАЛЬНОЕ ПРОТИВОРЕЧИЕ ТТИ СТЕ-ТАИ I


два других это Oifferent
pomu «.

"(час). «Фр» я <»л
возьмите A, B, C вместе
Форма Tiungte. "


a ((). "A ptimitin tentence CXF Язык I
occun .... **

Это тип Ienreneea:

]. «Aentence» docendo dudmua * октябрь в
эта книга.**

4. ** 1 т та недооценена в этом bocOi, что один опирался
обучая. "

5. «В одних и тех же руках. Aentrncn
Окунитесь в пукру и все противоречите друг другу. **

6. '' Подумайте о себе и сделайте это
mraniftgleat (что до, это не арендатор о!
и ту ^ язык). "

7. "empincalljr fatte aentenoe occura на ауди
и заправь ». (См.« P-eontraTalid », стр. 185.)

Aefltmcea всего Hittorp языка
и Iireranuc принадлежат здесь, etpeaallp Тоте из
Hitatp Aatftce, вызывая Maihematia и мета-
phpiica. Среди них оба acnttncca, которые
ertetelp цитировать tcRtethuis (Exainplea a (a) и aen *
tencra (^ empire a (^), 4-7), который предна
tpstax языка в кетдоне и амедмеи
alao определенный tyntheoc pmniuet, paraeularlp tuch as
Cntiore Focmulanona и theaea в Батии
логическая антъптиа {Примера $ 6) или опыта
(Пример 7).



ЧАСТЬ в

НЕОПРЕДЕЛЕННЫЙ ЯЗЫК H
A * ПРАВИЛА ФОРМИРОВАНИЯ ДЛЯ ЯЗЫКА II
§ 26 Символический аппарат языка II

XdiQguage I, с которым мы договорились до
настоящее, содержит только определенные понятия, м области математики *
Он содержит только арифметику натуральных чисел, и
что только в той степени, которая соответствует примерно
Фимтистская или мтуионистская точка зрения. Язык II включает в себя язык
Я, как юноша, все символы Я ел также символы
II, и все предложения I также предложения II Но
Язык II гораздо более выражен, чем Язык I Это
также содержит неопределенные понятия, он сводит все классические
математика (функции с вещественными и сложными аргументами, Iimitug
значения, исчисление бесконечно малых, теория агрегатов), и
Это, кроме того, предложения физики могут быть сформулированы.

Сначала мы указываем символы и наиболее важные
выражения языка II. Точные правила
для 3 S giTcn позже (§ 28) Готические символы

используется м синтаксис X ^ anguage 1 также будет использоваться здесь, вместе
с некоторыми дополнительными

В языке 11, м дополнение к ограниченным операторам Lan-
Guage I, мы также имеем vnbmUd операций вида (3), 3), и

(K3) - [Пример * (3 *) (Pnm (*)) ', sce §6]

На языке II, фу и пр новых синтаксических kmd встречаются, и
это dmded уровни и типы mto (§ 27) в предложении
fu (^ 1) = SIj, мы будем до сих пор называть STj argunuTii-expreznon,
и, далее, мы будем называть Si в 11 есть

ФАМ, который не только ^ Sfi состоит из нескольких терминов - так называемый
аргументы - но ^ также состоит из нескольких терминов - так называемые

Talue-tctms [а. S. «il 3>» • f »(3 i. 3!) = 3! 3 .. 3 я. <0 ^ м-

Позиция] Есть не только предикаты pt, но и предикаты
expreuioTu (разных типов), который может состоять из нескольких



$ 4 RW? T в. Иноминия: iancoace II

sjmbols, но irc использовал sj-ntactically точно так же как pr. Далее там
являются furciOT-evptnsions 5 «(из <hffer« он вводит), которые используются
синтаксически похож на фу (примеры будут приведены позже). Просто ты
one-sjTnbol выражение jj - это 3, поэтому pr - это a, а fu - фью.
Есть pt (и другие '^^ r), аргументы которых не 3
либо '•)? T или (того или другого типа); далее есть фу
(и другие гу) из которых аргументы и стоимостные термины не являются
но могут быть или 5 "(° f '' другой). Таким образом, аргумент

выражение выражения или выражение значения (синтаксическое обозначение ^
«3Ir3») состоит из одного или нескольких выражений форм 3. ^ r или
2u, разделенные запятыми.

В языке II существует несколько различных видов: не только
числовые значения 3 ('u *,' t> *, ... 'r'), но также и предикаты-переменные
p ('F', * G ',' / f ';' M '.' N ') и funcior-vanables
[Так же, как мы назначаем 3 для 33. так что мы назначаем p для pt и
UE f к фу.] NTuiables p и f (всех типов) также встречаются в
неопознанный оператор; (п); Qp): (Di Qf) *

На языке II символ идентичности * e 'используется не только
между 3 «d между S (здесь также при использовании между S это
обычно tmtten 's'), но также между (pt и между
[Примеры (для простейшего типа): «F,» P »эквивалентно в среднем
ing to • (*) (Pi (») sPs (*)) '; 'fujcfu' эквивалентен по значению
• (*) (fui (x) afu, (x)) '.] Weshall обозначает afTo-cji «ibbn * 0» 0'
на 91

В языке II, sentntiai ^ -mhols [^ atexncAen] fo также
происходят; это частично условные константы, то есть символы
которые используются в качестве сокращений для определенных предложений, и частично
Приговор Фан также происходит в операторах

формы (f) и (3f). Мы используем * c 'в качестве общего обозначения
для переменных четырех видов, которые мы упомянули,
а именно 3, p, f, f; все остальные символы называются corutoRlrQ).

Статья 27. Классификация типов

Каждый ^ r и, следовательно, каждый pt и все p, принадлежат определенному
тип. Далее, мы присваиваем тип всем 3, а именно типу 0.
У конкретного могут быть аргументы только определенных типов, и
3u может иметь только аргументы и стоимостные выражения определенных типов.
Для того, чтобы ^ i (? I,. (! Ij .. „9L) и могло быть



§ 27 КЛАССИФИКАЦИЯ ТТП 85

предложения, необходимо, чтобы Qti и ^ 'я принадлежал к одному
тип, а кроме того, что ТН * и должен принадлежать к тому же
тип (который может, однако, быть другого кмд, чем тот, к которому
принадлежит) и т. д. Для того, чтобы tfUi (STj,

и 2hii (? I'ii SIm) = 9liii + i S'm-Hi предложения,
3t и tH (от 1 = 1 до m + n) должны быть одного типа.
IS определяется типами аргументов (м, которые
номер и порядок должны быть приняты во внимание), тип
Определяется типами как аргументов, так и
оценочные термины

Тип выражения определяется в соответствии с
Следующие правила Каждые 3 (^ id, следовательно, также каждый 33) принадлежит
тип 0 Если n членов 9Itg имеют типы Ij, t, (в этом
порядок), мы назначаем тип / „SIrg [Символы

't' с sufEx сами по себе не являются синтаксическими обозначениями типов,
но синтаксические переменные такие] Если ^ gi принадлежит
В предложении? 3rj {5rrgi), тогда мы присваиваем тип (/ j)
Если tKrgi принадлежит к типу, а Sligx к типу ti m,
предложение 5Ui (3Irgx) = t! Irgt, тогда мы присваиваем тип (tj t *)
(Ihli и тип t, к выражению (SfU} (^ Tgi)


не 3, а px и фу только что упомянутых типов, так что для
Например, * M (Gr, 8uin) 'это предложение. Тогда' M 'принадлежит
тип ((0,0), (0,0 0))

Уровень * номер выражения i3 также определяется его
тип, м в соответствии со следующими правилами
число от 0 до 3-го * номер уровня? Irg равно
Наибольшее число уровня его членов
больше, чем у принадлежащего ему выражения aigument
Номер уровня гу больше, чем я самый большой из
два? & gbeIongmg toit Inaccordance в соответствии с нашими предыдущими правилами,
каждое обозначение типа, кроме запятых и двоеточий, состоит из
нулей и скобок Номер уровня легко получить из
обозначение такого рода, это наибольшее количество кастрюль
скобки м, в которые входит ноль обозначения типа К
готические символы, «5pt» и т. д., мы добавляем (как и прежде), где



86 ЧАСТЬ III. TKE INDEFUinR I.ANCDACE 11

необходимо указатели в правом верхнем углу для обозначения
число аргументов-термов, и далее, при необходимости, индексы
в верхнем левом углу для обозначения номера уровня.

Классификация типов, изложенная sbov'e, в своих основных пунктах,
так называемая простая расстановка типов, предложенная Рамси. Но
Здесь он завершается расширением не только на pt, но и на fu.


Кроме того, с его точки зрения предложения также подразделяются на типы,


сами к простой классификации типов.

Примеры, я. Gr принадлежит к типу (0, 0) (см. Выше), и, таким образом,
имеет номер уровня t; 'Gt *, таким образом, является * pt *, или, на словах, nvo-
Предикат Terraed первого le% -el.— a. Так как esxry 3 относится к


в любом случае, предложения в форме (jJ (pri (ii) 5Pfs (ii))
встречаются часто, тогда, в целях сокращения, это
стремясь ввести частичное свойство pr 'Sub' »или
подпункты ... "); определение следующее: 'Sub (F, C0 ^ (*)
[Р (а) 3G (х)]». Так как 'F' и 'G' в этом случае являются * pt * типа
(0), 'Sub' является 'pr * типа ((0), (0)). - 5. Пусть '(*) t (Pi (*) v
P »(x)) = P, (x) y быть управляемым. В соответствии с
логи теории агрегатов или clssscs, мы здесь обозначаем
P * (свойство или класс) как сумма Pj и P *. С целью
аббревиатуры, мы предлагаем ввести символ * sm 'в такой
таким образом, что выражение «sm (P„ P,) »означает сумму Pi и P ,,
и, следовательно, в случае id ^ en, u эквивалентного по смыслу * P »',
'sm (Pi, P ^' соответственно относится к тому же типу, что и
P »', т. Е. До (0). Вышеупомянутое доказуемое предложение может
сформулируем теперь более коротко следующим образом: «sm (Pi, Pi) = P» ».
Я ^ sm 'фу; каждый из двух аргументов как wxU как \ -alue-tcim


Пи 01 или другой тип речи; класс занятий;

см. § 37), так что «Cl (^ * - это предложение.« clsm (CI) »представляет
класс-сумма Cl; под этим подразумевается то свойство (или класс), которое
применимо ко всем тем числам, и только к тем, которые имеют



§27. КЛАССИФИКАЦИЯ ТИПОВ 87

хотя бы одно свойство, обладающее вторым свойством ^ Ievel Cl
примите 'Af * как' p типа ((0)), тогда определение Ac будет
foUows • clsm (Af) (a:) = (3 iO (Jlf (F) .F (*)) '* clsm (Af)' wa * Spt
типа (0), следовательно, 'clsm * la a * fu типа ({(0)) (0)) - 7 iJet
«scn (F, G)» означает наименьшее общее numtKr из двух про *
Pemes F и G, и пусть это означает 0 для случая m, который не такой
число существует Определение »как foUows 'scn (F, G} = (Kx)
(F (*). G (x)) * Каждый из двух аргументов 'sen' принадлежит
type (0) Значение * expressioa 'sen' (правый sidi * из
Уравнение) является 3-м 2, следовательно, принадлежит к типу 0 Таким образом, «сен»
IS * fu типа ({0), (0) 0), и 'sen (F, G)' также является 3 -
Дополнительные примеры будут приведены в § 37.


§ 28. Правила формирования числовых
Выражения и предложения

На основании вышеприведенных объяснений, которые были связаны
с материальными деталями, правилами формирования Лан *
gtuge n может быть официально установлен следующим образом
(Сравните аналогичные правила для Языка I, § 9)

Мы предполагаем ранее приведенные определения следующего
Concepu «связанные * и« свободные переменные »(теперь со ссылкой на все
0, а именно 3, p, f и f), 'open * and' dosed '(p 21),' определенная *
и «uidefiiute» (стр. 45), «описательный» и «логический» (стр. 25),
«33» и «Св.» (Стр. 26)

Выражение относится к типу 0 - m, в этом случае оно называется
Выражение nmnericsl (3) - когда и только когда оно имеет одно из
следующие формы 1 33,2 3 '. 3 (K3i) 3i (S) »(K3i) (®).

где 3i не встречается свободно m 3i> 4, где? ii принадлежит

к любому типу и HI, принадлежит к типу Ae (I, 0) и поэтому
JJu

В общем случае loUowing имеет значение true, если Wj принадлежит к типу ti.
и% к типу (/ jr,) - 10 в каком случае »^ привел функтор-
expression (gu) - Uien принадлежит к типу tj (но не только

m this case} Правило формирования (4), которое уже было дано
для 3 2 случая этого выражения Выражение типа

Форма (Ii), где любой тип, называется предикатом
выражение ($ r)

Регрессивные правила для 'именуемого ergument-expresnon' (или 'value-
Выражение *) (Srg ") - это то же самое, что? Irg * имеет одну из форм
3, ipr или 55 u An имеет вид? Irg ", 91 rg *, если Srgi и



88 ЧАСТЬ III. НЕОПРЕДЕЛЕННЫЙ ЯЗЫК рл

9Ir3, здесь принадлежат типы и tj соответственно, тогда? [Rgj,

9Игры принадлежат tj ^ pe Ij, # *.

Выражение называется предложением (S), когда и только ^ shen оно
имеет одну из следующих форм: фа; 2. Sfi - 5Ij, где STj и SIj
либо 3. iPr.QrSiiofthcsametype; ^. (S) или {S) oerfn (S);
4- (3i) 3i (®) Of (3 3i) 3i (S). где 3i не встречается свободно в
3i; 5- (o) (S) из (3 n) (S); час где принадлежит любому

типа и принадлежит к типу (1,) (и согласно
в общем

Si называется атомным атомом, когда у Sj есть один из
формы 51, pri (5l,) или fu, (? It) = 5fj. где pti не определено
'pr ^ и fui - неопределенное' fuj ,, и 51 ,, 5Ii и 91 * - аргумент-
Выражения которых все термины являются св. S, называется молекулярным
предложение, когда S, либо само атомарное предложение, либо формируется
ftotn. один или несколько таких символов отрицания и
соединение (и скобки).

Некоторые из синтаксических определений становятся проще, если мы не
рассмотреть весь язык II. но вместо этого только определенные
центральные языковые области II „II ,, .... которые образуют бесконечный ряд.
Что касается аппарата символов, предложений и производных,
каждый регион содержится во всех последовательных регионах, и я
содержится м II. В определенном смысле. Язык II представляет
сумма всех этих регионов. Происходит разделение на регионы
следующим образом. Не считая пр и фу, все символы
уже происходят во II, и, следовательно, в каждом регионе. Операторы с F
происходят впервые во II в. В IJ «pr» и «f« встречаются как
константы и как свободные переменные, но не как связанные mriables.
Кроме того, в области II («= 2,3, ...) pt и fu встречаются как константы
и как свободные переменные до уровня n, но как связанные переменные
только до уровня n - i. [Линия разграничения между II,
и дальнейшие области примерно соответствуют этому
элементарное и высшее исчисление функций твин Гильберта.]

§29. Правила формирования определений

На языке II туалет допускает только явные определения
не предусматривает никаких ограничений, так как, используя неограниченное количество операторов,

* {Примечание, 1935.) Теперь я предпочел бы признать regresnie defmitims
«фу во II, как в I. Таким образом, термин« определенный »(§43) будет



§ 29 FOIO.UTIOV ПРАВИЛА ДЛЯ РЕШЕНИЙ 89

каждое регрессивное определение может быть заменено
«Пусть fu» будет определяться посредством регрессивного определения, которое
Состоит из и € {. Из этих предложений мы строим
S3 и S | заменяя ФУИ на все, а затем мы определяем
fu "с помощью следующего определения ezpLcit (on '()' ss® P 94>
здесь используется только отношение к 3)

3 ») = (K3.) QW [() (S..S,) - (!. = F, (3 .. 3-))]

Тогда fUi ^ fUs доказуемо, и fu ^, таким образом, эквивалентно m
Следовательно, регрессивное дефинирование может быть заменено
этим определением

PrimidvesymbolsmLanguagell двенадцать логических констант,
а именно ню и одиннадцать отдельных символов (как м язык I,
seepp i6and23), 2 alio, 3 pTbandfUt, когда и как требуется, из
любой тип [V ', а также может быть введен как определенный символ
боб, но мы пбце их среди символов и состояния
их предложения как предложения pnmiLive, чтобы иметь возможность сформулировать
более поздние оставшиеся повторные предложения проще]

Формирование ruUsfor defmbons Каждое определение - это предложение
форма tZTi bISj, lii IS называется defimendum и d tSt defimeos
Символ, который нужно определить (только 33, pr, fu, serfn или fa)
происходит за этим, единственный символ, который может встречаться в
неравные значения как аргументы, запятые и брадиеты
Нет 0, который не встречается m не встречается свободно m Таким образом, a
определенный fa - это всегда сокращение для закрытого предложения
примеры определений, см. §§ 27 и 37]

Все определения приведены в общих чертах.
Выделить определенный символ Ox Occumng M предложение Si, с
Квалификация foUowmg, когда Ox - это пр или фу, elimmation
невозможно заработать на Sx> в том виде, в каком оно есть, если бык появился хотя бы один раз
м Si'Без последующего (то есть, либо в качестве аргумента
или как стоимостный термин, или вместе с '= *) Для того, чтобы избавиться от
эта трудность, мы можем преобразовать Si mto S3 m следующим образом
Таким образом, мы строим Sj из ^, заменяя Oxg на всех pbces в

имеют широкую протяженность. Доктор Тарски помешал мне
что исключение регрессивных определений составило бы даже * сумму ',
'prod' и почти все арифметические термины mdefimte —В случае
Требуемое исключение m RRi G 34 &) имеет большое определение
тогда, где это необходимо, должны быть преобразованы, м
указано в этом разделе, для определения exphat.



90 ЧАСТЬ III. Tire iD'Dmrre language n

который происходит в S, без наконечника \ truble c, из
того же типа (AP или F). ® ® * в противном случае происходят в tSj.

S, затем строится в виде:

Examp! Е. Пусть 'P |' определяется с помощью P, (x) h (Pi (*) *
п, (*))'. * P, 'не может быть немедленно устранено в * M (P «)' (© O-
Преобразуем в S »таким образом: '(*) (P' (*) Hrif *)) 3MfF) ';
'тогда возможны лунадинадоны: * {*) (F (*) H нС (*) »Pf (*))) 3bi


Б. ПРАВИЛА ТРАНСФОРМАЦИИ
ДЛЯ ЯЗЫКА II

§ 30, Примитивные предложения Ланкуак II


К переменной vahut переменной 3 относятся p или
давления, которые имеют тот же тип, что и \ Tmable (таким образом, к
диапазон значений 3 принадлежит 3) - S относятся к диапазону
значения \.

SimpU tvhttitutiim. «J» является синтаксическим описанием

выражение, которое получается из? I, когда p | заменяется на TFX на всех
места, в которых это происходит свободно в этом месте, может быть
нажатие из диапазона значений O}, который не содержит свободных
переменная, которая связана в одном из мест замещения в


5ufirn'ruti (»t Kith аргумент. Является синтаксическим

описание выражения S, построенное в следующем


манера. Px (9I ^ i) - предложение; условия Slrgi "e неравны
переменные, например, нет необходимости

происходят в Sj; с другой стороны, свободные переменные, которые не встречаются


в SIrgx может встречаться в <3x, но они могут быть только такими, которые не
связаны в местах замещения в (то есть, в тех
места, в которых p ^ происходит свободно в? r *). р, может не произойти ни на одном
место замещения без указания аргумента-
выражение. [Возникновение такого рода может при определенных обстоятельствах


обстоятельства должны быть исключены способом, описанным в § 29 -] Неравный
9Ирг может прийти после Px в различных местах замещения. Позволять
Аргумент-выражение? I „Sr„ .., 9I »следует за pj при некотором
stitution место. Тогда в этом phcc p, (9Ii, ... 91 *) должен быть заменен на



9 ^


§ 3® Примитивные предложения op ii

(ш *) ®

вид заработан во всех местах замещения

Пример Пусть 91 * будет • (*) (F (*, 3)) vF (0, *) v (3J?) (M (F)) '
Замена в 'fii' - это фу, заслуженное

out 'i' ”свободен только в H *« t первого и второго ocoirrence, поэтому
Таким образом, тот факт, что * F '
без 9Iig при его третьем и четвертом появлении не имеет значения
is 'u ^ fa (x) * На первом месте замещения мы должны заменить
«F (*, 3)» (j). то есть самим ^ Sj Тогда, на втором

место подстановки, мы должны заменить * F (0, ») 'на

IS 'usfu (O)' Результатом замещения является foUowmg
• (*) (a = fu (*)) v (u = fu (0)) v (3F) (M (F)) 'Тот факт, что
встречается переменная * x ', связанная с местом подстановки fint
свободно выраженное здесь выражение, не имеет значения, только
«Избыток *», который можно использовать, не должен быть связан ни с одним из
ти-штучки в ст *

Места Языка II С языка II мы
В нашем распоряжении есть фургоны F и P, мы можем во многих
случаи, чтобы сформулировать само примитивное предложение вместо схемы
примитивные предложения PSII х-3 и 7-14 соответствуют
схемы PSI l-ll языка I (§ 11), P511 10 и ll, являющиеся
продлен до новых кмдев Vanables

(U) PnxmtiTe предложения исчисление iententuil
PSHx. P ^ я '^ POQ)

psnx. (-P3p) 3p

PSnj. 6'3?) О ((? 3r) о (/. 3r))

PSn4. * 0> V9) = (~ .pDj)
psn5. (P.?)=~(~pv--})

PSnd. ((/•Os).(jDP))3(p5?)

(A) Pnrmtivc предложения операторов bmsttd sentmttal
PSn ?. (А:) 0 (е (х)) автофокусировка (0)
PSN с. Wji (fw) 3 [Wj- (fw) .fO'I)]
psno. Q *) у (FW) а ~ Wy (~ FM)


• (Примечание, 1935) В немецком онгме] GII4 (наш PSII4) работает
(p09) = (~ -pvg) По поводу изменения см. foomote на
P 32



ЧАСТЬ № ИНДЕТИНТ ЯЗЫК 11


92

(в) Примитивные предложения идентности.

PSnio. Каждое предложение вида Oi »Di

PSHit. Каждое предложение вида (di ^ o03 | ^ SiP

(d) Примитивные предложения an'Ameh'c.

PSni2. - (0 = * ')

PSnx3. (* '=>') 3 (* =>)

(e) Примитивные предложения K - <^ oiort.

psn> 4. G ((Ki) ^ [F Wl) 2 K ~ a »)> [F Ml • G (0)) V (3 *) J '
(Fm.W »[ '- (== *) 3 ~ FM1-ХО) 1

PSn IS. G ((Киль И.Ф. Wlls l (~ a «l [F (in. G (0)) V 0»)

(F (i). (,) »[~ (I = *) 3 ~ F (i)]. CM)]

(/) Пнмитивные предложения безусловных предложений.

PSU * <, каждое предложение fom (Dj)

PSn * 7. Каждое предложение вида fPj) (© i) ^ ©!

PSH 181 Каждое предложение вида (3 »i) (© 02 ~ (Oi) (-
PSN 19. Каждое предложение вида (Pj) Qr V Sj) 3 [fi V
; Там, где Ui не fj.

(g) Примитивное предложение полного побуждения.

PSN 10. [Р (0). W (F (*) Dr (»'))] 3 (») (F {X))

(A) Примитивное предложение JeJeefion.

PSn II. Каждое предложение вида ((pj [Pi (p *) 0 (3 Pi) [p »{t> i)]]»

<PJ (P.) Kp, (pJ. P, (p,) • Q pj [Pi (Pi)> Pi (p> W p
(P. ; = P.)]) 3 (3 PiKPi) (P. (Pi) 3 [(3 P>) iPi (Pi) • Pi M} '
M (»i) ((Pi (Pi) . Pi (Pi) . Pi (pO • Pi(Pf)l 3 (P, »Pi))]).
where Oj (and thus o, s^) is either a p or on f .

(i) Primitive sentences of rxlmrioifality.

PSB li. Каждое предложение формы FTI

(Pl) (Pl (Pl) 3PL (Р,)) 3 (р, = р)

PSn 13. Каждое предложение формы

(di) (t> t) - Cu «) (fr (ih. - = f 1 (Di... o„)) 3 (f I = f 1)

Переменные, которые представлены в схемах, могут принадлежать

§30 ПЕРВИЧНЫЕ ПРЕДЛОЖЕНИЯ № 93

любой тип, условие, что все выражение
должно быть предложение достаточно, чтобы seoire правильные отношения
друг к другу из типов различных переменных [Для м-
позиция, если, м PSII 21, Di принадлежит к типу (любого рода, кроме
0), то отсюда следует, что p *, Pj и P4 должны принадлежать типу (tj
и Pi к типу ((ti))] - PSII 4-6 являются заменами для определений
соединения tymboU 'V' и '=' (между (5) они
соответствовать RIae-e PSII 6 нужно только записать как
обратное следствие следует с помощью
PSII II - PSII 16 и 17 являются наиболее важными правилами для
неограниченный универсальный оператор, с помощью этих простых схем
подстановка и подстановка с аргументами соответственно
представляется возможным - PSII 18 заменяет определение exphat
неограниченный экзистенциальный оператор - PSII 19 делает возможным
называется shiftmg оператора umvcrsal - PSII 20 является Pnnaple
полной индукции, м. я ^ е я был сформулирован (РИ 4)
как правило, но здесь, с помощью неограниченного
оператор, он может быть сформулирован как примитивное предложение - PSII 21
PsneipU отбора И.С. Цермело (соответствует Расселу
Мультипликативная аксиома) в более обобщенной форме (применяется к
типов любого kmd), это означает «Если M является кла» (из
третий или более высокий уровни), а также классы, которые являются элементами РИ
не являются пустыми и взаимоисключающими, то существует хотя бы
один выборочный класс, H, из M - то есть, 3 класса H, который имеет
ровно один элемент м общий с каждым классом, который является
элемент М »Если это предложение применяется к числам в качестве элементов,
тогда это 13 доказуемо без PSII 21 (В таком случае это
Можно, например, построить селективный класс с помощью Takmg
наименьшее число из каждого класса, который является элементом Af)
Таким образом, установлено, что Cj и не 3, а
являются либо р или е - ^ Формулировка PSII 22 (м соединения
с PSII ii) влияет на результат, что два pr, которые являются ко-обширными
везде взаимозаменяемы и, следовательно, являются синонимами
все предложения языка 11 являются экстенсиональными по отношению к
(см. § 66) PSII 23 производит соответствующий результат для JfU It
Следует отметить, что уравнение вида 3i = 3s 1 =

или не означает, что два члена уравнений

эквивалентны т смысл два выражения эквивалентны т
значение (синоним), когда и только когда уравнение является аналитическим



ЧАСТЬ В. ИНДИНИТНЫЙ ЛАНКОЙС II


94

§30 PRIMITIVE SENTENCES OF n 93

any type whatsoever, the stipulation that the whole expression
must be a sentence is sufficient to seoire the correct relationship
to one another of the types of the different variables [For m-
stance, if, m PSII 21, Di belongs to the type (of any kind except
0 ), then It follows that p*, Pj and P4 must belong to the type (tj
and Pi to the type ((ti)) ] — PSII 4-6 are substitutes for defimtions
of junction tymboU ‘V’, and ' = ' (between ( 5 ), they
correspond to RIae-e PSII 6 need only be put down as an
impbcation, the converse implication follows with the help of
PSII II — PSII 16 and 17 are the most important rules for the
unlimited universal operator, by means of these schemata simple
substitution and substitution with arguments respectively are
rendered possible — PSII 18 replaces an exphat defimtion of the
unlimited existential operator — PSII 19 makes possible the so-
called shiftmg of the umvcrsal operator — PSII 20 is the Pnnaple
of Complete Induction, m Langu^e I it was formulated (RI 4)
as a rule of inference, but here, with the help of the unlimited
operator, it can be formulated as a primitive sentence —PSII 21
IS Zermelo’s PnneipU of Selection (corresponding to Russell’s
Multiplicative Axiom) in a more generalized form (applied to
types of any kmd whatsoever), it means “If M is a cla» (of the
third or higher levels), and the classes which are elements of RI
are not empty and are mutually exclusive, then there exists at least
one selective class, H, of M — that is to say, 3 class H which has
exactly one element m common with every class which is an
element of M ” If this sentence is appbed to numbers as elements,
then It 13 demonstrable without PSII 21 (In such a case it is pos-
sible, for instance, to construct the selective class by takmg the
smallest number out of every class which is an element of Af )
Therefore m PSII ax it is established that Cj and are not 3, but
are either p or f — ^The formulatioii of PSII 22 (m conjunction
with PSII ii) effects the result that two pr which are co-extensive
are everywhere mterchangeable, and therefore synonymous Thus
all sentences of Language 11 are extensional with respect to
(see § 66) PSII 23 effects a correspondmg result for the JfU It
should be noted that an equation of the form 3i = 3s 1 =

or does not mean that the two terms of the equations

are equivalent m meaning The two expressions are equivalent m
meaning (synonymous) when and only when the equation is analytic



PART in. THE INDEnNITE LANCOACE II


94

§ 31. The Rules of Inference of Language II

The rule* of inference of L»np»ge 11 *re very simple:

RIIx. RuleofDB/'fifalwn, G, is eaUed (iffrcaJle from Sj

ind Sj when St has the form St^Sf
RH 1. Rule of the imrrmal operator. C, is called directly de-
rirahle from Gj when St h“ the form (0) (Si)-
OnlyRI 3(aRlI i)oflhefourrulcs of inference of Language I
(§ 12) is here retained. R 1 x u repbeed by PSII 16 and 17 and
Rlla: from Sj, according to WI 2, is derivable (oi)(Si) or
(Pi)(S,); and from this, by PSII 16 or 17 arid RII x, are de-
rivable Of respectively. R 1 2 is repbeed by

PSII 4-6 5 RI 4 by PSII 20.

In the construction of a language, it is frequently a matter of
choice whether to give a certain rule the form of a Primitive
Sentenet or that of a Rule of Inftrenet. If it is possible without
too much complication, the first form is the one usxully chosen.

: In Language 1 the principle of complete induction can
only be formubted as a rule of inference; in Language II It may
be either a primidve sentence or a rule of inference. We have
chosen the former. Further examples emerge from a comparison
with other sj-stems, see § 33. J It is, however, incorrect to hold
that there is a difference in principle, namely that for the
establishment of a rule the language of sjotax (usually a word-
language) is necessary, while for the establishment of a primitive
sentence it U not. Actually, the Utter must also be formulated in
the language of sj'ntax, namely by means of the stipubtion '* ... b
tprimitive sentence” (or''...bdirectly derivable from the null-
class", compare p. 171).

The X.txan'dmvaiiim'^'dm'o^'U'^* proof, *demonstTdbU' have
the same definition here as they have in the syntax of Language I
(p. 29). If Oi,i)„...n„ are the free variables of Gj in the order of
their appearance, then *() {®i)* will designate the closed sentence
(t>i)(t>t) — (o«)(< 3 i): if » dosed, then ()(Si) b Gj itself.
Si b called refutable when ~()(G|) is demonstrable. Gj b
called xTSolable when Sj b either demorutrable or refiitable;
otheewbe it b called Irresolable. The terms * analytic’, * conse-
quence’, and so on will be defined later (§§ 34^,/).



§ 32 DERIVATIONS AND PROOFS IN H 95

§32. Derivations AND Proofs IN Language II

We will now give some simple theorems about demonstrability
and denvabilit^ m Lai^uage II The proof and derivation sche-
mata are shortened.


Theorem 32 x. Every sentence havmg one of the following forms
18 demonsirahle m Language II


(«) s.(5)dq%)(s.)

(oJ(~S,) = (~S.)(jj


Proof schema. FSII 16

(;)

(;)

(oO(~S,)3~(S.)(jj

w

(a), Sentential Calculus (cianspositioo)




(3)

(3). PSII 18

s.(”;)=(3n0(s.)

(4)

W ( 0 i)('^) 5 ( 3 ®i)(‘»i) From PSII 16, Theorem 1 a.
W (3 h) From PSII 10, RII 2, Theorem i h


Theorem 32^ In Language II ts denvabU


(a) from @| 3 €|i where

Ox <I<KS not occur freely m

s.

s.aCtp.)(s,)



Schema of denvation Premiss tH docs not 1

occur

freely m (pi.


(!)

(i), Sentential Calculus

~©.VS.

(2)

(2), RII 2

(».)(~s.ve.)

(3)

(3). PSII 19

~S.V(rJ(S,)

(4)

(4), Sentential Calculus

ei3(»J(so

( 5 )

(i) from where

Ox does not occur freely m

s.

QtO(S.)3S.


Schema of derivation Premiss t>i does not occur

freely m ( 3 j,


(0

(i), Sentential Calculus

©,V~Sx

w

(2), RII 2


(3)



96


p.uiTin. Tire iNDmsnx t-tvcxTACE n

(3) , PSII 19 e,V(Pi)(---Si) (4)

(4) , Sentential Calculus — (Pi) (~Sx)dS| (S)

(5) . PSlIiS ( 30 i)(®i) 3 ‘^

(f) from SjO(p)(Si)- SjOSi-

Schema of dent-ation. Premj«: l=i 3 (Pi)(Ss) (0

PSII 16 (Pi)(;*) 3 Sj ( 2 >

(i), (2), Sentential Calculus SjDSj ( 3 )

Theorem 32.3. In Language II are rsutz^aUy derivable:

(o) Sj and (t‘)(oi); hence also S, and {)(oi)- By RMs
and PSII 16.

Schema of denration. Premiss: (Pi)(nt)( 5 j) (*)

(1) , twice PSII 16 Si (2)

(2) , twice RII 2 lPt)(Pi)(®x)

§ 33. COMP.UUSON OF THE pRIMITn'E SdOTKCES
AND Rules of Lkncuace II with those
OF OTHER S^’STEMS

I. The method of ginnp sc)iene}a of primitive sentence* instead
of stating the pnmmve sentence* ihetnstlvw originated with ron
Neumann and has also been applied by both Gfidel

[UneKischeidbare] and Thtsti [ttldtrsf'TiuAsfr.J.

z. Sentential calculus. Russell [Priac. AfatA.] had five prirpitive
sentences ; these were reduced to four by Bernsys f^urfageaAaZAcJ].
Our system of three primitive sentences PSII 1-3 is doe to
Lukasiewicx

3. FuneUonal calculus. By this is usually understood a system
which corresponds approximately to our rules PSII (1-3). t 6 -i 9 .
and RII i and s. We will now compare these rules with the corre-
sponding ones in a number of other systems, with the object of
showing briefly that to the primitit^ sentences and rules of the
other systems wluch are not amongst those of Language II corre-
spond (on the basis of a suitably ^osen translation) demonstrable



relation in II. In the earlier systems (not only in those which are
mentioned here) substitutioQ with arguments was for the most part



§33 COMPABISOV OF II WITH OTHER SYSTEMS 97

admitted and undertaken m practice, apparently, however, exact
rules for carrying it out (see p 90) have never been stated

(a) Russell {[Pnne Math ] » 10, second version of the calculus of
functions) gives PSII 16 as a primitive sentence(« 10 i '(x)(F{x))0
/"(y)’) and not as a schema This necessitates a rule of substitution
which, however, is not ftmnulated but merely tacitly apphed
Further, PSII 19 is given as a pnmibve sentence (* 10 12), PSII 18
as a definition (a 10 01), and RII i and 2 as rules («i i, «io 11)
For our Theorem 3 2 36(*ii 2) Russell requires a primitive sentence
(«ii 07) which IS not necessary in 11

(&} Hilbert \Logik'\ Lke Russell, states PSII 16 as a primitive
sentence and adds the necessary substitution rule (x) Hilbert’s
second primitive sentence corresponds to our Theorem 32 ta
Hilbert gives three more rules Rule (P) corresponds to RII i, the
rules (y) to Theorems 32 2a and b PSII j8 is proved m Hilbert
(Formula 33a) and he obtains RII 2 as a derived rule (y')

(e) Godel lUnenitch^idbarei does not use the existential operator,
and therefore PSII 18 is not necessary G5del s schemata of primi-
tive sentences HI i and 2 correspond to PSII 16 and 19 RII/
and 2 are laid down as rules of inference (definiuon of ‘diwc( •con-
sequence ’)

(d) Tarski [tt'uiertpruehs/r} does not erect pnmibve sentences
for the calculus of funcoons but only lays down rules of inference
(Def 9 ‘consequence') 9(2) is a rule of subsutubon, subsbtution
with arguments is not admutted, so that PSII 17 disappears 9(3)
corresponds to RII i , 9(4) and 9(5) to Theorems 32 20 and e
respecuvely RII 2 is replaced by 9(5). and PSII 16 by 9(5) to-
gether with 9(2) Since he does not make use of an existenbal
operator, PSII 18 is unnecessary

4 AnthmetiC Like Peano {[FormuUure] II, § 2) we take ‘ 0 ’ and
a successor symbol (' ' ’) as primitive symbols We do not make use
of Peano's undefined pr 'number', because Languages I and II are
co-ordinate languages and consequently all expressions of the lowest
type are numerical expressions Therefore (1) and {2) of Peano s
five axioms are eliminated To his axioms (3), (4) and (5) correspond
PSII 13, 12, and 20 — On real numbers, see § 39

5 Theory of aggregates Smce we represent aggregates or classes

by pt (compare § 37), sentences contauung variables p correspond
to the axioms of the Theory of Aggregates — (a) An Axiom of
Infinity (Russell [Pmc Math ] II, p 203 , Fraenkel [Mengenlehre]
p 267, Ax VII, p 307) IS not necessary m II, the correspondmg
sentence^ is demonstrable 'The reason for this

IS that, in Peano’s method of designating numbers, given a numerical
expression an expression for the next higher number can be fonned.
(On this pomt compare Bemays [Phdosophtel p 364 )-~{6) To
Zeimelo’s Axiom of Selection (Russell \Prme Math ] I, pp ff

SI 8



PART III. Tire INDOTNITS LANGUAGE It


and [AffltA. PAi/.]; FrifnVel Ax. VI, pp. 483 ff.) corresponds
PSII ai. — (c)PSII aaisin Ano»io/;I*fmtJOTw/iI>’(FracnlcelDcf. a,
p. 27a; G 5 del [UntvUehfiJbarti Ax.V, i;Tarski [WiJfTtpnichsfr.']


spends approximately to FraenkcFs Axiom of Austondmmg, T,
p. 281)1$ not necessary in II, since, according to the syntactical rules
of definition, a pt* can be defined by every sentence having n free
variables, not excluding even the ao-called unpredicadve definitioDS
(concerning the legitimateness of sahich, ace § 44).— (/) Finally, let
us exaimne the axioms of Fracnltel whi^ have not previously been
mentioned ([Jlfengen/eAre] § 16). The Axiom of Determinatenesa
(Fnenlcel Ax. l)isin II a special case of PSII ix. Fiaenkel’s Axioms
of Patnng, of Summation, of the Aggregate of Sub-aggregates, of
Ausstmdenirig, of Replacement (II-V or V, and VllI) are not neces-
sary in Language II, because the aggregates (pr) postulated by these
axioms can always be defined. Predicate-functors for the general
formation of these aggregates can likewise be defined (compare the
examples 'am' and'eUm', pp. 86 f.).


C. RULES OF CONSEQUENCE FOR LANGUAGE II

§ 34a. Incomplete and Complete Criteria
OF Validity

One of the chief tasks of the logical foundation of mathematics
b to set up a formal criterion of validity, that is, to state the neces-
sary and sufficient conditions which a sentence must fulfil in order
to be valid (correct, true) in the sense understood in classical
mathematics. Since Language II is constructed in such a way
that classical mathematics may be formulated in it, we can state
the problem as that of setting up a formal criterion of validity for
the sentences of Language II. In general, it is possible to dis-
tinguish three Jdndt of eritaia ofvaUdily,

I. We may aim at discovering a Jfjmite eriterion of validity—
that is to say, a criterion of a kind such that the question of its
fulfilment or non-fulfilment could in every indmdual instance be
decided in a finite number of steps by means of a strictly esub-



§ 34® CRITERIA OF VALIDITY 99

Iished method If a cntenoa of this kmd were discovered we
should then possess a method of tohthon for mathematical problems ,
we could, so to speak, calcubte the truth or falsehood of every
given sentence, for example, of the celebrated Theorem of Fermat.
Some time ago Weyl ([PAiiuopfee] p 20) asserted — ^without, how-
ever, givmg a proof— “A touchstoneof this kmd has not yet been dis
covered and never will be discovered ” And accordmg to the more
recent findings of Godel \lJnentsdiaibaTi\ the search for a defimte
entenon of validity for the whole mathematical system seems to
be a hopeless endeavour Nevertheless, the task of solving this
so-called problem of resolution for certam classes of sentences
remains both an important and a productive one, and in this
direction many significant advances have already been made and
many more may be expected But if we seek a entenon which
apphes to more than a limited domam, then we must abandon the
idea of definiteness

2 We may set up a entenon of vahdity which, although itself
indefinite, is yet based upon definite rules Of this kind is the
method that is used in all modem systems which attempt to
create a logical foundation for mathematics (for example, the
systems of Frege, Peano, Whitehead and Russell, Hilbert, and
others) We shall designate it as the method of denvatton or the
d-method It consists of settmg up primitive sentences and rules
of inference, such as have already been formulated for Language
II The primitive sentences are either given as finite in number, or
they emerge by substitution from a finite number of schemata of
primitive sentences In the rules of inference only a finite number
of premisses (usually only one or two) appear The construction
of primitive sentences and rules of inference may be understood
as the defimtion of the term 'directly derivable (from a class of
premisses) m the case of a pnmrtive sentence, the class of pre-
misses 13 nuIL It IS usual to construct the rules m such a way that
the term 'directly derivable’ is always a definite term, that is to
say, that m every mdividual case it can be deoded whether or not
wt Vmt an mstance of a pimutive — 01 CpS the apphtalion

of a rule of inference, respectrvdy We have already seen how the
terms 'derivable', ‘demonstrable’, 'refutable’, ‘resoluble’, and
'irresoluble' are defined on the basis of this d method Smee no
upper limit to the length of a dcnvalion-cham is determmed, the



100


TART III. THE INDCnNITt tANCl’^CE 11

Icnns iTJcntionfd, althoujjh they are ba«ed upon ihc defirute teim
*<i\«cd.y dcrivabk\are them^h'es indefinite. It w-as at one tiroe
thought possible to construct a cmnplcie critenon of \nJidity for
classical mathematics with ilic help of a method of den«iion of
this kind; that is to say, it was bel)c\-ed. other that all Nalid mathe.
maucal theorems were already demonstrable in a certain ciistint;
sj-stem, Or that, should a hiatus be discottred, at any rate in tie
future the sjstem could be traiufomicd into a complete one of
the kind required by the addition of further suitable primitive
sentences and rules of inference. Now, hOTvettr, G^drl has shown
that not only all former systems, but all systems of this kind in
general, are incomplete. In every suffioenilynch system for which
a method of dem-ation is prescribed, sentences can be constructed
which, alUiough they consist of symbols of the system, arc yet not
resoluble in accordance With the method of the system — that is to
say’, are neither demonstrable nor refuiaWe in it. .;\nd, in par-
ticular, for evciy system in which mathematics can be fonnulated,
sentences can be constructed which art s’alid in the sense of
classical nuthematics but not demonstrable within the system.
In spile of this necessary incompleteness of the method of deiits-
lion fon ih» point, see \ 6o<f)* inetiod retains its fundamental
significance; for every strict proof of any sentence in any domain
must, m the last resort, make u’^e of it. But, for our parucular
task, that of constructing a romphir rrilmon of i-alidity for maihe-
maucs, this procedure, which has hitherto been the only one
attempted, is useless; we must endeasour to diseoN’cr another 'vay.

3. In order to attain completeness for our criterion we arc thus
forced to renounce dcfiniiencss, not only for the criterion itself hut
also for the individual steps of the deduction. (For a general dis-
cussion of the admissibility of indefinite syntactical concepts see
§ 45O ^ method of deduction which depends upon indefinite in-
diriduil steps, and in which the number of the premisses need not
be finite, we call a method of eomtijutnrt or a c~method. In the
case of a method of this kind, \ce operate, not srilh sentences but
with Sentential classes, trhich mxy also be infinite. We have
already laid down rules of consequence of this kind for Language I
(in § I4) and in what follows we shall state similar ones for Lan-
guage II. In this way a eompifte eriterion of voUdSty for
matict is obtained. We shall d^ne the term ‘anilyoc ’ in such •



lOI


§ 34a CRITERIA OF VALIDITY


way that it is applicable to all those sentences, and only to those
sentences, of Language II that are valid (true, correct) on the
basis of logic and classical mathematics We shall define the term
‘-contradictory’ m such a way that it apphes to those sentences
that are false m the logico mathematical sense We shall call Si
L-determtnaU if it is either analytic or contradictory , otherwise we
shall call it synthetic The synthetic sentences are the (true or false)
sentences about facts An important pomt is that Language II
eludes descnptive symbols and hence also synthetic sentences
As we shall see, this influences per tain details m the form of the
defimtion of ‘analytic’.

The foUowmg table shows which terms used m the two methods
correspond to one another


d terms

(depending upon the
me^od of derivation)
derivable
demonstrable
refutable
resoluble
iiresoluble


e-terms

(depending upon the
method of consequence)
consequence
analytic
contradictory
L-determinate
syntheuc


In every one of these pairs of terms with the exception of the
last, the d term is narrower than the corresponding c-term
The completeness of the entenon of validity which we intend
to set up, as opposed to that which is dependent upon a d-method,
will be proved by showing that every logical sentence of the system
IS L-determmate, whereas, m accordance with what was said
earher, no d-method can be so constructed that every logical sen-
tence IS resoluble


When Wmgenstem says [Traclafuj, p 164] “ It is possible to
give at the outset a aescnpQon of all ‘true’ logical propositions
Hence there can never be surprises in logic Whether a proposition
belongs to logic can be determined**, he seems to overlook the in-
definite character of the term ‘analytic*— apparently because he has
defined ‘analytic* (‘tautology*) only for the elementary domain of
the sentential calculus, where this term is actually a definite term
'The same error seems to occur m Schlick [Fundament, p 96] when he
says that directly a sentence is understood, it is also known whether
or not the sentence is analytic ’‘In the case of an analytic judg-
ment, to understand its meaning and to see its a priori vahdity are
one and the same process ” He tries to justify this opinion by quite



§34^ REDOCnON 103

to then Si and Sj — as may easily be established — are always
mutually derivable

Let * Si’ designate any sentence in question “STj results from
51i” means ”Si is transformed m such a way that the (proper or
improper) partial expression of Si, ?Ii, is replaced by ”

RK I. Every defined tymhol is ehmtnated with the help of its
definition (In Language II, all definitions are exphcit )

RR 2. Construction of the amptncUve tiandard form

a. (S2DS3).(Sa3®i) results from StsSs

b. '"-’SjVSj results from SfO^

c. ‘~»'Si«~^ results from '^(SjVSj)

d. ~SjV«««>Ss results from '—(^•Ss)

e. (SiVS3)*(^Vs«) results from SiV(S3«S4) or from

(Sj.SOvS,

; Si results from '«-Si

RR 3. Disjunction and emptstetum Here disjunctions and con-
junchoas not merely of two, but of many, terms arc meant, for
instance, (SiVSj)VS4 or SiV(SsVS4) is called a three-tcnned
disjunction havmg the three terms St, 64 The cancellation of
a term is understood to include the caoceliatioD of the appertaining
symbol of disjunction — or of conjunction-^d of the brackets
which thus become superfluous

a. If two terms of a disjunction (or of a conjunction) are equal,
then the first is cancellecL

b. If St IS a disjunction (or a conjunction) of which two terms
have the form St and ~ S3, then 91 (or 91 rmpectively) results
from Sf [91 is ‘0 = 0’]

c. If St IS a disjunction of which one member is 91, then 91
results from St

d. A term ~ 91 of a disjunction is cancelled

e. A term 91 of a conjunction is cancelled

; If St IS a conjunctioQ of which one member is '^91, then
~ 91 results from Sf

RR 4. Every limited 3»operator is elimmatcd with the help of
PSII9

RR 5« Equations

a. 91 results from 2ri=?Ii
3i=3t results from ^'=3*'



104 I’ART III- "rnE INDEFINnX LANGUAGE 11

c. o-'R results from nuw 3 i' =

d. 36 = 3 i results from 3 l = 3 b-

RR 6. Elimination of the tenlential variables f.
a. Let fi be the first free f in 6i:
from Sj.

RR 7. A K-operatof is eliminated:

a. NVhen it is limited, by means of PSII 14;

b. ^Vhen it is unlimited, by means of PSII 15.

RR 8. Let ( 3 j be a sentence svith a limited universal operator

(i,)3.(S.). . , , r

a. Let 3i not occur as a free variable in S*; Sj results irom (iS|.

b. Let 3 i be rm; froiii

c. Ut 3i have .he form 3.'; (5i)3.(S.)- 6 .( 3 ^,)
from S,.

•L (30 ( 3 :) (fO (3 h ) (3 3 ,) [- (f, (nu, j,) ^u) ^ (fi ( 3 »'*
fi(33.3i)')'''*“(fi(3i‘3j)*^)''®»] CTbis sen-
tence IS equivalent in meaning to (30 l(3x S 30 ^ defini-

tion of * Grgl ’ on p. 59, and the transfonnation described in § 29.)

RR9. Construction of the so-called standard form of die
functional calculus (see Hilbert [Logik'\ p. 63). Only unlimited sen-
tential operators now occur as operators. Such an operator is
called an initial operator of S, when either nothing, or only un-
limited operators occur before it in Sj (apart from brackets), and
its operand (apart from brackets) extends to the end of

a. Sj results from (oj) (S*) or from (3 o,) (S*) if t>i does not
occur as a free variable in Gj.

b. Let the first operator variable in Si which is equal either
to another operator variable, or to a variable which occurs as a
free variable in S„ be Thb operator variable together with all
variables which are bound by it (that is to say all variables t»i
which occur as free variables in its operand) are repbeed by



§ 34^ REDUCmON 105

variables which are equal to one another, but which are not equal
to the other vanables occurring in

c. (3 Dj) (~ Sj) results from ~ (Dj) (i^)

** (Pi) results from ~(3ni)(®s)

e. The first operator m ©i that is not an initial operator,
together with the appertammg operand-'brackets, is so transposed
that It becomes the last initial operator

A sentence is called reduced when none of the rules of reduction
can be apphed to it. The appbcation of the rules to a sentence
always leads by means of a finite number of steps to the ultimate
form, namely to a reduced sentence, this we call the reductum of
and the syntactical designation of It IS ‘®^S’

Theorem 34b.i. and ^©, arc always mutually derivable

Theorem 34b.2. If ©^ is reduced, then

A. ©1 has one of the following fonns i (i'i)(S8)or(30i)(©g),

where Dj occurs as a free variable in ©, and where ©* has one of
the forms i to 9 — 2 where has one of the forms 5

to 9 S,V©j, where each of the two terms has one of the
forms 2,3,5109 — 4 ©j«©j, where each of the two terms has
one of the forms 2 to 9 — 5 = ^0^ 3 3i

where at least one has the form d or e (see under B) —6 3 b “ 3
—7 ?Jriaipr* —8 5ui=gu, —9 ^tC^Irg) —10 31 — ii <^ 31 ;

only m the case of this form does 91 occur as a proper partial
sentence

B. Every 3 m ©j has one of the following forms

a nu — b 3i'i where 3i has either the form a or the form b
(a and b are ©t ) — c. 3i'. where 3i has one of the forms c, d or e
— d 3 — ® ( 9 lrg) — ^Every 3 i« Si has the form c or e

C. Every m ©^ is cither an undefined pit, or a p, or of the
form (ju(?Itg)

D. Every JJu m ©j is either an undefined fu#, or an f, or of the
form Su(^g)

Theorem 34b.3. If ©j is logical, reduced, and closed, then ©i
has one of the following forms i 9 I„(S*), where n>i,

31 , (« = 1 to n) IS either (o,) or (3 p»)» *ttd ©j contams no operators,
but does contain the free variables Oj, . o* , 2 31 , 3

Theorem 34b^. If ©j is logical and defimte, then ^©1 is
either 91 or ~ 9 i



X06 PAST III. TKE INDETINITZ LANCOACE II

Theorem 54b.5. If by the tppliatioo of i rule of xeducboa
(even out of turn, except in the of RR 90 * sentence of the
form Sj s Sj results from Sj, then is 3 t
Theorem 34b.6. Etery aiomie ^ut not every moleculir) sen-
tence is reduced (sec p. 88).

Theorem 34b.7. If Sj is reduced tnd contains no proper
partial sentence, no variable, and at the most one ’pr or one
*fu, but neither "pr nor •fu for « > i, then (Sj is an atonuc sentence.

§34c. Evaluation

We shall not define the term 'analytic* explicitly, but instead
we shall lay down rules to the effect that a sentence of a certain
form is to be called an analytic sentence when such and such
other sentences fulfil certain conditions — for instance, when they
are anal}*tic. We must do this in such a way that this process of
successive reference comes to an end in a finite number of steps.
We shall therefore proceed from a sentence to simpler sentences,
for instance from Si to or from a reduced sentence to
sentences which contain a lesser number of variables. If jj, for
example, occurs as a free variable in (5,, then we shall mil Si
analytic when and only when all sentences of the form

are analytic; thus we refer for instance from 'Px(*)’ to the sen-
tences of the infinite sentential class {‘P, ( 0 ) *,* P, (O')’, ' P, ( 0 ")’,...}.
In this manner, the numerical variable is eliminated. In the
case of a predicate- or functor-variable, however, the analogous
method docs not succeed; a fact which has been pointed out by
G6del. Let be, for example, ‘M(F)’ (in words: "M is true
for all properties”). Now, if from Sj we refer back to the sen-
tences M(Pj)’, 'M(Pj)', ind so on, which result from Sj by
substituting for * f"’ each of the predicates of the type in question
which are definable in II, in turn, then it may happen that, though
^ these sentences are true, ‘M(f)’ is nevertheless false-in so
ar as M does not hold for a certain property for which no predi-
cate ^ be defined in 11 . As a residt ot GSdel’s researches it is
certain, for instance, that for every arithmetical system there are
numerical properties xshieh are not definable, or. in other words,
indefinable real numbers (sec Theorem 6od.i, p 221). Ob-



§34^. EVALUATION IO 7

Tiously It would not be consistent with the concept of vahdity
of classical mathematics if we were to call the sentence “All real
numbers have the property M’* an analytic sentence, when a real
number can be stated (not, certainly, in the Imguistic system con<
cemed, but m a richer S 3 rstem) which does not possess this pro*
perty Instead we will follow Gddcl’s suggestions and define
‘analytic’ m such a way that ‘M(F)’ is only analytic if M
holds for every numerical property irrespective of the limited
domam of defimtions which are possible m 11

Thus, m the case of a p, we cannot refer to substitutions but
must proceed in a different way Let ‘F’ occur m as the only
free variable, a *p^ for instance Then we shall not examme the
defined predicates of this type, but instead all the possible valua-
tions (Beroertwigen) for By a possible valuatton (syntactical
designation, IB) for 'F' (i e a value assigned to 'F') we shall here
understand a class (that is to say, a syntactical property) of accented
expressions Now if 1Bi is a parocular valuation for ‘F’ of this
kind, and if at any place m 'F' occurs with Stj as its argument
(for example, m the partial sentence ‘/‘(O")’), then this partial
sentence is— so to speak — true on account of Si, if Sh *0
ment of Si, and otherwise false Now, by the evaluation of Si on
the basis of Si, we understand a transformation of Si in which
the partial sentence mentioned is repland by Ul if Sti is an ele-
ment of Si, and otherwise by '"'91. The definition of ‘analytic’
will be so framed that Si will be called analytic if and only if every
sentence is analytic which results from Si by means of evaluation
on the basis of any valuation for * F’. And Si will be called con-
tradictory when at least one of the resultmg sentences is a con-
tradictory sentence. We shall lay down analogous rules for the
other p-types

A valuation for a free will consist in a correlation by means
of which to every St an St is umvocally correlated In the case
of the evaluation of a sentence on the basts of a certain valuation
Sj for f„ we shall replace a partial expression fi (Stj) by that St*
which by means of iBi is correlated to St* We shall lay down ana-
logous rules for the other f types

Let ptj be descriptive , here a valuation of the same kind as for
a p is possible Here also. Si, m which ptj occurs, will be called
analytic if the evaluation on the Imsis of any valuation for pti



Io8 PARTUI. Tlin INDETINrra LANGUAGE II

leads to an analytic sentence. In contradistinction to the case of
a p, hortcvcr, S, wU here only be called contradictoty if the evalua*
tion on the basis of any valuation for ptj leads to a contradictory
sentence. For, in the case of a p. Si means : “ So and so is true for
every property ", and this is false if it does not hold for even one
instance. Here, in the case of the pr„ however, S, means: "So
and so is true for the particular property expressed by pti" where
we have a pr# and therefore an empirically and not a logically de-
terminable property; and this sentence is only contradictory —
that IS to say, false on logical grounds — if there exists no property
for which Sj la true.

On the basis of the foregoing considerations, we shall now pro-
ceed to lay down first the rules of valuation, VR, and then the
TuUi of evaluation, Ea'R. Later, in connection with these, we shall
formulate the definitions of ‘analytic’ and ‘contradictory’.
Symbols to which a valuation can be assigned are called eon-
valuable [ietcer/tarc] symbols (syntactical designation, ‘h’). The
convaluahle symbols in are all deacripthe pr# and fut, and
are also all j, p, and ( m those places where they occur as free
variables in Sj

VR I. As the valuation for a eonvaluable t^-mbol b,, any valua-
tion may be chosen which, m accordance with the following rules,
IS of the same type as b,.

a. A valuation of the type 0 is an (St.

b. A valuation of the type is an ordered n-ad of

valuations which belong to the types ti to respectively.

c. A valuation of the tj-pe (tj) is a class of valuations of the

type h

d. A valuation of the type (/, : rj is a many-one correlation
by means of which, to every valuation of the type L, exactly one
valuation of the type r, is correlated.

I’® n reduced sentence tathout operators', for all
. ° valuations be chosen according to VR x, and, in par-
t.cular, let equal valuations be chosen for equal symbols. Then,
y t e 0 owing rules, a univocnlly determined valuation results
lor every partial expression in S, of the form 3, ^Irg. ^r, or 5u.

B. nn Itself shall be taken as the valuation for nu.

b. Let Sti be the valuation for 3^ ^, then Sti' shall be taken as



§ 34 ^ E\AI,UATION 109

the valuation for 3i' (Thus, as the valuation for an 0t, the ;t
Itself IS alwa)’S to be taken )

c. Let the valuations Sj to S„ be assigned to the terms ?Ii to
of Slrgi Then the ordered n-ad S„1B2. shall be taken

as the valuation of ^Trgi

d. Let^li be an expression, 3. ^r, or gu, of the form 5U2 (^rgj) ,
and let the valuations 5Bi and be assigned to and
respectively Then that valuation which is correlated by to the
valuation shall be taken as the valuation for

According to these rules, the valuation of an expression % is
always of the same tj’pe as QIj itself

Examples 1 In connection with VR 1 a A 8 for a free s belongs
to the type 0, and is therefore an St, for example 0'f>’ — 2 In con
nection with VR 1 c A 8 for a *pt’, for example, for F’ in ‘F(*) ,
belongs to the type (0), and is therefore a class of Gt, that is to say,
a syntactical property of expressions which only applies to accented
expressions — 3 In connection with VR tb,c A 8 for a ‘pt’, for
instance, for ‘G’ in ‘G(x, >,*)’, belongs to the type (0,0,0) and is
therefore a class of ordered triads (or a three^termed relation) of
St — that is to say, a three*termed syntactical relation between
accented expressions — 4 In connection with VR le A 8 for a
*p:’, for example, for 'M' in ’M(F)' belongs to the type ((0)), and
18 therefore a class of classes of Gt —5 In connection with VR 1 d
A 8 for an ‘fu*, for instance, for */’ ‘/(*»y) = »' belongs to the

type (0, 0 0), and is therefore a correlation by means of which an Gi
IS univocally correlated to every ordered pair of Gt and is therefore
a many many one syntactical rebtion between Gt — 6 In con-
nection with VR 2 a, b, e Let the Gf *0’’ and ‘0’ be chosen as the
valuations for **’ and ‘y’ respectively, in accordance with VR i a
Then, m accordance w>th VR za, b,c, the expression ‘01 0,0'"' is
the valuation for ‘Ar.y.O'i’’ — 7 In connection with VRzd We
have already (p 86) considered the sentence ‘5m(FG)(x)'
( belongs to the sum of the classes F and G ') Instead of *sm’,
we will now put a variable ‘wi’ of the same type ((0),(0) (0))
‘»n{F,G)(*)’ As an example of SrUjf^xgi), let us lake from this
the ipr, ‘m(F, G)’, which has the fonn oi'd is of the

type (0) Let the class of the Gt from ‘O' ’ to ‘ O' ' ' ' ’ be chosen as the
valuation of ‘F’ (according to VR 1 c) and the class of the Gt from
‘0”l’ to ‘0"'"’ as the valuation of ‘G’ Then, according to VR ze,
thevaluauon 8i for ?Ir0i (‘F, G’, type (0),(0)) is the ordered pair
consisting of the two aforesaid classes in the aforesaid order For
Su* (1 e ‘m’), let 8, be arbitrarily chosen According to VR i d,



XI6


ro:T in. TKi isDmsTiT uivctucE n

ufusie tK«f 5 ^ IS chosen in ludi « wjy thtt the d*s$ cf i;t fctrn
* 0 ' ’ to * 0 ”"' ' « correlsted to (This irould, for instmte, cef»-
S 7 >ond to the consttnt *ss* u anloe for ‘ft*.) Then occonhasto
\’R 2 <i, this d*ss w the rtiuMian for * k {F, C)’.

Let Si be 1 reduced sentence tnihout cpcntors; thd Jet nha-
tions for ill b in Si be cboso iccorSiag to \Tl i lad ntedans
for further expressions be drtcjtained in acsordioce toth \'R 2 .
Then the evtluitjoa of Si, on the hi-sjsof theTaluiS 3 ans,ci)nas 3
in the trsnsfortnionn iccordin!* to the foBotring roles of eraltta-
tlan, EvR i, 2 . If a non-redaced sentence results front a tuns*
fonmtion, it must first be reduced and then transfortned faitita.

EtR I. Let a piniil acatcncc S, hare the form
and let the eiluafions for and ^jbe S?i and SSj, rcspisjattlT.
If S] IS an eleacBt of Sj then S, is rrplacrd by 91; eiertns* by
"-SPu

EtR a. Lei a pirtul sentence ©j bare the tuna Si®?!*, but
not 91; and let the reluifions Jar 3j and 3* be S\ and SSt
speedvdy. If SPi and SBj tre idenfical, Sj is ityiacsd by 91*.
otherwise by —91.

Theorem 34C>i. Let ©t be a reduced sentenre withoui open-
tors. The ereluidoa of ©i , on the biab of any rehixnnas fas the
b trhidi occur, leads in erery case^ in a finite nunabes of sttye, to
the final result; this is tither 91 or —91.— For erery Oj, and r
ocairting in ©i we hare i riluxtioa. Frren these rahiaocQS there
results a raJuition for erery 3 . ?!«>. Tt. and gti rrhidi ocoirs.
Thus etoy partial sentence of the fnrtn is rtplac*^

other by SI or by —91; and Ukewise erery partial soitcnceof Ac
form ?ri=«Is, sine* 3i and % hare the fonn 3 , or till. In
this Ttay, we get 3 ooncxtcnstion of 91 by means of

symbols of ncgslion, disjnncoon, and canjuncaon, from trlaci.
by &e apphcstian of RR 3 and 3 , ether 91 or —91 results-


§34^. DeFINITIOJI OF ‘ANALTTIC IN H* AJvD
* CONTRADICTORT IN II ’

The dcfiiutions of ‘endy^* and *r&Ktra;rfon’’ trith reference
to Language II are, as we hare already meniionoi. conaderahly
tnore complicated than they are with referttce to Xingnage L
On the basis of the foregcmjg sfipulafions concerning rrfuction



Ill


§ 34 ;/ ‘analytic’ AND ‘CONTRADICTORT’ IN II
and evaluation, these definitions for II can now be embodied in the
following rules DA 1-3 ('A* and *C’ are here used as abbrevia-
tions for “the necessary and sufficient condition under which
ill or Sj IS analytic” and “ contradictory”, respectively )

DA I. Definition of ‘analyde* and 'contradictory ' (m II) for
a sentential class iij We distinguish the following cases

A. Not all sentences of fti are reduced A (or C) The «‘l!»ng
of the reducta of the sentences of ftj is analytic (or contradictory,
respectively)

B. All sentences of i^i are reduced and logical. A Every
sentence of is analytic, C At least one sentence of Six is
contradictory

C. The sentences of Sii are reduced, and at least one of them
13 descnpttoe

a. An optfR sentence occurs mAi Let be the class which
results from Aj by replacing every sentence (5* by ()

(see p 94) A (or C) i^ is a^ytic (or contradictory,
respectively)

b. The sentences of Ax are closed A For every sentence
of Ai the logical sentence is analytic that results from

by replacing every descriptive symbol by a variable of the
same type, whose design does not occur m equal sj'mbob
bong replaced by equal variables and unequal symbols by
unequal variables C For the arbitrary choice of one valua-
tion for every descnptive symbol occumng in Ax (die same
valuation bemg taken for equal ^mbob) there is at least one
sentence m Ax which is contradictory in respect of this valua-
tion (see DA 3)

DAa. Defimtion of 'analytic* and ‘contradictory’ (m II) for
a sentence Si

A. ' 5 x^not reduced A(orQ is analytic (or contradic-
tory, respectively)

B. Sx 13 reduced and open, A (or C) () (Si) is analytic (or
contradictory, respectwely)

C. Si IS reduced, closed and logical

*. Si has the form (oi) (Sj) A S* is analytic m respect
of every valuation of Dj, C ^ is contradictory m respect of
at least one valuation of Oi



II2 PAW in. TltE INDiriNTrE lANCUACE ri

b. Si has the form QOi)(0,). A: Sj is Analytic iartspMt
of at least one tTJuation of p,; C: S. is contradictory in re-
spect of e\'eT>* s*alu3tJon of Dj.

c. Sihasthe form 91 or ^91- A: Form 91; C: Form ''*91.
D. Si >s reduced, elottJ and tJestriptirt. A (or C): the claw

{Si} is analjtic (or contradictory, rrspectivcly).

DA 3. Definition of “analytic (or contradictory) in respect of
certain STduations" for * reduced sentence Sj. (These terms only
serve as auxiliary* terms for DA t, 2. ‘A— S5i’ and ‘C-Sj’ here
mem; “necessary’ and sufficient condition under which ;1 is
analjtic (or contradictory, respectively) in respect of Si”, where
Si is a series of valuations, namely, that consisting of one valuation
for each symbolic design b occurring in Si (hence not for the
bound striables).

A. Si has the form (»,)(;,). A-Sj: for every valuation
Sjforpj, Si la analjtic ui ftspect of and 25». C-SBjtfotat
least one valuation for t*,, S* is contradictory in respect of Sj
and SSf

B. Sihastheform(3Pi)(Si). A-Siiforatleastonevalua-
tion IB, for n,, S, is analjtic in respect of S, and JB,. C -2?, : for
every valuadon 15, for n., S, is contradictory’ in respect of iS,
and S,.

C. Si contains no operator. A-SS, (or C-S5,): The result
of the evaluation of Si on the basis of jp, is 9^ (or <~9u re-
spectively).

Let Si (or Bi) be arbitrarily given; and let it be asled whether
Si (or 51i, respectively) is either analjtic, or contradict -ry, or
neither, i.e, sjuthetic. Then in the first place one and onlj’ one
of the rules DA is applicable (for DA 2 Ca-c this results from
Theorem 34 J.3). If tins rule is DA 2 O or DA 3 C the question
will be decided by nseans of this rule. Every one of the remaining
rules, on the other band, will refer back univocallj’ to 3 second
question concembg one or more other S or a 5t Thus for ;1
or ilj, the univocal result is a sequence of questions wWch is
alwaj3 finite and which terminates with one of those two final
rules,^ For an arbitrarily ^ea sentence or sentential class, a
suffiaent and necessary criterion for ‘analjtic’ — and likewise
for contradictory’ — can be formulated on the basis of this



§ 34 */ ‘ANALTTIC* AND ‘contradictory’ IN II 113

sequence of questions (An, example of this is to be found in
the proof of Theorem 34A t ) These terms are thus umvocally
defined for all cases by means of the rules DA. But there is
no general method of resoluhon for the mdividual questions, far
less for the whole entenon The terms ‘analytic’ and ‘contra
dictory’ are tndefimie

^^e have formulated the definitioa of ‘analytic’ m a word
language which does not possess a strictly determmed syntax.
The foUowing questions now present themselres i Can this
definition be translated into a strictly formalired syntax language,
Sj’ 2. Can Language II itself be used as the syntax language
for this purpose? Later we shall show (Theorem 60c i) that
for no (non-contradictory) language S can the definition of
'analytic in S’ he formulated mS itself as the syntax language
Hence the second question must be answered m the negative
On the other hand, the first question can be answered m the
*ffif®atrre provided that Si h» adequate means at its disposal,
especially ranables p and f of ceriam types which do not occur
mil

If We take as our object language not the whole of Language II
but the smgle concentnc regions (see p 88), then for our syntax
language we have no need to go outside the domain of II It is
true that the concept ‘analytic m II„’ is not definable for any n
m II, Itself as syntax language, but it is always definable m a
more extensive region (perhaps always m II,+i) Hence
every definition of one of the concepts ‘analytic m II,’ (for the
vanous n), and also every entenon for ’analytic m II with
respect to a particular sentence of II, is formulable m II as
syntax language

A certain pomt m the given defimtion of analytic m Il’may
appear dubious For the sake of simplicity we will consider the
corresponding definition of ‘analytic m II j’ Let a language S
be used as a formalized syntax language (for example, a more
extensive region of II, or II itself) Since m Ilxfree and un
defined occur, the defimtion of ’analytic m 11^’ (corre-
sponding to DA I Cb, 3 Cd) will contam phrases such as ‘for
®vcry valuation for a ”, this, accordmg to VR i a and
If, 13 the same as saying ‘‘for all syntactical properties of
accented expressions ” I<ow what 13 meant by this phrase



114 PART in. Tire iNOTriNrre i^NcuACt n

and how is it to be formuUtfd in the sjmboltc Unfruage S? K
we said instead mertly “for all syiitacticsl properties which are
definable in S then the definition of ‘onaljnic in II,’\rould
not effect what is required of it. For just as for cveo' language
there arc numencal properties which ate not definable in it (see
p, io6), so there are also syntactical properties which are not
definable in S. Thus it might happen that the sentence ‘Sj «
anilitie in Ilj’vvas true (analjTic) in the sj-ntax^languagc S, and
yet false (contradictor}) m a nclicr sjTitax.language S*, namely
if the phrase, "for all definable s}-ntactical properties,..*', con-
tained in the entenon for that sentence, although valid for all
the properties definable m S, was not valid for a certain propert}’
which is only definable m S'. Tlius the definition must not be
limited to the s}’ntactical properties which are definable in S,
but must refer to all s}TuacticaI properties whatsoever. But do
we not by this means arrive at a Thtonic absolutism of ideas,
that is, ax the ctmctpxion that the xotahty of all propertxts, ;wlodx
is non-dcnumerable and therefore can never be tshausted by
definitions, is something which subsists in itself, independent
of all construction and definition? From our point of vietv,
this metaphj'sical conception— as it is maintained by Ramsey
for instance (see Carnap p. loa)— is definitely ex-

cluded. We have here absolutely nothing to do with the meta-
ph}’8ic3l question as to whether properties exist in themselves or
whether they are created by definition. The question must
rather be put as follows; can the phrase “for all properties..,**
(interpreted as “for all properties what.«oevcr“ and not "for all
properties which are definable in S'')be formulated in the sjTn-
bolic syntax -language S?Thi3 question maybe answered in the
affirmative. The formulation is effected by the help of a uni-
versal operator with a variable p, i.e. by means of f®*"
example, ^hat this phrase has in the language S the meaning
intended is formally established by the fact that the definition
of ‘analytic in S’ is formubted in Uie wider syntax-language
S|, again in accordance with previous considerations (pp. lo6f.),
not by substitutions of the pr of S, but with the help of valua-
tions.) This is correspondingly true for the valuations of higher
types in the wtder language regions.



§34* ANALYTIC AND CONTRADICTORT SENTENCES II5


§ 34^. On Analytic and Contradictory
Sentences of Language II

Si (or ill) IS called L-determtnate if Sj (or jlj, respectivelj) is
either analytic or contradictory (or jli) is called synthetic if
Si (or Hi, respectively) 13 not L-determinate, and therefore is
neither analytic nor contradictory
Theorem 346.1. (a) Sj and ^Si are either both analytic, or
both contradictory, or both synthetic — (6) Likewise ©i and
( ) (®i) —(c) Likewise Si and {Si}

Theorem 3402. (a) If ©i is analytic, then '-'©i is contra-
dictory — (6) If ©1 13 contradictory and closed, then '*-'©1 is
analytic

Theorem 340.3 If every sentence of Hi is analytic, then Hj
also u analytic, and conversely

Theorem J4e 4. A Hf is canttzdictory d and only if at least one
sentence belonging to it u contradictory A H^ can be contra-
dictory even if no sentence belonging to it is contradictory (See
the remarks concerning Theorem 14 4 )

Theorem 346 5. A closed sentence ©j is analytic (or contra-
dictory) if (but not only if) the truth value toblc (| 5) of ©i, in respect
of partial sentences from which ©| is constructed with the help of
symbols of negation and junction, ahvays yields ‘T’ (or ‘F’,
respectively) for all admissible distributions of ‘T’ and ‘F’ In
this connection, a distnbution is called admissible if it always
assigns ‘T’ to an analytic partial sentence, ‘F’ to a contradictory
partial sentence and ‘T’ or ‘F’ to a synthetic partial sentence
Theorem 346.6. (a) ©jV©, is analytic if (but not only if) Sj
or Sj IS analytic. — (6) ©^V is contradictory if (and only if) ©1
and ^2 are contradictory
Theorem 340 7. SIi=?Ii is always analytic
Theorem 346.8. Let Hj be a sub cl^s of H* (o) If Hj is ana-
lytic, then Hi is likewise analytic — (A) If Hj is contradictory then
H* 13 likewise contradictory

Theorem 346.9. If Hi +H* is contradictory and Hi is analytic,
then ill IS contradictory

We have already seen that the concepts ‘demonstrable’ and
‘refutable’ do not fulfil the requirement that they constitute an



u6 PART in. THE INDEFUilTE LANCDACR II

exhausti\’e distribution of all logical sentences (wbltdi also include
all mathematical sentences) into mutually exdusive dassts. Tim
circumstance prorided the reason for the introduction of the
concepts ‘analjtic’ and 'contradictory'. We must now deterraine
whether such a classification is effected by these new concqjts; the
result of this test is given in TTieofeins to and ii.

Theorem 34e.ia. No sentence (and no sentential dass) is at
the same tune both anal)tic and contradictory. — A testing of the
single rules DA one by one shows that the conditions for 'analytic'
and those for ‘contradictory* are mutually exdusive in every case
provided that they ate mutually cxdusi\-e in that case to which
further reference is made, Itt the last stage, namdy DA a Cc or
3 C, they arc definitely mutually eadusist ; and therdore they are
so in general. [In contradistinction to the analogous tbeoiem
concemmg ‘demonstrable’ aod ‘refutable*. Theorem xo does not
require the assumption that Language JI is non-contradictory.]
Theorem Every logicd sentence is L-detenainale, that

is to say it is either analjtic or contradictory. (There is, however,
no general method of resolution.)— *Por the purpose of indirect
proof, let us assume that Si were both logical and synthetic. Then
according to DA *Si would be both logical and sj-nthetic;
and, sccordmg to DA 2 5 , ()(”s,) also would be both logical
and sjTitheric. Let this be Then Sj would be logical, reduced,
and dosed, and therefore, by Theorem 34 ft.3, it would h3\’e one
of the following forms: i. where

(i-^ to n) is either (p,) or (3P,), and S» contains no operators;
2 - 3 t; 3 - ~ 9 l. According to DAnCe, the forms and are
excluded here, since S, b supposed to be synthetic. Hence ^
JTOuId have the fiist-mentioncd form. Then, in accordance with
A 2 Ca and b, in respect of at least one series of >-aluations for
Di,...o,, S, must be neither analytic nor contradictory. The
e uahoa of Sj on the basts of such a series of valuations would,
^ ^A3 C, lead to a sentence which is neither 91

34 ^. 1 . that is impossible.

cw mg to Theorem 11 synthetic sentences are only to be
found amongst the descriptive aeniences.

Theorem 34ki2. If » definite G, is analytic, then it is also
demon5tiabIe.-{By DAa.^, Theorem 34^.4 and 346.1.) On the
other hand a defimte may be analytic without being demon-



§24^ ANALTHC AND CONTRADICTORY SENTENCES II 7
strable — ^Amongst the indefinite Si there are anal ytic ones which
are non-demonstrable, also some of the simple form pti(3i), where
pTi IS a definite pi] (compare the examples m § 36) In a case like
pri(fUi(nu)), where fu^ is any undefined fUb, is a defimte
Sb which IS analytic but not demonstrable
Theorem 346.13 Every definite St is resoluble, that is to say.
It IS cither demonstrable or refutable For this ageneral method of
resolntioa exists


§ 34/ Consequence m Language II

Two or more sentences are called incompatible with one another
if the class constituted by them is omtradictory, otherwise they
are called mutually compatible.

A sentence is (in material mterpretation) a logical consequence
of certam other sentences if, and only if, its antithesis is mcom-
patible with these sentences Hence we define as follows (si is
called a consequence of in II, if J^i + {'^()(®i)} >s contra*
dictoty Si IS called Independent of Aj, if Si is oei^er a conse*
qtience of Ai nor mcompatible with Aj We shall use the defined
terms not only m the case of a sentential Ax but also in the
case of one or more sentences (as premisses) For instance, we
aB Sj a consequence of Si and St if 0} is a consequence of

{S1.S1}

It happens sometimes that Si is a consequence of an infini te
sentential class Ai, without bemg a consequence of any proper
sub-class of Ai [Example Let pti be an undefined pTb, Ai be
the class of the sentences ptiCSt), and Si be pTi(ji) ] It is thus
essential that the definition of ‘consequence’, as opposed to that
of ‘denvable’, should refer not only to finite but also to infinite
classes

The concept ‘consequence’ is related, to the concept ‘derivable’
as ‘analytic’ is to ‘demonstrable’, that is to say, it is more com-
prehensive, but on the other hand it has the disadvantage of pos-
scssmg a much more complicated definition and a higher degree of
indefimtencss. ‘De riva ble* is defined as a finite chain of the re-
Is^on ‘directly denvable* ‘Consequence* might be analogously
defined as a rhain of a simpler ration ‘direct consequence’



PART in. THE IKDEFINITE LANCCACE 11


IlS

‘ Analytic’ would then be defined as * consequence of the sentential
null class’ and ‘contradictory* as ‘sentence of which every sen-
tence is a consequence '.In this way the definitions for Language I
were previously formulated (§ 14). In the case of the definitions
just given for Language II we took a different course, and for the
sake of simplifying the technical process first defined ‘analytic’
and * contradictory ’ and from them the term * consequence The
question now is whether the term ‘consequence* as so defined is
related to the terms ‘analytic* and ‘contradictory* in the way
described; that this is the case is expressed in Theorems 5 and 7.
Further, it must be shown that the relation ‘consequence’ pos-
sesses a certain kind of transitirity. 'Hiis would be obvious in the
case of the first meiliod of definition, but here the proof is not so
simple (Theorem S).

Theorem 34 f*x- If Sj « an element of ilj. then S, is a conse-
quence of Jlj. Si is ahvays a consequence of Sj.

Theorem 34f.a. If ft, 1$ analytic, and Si a consequence of ft,,
then Sj is also analytic. — fti + {— ’()(«,)} is contradictory;
therefore, by Theorems 34^.9 and 34Mf, '**'() (®i) ** contra-
dictory, and hence, by Theorem S, analytic.

Theorem 34f,3. If S, » contradictory and a consequence of
ft,, then ft, is also contradiaory. — According to Theorem 34e.x b
and ib, -^0 (S,) is analytic, and hence, by Theorem 34^*9, fti
IS contradictory.

Theorem 34f^. Let St be a consequence of S,; if S, is ana-
lytic, then St is likewise analytic; if St is conliadictory, then S,
is likewise contradictory.

Theorem 34f.5. If S, is a consequence of the sentential null
class, then S, is analytic; and cornTrsely. — This follows from
Theorem 34^.2.

Theorem 34f.6, If S, is analytic, then S, is a consequent of
every sentence; and conversely.

Theorem 34f.7. If ft, (or SJ is contradictory, then every sen-
tence is a consequence of ft, (or of S,, respectiwly); and con-
versely. — By Theorem 34r.8&. Converse by Theorem 3.

Theorem 34f.S. If ^ is a consequence of ft., and every sen-
tence of ftt is a consequence of ft„ then St i^ ^ consequence of ft,.

Proof. I.et ft4 be the class of the sentences () for c\-ery
e, of ft, ; likewise ft, for every S, of ft,; and let S, be ( ) C^S,>



119


§ 34/ CONSEQUENCE IN lANGUACE 11
Then S* and all sentences of 51 * and 5 ^* are reduced and closed
Let a senes of valuations for the b (here descnptive symbols) of a
sentence or a sentential class be designated by ‘IB’ withthccorre-
spondmgsuffii Assumptions i ftj+^/wO^Sj^Jiscontradictory,
hence also 2 For every 0 , of + (S.)j

13 contradictory, accordingly for every of ilg, is

contradictory Assertion (S,)} is contradictory, that

IS to say, ^4 + {--S' 0 ,} IS contradictory This, accordmg to DA i Cb,
means for any choice of Sj and Ss, cither ~ 0* or a sentence of
IS contradictory m respect of S, or S*. respectively For the
purpose of indirect proof, let us suppose the contrary, namely
Sj and 58 * are given m such a way that neither '*>'©4 nor any
sentence of 5I4 is contradictory in respect of S4+S4 Assumption

1 means for any S5 and SB*, either S* or a sentence of is
contradictory m respect of S85 or of S*, respectively Assumption

2 means for every of ftj m the case of any choice of S* and
S/, either or a sentence of ^4 ts contradictory m respect
of S/ or 104, respectively Hence, on our supposition, on the one
hand, for any arbitrary 0j, a sentence of say S7, would be
contradictory m respect of 0j, and on the other hand, as for every
€( of ilj, so also for ©, tn the case of an arbitrary 0; (contained

~07 would be contradictory m respect of 0 , But this is
impossible, for since 0, is closed, 0, and -*'07 cannot both be
contradictory m respect of the same valuation (see Theorem

34 « 6)

Theorem 34t9 (a) If 0^00, is analytic, then 0 , is a conse-
quence of ©4 — (6) If 0 j is closed and if ©^ is a consequence of
04, then SjDSj is analytic-

Proof of g a For a closed ©j the proof is simple For an open
the procedure is as follows Since ©^ o ©, is analytic, ( ) (-o' ©4 V
©*) also IS analytic, further, ~'()(0x)V() (©*) also (the proof
B too long to be given here) Acceding to TTieorem 34 e 2 i the
negation of the last-named sentence is contradictory, hence
0 ( 3 i)» — 0 (Gj) IS likewise contradictory, hence also the class
bence \Si, --'t') (©j'j) Tnereiore 0*

B a consequence of ©4

Proof of 96 {®j>‘^{)(iS*)} B contradictory, hence ©4*

'^0(^1) B also contradictory Smee this sentence is closed,
accordmg to Theorem 34e z & its negation is analytic, and con-



120 PART III. Tire INDEFINITT LAXCOACE Tl

sequently Si V () (Sj) b Hketrise aniljtic. Therefore, stn« Si
is dosed, '>»' Si V Sj is analytic, hence also Si D S*.

Theorem 34f.io. Si and (3i)^i) are consequences of the
dass of the sentences Si^gj|. — This corresponds to the rule
DC 2 for Language I (p. 38).


§ 34^. Logical Content


We call the dass of the non-analytic sentences (of II) wluch arc
consequences of Si or 5 \i (in II) the content of Si or i^i, rtspec*
lively (in II). (Forthereasonforthisdefuutionseepp.4if.) Let
eqvipoUtnt and ' tynonymoia' be given definitions for II ana-
logous to those for I (see p. 42). These formally defined terms
ttrrespond exactly to rvhat is usually designated in material
interpretation as ‘ equivalent in sense or ' equivalent in meaning’,
res^ctivcly, so long as ‘equh'alent in meaning’ is understood
as of equivalent logical meaning” and not as '‘designating
the same object”. In order that two object- (or number-)
designations SI, and SI, may be synonjrnous, SIi®?!, not only
must be true but must also be analytic. (See §75, examples 6

We say that S, or 51 , has nuU content if its content is the null
c ass. By total content, we understand the dass of all non-analvtic
sentences.


Theorem 34g.i. If two sentences are consequences of one
an^er. then they are also equipollent; and eonversdy.

eorem34gA If t'To scntcncesare equipollent, then theyare
sj-nonymous; and conversely.

Theorem 34g.3, (a) if jj analjtic, then Sf, and SI, are

synonjTnous. (i) If gi^ and SI, are synonjTnous, and if SI,=SI, is
a sentence, then this sentence b analytic.

Theorem 3454. If j, anajj^tje then S, and S, are

equ pollent; and convxrsely.

Theorem 345.5. If jj ^ yj

respecuvely) has null content; and conversely.

Theorem 34 e. 6 . If G, (or « J b contradictory, then G, (or
respectirely) has total content; and conversely.



§345 LOGICAL COSTENT I2I

Theorem 34g.7. Sj.Sj. S* and ^(Si, Sj, On} are equi-
pollent

Theorem 34g 8. The content of a disjunction is the product
of the contents of the terms of the disjunction — If the product
of the contents of several sentences is null (and consequently, ac-
cording to Theorem 8, the disjunction of the sentences is analytic),
we say that the sentences have mutually exclusive contents


§ 34A. The Principles of Induction and
Selection are Analytic

We shall now prove that the Pnnaplc of Complete Induction
and the Fnnaple of Selection are both analytic. These prmaples
are included amongst the pnrmtive sentences which were pre-
viously stated for Language 11 (PSII 20 and zi, § 30) By the
sample of the Principle of Induction, we shall show how the
cntenon of whether a certain particular sentence is analytic or not
IS developed step fay step fay mearu of the DA rules The proofs of
Theorems 1 and 2 are mteresting because they involve a funda-
mental question: m each one of these proofs, there is used a
theorem of the syntax language which corresponds with the
theorem of the object-language whose analytic character is to be
proved.

Theorem 34h.s. The Pnnaple of Complete Induction (PSII 20)
IS analytic.

Construction of the cntenon Let us call PSII 20 Si The neces-
sary and suiEcient criterion of the analytic character of ®i may be
transformed in the following manner, each step being univocally
established by means of the DA rules By DA 2^ the cntenon is
must be analytic Let this be We find S* by means of
reduction* _


must be analytic m respect of every valuation for F. By DA 3 if

for every valuation S. for ‘f. and for at least one valuation <8, for

‘v'.-(yU .TmustbeanalyticinrespectoflBiandS, ByDAyrJ:

in tie cie of every S. for ;f. for at least one valuation S. for



122


PART III. nin ISDEFINITT LANGUAGE 11


‘at’, and for every Sj for the operand which occurs in the
square brackets — let it be S| — must be analjtic in respect of
SSj, S,. and 2}j. By DA 3C: in the case of every Si for 'F\ for
at least one S* for and for every S, for ‘y the evaluation of
S4 on the basis of Si, Sj, and Sj must lead to Sfl. In this way the
criterion is constructed.


Proof that the entmon is fulJiHfd. Let Sj be ‘ '>«-F(0)VF(x)V
/•O’)', and Si be * --/'(0)V ,w/*(xi)vr(y)’; 6, is then e5.<=,.
Si is of the same tj^jc as 'F‘, i.e. of the type (0); and therefore,
according to VR 1 a and c it is a class of ;t. With regard to Si,
three cases arc to be distinguished: i. The St ‘0’ does not belong
to Si; 2. ‘O' and every other St belongs to Si; 3. *0’ belongs to
Si but an ;t exists — say ;ti — which does not belong to S}. —
I. In case 1, the evaluation of S|, independently of S| and Sj,
always leads to 91. For here, in accordance with VR a a and EvR 1,


‘F(0)‘ is replaced by and thus •~F(0)' leads to

from which, by reduaion in accordance with RRs/, 91 results.
Then, by RR 3 e, 91 results from (©, and from ;|, and hence, by
RR 3 a, also from S4.— 2. In case 2, S4 independently of S», for
any Sj, leads to 91. For, since every €t belongs to Si, ao also does
the valuaUon for 'y\ S,. Therefore, in accordance with EvR x,
the ^uation of ‘F(y)' leads to 91. Thence, as before, S4,iS,.S|
all yield 91. — 3. In case 3, it is possible to state, for any a SB*
such that the evaluation of (5, leads independently of Sj to 91.
Since, namely, ’ 0 ' belongs to 2J,, but St, does not, as step by step
St, a stroke ' • we pet an ei, such that it belongs
to S„ while Et,i does not. (In this inference, complete induction
u applied in the syntc-Ianguage.) Notv let ua take St, as S,
‘ nr'ti’ to accordance tvith EvR I,

h (a) will become SI, By VRah. S|,l ia the valuation for
a . Hence, according to EvR i. 'f (al)' becomes ~S1, and

hence ~F(ai)' becomes S|, from which wc get 31, And

• S. in SI.—The criterion

18 fulBlled m all three cases; and S. (PSfl ao) is accordingly


sentence of the form PSII at (Principle

of heltctton) IS analytic.

The proof is easy but too long to be given here in full. For the
sake of a fundamental queaUon wrhich is intolved, wc shall, how-



§34^ LNCUCTION ANTJSnKTtO'lANU.YTlC *^3

ever, at least mdiale Its form- Let us assume that Si s a sentence
oftheformPSIIai is then

(3 pO W Q(>.) Qp<) Qp.) Qp.) W Q p.) (Pi.) (”u) [s>] •

where Sj is

). .(~P,(P,)''

~{h=vd'> ~Pi(P.)V ~II.(P,.)'' ~Ps(Pm)''
~R(”ll)''(Pu=Pufl

;i IS a conjunction with 30 terms, every term of which is a dis-
junction having 4 or 8 terms- Lets, (»= t to ti) be the ii’aluation
for r, or p„ respectively According to DA, Si is analytic if the
foUowmg condition is fulfilled for every Si there is a Sj of a
kmdsuchthatjforeverySj thereisaSj Si Si.SjOfakindsuch
that, for every Sg, there ts a Sy of a kind such that for every Si»
and Sji the evaluahon of Sj based on Sj to leads to Let
Si he given arbitrarily We may classify the possibilities with
regttd to Si as follows Sj is either ntdlor it is not, Sj contains a
null class as an element or it does not, there are two classes be*
tongisg to Si and havmg an element m common, or there are not.
Then it IS easy to show that, in each one of these cases, the entenon
IS fulfilled. Here we shall only examine the most important case,
Bamely the last Sj and the classes belonging to 5 Bi arc not null
and no two of the classes belonging to ^ have an element in
wmmon Then — assuming that the fVi»opIe of SeUehon holds m
the syntax language — there is a selectrvc class of ©j, that is to ssy,
a dass such, that it has eactly ooe element 10 common with e% cry
class belonging to 18 i Let us take this selective class as SB5 Then,
as It B easy to show (classification of cases is e thcr an element

of Si or It IS not), the given cRtcnon can be fulfilled in every case
The Prinaple of Selection itself is used in the foregomg proof
It must be noted however, that this principle does not appear here
^ a sentence of the object language, but as a senttnee of the jynr<iv-
tkich ire me in our tyntocfical treoesUgattons It is dear
^t the possibility of pronog 3 certam syntactical sentence de-
pends upon the richness of the syntax language which is used, and
*®Pcciany upon what is regarded as valid m this language In the
present case, the situation 19 as follows we can work out m our
^tax language S (for which we have here taken a not stnctly
®ttnaincd word language) the proof that a certam sentence, Sj,



124 PARTIII. niE INDEFINITE LANCDACE II

of the object-language 11 is analytic, if, in S, have a certain
sentence at our disposal, namely, that particular sentence of S
which (in ordinar)’ translation) b translatable into the sentence
Si of II. From this it follows that our proof is not in any way a
circular one. An exact analogue holds for the application of the
Principle of Induction of the syntax-language in the proof of
Theorem i. The proofs of Theorems x and 2 must not be intei'
preted as though by means of them it were proved that the Prin«
ciple of Induction and the Principle of Selection were materially
true. They only show that our definition of ‘analytic’ effects on
this point what it is intended to effect, namdy, the diaracterizadon
of a sentence as analjtic if, in material interpretation, it is regarded
as logically valid.

The question as to whether the Principle of Selection should he
admitted into the whole of the language of science (including also
all syntactical investigations) as logically valid or not b not decided
thereby. That is a roattcf of choice, as are all questions concerning
the language-form which b to be chosen (cf. the Principle of
Tolerance, 1 17 and 1 78 ). In view of our present knowledge of
the syntactical nature of the Principle of Sdection, its adimssion
should be regarded as expedient. The fact that by means of its
admission the construction of the mathematical calculus b ob-
viously considerably simplified speaks for it Against it, there is
hardly anything to be said, so long as the existence of any con-
tradiction in it has not been proved (and seems, on the contrary',
lughly improbable).

§ 34*- Language II is Non-Contradictory

We have already attempted to represent the inexact concept of
logical validity (in II) by mcans-of two different terms: the d-term
‘ dmonstrailf' and the c-term ‘ano/yftV’. The relaoon subsisring
between these two tenns must now be examined more closely.
We shall show that the second term b an extension of the first:
every demonstrable sentence is analytic, but not conversely. In
the same way we shall show that if <4 is derivable from Sj is
also always a consequence of Slj. In connection with this, we shall
show that I^anguage II is ttonramtradictor ^' — that is to saj', that
two sentences and '—Si ate never demonstrable in II.



125


§34* II IS NOV-COVTRAOICTORY

Iq Older to sliow that every demonstrable sentence is analytic
(Tbeorem 21) we must prove that every one of the pnroitive
tences PSIl 1-23 of Language 11 Q 3°) “ analytic. The mdmd^
pramave sentences will be tested one after the other m the fol-
bvnng pangcaphs (Theorems 2-14)

Theorem 34L1. All sentences which are demonstrable in the
ordinary sentential calculus— hence, for example, the PnnapU
of Exchded MtdMe, the Principle of Contnidiction, and the
Ptiflaple of Double Negation — arc analytic.— This follows from
RR2,3

Theorem 34I.S. The pnrruuve sentences PSII i-^ axe
lyta— This follows from Theorem 1.

Theo r e m The pnnuUve sentences PSlI 7-9 are ana-
lytic.— This follows from RR 8 h, c, 4 and Theorem 34^5
Theorem 34-4- EverysentenceoftheformPSII rots analytic.
—Thu follows from RR 5 a.

Theorem 341$. Evetyseoteoceof the form PSlI 11 uanalytic.
—The proof is a simple one based upon a differentiation of cases .

Si aad Ot either have or have not the tame valuation.

Theorem 341,6. PSU ta is arulyuc.— This follows from
RRir, 2/

Theorem 34L7. PSII 13 is analytic. — ^This follows from

Theorem 341,8. PSIl 14 and 15 arc analytic.— This follows
fioia RR ya, d, and Theorem 346 j
Theorem3^9« EverysentenceofthcfonaPSlI 16 is analytic.
Proof By (partial) reduction we get

Thu is analytic since the operand is analytic m respect of at least
one valuation, Sj, for Ui, utaamudi as or any arbitrary valua-
t»a for may be taiea as

Theorem 34I-10. Every logical sentence of the form PSU 17 is
^ync.— The primitive sentence PSIl 17, the PnnapU of
ubrnmiion mth argumtnts, represents one of the critical points
m the Ic^ico-mathemancal system, especially m the case where
so-called tvrplut vanabUi occur

Lct(S,bea«~ — * . . . r .

Ist^iheoi.c^ Ot . - . . . . .

"mblo (which do nc ; _; s„ ab) b, conUms the



126 PART in. Tire INPERNITE LANGUAGE II

surplus free variables (by surplus \*ariables are under-

stood those that do not occur in ?lrg,). Let the ^•ariables which
occur as free \Tiriables in Sj, in addition to Pj, be In

order to show that S, is analytic, we will show that ^S, is analjtic
in respect of any given series S of valuations for the t-ariables
V ®y partied reduction we get for
^ [S.v Si], <vhcrcm S. B (3 p,) (~S,) luid S. is .

Two cases may be distinguished:

1. Let there be a valuation for such that is

analytic m respect of 2 ), and S. Then, according to DA3B,
(3 Pi) IS analjLic in respect of S; hence, so also is '‘Sj,
and further "Ss

2. Let there be no valuation for p^ of the kind described. Then,
for every arbitrary valuation ©, for Pi. ^ ( — • Si) is not analytic in
respect of ©, and S, and therefore, since it is logical, in accordance
with Theorem 34e.11, it is contradictory. Thus, is analytic
m respect of S. and S. Now, on the basts of the given valuations
S we wall choose a certain valuation IB, for p, in the following
manner According to VR i e, a possible \‘aJuation for p, is a class
of possible N-aluations for 2lrf|,: now let S, be determined by the
condition that a possible valuation for Jlrfli shall be an clement
of ©1 if, and only if, is analytic in respect of ©; and S.
p, is always followed m S| by an argument-opression. Let a
certain partial sentence in ©1 containing p, be PiCJti.SI,....?!*).
Assume that ©' is the series of valuations for 21 ,,.. . 2 ft which,
according to VR 2, result from the valuations S (of which here
only the valuations for the free s-ariables occurring in 2f„.,.2r»
come into consideration); here, when partial sentences occur in
those expressions, we take 9 } as the valuation for an analytic partial
sentence and ~»<91 as the \-aluation for a contradictory' partial
sentence. Then ^p, ( 9 I„. ..?!*), since it is logical, is either analytic
(Case a) or contradictory (Case b) in respect of S, and ©'. In
Case a, according to EvR i, S' is an element of ©,; in Case b it
is not. Now, S' is also a possible valuation for 2Ir0i. In Case a,
in accordance with our choice of ©,, is analytic in respect of
S' (for D,, ...Di) and S; in Case b, it is contradictory. Thus, in



127


§ 34* n IS NON-COVTRADICTORY


Case a, j ^ analytic m respectof S, and m Case b

It IS contradictorj — Sj is obtained from Si by replacing, at the
substitution positions, a partial sentence of the form Pi (%, 31*)

by the corresponding partial sentence j j As we

have already seen any two correspondmg partial sentences of this
kind are either both analytic or both ointradictory m respect of
Si and ® Hence, if ^Si is analytic m respect of Si and ©, then
is also analytic m respect of S It has been shown earher
that ^Si 13 analytic m respect of S and every arbitrary valuation
for pi, and therefore it is also analytic m respect of S and Si


Accordingly is analytic tn respect of S, and hence so also

IS

Theorem 34I ii. Every sentence of the form PSII 18 is
analytia— This follows from RR9f, 2/, and Theorem 34^ 5
Theorem J4LX2 Every sentence of the form PSII 19 is ana-
lytic — By means of partial reduction, we get




the rest of the proof ts analogous to that of Theorem 9
Theorem 34L13 Every sentence of the form PSII 22 is
analytic.

Proof Let (Sj have the form PSII 22 ^Si is
(3»i)[(Pi(<'>)''P,(ii,)v(p,=pJ).(~r(|!,)v~p,(d,)v(p, = p,))]
For this to be analytic, there must exist for any arbitrary valua
tioas Si and S* for Pj and respectively, a valuation S3 for p,
such that the evaluation of the operand on the basis of these
valuations leads to 91 By means of a classification of cases, it is
easy to show that this condition is fulfilled

TTieorem 341 14 Every sentence of the form PSII 23 is
analytic.

Proof Reduction leads to

Qt.)

For this to be analytic, there must exist for any arbitrary valua
tions for fi and f, a senes of valuations for Pi, p„ such that the



12$ PART in. Tire isnmNiTt iANGr.\CE ii

evaluation of the operand leads to Su It is easy to dsmon'tnte
that this condidon is fulfilled. If any arhitraiy ^'alu2tioas for
and f, ire given, then either they agree \rith one another or they
do not. In the first case^ the second term of the disjunction, and
hence, the whole operand, hecomes $ 1 . In the second case, we
tale a series ol vduations for p„ ...p„ such that, with it, by means
of the valuations for j, and Ij, two differtnt A’aluatioos are corre-
lated. Then the first tenn of the disjunction becomes ?i and hence
the whole operand becomes

Theorem Every logical primitn't sentence of II is

analytic. — This follows from Theorems 2-14, 34A.1 and 2,

Theorem 34I.1 6, If Sj is analytic, then is also analytic.

Theorem 3.^17. Every pnmitive sentence of II is analytic. —
This follows from Theorems 1$ and 16.

Theorem 34l.t3. defiiution in II ia analytic.— By RR i

and Theorem 34 e.?.

Theorem 34I.1 9. («) Sj is a consequence of Si and Sj 3 S*.—
(6) (®)(®i) js a consequence of Sj.

Theorem 341,30. If, accordingtoRII land 3(^31), Sitsdirectly
derivable from Si ot from Sj and St, then S* is a ennsequenee
of 6} or of Si and S*. respectivtly. This follows from Theorem 19.

Theorem 3^1, Erei3-<InBm»iro.*'iearnrrjiff (La II) w
—From Theorems 17, iS and 20, and Theorem 34/— The con-
sersc is not true (example; Theorems 36.2 and 5). (See the
second diagram on p. tSy.)

Theorem 34 L a a . If 0„ « JrnrabU (in II) from Si, Sp
then S, is a consr^uairt of Si,™S„. — ^This follows from
Theoreirjs 17, 18, 20, and 34/.S,

Theorem 3-0,33. ^.-91 is not demonstrable in II.— This
follows from Theorem 21 and DAnO.

A language S is called contradictory if every sentence of S
is demoiistrahle in S; otherwise h u called non-contradletory.
(See § 59.)

Theorem 34^34. Language II (as the system of the d-rules
FSII 1-23 and RII j-2) is a non-tontra;ciory language. — -niis
follows from Theorem 23.

Hilbert set himself the task of proving “with finite means’* tie
non-^KtraJktoriMt of cktwal nathnnatics. ^\’hat is meant by



129


§ 34* n IS NON-<»NTllADICTORY
‘finite means’ is not stated exactly in any work of Hilbert’s which
has been pubhshed up to now (including [Grundl 1934]), but pre-
sumably what we call 'defimte ^ntactical concepts' is intended
Whether with such a restriction, or anythmg like it, Hilbert’s aim
can be achieved at all, must be regarded as at best very doubtful
m view of Godel’s researches on the subject (see § 36) Even m
the achie^ ement of the partial results which are attainable, there
are very considerable difficulties to be overcome The proof which
we have just given of the non-contradictonness of Language II,
m which classical mathematics is included, by no means repre-
sents a solution of Hilbert’s problem Our proof is essentially de-
pendent upon the use of such syntactical terms as ‘analytic’,
which are indefinite to a high degree, and which, m addition, go
beyond the resources at the disposal of Language II Hence, the
significance of the presented proof of non-contradictonness must
not be over-estimated Even if it contains no formal entire, it
gives us no absolute certainty that contradictions m the object-
language II cannot anse For, since the proof is earned out m a
syntax-language which has ncher resources than Language II, we
are m no wise guaranteed against the appearance of contradictions
m this e7ntax>hoguage, and thus m our proof

§ 35. Stntactical Sentences which
Refer to Themselves

If the syntax of a language is formulated m that language itself,
then a syntacUcal sentence may sometimes speak about itself, or
more exactly, it may speak about its own design — for pure syntax,
of course, cannot speak of mdividual sentences as physical thmgs,
butonlyof designsandforiDS For instance, “asentence

of the design is closed (or open, demonstrable, synthetic, and
the like)”, and here itself possesses the design which is de-
scribed in It For every syntactical property, it is possible so to ;”
construct a sentence that it attributes to itself— whether nghtly or
wrongly— just this property We shall state the method of domg
this, since it leads to important consequences for the questions of
the completeness of languages and the possibihty of a proof of
non-contradictonness We have already formulated the syntax of
Language I m that language ttsdf In the same way the syntax



130 PART in. THE CCnETOflTB LANCCACH H

of Tjtnmiapt 11 CCJI he farrailaiti in II itself, »nd to tn even Trider
extent, since in Jl indefinite synttcdol concepts ctn

slso be defined. Our funhcr bvestigitions wHl bsve reference to
Lingtiage 11 , but they can easily be transferred to Langxixge I,
since in them tve use only definite symbols of the Knds whidi have
already occurred in I.

’str(n)' means: **lhc ^*; 1 , which has the value [For

example, str(4) « Resresare definition:

str(0)5srtihe(4) (i)

str(ij')**ys(str(«), reihe(i4)] (2)

Let any syntactical property of exptreaons be chosen — ^for
instance, ‘descriptiTe’ or ‘oon-dcmonstTahle fm IT)*. Let Sj be
that sentence with the free variable ‘jc* (for whidt we wQl ttte the
term-nutnher 3) which expresses this property [in the examples:
*DesirA(x)', ‘—BewSatjIIfr,*)*; compare p. 76]. Let ^ be
that sentence which results fromtSi if for ‘subsi [*,3,str(x)]’
is substituted. [In the second ewn ple, ;j, is *«^BewSstdI
(p,8ubst[*,3,8tr(*)l)’.] By mesTis of the rule which has been
stated earlter (p. 6S), the term^number for et'Ciy defined symbol
is univomlly determined. Thus, if ;1 is pven, the series*Qumber
of 6, can be calculated ; let it be designst^ by * b* (‘b * u t defined
53). Let the ®^scntence 8ub$t[b,3,8tr(b)] be ;,; thus S, is the
sentence which results from ( 5 j when the ;t with the value b is
substituted for ' It b easy to see that, syntactically interpreted,
;, means that ;1 itself has the chosen syxitactical property.

TVe will explun this point by the example of the proper^ *DOn-
dcmonstrable (in II)\ Here instead of ‘I©,’ we will writ e *G’,
[This sentence forms the analogue in II to the sentence con-
structed by Godd \lJnniUtheiihGrr\, the only difference being
that in it we use a free instead of a bound variable.} Let b, be the
series-number of the Sj (given above) of this example. str(b^)
is an to mahe the following discussion clearer, we will

indicate th'is ;t by *0n..* (thb El consbts of ' 0 * and accents
and is thus far too long for anyone to write out in full). Hence,
0 'i«» = bt. Let G be the sentence wluch has the series-number
sub 5 tlb,. 3 ,str(b 01 (or subst[0n..,3,5tr(0n..)]). Hence, G b
the sentence whi<i r«ults from E, if ‘On..’ is substituted for
***; G b accordingly the sentoice ’'-'BewSatin(r,substIO''--,



§35 SENTENCES WHICH SEPZR TO THEMSELVES I31

3,str(0'>*»)])’ In this way, we have determined the wordmg of
(5 Syntactically interpreted, it means that that sentence which
has the senes number subst[0“»*,3,str(0*'»-)l ^ ^^t demon-
strable But that sentence la <5 itself Thus <5 means that ® is
not demonstrable.

Inadentally, it is to be noted that a sentence of descnpUve
syntax can refer to itself m an even more direct maimer, namely,
not merely to its design but also to itself as a physical thmg con-
sisting of prmter’s ink. A sentence which occurs at a certam place
can, m material mterpretation, mean that that sentence which
occurs at that place, 1 e itself, possesses such and such a syn
tactical property And here it ta even easier than in the case of
sentences of pure syntax to construct for every given syntactical
property a sentence which — whether nghtly or wrongly — attn
birtes that property to itself Suppose the property m question
IS expressed by the pt ‘Q*. then the sentence ‘Q[ausdr(b,a)]'
means '*Tbe expression occumng at the positions a to a + b hu
the property Q” (compare p 78) \ExampU At the places a to
a4-8 (mdicated, say, by numbered positions on a piece of paper)
let the sentence *OeskrA[ausdr(8,a)]’ occur Syntactically
mterpreted, €1 means that the expression which occurs at the
places a to a+8 is a descriptive exptession. But this expression
IS Itself, Inadentally, is true (empincally valid) smce (9^
contains the fu^ *ausdr’ ]

^ §36 Irresoluble Sentences

We will now show (following Godel s hne of thought [Unent
scheidbare^ that the sentence (5 constructed m the preceding Section
IS xrresohible tn II r

We have built up Language II m such a way that the syntactical
rules of formation and transfonnation are m agreement with a
material mterpretation of the symbols and expressions of II which
we had m view [From the systematic standpomt, the converse
relationship holds logically arbitrary syntactical rules are laid
down, and from these fbrmaT rules the mterpretation can be
deduced. Compare § 62 ] In particular, the dehmtion of
‘analytic (m II)’ is so constructed that all those sentences and
only those sentences which are fiscally valid m thei^ material



PART nl. THE ECDETIKm: LANCTACE 11


132

iatcTpretation »re called anilyttc. Funhex, xn the cncstrucjjon of
the srithmetirwJ syntax of 1 a» 1 (D 1-J25), we proceeded ia sudi
a way that a sentence of tins syntax — and hence a syntacocalJT
inteiprctable, logical sentence* naaaely, an arithraetjcal sentence
of 1 — turns out to be true arhhmeticiHy when and only when on
a syntactical bteipretanon it xs a true syntacxical sentence, per
instance: ‘BewSatx(a,b)* xs arithmetically true when and only
when a is the series-series-number of a proof in acsordince with
the rules laid down, and b the series-number of the last sentence
in this proof.] Now let us suppose that xn the same way the
arithmeriredsmtaxof II IS stated in II. pot instance** BewSattll
(r,x)' is defined so that it means: **f is an ^^'pioof of the ^sen-
tence Here, ‘BewSitsir is a definite pr.] Then a syn-
tactically interpretable arithmetical sentence of Language 11 trill
here be logically rahd, and therefore also analytic, when and only
when, tnatenally interpreted, it turns out to be a true syntactical
sentence. Thus wc have here a shorter method (which is beesuw
of Its danty, easy to use) of protmg with rtspeut to certain Sj
(a proof which xs otherwise eery tedious) that they are analytic
(or eontndictory); this proof arises from a non-fortnal conridera*
tion of the truth or falsity of the sentence xn question tn is
syntactical interpretation. [In the abete example: if weeanvbow
that the * ,s a prtiof of the ^sentence b, it is

therdiy demonstrated that the sentence *BewSatslI(a,b)’ is
analytic in II.]

<5 was the sentence * ~BcwSatrII(r,subst [...])*; for the saic

of brerity we will write here *subft[ ]* instead of *subst[0”*‘,

3,str{0"«*)3'. The senes-number of G was suhst]..,].

Theorem 36.x. If Language 11 is xion-contraihctDTy, © is not
demonstrable in II. — Suppose that there were an ssspn^if a of O.
Then the sentence of II which means this, namely ‘BewSitzII
(a,subst[...])’, would be true, and thus analytic, and, since it is
definite, also demonstrable. Now if G were demonstrable, so also

m>dd be ©(.'.j. wKch is ‘~BewS,tt^(^stlbst Em

xlfis sentence is the negation of the previous sentence. Thus II
would be contradictojy.

Theorem 36.S. © isBofdfjnowff^ci'Zeinll. — From Theorems x
and 341.24.



133


§ 36 IRRESOLUBLE SENTENCES

Theorem 36 3. (5 is not tefuUtbU m II — Suppose that (5 were
refutable, and therefore (compare p 94) ‘~(r)(~BewSatzII
(r,subst[ ]))’ demonstrable Then ‘(3r)(BewSatzII(r,subst
[ 1 ))’ would be demonstrable, and, by Theorem 34121, ana-
lytic, and therefore true, that means that a proof for the sentence
with the senes number subst[ J would exist, and therefore for
© But according to Theorem 2 this is not the case

Theorem 36^. © is irresobtble in II — By Theorems 2 and 3

Theorem 36,5. © is analytic — In syntactical mterpretation,
© means the same as Theorem 2, is therefore true, and conse-
quently analytic. Thus © 13 an example of an analytic but non-
demonstrable sentence of II (see diagram, p 185) Every sentence
of the form © where is ‘r*, ts analytic and defimte and

therefore, according to Theorem 34e iz, also demonstrable, but
the universal sentence © itself is not demonstrable

Let be the closed sentence ‘( 3 *)(r)(^BewSatzII(r,*))’
In syntactical interpretation it means that there exists m 11 a non*
demonstrable sentence and that therefore Language 11 is non*
contradictory

Theorem 3$ & is analytic.— 2 B]i is true, according to
Theorem 341 23

Theorem 36.7. SBjj is not demonstrable m II —Theorem 7
can be proved by applying the proof given by GSdel {[UnenU
schadbare] p 196) We will indicate the argument very briefly
The proof of Theorem 36 i can be effected by the means at the
disposal of Language II, that is to say, the sentence 2 B|| 3 © 13
demonstrable in II Now were 2 Bn demonstrable, then, ac-
cording to RII I, © would also be demonstrable But this, by
Theorem 2, is impossible The non-eontradiettmnesx of II cannot
be proved by the means at the disposal of II SBh is a new
example of an analytic but at the same tune non demonstrable
sentence

Theorem 7 does not mean that a proof of the non-contradic-
tonness of II would not be possible at all, indeed we have already
indicated such a proof The theorem means rather that this proof
jj e-sJy noJb the sescamts of 3 sjcotaT SonmiiaSsd sa a

language ncher than II Tlie proof which we stated earher
makes a very essential use of the term ‘analytic (m II)’, but this



r.«T in. TOE caJErrcm: L-cccavcs it


*34

term (as ive shall see liter) cannot be defined in any syntax
formulated in Lanfuage 11 .

Corresponding results are tiue/or Ler^y^e I eJsox if is the
analogously constructed sentence to in I '«-BeteSatz(r,subst
[...)) •'), Cij IS analjtic but irresoluWe in Language I. Let SD, be a
sentence of I which approximately cofiesponds to the sentence
2Bn (such as ‘ "-BcwSau(r,e)*. where c is the series-number of
— 91). Then 2B, is analjtic but irresoluble in I. The non-con-
tndictonness of Language I (the non-dcmonstrability of some
sentences in 1) cannot be proetd by the means at the ^sposaJ of I.

The fact that the fion-contradictoriness of the language cannot
be prored in a sj-ntax which limits itself to the resources of that
language is not due to any particular wtaincsses in Languages I
and II. This propert)*, as GSdel has shown, is

an attribute of a large class of languages, to whidi belong all the
systems knoten hitherto (and possibly all sjstems whatever) which
contain within themselves the arithmetic of the natural numbers.
(On this pint eorapire also Herbrand [.VenvofslrAf.) pp. 5 f.)


D. FURTHER DE\’ELOrMENT OF L.ANGUAGE II
I 37. PRED1C.KTES AS Cl.\SS-SyMBOLS

Frege and Bussell both introduce f/aM-e-tprrfnoer in sudi a
way that, from e>-ery expression which designates a property (for
instance, from a pr* or from a so-alled one-termed sentential
function — that is to say, a sentence ha^•ing exactly one free ran*
able) a class-expression is constructed which designates the class
of those objects possessing the propitj* in question. In Language
II tve do not intend to introduce any special class-expressions; in
their pl-ice we use the predicates themselves. In what follows we
shall indicate how a shorter xnethcK] of tvriting can be introduced
in which arguments and operators can, under certain carcum*
stances, be left out. The result of this is a symbolism that is per-
fectly analogous to Bussell’s symbolism of dassrs. A sentence in
this sj-mbolism can be paraphrased in the tvord-language in terms
either of “properties” or of “dasses”, as one wishes,

A propertj’ (or dass) is called Rttl/ (fivr] when it does not apply



§37* PREDICATES AS CLASS-STMBOLS 135

to (or contain) any object whatsoever , and tatwersal when it apphes
to (or contains) every object. Thus our definitions are as follows

Def.37-** Leerto>(F)s~(3*)(F(*))

Def;37^ Un^j(fOs(*)(F(*))

Analogous de^tions can be framed for other types, the type
of the argument (here *(0)’ for *F’) may be attached m the form
of a suffix, for instance

Def.37.3- Leerpt (D(F) = ~Q*)( 3 y)(F(*,y))

Now with the help of the symbols of negation and of junction
we will form some combmed
Orf’-s?* (~f)(») = ~F(»)

Drf.37-5. fFVG)Ws(F(«)VC(,))

Drf.37«. (F.G)(»)e(F(«).GW)

Corresponding defiruuons may be framed for any other types,
including many-termed pr Analogous can be constructed
with the help of the other junction symbols, they are, however,
seldom apphed m practice

We define the pt ‘ and *V’ for the null properl y and for the
umvmol property as follows

Def.37.7* A](x) = — (x=*)

DeC.37'8’ Va(x)s(x*x)

Corresponding defimuons can be framed for all the remaimng
pt'types m which the desigaation of the type of the appertaining
IS attached as a suffix

Theorem 379. *(F=G)={*)(F(x)sC(x))’ is demonstrable
(with the help of PSII as and 11) — Analogously we now define
as follows

Dct37io. (FcC)=(x)(F(x)oG(*))

Corresponding definitions may be framed for any two pr of the
same type, and therefore, speofically, also for many-termed pr

[According to the previously stated syntax of Language II,
instead of ‘ F V G ’ we should wnte ‘ V (F, G) ' or * sm (F, G) where
‘sm* (as m the example, p 86) is an fu of the type ((0),(0) (0))
or, m general, of the type ((/),(<) (*)) for any type t whatsoever
And instead of *Fc G we should wnte * c (F, G)’ or ‘ Sub (F, G)’,
where ‘Sub’ is a pr of the tj-pe ((0),(0)) (compare p 86), or,
mwt , "ti tilt VjT* urAt

*FvG’ and *FcG’ m order not to deviate too far from the usual
Russellian symbolism ] According to Theorem 9 and Def 10,



136 TART III. Tin- ISDEFIKITE L.VCCU\CE n

for a sentence of the form (Di)(i>i)-«(p,)Cpri(Pi....P«)-
prj{Pi, — P«))t\c can al\vaj3 write prispr*; and for a sentence of
the form (p,).. (pj(rri(P„-.P«)opr,(P„...P,)) ^

\mte pr,C pFf For this modeof sj-mboliration srithout arguments,
two different translations into word-language are possible. For
instance, let ‘P’ and 'Q' be pt*; then we can translate ‘PcQ’ as:
“The property’ P implies the propertj’ Q", or, if we wish, as:
“the class P is a sub-dass of the dass Q*'; correspondingly “sub-
rdation", when it is a question of many-termed pr. Further, we
can interpret the 'PvQ' when it is used without arguments as
the “ pjm of the dasses P and Q and * P • Q ’ as the "f^roduct of
the dasses P and Q"; analogously also the “sum” and “product
of relations” in llie case of many-termed pr. ‘A’ and ‘V’ used
unlhout arguments can be interpreted as "nuU clast" and "vis-
rrrij/f/a«” (or as “null relation” and “universal relation", rr-
speetiN cly) As an example of an application of the dass sj-mbolism,
the .\xiom of Selection PS 11 21 may be used (the p which occur
are to be taken from suitable tj^pcs of at least the second order):

[(.Vc~L«r).(F)(Oa.U<F).M(C).~L«r(f.C)]3

Hereby ' Al ’ (“cardinal number t ") « to be defined as follows
(compare § 38^)’

Al(f)EQx)0)(Fb)=b' = «))

The mode of sjTnboliration whose introduction is indicated in
the foregoing is completely analogous to Russell’s sjTnbolism of
dasses; the whole theory’ of dasses and relations of the [iVtmr.
MaOt.\ can easily be put into this simplified form. But we shall
not go into this here, as it raises no further fundamental problems.

§ 38. Tiie Elimination of Classes

The liistoncal development of the use of dass symbols in
modem logic contains several noteworthy phases, the examination
of whidi IS fruitful for the study even of present-day problems. We
select for our consideration the two most important steps in this
development, which are due to Frege and Russell. Frege {GrunJ-
gfsetzf] was the first to give an exact form to the traditional dif-
ferentiation between the content and the evfojt of a concept. Ac-
cording to his view, the content of a concept is represented by the
sentential function (that is to say, by an open sentence in w hich the



137


§ 38 ELIMINATION OF CLASSES

free variables serve to express mdeternunateness and not univer-
sality) TTie extent (for instance, m tbe case of a property concept,

1 e of a one-termed sentential function, the corresponding class) is
represented either by a special expression containing the sentential
function, or else by a new symbol which is introduced as an ab-
breviation for this expression An identity-sentence with class
expressions here means the coextensiveness of the corresponding
properties (if, for instance, ‘kj* and ‘k*’ are the class symbols
belongmg to the pt ‘Pj’ and *P**, then ‘ki=k|’ is equivalent in
meaning to ‘ (*) [Pi (*) = Pi (x)] *) Later on, Russell proceeded m the
same manner FoIIowmg the traditional modes of thought, how-
ever, Frege made a mistake at a certain point, and this mistake was
discovered by Russell and subsequently corrected

It was a decisive moment m the history of logic when, m the year
1902, a letter from Russell drew Frege’s attention to the fact that
there was a contradiction m his system After years of laborious
eSort, Frege had established the sciences of logic and anthmetic on
an entirely new basis But he remained unknown and tmacbiow-
ledged The leading math ema ti cians of his Ome, whose mathematical
foundations be attacked with unsparing criticism, ignored him His
books were not even reviewed Only by means of the greatest per-
sonal sacnfiees did he manage to get the fust volume of his chief
work [Grundgetetze] pubLshe^ m the year 1 893 The second volume
followed after a long mterval m 1903 At last there came an echo—
not from the German mathemanaans muchless the German philos^
phers, but from abroad Russell in England attributed the greatest
importance to Frege’s work. In the case of certain problems Russell
himself, many years after’ Frege, but sbll m ignorance of him, had
hit upon the same or like solunons, m the case of some others, he
was able to use Frege's results m his own system But now, when
the second volume of his work was almost printed, Frege learned
from Russell’s letter that his concept of class led to a contradiction
Behind the dry statement of this fact which Frege gives in the


with the problems of the extension of concepts, of classes, and of
aggregates— amongst them both Dedekmd and Cantor
The contradiction which was discovered by Russell is the anti-
nomy which has smee become famous, namely that of the class of
those classes which are nox members of themselves In his Ap-
pendix, Frege examined vanous possibilities for a way out of the
difficulty, but without discovering a suitable one Then Russell, m
an Appendix to his work [PnnapUi\ which appeared in the same
year (1903), suggested a solution m the form of the theory of types,
according to which only an mdividual can be an element of a of

the first level, and only a class of the «th level can be an element of



PART in. Tire INDOTNITE LANGUAGE II


138

a cits J of the n + ith level. Accordii^ to this theory, a sentence of the
form'A«A'or*— (fc«*)’»ne5thertr«enorfalsc;itismcrelyineaning.
less. Lsteron Russell showed that this antinomy can also besoformu-
lated as to apply not only to dassea but to properties as well (the
antinomy of ‘ imprtdicable *, see § 600). Here, also, the contradiction
is eliminated by means of the rule of types; applied to pt* (as sym*


But on tsv’o points — like traditional logic and Cantor's Theory of
Aggregates — he made errors, which were corrected by means of
Russell's rule of types. It is because of these errors that, in spite of
the perfectly correct classification of functions, the antinomies
arise. Frege’a first error consisted m the fact that in his system all
expressions (or more exactly, all expressions which begin with the
assertion symbol) are either true or false. He n-sa thus obliged to
count as false, expressions m which an unsuitable argument was
attnbuted to some predicate. It was Russell who first introduced
the triple classification into true, false, and meaningless expressions
— a clasatfication which wss to prove so important for the further
development of logic and its appbcation to empirical sdenee and
philosophy. According to Russell, those cipmsions which have
unsuitable arguments are neither true nor false; they are meaning*
less (in our terminology: they arc not sentences st all). When this
first error of Frege is corrected, then the antinomy of the term
'impredicable' can no longer be set up in his system — for the de-
finition would have to contain the contra-syntactical expression
‘F(F)'. 'The antinomy which relates to classes, however, can still be
constructed in his system. For Frege made s second mistake in not
applying the type-classification of the predicates (sentential func-
tiora), which he had constructed with such insight anddirity, to the
dasses corresponding to the predicates; instead of that, he counted
the classes— and similarly the many-termed extensions— simply as
individuals (objects) quite independently of the level and kind of
the sentential function which defined the class in question. And
even after the discovery of the contradiction, he still thought thst he
need not alter his procedure (Vol. It, pp. 254 f.), because he beliewd
the names of objects and the names of functions to be differentiated
by the fact that the former have a meaning of their own while the
latter remain incomplete symbols which only become significant
after being completed by means of other symbols. Now, since Frege
held the numerals ‘O', ‘ 1 *2*, etc., to be significant in themselves,
and since, on the other hand, he defined these symbols as class
symbols of the second level, he was compelled to regard class



*39


§ 38 EUMEtATlON OF CXASSES

symbols, 83 opposed to predicates, as mdiTidual names Today we
have the tendency to regani aU the partial expressions of a sentence
which are not sentences in their turn as dependent , and to attribute
independent meaning at most to sentences

In order to define a car dinal number in Frege’s sense without
makmg me of classes, we have only to replace Frege s class of pro*
pemes by a property of properties (designated by a *pt) It is re-
maiiable that Frege at an earlier stage expressed this view himself
({Qnadlagen} 1884, p 80, Note) “ I think that [m the defimtion of
•cardinal number'!, instead of ‘extent of the concept’ we might
say simply ‘concept’ But then two kinds of objections would be
raised I am of the opinion that both these objecnons could be
removed, but that might lead too far at this stage ’ Later he
apparently abandoned this view altogether Then agam — as it
appears when one looks back — ^Russell seemed to be very close to
the decisive point of abandoning tdasses altogether While for Frege
It was important to mtroduce the class symbols as well as the pre-
dicates — since m his system they obey different rules — the whole
question had a different aspect for Russell In order to avoid
Frege's error, Russell did not adopt the class symbols as m-
dividual symbols but instead he divided them mto types which
correspond exactly to the types of the predicates But by this means
a quRe unnecessary duplication was introduced Russell himself
recoguaed that itwas of DO importance for logic whether “classes ’

thato to say, anything wlucb is designated by the class symbols
-- reaQy «Enst * or not ( ‘no-class theory ^ 'The further develop-
ment proceeded ever mote definitely in the direction of the stand-
point that class symbols are superfluous In connection with
Wittgenstein’s statements, Russell himself later discussed the view
that classes and properties are the same, but he did not as yet ac-
knowledge It (192s [fttnc Math.} 2nd edition of Vol 1) The
whole question is connected with the problem of the Thesis of
Eitenaionahty (see § 67) Behmann [togiA] mtroduces the cl^
symbolism merely as an abbreviated method of writing m which the
predicates are given without arguments he insists, however, on
differenUatmg between extensional and intensional sentences hold-
ing that this method of writing is only admissible for the former
^on Neumann [B« etst?teone} and Godel {Unentickeuibare} do not
even symbolically make any difference between predicates and the
correspondmg class symbols m the place of the latter, they simply
use the former The critique of Kaufmann {l^Uneitdltcke] [Bemer-
concerning Russell s concept of class is also worthy of note
But this tntiasm is reaLy directed less against the Russellian system
Itself than agamst the philosophical discussions by Russell and others
of the concept of class, which do not pn^ierly belong to the system

We win summarize bnefiy the development which we have
JtJst been considering Frege mttoduced the cTbss expressions m



1^0 P.«T III, Tirz INDEFINnT L-CCCO^CE II

order to h3\-e, besides the predicate*, something which could be
treated lihe an object-name. Russell recognired the madmlssi-
bilitj- of such a treatment, but, nevertheless, retained the class
eipressions. The former reason for their introduction having been
remo\'ed, howevtr, they are now superfluous and therefore have
been finally discarded.


§ 38/?. On Existence A^umptions in Logic

If logic is to be independent of empirical knowledge, then it
must assume nothing concerning the raarfenre cf oi-jects. For this
reason Wngenstcln rejected the Axiom of Infinity, which asserts
the erdstenee of an infinite number of objects. .\nd, for kindred
reasons, Russell himself did not include this axiom amongst the
primitive sentences of his logic. But in Russell’s sj’stem [fVmr.
Math.] as well as in that of Hilbert acnteaces such as

'(3*)(f(*)V-.f(x))' and '(3»)(r=*)’, and ethers Uketheto,
in which the existence of at least one object is stated, are (logi-
cally) demoostrable. Later on. Russell himself criticised this point
([A/ctA. i’Ati], Chap, xvitt. Footnote). In the abot-e-mentioned
sj”Btcms, not only the sentences which are true in e%’ery domain,
independently of the number of objects m that domun, but also
sentences (for example, the one just given) which are true, not in
every domain, but in every tton-rtrpty domain, arc demonstrable.
In practice, this efistinction is immaterial, since we are usually
concerned with non-empty domains. But if, in order to separate
logic as sharply as possible from empirical science, we intend to
exclude from the logical system any assumptions concerning the
existence of objects, we must make certain alterations in the forms
of language used by Ru««a and Hilbert.

We may proceed somewhat as follows; No free variables art td-
itutTed in sentences and therefore universality can only be expressed
by means of universal operators. The schemata of primitive sen-
tences PSII i5 and 19 are tvtained (see § 30); PSII 16 and 17 are
replaced by rules of substitution: (pjJ can be transfoirned into
(Pi)C®i) Wto disappears; but

certain other rules must be laid down instead. In the language thus
altered, when an object-name such as *a' is piven, 'P(b)* can be
lerived from ’{aKPCx)}’; and apain. ‘axlCPfr))' from 'Pfa)'-



§ 38(2 EXISTENCE ASSITUFTIONS IN LOGIC 14I

The important point is that the existential sentence can only be
derived from the tmiversal one ^hen a proper name is available , that


use a proper name

In our object languages I and 1I» the matter is quite dif-
ferent owmg to the fact that th^ are not name-lcrnguages but
eodriinate languages The expressions of the type 0 here designate
not objects but positions ITie Axiom of Infimty (see § 33 , 5 a) and
sentences like ‘( 3 *) (*=x)’ are demonstrable m Language 11, as
are similar sentences in Language I But the doubts previously
mentioned are not relevant here For here, those sentences only
mean, respectively, that for every position there is an immediately
succeeding one, and that at least one position exists But whether
or not there are objects to be found at these positions is not
stated. That such is or is not the case is expressed m a co-ordinate
language, on the one hand, by the fact that the fue at the positions
concemed hare a value which appertains to the normal domain,
or, oa the other, by the fact that they have niercly a trivially
degenerate value But this is awed not by analytic but by syn-
thetic sentences

Example In the system of the pkysteal laiiguage, the sentence
which states that quadruples of real numbers (as quadruples of co-
ordinates) exist is analytic In its mateml mteipretabon it means
that Spatio-temporal positions exist Whether somethmg (matter or
an electro-magnetic field) is to be found at a particular position is
expressed by the fact that at the position in question the value of the
density — or of the field vector, respectively — is not aero But
whether anything at all exists — that is to say, whether there is such
a non tnvially occupied posiUoo — can only be expressed by means
of a synthetic sentence

If it IS a question not of the existence of objects but of the
existence of properties or classes (expressed by means of predicates),
then it u quite another matter Sentences like ‘( 3 F)(F=F)’
("There exists a property (ot class)”) and ‘(3F)(Leer(F))’

domam, mcluding the null domain, they are aLo analytic and
logically demonstrable in the aforesaid system without existence
assumptions.



142 PART HI. THE INOOTNTre LANCDACE H

There are, howe\er, also sentences about the cdstence of pro-
perties the legitimacy of which b disputed; the most important
examples being "the Axiom of ReducibiUty and the Axiom of
Selection. We need not here go into the question of the Axim of
Redudbility. In Russell's form of language, it was a ne ces sa r y
axiom on account of his branched classification of types (see
p. 86 ); but in Language II it is superfluous. [On the Axiort of
Comprehension, nhich is closely related to it, sec §33.5^*]
called .^Ixion of Selsetion (PSII 2 i) maintains the existence of a
selective class even in those cases where no such class can be
defined; and it is therefore a so-called pure (non-construedve)
existence statement As such it b rqected by Intuitionism. In
Language II we have stated h as a primitive sentence, and we
regard the question of its assumption as purely one of expedience
(sec pp. 97 f.). That is true not only within the bounds of the
formalistic xnew of language as a calculus but also from the stand-
point of roaterial interpreution. For, m such an iniwprrtatioa,
only the atomic S» are given a meaning directly; the remaining
then acquire one mdirectly. The Si (and irith them all sen-
tences of mathematics) are, from the point of view of znaterul
interpretation, expedients for the purpose of operating with the
Sfc. Thus, in lijnng down an 6 | as a primitive sentence, only use-
fulness for this purpose is to be taken into consideratioa.


§386. Cardinal Numbers

In the material interpretation of Languages I and II, the 3
to be interpreted for the most part as designations of positions or
of values of an fuj. Concerning the possibility of formulating
statements of cardinal numbers ("There are so and so niany ...”)
we have so far said nothing. We will now proceed to show several
possibilities of doing so, whidi lie partly within and partly without
the syntactical framework set up for Language 11.

The first method consists in defining erery f^rHinal nmnber
(Anzahl) as a *pr. For example, 'A5(P)' (where ‘A5’ counts as
one sjTnbol) means: "The property P has the cardiibl number 5 ,
that is to say there are exactly 5 numbers (positions) which have
this property." Taking as an aunliiry terra ‘Am5{P)’ {MtndesU



§385 CASDIKAX. NUMBERS I43

Aluahl , ‘ AmS ’ is one symbol) which means “ There are at least
5 numbers which have the property P”, we dehne as follow^
Aml(F)a(3x)(FW)

Ana(F)=Q.)(a5.)(~.'(«=3.).P(»).F{y))

Am3(F) = (3.)(35r)Q.)(~(«=,).~(.=,).~0'=»).F(«)
.F(y).FM)

and so on On the basis of these minimum numbers, the exact
numbers are defined

AO(P)= ^Aml(F)

A1(F)= (Aml(F).~Am2(F))

A2(P)h (Am2(J0.~Am3(F))

and so on.

These defimtions of the carduial numbers correspond to those
of Frege and Russell, only here the second level classes are re*
placed, for the reasons discussed m § 38 , by second level predi-
cates These *pr are here not written, as in Russell, simply as ‘O’,
‘ 1 *, and so on, because we already use these symbob m our
languages as symbols of the tyj^ 0 , and therefore may not use them
also as symbob of the type (( 0 ))

The seeond Truthod employs specul aumber-operators which
were not provided for m the previously stated syntax. Here, for
example, ‘(0>“3*) (P{*))’ means “There are exactly 3 numbers
(or positions) having the propiettyP” [*«' m '(«3*)’ is not an
operator-varuble and is not bound ] In this case we can either,
on the lines of the first method, define every mdividual number-
operator, or, more simply, construct two pnmtive tmtenea to
represent a general regressive definition

(0 (03«)(F(»))3~a«)(FM)

( 2 ) («'3«)(FW)3Q«)(3 G)[FW-W (GWs [Ftv).

~Cy=«)]).(»3*)(GW)]

TTie third method expresses “There are 3 “by means of
‘Anz (3 P)* As in the second method, analogous prumtive sen-
tences can be constructed for the pr ‘Anz’

Tht fourth method is perhaps the most useful It is like the first,
but m the place of a *pt it uses a *fu, and writes *anz(P)= 3 ’
As m the second method, two prumtive sentences which take the
pbce of a regressive definition can be constructed for the functor



PART in. TlIEINDEFINrre LANCUACH II


144

‘anr’ of the type (( 0 ) : 0 ). But instead of the primitive sentences,
an explicit definition can also be constructed (according to the
method stated on pp. 88 f.):

a..z(F)=(K<.)a/)CG)[([/(C)= 0 ]E~( 3 ,)[G(.)]).

~(y = »)l).C/m = «)]).(t>=/(F))]

In a precisely analogous vray an ".'fu ‘anz’ of the type ((/i): 0 )
can be defined for the "pt‘ of the type (tj) and n> i.

A definite cardinal-number term referring to a limited domain
can similarly be introduced in accordance with the four methods
just given. The sentence: “There are 3 places up to the place 8
which have the property P” may be expressed, for example, as
follows: ‘A3(8.P)*.-a. ‘( 33 ^) 8 (PW)‘-- 3 - ’An2(3,8.P)\

— 4. ‘an2(8,P)s3’.

All the cardinal-nuiTibef terms which have been mentioned can
be applied to logical as well as to descriptive properties (for
example, to the number of the prime numbers less than 100, as
well as to the number of red positions).

§38c. Descriptions

By a detcriptian we understand an expression which (in material
interpretation) does not designate an object (in the widest sense)
by a name, but characterizes it univocally in a different way,
namely, by means of the statement of a property which belongs
only to that object.

ExampUt. Description of a number: “The smallest prime num-
ber which is greater than 20“; of a thing: “ The son of A” ; of a pro-
perty : “The logical sum of the properties P and Q”. In the word-
language a description is effected by the use of the definite article in
the singular number (“the so-and-so”).

Profiting by the attempts of Frege and Peano, Russell has pro-
duced a detailed theory of descriptions: [iViRC. VoL I,

pp. 66 ff. and 173 ff.; and [Math, Phil.}.

. Following Russell’s method one could (in an extension of the
syntax of Language II) symbolize a description with the help of
a special deseriptional operator ‘1**. “That number (or position)
which has the property P“ would then be written as follows:
'(tx)^(*))’. We call a description of this kind an empty or a


MS


§ 38 c. descriptions
umvocal or an amhtguous descnpbon, respectively, i; there is no
number, or exactly one number, or several numbers havmg the
property A numerical description is used like a 3, for example
as an argument. ‘Q[(»*) (P(*)) 3 ’ means “The number having
the property P has also the property Q “ This sentence is to be
taken as true when, and only when, the description is umvocal and
the descnbed number has the property Q It is obviously neces-
sary to make clearly recognizable the partial sentence (narrower or
ivider) which is to express the property to be ascribed to the de-
scnbed object This can be done (as by Russell) by means of an
auxiliary operator the whole descnption (consisting of desenp-
tional operator and bracketed operand) is put m square brackets
in front of tbe partial sentence in question In accordance with
the matenal interpretation previously given, we can now construct
the followmg schema of primitive sentences which appbes to
dcscnptions of any type w^tsoever ( 3 , ^pr, or gu)

K'uO IWi K'Pi) (pt,(ti,))l]= [(i3 D,) (pt, W).

(P.)(Pti(Pi)=PPs(PO)l

The necessity for the use of tbe auziltaiy operator may be seen
by 8 companson between the following two seateaoes [analogy
the necessity of the umversal operator m order to be able to dif-
ferenuate between ( 3 i)(-^( 3 i)and '^(3i)(®j)]

[(,*)(P(»)) 1 [~Q[(.»)(PM)J] (.)

~[('*)(PW)][QK'»)(PW)]] W

(x) means “ There is exactly one P-number, and every P-number
(and therefore this one) is not a Q number”, (2), on the other
hand, means “It is not true that there is exactly one P-number
and that every P-number is a Q-number” If the description is not
umvocal (that is to say, if there are either no P numbers at all or
several P-numbers) then (i) b false but (2) is true To simplify
the symbolism it is possible (as Russell does) to rule that the clumsy
auxihary operator may be left out when its operand is the smallest
partial sentence m which the description m question occurs In
this case, for mstance m (2), we speak of a “primary occurrence”
0/ the descnption, o&ersnse, for mstance m {1), of a “secondary
occurrence” Accordmg to this rule, (2), but not (i), may be
written bnefly thus ‘ ~ [Q [(» *) (p (»))] ] ’

Descnptions are expressions of a special kmd which cannot m



146 PART in. THE IVDETINITE LANGUAGE II

all be treated in exactly the same manner as the other ex-
pressions ( 3 , ipr, or gu) of the type concerned, ^\'hile, for in-
stance, according to PSII 1 6 . (j,) (pr* (3i))3 pr, (3i) ‘S true for every
ordinary 3 n not always true when a numerical description is
used for 3 i- For example, the sentence (3i)(pri(3i))3PrsK’3i)
(Pfi ( 3 i))] falsified on account of the fart that ^e descrip-

tion IS not univocal. The sentence which here holds in its place is:
(30 (pri(30)3 [(13 3O (pri(3i))3Pr.t('3i) (PPi (SOM : stntcnce
is demonstrable with the help of the schema of primitive sen-
tences already given.

If we wish to use definite descriptions, we must write the
descriptional operator with a limit; *{»*)5 (P(*))’ then means:
“That number up to 5 which has the property P.”

The K-operator is a descriptional operator of a very special
kind; and the clumsy auxiliary operator is not necessary for its
use. The K-descriptions, since they are aUvaj-s uni>cal, can be
treated like ordinary 3- This univocality is, however, only achieved
by laying down the convention that when no number exists which
has the property m question, the v'alue of the description is zero.
Herein lies the disadvantage of the K-operator; however, it might
prove expedient in many cases. The K-opemior itself is only ap-
plicable to numbers; nevertheless, wHh iu help very often pr and
fu of higher levels can also be defined. Let and ‘g' be and
let ‘Q’ be a *pr* of the type ((0:0), (0:0)) (so that *Q(/,f)’ is a
sentence). Suppose that we vnsh to define the functor ‘ k ' so that
‘k(g)’ b equivalent m meaning to “that functor / for wWch
Qiftg) is true". The definition can make use cither of an
ordinary descriptional operator (with an operator variable f):

or else of a K-operator (with an operator-variable 3 ):

tCs)(«) = (Kj)Q/)[Q(/'.s).(:,=/(,))].

If the first definition is setup, then the defined symbol ‘k* cannot
be used everywhere like an ordinary fu of the type in question ;
this disadvantage does not occur in the case of the second
definition.



§ 39 Real Nxjmbers

The real numbers, together with their properties, relations, and
functions, can be represented within the framework of the given
syntax of Language II If a particular (absolute) real number
consists of the integral part a and the real number b {<!) this
number can be represented by means of a functor ‘k’ which is
defined so that k(0) = a, and, for b> 0, k{«)=0 or 1 respectively,
accordmg to whether at the nth place in the development of the
dual fraction of b, ‘0’ or ‘1’ occurs In order that the develop-
ment of the dual fraction may be umvocal, we exclude those dual
fractions m which, from some pomt onwards, only ‘0’ occurs
The real numbers with sign (positive or negative) can be repre
ented m a like manner

The method of representatioo of real numbers indicated here was
stated by Hilbert [Gruruilagen 1923) (see also von Neumann
[Beueuth ]) Hilbert has planned a construction of the theory of
real numbers on this basis, but up to now he has not produced it

A real number is thus represented by means of a of the
type (0 0), tve shall designate this type briefly by 'r' Then »
property (or aggregate) of real numbers (for example, “algebraic”
or “transcendental" numbers) is expressed by means of a of
the type (r), a relation between two real numbers (for example
“is greater than" or “is a square root of ’) by means of a *pT* of
the type (r, r) , a function of a real number (such as “ square root”
or “sine”) by means of a *fu*of thetypc(r r), a function of two
real numbers (for instance “product"or“power”) by means of a
*fu* of the type (r, r r), and so on Theanthmcticalegaairyoftwo
real numbers fUi and fu* is expressed by fUi =fUa, for this sentence
(accordmg to PSII 23 and ii) is true when and only when the
values of the two functors agree for every argument, and therefore
when and only when the two dual fractions comade at all places
As opposed to the equahty of two natural numbers (represented
by Si), the equahty of two real numbers, even when they are
stated in the simplest possible form, is, m general, mdefinite
— smee It refers back to an unlimited universality A complex
smnibeT J5 AO Ard^.red pstr d rea^ and ihas aa

of the type r,r, a function of one or two complex numbers is a
*fu of the type (r, r r,r) or (r,r,r,r r,r) respectively



PART lU. Tire INDEFINITE LANGUAGE II


148

In this way all the usual concepts of classical mathematics
{Analysis, Tfitory of Funttions) can be represented, and all the
sentences which have been constructed in this domain can be
formulated. The usual axioms of the arithmetic of real numbers
need not be set up here in the form of new primitive sentences.
These axioms— and hence the theorems derivable from them— arc
demonsirabk in Language II.

It ill now he shown very briefly hotv the most important logical
kinds which arc distinguishable with respect to sequences of
natural numbers, and therefore also vsith respect to real numbers,
can be represented by means of syntactical concepts. First we must
distinguish betw een a sequence given by means of a mathematical
laiv and one given by a reference to etpenence. In the representa-
tion by means of *{11', this difference is expressed by the difference
between fU( and fiij Thus the term “sequence of free selections”
{frete Walilfol^e) of Drouwer and Weyl is represented by the
syntactical term ‘fus’- The regular sequences can be divided into
those that are calatlabU (see Examples 1 a and b) and those that
are tnealculable (Example 2). Syntactically this difference is
characteniable as the difference l^tween definite and indefinite
fui; for the former, by means of a fixed method, the value can be
calculated for any position; for the latter, in general, this is not
possible. In the case of sequences determined by reference to ex-
perience, we can diffcreniiate further into: 1. Analytically regular
sequences; in the case of these, the reference to experience is not
essential, since it is equivalent in meaning to a certain mathe-
matical law (Example 3) — 2. Empirically regular sequences;
although the determination of these cannot be transformed into a
law, yet they have the same empirical distribution of values as an
analytically regular sequence — whether by chance (Example 411)
or in conformity with a natural law (Example 4i). — 3. Irregular
or unordered sequences; for these there is no mathematical law
which, even in a merely empirical way, they could possibly obey.

For an fUj fUj, these three kinds are to be characterized syn-
tactically in the following manner: i. There is an fUi fuj such that
fUg is synonymous with fit,, and therefore such that fuj = fUj is an
analytic sentence.— 2. There is an fut fUj such that fu, = fUj is a
synthetic but at the same time scientifically acknowledged sen-
tence (that is to Say, in Language II it is a consequence of scientifi-



149


§ 39 NUMBERS

cally acknowledged premisses, m a P language it is P-valid (com-
pare p 184)) — 3 Condition 2 is not fulfilled [For all three
concepts a further classification maybe made according to whether
the mathematical law m question is calculable or not, that is to say,
whetherthefuiconcemedisdefimteornot ] It is to be noted that,
m the definition of the concept of the unordered sequences, the
kind of laws which are to be excluded must be stated, or, more
exactly, m syntactical termmology, the rules of formation for the
defimtions of the fUi which are to be excluded must be stated, for
example by means of reference to a certain language [E g , let a
sequence fu^ be called unordered in relation to Language II if
there is no fui fu, definable m II such that fUi=fUt is vahd m a
non-eontradictory langu^ which contains II (Example 5) ] The
same holds good for the term “irregular collective” m von Mises’s
Theory of Probability

Examples s CaieuLibU regular sequences (0) The recurring dual
fraction with the period ‘oti’, (ft) the dual fraction for w —
2 InealeuIaiU regular sequence 'k|’ let ki(n) be equal to i if a
Fermat equaaon with the exponent n exists, and otherwise let
ki(«) be equal to 0—3 Afutlytieally regular sequence letka(n)
be equal to m if the nth cast of a certam dice shows an m , our de


shows by chance alternately either a 3 or a 4, (Of course this can
never be completely escabhshed, but it is conceivable as an assump-
tion ) (ft) Let k« (r) be equal to 1 when a certam compass needle,
used as a roulette pointer, m the position of rest after the nth play
pomts to the South, and equal to z when it pomts to the North.
According to natural laws, k«=k( is vahd — 5 Sequence ‘k,’,
imordered m relation to Language 11 let ki(n) be equal to 1 when
n 13 a senes number of an analytic sentence of II, and otherwise
equal to 0 Since 'analytic in IF is not definable m II (see p 219)
there is no fui m II which has the same distribution of values as ky


§ 40 The Language of Physics

Since, m Language II, not only logical but also descriptive
symbols (pt and fu) of the various types may occur, there is a pos-
sibihty of representmg physual concepts A ph}’Sical magnitude
(of a state or condition) is an fuj, the argument-expression con-



150 PART III. TIIK INDEFINTTB LANGUAGE II

tains four real numerical tipresaons, namely, the time-space
co-ordinates; the ^alue-esp^ession contains one or more real
numerical expressions (for instance, in the case of a scalar, one;
in the case of an ordinary vector, three). A set of four co-ordinstes
is an expression of the type r,r,r,r; we will designate this type m
a shorter way by ‘q’. [Sxiwtpfet.* t. “At the point k|,k{,k5, at the
time k*. the temperature is k,” may be expressed e.g. as follows:
'tcmp0c„k„k„k4)sks', where 'temp' is a of the type (q:r).
2. " At the space-time point k„k^fca.kj there is an electrical fidd
with the components kj,k«,k,” may be expressed, say, by
‘el(k,,k4,k„k4) = (kj,k,.k,)*, where ‘cl’ is a »fu* of the type
(q:r.r.r).]

An empirical statement does not usually refer to one individual
space-time point, but to a fimtt spoee-timt domain. A domain of
this kind is given by means of a *pr* of the type (q) — namely, by
means of a mathematical (ptj) or a physical (ptb) property which
belongs to all the space-time points of the domain in question and
only to those. A magnitude which is referred, not to individual
8p3ce*Ume points but to finite domains (for instance: tempera-
ture, density, density of charge, energy), can thus be represented
by means of a *f«jt whose argument is a pr of the kind stated; in
the case of a scalar, the type is ((q):r); in the case where there »re
several components, it is ((q):r, — r). \prppnly of a domoiB i*
represented by means of a ^pr^* of the type ((q)); the argument is
again the pr which determines the domain. TTie majority of the
concepts of everyday life, as well as those of science, are such
properties or relations of domains. [Eramyifej; i. Kinds of things,
such as “horse"; “In such and such a place is a horse" means
“Such and such a space-time domain has such and such a pro-
perty." — 2. Kinds of subnaticcs, such as “iron". — 3. Directly
perceptible qualities, such as “warm", "soft”, "sweet”,— '4
Terms expressing disposittems, such as “breakable". — 5. Con-
ditions and processes of all kinds, such as “storm", “tjphus".]
It follows from all these suggestions that oU the smfenees of
physics can be formulate J in a lan^age of the form of II. To this
end it is necessary that suitable fUb and ptb of the tjyves given
should be introduced as primitive terms, and that, with their help,
the further terms should be defined. (Concerning that form of
the phj'sical language in whkh ^mhetic physical sentences also —



§ 4® THE LANGOACE OF PHYSICS 1 5 1

for example, the most general laws of nature — are laid down as
pnrmtive sentences, see § 82 )

According to the thesis of Pkynealim, which will be stated later
(p 320) but which will not be established in this book, all terms of
science, mcludmg those of psychology and the social saences, can
be reduced to terms of the physical language In the last analysis
they also express properties (or relations) of space-time domains
[Exampla “A is fonous” or “A is thinking” means ‘‘The
body A (1 e such and such a space tune domain) is m such and
such a state *’ , ” The society of such and such a people is an economy
based on a monetary system” means “In such and such a space-time
domain, such and such processes occur For anyone who takes
the point of view of Physicalism, it follows that our Language II
forms a complete syntacucal framework for science

It would be a worth-while task to mvestigate the syntax of the
language of physics and of the whole of saence in greater and more
exact detail, and to exhibit the most important of its conceptual
forms, but we caxmot here undertake such a thing



PART IV

GENERAL SYNTAX

A OBJECT-LANGUAGE AND SYNTAX
LANGUAGE

We have now constructed the syntax of Langtiages I and II and
have thereby given two examples of tpeacd syntax In Part iv we
shall undertake an mvestigation of general syntax — that is to say,
of that syntax which relates not to any particular mdividual Ian
guage but either to all languages m general or to all languages of a
certam kind Before we go on, m Dtvrsion B, to outline a general
syntax applicable to any language whatsoever, we shall first set
down, m Division A, some prelimmaiy reflections concenang the
nature of syntactical designations and of certain terms which occur
u syntax

§41. On Syntactical Designations

A dengnatson of an object can be either a proper came or a
descnpaon of that object The evident necessity of keepmg in
mind the distinctioa between a designation and the object desig-
nated thereby (for instance, b e twe en the word ‘Pans* and the aty
of Pans), although frequently emphasized m logic, is not always
observed in practice If the object designated is such a thing as a
town, and the designation itself a word (cither spoken or written),
the distinction is obvious And for precisely that reason, m such
mses failure to differentiate between the two does not lead to any
harmful consequences

If mstead of "‘Pans’ is bi syllabic” we wnte "Pans is bi-
syllabic”, the method of wntmg is mcorrcct, because we are usmg
the word ' Pans ’ m two different senses , in other sentences as the
designation of the cit>, and m the sentence m question as the
designation of the word ‘Pans* itself [In the second use, the
word ‘Pans’ is autonymous Sec p 156 ] Nevertheless, m this
instance no confusion will arise, since it is quite clear that the sub-
ject here is the word and not the OQ



part IV. GENERAL SYNTAX


*54

It is another matter when the designated object is itself a
linguistic expression, as is the case with syntactical designations.
Here a failure to pay attention to the distinction leads very easily
to obscurities and errors. In meta-mathcmatical treatises— the
greater part of the word-text of mathematical ^vriti^gs is meta-
mathematics, and therefore syntax— the necessary distinction is
frequently neglected.

If a sentence (in writing) refers to a thing— my writing-table, for
instance — then in this sentence a designation of the thing must
occupy theposition of the subject ; one cannot simply place the thing
itself— namely, the writing-table— upon the paper (this could only
be done in accordance with a special convention; see below). In
the case of a writing-table, and perhaps even of a match, this seems
self-evident to everyone, but it is not so self-evident when we are
dealing with things which are especially adapted to be put on paper,
namely, with written characters. For example, in order to say that
the Arabic figure three is a figure, oneoft en writes something of this
kind: “3 is a figure.” Nowhere, the thing itself which is under
discussion occupies the place of the subject on the paper. The
correct mode ofwriiing would be:" A three is. . . "or"'3’ls..

If a ttntenee is concern/d toitk an txpression, then a designation of
tins expression — namely, a syntactical designation in the syntax-
language — and not the expression itself, occupies the place of the sub^
ject in the sentence. The syntax-language may be cither a word-
language or a symbol-language, or, again, a language composed of
a mixture of words and symbols (for instance, in our text it con-
sists of a mixture of English words and Gothic symbols). The most
important kinds of syntactical designations of expressions are
enumerated below:

A. Designation of un expression as an individual, spatlo-
temporally determined thing. (Occurs only in descriptive syntax.)

1. Name of an expression. [Occurs very seldom. Example:
‘the Sermon on the Mount” (which can also be interpreted as a
description).]

2. Description of an expression, [Example; “Caesar’s remark
on crossing the Rubicon (was heard by so-and-so).”]

3. Designation of an expression by means of a like expression



*55


§ 4* O'^ SYNTACTICAL DESIGNATIONS

m inverted commas \Examples “the saying ‘alea lacta est’”,
"the mscnption 'nutnmentum spintus’ ”]

B Designation of an expresnomd design {^ct ^ 15)

1 Name of an eipressional design (eg of a symbolic design)
Examples “A three”, “omega”, Lord’s Prayer”, “Fer-
mat’s Theorem” (which can also be mterpreted as a description),
‘ mt”, “?l” ]

2 Descnphm of an ezpresstonal design by means of the state-
ment of a spatio temporal position (indirect descnption, so-called
ostension, see p 80) [Examples " Caesar’s remark made at the
Rubicon (consists of three words)”, “ausdr (b, a)” (see p 80) ]

3 Descnption of an ezpressional design by means of syntactical
terms [^amples “The expression which consists of a three, a

plus symbol, and a four", ]

4 Designation of an expresstoaal design by means of an ex-
pression of this design in inverted commas [Examples “‘3’",

“*3-1-4’”, “‘alea ucta est* (consists of three words)” ]

C Designation of a more general form (that is, a form that can
also apply to unequal expressions, see p 16)

1 Name of a form (for instance, of a kind of symbol) [;x-
amples “variable”, “numerical expression”, “equation”, “n”,

2 Desenptum of a form [Examples “An expression consisting
of two numencal expressions with a plus symbol between them ” ,

•; 3 = 3 ”]


3 Description of a form by means of an expression of this form
in inverted commas together with a statement of the modifications
permitted [Example “An expression of the form ‘x=y’, where
any two unequal variables may occur m the places of ‘x’ and
>•"]


It is frequently overlooked that the designation of a form with the
help of an expression m inverted commas leads to obscunties if the
modifications pemutted are either not given at all or are given m-
cxactly For instance, we often find * For sentences of the form
‘ (*) (p V F (x)) ’ so and so holds ”, which leaves open such questions
as the following Is it necessary for the I *p * to occur m the sentence.



1^6 TART rv. CIN131.U. STNTAX

or may any \ ocnir in its placr. or any sentence? Must the y ‘F*
occur, or may any p take its place, or any pr? Or, acain, “ w piaa
of may wc have any sentence with the one tree Tanahle x.

Of cx-en With sevetal free ranaWes? This formulation is actnrflinpiy
obscure and ambieuous (epnte apin from the fact that the
commas are usually left out altoecther, and that tc^* oftra
sentence is emnen instead of “for sentences of the form... >

§ 42. On the Necessitv of Distinguishing
BETW'EEN an Expmssion and its Design.ation

The importance of ArtrKjiriifcmf clfsrh' f>rftrwn mj c^toss3»
anS it: xynt^hrxtj <?«tjjubon will readfly be seen from such ex-
amples as the foUotrint^; if, in the fiiT sentences bdow, instead
of the expressions ‘«u’, “to**, *omepa*, ' ‘ome^a”, *“ome^ ,
tee were tn every case to use the word ‘omega*, a very serious
confiKion would ensue :

(t) eu 15 an ordinal type.

(5) ‘«i»’ is a letter of the alphabeL

(3) Omega is a letter of the alphabet.

(4) ‘Omega’ is not a letter of the alphabet but a word of five
letters.

(5) The fourth sentence is not eoncemed with omega and there-
fore not with ‘«u’, but with ‘omega*; hence in this sentence it is
not, as in the third sentence, ‘omega’, bat “omega” which
occupies the place of the subject.

Since the name of 1 pven object roiy be chosen arbitrarily. 1^ t*
quite possible to take as a name for the thing, the thing itself, or, as t
name for a kind of thing, the things of this Idnd. \t*e can, for instance,
adopt the rule that, instead of the word * match', a match shall alw ays
be placed on the paper. But it is more often a lingiustic ea^xresunn
than an extra-lmgmstic object that is used as its own d««gnsdo^
We call an expression whidi is used in this way CKronvmaw. In this
case the expression is used in some places ais the desicnadon of its^
and in others as the derignadon of something else. In order to
obviate this ambiguity of ^ expressions which also occur autony-
tnously, a rule must be laid down to determine under whst con-
ditions the first, and under what the second, interpretation is to be
Xsken. Example: We have used the symbols ’ * V, * 5= and so

forth somebmes as autonymous amd sometimes as non-autonymous
symbols, but we have at the same rime stipulated that they am
autonymous only ohen they occur in an expression contaihing
Gothic symbols (see p. 15), Counler-exarrp^e: Formulations of the


157


§ 42 AN EXPRESSION AND ITS DESIGNATION

following kind are frequently found “We substitute a+3 for x, if
a+3 IS a prime number, ” Here the expression ‘a+3’ is used
autonymously in the first case and non autonymously in the second,
namely (toput it m the material mode of speech), as the designation of
a number For this, no rule is given The correct method of writing
would be “We substitute ‘a+3’ for ‘x’, if a+3 is a prime num-
ber, ' On the employment of autonymous designations in other
systems, see §§ 68 and 6g

Sometimes (eten by good logicians) an ahbrevtali<m for an ex-
pression IS mistaken for a designation of the expression But the
difference ts essential If it is a question of an expression of the
object-language, then the abbreviation also belongs to the object
language, but the designation to the syntax-language The mean-
mg of an abbreviation is not the original expression itself, but the
me anin g of the ongmal expression

Examples If we write 'Const* as an abbreviation for ‘Con-
stantinople ’ this abbreviation does not mean the long name, but the
city If ‘ ’ IS introduced as an abbreviation for * i + 1 ’, then ‘ i + 1 ’
IS not the meaning of ‘a , but both expressions have (m the
material mode of speech) the same meaning — that is (formally ex-
pressed) they are synonymous An expression may be replaced in a
sentence by its abbreviation (and conversely), but not by its designa-
tion The designation of an expression is not its representative, as
an abbreviation is Very often obscurities ensue because a new
symbol is introduced in connection with a particular expression
without Its being made clear whether this symbol is to serve as an
abbreviation or as a name for the expression And sometimes the
confusion which results ts impossible to eradicate, because the new
symbol is used m both senses, now in the word text as a syntactical
designation, and now m the symboLc formulae of the object-
language

Possibly many readers will think that, even though, stnctly
speaking, it is necessary to distinguish between a designation
and 3 designated expression, yet the ordmary breaches of this rule
are harmless It is true that this is often the case (for instance, m
the example gisenaboveof ‘a+3’), but the constant common dis
regard of this distmction has already caused a great deal of con
fusion It IS this disregard which is probably partly responsible
lor the lact that so much uncertain^ still exists concerning the
nature of all logical mi estigations as syntactical theories of the
forms of language Perhaps the confusion between desigiution and
designated object ts also to blame for the fact that the fundamental



PART IV. GENERAL SYNTAX


»sS

difference beW’cen the sentential junctions (e.g. implication) and
the syntactical relations betviecn sentences {e.g. the consequence-
relation) is frequently overlooked (sec § 69). Similarly, the ob-
scurity in the interpretation of many formal sj'stems and logical
investigations may be traced back to this. We shall come across
various examples of such obscurity later.

Frege laid special emphasis on the need for differentiating be-
tween an object-symbol and its designation (even m the witty but
fundamentally senous satire [ZaUen]). In his detailed expositions
of his own symbobsm and of arithmetic, he always maintained this
distinction very strictly. In so doing, Frege presented us with the
first example of an exact syntacucal form of speech. He does not use
any special symbolism as hia syntax-language, but simply the word-
language. Of the methods mentioned above he uses for the most
part A3, B4, and C2 — expressions of the symbolism in inverted
commas, together with desenpuons of forms with the help of the
word-language. He says {\Grundgestt2e\, Vol. I, p- 4)5 “Probably
the constant use of invened commas will seem strange ; but by means
of these I differentiate between the cases in which I am speaking
about the symbol itself and those in which I am speaking about its
meaning. However pedantic this may appear, I hold it to be neces-
sary. It IS remarkable how an inexact method of speech or of writing,
which may have been adopted originally only for the sake of
brevity and convenience, with full awareness of its inexactitude, can
m the end confuse thought to an mordinate degree, once the con-
sciousness of Its inaccuracy has vanished.''

The requirement laid down by Frege forty years ago was for a long
time forgotten. It is true that, on the whole, as a result of the works
of Frege, Peano, SchrSder, and particularly of Whitehead and
Russell [Pnne. AfaiA.], an exact method of working with logical
formulae has been developed. But the contextual matter of nearly
all logical writings since Frege lacks the accuracy of which he gave
the model. Two examples may serve to indicate the ambiguities
which have arisen in consequence of this.

Example 1. In the text of the majonty of text-books and treatises
on logistics (Russell’s [Pnne, Math.}. Hilbert’s [Logik], and Carnap’s
ILogistik] amongst them) a sentential variable is used in three or
four different senses: (i) As a sentential variable of the object-
language (as an (, for instance j'p*). (a) As an abbreviation (and thus
a constant) for a compound sentence of the object-language (as a
wnstant fa, for instance: 'A'). (3) As an autonymous syntactical
designation of a sentential variable d’). (4) As a syntactical deig-
nation of any sentence (‘ S ’). Thus in many cases it is not possible
to arrive at the correct way of wniing by merely adding inverted
commas. The usual formulation: "If p ia false, then for any 9,
p 09 is true" cannot be replaced by‘‘If‘p’is false, ...’’; for *p’is cer-



§42 AN EXPRESSION AND ITS DESIGNATIO> I59


tainly false {by substitution every sentence is derivable) We must
write either “If ‘A’ is false, then for any ‘B’, ‘AoB' is true”,
where ‘A’ and ‘ B ’ are abbreviating cmutants of the object language
(m this case with meanings left undetermined) , or “ If Sj is false,
then for any ;» the implication-sentence of Sj and S* is true ” If
suitable conventions are established (as on p 17) then, instead of
“the impbcaaon-sentence of Sj and < 3 i”, we may here write more
briefly

Example 2 In a treatise by a distinguished logician, the foUowmg
sentence occurs formula which results from the

formula a when the variable x (if it occurs m a) is replaced throughout
by the combmation of symbols p ” Here we are from the beginning
completely uncertain as to the interpretation Which of the symbolic
expressions m this statement are used as autonymous designations,
and are accordmgly to be enclosed m inverted commas if the cor-
rect mode of expressing the author’s meaning is to be achieved?
At first we shall probably be mclined to put ‘a\ and ‘p’ m m-
verted commas, and, on the other hand, to interpret as a

syntactical mode of writing, and therefore not to enclose it as a
whole in mverted commas, but only its component letters
(This would correspond approximately to our own
formula ‘Si^^‘)’or,mofeclosely,to Buttheoccur-

rence of thephrases “ the combinationof symbols p’ and “if x occurs
in a " rules out this interpretation , for 'p' is certainly nocombinanon.
and obviously 'x' does not occur in ‘c’ Perhaps *x’ only is autony-
mous, while ‘p’, 'a', and ‘ (for which we should then have to

write ‘^,^,^a’) are not to be taken as autonymous syntactical
designations? But opposed to this possibibty is the circumstance
that in the symboLc formulae of the object-language which is dealt
with m the treatise, ‘p’ and *a’ and even* o’ occur (for instance,
in the axiom ' (x) a 3 a *) Possibly all the symbobc symbols and


expressions — not only in the sentences of the text but also m the
symbobc formulae of the system, are intended as non-autonymous
syntactical designations? In that case the way of writing that
sentence of the text was legitimate, and the axiom referred to
would correspond to our syntactical schema PSII 16 But, on the
oiherJhanct- jjiisjisnnrfas;»r;v«««MaJe with ibe jesrxif the
the treatise as it stands We do not know to which object-language all
the formulae, as syntactical formulae, are to refer For our context
here it is a matter of no importance which of these diiferent in-
terpretations IS mtended Our object » only to show what con-



r.\KT tv» CDCrRAL FTKTM


)6o

fusions an^e xchcij it is not mid( dearschctbcx an eipressicffl Woncs
to the object-IsnsuaiT or JS a syntactical desienation, and. if tie
litter, whether it is axitonytniMS or not,

Frege's demand for the maintenance of the distinctina between a
designation and a designated eiprcsacm is, as fir as I laiow. stnctJy
fulfilled only m the emnncs of the Wamiw sdiool (luilasiewica.
Leimewsb, Taisia, and theat pupils) who ha\T ccnsQO’U'ly taitn
him as their model. These l.-»oQans maVe u^e of tpeoal syn-
lacacil symbols. This method has great advantages, although (as
Frege's own example sho'ws) « is ntvt essential for conermess. The
clear sjrfvholic separation of objert-symboU and syntax 'Symbols does

notmerely faditaTC correct formulation, but. in the case of the Warsatr

logicians, has been further justified hy the fruitfulness of thrir m*
vestiganons, which hat'e led to a plenitude of important results. The
use of spetaal syntactical symbols tnthin the wotd-text ought, in the
majority of cases, toprm'e by far the most productive mtthcid; for it
»s both elastst and easily comprehensible, as well as suSaentJy exact.

[This method IS appiedmthe text of tbeprtsent work :word-lineuare

ctanhined with Gothic symbols. The employment of Gothic letrers
by Hilbert and of heavy pnm by Chur^ are ptehminary steps in
this direcQon.] In speaal ca-sos. it may appear deairable to syw
bolise completely the sentences and defirunons of syntax and thu*
to ehminate the tvord'linguace altogether. By this means *n in-
creased exaetnes* is attained, albeit at the cost both of facility »
treatment and of comprehensibibty. Cortspletely symbolised tyi**
tacQcal defirutions of ^s land are used by Le^ewsU and G^el*
In his [\rv« Syrfin] Lefniexesla takes as objeet-lantTuage the sen-
tential cslculu* (with juncnon-vanables m operators as weB), and in
[OnroJocie] the system of the s-sentences. As syntax* Isnguage.he uses
the symboLsm ci Russell, which. hoTVexer, is only intended to serve
as an ahbreviatjon for the word-language. GSdel [l/neittrcketibgre]
takes as object-lancuage the anthmedc of the nsnual numbers in a
modified form of the Russdian symbobsm; as syntax-language, he
uses the symbolism of Hilbcru (We havt also applied this more
euctmethodintheformalconstTUCtionof Part II, where Language
I is at the same time both objcct-laruruage and syntax-language-)

§43, On ' tHE AdMIBSIBILTTY OF IbTJEFlSTTE TEBMS
We have csiled a defined symbol of Language II dffiaJe when
no unrestricted operator occurs in the chain of its definirion*;
otherwise, inJe^red *5)- IfpriKadefinite^pTilhenthepropertT

which is expressed by means of pij is resoluble; every sentence ci
the fomi pr^ CSrsO iu whidt the arguments are definite 3 — ^
simplest case, accented expreMaons—can be dedded according to a
fixed method. For an indefinite pij this docs not hold in general.



§43 AD\aSSIBILrrT OF INDEFINITE TERAE l6l

For certain indefinite pti vre are sometimes able to find a synony-
mous defimt^ pti and by this means a method of resolution But
this IS not possible m the majority of cases

Examples We can represent the concept ‘prime number’ by an
mdefimte pt'Pnmi’ as well as by a synonymous definite pt *Pnm*’
For example, we may define as follows (compare D ii, p 6o)

‘Pnmi(*)=[~(x=0) (xs=i).(«) ((«=!) V

(l•=x)V~Tlb(x, «))]’,

and m the same way for ‘Pnm*’, but with the restricted operator
*(u)*’ instead of ‘(b)’ Then ‘Pnmi=Pnmj' is demonstrable, and
thus the two pt are synonymous On the other hand, for the m-
definite pt ‘Bewhll’ defined in If (where ‘Bewbll(a)’ means, in
syntactic^ mterpretation “The SNscntence a is demonstrable m
II”, see p 7S)j no synonymous definite pt is known, and there is
reason to suppose (although so it has not been proved) that no
pr of kind exist. (The discovery of such a pt would mean the
discovery of a general method of resolution for II, and thus also for
classical mathemancs )

The lack of a method of resoIutiOQ for lodefioite terms has in-
duced man y logicians to r^ect these terms altogether, as meanmg-
less (e g Poincare, Brouwer, Wittgenstem, and Kaufmann) Let
us consider as examples two mdefimte ^pi], ‘Pi’ and *P,’ (in II,
for example), which, by means of a defimte *pif, ‘Q’, may be de-
fined m the following manner;

P.W=(3y)(Q(*.>)) (>)

P.W=W(Q(».y)) W

The logicians refe r red to argue roughly as follows: the question
whether, for instance, ' Pj ( 5 )’(or ‘ Pj ( 5 ) ’) is true or not , is meanmg-
less, inasmuch as we know of no method by which the answer may
be sought, and the meaning of a term consists solely in the method
of determination of Its appLcability or non apphcabUity To this it
may be rephed it is true that we know of no method of searchmg
for the answer, but we do know what form the discovery of the
answer would take — ^that is to say, we know under what conditions
we should say that the answer had been found. This would be the
case, for sample, if we discoiered a proof of which the last sen-
tence was ‘ Pi ( 5 ) ’ ; and the question whether a given senes of sen-
tences IS a proof of thi-t kmd or not is a defimte question. Thus
there exists the posiUnbty of the dueooery of an anstser, and there
appears to be no cogent reason for rqectmg the question.



§44 AD^^SSIBIL^^Y OP IMPRZDICATrVE TERMS 163

It IS defined (or can only be defined) with the help of a totahty to
which It Itself belongs This means (translated mto the formal
mode of speech) that a defined symbol is called impredicative
when an unrestncted operator with a vanaole to whose range of
values belongs, ocaira m its chain of definitions Example [(3)
serves only as an abbreviation^

M(F.»)s [(F(7).WtFW3F(j')])3f W] (3)

P,M = (F)[M(F.»)] (4)

[‘Pj(c)’ means “c possesses all the hereditary properties of 7
As opposed to ‘Pi’ and *P,’ (Examples m §43), ‘Pj’ is not only
ind efini te but impredicativc as well, smce it Is of the same type
zs‘F' Now, against the admissibihty of such a term, the following
objection is usually advanced Assume that a concrete case is to
bedeaded,suchas‘Ps(5)’,ie ‘(f)[M(F,5)]’ For this purpose
It must be determined whether every property has the relation M
tc 5 , It must also be known, it u said, amongst other th ing s,
whether this is true for P,, that is to say, whether ‘M(P,,5)’ is
true But this, according to {3), is equivalent m meaning to
‘(Pj(7)« )3Ps(5)* In order to find out the truth^raiue of this

unphcatioQ, the values of both members must be established, and
hence also that of * Pj (5) ’ In short, m order to deter min e whether
‘p3(5)’ is true, a senes of other questions must be answered,
amongst them whether ' P) (5) ’ is true. This is said to be an obvious
cucle,therefore'P3(5)’ IS meaningless and consequently ‘P,’ also
This form of argument seems, however, to be beside the point
(Carnap [Logizwrtusy^ in order to demonstrate the truth of a uni-
versal sentence, it is not necessary to prove the sentences which
result from it by the substitutioa of constants , rather, the truth of
the universal sentence is established by a proof of that sentence
itself The demonstration of all individual cases is impossible from
the start, because of their infinite number, and if such a test were
necessary, all umreisal sentences and all mdefinite pi (not onl) the
impredicative ones) would be irrcsoluble and therefore (by that
argument) meaningless As opposed to this, in the first place, the
construction of the proof is a finite operation, and m the second
place, the possibihty of the proof is quite independent of whether
the defined symbol occurs amongst the constant values of the
variable m question. In our example, ‘M(Pj, 5)’ can be resolved



TART IV. CENHIAL SYNTAX


before we resolve ‘P, (5) —for ‘ (P„ 5)’ can easily be proved.

For the purposes of abbreviation, we define as follows:

Then first

; ~ [(i'.(7)-fa')tr.W3P.(>o])3r.(s)]'
is demonstrable; and next, from this,

*-M(P*.s)'. ‘~(F)[M(f.S)]’.

and consequently ‘ ~Pj(s)'; and similarly for every 33 from *0’ to
‘6’ in place of 's'. Further, 'Pjfg)' is easily demonstrable, and
similarly for every 33 from ‘7* onwards.

In general, since there are sentences with unrestricted operators
which are demonstnble, thne i* alwayt the possibility of coming
to a deeiston or to whether or not a certain indefimle or impredi-
cative tervi is applicable in a particsilar indiddual case, even
though we may not always have a method at hand for arriving at
this decision. Hence such terms are justified even from the stand-
point which tnakes the admissibility of any tenn dependent on the
possibiU^ of a decision in every individual case. [Incidentally, in
my opinion, this condition is too narrow, and its necessity is not
convincingly established.]

The proper way of framing the question is not “Are indefinite
(or impredicative) symbols admissible?" for, since there are no
^ what meaning can ’admissible’ have
here ? The problem can only be expressed in this way: " How shaU
we instruct a particular language? Shall we admit symbols of
^M^quences of either pro-
;vT • I *1 ® question of choosing a form of language

that IS, of the establishment of rules of syntax and of the in-
vestipuon of the consequences of these. Here, there are two
pnncipat points to be considered: first of all, we have to decide
"f unrestricted operators are to be admitted, and
/r universal predicaie-variablcs arc to be

,dm,M for ,,p„. We .rill call p.

D belong to the range of values of

h Wh« ,s to aay, eaj be aubaUtuted for pj. I„ II all p ate pni-

other hrr’-'^tL-”' “ ""“y *>' aubatituted. On the

> 5 ^’' (»). ky 'he branehed
types. 13 divided again into sob-types, in aueh a way that



§ 44 » ADMISSIBILITY OF IMPREDICATITE TERMS 165

for a particular p only the pt of a particular sub-type may be
subs ti t u ted — -i If the first point is decided in the negative and
unrestncted operators are eicloded (as, for instance, m our
Language I), then all the indefinite and consequently all the im-
predicatrre symbols are excluded If, however, we admit the un-
restricted operators, then the definiens of an ind efini te definition
(compare Examples (1) to (4)) is m accordance with the rules of
syntax, but then it is natural to admit the definiendum as an
abbreviation for the definiens — 4 The unpredicative definitions
of pt of any types whatsoever can be excluded by deciding the
second point m the negative, and so not adnuttmg umversal
variables for these types [In this way Russell rejects all umversal
p, and Kauiinann all p in general ^ If, however, we admit uni-
versal p and, moreover, admit them also m operators, then the
definiens of an unpredicative defimdon (compare Example (4)) is
10 accordance with the rules of syntax. But then, again, it is
satoral to admit the defimeodum as an abbreviation for the
definiens In any case, the material reasons so far brought for-
ward for the rejection either of indefinite or of unpredicapve terms
are not sound. We are at liberty to admit or rqect sudi definitions
without givug any reason. But if we wish to justify either pro-
cedure, we must first exMbit its formal consequences

§ 45. Indefinite Terms in Syntax

Our attitude towards the question of lodefimte terms conforms
to the prmaple of tolerance, in constructmg a language we can
either exclude such terms (as we have done m Language I) or ad-
mit them (as m Language II) It is a matter to be decided by con
vention If we admit mdefimte terms, then stnet attention must
be paid to the distmction between them and the definite terms ,
especially when it is a questiOQ of resolubility Now this holds
equally for the terms of syntax. If we use a defimte language m the
fonnahxation of a syntax (e.g Language I m our formal construc-
tion), then only definite syntactical terms may be defined- Some
important terms of the syntax of transformations are, however,
mdefimte (m general) , as, for instance, ‘ derivable ’, ‘ demonstrable
and a fortwn ‘analytic’, ‘contradictory’, ‘synthetic’, ‘conse-
quence’, ‘content’, and so on. If we wish to mtroduce these



l66 TART IV. GENERAL SYNTAX

terms also, we must employ an indefinite sjTitax-linguage (such as
Language II),

In connection with the use of indefinite sj-ntactical terms in the
construction of a particular language, we must above all differ-
entiate the formation and the transformation rules. The task
of the formation rules is the construction of the definition of
‘sentence*. This is frequently effected by defining a term 'ele-
mentary sentence’, and determining several operations for the
formation of sentences. An expression is then called a sentence
when it can be constructed from elementary sentences by means
of a finite application of sentence-forming operations. Usually
the rules are so qualified that not only the terms ‘elementary
sentence ‘ and ‘sentence-forming operation' but also the term
'sentence’ is definite. In this case it can alwaj'S be decided whether
a particular expression is a sentence or not. Although the adoption
of an indefinite term ‘sentence’ is not inadmissible, it would in
most cases be tnexpedienc

of ‘sentence* os an indefimte term: (i) HeytinsIWotA. l]
p. 5: the definition of 'sentence* (there ‘expression’) is by rules 5.3
and 5.33 dependent upon the indefinue term ’demonstrable’ (there
correct'), and is thus itself indefinite. (2) DQrr [Z-nW*] p. 875
whether a certain combination of two sentences (‘general value’ and
‘principal value of the remainder’) is a sentence or not (there
significant* or ‘meaningless *) depends on the mith-valuea of the
two sentences; here therefore the term ‘sentence’ is not orily not
logically definite, but is moreover descriptive (i.e. dependent on
s^theoc sentences). — If, in ; language (e.g. in Peano), conditioned
definitions are admitted (S,o( 9 I,=in»). where SI, is the de-
muendum), ^en the term ‘sentence* is in general not logically
defimte. An indefinite term 'sentence* would perhaps be least open
to objection if it referred back to definite tetins, ‘elementary sen-
tcnce and •sentenee-forminB operation’. Von Neumann ([Be-
wmtA.j p. 7) holds that the definiteness of the term ‘sentence’ is
useless*'^*'^'**’ otherwise the system is "incomprehensible ted

• terms concerning transformations, namely * de-

nva e and demonstrable*, arc indefinite in the case of most
an^ages ; they are only definite in the case of very simple systems,
for instance iri that of the sentential calculus. Nevertheless, we
ran formulate the rules of transformation definitely, if, as is usually
one, we do not define th(»e terms directly but proceed from the



§45 rNDEHNITE TERMS IN SYNTAX 167

definition of the definite terms * dire^y denvable ’ (usually formu-
lated by means of rules of inference) and ‘primitive sentence’
[Here ‘primitive sentence’ can be represented as “directly de-
nrable from the null senes of premisses”, the definitions can be
taken as prumtive sentences of a particular form ] ‘Denvable’ is
determined by means of a finite chain of the relation ‘directly
denvable’, ‘demoastrable’is defined as "denvable from the null
senes of premisses” With the term ‘consequence’ (which has not
been defined m the languages m use hitherto), it is another matter
Here the rules are mdefinite even if they first define, not ‘ conse-
quence’, but only ‘direct consequence’ (as, for instance, those for
Language I m § 14)

B THE SYNTAX OF ANY LANGUAGE
(a) GENERAL CONSIDERATIONS
§46 Formation Rules

In this section we shall attempt to construct a syntax for Ian-
guaga in general, that is to say, a system of definitions of syntacucal
terms which are so comprehensive as to be appbcable to any
language whatsoever [We have, it is true, had chiefly m mind as
examples langu^es similar m their pnnapal features to the usual
symbohc langui^es, and, m many cases, the choice of the definitions
has been influenced by this fact Nevertheless, the terms defined
are also applicable to languages of quite different kinds ]

The outline of a general syntax which follows is to be regarded as
no more than a first attempt Tlie definitions framed wUl certainly
need improvement and completion in many respects , and, above all,
the connections between the concepts will have to be more closely
investigated (that u to say, further syntactical theorems will have to
be proved) As yet there have been very few attempts at a general
syntactical mvesngation , the most important are Tarski’s [Methodo-
logtel and Ajdukiewics's {SpTaefte\

By a language we mean here m general any sort of calculus,
that 13 to say, a system of formation and transformation rules con-
cerning what are called expressions, i e finite, ordered senes of cle-



t68


PART nr. General syntax


merits of any kind, namely, what arc called tymhoU (compare §§ l
and a). In pure syntax, only syntactical properties of expressions,
in other words, those that are dependent only upon the kind and
order of the symbols of the expression, are dealt tvith.

As Opposed both to the symbolic languages of logistics and to the
strictly scientific languages, the common word-languages contain
also sentences whose logical character (for example, logical validity
or being the logical consequence of another particular sentence, etc.)
depends not only upon their syntactical structure but also upon extra-
syntactical circumstances. For instance, in the English language, the
logical character of the aentenccs ‘yes' and ‘no’, and of sentences
which contain words like *he’, ‘this’ (in the sense of "the afore-
mentioned ’’) and so on, is also dependent upon what sentences have
preceded them in the same context (treatise, speech, conversation,
etc ). In the case of sentences in which words like* I’, ‘you*, ‘here’,
‘now’, 'to-day', 'yesterday', ‘this’ (in the sense of “the one pre-
sent ) and so forth occur, the logical character is not only dependent
upon the preceding sentences, but also upon the extra-linguistic
situation— namely, upon the spatio-temporal position of the
speaker.

In what follows, we shall deal only with languages which contain
BO txprtttiont dtptndtnl upon extra^linguiitie /attort. 'The logical
character of all the sentences of these languages is then invariant in
relanon to spatio-temporal displacements ; two sentences of the same
wording Will have the same character independently of where, when,
Of by whom they are spoken. In the case of sentences having extra-
s^tacticil dependence, this invariance can be attained by means of
the addition of person-, place-, and tune-designations.

^ In the treatment of Languages I and II we introduced the term
consequence’ only at a late stage. From the syitematic standpoint,
hoaever, it is the beginning of an syntax. If for any language the
term 'consequence' is established, then everything that is to be said
concerning the logical connections seithin this language is thereby
determined. In the following discussion we assume that the trans-
fomatlon rules of any language S, i.e. the definition of the term
direct consequent in S’, are given. [For the sake of brevity in
the tse of syntactical terms, we usually leave out the specification
m or of S .] We shall, then, show how the most important
syntactical concepu can be defined by means of the term ' direct conse-
quence . In this process it will become clear that the transforma-
Uon rules determine, not only concepts, such as ‘valid’ and 'con-
tra-valid', but also the distinethn between logical and descriptive
symbols, between variables and constants, and further, between logical



§ 46 FORMATION RULES


169


and extra hgual {phyncdC) txansjormattm rules, from which the
difference between ‘vahd’ and ‘analytic* arises, also that the dif-
ferent kinds of operators and the various sentential connections can
be characterized, and the existence of an anthmetic and an in-
fimtesmal calculus in S can be determined

As syntactieal Gothic symbols, we use (as previously) ‘a* for
symbols, ‘S’ for (fimte) expressions, ‘ft* for (fimte or mfinite)
classes of expressions (for the most part, of sentences) AU further
Gothic symbols in the general syntax (even those used previously
m I and II) are defined m what follows We say of an expression

that it has the form STj when it results frotnSIi by the replace-

ment at some place m lllj of a partial expression ^ by ?Ij (On the
difference between replacement and substitution, see pp 36 f)


We resUnct ourselves to finite expressions only because, up to
now, there has been no particuiar reason for dealing with tnfimte
expressions There is no fundamental objection to the introduction
of infinite expressions and sentences The treatment of them in an
anthmetized syntax is quite possible While a finite expression is
represented by a senes of numbers which can be replaced by a single
senes-number, an mfirute expression would have to be r e pre s ented
by an infinite senes of numbers or a real number Such a senes is
expressed by means of a (definite or indefinite) functor According
to what was said previously (§ 39) we can speak not only of infimte
expressions which are systematically constructed, but also of infinite
expressions which are not detennmed by any mathematical law
An fui corresponds to the fonner, an fu^ to the latter


We will assume the defimtion of 'direct eomequeTtee' to be stated
m the following form " is called a direct consequence of 51 ^ m
S if (1) HTj and every expression of 5 li has one of the following
forms , and (2) and fulfil one of the following conditions
” The definition thus contains under (1) the formation rules
and under (2) the transformation rules of S Now we call % a
sentence (S) if ^ has one of the forms under (x) Those a that are
€ are called sentential symbols (fa)

and ^ (an a is also an ^ are said to be syntactically related

whes asDsts aa St jsmA tlaS 8^ as ^ sad ^ 13 **) ^


asentence Tworelatedexprcssions^Itand^^arecalledlsogenoas
if for any ©j, Sj j^J and Si sentences. A class Sii of



* 7 °


P.UtTIV. CESniALSTXTAX


expressions is called a {renas if ct «7 two cxpresaoas cf 51, are
isogenoxis, and no expression of 51, is isogenous with an erpfc»on
which does not belong to 51,. plelatedness is a sunilaritT (on
these and the following terms see Carnap P-4S) ; fiinher,

isogeneity is transitiTe, and therefore an equality; the genera are
the abstractive classes with respect to isogeneity; hence different
genera hare no members in common.] The sub-<lass of a genus
of expressions wluch contains aQ the symbols and only the symbeJs
of this genus is called a symbolic genus. Every SI of S belongs to
exactly one genus; if the genus of 21, is {21,},' so that 21, is not
isogenous with some unequal 21, then 21, is called isolated. Two
expre^onal genera or two symbolic genera are called related when
at least one expression of the one is related to one of the other; in
this ease every expression of the one is related to every ciprcssioa
of the other.

In what follows, definitions of further syntactical formation
terms will result from the transformation rerms.

Exampla: In I and 11 every i b isolated; for not

a sentence. In Kilben’s symbolism also, every } is isolated; here,
(3i)(pri(!i))j^^ for unequal and 3 , is not a sentence.
Inland II aUcDnstantJitojretherfontiBsenus. On the other hand,
m 1 and 11 j, and nu. for example, are reared but pot isogtaoui,
sina in an operator J, cannc»t be repbeed by nu.

The ?>t or 2u of any type fin II are to be divided into two related
genera : that of the p (or 0 of f and that of the remaining (or Ju,
respectively). Thus the rt (or fa) of f are to be divided into two
related sjmbolic genera: that of die p (or 0 of * and that of the
constant pr (or fu, respectively) of U

§ 47 . Tr.\nsfor.\ution Rules; D-TtR-MS

We will now assume that the transformation rules of S which
hare been given in one waj* or another are oarened into the form
previously indicated of 3 definition of 'direct consequence in S’.
It mai.es no difference in what terminology the rules were origin-
ally stated ; all that is necessary’ is that it be clear to what forms of
expressions the rules are in general applicable (which gives os the
definition of ‘sentence’) and under what conditions a tiansfonna-
tion or inference is pennitied (which gives us the definition of

direct consequence").



§47 TRANSFORBiIATION ROLES, D-TERMS I 71

For instance, instead of ‘diRct consequence’, we frequently have
the terms ‘derivable’, ‘deducible*, ‘inferable’, 'results from’, ‘may
be concluded (inferred, derived ) from’, etc , and, mstead of
‘direct consequence of the null class’, it is customary to find
‘primitive sentence’, ‘anom’, ‘true*, ‘correct’, ‘demonstrable’,

‘ logically valid ’, etc We shall assume that even those rules con-
cerning symbols of S, that are usually designated as defirutions, are
mcluded m the rules concerning ‘direct consequence’ (for instance,
as pmmtive sentences or rules of inference of a special kind) , the
defimnons can either be fimte m number and stated singly, or un-
limited m number and established by means of a general law (as,
for example, m I and If)

The second part of the definition of ‘direct consequence* con-
sists of a senes of rules of the following form "01 IS a direct con-
sequence of the sentence-class il, if (but not only if) 0^ and
have such and such syntactical properties ’’ We will extend this
senes by means of the following rule (which sometimes already
belongs to the onginal senes) " 0| is always a direct consequence
of {0j} ’’ We call the rules of the whole senes rules of consequence,
or, bnefiy,c-nile3 Those in which the properties stipulated for 0^
and are definite we call rules of dematum, or, bnefiy, d>ru]es
01 is called dtrectly demabte from Aj if Aj and 0] satisfy one of the
d rules 0^ IS called a pnmxtsxie sentence if 0) is directly denvable
from the null class A finite senes of sentences is called a deriva-
tion with the premiss class ft, if every sentence of the senes cither
belongs to ft, or is directly denvable from a class ftj, the sentences
of which precede it m the senes A denvation with a null premiss-
class IS called a proof 0, is called denvable from (or a d-conse-
quence of) the sentential-class ft, if 0, is the last sentence of a
denvation with the premiss-class ft, 0, (or ft,) is called demon-
strable (or d-vahd) if 0, (or every sentence of ft,, respectively) is
denvable from the null class and is therefore the last sentence of a
proof 0, (or ft,) IS called refutable (or d-contravalid) if every
sentence of S is denvable from {0,} (or ft„ respectively) 0,(or
ft,) IS called resoluble (or d-detenmnate) if S, (or ft,, respectively)
IS either demonstrable or refutable, otherwise irresoluble (or
d-indeterminate)

LaA A, tc vtt ’/ai-gtsv xVaa tS -syirfecffl m S Vcfvm g itttuwmg
properties The symbols of ft, can bcarranged (not necessarily uni
vocally) in a senes If o, belongs to ft,, then there is by the d-rules
a definite direction for constraction (m an anthmetized syntax,



172 PART IV. GENERAL SYNTAX

that means a definite synuctical functor), according to which, for
every sentence Sj in which o, occurs, a sentence Sj can be con-
structed such that Sj does not contain fli, but only sjTnbols which
either do not belong to or which precede Qj in that series, and
such that and Gj are derivable from one another. We call such
a direction a definldon of and the transformation of Gi into G|
the elimination of Qj. We call the symbols of 51 ^ defined, the
others undefined.

We divide the syntactical terms into d-terms and c-terms,
according to whether th«r definition refers only to the d-rulcs (as
for instance in the preceding definitions) or to c-rules in general.

§48. c-Terms

We shall now define a number of c-terms, beginning with ‘ con-
sequence’, one of the most important syntactical terms. In what
follows the Si are always sentential classes. Gj is called a conse-
quence of III, if Gi belongs to every sentential class satisfying
the following two conditions: i. Hi is a sub-class of 2. Every
sentence which is a direct consequence of a sub-class of Sit belongs
to H<.* Sit ^ called a consequence-class of Hi if every sentence
of Ht is a consequence of Hi. If d-rulcs only are given, then the
terms ‘derivable’ and ’consequence’ coincide; and if the term
‘ direct consequence ' already possesses a certain kind of transitivity
then it coincides with ‘consequence’.

What has previously been said in (he case of Language I holds
in gcncnl for the fundamental difference between ^derivable in S’
and ‘ consequence in S' (see pp. 38 f.), and analogously for every pair
which consbts of 3 d-term and its correlative c-term; compare the
second and third columns in the survey on p. 183.

In almost all knoxen systems, vnly definite rules of transformation
are stated, that is to say, only d-rules. But we have already seen
that it is possible to use also indefinite syntactical terms (§ 45). We
shall therefore admit the possibility of laying down indefinite
transformation rules and of introducing the c-terms which arc
based upon these. In dealing vrith the syntax of Languages I and

* (Note, 1935.) The above definition of ‘consequence’ is a cor-
rection of the German original, the need for which was pointed out
to me by Dr. Tanki.




173


§48 C'TES&IS

II we have come to recognize both the importance and the fer-
tility of c-terms (such as ‘consequence’, ‘analytic’, ‘content’,
etc.) One important advantage of the c-terms over the d-terms
consists m the fact that with their help the complete division of Sj
mto anal ytic and contradictory is possible, whereas the corre-
spondmg classification of €i into demonstrable and refutable is
mcomplete.

Only d-rules arc given in the systems of Russell [Prmr Math ],
Hilbert \LogiK\, vo" Neumann [Bettettth ], GSdel [UnenUchetdbare],
Tarsb ]WidertpruchsfT'\


the form is demonstrable, thea(3i)(®i)™*y^® laid down as

a pnmitive sentence ” Hilbert calls this rule a “new finite rule of
inference” What is to be understood by ‘finite’ is not precisely
stated, according to indicatiom given by Bemays [PMosephe]
P 343> It means about what we mean by ‘definite’ Tbs rule is,
however, obviously mdefinite Its formulation was presumably
motivated by the mcompleteness, indicated above, of all antfameocs
which are restncted to d-rules The rule given, however, which
refers only to numerical variables 3, is not sufficient to secure a com-
plete classification

Herbrand \Non~«mtrad'] p 5 makes use of Hilbert's rule, but
with certam restrictions; G^ and the definitions of the fu which
occur m Gi must not contain any operators

Tarski discusses Hilbert’s rule (“Rule of infinite mductiOQ”
[TTidcripnicAr/r ] p iii) — hehimselfhadprenouslyfigzyllaiddown
s similar one — and nghtly attributes to it an “jnfiiutist character”.
In his opinion “it cannot easily be harmonized with the mte r pret a -
tion of the deductive method that has been accepted up to the
present”, and this is correct m so far as this rule differs funda-
mentally f ro m the d-rules which have hitherto been exclusively used
In my opinion, however, there is nothing to prevent the practical
apphcation of such a rule

In Language 1, DC 1 refers back to the definite rules PS x-11
and RI 1-3, DC z is indefinite

Rj IS called valid if is a clzss of consequences of the null class
(and hence of every class) [We do not use the term ‘analytic’
fiere because we wish to feave open tfic possibility that S contains
not only logical rules of transforxnzaon (as do Languages I and II)
but also physical rules such as natural laws (see § 51) In relation
to languages like I and II, the teims ‘valid’ and ‘analytic’ com-



174 CENER.4L SYNTAX

cide.] i^i is called controvalld if everj* sentence is a consequence
of ill is called determinate if ilj is either valid or contra%-alid ;
otherwise indeterminate. In a \TOrd-language it is convenient in
many cases to use the same term for properties both of sentences
and of classes of sentences. We shaU call a sentence Sj valid {or
contravalid, determinate, or indeterminate) if {Sj} is t’alid (or con*
travalid and so on, respectively). And we shall proceed in the same
way tvith the terms which arc to be defined later.

Theorem 48.x. Let be a consequence-class of if jlj is
valid, ilj is also valid; if ilj is contravalid, so also is jl,.

Theorem 48a. Let Gi be a consequence of Sj; if Si is t'alid,
S] IS also valid; if Sj is contravalid, so also is S|.

Theorem 48.3. If every sentence of 51i is valid, is also valid;

and conversely.

Theorem 48^ If at leastonesentenceof is contravalid, then
is contra%*alld; the converse is not universally true.

Two or more sentences are called incompatible (or d*lncom-
patible) tvith one another if their class is contravalid (or refutable,
respectively) ; othenvise they are oiled compatible (or d*com*
paiible). Two or more sentential classes are called incompatible
(or d*incompatible) with one. another if their sum is contra\'alid
(or rcfuUble, respectively); othenvise they arc oiled compatible
(or d*compatible).

51, is oiled dependent upon 51, if 51, is a consequence-class of
il„ or is incompatible with 51,; othenvise it is called independent
of 51,. 51j is called d-dependent upon 51, if either every sentence of
51, is derivable from 51,. or 51, b d-incompatible with 51,; other-
wise it is called d-independent of 51,. (The definitions are ana-
logous for S, and G,.)

Theorem 48.5. If 51, is dependent (or d-dependent) upon the
null class, then 51, is determinate (or resoluble, respectively); and
conversely.

We say that there u (mutual) independerue within 51, if every nvo
sentences of 51, are independent of one another. And we say that
there is campUte independerue within 51, if every proper non-null
sub-class of 51, is independent of its complementary class in 5ti.

Theorem 48.6. If 51, is not contravalid and Is not a consequence-
class of a proper sub-class, then there u complete independence
within 51,; and conversely.



175


§ 4^ C-TESMS

ill 13 called complete (or d-complete) if eveiy R (and conse-
quently every S of S) is dependent (or d-dependent, respectively)
upon Rii otherwise it is called incomplete

Theorem 48.7. If 51 ) ^ complete and is a consequence-class of
Aj, then Ai also is complete

Theorem 48.8. If the sentential null class is complete (or d-
complete) m S, then every A m S is complete (or d-complete,
re sp ec ti vely)

The arrows m the table on p 183 mdicate the dependence be-
tween the defined d- and c-concepts Although the d-method is the
fundamental method and the d-terms have the simpler defimtions,
yet the c-terms are the more important from the standpomt of
certam general considerations They are more closely connected
with the matenal mterpretation of language, and this is shown
formally by the fact that simpler relations obtam among them.
In what follows we shall be dealing pnnapally with the c-terms,
and shall only state the corresponding d terms occasionally (if 00
special term is given, one is constructed from the c-term by pre-
firmg a * d- *)


§49. Content

Bythecontentof Ai(orof< 3 j,cf p 174) m S, we understand
the class of the non-valid sentences which are consequences of Ai
(or Sx, respectively) This defimtion is analogous to the previous
defimtions for Language I (p 42) and Language II (p 120); it
must here be noted that m Languages 1 and II ‘vahd’ coinades
with ‘analytic’

Other potnbUiUes of defimttort Instead of the class of the non-
valid consequences, one might perhaps designate as ‘content’ the
class of all consequences As opposed to this, our definition has the
advantage that by it the analytic sentences in pure L-languages (see
below) such as I and II have the null content. Again, it might be
possible to take as ‘content’ the class of all mdeterminate conse-
quences, or even the class of aD non coniravalid consequences Let
S be. a non-descnptive language (such as a mathematical calculus)
Then, in S there are no mdeterminate (or synthetic) sentences In
this case, on the basis of our definition, the analytic sentences are
equipollent, and similarly the contradictory sentences , but there is
not equipoUence between the two On the basis of either of the above
defimtions, on the other hand, all sentences would be equipollent.



176 PART IV. GENERAL STNTA 3 C

though they differ essentially from one another in that only analytic
aentencea are consequences of an analytic sentence, but all sentences
are consequences of a contradictory sentence. AjduUemcz gi\es
a formal definition of ‘sense* which is worthy of note. It differs
considerably front our definition of ‘content*, for, according to it,
the term ‘equivalence of sense ’is very much narrower than our term
‘cquipollencc*.

and are called equipollent when their contents coincide.
If the content of ilj is a proper sub-class of the content of 51,, then
51j is called poorer in content than and 51, richer in content than
51,. We say that 51, has the null content if the content of 51, is
empty, i.c. the null class. We say that 51, has the total content if the
content of 51, is the class of all non-valid sentences. Two or more
classes are said to have exclutice contents if their contents have no
member in common. All these terms are also applied to sentences
(see p. 174). We say that a mutual exclusiveness in content subsists
in 51, if every two sentences of ft, have exclusive contents.

Theorem 49*t< If ft, is a consequence-class of 5^,, then the
content of 5i, is contained in that of ft,; and conversely. In the
transition to a consequence, an increase in the content never oeevrs.
It is in this that the so-called fauto/ogical charaeitr of the const-
cpsenet-rtlation consists.

Theorem If ft, and 51, arc consequence-classes of one
another, then they are equipollent-, and convtrsely.

Theorem 49.3. If 51, is a consequence-class of 51,, hut ft, not a
consequence-class of 51,, then 51, is richer in content than 51,; and
conversely.

Theorem 49.4. If 51, is rofid, then ft, has the null content ; and
conversely.

Theorem 49.5. If 51, is contravalid, then 51, has the total content',
and conversely.

Theorems i to 5 hold likewise for G, and Sr

51, is called perfect if the content of 51, contained in 51,.
According to this, every content is perfect The product of two
perfect classes is also perfect; but this u in ceneral not true for the
sum.

?ti is said to be replaceable by ^ if S, is always equipollent to
*=1 ^0^ • % “I'd 91, arc called synonsmious (with one another) if
they are mutually repbccable. Only expressions of the same genus



§49 CONTENT 177

can be syiion3rmous [If 2 Ii la replaceable by ^ it is usually also
synonymous with 21, j

2 Iji IS called a prmapal expremon if 9 , is not empty and there
exists an expression which is related to, but not synonymous with,
9 , We count 2& prmapal symhoU, first, crery symbol which is a
prmapal expre ss ion, and, second, symbols of certam kmds which
will be described later on (c.g 91 , n, *0, pr, bI, 33), the rest of the
symbols are called luijiduay ^fmbols \ExampIe The prmcipal
symbols of Language II are the fa, 33, pr, fit, BCifn, and

* = ’, *>’, ‘3’ (by the defimtions of general syntax, is a nf,

* =5 ’ a pr, a 3fu, the null expression is related to ‘3' but is not
synonymous with it) The remaining symbols are subsidiary
symbols, namely, brackets, commas, and 'K‘ (because m II there
are no numerical operators other than the K-operators) ]

§50. Logical AND Descriptive Expressions;
Sub-Languages

If a material mterpretatioo is given for a language S, then the
symbols, expressions, and sentences of S may be divided mto
logical and desmptive, 1 e those which have a purely logical, 01
mathematical, meaning and those which designate something
extradogtcal — such as empirical objects, properties, and so forth
This classification is not only mexact but also non-formal, and
thus 13 not apphcable m syntax. But if we reflect that all the con-
nectioos between logico-mathematical terms are independent of
extra-hnguistic factors, such as, for instance, empirical observa-
tions, and that they must be solely and ompletely determmed by
the transformation rules of the language, we find the formally
expressible distmguishing peculian^ of logical symbob and ex-
pressions to consist m the fact that each sentence constructed
solely from them is determinate. This leads to the construction of
the following defimtioo. [1 he defimtion must refer not only to
symbob but to expressions as weU, for it is possible for Oj m S to
be logical m certam contexts and descriptive m others ]

Let be the product of aQ expressional classes of S,
which fulfil the followmg four conditions [In the majority of
the usual language systems, there exists only one class of the
kmd this is then ili ] i. If 9 i belongs to ft*, then 9 j is not



1^8 PART IV. GENERAL SYNTAX

empty and there exists a sentence which can be sub-divided into
partial expressions in such a way that all belong to and one of
them is Dlj. 2. Every sentence which cai{be thus sub-divided into
expressions of ft, is determinate. 3. The expressions of ft, are as
«ma11 as possible, that ia to say, no expression belongs to 51 , which
can be sub-divided into several expressions of ft,. 4. 51 , is as com-
pTthtnswt as posable, that is to tay, it is tiot a ptopec suh-dass of
a which fulfils both (x) and (2). An expremon is called logical
(2Ii) if it is capable of being sub-divided into expressions of ftj;
otherwise it is called descriptive A /an^jc is called hgual
if it contains only Oj; otherwise detcriptive.

With a language which is used in practice — for instance, that of
a particular domain of science — it is usually quite dear whether a
certain symbol has a logico-mathematical or an extra-logical, say a
physical, meaning. In an unambiguous case of this kind, the
formal differentiation just given coincides with the usual one.
There are occasions, however, when a mere non-formal considera-
tion leaves it doubtful whether a symbol is of the one kind or the
other. In such a case, the formal criterion helps us to a dear de-
cision, which on doser examination will also be found to be
materially satisfactory.

Example: Is the metrical fundamental tensor by means of
which the metrical structure of ph)rsical space is determined, a
mathematical or a physical term? According to our formal criterion,
there are here two cases to be distinguished. Let S, and be
physical languages, each of them containing not only mathematics
but also the physical la^^'s as rules of transformation (this will be
examined more closely in § 51). In S, a homogeneous apace may be
assumed: ‘g^’ has the aamc value everywhere, and at every point
the measure of curvature is the same in all directioru (in the simplest
case, 0 — Euclidean structure). In S,, on the other hand, the Ein-
ateinian non-homogeneous apace may be assumed: then ‘g^’ has
various values, depending upon the distribution of matter in space.
Tpiey are therefore — and this is an essential point for our differcntia-
rion — not determined by a general law. ‘g„’ is thus a/ogica/ijsnAo?
in S, and a deseriptiie symbol in S*. For the sentences which give
the values of this tensor for the various apace-time points are in S,
all determinate; and on the other hand, in S, at least part of them
arc indeterminate. At * first glance, it may appear strange that the
fundamental teruor should not have the same character in all
languages. But on closer examination we must admit that there is
here a fundamental difference between S, and S,. The metrical
calculations (for example, the calculation of a triangle from suitable



179


§ 5° LOGICAL AND DESCRIPTITE EXPRESSIONS

dettnninations) are made m by means of mathema&cal rules
which. It IS true, in some respects (for instance, m the choice of the
nhie of a fundamental constant such as the constant curvature of
space) are based on empirical obseivaboas (see f 8r) But on the
other hand, for such calculations m Sy empirical data are regularly
required, namely, data concerning the distribution of theTalues of
the fund^ental tensor (or of the density) m the space^time domim
m question.

Theorem 50.x. Every hgieal satteitce is dete mina te, every m
determinate sentence is descnptiie With the given form of de-
finition for ’logical’ this follows directly If 'logical expression’
IS defined m some other way (for instance, by the statement of the
logical pnmrtxve svmbots, as in Languages I and 11 ) then the de-
finitions of the terms * valid ’ and * contra^'ahd ’ (which m I and 11
coinade with ‘analytic’ and ’contradictory’) must be so contrived
that every Si is determinate

Theorem 50,4. (a) If S is logical, then every ft m S is deter-
mioate, and conversely (i) If S ts descriptive, then there is an
indeterminate ft m S , and conversely

Sj IS called a sab-langusge of if the following condihons hold
X every sentence of S, ts a sentence of S^, 2 if ft, is a conse-
quence^<Iass of ft, m then it ts likewise a coosequence-dass of
ft, m S, S, IS called a eonservattve mh-language of S, when, m
addition 3 ifftgts a consequence-class of ft, m S,, and 51 , and ft,
also belong to 5 „ then is also a consequence-class of ft, in S,
If S, ts a sub-language of S, but not S, of S,, then S, is called a
pnper rjb-lmgaa^ of S, By the topical rub-^anguas* of S, we
understand the conservative sub-language of 5 which results from
S by the elimination of all the de scrip tive sentences.

Let S, be a sub-language of S,, and ft, and 51 , sentential classes
of S,. The table on p 225 states under what condmons a syn-
tactical property of ft,, or a relation between ft, and 51 ,, which
obtains in S., obtains also m S, (rubnc 3), or conversely (njbnc 5)
Thus, for example, we can see from the table that if ft, is vahd m
S,, then It 13 also valid m S, , if ft, is vahd in S, and S, is a con-
servative sub-Iang^^ of then ft^is also vahd in

ExoH f le I IS a proper cooservatiTe sub-language of 11 Let I' be
the language which results from 1 if un r e str icted operators with 3
are admitted, then I u a pr^ier sub-language of I although both
languages possess the same symbols.



§51. LOGICAL AND PHTSICAL RULES l8t

IS a descnptive primitive sentence of tbeJtmd PSI i But Si is
obviously true m a purely logical Way, and we must arrange the
further definitions so that Si is counted amongst the L-rules and
IS called, not P-valid, but analytic (L-valid) That Si is logically
true IS shown formally by the fact that every sentence which results
from Si when ‘Q’ is replaced by any other pt is likewise a
primitive sentence of the kind PSI i ] The example makes it clear
that we must take the general replaceability of the STs as the de-
finitive charactenstic of the L-ruIes

Let Sj be a consequence of Ai m S Here three cases are to be
distmguished i 51 i and S^ are logical 2 Descnptive expressions
occur m Sii and m Sj, but only as undefined symbols, here
two further cases are to be distinguished z a for any 5I3 and Si
which are formed from fti (or S*) by the replacement of every
descnptive symbol of fti (or S* respectively) by an expression of
the same genua, and specifically of equal symbols by equal ex-
pressions, the following is true Siisaconsequenceof the

condition mentioned is not fulfilled for every As and St 3 In A|.
and 0, defined descnptive symbols also occur, let A] and ^ be
constructed from Ai (or S} respectively) by the ehmination of every
defined descnptive symbol (mdudiog those which are newly mtro-
duced as the result of an elumnauon) , 3 a the condition g;iven in
2 a for Ax and Sj is fulfilled for Aj and , 3 & the said condition
13 not fulfilled In cases i, 20, 347, we call Sg an L-conse-
qnence of Ax , m cases 2 3 we call Sg a P-consequence of Ax

Thus the formal distinction between L- and P-rules is achieved

If S contains only L rules (that is to say, if every consequence m
S is an L-consequence), we call S an L-language, otherwise, a
F-language BytheL-sub language of S we shall mean that sub-
language of S which has the same sentences as S but which has as
transformation rules only the L-rules of S

Theorem 51.1 Every logical language is an L-language. The
converse is not alwaj’s true

The disbneiton betzceen L- and P language! must not be confused
with that between logical and descnptive languages The latter is
dependent upon the symbobc apparatus (although only, it is true,
upon a property of the symbobc apparatus which appears m the
transformation rules), the former on the kmd of the transforma-
tion rules Languages I and II are, for example, descnptive



part IV. CESTRAL SYNTAX


iSz

languages (they contain as is shown by the occurrence of in-
determinate, namely, synthetic sentences), but they are L-lan-
guages : every consequence-relation in them is an L-consequence ;
and only analytic sentences are valid in them. Sinularly, the dif-
ference between the L-sub-language of S and the logical sub-
language of S is to be noted. For instance, if S is a descriptive
L-language (like I and II) then the L-sub-language of S is S itself,
but the logical sub-language of S is a proper sub-language.

§52. L-Tmis; ‘Analytic’ and ‘Contradictory’

To the previously defined d- and c-terms we now add L-terms
(to wit, L-d-terms and L-c*tenns). If in the L-sub-language of S,
ill has a particular (d- or c-) propert}', we attribute to it in S the
torresponding L-propertj*. For instance. Si is called L^dmm-
in S if Sj is demonsttaWe in the L-sub-language of S.
ftj is called the L-nntent of ft, in S if is the content of in the
L-sub-language of S, and so on. Instead of ‘ L-valid ‘ L-^ntra-
valid', and ‘L-indetenntnate*. we shall usually say ‘aBaljTie',
‘coatradlctoi^’, and 'synthetic’. In the table which follows
(p. 183), the correlative terms are placed on the sameline. An arrow
between two terms shows that one may be inferred from the other.
[Example: If (Sj is L-dcri%’able from Si then it is also derireble
from Si; and if deri\-ablc from S|, then also a consequence of Si-
Between an L-d- and an L-c-term, ihe inference alwaj's holds in
the same direction as between the correlative d- and c-terms.]
Here again the d- and L-d-tenns are more fundamental for the
method of proof; on the other hand, the c- and Li-c-terms are the
more important for many applications.

Since I and II are L-langusges, in iheir case every syntactical
term coincides with the correlative L-term (for instance, 'de-
monstrable* with 'L-demonscrablc', ‘consequence’ svith ‘L-conse-
quence’, ‘valid’ with ‘analytic’, ‘content’ with *L-content’, and so
on). The L-d- and L-c-terms which were previously defined for I
and II agree with ’hose now defined, even where the eaiber de-
finition has quite a different fonn (ms for example in the case of
‘analytic in II’).

*nieorem 5a.r. (a) Every analytic sentence is valid. (A) Every
valid logical sentence is analytic.— Regarding (A) : Let S, be a valid



§52 L-TERMs; ‘analytic* AND ‘contradictory’


I I

a i ;=;

' IsS I 5 l&l 1-1.1111 I

S-s§a-2SsS..S68E3,'c|

8t?-S7E8-SsSSS?S.I

JidiJiAj JJ J J J J J J Ji

*<*


I 1 I E I f-sl S-a-f s I
I’ 1 1 S ills-li ^ II 1
is lIlAStl I I I 111

t t t t i t * t * t +


I S


S E
' S’!
S j=
8 S
^ *0


Sii«’2g'225

I I S Eli 111
illiitlii

SSS'O'O'O'O'a’a


t t t f * t * t I t I


n - - E ® S
J 3 &• 5 •« g « §•

— - ^j3Ec.Ec.e.S

g§23c8Eg.-SB8

cBasssS-Ssss


' s — ~



184 PART IV. GENERAL S^WAX

Si- Then 0 j is a consequence of the null class, and hence an L-

consequcnce of it, and therefore analytic.

Theorem 52.2. (a) Every contradictory sentence is contravalid.
(fc) Every contra^-alid logical sentence is contradictory.— Regarding
(i): Let Sj be a contravalid Sj. Then every sentence is a conse-
quence of Si- Therefore, in the first place, every Sj, and in the
second place, in the case of all Si,, every S® transformed according
to rule 2 a or 3 a (p. 181), is a consequence of Si. Hence every sen-
tence is an L-conscqucncc of Si- Therefore, Si is contradictory.

Theorem 52.3. Every logical sentence is L-determinate; there
are no tynthetic logical tentencce. This follows from Theorems
50.1, 52. li and zb.

Theorem 52.4. If every 8cnten<» of ftj is analytic, then Ri is
analytic; and conversely.

Theorem 52.5. If at least one sentence of R] is contradictory
then Ri is contradictory. If Ai is logical, then the converse is also
true.

Theorem 52.6. Let ^ be a consequence of Siy (a) If Rj U
analytic, then S, is also analytic. (8) If <9, is comndictory, then
Ri is also contradictory.

Theorem 52.?. If ( 3 | is an L-consequenceof the sentential null
class (and therefore of every class), then Si is analjtic; and con-
versely.

Theorem 52.8. If 51 j is contradictory, then every sentence is an
L-conscquence of Rjj and conversely.

Theorem 52.9- The L-content of 51 , is the class of the non-
analytic scnicnces that are L-conscqucnces of 51 i.

The ordinary concept of the equivalence in seme ol two sentences
is ambiguous. We represent it by means of two different formal
terms, namely, equipollencc and L-equipoHence. Analogously, wc
replace the ordinary concept of the equivalence in meaning of two
expressions by two different terms, synonymity and L-synonjinity.
(Compare § 75 : Examples 6-9.)

The L'lerms are obtained by restriction to the L-rules of the lan-
guage. For some of these terms, we will define corresponding P-
terms.Thesearecharacterizedbythcfactthat,forthem,theP'ru!es
also are taken into account. In L-languages, they are empty. Sfls
called a V-eomequenee of 51 i if is a consequence, but not an
L-consequcncc, of 51 i. 51 i (or SO « P-vaUd if it is valid but not



§ 5 * L-TEKMS, ‘analytic* AND ‘contradictory’ 185

analytic 5 ^^ (or Sj) is P-cootravalld if it is contravalid but not
contradictory and are 'P-eqmpoIUnt if they arc equipdllcni
but not L-equipoUcnt and % are P-synor^oui if they are
synonymous but not L«synonymous In what follows we shall
imke very little use of the P-tenns
For a P-language we get the following classification of descnptive
sentences (for the G(, see p 210)


(d>tenns } demonstrable irresoluble refutable


(P-terms )
(L-terms )
(c terms )


(L-vahd)

analytu

valid


P-contra-

valid


(Ifcontravalid)

emtradutory


mdetenmnate contxavabd


For an L*language (such as I and II) the classiftcation of the de*
scnptive sentences is simpler, since the c> and L-c«terms coincide

(d terms ) demonstrable itresoluble refutable


(c« and L* valid indeterminate contravabd

terms ) analytte tynthet contradictory

Examples Assume that S is a P-Ianguage with Enghsh words used
in their ordinary meaning Let the most important physical laws be
stated as pnmitive sentences of S LetGibe ’thisbodyaisofiron’,
Si ‘a IS of metal', 6, ‘a cannot float on water’ and are
consequences of €1, and specificalJy, €| is an L-consequence, but
Sj IS not, and is therefore a P-consequence Let S4 run ‘In this
vessel b of volume 5000 c c. there are a grm of hydrogen under
such and such a pressure’, ©» * In b (of volume 5000 c.c) there
are 2 gim of hydrogen at such and such a temperature ’ €4 and 0f
are consequences of one another, and, speafically, P-consequences,
since each of these two sentences can be inferred from the other by
means of the physical laws 64 and €4 are equipollent, but not L*
equipollent, and therefore they are P-equipollent If in the matenal
mode of speech we ask whether ^ (like Sj) is implicit in Sj and
whether ;4 and Gt mean the same or not, these questions are
ambiguous The answer is dependent upon what is legitimately
presupposed in ‘being impbcit m’ If we assume only logic and



PART IV. GENERAL SYNTAX


l86

mathematics, then the questions are to be answered in the negative ;
but if we assume the physical laws also, then they must be answered
in the affirmative. For instance, m the latter case S* and G, mean
the same to us even if we know nothing more about the described
volume of gas. The material difference between die two assumptions
corresponds to the formal difference between equipollence fin a
P-language) and I^quipoHence.

The view that the terms ‘analytic' and ‘contradictory* are purely
formal and that analjtic sentences have the null content has been ci-


self-evident”. Later, Wittgenstein made the same view the basis of
his whole plulosophy. “ It is the characteristic mark of logical een«
tences that one can perceive from the symbol alone that they are
true; and this fact contains tn itself the whole philosophy of logic*'
{[Traetatuslp. 156). Wirtgenatem continues: “And so jso it is one of
most important (acts that the truth ot falsehood of non-logical
sentences can not be recognized from the sentences alone.*' This
statement, expressive of Wittgenstein’s absolutist conception of
language, which leaves out the conventional factor m language*
construction, is not correct. It is certainly possible to recognize
from its form alone that a sentence is analytic; but only if the
tyntactical rules of the language are given. If these rules are given,
however, then the truth or falsity of cenain synthetic sentences—
namely, the determinate ones — can also be recognized from their
form alone. It is a matter of conventiem whether we formulate
only L*rules, or include p.niles as well; and the P-ruleS can be
formulated m just as strictly formal a way as the L*ru 1 es.


(6) VARIABLES

§53. Systems OF Levels; Predicates
AND Functors

By a system of levels in S,wc understand an ordered series
of non-empty classes of expressions which fulfil the six conditions
given on p. 188. Since the number of the expressions of a lan-
guage b, at the most, denumerably infinite, the number of the
classes of fRj b likewbe at the most denumerably infinite. These
classes we call levels ; let them be numbered with the finite— and,
if necessary, also with the transfinite— ordinal numbers (of the



§53 SYSTEMS OF LEVELS, PREDICATES AND FUNCTORS 187

second number-class) level 0 (or the zero level), level i, 2, w,
ftj + 1, We shall designate the expressions which belong to the
classes of Sl^ by ‘ Stu ’ [•5fti/e] , and, specifically, those which belong
to level a (where ‘ ct ’ designates an ordinal number) by * “Stu ’ [For
the sale of brevity, the phrase “ m relation to ’* is omitted here
and also m the case of the other defined words and the Gothic
designations which follow ] We count all the symbols Stu as
pnncipal symbols

An ordered senes of wi + i expressions 2I«+x (which

may also be empty) is called an ezpresslonal firaisework (^g)
lAusi^tuksgeruit ] — more precisely, an m termed expressional
framework (?Ig") — for a particular expressional form if there
exists at least one expression ^ of this form which can occur as a
partial expression m a sentence and is composed of the expressions
®i« of the framework, say 9I91, together with m principal

expressions Uli.Slj, 51,1,, in altemaung order Thus has the
form The expressions S'j, ^ are

called the fint, mth argument of ^gj in ^ , the senes which
they fonn (m the correct senal order) is called the m-termed
argomeot-series (^Ixg or, more precisely, SIrg") of 2Ig, m 21, 91,
IS also designated by ‘9Igi(9Ii. ?C)’» ^ senes of

thosearguments, by * 2(gi (9Irgi)’ 91, is called a full expression of
^9i ® that 9Ig” and 9Ig” have the same course of values if

every two hill expressions of 9lgj and SIg, containing the same
9hg are synonymous

The STg" for the form S are called m-termed sentential frame*
works (;g, <cg**) [5atr^«Ti«t] This is the most important kmd
of 9Ig A full expression of €g, is an €, it is called a full sentence
of Sgj Sg” IS called coextensive with Sg” if every two full sen-
tences of Sgi and Sg- contaimng the same 2lrg axe equipollent.

Theorem 53 I. If Sgi and Sg- have the same course of values,
then they are coeitensne, the converse is not always true (com-
pare, however. Theorem 65 4 A)

Let 9Ig” be composed of “Stui with or without subsidiary
symbols. Jet 2!, be the hill expression 91gi(2Irgi), let here every
argument, as well as 21, Itself, be either an S or a^Stu witb^<(r.
Then 21, is also called a full expression of StUi , 2Irgi is also called
an argument senes of StUg m 21,, Stu, is called (m 21,) m-
termed (Stu**), we then designate 21, also by ‘StUi(2hrg,'' If m



iSS PARTW. GESIKAL SYNTAX

this case is in Sg, and therefore SI, an G, then Ghij is called
a predicate-expression a sj*mbol Spr is called a

predicate (pr, pr", “pr). On the other hand, if %, is an Gtu, then
StUi is called a functor-expression (gu.tJu", *5'0l ® symbol 5u
is called a functor (fu, fu**,"fu). and which are isogenous
and thus of the same le\-el, arc called ncstenstrf if the corre-
sponding Gg are coextensive. We say that gUj and which
arc isogenous and thus of the same le\'el, have the tame ecurse of
values if the corresponding ?Ig haw the same course of values.
The ®Gtu arc called individual expressions and, as sj-mbols, in-
dividual sjmbols.

Theorem S3A (a) If fpii and are synonymous, theyare also
coextenshx. (i) If {jUi and git, are synonjmous, they hsTC the
same course of ^'alues. The converse of either is not alwaj'S true,
(Compare, howc^•cr, Theorem 66.1.)

SpTi and ^r, are sjmonjTnous oijy if every sentence G, is
equipollent to O'* other hand, they are coextensive if

merely for ettry full sentence G, the same condition is fulfilled.
It is possible for * P ' and * Q * to be coextensive but, for a particular
*pt ' M *, the sentences * M (P) * and *M (Q) * net to be equipollent, so
that *P’ and ‘Q* are not s>'non)mous. (In this case, *M(P)* is in-
tensional in relation to 'P'. See §66.)

Cofufiftbnr; (i) An Gtu is not an G. (i) If ?Ij is isogenous with
an “Gtu, then % also is an “Gtu. (3) Every "Gtu where b >0 is
either a or an gu. (4) For every ®Gtu„ there exists a with a
full sentence of which Gtu, is an argument. (5) Let GtU, be an
"etu where a is greater than 1, and which is therefore either a ipr
or an gu. (a) There exists a greatest ordinal number less than s,
say ^(so that B=j 5 + 1); then for that orgu Gtu, there exists a
full expression ?I, such that one of the arguments or ?!, itself is a

ctu. (b) There is no greatest ordinal number less than a (for
instance, where a^oi); then for ei-ery ^ which is less than a there
is a y such that ^<y<o, and a full expression ?I, for Gtu, such
that one of the arguments or % itself is a yGhi. (6) 9\, is as great
in extent as possible, that is to say, the class Gtu in relation to 91 ,
is not a propver sub-class of the class Gtu in relation to a scries
91 , which likewise fulfils conditions (i) to (5).— 91 , is called a
suitahle argument in general (or for the flh argument-place) for



§53 SYSTEMS OF LEVELS, DEDICATES AND FUNCTORS 189

® 9 n or G^ter also for Sflti orSIfUj), if there exists either a
full expression or a full sentence m which occurs at some argu-
ment-place (or at the rth place, respectively)

Examplei In Language II (as in all the usual languages with
higher functional calculus) there is exactly one system of levels To
this the 3 belong as *6tu, and also the $r and {ju Thetemis‘$r’
and ‘ gu’ which are defined here in general syntax are, however,
wider than those previously appbcd in Language II According to
thencw terms, thentr!nare*pi*,‘— 'isa’pr*,*'’ isa'fu* Further,
‘ = ’ IS a pi*, let It be prj, it is an "pt since for every mtegern{>0)
there exists a full sentence pt,(’pt, “pt) (e g ‘P=:Q') If we were
to specify that the symbol ‘C’ for the different types (Def 37 10)
should not be furnished with the corTe«ponding type index, but that
It should be used for all types of ^pt irrespectively, then ‘c’ would
also be an “pt* Under like conditions 'v’ m ‘FvG’ (Def 37 5)
would be an "fu*

In [Prme Math] Russell has used the symbol ‘ C’ and many
others with arguments of any (finite) level whatever, so that,
according to our definition, they belong to the level to Russell does
not, however, attnbute a transfinite level to these, but interprets


§ 54. Substitution ; Variables and Constants

What u a vanabU^ It has long been recognized that the old
answers “a varying magnitude” or ‘‘a varymg concept” are m-
adequate A concept, a magnitude, a number, a property — none of
these can vary (although a thing can, of course, have different pro-
perties at different tunes) A variable is, rather, a symbol with a
certain property But what property? The answer “a symbol
with a vaiymg meaning ” is equally madequate For a variation m
the meaomg of a symbol is not possible within one language, it
constitutes the transition from one language to another More
correct is another answer which is frequently given “A symbol
with a deteimmed meanmg is a constant, and one with an unde-
termmed meanmg is a varuble ” But even this is not quite cor-
rect. For It is possible to use constants which have undetermined
meanmgs, these differ essentially from the variables m that they
do not permit of substitution

Examplet In a name-language, in addiuon to names with de-
temuned meanings, such as ‘Prague’, names with undetermined



PART GENERAL SYNTAX


190

meanings, such as *a’ and ’b’, may also be used. If ‘Q’ is a constant
pt (whether of detentxined or undetermined meaning makes no dif-
ference), then from ‘Q(*)' the sentences 'Q (Prague)’, *Q(a)’,
‘Q(b)’ and so on are dens'able, but they are not deris'able from
*Q(a)’. This shows that while'*’ is a sTOablc, ‘s’, in spite of hiving
an undetermined meaning, is a constant. In material interpretation:
'a’ designates a certain thing; it is merely not stated for the moment
(but may, however, be stated later) what thmg it designates. In the
examples to be found in this book, constants tvith undetermined
mcamngs ha\x frequently beenused;foreiamp1e,‘a’,*b’onpp.t2f.,
'P' and ‘Q’ m many places, such as pp. 25 and 47. The difference
between the variable 'p ' and the constant of undetermined meaning
‘A’ is brought out espeaally clearly in the examples on p. 158.

Vanablts and mnstanU are dutin^isked from one anotJjer by their
syntactical character; variables are the sjTnbols of S for which,
according to the rules of transformation of S, under certain con-
ditions, ruiriituhon is permissible. This rough distihetion is true
for all the ordinary’ sjmboUc languages. The exact definition of
* variable however, cannot be $0 simple, inasmuch as it must take
into account the various possible kinds of substitution, and es-
pecially the three pnntipal kinds— substitution for free vambles,
for bound variables, and for constants.

W. V. Quine (in a verbal communication) has shown that it is
possible to use an operalor-eonstant instead of an opefator-s-ariable.
Instead of '(*)(*=*)' we can, for example, write '(0)(0s:0)’.
InadentsUy, wc can extend this method so that a language (even a
language which includes both arithmetic and infinitesimal calculus)
contains no variables at all. For instance, in Language 11 we may,
to begin with, construct a Language IF in which no free \-ariablea
appear in sentences. Here PSII 16 and 17 have to be replaced by
rules of substitution; (tij)(©i) may be transformed into
•nd (p,) (©,) into G, WI X drops out ; but several new

rules must be formulated. 11' is then constructed from IT by writing
instead of a bound Vi, in the operator and in the places of sulwtitution
in the operand, tome expression or other from the range of values of
0,. [In II', as opposed to the usual languages, related sjmboh are
always isogenous.] In the symbolic languages hitherto in use,
substitution for constants does not occur. Languages of the kind



§54 SUBSTITUTION, VAMABLES AND CONSTANTS I 9 I
‘a=a’ IS denrable from‘a:=x’,m II from *(*)(*=*)’, aod m II'
from‘(3)(3 = 3)’

We say that substitution occurs m S when there are expres-
sionsmS — we call them vatiable- cx p i 'c a sions (2?) — to which what
now follows 13 apphcable, and which, in particular, fulfil the con-
dition given below, p 195 [This condition can at that pomt be
formulated more simply with the help of the terms which will by
then have been defined ] [To facihtate the comprehension of
what we are about to say, it should be noted that m the ordinary
symbohc languages, all IQ are q^tnbols, and, specifically, variables ]
To every 5D, say JOj, there 13 correlated a class (which may also be
empty) of expressions which we call operators (Op), or, more pre-
cisely, operators with IDj (Opcj) Let Opj be an Opcj , then there
IS correlated to Opj a class of prmapal expressions which we call
tnbstitutioQ-vslues of IBi tn relaium to Opi , this class cootams at
least one expression which is not synonymous with SQj Further,
to 3)i Itself IS correlated a class of pnnapal expressions which we
call tuhtMuUon-valuet for /ree1D|,thiscl^, when it 1$ not empt^,
contains at least two expressions which are not synonymous with
one another Let 5li be that class to which belong all substitution-
values for free 93i and all substitution-values for STj in relauon to
some CpSj, together with all expressions which are isogenous with
one of the above We call the expressions of 51] the values of S3]
;Op]^ IS called ttn&rm/rd if every value of ^ is also a substitution-
value of 50] in relation to Op], otherwise, hmtUd

Let SI] be a full expression of and specifically either an S

or an €tu , and let S^ be constructed from SI] by replacing every
argument SIj (1 = i to m) by a Sj to the values of which SIj belongs,
SI] being so qualified that it can occur as a partial expression m
a sentence SIj is then called an m-teimedexpresslonal function
(SIfu, Slfu"), 83^ IS called the tth argument m S^ An Slfu" is
called improper when « = 0 , proper when m > 0 If SIj is here an
then S^ IS called an m termed sentential function (Sfu, Sfu”)
The Sfu constiTute the most important kmd of SIfu

The difference between tenlential framework, tententtal function, and
predicate-exprestum should be carefully noted, smce, owing to the
fact that the term ‘sentential function* u used m all three meanings,
this difference is often disregarded. Exampla of Sg m II (here we
separate the expressions of the expressional senes by dashes)



152 r.wn\ CENTRAL STXTAI

‘P(3,— )vQ(— )*, ‘Q(— )’ [but liso ’i-K) — (B>’ (-willi liit iipj-
tnent * v •) and ' { — x){P (i)) ' (» which ‘ 3 ’ and the null eipr&sian
are the suitable ai^rumentsjj; examples of Gfu: *P(3i»)vQ{x)’,
‘Q(x)'; examples of ‘Q* also *sm(P,Q)’— see p. S6].

The differences between the remaining 3 g, the remainim- ajn, and
the gu are analogous. The only reason why we m\Bt also deal with
the Sg and the Sg in addition to the Sfu and the Sfu is that it
cannot be generally assumed that there are in every language variable-
expressions for the aigomeirts concerned.

The ;fu* are S. In 1 and II, all the efu are ;, and, specifically,
ffie proper Sfu are open S, and the improper Sfu are dosed E.
In the majority of the usual symbohe languages, all the Efu are B;
m many of them, however, the rules which gosT i n this pemt are
not clear.


Let Cpi occur at a certain pbet in Sj; then, to this Cpj is
correlated by means of definite rules of formation (which. He
all rules of formation, are contained in the rules of trans-
formation; see above), a paruil expression 2Ifu of Sj consist-
ing of Opj, an EfUj. and sometitnes subsii^aiy symbols as
well; €fU| is called the operand of Cpj (at this place) in Si-
[Usually, SfU| here comes after ;!P); and sometinies the begin-
ning, the end, or both, of the operand efuj is indicated by means
of spedal subsidiary symbols (for example, by bracLeta in I and II
and by dot-symbob in Russell) as well as by Cpj.] We designate
SrfUjalsoby‘Cpi(cfUi)’. If Sfu, can be an operand belonging
to Op, — that is to say. if there exists an ?Ifu of the form Cp, (efu,)
— we call Sfu, operable in rdation to Cp,. SP, is called bound in
91, at a particular place if this particular place belongs to a pardal
expression of 91, which has the form Cpjr, («fu)l ;“4 speci-
fically, it is called luiutedly (or tmlimitedly) bound if Cp, is
limited (or unlinuted, respectively). If 2^ occurs in 91, at a
pbee at which S, is not bound, then ^ is called free at this place
in 91,. The places at which 5?, occurs freely in^ art called »5-
jn'ftif)on-pZaf« for 21, in 9^ We designate by ' 9IfUj

pression which results from 91fui on replacing 21, by 91, at ill sub-
stitution-places in 9Ifuj; here 91, must be a value of 21,, and there
must be no 21, which occurs freely in 91, and is bound in 9IfUj at
one of the substitution-places for 55,. [If 91, does not fulfil these


conditions, or if 21, docs not occur freelv in 9lfu„ then ‘9Ifu,


(^V

\%)



‘93


§54 SUBSTITUnO*!, VARIABLES AKD COVSTASTS I


designates 'fflfui' itself] We call a variant of 3lfUi

(m 3Ji) A sentence of the form is called a variant of


6fUi in relation to if SfUj is operable m relation to Opi and

a substitution value of IBi m relation to Opx
We distmgmsh between two different kinds of operators
sentential operators and descnptiona] operators If Opi (SfUi) is
an Gfu, say SfUj, then Cp, is called a sentential operator in SfUj ,
and if every expression of the form Opj (Sfu) is an Spt then
OPx IS called asentential operator Assume that Opi (6fUj) is not
an Gfu, and is hence another 9fu, say ^fUj . then SIfu j is called a
descnpttanal function, or, if it is closed, a description A description
13, accordmgly, always an Gtu Opi is then called a descnptional
operator m tKfu,, and if every expression of the form Opj (Sfu)
IS a descnptional function, Op^ is called a descnpuonal operator
Let Gi be CpitJt (‘2fUi), Opi is accordingly a sentential operator
m S, If, here, occurs freely in SfUj, and if every variant of
GfUj to relauoQ to OPi is a consequence of Gj, then OPi is called
a universal operator m Gi If OpiSj is a universal operatorin every
sentence of the fonn Opi (Gfuj), where GfUj is any Gfu in which
IDi occurs freely, then Cp^ is called a umversal operator


Let Bi occur freely m G( , then, if every variant j , where


91, IS any substitution value whatsoever of free 3J2> a conse-
quence of Gx, we say that m G, there exists substitution for free
Bx If in every sentence m which B, occurs freely, substitution for
free Bx exists, then we say that there exists (m S) substitution for
freeBj

For the foregoing definitions, beginning with 'S it is required
that the following condition be fulfilled namely, for every Bx
there is at least one Gi such that either there exists svhsUtutum for
free B, m or Gi has the form Opipj (Gfu,), where Bx occurs
freely m GfUj and Opi is a vmvenal operator m (Sj

9Ix IS called a luhstituiion-value of B, if at least one of the
^:^hmagcoadcQaascsfslS!ixl (c) Tbero cscaS stcAsCrfcrti o iT ;jr
free Bx, and is a substitution value for free Bx , (a) There exists
m S a universal operator Opij^ and Bx is a subsutution value in
relation to Cpi



PART IV. GENRR.^1. SYNTAX


194

If 2?i occurs freely in <3j, but if at the same time there exists no
substitution for free in Si, then we say that S5i is comUat in Si
(in the usual languages this does not occur). If Si is constant in
every sentence in which it occurs freely, and if at least one such
sentence exists, then we call Si eotatant. If Oi is a S and constant
(either in St or generally), then we call Oi (either in Si or gener-
ally) a tan'abU-consfanf, tf Qi is a S and constant in no sentence,
then Qj is called a variable (©). AUs)'mbo\s which areS, and hence
all D also, are counted amongst the principal symbols. If Oi is not
a D (and hence either not 2 S) at all or a S which is constant in at
least one sentence), Oj is called a constant (T). If Ii is an *Stu,
then Ii is called a constant of the level «{*f)-

Si is called open if there exists a Sj such that it occurs freely in
Si and there is substitution for free Si in Sil otherwise. Si is
called closed. An which is not an S is called open if there
exists a 5Bi and an Si such that 9i is a partial expression of S],
tSi occurs at a place in at which it is free tn Sj, and ihett
IS in Si substitution for free S,; otherwise, 21, is celled closed.
If no substitution for free S exists m S, then all 21 are closed;
S is then called a closed lanpsage^system.

Example of o eleifd lansuojr-^-Mm? U', p. ipo.

A languagt’S^-stem tnthaut venable-expTtstiortt can easily be
constructed ; obviously such a sj-stem is also a dosed sj'stem. An
example is afforded by Language Ij, which is constructed in the
following way as a proper conservative sub-language of I. Sjuibols
of Ik arc the I of I. The 3 («nd S) of fk are the 3 (“nd S, respec-
tively) without v of 1. As achernata of primitive sentences, PSI 1-3
remain unchanged, PSI 4-6 and n drop out, PSI 7-10 are replaced
by the foUosnng: 7. 3, = 3,. 8. {3i=3t)3(Si3G,[^^.
9. -(nu=3,«). *0. (3i' = a')3(3i=3*)- Of the rules, RI 2 and
3 remain unchanged; RI 1 and 4 drop out. The definitions are not
formulate as aentencea, but as syntactical rules concerned with
synonymity. All the definitions in 1 can be correspondingly trans-
ferred to Ik. For insttner. in place of D 3 (p. 59) the rule is
given: “ If fUi is ‘prod’, then for any 3t. fui(nu, 3») i* lynonymous
with nu. and for any 3, and X fu,(3i',3,) is tynonymous
with fiit(fui(3i.^,3i]. where fu,. is ‘sum’.” To a syntactica!
sentence concerning an open sentence of I, there corresponds a
syntactical sentence concerning sentences of Ik of a particular form.
For instance, the sentence: “ E\ery sentence of the form f u, (3i. 30
3i)» *here fui is ‘prod', ia demoiutrable in Ik” corre-



§54 SUBSTITUTION, VARIABLES AND CONSTANTS X95

spends to the sentence '“prod(* >)=prod(y,ar)’ is demonstrable
in 1 In this way, arithmetic can be formulated m Ik It must
nevertheless be noted that here S are only given up m It itself, for
the syntax language, on the other hand S are necessary in order to
formulate the priimtive sentences and rules as general stipulations

If IS closed and contains no €fu (and hence no S) as proper
parts, then we call Si an eUmentaiy sentence In an elementary
sentence, neither d, Cp, nor 95f (§ 57 ) occur

If 0 is an fa, then t)i is called a sentential variable (f) If all
substitution-values of Di (m Si or in general) are ^r, then Di is
called a predicate variable (p) (m Sx or m general, respectively),
if all substitution-values are ^r", then 0 i is called a p” The same
applies to the gu funetor-vanable (f, f") Let all the substitution-
values of S5i (in Si or m general) be *Shi Then SOi is called (in Si
or in general) an “35 (correspondingly “o, “p, “f) A "o is called an
individual variable, a “t an individual constant Let all the substitu-
tion-values of S5i (m Si or in general) be Stu, but of vanous levels ,
then !iDi is called (m Si or in general respectively) an '“’25 if for
every p<a there exists a y such that fiSY<a, so that at least one of
the substitution values (in St or m genera!) belongs to the level y,
but none to the level « or to a higher one [According to this, for
example, in the Sfu, 25ri (pi), Pi is an '"’p if and only if Spri is an
"ipr “25i IS not necessanly an Stu, ”55i is an StU (m Si or m
general) and, more precisely, an “Stu if and only if 25i occurs
freely (in S; or m at least one S respectively) An '“’ll) is not an
Stu

Examples i Languages I and II All 2J are v are the 3
Substitution-values for free 'x’ are the 3 substituUon values for
‘x’ m relauon to ‘( 3 *) 2 (P(*))’ «« the 3 which are synonymous
with ‘O’, ‘1’, or ‘2 Every p (or f) is an Gtu of a certain level,
values and substitution-values are all (or gu, respectively) of the
same type €fu are the @ Every € is operable in relation to every
operator Substitution for free o *P( 3 ) is a consequence of
‘P{x)’, for bound 0 'P( 3 )’ is a consequence of ‘(x) 5 (P(j;))’
Sentential operators are the universal and existential operators,
descnptional operators are the K-operators — z In Russell’s lan-
guage, there are descriptions which ate *5tu, and also descriptions
which are For instance, *:f(P(x))* is a class-expression and
thus a it IS a desenpUon with the descnptional operator

Correspondingly, 'HP’ is a descnptional operator for a



196


FART rV. GENERAL SYNTAX


§ 55. Universal and Existential Operators

We shall first discuss the subject in the material mode of speech.
Let a domain contain m objects, and a certain property be attributed
to each one of these objects by mcansof the sentences Sj.Sj, ...G„
respectively. Now if G« mearii at least as much as the sentences
Si to Sb, taken together, we may call S* a corresponding uni«rsal
sentence in the mder sense; and specifically, if ;„ does not mean
more than all the indiridual sentences put together — that b to say,
if it means exactly what they do — a proper universal sentence. If
the universal sentence is constructed with a universal operator,
then the dosed variants of the operand are the corresponding in-
dividual sentences. We therefore define as follows: a universal
operator Opi (restneted or unrestricted) b called a proper tori-
cersal operator if every dosed sentence of the form Opi (SfuOi
any Gfu; whatsoever, is a consequence of (and hence equipollent
to) the class of dosed variants of Sfu, in rdation to Op,;
othenvise it b called an improper umversel operator (namely, if
there exists a dosed sentence Op,(€fui) which b not a conse-
quence of the dass of the dosed variants of (5fU| in relation to
OPi).

An existential sentence follows from every one of the individual
corresponding sentences. Materially expressed, its meaning b con-
tained in the meaning of each of the individual corresponding sen-


define in the following manner.

Let S, be (GfuJ; Op, b accordingly a sentential operator
in 3,; if here SJ, occurs freely in Gfu, end if G, b a consequence
of every variant of Gfu, in relation to Op,, then Op, b called an
exbtentbl operator in G,. If Op,|,| b an existential operator in
every sentence of the form Op,(Gfii), where Gfu, b any Gfu in
whi(* 3)i ocrars freely, then Op, b called an existential operator.
[This definition is analogous to that of ‘universal operator’ on
p. 193 .] Let Op, bean exbtential operator. Ifthe content of each
closed sentence of the form Op,(Gfu,) coinddes with the pro-
duct of the contents of the dosed variants of Gfu, in relation to
Op,, then Op, is called t proper existential operator; otherwise it



197


§ 55 UNIVERSAL AND EXISTENTIAL OPERATORS
13 called an improper one (namely, if there exists a dosed sentence
OPi(®fUj) whose content is a proper sub-class of the product
of the contents of the closed variants of SfUj in relation to Opj)
Examples Universal operators occur in the languages of Frege,
Russell, Hilbert, Behmann, Godel, and Tarski (see § 33) , they have
m the majonty of cases the form (d) Existential operators also occur
in each of these languages, m those of Russell, Hilbert, and Beh-
mann, some are Simple (for example, formed either with ‘ 3 ' or with


also an existential operator ) In the languages mentioned, the simple
universal and existential operators are unlimited, but it is also
possible to construct limi ted operators [such as ‘(x)((x<3)D’
and ‘Qx)((x< 3)«’] In Languages I and II there are also limited
operators which are simple, that la to say, which contam no partial
sentence

In Languages I and II the universal operators with 3 are proper
universal operators For not only is every sentence — and hence
every clos^ sentence— of the form pri(3) > consequence of
(}i)Cp*i(Bi))> conversely, this umversal sentence is also a conse-
quence of the class of those closed sentences (by DC 2, p 38) and
therefore equipollent to it In the other languages which we have
mentioned, on the contrary, the same thing is not true for the uni-
versal operators with *0 or with 3 (unless Hilbert’s new rule is laid
down, seep 173), hence these t^rators are imprt^er

The universd and existential operators of higher levels — that is to
say, with p (or f) — are apparently improper in the majonty of


number properties which are definable m III but, on the other hand,
not to all the number properties which are indefinable in Hi (see
p 106) Then {Pi)(pti(Pi)) is contradictory, the class of all dosed
vanants of the operands, however, analytic, and hence this contra-
dictory sentence carmot be a consequence of it Further, on the
same hypothesis QPi)(''-pTi(pJ)»anaIytic, here all closed vanants
of the operand are contradictory, the content of the existential sen-
tence 13 null and the product of the contents of the vanants is the
total content , therefore the former is a proper sub-class of the latter

Let S 3 i occur at a certain pbee in Si, and let it be either free or
bound by Opi Let R, m the 6rst case be the class of substitution-
values of a free IBi , and m the second case the class of substitution-



PART nr. GENERAL SYNTAX


Values of ©1 in relation to Oft. Let Ri be subdivided into the
largest sub-classes (non-empty) of expressions sj-nonjTnous with
each other. We call the number of these sub-classes the vanaBj'/jfj’*
numhtT of ©1 at the place in question in Sj; in the case of a finite
(or infinite) number wc speak of fim'te or infinite variahility re-
spectively. We say that ©, at a certain pbee in Sj has ih^iVe
universality if ©i has infinite variability at that place, and is there
either free or bound by a universal operator.

Examples: **’ has in ‘( 3 a)s(I*(v))’ the \-ariability-numbcr 6;
in ‘P(x)’ and in ‘{i.)(P(a))’ it has both infinite variability and
infinite universality. In a sentential calculus of the usual form,
with only free (, and no constants |a, every sentence is either analj-tic
or contradictorj* Thus every 1 there has the vanability number a.
The same is true even when we introduce universal and existential
operators; the f are then unrestrictedly bound but have only finite
variability.

We call Sii a greatest definite cxpressional class if the following
condition* are fulfilled : (t) For every % of there is a sentence
which is capable of being sub-divided into expressions of of
which « one; (a) If Sj is detenrunate and capable of being sub-
divided into expressions of and if 3 | contains no expression
with infinite variability, then Sjis resoluble; (3) fljis not a proper
sub-class of an cxpressional class which likewise fulfils conditions
(r) and (2). We call the product Jl, of all greatest definite
classes of expressions of S the definite expression-class of S. Sj is
called definite if it is capable of being sub-divided into expressions
of Jljand contains no expression having infinite variability; other-
wise, indefinite. [The terms ‘definite’ and ‘indefinite’ hereby de-
fined are themselves indefinite. Before, in the syntax of I and ll,
we defined the terms ‘definite’ and ‘indefinite’ as definite terms;
such definitions cannot be formulated generally, but only spccifi*
cally, for particular languages — that is, if they are to express
approximately the meaning which is intended (cf. § 43). TTie terms

definite’ and ‘indefinite* as defined here will not be used in what
follows. If in general 8}7ita.x the word ‘definite’ or ‘indefinite*
occurs in relation to the sjTitax-languagc (as, for instance, on
p. 171), we may look upon l.anguage II (or some kindred lan-
guage) as the syntax-language and take the earlier definition of
‘definite’ (§ 15).]



§56 BANC; 19^

§56. Range

(Compare the addition at the end of § 57)

We have called complete if every sentence is dependent upon
A complete R leaves, as it were, no question open , every sen-
tence IS either affirmed or denied {though not, generally, by a
defimte method) If is contravalid, then is complete in a
tnvial sense every sentence is at the same tune affirmed and denied
We will call apremtst dost if ts complete but not contravalid,
and if there exists no complete class which is a proper sub-class
of ill

Theorem 56.T. (a) If S is inconsistent (§ 59), then there are no
premi5S<Iasses in S (&) If S is consistent and logical, then the
empty sentential class is the oiJy premiss-class (c) If S is de-
scnptive (and therefore consistent), then every premiss class is
both non-empty and indeterminate, and every one of its sentences
is indeterminate

Theorem 56,2. Two non-equipoUent premiss classes are always
incompatible with one another

In material interpretation, every non-empty premiss class re
presents one of the possible states of the object-domam with which
S IS concerned il, is called a premiss class of ilg— in the sense of
a correlate of ‘consequence-class’ — if il| is a premiss-class and ilg
a consequence-class of il| That ilj is a premiss class of 3 ^ means,
in matenal intejpretauon, that ilg is one of the possible cases in
which Sg IS true By a range weunderstand a class OTg of premiss-
classes such that each class which is equipollent to a premiss-class
belongmg to OTg, belongs also toTOg By the range of Jig we under-
stand the class of premiss classes of Jig That URg is the range of Sg
means, m material interpretation, that 2Jlg is the class of all possible
cases m which 3 i is true , in other words, it is the domain of possi-
bilities left open by S,

Herein lies the reason for the choice of the tenn ‘ranee’ (‘Slnel-


By the total range we understand the class of all premiss-classes
The terms ‘range’ and ‘content’ to some extent exhibit a



200


PART IV. GENERAL SYNTAX


duality, as is shown, for example, by the following theorems
(3 to 6) which are analogous to theorems 49.1, 2, 4 and 5.*

Theorem 56.3. If 5I]isaconscquence-das3ofII|,therangeofiI|
is contained in that of 51..

Theorem 56.4. If 5Ii and 51* are consequence-classes of one
another, they have the same range.

Theorem 56.5. If is valid, the range of 5Ii is the total range.

Theorem 56.6. If 5Ii is the range of 5Ii is null.

Theorems 3 to 6 hold correspondingly for Sj and Sj.

Theorem 56.7. (a) The range of 5Ii + 5Ij is the product of the
ranges of 51, and 51j. (A) The range of 51, is the product of the ranges
of the individual sentences of 51,.

By the supplementary range of 51,, we understand the class of
premiss-classes which are not premiss-classes of 51,. The supple-
mentary range of 51, is always also a range; but it is not alnaj-s the
range of a 51. If the supplementary range of 51i is the range of 51^
then we call 51, a contra-elast to 51,. Correspondingly, S, is called
a tontra-senience to 3, if {3,} is a contra-class to {3,}. If 3, «»
a conira-sentence to 3,. then S, is likewise a contra-sentence to
<3*. If S, is a contra-sentence to 3„ then, in material interpreta-
tion, S, is true in all the possible cases in which 3, is false — and
onlv in these; thus, 3, means the oppotiVe of 3,. If, in S, there is
no negation, then, as a substitute for 3„ we can take a contra-
sentence to S„ or a contra-class to {S,}. In case neither exists,
then there is no substitute for -'>'3,, but there is a substitute for
the range of <^3,, namely, the supplementary range of 3,, there
being always exactly one such range. — ^The terms ‘ range ' and 'sup-
plementary range’ will make it possible for us to characterize the
individual sentential junctions.


§57. Seoto^tial Junctions

If there is a full sentence 3, of in which all n arguments
are 3, then called an n-termed sentential junction in 3,.

• (Aofe. I93S.) It is, howe\-cr, to be noted that the converses of
iheorems 3-6 do not generally hold; this fact has been pointed out
1° .TJ ascertain the exact situation a further de-

‘ . ' ; lar, it would be worth

_ ;* ; ' • ’range’ which secures



201


§ 57 SENTENTIAL JUNCTIONS
If S3"> With, n arbitrary sentences as ailments, constitutes a full
sentence, then 6g" is called an n termed sentential junction
(33f, 2}f") If IS composed of and possibly subsidiary
symbob as well, is called an « termed sentential predicate-
expression, if Q] is a sentential predicate-expression, Oi is called
a sentential predicate, or a junedon-syinbol (i)I,dI'*) A of is,
accordipgly, a *pr" to which sentences are suitable as arguments

In order to prepare for the definitions of particular kmds of junc-
tion, we will proceed in a way that is dependent upon the method
of the ralue-tables (see § 5), but without assuoung that S contains
3 negation Let us consider a value table for, say, three members.
Si, Sj, and S, The second row runs ‘TTF’, and to the case
designated by this row corresponds the sentence Si»St»~S3
Let S, be any junction sentence SJIi(Si, Si,Ss) For this the
column m the value-table may be stated, m the second row it is
occupied either by ‘T’ or by ‘F’ ‘T* would mean that S4 was
true m the second case, and that, accordingly, S4 was a conse
quence of Sj » Sj* — Sj , ‘ F’ would mean that ®4 was false in the
second case, and that therefore <*>’(54 was a consequence of
W« want now to express these relations without
making use of negation, and this u possible with the help of the
ranges IV e will (in this section only) designate the range of @1 by
‘[SJ* and the supplementary range of Sj by ‘ — [SJ* Si»<34
has the same content, and thus tbesame range, as Hence,

according to Theorem 5676, [S|»< 3 J is the product of [SJ
and [SjJ VVe replace the range of by — hence we
replace by the product of the classes [SJ,

[SJ, — [S3] That 134 (or is a consequence of this con-

junction 13 (accordmg to Theorem 56 3) expressed by the fact that
[Si^Si^-^SJ IS contamed m [S4] (or m — [64], respectively)
On the basis of the foregoing conclusions we can now state the
following definitions

LetSj,S3, S, be n closed sentences VVe construct (accordmg
to the rows of the value tables) the m(=2") possible senes
9^1.9^*, 9 l„ of ft ranges each, where the ith (t= i to n) range is

either [SJ or — [SJ The suffixes of the 91 may be determined
according to a sort of lexicographical arrangement of the ranges
if 91 * and Sfij agree m the first i-x serial terms (ranges), while the
Ith term of 9 J* is [ 3 J and of 9 i| o — [SJ, then 9 l» must precede



202


PART TV. GENERAL SYNTAX


SRi, that is to say, A must be less than /. We trill now construct a
senes of m ranges, 3Ki to 31l« (which likewise correspond to
the rows of the table, namely, to the conjunctions), in such a
way that, for everj' k (k=i to m), SDlt is the product of the
ranges of the senes Kj. If, for a certain S3fj and a certam k
(i;ksm) and n arbitrary closed sentences Sj, ... the class SRj,
constructed for Sj, ... S* in the way already described, is always
a sub-class of [2.'li(Sj,...S,)], we say that the Mh characteristic
letter for Sfj is ‘ T If, on the other hand, for any closed Sj, ... ;«,
911jc IS alwaj-s a sub-class of — [3f,(Sj,... S,)], we say that the
Ath charactenstic letter for SI, is ‘ F *. If neither of the two con-
ditions IS fulfilled, then does not possess any Ath character-
istic letter. If 3?fi possesses a characteristic letter for every
* (A « I to m), we call the series of these m letters the characteristic
of 2.II1 — Let Sj, ... be n dosed sentences of any kind ; SHj.—SlIa
the ranges which are constructed from these in the manner stated.
Then every premiss-class of S belongs to exactly one of these
dosses 3)?. For any S51J which possesses a characteristic,
[Sli («|, ... S,)] is the sum of those 3)1, for which the Ath cha-
racteristic letter is 'T'. — For the 231" there are 2** possible cha-
racteristics.

With the help of the chancteristic we arc now in a position to
define the various special kinds of junctions 5 we will restrict our-
selves here to the most important of these. Wc call a SF with the
characteristic ‘FT’ a proper negation, and a SI* with thcchanc-
terisiic "nTF’ (or ’TFFF’, ‘TFTT', ‘TFFT*, ‘FTTF’) a
proper disjunctloa (or conjunction, implicotion, equivalence,
exdusive disjunction, respectively).

If for every S„ Sl|(S,) is incompatible with 0,. then SIi »
ca ed a ftegalton. SIi is called a disjunction if for any G, and Gj
w alsoevcr, Sf,(>c,,G.) is always a consequence of Gj and a
rons^quenec of S.. SI5 is called a conjunction if. for any G, and
v=i. and Gj are alwajs consequences of Srj(G,.G.). Si! «*
called in implication if, for any e, and G.. G. is alvvajs a conse-
quence 0 (ci.SI^fd.Ss)). If a junction of these kinds is not a
proper one, we call it i«proprr. If for one of the junctions men-
tioned there exists a junaion-svinbol. w e call it a symbol of nega-
tion (proper or improper) or a sj-mbol of disjunction, etc., re-
spectively. •'



§ 57 SZSTENTTAI. JUVCTIONS 203

Ttcorem 57 . 1 . If 23fi is a negation, then for any Sj every sen-
tence IS a consequence of — The class here mentioned

13 contravalid-

Theorem 57 ^. If S3fi is a proper negation, Sfj a proper dis-
junction, and SJfa a proper conjunction, then for any Si the fol
lotnng 13 true (a) If Si is dosed, ®fi{Si) is a contra-sentence to
Si. (i)*. (0 33fs(Sfi(Si).S,) ts valid, (d) Sfs(Sfi(Si),Si) is
contravahd. Accordmg to (c) and (<f), the prmapUs of traditional
logic such as those of excluded middle and of contradiction are vahd
in every language S for the proper junctions, if such occur m S
Theorem 57 3 . If 231, has a characteristic, and if is co-
extensive with SI,, then 25^ has the same characteristic.

Examples The junctions which are designated as ‘negaSon’ are,
m the majonty of systems (for instance, in those of Frege, Russell,
and Hilbert, and ta our own Languages I and II) negations m the
sense here defined In I, ' ~’Pnm(x)' is not a contia-sentence 'o
*Pr im (a)*, both sentences are contradictory, and their range is thus
nuIL lo spite of this, ‘ — ‘*«'Pnm(a)* is equipollent and equal in
range to ‘Pnm(x)’ I; ‘Q’ is an undefined pi$, then in II there
e asts a conaa-sentence to *Q( 4 r)’, namely ‘ ''>'(x)(Q(x))* In I, on
the other hand, there ts neither a c on tra sentence nor a contra-class
to ‘ Q (x) % but there is a supplementary range

In the systems of Russell and of Hilbert and m our own i^anguages
I and n, * V’ IS a rymbol of proper disptnciion In Hilbert’s system,
the junction which consists of three null expressions is also a proper
disjunctioa (S, is equipoUent lo S, V ;j) In the English lan-
guage the connects es ‘either or’ (21, is empty) — as also ‘aut
aut’ m the Latin language — consutute a proper exclusive disjunc-
noQ. Hilbert’s symbol *&’ and the of 1 and II and Russell s
system (and in the latter also the many-pomt-symbols) are symbols
of proper ccT^smction In Russell and in I and II, ‘3’ is a symbol
of proper vrplication as is also Hilbert’s * -; ’

In the systems of RusseD and Hilbert and Languages I and II, all
d! have a characteristic , but in those of Heytmg and Lewis of with-
out a ch a r acte ri stic also occur For instance, Heyting’s symbol of
negaUon (we will here write it thus *— *) is a symbol of improper
tKgarwft, without a charactensdc €,and — S, are certainly alwavs
incompatible with one another, but — -S, is not alw a y s a contra-
sentence to <S, ;, and — S, possess the common consequence

S,v — which IS in most cases tMt valid but indetemunate

S, IS not generally equipoDeni to S, In Lewis’s system, the

symbol of Strict imphcatioR m a symbol of improper implication.


• Omitted m this cdiuon.



PART IV. GENERAL SYNTAX


204

Without a chararteristic (see § 69), (Concerning the intensiondity
of the ol which have no characteristic, see § 65.)

Let OpiBj be a universal operator, and an existential

operator-, let the substitution-values of S)i be the same in relation
to Opi as they arc in relation to Dp^; and let SBIi be a ne^afitm. We
callOpi, Op2, and ^tiossociated, if for every SfUj which is operable
in relation to Opi and Op*. S 5 fi( 0 pi(SfUj)) is equipollent to
Opj (Sfj (SfUi)). If both the operators as well as the negation are
proper, then they are also associated.

Example: In II, (pj) and (3 p,) are certainly improper; but these
operators and are associated, since ~(Pi)C®fUi) ** always
equipollent to (3 Pi) Gfu,).


(Addition, 1935.) Since the concept of ‘range ’as defined above
does not always fulfil the requirement of duality (see footnote,
p. 200), the definitions of the sentential junctions based upon
this concept are not always in accordance with the usual meanings
of the junctions as laid down by thetnith-value-tables. Dr. Tarski
has found simpler definitions of the sentential junctions which do
not make use of the term 'range'. It is possible to proceed, for
instance, as follows. We say that the relationship of negation sub-
sists between and if 51, and il, are incompatible and haw
Mclusive contents, stands In the relationship of disjunction to
ft, and ft, if the content of ft, is the product of the contents of ft,
and ft, (compare Theorem 34^.8). ft, sunds in the relationship
of conjunction to ft, and ft, if ft, h equipollent to fti+ft,. ft,
stands in the relationship of implication to ft, and ft, if the fol-
lowng ^o conditions are fulfilled: 1. ft, is a consequence-class
° 1 + ft, 5 2. if ft, is smaller in content than ft„ then 5 U is not

a consequence-class of ft, + 51 , (compare §6j, paragraph i). ft,
stands in the relationship of equivalence to ft, and ft, if ft, stands
n “‘^bonship of implication both to ft, and S^, and to ft,
r <n»i fi- ^ * proper negation if, for every closed

stands m the relationship of negation to S,. We call

alwaj™ stands in the relationship of disjunction to S, and 6,. The
remaining junctions arc to be timilarly defined.



§ 58 ARrnDrimc


205


(c) ARITHMETIC, NON CONTRADICTORINESS,
THE ANTINOMIES

§58. Arithmetic

Let ^ be an “3tu, (JUj an and Sx the infinite senes of

expressions eonstmcled in the foDofmog manner the first teim is
%), and for every « the (« + i)th term is the full expression of gUi
with the nth term as argument. 31, has accordingly the form 31o ,
tjUxC^o). (3Ui(2to)). 5«i(^) If every two dif-

ferent expressions of 31, are isogenous (hence each one an “Stu)
but not synonymous, we call 91, a manerual expresnm smes or
3-5enes The expressions of S, and those synonymous with them
are called numerical expressions (3) of 5R, Those 3 which are
synonymous with 3I|, are called null-expressions, or 0-3, of 91, ,
those which are synonymous with Stti(%) are called 1-3 of SRj,
etc. A 3 which is synonymous with 5u(3i) » called a tuecessor-
expremm of 3i [These and the foUowug terms are always re-
lated to a particular 3*scne3 91,, for the sale of brevity, the phrase
"of 91, " or *'m relation to 91,” will usually be omitt^]

If 0, is a 3> tt IS called a numeral (^) If o, is a 0-3, it is called
a zero-symbol (rm) SI, is called a numerical 31 if the 3 belong to the
substitution-values of Si If t), is a numencal 31, then n, is called
a mmerKal variable (3)

If, for Sg" (or 33r^), there exists a fall sentence with only 3 ss
arguments, ^en Sg, (or ts called a numencal Sg (or 93t)
If for there exists a fall expression such that thia expression
Itself and all the arguments are 3» 'hen is called a numencal
5u If pr, (or fuj IS a numencal ^r or gu, then pr, (or fuj is
called a numerical prcdlcate(3pi) (or anomericalfunctor (sfu),
respectively).

^1 (or “ called a juw-3g (or -^h, respectively) for the
Wi place (k = i, 2 , or 3) if for any m and n whatsoever, the fol-
lowmg IS true if 3i » an ffi-3, and 3» an a-3» ihen the full sen-
tence of 0g, (or of 35rj, respectively) m which 33 « the Mh a^-
ment and 3i and 3j the two other arguments, is valid when and
only when 3s is an (m +n)-3 S'** “ called a rwn-lju provided
that (IfUi IS a numencal Qu and the following is true for any m and
n if 3, u an m.3 and 3, an «-3, then 5ui(3i, 30 IS an (in+n)-3



206


PART IV. GENERAL SYNTAX


;Pro«/mrr-;fl’, ‘-Su’ arc analogously defined, where 33 .

or t5Ui(3i, 3 j) respectively, is an (in.«)>3. If gpti is a sum-lpc
(or product -^t), gpr^ is called a sum-predicate (or product-predicate,
respectively). If gfUi is a sum*gu (or product-gu), is called a
sum-functor (or produel-functor, respectively). It will readily be
seen that in a simibr way all the other arithmetical terms which
occur m the arithmetic contained in S can be syntactically charac-
terized; that IS to say, those kinds of © 9 , ^r, or iJu to which a
particular arithmetical meaning belongs can be defined. \Ve shall
content ourselves here with the foregoing examples.

Wc say that S contains an arithmetic if, in S, there U at least
one 3*seric3 9li, one sum-0g, and one product-S9. relation to
31^. Let S contain an anthmetic in relation to 9lj, If an ©j and a
fQ, exist such that for every 3 of ^1 there is a synonymous sub-
stitution-value of 2)1 in ©, and that Sj m ©j has infinite univer-
sality, then we say that S contains a general arithmetic (in relation
to %).

3 i 3s S'fs called eorretpon^mg 3 ” exists

such that 3i *8 an «-3 of IR,, and 3s '* an ”*3 of
Jlj may belong to difTerent levels, and even to difTcrent languages.
We say that two numerical ©g" (or two numerical fpr") (in one or
two languages) have a corresponding extent if every two fuH sen-
tences of them with corresponding 3 as arguments arc either both
valid or both conttavalid.

If S contains an arithmetic, then it certainly contains expres-
sions which can be interpreted as designations of real numbers,
namely the numerical Gg'; and further, it may contain numerical
iPr* and numerical of which the full expressions are 3
§ 39). We will call Slj a 51 for real numbers if there are infinitely
many numerical Ipr* (or numerical Qu' of the kind mentioned)
which belong to the substitution-values of Sj. If 2lj is a 21 for real
numbers and if 21 , in Gj has infinite universality, then we call ©1
a universal sentence coneersirng real numbers. The arithmetical
equality between two real numbers that are represented by two
©g* (or two Ipri) in relation to the same 3 '*<rics 91i finds its
syntactical expression in the cocxtetisivcncss of the two Gg (or
2Jr, respectively), or In the case of gu in the equality of the course
of values. If, however, it is a question of different 9{j and 91, which
may also belong to different languages, the arithmetical equality



207


§ 5^ ARiTiDirnc

will be represented by correspondence in extent. In tbis way, real
numbers of various languages can be compared with one another,
an expression can be characterized as being the expression of a
particular real number {for example ‘jr-expression in relation to
31i’) We can easily see how it may be syntactically determined
whether a dyyrrrnftj/ and integral calculus a theory of funetums
of more or less wide extent is contamed m S We shall not go any
further mto this question here

Examples r Language I The following senes are examples of

a-senes *0’, ‘O' *, *0H’, . ‘O’, ‘0»’. , .51, ‘3’.

‘3'’. . 51, *0. ‘nf(0)’. ‘nf(nf(0))’, , % ‘y. ‘fakfs)’,

‘fak(fak(3))’. The fu of I are jfu in relabon to each of these
senes , and moreover ‘ ’ is also a jfu m relation to each of these
senes, and specifically it is the senes-fonnmg jfu m 51, ‘sum’ is a
sum fu, ‘prod’ a product-fu. Language I contains a genera] anth-
metic inasmuch as there are sentences with free 3 in it Real num-
bers can be represented m Language I by means of pt‘ or fu', there
ta, however, no 31 for real numbers and no 3jt for teal-number
arguments

z Language ll (see § 39) Here also, the aforesaid senes
51x> 51, are 3*sen«a, but there are also others of quite different

kinds The *pr can be used as pi of real numbers Since there are
'p, 'p, and *f banng infiniCe umveisalitT, there are consequently
universal sentences concenung real numbers and functions of re^
numbers, etc.


§ 59 . The Non-Contradictoriness and
Completeness of a Language

S 13 called contradictory (ord inconsistent) if ^ery sentence of
S IS democstiable , otherwise, non-contradictory (or d-consist-
cnt) [It IS to be noted that the term ’ contradictory ’ when applied
to sentences (German kontradskionseh) is an L-c term (see § 52),
but when apphed to languages (German tctderspnchsToH) is a
d-term and not an L-tena- ] The foUowmg c tenns correspond
to these d-tcrms S is called inconsistent if every sentence of S is
vahd , otherwise, consistent. If theL-sub-Ianguage of S is contra-
dictory (or non-contradictory, incoosistent or consistent), then S
13 called L-contradictory (or L-non-contradictorj, etc., re-
spectively) The relations between the defined d-, c-, and L-terms
are mdicated by the arrows shown m the table on p 210

Theorem 59J. If 5 i3contradictory(ormconsistent), thenevery



20S


PART IV. GENERAL Sl’NTAX


and every G is, at the same time, both demonstrable and re-
futable (or valid and contravalid, respecti\-ely) ; there are no
irresoluble (or indeterminate) 51 or G.

Theorem 59.2. If, in S, there is a 5^ or an S which is either
non-demonstrable (or non-valid) or non-refutable (or non-
contravalid, respectiiely), then S is non-contradictorj* (or con-
sistent, respectively). By Theorem 1.

Theorem 59.3. If, in S, there is a 51 or an S which is at the same
time both demonstrable and refutable (or valid and contravalid),
then S is contradictory (or inconsistent, respectively); and con-
versely.

Theorem 59^. If S contains the ordinary sentential calculus
(with the negation ‘ '»• ‘) then in S every sentence is dcrit'able from
Gj and '—Gi- In I and II this isarriitd at ivith the help of PSI 1
and PSII r, respectively.

Theorem 59-S* If S contains the ordinary sentential calculus,
then S is contradictory when and only when an Gi exists such tbit
Sj and — G, are demonstrable in S. By Theorem 4.

Theorem 59*& If S contains a negabon SlTj, then S is incon-
sistent when and only when an G| exists such that and S3Ii(Si)
are valid in S. By TTieorem 57.1.

The definitions of 'contradictory* and *non-contradictor)*'
correspond (as Theorem 5 shows) to the ordinary use of language
without, however, negation being assumed. (Sec Tarsii
[Methodolosii] I, p. 27 f., and Post [Introduetion].)

A non-contndictorj’ language may nevertheless be inconsistenL
For although it contains no d-contradiction, it may still contain
a c-contradiction, that is to say, a contradiction which depends
upon the c-rules only. This is the reason for introducing the
narrower term ‘consistent’, which applies only to languages that
contain no contradictions of any sort.

Theorem 59.7. If S is inconsistent or contradictory, then it b
true that: (a) ei-ery two sentences of S are equipollent; (i) ei-ery
two expressions of S which are isogenous are synonymous.

Theorem 59-S. If S is inconsbteni or contradictory, then Scon-
tains no 3-seriea and therefore no arithmetic. — By Theorem 7^5
different terms of a 3-*etie3 arc not sj-nonj-mous.


E^omf>teofanon~amtTadietorybutinnmsutfntlanpuare, Let S(U|
be [(*>0).G->0).(T>0).(«>a)Jo(;^+^,.:^s.).. Let S. be



§59 ^0'^-C0NTRADICr0IUNESS AND CX3MPLETENISS 209

()(€fui), in Other words, Fermat’s theorem. Let every closed
logic^ variant of SfUj be dem<Histiable m S (hence, for every indi-
vidual set of four positive integers Fermat s property can be demon-
strated) Let Gj Itself be analytic but non-demonstrable, 1 e let S
con tain an indefimtc rule analogous to DC 2 (p 38) by which
IS a direct consequence of the class of those vanants Further, let
the sentence ~ Gx (although possibly contradictory m classical
mathematics) be demonstrable in S (for instance, laid down as a
primitive sentence, other sentences such as are d incompauble with
It being cancelled) Then S is inconsistent (and moreover L-incon-
sistent) At the same time, however, S may be non-contradictory,
smce Gi and ~ G^ are not both demomoable There is mdeed no
d-contradiction here but there is a c-<ontradicnon — namely, that
between the class of those vanants and ~ Gi This c contradiction
IS evident in the ordinary material mteipretation the demonstrable
sentence ~ G^ means that not all sets of four positive integers have
Fermat s property while for every such set a demonstrable vanant
occurs, which means that this tpiadruple has Fermat s property
But the c-contradiction, the inconsistency, is also purely formal
without any reference to material interpretation the class which
consists of those vanants and —6} contams only demonstrable
sentences but is nevertheless coooadictory, that is to say, every sen-
tence is a consequence of it , hence every sentence of S is at the same
time both analytic and contradictory

For such languages ss have no other 9 than the 3 our term ’con-
sistent’ corresponds to CMets tetm IVnenUehetJbare] p 187, *«>-
non-contradictory’, see also Tarski {Wuiertpruehsfr )

The language S is called complete (or d-complete) if the sen-
tential null-class (and hence, accordmg to Theorem 48 8, every Jl)
IS complete (or d-completc, respectrs'ely) , otherwise, mcompkte
(or d incomplete) The language S is called determinate (or
resoluble) if every R (and hence every S also) is determmate (or
resoluble, respectively) in S, otherwise, indeterminate (or ure-
soluble) The corresponding L-terms {* L-complete ’ and so on) are
only attributed to the language S when the ongmal term ( com-
plete ’, etc.) IS attributable to the L-sub language of S

Theorem 59 9. If S is complete, then it is detenmnate, and
conversely By Theorem 48 5

Theorem 59 10. If S is complete, then it is logical, and con
versely By Theorem 50 J a

Theorem te. If S ts co tnpJrte , c&en rc ts Z,-cqinpfetg, sad
conversely By Theorems xo and 51 1
Theorem 59 12. (a) The terms ‘complete language’, ‘L-com-
plete language’, ‘determinate langiuage’, ‘logical language’ co-



TART IV. dNnUl. SYJfT.«


sto

iodde. (A) The terms ‘incomplete Iiagtiage*, 'L-incomplete lia*
guage’, ‘indeterminate language’, 'descxiptitT language’ coindde.

Theorem 59.13. If S is d-complcte, then it is resoluble; and
conversely. By Theorem 4S.5.

For the d-tenns, no a’alid theorems arralogous to Theorems ii
and ta exist

Theorem 59.14. (a) If S is contradietorj*, then S is both d*
complete and complete. (A) If S is inconsistent, then S is com-
plete. By Theorem i.

How the properties of languages here defined are tiansfenrd
from one language to another can be seen from the table on
p. 225 (D). The relation of the terms to one another is indicated
by the arrows in the table below (as on p. 1S3).


Propfrtift of loji^a^es

L-d-terms; d-terms: tsterms: L-e-trrtns;

L><ontr»<ijctory -* cotitndictofy -• loconuittnt *• I..5neoa*istent

non-conti*. ,

H.einrv —conusieat -• L-conurttot


non'<Qnti*'

ieiory *“ dtcwr>‘

L.d-cmBpUt< \ / d<ompt«w
L-rrtdluble j resoluble


feomrlrtt \ /L-eomplrte
1 detrmuntte r**\L-d«tiTnSn»te

(leptal J

( iBcomplete "I f Iriaeomplete

lodeier- [ ** | (Hadetermiiuw)

minaie I {, tratbedc


L-d.incomnletel (6-
L.irre«olub1e j ^\ir


f iBcomplete \ / 1/

ladetrr. | (I

minaie {,

_de»tnpti\-e J


We shall see that everj’ consistent language which contains a
general arithmetic is irrcsoluble. Only ptxircr languages art re-
soluble. for example, the scnlenlial calculus. A richer language,
though not resoluble, can yet be determinate and complete, pro-
tided that sufEdent indefinite rules of transformation are laid down.
This is the C3<e, for instance, trith the logical sub-languages of I
and II. Forsuchan irrrro/aWeAvIfomp/err language, the following
classification of sentences holds; it is at the same time the classi-
fication of the logical sentences of any irrcsoluble language what-
soever (forihe classificationofthedescriptive sentences, seep. I Sf):
(d-terms:) demonstrable irresoluble refutable


contravalid

eontradUiory



6oa THE ANTDTOMIES


211


§ 6oa The Antinomies

In investigating the non-contradictonness of a language, the
first thing to be asked is whether the familiar so>caiIed anti-
nomies or paradoxes which appeared in earher systems of logic and
of the Theory of Aggregates have defimtely been elimmated This
pomt IS an especially cntical one when we are concerned with a
language which is nch enough to formulate, to any extent, its own
syntax, whether m an anthmetized form or with the help of special
syntactical designations The syntactical sentences may sometimes
speak about themselves, and the question arises whether this re-
flcxivencss may not possibly lead to contradictions This question
IS significant because it is not concerned with calculi of a specially
constructed kmd but with all systems whatsoever which contam
anthmctic. We shall now mvestigate this question and m doing so
we shall avail ounelves of the results obtained by Godel

e shall follow Ramsey’s example in dividing the antinomies
into two kinds, and we shall see that those of the second kind are
the ones which come into consideration for our mquiry These will
therefore be exammed more closely In the examples we propose
to use partly the word language and partly a symbolism similar to
that which was used m Language II, for the syntactical designs
tions we shall employ in some cases Gothic symbols, and in others
inclusion m mverted co mmas Let us consider, to begm with, the
following two antmomies

1 Russell’s antinomy [/Vmc AfalA I] , [MatA Phil ] We de-
fine as follows a property is called tmpredieable when it does not
apply to Itself Expressed in symbols *‘Impr(F)~<— »F(f)” If
m this case we substitute ‘Impr’ itself for ‘<F’, we get the contra-
dictory sentence “Impr(Impr) = <'«'Impr(Impr)”

2 Grelluig*5 antinomy Defimtion in alanguagewhichcon
tains Its own sjmtax, a s^mtacUcal predicate (for example, an
adjective) is called futerological if the sentence which ascribes the
property expressed by the predicate to the predicate itself is false
If, for instance, ‘ Q ’ is a syntactical predicate, then " Het { Q *) =
'^Q(‘Q’)” IS true (The fundamental difference between this
antmomy and the foregomg, which is disregarded m many pre-
sentations, 13 to be noted, namely, that here the property Q is
attributed, not to the property Q but to the predicate, i e the



212


TART IV. GENTRAl. S\'NT.\X


symbol ' Q ’.) Examplf: the adjective 'monosyllabic ’ is heterological
because ‘monosjUabic* is not monosyllabic but penta-sjUabic.
Now, if instead of the predicate ‘Q*, uc take the predicate *Het’
itself which has just been defined, we get, from the definition as
stated, the contradictory sentence “Hct(‘Het’)e'— 'HetCHet’)”.

In order to a\'oid antinomies in his language, Russell set up a
complicated rule of tj-pes, which, particularly in the theory of real
numbers, gaAe rise to certain difficulties, to overcome which be
found It necessary to state a special axiom, the so-called Axiom of
Reduabihty. Ramsey {[Foundations] Treatise l, 1925) has shown
that the same object may be attained by a far simpler method. He
discovered, namely, that it is possible to differentiate between two
kinds of antinomies which may be designated as logical (in the
narrower sense) and Simtactical (the latter are also called linguistic,
epistemological, or semantic). Example (i) belongs to the first
category’ and (a) to the second. Following Peano, Ramsey pointed
out that the antinomies of the second kind do not appear directly
in the sj-mbohe sjstem of logic, but only in the occompanj-ing
text; for they are concerned with the expressions. From this fact
he drew the pnctical conclusion that in the construction of a
sjTnboIic system it is not necessary to tale note of these syntactial
antinomies. Now since the antinomies of the first kind are already
eliminated by the so-called simple rule of types, this is sufficient;
the branched rule of tj’pcs and the axiom of reducibUity wluch it
necessitates arc superfluous.

On the basis of the simplr rule of typot (as in II for instance) the
t>T>e of a predicate is detennined by the type of the appertaining
a^ments alone. On the basis of RusseU's branched rule of type*,
the form of the chain of definiaona of a predicate is also a factor
m detefrnining its tjTie (for irutance, whether it is definite or not).

ut the simple rule of types is sufficient to determine that a predicate
_ f* other than that of the appertaining


«Me. In the same way, the other wcU-known antinomies of the first
land are obwated by means of the twnplc rule of t>-p«-

The problem of the riTiter/ira/ mtwomies, howettr, obviously
rwpi^rs when it is a question of a language S in which the s)-niai
of S Itself can be formulated, and therefore in the case of ever)’



213


§6oa THE ANTINOMIES
language which contains arithmetic There is a prevalent fear that
with a syntax of this sort, which refers to itself, either contradic-
tions similar to the syntactical antmomies will be unavoidable, or
in order to avoid them, special restrictions, something like the
“ branched ” ruleof types, will be necessary A closer mvestigation
Will show, however, that this fear is not justified

The above-mentioned view is held, for mstance, by Chwistek
He had already, before Ramsey, had the idea of statmg only the
simple rule of types, and thus tendering the axiom of reducibiLty
unnecessary Later, however, he came to the conclusion that with
the rejection of the branched rule of types the syntactical anti-
nomies — that of Richard, for example — would appear (see Chwistek
[Nom Grundl ]) In my opinion, however, the mdispensabiLty of
the branched rule of types m Chwistelds system is due only to the
fact that he uses the autonomous mode of speech for his syntax (the
so-called Semantics) (see § 6S)

Apart from Grelling’s, the most important example of a syn-
tactical antmomy is the one which was already famous in antiquity,
the anttnomy 0/ tfu liar (for the history of this see Rustow)
Someonesays “I am lymg”,or moreexactly “I am lying m this
sentence", m other words "This sentence is false " If the sen-
tence IS true, then it is false, and if it is false, then it is true

Another antmomy which belongs to the category of the syn-
tactical antmomies is Richard's (see [Pnnc Math ] I, 61, and
Fraenkel [MengenUkre] p 214 fl ) In its onginal version it is con-
cerned with the decimals definable m a particular word-language
It can be easily transferred to in the following manner Let S
be a language whose syntax 1$ formulated in S In S there are at
most a denumerable number of jpt which are definable Therefore
we can correlate univocally a natural number with every such 3pt^
(for mstance, by a lexicographical arrangement of the defimtion-
sentences or, 10 an anthmetized syntax, simply by the term-
number of the 3pi^) Let 'c' be a numerical expression, we will
call the number c a Richardian number if c is the number of a
3Pt*, say ‘P’, which does not appertam to the number c, so that
*P(c)’ IS false (contradictory) Accordingly, the adjective
* Richardian* is a defined 3pt*, and thus has correlated with it a
certam number, say b Now b must be either Richardian or not
If b IS Richardian, then, accordmg to the defimtion, the property
havmg the number b does not appertam to b, therefore, m this



214 PARTIW CEX131.U. stnt.\s

case, in contradiction to our assumption, b is not Richardiin.
Hence b must be non-Richardian. b must leaw the definitioa of
'Richardian' unfulfilled, and therefore must possess the property
ha\Tng the number b ; that is to say, b must be Richardian. This is
a contradiction.

It is characteristic of the syntactical antinomies mentioned that
they operate with the concepts 'true* and ‘Wse’. For this reason
we will examine these concepts more closely before considering the
S}-ntactical antinomies any further.

§ bob. The Concepts * True ’ and ‘ False *

The concepts ‘true’ and 'false* are usually regarded as the
principal concepts of logic. In the ordinary word-languages, they
are used in such a way that the sentences ‘Sj is true’ and ‘Sj i*
false' belong to the same language as (©j. This euslOTtiary usa^e t>J
the terms 'Ove' and 'false' leads, hozrever, to a eontredietion. This
will be shown in connection with the antinomy of the liar. In order
to guard ourselves against false infertnets, wit proceed in a
strictly formal manner. Let the syntax of S formulated in S con«
tain three sjntactical adjectives, ‘51’, ‘ 2 B’, * 5 ’, concerning which
we will make only the following assumptions (^’ 1 - 3 ). In these,
u e shall write the sentence: '‘UI, has the property 91*' in an abbrevi-
ated form, thus: ‘31(91,)’. If ‘9l(?IJ’ is interpreted as "Six « «
non-sentence ”, ‘2Q (Six) ' as: "The expression Six is a sentence, and,
specifically, a true sentence”, and *5(91,)' as; "51, is a sentence,
and, specifically, a false sentence”, then our assumptions V 1-3
are in agreement iviih the ordinary use of language.

V I. EvcryexpressionofShaseiactlyoneofthethreeproperties

Vac. Let *A' be any expression whatsoever of S (not: "desig-
nation of rm expression”); if aBCA*), then A. [For instance; if

this tree is high” is true, then this tree is high.]

\zb. If A. then SC ('A*).

V 3 . For anySI,, the expressions ‘91(91, )', ‘SC(9Ii)*,
do not possess the property^l (hence, they do possess either SB
or 5 , according to V i).

From V 1 and ai it follows that:

If )» not 2B{‘A^, and therefore not A. (a)



2*5


§6o5 THE CONCEPTS ‘true’ AND ‘false’

From V i and 2 a it follows that

If not A, then not 2B (‘A*), and therefore Jy (‘A’), or 91 (‘A’) (5)
Now in analogy with the assertion of the liar, it is easy to show
that the investigation of an expression 21, with the text * 5 (21,) ’ leads
to a contradiction The fact that an expression is here designated
by a symbol (namely *l^’),which itself occurs m itself, easily has
a confusmg effect. But we can also establish the contradiction
without this direct reflexive relation, it is not, as is so often be-
lieved, the reflexiveness which constitutes the error upon which
the contradiction depends , the error hes rather m the unrestncted
use of the terms ‘true’ and ‘false’ Let us examine the two ex-
pressions ‘ and ‘2B(2y’ Obviously these are expressions,

at worst non-sentences We arc entirely at liberty as to which ex-
pressions we choose to designate by ‘21,’ and *212’, let us agree
that

(a) 21, shall be the expression ‘SD(2y’, (A) 21, shall be the ex-

-SW (6)

(Here, as can be seen, no designation of an expression occurs m
the expression itself )

According to V 3

Either SDC8(SIJ’)or gCBW). (7)

We first make the assumption 935 (‘ 5 (^i)’) From this, m

accordance with Van, it would foUow that 5 (2IJ This, ac-
cording to (6a) is 5(2Bpy)» froni which, according to (4),
would follow not 2B(2y This is, by (6i) not 215 (‘^(Sfi)’)
Our assumption leads to its own opposite and is therefore refuted
Thence, according to (7), it is true that

5CS(5tJ’> (8)

From this, by (4), follows

“ot S(2Ii) (9)

This, according to (6 a) is

notgC38(Sy), (,o)

BjV3

acCJiiW)™ S(2I!W)- (II)

From (to) and (it)


(12)



2i6 part rv. CENBLU. STNTU:


Thence, in accordance with V zat


SBCsy.

(;3)

From (S) and (6 ft):


S( 3 J.

(;4)

Therefore, m accordance with V i;


not 2 D(?L).

(;5)


{13) and (15) constitute a contradiction.

This contradiction only anscs tvhcn the predicates 'true' and
‘false* referring to sentences m a language S are used in S itself.
On the other hand, it is possible to proceed without incurring any
contradiction by emplojmg the predicates ‘true fm S,)’ and ‘false
(in Sj) ‘ in a sj-ntax of Sj which is not formulated in Sj itself but in
another language S.. Sj can, for instance, be obtained from Sj by
the addition of those ttvo predicates as new primitive symbob and
the erection of suitable primitive sentences relating to them, in the
following way: j. Every sentence of Sj is either true or false.

2. No sentence of Sj is at the same lime both true and false.

3. If, in S„ < 3 j is a consequence of il,. and if all sentences of are
true, then G, is likewise true. A theory of this kind formulated in
the manner of a s}-nm would nevertheless not be a genuine s)"ntix.
For truth andfahthoodarertotpropfrt^ittaetiealpnpntitiiVihethtT
a sentence is true or false cannot generally be seen by its design, that
is to say, by the kinds and serial order of its symbols. [This fact
has usually been o>‘crlooked by li^cians, because, for the most
part, they have been dealing not with descriptive but only with
logical languages, and in rebtion to these, certainly, ‘true’ and

false* coLnade tvith ‘analjtic* and ‘contradictory’, respectively,
and are thus sj-nlactical teiros.]

Even though ‘true* and ‘false’ do not in general occur in a
proper syntax (that is to say, in a syntax which is limited to the
dcsign*propenies of sentences), yet the majority of ordinary sen*
tences which make use of these words can be transbted cither into
the object-language or into the symtax-language. If Gj is ‘A’, then
Gj is true’ can, for example, be transbted by ‘A’. In logical in*
vestigation, ‘true’ (and ‘false’) appears in two different modes of
use. If the truth of the sentence in question follows from the rules
of transformation of the bnguage in question, then ‘true* can be
transbted by *«lid’ (or, more specifically, by ‘analytic’, ‘de-



217


§6o5 THE CO’WEPTS ‘true’ AND ‘false’

monstrableO and, correspondingly, ‘false’ by ‘contravabd’ (or
‘contradictory’, ‘refutable*) ‘True’ may also refer to mdeter-
mmate sentences, but m logical investigations this only happens in
the conditional form, as, for example ‘ If Si is true, then Sj is true
(or false, respecdvcly) ’ A sentence of this kmd can be translated
mto the syntactical sentence ‘ ^ consequence of Si (or is m-

compatible with Si, respectively) ’

§ 6oc. The Syntactical Antinomies

We will now return to the question whether, m the formulation
of the syntax of S in S, contradictions of the kind known as
syntactical antinomies may not anse if, in the ordmacy phrasing
of these antinomies, ‘true’ and ‘false’ are replaced by syntactical
terms m the manner indicated above

Let S be a non contradictory language (and, further, a con
sistent one), which contains anthmetic, and hence an anthmetized
syntax of S itself also Then a certain method exists whereby it is
possible to construct, for any and every syntactical property
formulable m S, a sentence of S, Si, such that Si attributes this
property— whether nghtly or wrongly — to itself TTus has already
been shown in the case of Language 11 (see § 35) Now, by means
of a construction of this kind, we will try to restate the antinomy
of the liar It consists of a sentence which asserts its own false-
hood

First, let us replace ‘false’ m this antmomy by 'non-demon-
strable’ If we construct a sentence of S, Si, which asserts of itself
that It IS non-demonstrablc m S, then we have m Sj an analogue
to the sentence @ of Language II which has already been dis-
cussed (and to the sentence ©i of Language I) Here no contra
diction anses If Sj is true (analytic) then Si is not false (contra-
dictory), but IS only non demonstrable in S This is actually the
case (see Theorem 362) The properties ‘analytic* and ‘non-
demonstrable’ are not mcompatible

Via, 'IS. ‘ Cdat’ ty * vetitaisJA’ ’i/t . ^ Si*

bar Assume that a sentence, Sj, is constructed m S which asserts
that S* is Itself refutable (m S) Sj is then an analogue to the
assertion of the bar ^\e will now obsen e whether the contradic-
tion anses m the ordinary way First let us assume that Sj is



2lS P.UtT IV. asJERAL STSTAS

2ctually refutable. Then Sj toU be true, and therefore tnalj-uc.
On the other hand, however, ever refutable sentence is contra*
dictoiy, and hence not an3}3r6c. Therefore the assumption is a
false one and Sj is non*refutable. From this no contradiction
follows. Sj is actually non-refutable; since ;. means the opposite
of this, Sj is false, and is therefore contradictory. But the pro-
perties *non-refutable’ and ‘contradictory’ are quite conastent
withone another (see the diat:iam on p. 210); for instance**— (){S)
possesses both.

The iropos^ility of reconstructing the antinotny of the bir
with the help of the terms ‘ non-dcmonstrable ’ or ‘ refutable’ is due
to the fact that not all analytic sentences are also demonstrable, and
sirtularly not all contradictory sentences are also refutable. But
what would happen if we were to use in place of ‘true ’and ‘false’
the syntactical terms ‘analytic’ and ‘contndicJOfy’? Like ‘true’
and ' false these two terms eoasPtute a complete classificadon of
the logical sentences. It is easy to show that we can construct
contradictions if we assume that ‘analytic fin S)‘ and ‘contra-
dictory (in S) ' are defined in a syntax which is itself formulated in
S. We could then, of course, construn a logical sentence Sj which,
in matenal interpretation, would mean that Sj was contradictory.
Sj would correspond exactly to the assertion of the liar. Since it
would be a logical sentence. Sj would be other analytic or contra-
dictory. Now, if Sj were contradictory, < 5 , would be true, there-
fore analytic, therefore not contradictory. Hence, Si would have
to be non-contradictory. But then Sj would be false, and there-
fore contradictory — which would be a contradiction.

On the same assumptions it would be possible also to construct
Grelling’s antinomy. Let us state the procedure for I.anguagc 11 .
Assuming that a predicate 'An* is defiruble in II in sudi a way
that ‘An(x)’ means: “The * is analytic (in II).’*

‘Hctcrological’ could then be defined as follows: *Hct(r) =
— An (subst (x, 3 , str (x)]) Let * Hct (x) ’ have the series-numte b.
Then it b easy to show that, for the sentence ‘Hct(b)’, «ther
Bssumpnon — that it is analjdcal or that it is contradictory — leads
to a contradiction.

We h3^c seen that if ‘analytic in S’ is definable in S, then S
contains a contradiction; therefore we arrive at the following
result:



219


§ 6oC THE SYNTACTICAL ANTINOMIES
Theorem 6oCiX. If S is consistent, or, at least, non-contra-
dictory, then ‘analytic (m S)’ u indefinable m S The same thing
holds for the remammg c terms which were defined earher (in so
far as they do not coincide with d-teims), for instance, ‘valid’,
‘consequence’, ‘equipollent’, etc But it is not true for every
c-term which does not comade with a d-term

If a sj’ntax of a language is to contam the term ‘analytic
(m Sj)’ then It must, consequently, be formulated m a language Sg
which IS richer m modes of expression than Sj On the other hand,
the d-term ' demonstrable (in S])’can, under certam circumstances,
be defined in , whether that is possible or not depends upon the
wealth of modes of expression which is available m Sj With
Languages I and II the situation on this point is as follows
‘analytic m I' is not definable m I, but it is definable in II,
* analytic m II ’ is not definable mil, but is only definable tn a still
ncher language ‘Demonstrable ml’, because it is mdcfimte, is
not definable in I, but ‘demonstrable in IT can be defined m II,
namely, by means of ‘ Q r) [BewSatxII (r, »)] ’

The foregoing reflections follow the general lines of G5del’<
treatise They show also why it is impossible to prove the non
cootradictormesa of S m S Closely related to Theorem i is the
following theorem (a generalization of Theorem 36 7, sec Godel
[Vnentscheidbare], p 196; Godel mtends to give a proof of this
generalized theorem m a contmuation of that Treatise)

Theorem 6oc,3. If S is consistent, or at least non-contradictory,
then no proof of the non-eontradictonness or consistency of S can be
fomvlated tn a syntax ichuh uses ordy the means of expression tchuh
are available tn S

The mvestiption of Richard’s antinomy (p 213) leads to a
similar conclusion Assume that m S there is an by means of
which a unn*ocal enumeration of all the jpt* which are definable
m S rmght be constructed This could be effected, for example, by
means of an fu^ such that every full expression fUi(3pt*) was a 3
We will use the symbolism of II and write fu^ ‘num’.

The unnocahty of the numhenng is assumed

(num(F) = nuni(G))D(x)(F{x) = G(x); (i)

With the help of ‘ num ‘ Rj * (“ Richardian ”) muld now be detmcd .

Ri (x) H (F) [(num (f)= *) 3 ~ ^ (*)] (2)



220


TART IV. CENER.U. SYNTAX


Since ' Ri ’ is a 3pr', il has a certain particular number designated
by ‘num(Ri)’. We assume first that the number of *Ri’ is itself
Richardian: ‘Ri[num(Ri)]’. Then if we substitute in (2)
‘num(Ri)’ for ‘.v*, and ‘Ri’ for 'F', * »«- Ri [num (Ri)] ’ easily
follows. Since our assumption leads to its opposite, it follows that
it is refuted; and therefore it is proved that

'>'Ri[num(Ri)], (3)

From (i):

(num (F) =num (Ri))D (-~F(num (Ri)) 5 Ri [num (Ri))).

(4)

From (3), (4):

(num (F) = num(Ri))D — F [num (Ri)]. (5)

From (2);

(F) [(num(F) = num(Ri))D<«->F{rum-(Ri)))D Ri[num(Ri)].

( 6 )

From (s), (6):

Ri(num(Ri)). (7)

The pfo^'ed sentences (3) and {7) contradict one another; S is

therefore contradictory. Thence follows:

Theorem 60C.3. If S isconsi$tent,or8t least non*contndictorj’,
then it is not possible to construct in S either an mg or an 0 U by
means of uhich a uni\-ocal enumeration of the jpr* * * * of S could be
constructed. — Although the aggregate of the 3pt‘ which are de-
finable in S is a denumerable aggregate, in accordance with this
Theorem an enumeration of them carmot be effected with the
means available in S itself. [The condition in this Theorem is only
added for the purpose of facilitating understanding; if S is in-
consistent, then in S no univocal enumeration of a number of
objects is possible at all, since no (non-sjmonjinous) 3 ***
as-aibble.]

§ 6 od. Every ARinfMETic is Defectat

Let Sj contain an arithmetic (in relation to a certain

and let the real numbers be represented in Sj by 3fu‘. Let Sj be

a conservative sub-language of S,. and lei the arithmetized sj-nux

of Sj be formubled in Sj. We will show that with the help of the

ariihmetico-syntactical terms of Sj, as referred to Sj, a 3fu* can be



221


§ 6 od EVERY ARITHMETIC IS DEFECTIVE
defined m Sg for which there a no jfu* m Sj having the same course
of values , this is true for evety language Sj, however nch it may be,
if we take a sufficiently nch laogtmge as Sg We define the jfu'
‘k’ m Sg m the followmg way i If * is not a term-number of a
3fu» of Sj, then k (x) = 0 , a If x is a term-number of a jfu* of Sj,
let us say ‘h’, then k(*)=h(x)+ 1 Then every jfu* of Sj deviates
from ‘ k’ for a certain argument (namely, for its own term-number) ,
and therefore in Sg there is no jfu* having the same course of
values as ‘ k ’ In other words a real number can be given which is
not equal to any real number definable m Sj (see p 206)
Theorem 6ocLi. For every language S a real number tehieh
cannot be d^ned tn S can be gtven
The above definition of *k’ corresponds to the so-called
diagonal method of the Theory of Aggregates Theorem i corre-
sponds to the well-known theorem of the Theory of Aggregates
which states that the aggregate of the teal numbers is a non-
denumerable aggregate (On the concept of the non-denumerable
aggregates see, however, § 71 d ) On the other hand, the above
line of thought also corresponds to Richard’s antinomy.

We will now sutnnurue briefly the results of this investigation
of the syntactical antinomies Let the syntax of a language S be
formulated m S The reconstruction of the syntactical anlmomiea
by means of terms which are defined m S (for mstance, m Lan-
guage II, *non-dcmonstrable m II ’ or * refutable m II’) does not
lead to contradictions, but it opens the way to the proof that
certam sentences are non-demonstrable or irrcsoluble m S With
the help of other terms (for instance, ‘analytic’, ‘contradictory’,
‘consequence’, ‘correlated number’, ‘term-number’) the recon-
struction of the syntactical antinomies is possible This leads to the
proof that these terms (of which the definitions have up to now
only been formulated m words and not within a formalized system)
cannot be defined m S, if S is consistent, or at least non-contra-
dictory. Since terms and sentenexs of pure syntax are nothmg
other than syntactically interpreted terms and sentences of arith-
metic, the investigation of the ^tactical antinomies leads to the
conclusion that eiery anthmetu which is to any extent formulated
in any language « necessarily defective in two respects

Theorem 6o<L2. For every anthmetical system it is possible to
state (a) indefinable anthmetical terms and (i) mesoluble anth-



P.«rr IT. dNOLU, SI'NT.«


rviucJ srrJwts (Godd [t 7 BO!»*«! 3 SaT-p]). In connection iritli
(o) see Theorems 6oc.i, 3, 6odlx. In connection with (i) see
Theorem 60^.2; further urcsolable sentences analogous to G in
II and Oj in I (see § 36) can be constructed.

This defectiveness is not to be understood as it there were, for
instance, ariihmeocal terms trhidi could not be formally (i.e. in a
calculus) defined at all, or anihmerical sentences whidi could not
be resolred at all. For every term which is staled in any un-
ambiguous way in a word-language, there exists a formal defiiu-
tion in an appropriate language. Every arithmetical sentence
which IS, for instance, irrcsoluble in the language Sj is yet de-
terminate in Sj; in the first pUee there exists a lidser syntax-
language Sj, within which the proof cither that Sj is analytic or
that ;j IS contradictory can be stated; and secondly, there exists
an object-language S, of which S, is a proper sub-language, such
that ^ is resoluble in But there exists neither a language in
which all anihmetical terms can be defined nor one in which all
anthxnetical sentences are resoluble. [This is the kernel of truth
in the assertion made by Brouwer { 5 /rar.^], and, following him,
by Hejting p. 3, that mathematics cannot be completely

formaliaed.] In other words fverythix!^ r-Mhfnatirel eex he
^ormakzti, tul maVtemafia rcwiof he exhmuud by one it

requires an infinite series of ettf richer languages.

(d) TRANSIATION AKD INTERPRETATION

§61. Translation FROM One L.\n*gu.\ce
INTO Another

We call a syr.laefiral rorrvlation between the sjTitictical ob-
jects (31 or R) of one kind and those of another when Cj b a many-
one relation by means of whidi exactly’ one object of the second
kind b correlated to et-ciy object of the first, and every object of
the second kind to at least one of the first. The SI (or 51 ) which b
correlated to SIj (or Rj, respectively) by means of Oj b called the
G,-«>rrfZflIe of SI, (or of R,), and b designated by ‘C, [SIJ ' (or
* J ')• Herein the following condition is assumed j if SI« has
no direct C, -correlate but can be auWitided into the expressions
which have such Correlates, then Ci[^ b equal



§6l TRA^*SLA;nO^ FROM ONE lANGCAGE INTO A^ OTHER 223

to the expression composed of QjPIj],Qjpy, The

fJaas which contains all and onfy the Qi-correlates of the sen-
tences of Ri IS designated by According to this, the

correlates of sentences are also detcinsmed by means of a correla-
tion between expressions, and the correlates of sentential classes
by means of a correlation between sentences [In a formalized
syntax, Qj can, for instance, be either an Sg*, a an Sg*, or an
] We say that a certain syntactical relation is transformed mto
a certain other one by means of Q, if, when the first relation sub-
sists between any two objects, the second suEjsists between the
Qi-correlates of these objects

A syntactical correlation, Q^, between all sentential classes (or
all sentences, or the expressions of an expressional class Ai, or all
symbols) of Sj and those of S], IS called atrassformanceof Shinto
Sj in respect of classes (or of sentec<^ Or expressions, or symbols,
respectively) provided that, by means of Q^, the consequence rela-
tion in Sx IS transformed into the consequence lelaDon in S^. For
5^ It u assumed that no expression of Ri, but every sentence of
wbch does not belong to Ai. is umvocally analyzable into several
expressions of Ai is called a transfomanee of into Sj if is a

transformance of Sj mto S} of one of the kmds mentioned. ‘ h-trara-
farmanee in respect of classes (sentences, and so on)' is analogously
defined, the requirement m this case being the maintenance of
the relation 'L-cocsequence*

Theorem 6z J. If 2 i Q a transfoimance of Sj into Sj, then Qj
IS also an L-transfonnance of Sj mto S,.

'nieorem6u. If Qi is a transfoimance of mto S^m respect of
sentences, then by Qj the consequence relation between sentences
m Sj 13 transformed mto the consequence relation between sen-
tences m S,. The converse is not universally true.

A transformance of Sj into S, is called reversible when its con-
verse (that is, the relation subsistmg m the reverse direction) is a
transformance of S, into Sj , otherwise vrevernble

Theorem 613. Let a transformance of Sj mto Sj, if Qj is

reversible, then ^ one-one lelation. The converse is not um-
versally true.

Example of an tTretmihle tzansfoimance m respect of sentences
the transformance given by Lewis [Logte], p 178 of his systan of
strict implicatioa (without die eiKti^nnal postulate) into the ordinary



224 P.UJT IV. CENOUL STTfTAX

sentfntiil calculus. In this case, the correlate of the three sra*
tcnces of the first system, 'A*, *M(A)’, and * — RI(— A)’ (writing
•M' instead of the symbol of possibility) is the same sentence, ‘A’.
The transformance is thus not a one-one relation and is therefore
irreversible.

If there exists a transformance (in respect of classes, etc.) of Sj
in Sj, then Sj is called tranifomahlf (in respect of classes, etc.) in
Sj. If Sj is reteisibly transfoimabte in Sj in respect of symbols,
then and Sj are called Isomorphic.

Theorem 61.4. If in Sj there isaralld sentence Sj and a contra-
s-alid sentence S., then any language is transfonnable into S, in
respect of sentences. — It is possible, for example, to take Sj as the
correlate of every contravalid sentence of S„ and Sj as the corre-
late of every other sentence. This Theortirj shows the compre-
hensiveness of the concept of transformability; the concept of
reversible transformability is a much more restricted one, and
that of isomorphism more restricted still.

Theorem 61.5. Let Sj and S, be isomorphic with respect to the
correlation Qj. If Oi is a pi* in Sj with a characteristic, then
is a til* m S| with the same characteristic. If, for instance,
Cl is a symbol of proper negation (or of disjunciiorj, and so on),
then [4J is likewise a symbol of proper negation, etc.


Cl


Let O, be a transformance (in respect of classes, etc.) of Sj into
Sj5 and let Sj be a sub-language of S, (sec dbgram). Then Oj is
called a translation (in re s pect of etc.) of Sj into S*; and

Si is called tratulataile (in respect of dasses, eta) into The fol-
lowing table gii-cs for a number of syntactical properties and rela-
tions of ft or S (column i A), and for a number of properties of
languages (column 1 B), the conditions (sufficient but not neces-
sary) under which they are transferred from one to another of the
three languages between which the gii’cn relation holds: in the
direction of the transformance, that is, from fti to Q, [ftj (column
2); and conversely (6); from the sub-language to the total language







Complete (L*compIetc, deter-
minate , L-determmate , logical)
Incomplete (L-mcowplete, in-
dctcTTtwnatc , L-mdetetrovnate ,
descriptive)



226


P.UIT nr. CENQtKL SYNT-CC


(3) ; and corntrsely (f); in the direction of the translation (4) ; and
con\crsely {7).

laltons for the conditions:
gen • generallj , i.e. in all cases;

L where C, is an L-rram/onn<7nff;

R- where C| is a rnYrsihlf transfonnance;
c: where Sj is a eonjmwfitv sub-language of S, (see p. 179):

r; where Sj is a nffiaently ncJt sub-language of S,. namely,
a language containing either a R which is contrat'alid in
S], or all the sentences of Sj.

The conditions p\en in the table in sqvare bracketj refer to the
L-term which corresponds to the term occurring in column (i).

Exampln • If Ri is s-alid in Si, then it is also salid in Sj. If R| is
analytic in Sj, then it is also analytic in S|, provided that Si is a re-
.•ersible L/-transformance. If Sj is mconsisient, then S* is also in-
consistent, proMded that Sj is a consersinve sub-language of Sj.

Since everj’ transfonnance is at the same time a translation
(namely, into an improper sub-Ianjpiage), the following theorems
and dedmtions on aI>o be referred to transformaners.

Theorem 61,6. If S| is translatable into St in respect of
symbols, then it is also translatable in respect of expressions; if
in respect of e-xpressions, then also in respect of sentences, and
conversely ; if in respect of sentences, then also in respect of classes.

Let S, and Cj be translations of S| in S.. We say that and Ct
eanaJe tn content if, for ever)' 51 , in S,, O, [AJ and 0 ,[ 51 il are
equipollent in S,.

Let S, and S, be sub-languages of S,; and let Oj be a tiatsslation
of S, into Sj. If in this case, 51 , and Q, [ 51 J are alwap cquip'ollent
in S3, we call Qj an equipollent translation in respect of S-. Ana-
logously, * L-equipollcnt translation’ is defined by reference to
‘ L-equipollcnt classes'. Further, ifC, is a translation in respect of
SJ mbols or expressions such that ?l, and O, [?!,] are alwaj-s sjTiony-
mous in S,, w e call Q, a synonj'mous translation in respect of S,.

sjnonjTnous translation is also an equipollent translation.

Theorem 61.7. If S, is a conser>-ali\-e sub-language of S, then
the equalitj- of sjTnbols represents a sjTionjmous translation of S,
in S, in respect of S,.

Example! ;tA:t T be the sub-language of 1 constructed by means of
eliminating the \ariab Jes. Then I’is synonymously transbtable into I
bymeansof the equality of symbols. Again, I is transistable into Tin



§6l TRANSLATION FROM ONE LANGUAGE INTO ANOTHER 327

respect of classes, although I' is a proper sub-language of I If, for
instance, Si is an open sentence of I with exactly one free variable,
3i, then the class of all sentences of the form Sj niay be taken
as the correlate of { Si] This translation is equipollent in respect of
I There is no equipollent translation of I into I in respect of sen-
tences, this example therefore demonstrates the importance of the
concept of UaTulation in respect of classes

I^t Si be the intmtiomst sentential calculus of Heyting \Logtfi \ , and
let S3 be the ordinary sentential calculus (that of Language 11 , for
instance) The ordmary translation, Q|,of Si into S3(that is to say, the
translation in which the symbol of negation is the Qi-correlate of the
symbol of negation, the symbol of disjunction the Qi-correlate of the
symbol of disjunction, and so on) is a transformance of Si into a
proper sub-language, Sj, of S3 This transformance is one in respect
of symbols (if, m Si and S3 we insert all the brackets as in Language
II) S3 13 & proper sub-language of S3, since, for instance, ‘p v ~p’
1 ' * ’ ^ « N • • «• • - t.

1 < . .

(

translation was onginated by Glivenko, G&del gives another trans-
lation m connection with it (.[Koll 4), p 34) ]

On the concept of translauon, see also Ajdukiewicz

§62. The Interpretation OF A Language

^^^len only the formal rules of a Uncage, for mstance our
Language II, or the Latin language, are known, then, although it
IS possible to answer syntactical questions concerning it — to say,
for example, whether a given sentence is  or contravalid,
descriptive or existential, and so on — it is not possible to use it as
a language of commuiucatioo, because the interpretation of the
language is lacking Tbcrearetwowaysinwhich anyone may learn
to use a language as a language of communication the purely
practical method which is employed in the case of quite small
children and at the Berlitz school of languages, and the method of
theoretical statements or assertions, such as is used, for instance,
in a text -book without illustrations In the present work, by the
interpretation of a language we shall always mean the second pro-

will these interpretative statements take? To give an illustration,
w hen we wish to state what a certam Latin sentence means m English
we shall do so by equating it with another sentence which has the



PART IV. CENER.\L SYNTAX


same meaning. Frequentlythcsecondsentenccvnll likewise belong
to the Latin language (for example, whenever we explain a newword
by a familiar sj-nonjTn); usually, howcv er, it will be a sentence in
English, but it may also be a sentence in any other language, such
as French. The interpretation of the expressions of a langua-e S,
is thus given by means of a tranilation into a language S., the state-
ment of the translation being effected in a sj-ntax-language S,;
and It is possible for two of these languages, or even all three, to
coinude. Sometimes, spccia) conditions are imposed on the trans-
lation — for instance that it must depend upH>n a reversible trans-
formance, or that it must be equipollent in respect of a particular
language, and so on.

The interpretation of a language is a translation and therefore
something which can be formally represertitdx the construction and
examination of interpretations belong to forma! sjmtax. This
holds equally of an interpretation of, say, French in German when
what is required is not merely some kind of transfornunce in
respect of sentences, but, as we say, a rendering of the sense or
meaning of the French sentences. We have already seen that, in
the case of an tndh'idua) language like Gennan, the construction
of the 8>Tita.x of that language means the construction of a calculus
which fulfils the condition of being in agreement with the actual
historical habits of speech of Germsn-spealcing people. And the
construction of the calculus must take place entirely within the
domain of formal sjntax, although the decision as to whether the
calculus fulfils the given condition is not a logical but an historical
and empirical one, which lies outside the domain of pure sjmtax.
The same thing holds, analogously, for the relation between
two languages designated as translation or interpretation. The
ordinary requirement of a translation from the French into the
German language is that it be in accordance with sense or meaning
— which means simply that it must be in agreement with the his-
torically known habits of speech of French-speaking and German-
speaking people. The construction of c%*cry translation, and thus
of every so-called true-lo-saue translation, also takes place nitkin
ihe domain of formal nTifox'— although the decision as to whether a
proposed translation fulfils the given requirement and can thus be
called truc-to-sense is an histoiica], cxtra-sjTitactical one. It is
possible to proceed in such a way that the exlra-sj-ntactical re-



229


§ 62 THE INTERPRETATIOX OF A UOvCCAGE

quirement is here of the same kind as in the first case, namely, is
concerned with the agreement of a syntactically constructed cal
cuius With a certain historically given language e first stipulate,
for instance, that the Frentdi language be represented, say, by the
calculus Si, the German language by Sj, and, further, that the
language which consists of the French and German languages as
sub-languages be represented bj the calculus Sj, of which Sj and
Sj are sub-languages Then a syntactically given translation, Qi,
of Sj into §8 IS true-to*sense if it is eqmpollent m respect of S3
Under certam circumstances it will be required m addition that
Qi be a sjTionjTnous expresstonal translation in respect of Sj

Sometimes the interpretation of a language Sj m relation to an
existing language S. is given by constructing from Sj a more com
prehensiv e language S3 bj means of the addition of a sub language
which IS isomorphic or even congruent with The interpretation
of a symbolic calculus — such as a mathematical calculus — on the
basts of an existing saeotific bnguage, is, m particular, often
effected in this manner

ExantpUt If, for instance, the sy'stem of the calculus of vectors is
first constructed as an uninterpreted mathemaucal calculus, the
mterpreuuon can be performed in such a manner that the original
language of physics ts extended by the inclusion of the calculus of
sectors Because the vectorsymbolsareusedin the new language m
conjtinction with the other linguistic symbols, they have thcmsehes
gamed a meamng withm the physical language In the same way,
any sj-stem of geometncal axioms can first be gtien as an isolated
calculus and the various possible mterpretations may be represented
as different translations mto the language of phj'sics (see § 71 e) Ifm
this case the tcrmmology of geometry 1$ retamed, then it is a ques-
tion of a translation into a congruent sub-language of a new language
constructed from the old language of physics by the inclusion of
geometry

In order to establish a paiticular mterpretation of the language
83, that IS to say a particular translation of into S., it is not
necessary to give the correlates of all symbols or of all sentences of
Sj It u sufficient to state the correlates of certain expressions, m
many cases, for example, it is suffiaent to state the correlates of
ceitam descriptive sentences of a simple form, m which not even
all undefined symbols of need occur In this way, m connection
with the transformation rules of Sj, the whole translation is uni-
vocally detennmed, or, more exactly, any two translations which



2j0


r^RTrV. GEJJTRXLSYNTCC


ha\e tho«e correlates in common coincide in content. It is eus*
tomary in the construction of a syml'olic language, particularlT in
iogi^iics, to !?\-e an interpretation by means of an expository text,
and hence by means of a translation into the ordinary word*
languaj^. And generally it is also customary to state many more
correlates than are necessary. This is certainly useful for fadli*
tating comprehension; and in introduang Language I we hare
proceeded in this manner ourseh-es. But it is important to realire
that interpretatire staterncnts of this kind are m most cases over*
determmed


Exar^filn: i. Let the dnmj'ttve Lcncswce II contain one-termed
predicates 'Pj', ...'Pi' as the only undefined descnpure sjwibcls.
Then for a complete interpretation of Lanctiage 1 1 such stipulations
as the followmg are sufficient- (i) 'O’ shall designate the miual
position, and an St 'O** '’with n dashes, the (rs+ i)st position m
such and such a senes of positions, (a) ‘P,’ shall be equivalent in
meaning to ; red ‘Pk’to’blue*. (3I an atomic de^cnptire sen*
t«ee of the form rr,(St,>. where pr, » an undefined prs. shall mean
that the position desicnated bv St, has the property designated by
Ptj. In the sentences for which the translanon is hereby determined,
no defined si-mbols of any kind whatever ocrur ; further, no varjihles
(P« *• }. f), hence no operators and, finallv, none of the undefined
logical constants ‘ ‘3 * K v 3 ’. In spite of this,

the interpretation of all the remaining sentences of 11 ia also de*
tennined hy the abore supubnons; that is to say, for the correlate
of any other sentenre of Lancuage 11. the only choice is between
equipollent sentences of that sub-bncuage of English into which
Languas:e II is rerersibly transfonned. Thus 'P,(0)vP,(0')*, for
ms^«, must be translated into: ‘the first or the second position.

ifo a sentence which is equipollent to this
one). Or again, for example: *(*)(r,(x))‘ must be translated into
all positions are red'; for it follows from the transfonnation rules
o L.anguage 11 that * V b « s>-mbo) of proper disjunction and ‘(ar)’
a proper universal operator.

^ sub-language of H. II| is to be inter-

f - . , ^ jufans of a transUtion in respect of expressions into a
suiuWe other ImjrujK, ; imd. by thi, u«.sI«ion, . comlite i. to bo
"""X *“; "qoitnoeht it immded to
“ 'he ordinary sense; if the require-

ment IS not stated, then the Irhial translation mav be taken in which
^e correlate of eveiy analytic sentence is 'OaO', and the correlate
« er>- contradictory sentence * ^(0= 0) J Wc shall only gii c the
correlates of two s>-mbob: 'O' wai be translated into 'Q* and
It , k-’i-* *'’'®‘*'^.'’’’*’^‘"*f'PrctationofihcwholeofLanguag<
III. "hich contains classical mathematics, is established.



23:


§ 62 THE INTERPRETATIOM OF A LANGU\GE

From the standpoint of inte r pre ta tion, it is charactenstic of the
undefined desmpitce tymboh that thetr interpretation, even after
that of the other symbols, is still oTbtirary wthin a wide domain
(arbitrary, that is to say, when we merely consider the syntax of
the isolated language, the choice can then be limited by further
conditions) Thus, for instance, it is not determined by the trans-
formation rules of Language II and the interpretation of the other
symbols, whether ‘P,’ is to be interpreted, say, by ‘red’ or by
‘green*, or by the designation of any other property of positions
In most of the symbolic languages even expressions which are
mterpreted by their authors as logical belong to the descriptive
expressions as understood in general syntax The majority of the
usual systems are interpreted by their authors as logical languages ,
but since commonly only d-rules are laid down, these languages
are for the most part indeterminate and therefore descriptive In
consequence, for certain expressions of these languages, even if the
other expressions are mterpreted according to the statements of
the authors, mterpretations are possible that are essentially different
from one another

Example The umtersal cperator tstth a nwreneal vanable 3 is a
proper umvenal operator m Languages I and II but m the usual
languages — for mstance, m IfVme Moth }— it is an improper one
(seep 197), because these languages contain only d rules "Thus, m
the usual languages there are sentences that are indetennmate, and
therefore designated by us as descnpCive, although they are inter-
preted by their authors as logical sentences In order to remain
withm the framework of our own symbohsm, mstead of considering
one of the earlier sj-stems, we will consider that of Language IIj
which results from II by [mutation to the d rules (but which must
contain all the definitions stated previously in Language I) The sen
tence 15, n hich is uialyoc but irresoIuWe mil (§ 36) is thus in Ifd
an mdetennmate sentence The umtersal operator (5) in both [Prmc
Math \ and He u not logical but Jescnptne By this nothmg is said
against the usual translation, in which the correlate of (5 is a logical


example Let the *pi*pr, and ptjbe the only undefined desenptne
symbols of 11^ and 11 \Ve will mteipret Ilo b> means of two dif-
ferent trarislations into II, C^and Qx. Pori^iand Cj, we determine



TART IV. GENERAL S^liTAX


first, that the correlates of all sentential junctions shall be these junc-
tions themselves ; second, that the correlates of all atomic sentences
shall be these sentences themsehes. Hence the correlates of all
molecular sentences are also these sentences themsehes. We will
now show that Cf, and Cj may nevertheless still be essentially dif-
ferent from one another — that is to say, tliat they need not coincide
m content. Let the ;5, -correlate of every sentence be that sentence
Itself; this is the ordinary interpretation, in which the improper
universal operator (jO of I!,i is uitctpretcd by means of a proper
universal operator of II. Let Sj be (3i)(pri(ji)); and let
be ( 3 ,) (pr,( 3 i))*pr,( 5 ). This sentence is (m II) obviously richer in
content than 0| [Si], namely, S, itself. Let S* be (3 3iKPh(3i))l
and let be (3 3 i) (pt,( 3 ,)) v-- pr.fs); tHis sentence is (in 11)

obviously poorer in content than namely, St itself- It can

easily be shown that Cj is really a translation (althouch not an equi-
pollent translation m respect of II), that is, that by means of Cj the
consequence-relation m I Id is transformed into the consequence-
relation in II. Tor cvamplc, let 0> be pri{0fi), then Sa ** a conse-
quence of Sj. and Sj a consequence of ;*; correspondingly,
Q, { Sj] — i.e 0j — is a consequence of the previously given C} [0il ,
and the given Cj(Ss] is a conscquertce of 'ie- ®*- The

reason why, in addition to the ordmao mierpretation, the essen-
tially diflercnt interpretation Q„ which interprets ihe universal and
existential operaton descriptively, is also possible, is that the trans-
formation nilcs of lid only determine that eveo' sentence of the
form pr,(St) IS a consequence of ( 3 i) (pr,( 3 ,)), but do not determine
whether this universal sentence is equipollent to (as in the usual
interpretation QO, or richer in content than (ns in the ease of I5|),
the class of sentences of the form pri(0t).

Other examples of descriptive symbols that are interpreted by
their authors as logical are the tntenuonal tmtential jurictiotit treated
of by Lewis and others. (There are, however, also intcnsional sen-
tential junctions that are logical.)

Let S be a descriptive language for which an interpretation has
been given in the ordinary way in the vvorvls of an expository text.
In judging of this interpretation we must, then, distinguish (as the
example just examined shows) between interpretations by means of
descript iv cevpressions and interpretations by means of logical ones,
(i) Inlcrprciations by means of descriptive expressions generally
yield something new which has not already been given in the con-
struction of the calculus; they are (to a certain extent) necessary
for the establishment of an interpretation of the calculus,
(s) Suppose the expression 91, of the calailus is interpreted by a
logical expression of the word-language. I lerc, there are tw o cases
to be distinguished, (an) 91, is a logical expression in the sense



233


§ 62 TIIE INTERPRETATION OF A LANCtAGE
understood in general syntax , then that interpretation may already
be implied in the other interpretations, and if so it only serves as
an explanation, which is theoretically unnecessary, but which
facilitates understanding {zb) is a descriptive expression in the
sense understood in general syntax (for example the universal
operator in Ild) Then the interpretation of by means of a
certain logical expression can be replaced by the erection of suit-
able c rules for S, with the aid of which 91, becomes a logical
expression of the kind intended [Taking our example let lid be
expanded by indehnite c rules to 11 , then, m accordance with the
intended interpretation, ( 5 ) will become a proper universal
operator ]

General sjntax proceeds according to a formal method, that is
to say, m the investigation of the expressions of a language it
considers only the order and syntactical kind of the symbols of an
expression We has e already seen that this formal method can also
represent concepts which are sometimes regarded as not formal
and designated as coneepis of mtamng (or concepts of a logic of
meaning), such as, for instance, con$e<|uence-relation, content,
relations of content, and so on Finally vie have established the
fact that even the questions which refer to the interpretation of a
language, and which appear, therefore, to be the very opposite of
formal, can be handled withm the domain of formal syntax
Accordmgly, we must acknowledge that all questions of logic
(taking this word in a s ery ividc sense, but excluding all cmpincal
and therewith all psychological reference) belong to syntax As
soon as logie « formulattd in an exact manner, it turns out to be
nothing other than the r>7ilax ather of a particular language or of
languages in general


(0 EXTENSIONALITY

§ 63 Quasi-Syntactical Sentences

\\ e are now going to introduce a number of concepts which arc
nccessar} for the discussion of the problem of cxtensionality, for
the logic of modalities, and, later on, for the analj-sis of philo-
sophical sentences We shall first explain these concepts in an in-
formal and mcxact manner Let B ^ a domain of certain objects



PART IV GENERAL S\'NTA.\‘


234

whose properties are described in the object -lanjiiage S,. Assume
that there exists in reference to B an object-property Ej, and in
reference to Sj a syntactical property of expressions E,, such that
alwajs and only when Ej qualifies an object, E. qualifies the ex-
pression which designates that object. \Vc shall call Ej the sj-n-
tactical property correlated to Ej. E, is then a property which is,
so to speak, disguised as an object-property, ‘••it which, according
to its meaning, is of a syntactical character; we therefore call it a
quasi-syntactical property (or sometimes a pseudo-object-pro-
perty). A sentence which ascribes the property E| to an object c
IS called a quasi-symtactical sentence; such a sentence is trans-
latable into the (proper) syntactical sentence which ascribes the
propertj ;) to a designation of c

Examples: i. Irrtflemeness Let S, be a descriptive L-Ianguage
(like I and 11) with a symbolism similar to that of II, but a name-
language, and let it be concerned with the properties and relations
of the persons living in the district B on 0 certain day. *Shav(a,b)’
will mean ‘a shaves b' (on the day in question). We define the
*pi*‘ Irr' as follows- ‘ Irr(F)s(*)(— E(ar,»))’, or, in words: *a re-
lation P » called iTTeflexive when no object has ihis lelation to itself’.
'lrr(Shav)’ is thus equipollent to •(*)( — Shav(r, a))’ (6,). Sj
means that in D, on that particular day, no one shaves himself;
whether that is the case or not cannot be deduced from the trans-
formation nilea of S,; S, is s>nihetic. Let Sj contain, in addition,
the ‘pt* ‘Llrr’: ‘Llrr(P)’, or, in words, *P is L-irreflexive (or
logically irreflexive) means that P is irreflexivc by logical necessity,
that IS, ‘Llrr’ must be so defined that ‘Llrr(P)’ is only onaljiic
when '(*) (— -Pft, t))’ is analytic, and othemise it is contradictory.
Then *LIrr(Shav)' is contradictory, since Sj is not analytic but
synthetic. Let ‘ Broth ’ be so defined that ‘ Broth (a, b) ' means '* a is
a brother of b". Then ‘Irr{Broih)* is analj-tic, and consequently
Llrr (Broth)’ is also analytic. ' Iir' and ‘ Llrr* are predicates of Si.
Let the syntax-language S,of S* be a word-language; we now define
the predicafe'L-irTcfloxive’ in S, as follows: a two-termed predicate
pri of Sj IS called L-irrcflexivc when (o,) pr, (e,. p,)) is onabti'c.
According to this, ‘Sliav’ is not L-irreflexive, but * Broth’ is. In a
language which contains both S| and Sj as sub-languages, for any
predicate ‘P’. the sentence ‘LlrrfP)* is always equipollent to the
sj-ntactical sentence “‘P’ is Hrreflcxivc’*. ’L-irrcflexivc* is the
8>-nt3ctical predicate which is correlated to the predicate ‘Llrr*.
‘Drr’ is a — • , . - •

‘Irr*isnot. ,

correlated s; ;

both sentence are analytic. The same is true of ‘ — I.IrrfShav)*
and ’“Shav* is not L-irreflcxive'*. On the other hand, there arc no



§63 QUASl-SYNTACnCAL SENTENCES 235

syntactical sentences correlated tothesyntheticsentences Iir (Shav) ’
and ' ~IrT(Shav)’, therefore, these are not quasi-syntactical sen
tences (Concerning ‘L-irreflcxive’, see §716)

2 Implication In the descnptise L-language Sj we shall write
‘Imp(A,B)’ instead of ‘AoB’ Further, let there he introduced
into Sj (by defimoon or hy primitive sentences) a predicate ‘ Limp ’
such that, for any closed sentences ‘A* and ‘B“, ‘ Limp (A, B)’ is
analytic if, and only if, ‘ Imp (A, B) * b analytic , otherwise it is contra
dictory Let ‘Aj’ and ‘Bj’ be two dosed sentences such that 'Bj’
IS not a consequence of *Aj’ Then ‘ Imp (A|, Bj) ' is not analytic, and
therefore ‘LInip(A, Bj)’ is contradictory Now let ‘Aj’ and ‘B.’
be two closed sentences of whidi ‘B,’ is a consequence of ‘Aj
Then ‘Imp(A,,Bi)’ is analytic, and consequently ‘LImp(Aj,Bi)

IS also analytic. Let the syntax-language of Si be the ordinary
word-language Then, m a language which contams Si and Sj cs
sub-languages, for any two closed sentences, ‘A' and ‘B’, 'Limp
(A,B)* IS always equipollent to the syntactical sentence “B is a
consequence of 'A'” ‘ Limp’ is thus a quasi-syntactical predicate
of Si to which the syntactical predicate ‘consequence’ is correlated
As opposed to this, ‘Imp’ is not quasi-syntactical To the quasi-
syntacucal sentence *LImp(Ai,Bt)’ is correlated the syntactical
sentence “'Bi' is a consequence of ‘At”’, likewise, to Ae quasi-
syntactical sentence ‘ -^LlmpfALBi)’ is correlated the synuctical
sentence “‘Bi’ is not a consequence of 'A|’” On the other hand,
there are no syntactical sentencea correlated to the synthetic sen-
tences ‘ Imp (A,. B,)’ and ‘ '—Imp(Ai,Bj)’. consequently these sen-
tences are not quasi-syntactical llie relations in this example, to
which we shall return later in a discussion of the logic of modalities,
are completely analogous to those of the first example

We now pass from the informal and mexact to the syntactical
discussion of these concepts Let S] be any language , and let Sjbe
a logical language Let be a one-one syntactical correlation be-
tween the expressions of and the expressions of a class ua S{,
and let the expressions of 5 Lbe “Stu which are all isogenous with
one another Then we shall call S{ a syntax-language of (with
respect to Dj); and we shall call QipIJ the jyntaetical designation
of (with respect to Qi) The 0 g, or ^r, of S* to which the ex-
pressions of are suitable as arguments we call syntactical Sg,
or (with respect to Qj) If the expressions of ilj are numerical
expressions, we call S3 an antkmetized syntax-language If 83 is a
-zIl h VKP/gifflge 5,3, Tit •sstj ^tk*L S>3 toRtawa o sywtas ^
Si (with respect to Ci)

An lcg",^gi, of Sj is called a ^lum-syntactical ;g when there
exist an S3, a C<. and a logical €Q*,Sq., which fulfil the foUowmg



PART IV. GENERAL SATSTAX


236

conditions: Sj is a sub-language of Sj; contains a syntax of Sj
mth respect to Q| ; if Sj is any full sentence of Sgj in Sj, such as
in nhich the arguments are not S, then Sj is
equipollent in S, to ...Qi [?!;]); 1ft this be

©j. ©1 IS then called quasUsyntoctlcnl m respect of
Sj is called a synlactieal correlated to Sj (with respect to

Qj); ©Qj is called a synlacltcal Gg correlated to ©g, (with respect
to Qi). These definitions also hold for in the place of

eoi.eGs-

Let Sqj be a sjTitactical ©g which is correlated to ©g, with
respect to Qj. Let Sfii, have the form ©gifWi, where at

least one of the arguments is a let Sfiij have the form
Sgj(5l't....?r,), where 31', ( 1=1 to n) is D,[3I.l if SI. is nota S;
if SI, is a S3,, let SI', be a S3 of Sj, to the substitution-values of
which belong the Qj-corfebles of the substitution-values of 23j.
Then we call ©fuj a s)-nlacucat ©fii correlated to ©fiij (w ith respect
to Qj). Let Gfii, be a sj^itactical ©fit correlated to ©fii,. Let ©|
be constructed from ©fu by means of operators, and similarly ©j
from Sfuj by means of corresponding operators. Then we say that
©I IS a sjntactical sentence correlated to ©,. Every sentence that
contains a quasi-s^mtactical ©g, ipr. or ©fit, is called a quasi-
syntactical sentence. For compound quasi-sjmtactical sentences,
the correlated syntactical sentences are constructed in a manner
analogous to the simple cases here described.

Esamfile: Let *r,(F)' and *P,(F,h)’ be quasi-s>’ntactical Gfu in
S,. Let the correlated syntacticnl Gfu be ‘Qi(r)’ or 'Qjfr.jO’.
respectively. Then the syntactical sentence ‘(x) [Q, (*)0(3J')
(Qtfx.y))]’ is correlated to the quasi-syntactical sentence
•(nfP.IFlSfatilCPsfF.w))]’.

'The difference between the quasi-s}-ntictical sentences and the
others is connected with the difference betw ecn 8>'ntactical concepts
and the concept ‘true’. If one were to take ‘true’ as a sjTitactical
term, then every sentence whatsoever, G,, in relation to every partial
expression, 9I„ would be quasi-s}-ntactical. For Si is alwa>-s equi-
pollent to the sentence ‘ ?I, is such that S, is true.’ If Sj is a hs>eal
language, then, with respect to S,. ‘true’ coincides with '8nal>nic'
(that is to say, there are here no synthetic sentences; see 'Theorem
52 . 3 ). Consequently, in this case, tlic concept ‘quasi-syntactical*

Uist pr|(^i) IS analytic'; for either both sentences are analytic or both



237


§63 QUASI SYNTACnCAL SENTENCES

are contradictory Therefore, prx(3i) w a quasi syntactical sentence
in respect of 3i But m relation to the descriptive Language I this
IS not the case If fUi is an undefined fu*, then pti (fu, (nu)) is syn
theoc, and therefore not equipollent to the syntactical sentence
‘ fuifnu) IS such that pti (fu,(iui)) is analytic , for the latter is con
tradtctory When, in what follows, we establish the fact that certain
sentences m certain languages are quasi-syntactical sentences, this
means that they are still quasi-syntactical even if we expand the
language so that it becomes descnpUve (and m such a way that
descriptive arguments for the positions in question exist) [Later,
for mstance, we shall assert that the of the logic of modahnes are
quasi-syntactical , by which we mean also to mamtain that they are
soil quasi syntactical even if we extend the calculus of the logic of
modabues by admitting synthetic sentences, also, as arguments For
the consequence-predicate of the logic of modabties (e g for the
symbol of strict impbcation and similar ones) this is shown by the
example * Limp* on p 23s ]

§ 64 The Two Interpretations of Quasi-
Syntactical Sentencfs

L>et the sentence (oiof the form <991(^1) be quasi-syntactical, and
let the sentence Sjoftbe form €9t(^beacorreIated,and hence an
equipollent, syntactical sentence We will distinguish two possible
interpretations which might here be intended (This is only a
material, non-formal mvestigation which serves as a preliminary
to the formal definitions ) In both, ^is interpreted as a syntactical
designation of the expression and Ggj ^ 2 designation of a
syntactical property of expressions The two cases to be dis-
tinguished are as follows (i) where €91 is taken as equivalent m
meaning to Sgj, and (*) where it is not In the case of the first
mterpretation, EgxaswcUasSgjdesignates a syntactical property,
since Si and Sj are equipollent, the equivalence in meaning of the


l\

term ‘autonymous’ has already been explamed on p 156, its
stnctly formal definition will be given later ] In the case of the
second mterpretation, Sgi designates not a syntactical property
but an object-property, which is attnbuted to the object designated
by ?Ii (not to the expression Bj) m the sentence Si We will m
general assign to the material mode of speech any sentence which



PART rv. GENERAL SYNTAX


238

(like Si in the second interpretation) is to be interpreted as at-
tributing to an object a particubr property, this property being
quasi-sj-ntactical, so that the sentence can be translated into
another sentence w hich attributes a correlated sjntactical propert)*
to a designation of the object in question. In contrast oath the
material mode of speech of the quasi-syntaetical sentences of the
second interpretation \vc have the formal mode of speech of the
sjTitactical sentences.

Example, j Quasi-sjntactical sentences ; (a) autonjmous mode of
speech, “Fite is a number-word*’; (fc) material mode of speech,
“ Five is a number.” 2. Correlated simtactical sentence : “ ‘ ’ »

a number-word.” (For the sake of simplicil>‘, in 10 and 2 we have
taken as pr that are equivalent in meaning the same w'ord, ‘number-
word ’.)

Our task now 1 $ to represen/ fomoUy the differenro between
the two interpretations that has just been indicated matcnally.
WTiich formal syntactical properties of Sg, and Sgj correspond to
the fact that is intended as equivalent in meaning to Spi and
thus as a designation of a syntactical property ? It is not neeessarj*
for Sgi and Sg, to be sj non> mous (or L-sjmonjmous) • for it may
well be that, m spue of their equivalence in maning, we intend to
admit only Sfli with autonj-mous arg:uments, and not (Sgj. In this
case <2gj(5lj) would certainly be equipollent to Sgi(?Ii); but
Sgj(2I,) would not — for it need not be a sentence. Dut if Ggi is
intended to designate a sj-ntactical property, and, further, the same
syntactical property as Sg-, then Sgi (?lj) is equipollent to SSiCJIi).
On the basts of this prclimmar)- consideration, we formulate the
following formal sjntactical definitions (for the sake of simplicity
we do so in relation to Sg’ ; the definitions for the case of tw o or
more arguments arc analogous, likewise those referring to ^r).

Let the sentence Si of Sj have the form Sgi(5I,) and be quasi-
S)-ntactical in relation to HI, ; and let 9li not be a S. Let S, contain
both S, and a synUx of S, with respect to O,. Let Sg, (CiPIi))
be a sj-ntactical sentence of S, correlated to S, with respect to O,.
Two cases are to be distinguished: ( 1 ) Sgit^^iI^liDis a sentence
of Sj and, in S,, is equipollent to (cgj (Ci fJIil); likewise, for everj'
which is isogenous with is equipollent to

‘Sfls (^i Plj)- fhis case we call % autonymous in ©1 (with
respect to Qi), and €1 a sentence of the outoitymous mode of



§ 64. INTERPRETATIONS OF QOASI-SYNTACTICAL SENTENCES 239
Speech (with respect to Qj) (2) The given condition is not
fulhlled In this case we saythatSibelongs to the material mode
of speech (with respect to Qj) Let Ot he a translation of into
Sj m respect of sentences, further, let the Qg-correlate of every
quasi-syntactical (with respect to Qi) sentence of Sj be a syntactical
sentence correlated to It wthrespeetto Qi, andlettheCj-correlate
of every other sentence be the sentence itself The Qj'^tanslation
of the sentences of the material mode of speech into correlated
syntactical sentences is called a translatton from the matenal into
the formal mode of speech

It i« to be noted that the differentiation betivcen autonymous
and matenal modes of speech is concerned with interpretation
This means that this differentiation cannot be made m relation to a
language tihich is given as an isolated calculus without any m-
terpretation But it does not mean that the distinction lies outside
the domain of the formal, m other words, of syntax For, even the
interpretation of a language can be formally represented and thus
be incorporated in the syntax. As we hai e seen, the interpretation
of a language m relation to an assumed language can be
formally represented either by the translation of Sj into S*, or by
the mcorporation of as a sut>*Ianguage in a third language Sj,
which is constructed from the language S{ by extension If is a
quasi-syntactical sentence of S„ and if the mterprctation of is
formally determined by the fact that Sj is a sub-language of a
language Sj which contains also the syntax of Sj, then, according
to the definitions just given, it can be detemuned whether Sj be-
longs to the autonymous or to the material mode of speech But in
practice we are frequently not in a position to make this distinction
wnth accuracy , namely, where it is a question of a system which
another author has constructed without giving either the trans-
lation of S2 into, or Its incorporation in, another language also con-
taining the syntax of If in such a case no mterpretation what-
soever IS given, then the distinction disappears In the majonty of
calculi which have been constructed up to the present, although an
mterpretation has been given, it usually not been done by
means of strict syntactical rules (cither incorporating m, or trans-
lating it mto, some other formally established language Sj), but
only by matenal explanations, that is to say, by the translation of
sentences of mto more or less vague sentences of a word-



P.U5T TV. CESERAL STXT.«


240

language. If undertake on the baas of such explanations a
translation of Sj into a formally established language S., we can, at
most, suppose that what nas meant by the author has been more or
less accurately expressed, that is, that we ha%-e proposed a trans-
lation which debates less or more from that which the author him-
self would hare proposed as a translation of Sj into S*. WTien in
what follows we attribute certain sentences of the calculi of other
authors, or of the word-language, cither to the autonjmous or to
the material mode of speech, it must be noted that this is not in-
tended as an exact and final classification; in the case of the sen-
tences of other calculi, the differentiation depends upon the in-
terpretatirc explanations gitxn by their authors, and in the case of
the sentences of the word-language upon consideration of the
ordinary \>at of language. On t!ht other hand, the dtcasvon that
certain sentences are quasi-syntacticil (not genuinely sjtitactical)
can be made with the same degree of exactitude with which the
language in quesQon is itself constructed; in this we need take no
heed of interpretation, whether girtn msterially or formally.

§ 65. Extensionality in Rel.\tion to
Partial Sentences

By way of preparation for the definition of cxtensionalitj', we will
first examine the definition that has been usual hitherto. An ixfuj
with one I’ariable f, is commonly called citcnsional (or a truth-
function) in relation to f„ if for any Sj and Sj uhatsoettr, haviog

the same trulh-valuc, and have the same

truth-value. If, for instance (in a symbolism like ihSt of II).
‘T(/')’ is an Gfu of the kind in question, then *T(p)‘ is called
extensional, if ‘(fS9)0 (T(p)5T(9))‘ (Sj) is true. We must
formulate this definition differently; we do not use the term ‘true*
becauseitisnotagenuinesyntactiedterm; further, we will not make
the limiting assumption that sentential variables and symbols of
proper equivalence and implication exist in S. Since Sj must be
not only true (mdetermiiutely) bat valid, we can replace the given
condition by the following; for any closed sentenca whatsocicr,
aay ‘A’ and 'D*, *T(A)hT(B)* (S*) must be a consequence of
* A 5 B ' (Si). The implication having been diminated, we will now



§ 65 EXTENSIONALITY in RCtAtlON TO PARTIAL SENTENCES 24I

elinunate the eqmvaleoce also S| has the property that ‘ B ' is a
consequence of Sj an<J ‘ A’, and* A* a consequence of Sjand'B’
Further, Sj is the poorest in content of the sentences having this
property, that is to say, if any 5^^ likewise possesses the property m
question, then Sg is a consequence of hence, S3, if a conse*
qucnce of Sj is also a consequence of These considerations
lead us to the statement of the following defimtions.

In analogy with the previously defined concepts — absolute
concepts, as it were — of the equipollence of two (or two S), the
coextensiveness of two 0g (or Sfu or ^r), the synonymity of two
51, and the identity of the course of values of two 5lg (or ?Ifu
or lyu), we iviU now define the correspondmg relative terms in
relation to a sentential class and are called equipollent (to
one another) m relation to 51i if Gg is a consequence of
and Qi a consequence of Gg^ and Gg^ are railed

eoextensne (with one another) in relation to it, if every two full
sentences with equal arguments are equipollent in relation to A, ,
similarly for two Gfu or two (isogenous) Two isogenous ex-
pressions 51, and arc called synor^ous in relation to 5t„ if every
S, 13 equipollent to ®i^^J and to relation to it. We

say that 5lg, and Hlg, ha^ e the same course of vahus tn relation to it,
when every tno full expressions with equal arguments are synony-
mous in relation to it, , likewise for two 51fu or (jU
Theorem 65 1, (o) If two Gate equipollent, then they are also
equipollent in relation to every it- (i) Analogously for co
extensiveness (c) Analogously for synonymity (d) Analogousl)
for identity of the course of values
Theorem 65.2. (0) If G, and Gg are equipollent m relation to a
vahd it, then they are equipollent (absolutely) (6) Analogously
for coextensiveness (e) Analogously for synonymity (d) Ana-
logously for identity of the course of values
Extensionality in relation to partial sentences G, is called ex-
tend onal m relation to the partial sentence Gg if for any closed G3
and any it, such that G, and ^ arc equipollent m relation to it,,

S, and always equipollent in relation to it. An Sg,

to which S are suitable as arguments is called extensional if, for any
closed G, and Gg and any it, such that G, and Gg are equipollent



PU?T IT. CENEKAL STNT.M


in relation to 51 i. Sgi(Si) and Sgi(S.) art always equipollent in
relation to Siy Cbrrespondingly in the case of an Sfu or to
which S are suitable as arpunents. If every sentence of S is et-
tensional in relation to every dosed partial sentence, then we call
S rxifKSJOfu:} in rrlabon to partiaJ serJrvts. ’Isrensiona] ’ is to
mean the same as ‘not cittnsional* (in the diiferent connections),
[‘Intensional’ as we use it means nothing more than this and in
pardculir it means nothing He ‘rdated to sense', etc.; in many
authors the word has a meaning of this Idnd, or even a mixture of
the two meanings (see § “t). J

Theorem 65.3. If S is eitcnsional in relation to partial sen-
tences, then all Sg, ;fu, and of S to which ; are suitable as
arguments arc eitensional.

Theorem 65>4> Let S be exlcnsional in respect of partial sen-
tences. (a) If two dosed ; arc equipollent in relation to then
they are also sjmonjTDOus in rdadon to Siy (fi) If two dosed ; are
equipollent, then they are also sjmonymous. (c) If two closed
whose arguments are ; are coextensive in relation to 5 ^ j, then they
are also sjtionjTnous in relation to ft,, (d) If two dosed whose
arguments are ; are coexterumT, then they are al-so synonymous,
(r) If tn*o dosed Qu whose arguments are S hare the same course
of values in relation to ft,, then they arc also synonymous in relation
to fti- (/) If two dosed yu whose arguments arc ; have the
same course of values, then they art also synonymous.

Tbearem 65.5. .Senrmria/yitJjfft'oiu. If a S/t or a t! possesses a
characteristic, then it is cxtatsionaI;and conmrscly. — ^Tbus, proper
negation, proper implication, etc., are extensional.

Theorem 65.6. If S is extensional in re s pect of partial sen-
tences, then all 10! are extensional.

Theorem 65.7. Let ! 0 l, be s /roprr ffiaralfficr in S. Then it is
true that: (a) ;, and ;, are alwaj’s equipollent in relation to
S 3 !i(Si»S 0 - (^) S is extensional in relation to partial sentences
if, and only if. for any dosed and

is always a consequence of Sf,(S,.S0- (0 Further, let S!, be
a proper imphcatioo in S; then S is extensional in relation to
partial sentences if, and only if, for any dosed ;,, ;,, and ;^

Si;(si.(S.,SJ, b .-did.



§65 ETfENSIONAUTTIN RELATION TO PARTIAL SENTENCES 243

Tlieorem 65.8. Let ‘ ’ be a symbol of proper equivalence m

S (n) If, for closed Sj ^ ^=*93 u a consequence of SKi,
but Sj = Sj 13 not a consequence of then Sj 13 mtensional

in relation to Si (4) Ift for two closed sentences G* and Ss,
01=03 13 valid but is not valid, then 13 in-

tensional in relation to Gi

Ggf 13 called an Gg of tdtnMy, if every two possible closed a^u-
ments SIj and ^ are always synonymous m relation to Ggi (2ti, 3Ii)
An Gg of identity, Ggj, is called an Sg of proper identity if for
every two possible closed arguments ?Ii, SIj, which are synonymous
m relation to Sli, Ggi(^i,^Ij) i3 always a consequence of ftj,
otherwise it la called an Gg of improper identity If Sg^ is an Gg
either of proper or of improper identity, then Ggi(?Ii,?Ij)is called
a sentence of proper (or improper, respectively) Identity (or an
equation) for IQi and A ptf is called a symbol of proper or
improper identity (or predicate of identity, or symbol of equahty)
m general, or for all expressions of the class SKi, if the sentence
pti(3ri,3Ii) IS a sentence of proper (or improper, respectively)
identity for and ?Cj for any closed ?I or any closed ?I of
respectively (S may, for instance, contain different symbols of
identity for 3» ®. )

Theorem 65 9. Let be a sentence of identity for the dosed
expressions and % (a) and ^are synonymous m relation to
Gi (4) If Gi is valid, then and  are synonymous (absolutely)

Theorem 65 xo. Let S be extensional m relation to partial
sentences (a) If SJIi is a proper equivalence, then for any two
closed sentences Gj and Gj, Sf^CGi.G*) is always a sentence of
proper identity for Sx and G, (4) A symbol of proper equi-
valence 13 a symbol of proper identity for sentences

§ 66 ExTENSioNALirv IN Relation to
Partial Expressions

Here we shall agam start from the usual definition (usmg the
symbolism of II) It is customary to call an GfuJ with a variable
Pi, say ‘M(F)’, extensional in relation to 'F' if

•(<)(FW = G(.))3 0«(iOaM(G))’


x6-2



TART IV. GENERAL SYNTAX


244


is true. We can, as previously, alter the formulation of the con-
diUon thus: for any 'Pi' and ‘P*’, ‘M(P,) = M(P,)’ must always
be a consequence of With this as a basis,

we now give the following definitions.

Extetisionahty in relation to partial expressions. Let ^rj occur in
©i; 61 is cuilled extensional in telation to if for any dosed
•^r„ and any ilj such that and are coextensive in relation


to Ri, ©1 and J ^ aUwys equipollent in relation W Ri-

Let 5 tii occur in ©t ; S| is called extensional in relation to
for any closed and any Rj such that gili and gitj have the

same course of values in relation to R,,S, and Si arccqui-

U>'d


pollent in relation to Rj. If G, u extensional in relation to all the
dosed S, ^r, and 5*1 which occur in Sj, Sx is called extensional.
An ©fl„ to »hich ipr, Of *3 suitable ns arguments, is called
extensional if every full sentence of Sgi with closed arguments is
extensional in relation to every argument. Correspondingly for
every Sfux or ^fx to which 5 u, or S are suitable as arguments.

If every sentence of S is extensional m relation to every dosed
partial expression ipr (or gu) then S is called extensional in relation
to ipr (or gu, respcciivdy). If S is extensional in relation to partial
sentences, to ^r. and to gu, then S is called extensional.

Theorem 66.1. (o) If S is extensional in relation to Ipr, then two
closed ^r which arc coextensive (absolutely or in relation to Rx)
are always (ibsolutcly or in relation to Rj, respectively) synony-
mous. ( 6 ) If S is extensional in relation to gu, then ttvo dosed gu
which have the same course of values (absolutely or in relation to
Ri) ate always (absolutely or in relation to Rx, rcsp^ivcly)
synonymous.


Examples: The languages of Russell and of Hilbert and our onn
Languages 1 and II are extensional in relation to partial tenieneet.
That is shown, for instance, by the criterion of Theorem 65.7 c (cf.
Hilbert [Logik], p. 61). The symbols of equivalence in these lan-
guages are aymbols of proper equivalence and hence, according to
Theorem 65, lo^, they are also aymbols of proper identity for G.
TTie fortn of the language swll be simpler if only one symbol of
identity is used (ss in I and II, and in contrast with Russell and
Hilbert), the same for 6 as for 3 . *91 and so on. If from Russell's
language R we construct a new language R', by extending the rules
of formation to admit of undefined pra with S as arguments, then



§ 66. ErTENSIONAUTT IH RELATION TO PARTIAL EXPRESSIONS 245

R' IS no longer necessarQy extensional tn relation to partial sen-
tences, m order to guarantee exteosionality here also, we can pro-



II in the same way, then it la extensional in relation to partial sen-
tences Here no new primitive sentence is necessary, since we use
the symbol of identity aa symbol of equivalence, so that the above
sentence of implication u demonstrable

Languages I and II are also txtennonal in general In II the ex-
tensiooahty in relation to ^r and Su >a guaranteed by PSII 22 and
23 (see p 92) In the case of the other languages, the question of
extenstonahty in relation to and Su can only be decided after
further stipulations have been made, espeaally regarding what
undefined “prj (for «> t) are to be admtted
The languages of Lewis, Becker, Chwistek, end Heytmg are
trttamonal, for partial sentences as well as for the rest (see § 67)


§67. The Thesis OF Extensionality

Wittgenstein ([rrartafui], pp 102, 142, 152) put forward the
thesis that every sentence is **a truth*function of the elnsentary
sentences’* and therefore (m our temunology) extensional tn re-
lation to partul sentences. Following 'Wittgenstem, Russell
{[htroduettm], pp 13 ff , [ftw Math ] Vol J, and edition, pp
xrr and 659 ff ) adopted the same view with regard to partial
sentences and predicates , as I also did. but from rather a different
standpoint (XAufhau\, pp 59 ff) In so domg, however, we all
overlooked the fact that there is a mulupliaty of possible languages.
Wittgenstein, espeaally, speaks contmuallj of "the” language
From the point of neiv of general syntax, iC is evident that the
thesis 13 mcomplete, and must be completed by stating the lan-
guages to which It relates In any case it does not hold for all
languages, as the well-known examples of mtensional languages
show The reasons given by Wittgenstem, Russell, and myself, m
the passages ated, argue not for the necessity but merely for the
possibility ofan extensional language For this reason we will now
formulate the them of extermandhty m a way which is at the same
time more complete and less ambitious, namely a umvendl
Lmguage of saerue may bt exteymonal, or, more exactly for ev ery
given mtensional language Sj, an extensional language S, may be
constructed such that Sj can be translated mto S, In what follows



246 PWT rV. CENER.U. SYNTAX

we shall discuss the most impoitant examples of intensional sen-
tences and demonstrate the possibility of their transbtion into
extensional sentences.

Let us enumerate some of the most important examples of i«-
tensional sentences. ‘A’ and ‘B* are abbreviations (not designations)
for sentences, e.g. ‘'It is raining now in Paris", etc. t. Russell
([Pn’fic. MalA.], Vol. l, p. 73 and [Math. Phil.'], pp. 187(1., and
similarly Behmann [Logtit], p. 29) gives examples of approximately
the following kind* "Charles says A", “ Charles behcres A”, "it is
strange that A ”, “ A is concerned with Paris ", Incidentally RusseU
himself later, influenced by'VVittgenstem’s opinions, rejected these
examples, and asserted that their intensionality was only ap-
parent ([/Vine. A/alA.], Vol. t. and edition. Appendix C). We
prefer to say instead that these sentences are genuinely intensional
but are translatable into extensional ones. 2. Intensional sentences
concerning being-contained-in and substitution in relation to ex-
pressions : " (The expression) Prim (3) contains (the expression) 3 " ;
"Prim (3) results from Prim (x) by substimting 3 for r”. Sentence*
of this kind (but written in symbols) occur in the languages of
Chwistek and Heyting. 3. Intensional sentences of the logic of
modalities: "A is possible”; "A is impossible”; “A is necessary”;
" B is a consequence of A ” ; " A and B are incompatible Sentences
of this kind (m ij-mbols) occur in the aj-stems of the logic of modali-
ties constructed by Lewis. Becker, and others. 4. The following
intensional aentences are akin to those of the logic of modalides:
"Because A, therefore B”; "Although A, nevertheless B”; and the
like. That any sentence ^ of the examples given Is intensional in
relation to ‘A’ and 'B' follows easily from the criterion of TTieorem
65.80. If, for instance, 'A' is analytic and 'C is s}Tithedc, then
_P' ‘f • consequence of* C’; but the false sentence "A is neces-
sary =;C is necessary” is not a consequence of 'C. These examples
Will be discussed in greater deuil in what followa.

The above examples appear at first glance to be very different
in kind. But, as a closer examination will show, they agree with
one another in one particular feature, and this feature is the reason
for their intertsionaiity: ail these sentences are quasi-syntactieal sen-
tences and, in particular, they arc quasi-syntactical with respect to
those expressions in relation to which they are intensional. With
the establishment of this characteristic, the possibility of their trans-
lation into an extensional Janguas;e is at once given, inasmuch,
namely, as every quasi-syntactical sentence is translatable into a
correlative syntactical sentence. That the 83-ntax of any language
(even an intensional one) can be formubted in an extensional
language is easy to see. For arithmetic can be formulated to any



247


5 67 THE THESIS OF ECTENSIONALITT

desired extent m an extensional language, and hence an anth-
metized syntax also Incidentally this is equally true of a syntax m
axiomatic form

^Vhat we have said holds for all examples of mtensional sen-
tences so fat known Since vre are ignorant of whether there exist
mtensional sentences of quite another kmd than those known, we
are also ignorant of whether the methods described, or others, are
apphcable to the translation of all possible mtensional sentences
For this reason the them of extensionahty (although it seems to me
to be a fairly plausible one) is presented here only as a supposition

§ 68. Intensional Sentences of the Autonymous
Mode of Speech

Some of the known examples of mtensional sentences belong to
the autonymous mode of speech When translated into an ex-
tensional language, they are transformed mto the correlated syn-
tactical sentences We will first of all examine the converse process,
namely, the construction from anextensional syntactical sentence of
an mtensional sentence with an autonymous expression. By this
means the nature of these mtensional sentences will become clear

Let and Sj be extensional languages, and let Sj contain
as a sub language and the syntax of by virtue of Qj Let be
an ®, ^r, or 5 u of Sj, and (in S,) have the form ipr*(Qi plj)
In matenal interpretation Gj is a syntactical designation of
^i» Sa ascribes to a certain syntactical property expressed by
Spr* >s m general not a sentence of S, Now, out of Sj,

we construct an extended language S, (that is to say, Sg is a proper
sub-language of Sj) The rules of formation are extended as
follows m S3, for every 9 ^ which 1$ isogenous with m Sj,
a sentence, and hence ^rg( 9 Ii) also (let this be Sj);
further, the rules of transformation are extended as follows m S3,
for every 9I3 which is isogenous with 91 , in S,, $tg(9l3) is equi-
pollent to ipr3(QiPl3])i and therefore S, is also equipollent to
? 5 tg(Q,piJ)(tlus IS Sg) Then, according to the entenon given
carher (p 238), 91 , is aulot^ous m S, A sentence which is
formulated like S, is in general mtensional m respect of 91 ,

Example Let S, be I As syntax-language m S, we will take the
word-language Let the S,-coiTeIate3 (the syntactical designations)



PART IV. GENERAL SYNTAX


248

be formed with inverted commas. Let ?Ij be *0'i = 3’, and ic.
cordingly ?!,, ''0'l = a’'. Let ^be**0'l = 2’ is an equation’. Thra
€, is ‘0" = 2 is an equation’. For S, we stipulate that Sjand S|be
mutual consequences of one another; and likewise, corresponding
other sentences with the same ?lr. Then 'O’* = 2' is autonjtnous in
Si, and, according to Theorem 65.S6, Gj is intensional in relation to
*0” = 2’. For let % be ‘Prim(3)'; then SjSSIi is analytic but
; Pnm (3) is an equation’ (64), because it is equipollent to "Prim (3)’
IS an equation’, is contradictory; hence, since Gi is analjtic, Si=Si
IS contradictory.

Now some of the examples of intensional sentences previously
mentioned have the same character as the intensional sentences
constructed in the way here described: their intensionality is due
to the occurrence of an autonjmous expression. We trill dte some
examples of this, at the same time giving the correlated syntactical
sentences. The latter may belong to an extcnsional language.
[Sentences i h and 2 b belong to descriptive syntax, 3 ^ 4 ^

5 J to pure synta.x. The preceding investigations and definitions
have ail been given in relation to pure syntax only; they may,
however, be correspondingly extended to apply to descripthe
syntax.] To interpret these sentences as belonging to the autony*
mous mode of speech seems to me to be the natural thing, espe*
cially in the case of 40 and 50. However, if anyone prefers not
to ascribe one of them (say 2 o or 3 o) to the autonymous mode of
speech, he is at liberty to do so; the sentence in question will then
belong to the material mode of speech. The only essential points
are: (i) these intensional sentences are quasi-syntactical; and
(2) they can (together with all other sentences of the same lan-
guage) be translated into extensional sentences, namely, into the
correlated syntactical sentences.

Intensional sentences
of the autonymous mode
of speech

Let ‘A ’ be an abbreviation (not 1

1 a. Charles says (writes, reads)

A.

2 a. Charles thinks (asserts, be-

lieves, wonders about) A.

[Of the same kind is the following: "it is astounding that. . ; "• that
IS to say: "many wonder about the fact that. .


I Extensional sentences

of syntax

designation) of some sentence.
I 16, Charles saj’S ‘A’.
xb, Charles thinks ‘A’.



§ 68 INTE^SIO'^AL SENTENCES OF THE AUTOVTMOUS MODE 249

3a A has to do With Pans 3S ‘Pans’ occurs in a sentence

which results from ‘A* by
the elimination of defined
symbob

4a Pnm (3) contains 3 46 *3’ occurs m 'Pnm(3)’

5 a Pnm (3) results from gfc ‘Pnm (3)’ results from

Pnm (it) by the substitution ‘ Pnm (x) ’ by the substitu-

ofgforx I non of ‘3’ for ‘x’

We have here mterpreted the previously mentioned (p 246)
examples of mtensional sentences put forward by Russell, Chwistek
andHeyting as sentences of the aulmymous mode of tpeeeh Thisin-
terpretatton is suggested by the relevant mdicanons given by the
authors themselves Russell’s sentences are already presented m the
word language, and for the sentences of Chwistek and Heyting
which are formulated m symbols, the authors themselves give para-
phrases in the word-language corresponding to 411 and 5a

Chwistek’s system of so-called semantics is, on the whole, dedi
cated to the same task as our syntax But Chwistek throughout em-
ploys the autonymous mode of speech (apparently without bemg
aware of it himself) He uses as the designanon of an expression
with which a sentence of semantics is concerned either this ex-
pression Itself or, altemauvely, a symbol which is synonymous with
It (and IS thus, onginally, not a designation but an abbreviation for
It) As a result of the employment of the autonymous mode of
sneech many sentences of Chwistek’s semantics are mtensional
Because of this, he has come to the conclusion that every formal
(Chwistek says ‘'notninalisQc”) theory of Imguistic expressions
must make use of mtensional sentences This view is refuted by the
counter-example of our syntax, which, although strictly formal, is
consistently extensional (this is most clearly seen m the formalized
syntax of I m I, m Part II) The fact that Chwistek believed himself
forced to abandon the simple rule of types for his semantics and to
return to the branched rule (see §6oa), was also, m my opinion,
only a consequence of his use of the autonymous mode of speech
Heyting gives as the word-translation of certam symbolic ex-
pressions of his language “the expression which results from a
when the vanable x is replaced wherever it appears by the com-
bmauon of symbols p ([MarA l], p 4) and “g does not contam
([AfatA i], p 7) Such formulations, like our examples 40 and 5a,
belong, without any doubt, to the autonymous mode of speech
But even the sentential calculus of Heyting’s system [I>ogiA] contams
mtensional sentences, sentential junctions which can be shown to

make it natural to suppose not only that the whole system can be
translated by us mto a system of syntactical sentences, but also that
this was in a certain sense the author’s mtention “ In a certain
sense” only, because the distinction between the object- and the



250


PART IV. GENERAL SYNTAX


synta.t-hnguage 9 is nowhere explicitly made; so that it is not even
clear which language it is whose syntax b supposed to be represented
in the system. According to [GrundUgung], p. 113, the assertion
of a sentence (which is formulated symbolically by placing the
symbol of assertion in front of the sentence) is “the establishment
of an cmpincal fact, namely the fulfilment of the intention expressed
by the sentence" or of the expectation of a possible expcnence.
Such an assertion may mean, for example, the historical circum-
stance that I have a proof of the proposition in question lying in
front of me. According to this, the assertions in Heyting’s system
should be interpreted as sentences of descriptive syntax. On the
other hand, Gddel (/Tof/o^uium 4), p. 39, gives an interpretation of
Heyting's system m which the sentences of the sj’stem would be
purely syntactical sentences about demonstrability; “A* is de-
monstrable ’ is formulated by means of ‘ BA ’, and consequently in
the autonymous mode of speech.


§ 69. Intensional Sentences of the Logic
OF Modalities


We shall now give some further examples of inlnaional sen*
tenccs together with their tramUtion into txteraional synlaetieal
senteneet. By means of this translation the intertsional stntnces are
shown to be quasi^tyntcetical. Sentences la to 40 contain terms
that are usually known as modalitits [‘possible’, ‘impossible’,
‘necessary’, ‘contingent’ (In the sense of 'neither necessary nor
impossible ’)]. Sentences 5a to ja contain terms that are similar
in character to these modalities, and are therefore treated by the
newer systems of the logic of modalities (Lewis, Lukasiewicz,
Becker, and others) together with them. In these systems, the
modal sentences arc symbolically formulated in approximately the
same way as our examples 16 to 7A. Examples 8 a are intensional
sentences of the ordinary word-language which we add here be-
cause, as the syntactical translation ahows, they are akin to the
modal sentences. ‘ A ’ and ' B 'are here sentences — i.c. abbreviations
(not designations) of certainsentcnccs(such as synthetic sentences)
cither of the word-language or of a symbolic language.


Intensional sentences of the
logic of modalities


I o. A is poj-

aa. A.~A ii
impossible.


zb. I(A.— A);
~I’(A.~A).


Exlensional sentences of
le. ‘A’ is not contradictory,
ar. * A. ~A’iscontradictory.



§6q INTESSIOSAL SENTENCB — LOGIC OF MODALITIES 25I


3^ ‘A v^-A’ IS analytic.

4c ‘A ’ 13 synthetic (‘A’ is
neither analytic nor contra-
dictory, neither ‘A’ nor
‘ '—A' IS confradictory )

5r 1$ an L-consequence
of ‘A’


6c ‘A’ and ‘B‘ are L.-equi-
poUent (1 e mutual I.>-conse-
quences)

7c 'A* and ‘B’ are L-com-
panble (‘—B* is not an
L-consequence of ‘ A’ )

8c 'A’ IS analytic, ‘B’ is an
L-consequence of ‘A’, ‘B* is
analytic (* A’ u valid, ‘B’ is
a consequence of ‘A’, ‘B' is
valid )

Since the terms used m the logic of modalioes are somewhat vague
and ambiguous it is also possible to choose other syntactical terms
for the translations , tn a c, for instance, instead of ’contradictory' we
may put * contravalid ‘ L-refutable’, or ‘ refutable ’ Similarly in the
other casea, instead of the L-c term we can take the general c-texm,
the L-d-term, or the d-teno With regard to 8 c, in the majority of
cases the general c-tenn (or the P-term) is perhaps more natural as
an interpretation of 8 a than the L-tenn. The difference between the
so-called logical and the so-called real modalities can be represented
in the translation by the difference berveen L- and general c-terrtis
(or even P-terms)

9a A is logically impossible 1 pe *A’ is contradictory

roa A is really impossible j ioC| *A' is contravalid

I loct *A' IS P-contraialid

The translation of lOo depends upon the meanmg of ‘really im-
possible* If this term is so meant that it is also to be applied to cases
of logical impossibility, then the translation locj must be chosen,
otherwise ioC| Analogous transbaons may be given for the three
other modalities — for*Jogically(or“reaIly”, respectively) possible’,
‘necessary’, and ‘contingent’

That sentences 10 to loa and th to 76 are tntensvmal is easily
seen. [Example Let *Q’ be an undefined ptc, and ‘ = ’ a symbol
of proper equivalence Let S, be 'Pnm(3)“Q(2)’, S, be
' Pnm (3) IS necessary’, and ^ ‘Qfz) 15 necessary’ ThenS,= l=j
cannot be a consequence of S, (for is synthetic, Sj analytic, and
S, contradictory, and hence is contradictory) 'ITierefore

(by Theorem 65 76) ®i is mtenstonal in relation to ‘ Pnm (3) ’ ]


3fl Av~A is
necessary
40 A IS con-
tingent


S a A stnctly
implies B, B
13 a conse-
quence of A-
6<j A and B
are stnctly
equivalent


36 N(AV~A),
~P~(Av~A)
4;. ~N(A).~
1 (A),

P(A).P(-A)
56 A<B


6l> A=B


•jh C(A,B),
~(A< ~B)


70 A and B
are compat-
ible

8a Because A, therefore B, A,
hence B



252 r.utT nr. cenbliu. stkt.w

Since the sentences here sre qussi-syntacticai, to an
interpret them as sentences other of the autonymous or of the
material mode of speech. In the case of the sentences of § 6S, the
a-erbal fonnuUtjons, or the verbal paraphrases given by the
authors, suggest interpretation in the autonymous mode of speech.
On the other hand, in the c«c of the symbolic sentences ihvoyb,
it is not dear which of the two interpretations is intended — in spite
of the fact that paraphrases (of the same kind as sentences t c to
7 fl), and sometimes even detailed material explanations as well, are
given by the authors. In relation to a particular example, the
deasii-c question (as formulated in the material mode) is the fol-
lowing: Are *I(A)‘ and ‘A is impossible’ to refer to the sentence
* A‘, or to that which is designated by ’A’? In the formal mode:
Is “A’ IS impossible’ also to be a sentence? [If so, it must un-
doubtedly be equipollent to ‘ A is impossible.’] If the answer is in
the affirmatw-e, then ‘I(A)‘ and ‘A is impossible’ both belong to
the autonjTnous mode of speech; if in the negatiit, then they
belong to the materul mode of speech. The authors do, it is true,
say that the sentences of modalit)’ are concerned with propositions,
but this assertion ivould decide the question only if it xrert quite
dear what was meant by the term ’proposition’. We will discuss
the two possibilities separately.

!. Suppose that by the term ’proposition’ the authors mean
what we mean by ’sentence’. Then the term ’proposition* is a
sjTitactical term, namely, the designation cither of ccruin phj'sical
objects in descriptive syntax or of certain e.Tpressional designs in
pure syntax. Then *A is impossible* is concerned with the sen-
tence ‘A’, hence is equipollent to "A* is impossible’, andbdongs
to the autm^-moiu node 0/ jpeecA. In this osc the iniensionality of
the modal sentences does not depend upon the fact that the}’
speak about expressions (in the examples, about sentences, in
other cases, also about predlate-expressions) but upon the fact
that they do so according to the auion}'mous and not according
to the SjTitactical method.

2. Suppose that by a * proposition * the authors mean not a sen-
tence (in our sense) but that which is designated by a sentence.
[For instance, in Lewis’s [L<5ic], pp. 472 ff., the distinction be-
tween 'proposition* and ‘sentence’ is possibly to be understood
in this way.] We will leave aside the question of what It is that is



§69 INTZNSIONAL SENTENCES — LOGIC OF MODALITIES 253
designated by a sentence (some people say thoughts or the content
of thoughts, others, facts or possible facts), it is a question that
easily leads to philosophical pseudo-problems So we shall simply
say neutrally "that which is designated by a sentence" In this
mtcrptetalioo, the sentence *A u impossible’ ascribes unpossi-
bihty not to the sentence ‘A* hut to the A which 13 designated by
the sentence Here the impossibiti^ is not a property of sen-
tences “A‘ 13 impossible’ is not a sentence, it is therefore a
case not of the autonymous but of the maienal mode of speech.
‘ A is impossible’ ascribes to the A which is designated by the sen-
tence a quasi-syntactical property, instead of to the sentence
*A’ the correlated syntactical property (here ‘contradictory’)
[In this example, the second interpretation is perhaps the more
natural It is the only possible one in the case of the formulation
‘the process (or state of affairs, condition) A is impossible', see
§ 79, Examples 33 to 35 On the other hand, we are perhaps more
inclined to relate a sentence about the consequence-relation or
about denvability to sentences rather than to that wbch is desig-
nated by them, and accordingly to choose the first mterpreution ]
We shall see later that, m general, the use of the mater^ mode of
speech, though it is not madmissible, bruigs with >t the danger
of entanglement m obscurities and pseudo-problems that are
avoided by the appbcation of the formal mode So also here, the
systems of the logic of modabties are (on the whole) formally
correct, But if they are (m the accompanying text) interpreted m
the second way, that is, in the matenal mode of speech, then
pseudo-problems easily anse This may perhaps explam the
strange and, m part, uomtelligible questions and considerauons
which are to be found m some treatises on the logic of modalities
C I Lewis was the first to pomt out that m Russell’s language


of logic and of mathematics, that in it necessarily valid sentences can
be proved and a sentence which follows from another can be denved
from the former

Although Lewis’s contenaon is correct, it does not exhibit any
lacuna tnlAwi Russell’s language The requirement that a language be
capable of exp ressmgnecessity,possibility,theconsequencc- relation,
etc., u m Itself justifiable , it u fulfilled by us for instance in the case



254 part nr. GENERAL SYNTAX

of our Languages 1 and II, not bymeans of anything aupplementaiy
to these languages, but by the formulation of their sj-ntax. On the
other hand, both Lewis and Russell — they are agreed on this point —
look upon the consequence-relation and implication as terms on the

- • • • . P...1 -.1. .t.


implication without charactenstic). This is intended to express the
consequence-relation (or derivability-relation), that is to say, in
Lewis’s language, ‘ A < B ’ is demonstrable if * B ’ is a consequence of
‘A’. Lewis rightly pointed out that Russell's implication does not
correspond to this interpretation, and that, moreover, none of the
so-called truth-functions (in our terminology; the cxtensional sen-
tential junctions) can express the consequence-relation at all. He
therefore believed himself compelled to introduce intensional sen-
tential junctions, namely, those of strict implication and of the
modality-terTTU. In this way his system of the logic of modalities
arose as an intensional extension of Russell's language. TTie system
IS set forth by Lewis in [.Sun<ey),pp. 291 ff., foUoi^ing MacColl, and
later presented in an unproved form in (Logic], pp. 122 ff., profiting
bytheresearchesof Decker andothers.ToRus$eU's$ystem are added,
as new prinutive symbols, symbols for * possible ’ and ' strictly equi-
valent and with the help of these, * impossible ' necessary ' strict
implication', ‘compatible’, etc., are defined. Similar systems have



[Aussagettkalkal]). In [MthnoertigeJ he interprets the sentences of
the three-valued calculus by a translation into the modal sentences ;
these are, as are Lewis’s, formulated in accordance with the quasi-
syntactical method.

It is important to note the fundammtally dijjrrent nature of im~
plication and the eonsejaence-relation. hlaterially expressed: the
consequence-relation is a rebtion between sentences; implication
is not a relation bettcecn sentences. [Whether, for example, Russell’s
opinion that it is a relation between propositions is erroneous or
not, depends upon what is to be understood by a “ proposition”.
If we are going to speak at all of 'that which is designated by a sen-
tence then implication is a relation between what is so designated ;
but the consequence-relation is not,] *AdB’(Si) — as opposed to



§ 69 INTENSIONAL SENT EN CES — LOGIC OF MODAUTlES 25$

the syntactical sentence “B’ is a consequence of *A” (Sj) —
means, not somethmg about the sentences ‘ A ’ and * B but, with
the help of these sentences and of the junction symbol ‘ D some*
thing about the objects to whidi *A* and ‘B’ refer Formally ex-
pressed ‘o’ IS a symbol of the object-language, and ‘conse*
quence’ a predicate of the syntax-language Of course, between
the two sentences Si and Si there is an important connection (see
Theorem 14.7) Sj cannot, however, be inferred from Si but only
from the (equally syntactical) sentence ‘Sj is vahd (or analytic)’
The majority of the symbohe languages (for example, Russell’s
[fVrnc A/art ]) are (after a suitable extension of the rules of m-
ference) logical languages, and therefore contam no mdeterminate
sentences Hence, in these sj-stems, S* can be inferred from Si
This explains why the sentences of implication are m general
erroneously mterpreted as sentences about consequence-relations
[This IS one of the points which shows clearly how unfortunate it
u that the indetercnmate sentences have, for the most part, been
disregarded m logical uvestigations ] The relahon of the tn-
ienstonal tymboh of imphcabm in the ^sterns of the logic of modali-
ties, for instance that of the symbol of strict impbcation to ‘3 ’ and
to ‘consequence’, will become clear with the aid of the earUer
example on p 235 , this relation corresponds exactly to that sub-
sistmg between ‘Limp’, ‘Imp’, and ‘consequence’ [We can
Ignore here the differences between the intensional imphcations m
the various systems, they correspond to the different defimtioi,s
of the syntactical concept of ‘consequence’ ]

Russell’s choice of the designation ‘implication ’for the sentential
junction with the characteristic TFI'l’ has turned out to be a very
unfortunate one The words ‘to imply’ in the English language
mean the same as ‘ to contain ’ or ‘ to involve ‘ ^Vhether the choice
of the name was due to a confusion of implication with the con-
sequence-relation I do not know, but, xn any case, th»s nomen-
clature has been the cause of much confusion m the minds of many,
and It IS even possible that it » to blame for the fact that a number of
people, though aware of the difference between implication and the
consequence-relation, suU thmk that the symbol of implication
ought really to express the comeqiKnce-relation, and count it as
alaiiureonfhepartol'ftus symDo'k'&iatit doesnot ctoso liwe'have
retamed the term ‘implication’ m our system, it is, of course, m a
sense entirely divorced from its original meaning, it serves m the
syntax merely as the designation of sentential junctions of a par-
ticular kmd



256


PART IV. GENERAL SYNTAX


§ 70. The Quasi-Syntactical and’ the Si'ntactic.^l
Methods in the Logic of Modalities

All the foregoing sj-stems of the logic of modalities (mthin the
province of modem logic, in simboUc language) have, it seems,
applied the quasi-s^ntactttal mtlhod. This is not a matter of con-
scious choice between syntactical and quasi-s>*ntactical methods;
rather the method applied is held to be the natural one. .All in-
tensional sentences of the prenously exisung sj-stems of the logic
of modalities are, in any case, quasi-sj-ntactical sentences, inde-
pendently of which of the mo interpretations earlier discussed is
intended or (by a suitable tneorporauon in a more comprehensive
language) earned mto effect. [Inadentally, it should be noted that
for each of the systems one of the mo interpretations can he
arbitrarily chosen and earned out, provided no attention is paid
to the authors' mdications regarding interpretation. Accordingly,
It IS, m particular, possible to interpret every sentence Sj of the
logic of modilrtiea that ts tntetistonal in respect of a partial ex-
pression ?Ij, tn such a way that 9 l| is autonjinous in Sj.] Evfry
intentional tyttem of the hpt of modahues (and that even when
synthetic sentences are ad^tted as arguments) ean he tramlaud
into an exttnsional ntaeticol lan^age, whereby every intcnsional
sentence, since it is quasi-syntactical, 1$ translated into the corre-
lated syntactical sentence. In other words: rynfox already eon-
tains the ukole of the lope of modahlirsy and the construction of
a special intensional logic of modalities is not required.

^\’hcthcr, for the construction of a logic of modalities, the quasi-
sjutactical or the syntactical method is chosen is solely a question
of expedience. We will not here decide the question but will only
state the properties of both methods. The use of the quasi-
syntaetical method leads to intcnsional sentences, while the 8)n-
tactical method can also be carried into effect in an extcnsional
language. In a certain sense, the quasi-sjTitactical method is the
simpler; and it may be that it will prove to be the appropriate one
for the solution of certain problems. It will only be possible to
pronounce judgment on its fruitfulness as a whole when the
method is further dev eloped. Hitherto, if I am not misialcn, it has
in the main only been applied to the domain of the sentential



257


§ 70 METHODS IN THE LOGIC OF MODALITIES

calculus which, on account of the resolubility of its sentences, is
quite a simple one (see Party [KbZl ] pp t5f) It cannot be said
that the logic of modahties does not necessitate any syntactical
terms and is therefore simpler For the construction of every
calculus, and therefore also of the logic of modalities, a syntax-
language IS required in which the statement of the rules of m-
ference and of the primitive sentences ts formulated (see § 31), it
IS usual simply to take the word language for this purpose Now,
as soon as this syntax-language is obtained, everything that it is
desired to express by the sentences of modality — and, in general,
far more — can be defined and formulated within it. That is the
reason why we have here given preference to the syntactical
method It is however, in any case, a worth-while task to develop
the quasi syntactical method m general, and its use in the logic of
modahues m particular, and to investigate its possibilities m com
panson with the syntactical method

Even if m the construction of a logic of modalities we wish to use,
not the syntacucal but the ordinary method hitherto employed,
the realization that this method 1$ a quasi-synuctical one can help
us to overcome a number of uncertainties These, for example,
hare manifested themselves at vanous pomts in the fact that,
wishing to start from evident axioms, logicians have found them-
selves iQ doubt about the evidence of certain sentences , it has even
happened that sentences which had previously been individually
regarded as evident have turned out later to be mcompatible As
soon, however, as it is seen that the concepts of modahty — even
when they are formulated quasi syntactically — are concerned with
syntactical properties, their relativity is recognized. They must
always be referred to a particular language (which may be other
than that m which they are formubted) In this way the problems
regarding the evident character of tibsolute relations between the
modality-concepts disappear

§71. Is AN Intensional Logic Necessary’

Some logicians lake the view that the ordinary logic (for m-
stance, that of Russell) is defiaait in. some respects and must there-
fore be supplemented by a new logic, which is designated as m-
tensional logic or the logic of meaning (eg Lewis, Nelson



258 PART IV. CESER.U. STTJTAX

[Interuional], Weiss, and Jorgensen [Zitle], p. 93). Is this require-
rnent justified? A close examinadon shows that two different
questions, tshich should be treated separately, are here involved.

1. Russell’s language is an extensional language. It is required
that It be supplemented by an intfnsional language for the purpose
of expressing the concepts of modality (‘ consequence Vn^^cssary’,
etc,). We have dealt with this question before, and have seen that
the concepts of modality may also be expressed in an extensional
language, and that their formulation only led to intensional sen-
tences because the quasi-syntactical method tvas used. Neither for
an object-language concerned with any domain of objects nor for
the syntax-language of any object-language is it necessary to go
outside the framework of an extensional language.

2. As opposed to the ordinary formal logic, a logic of content or
a logu of meaning is derrunded. And, further, it is believed that
this second requirement also will be fulfilled by the construction
of an intensional logic of modalities; thus it often happens that the
designations ‘intensional logic* and 'logic of meaning’ are used
synonymously. It is thought, that is, that the concepts of modality,
since they are not dependent merely upon the t^ulh-^;aIues of
the arguments, are therefore dependent upon the meaning of the
arguments. This is often especially emphasixed in connection with
the consequence-relation (e.g. Lewis (SwrryJ, p. 328: “Inference
depends upon meaning, logical import, intension”). If all that is
meant by this is merely that, if the meanings of two sentences arc
given, the question of whether one is a consequence of the other
or not is also determined, I will not dispute it (although I prefer to
regard the connection from the opposite direction, namely, the
relations of meaning between the sentences are given by means of
the rules of consequence; see § 62). But the decisive point b the
following: in order to determine tcheiher or not one sentence is a
consequence of another, no reference need be made to the meaning of
the sentences. The mere statement of the truth-values is certainly too
little; but the statement of the meaning is, on the other hand, too
much. It is sufficient that the syntactical design of the sentences be
given. All the efforts of logicians since Arbtotle have been directed
to the formulation of the rules of inference a formal rules, that is to
say, as rules which refer only to the form of the sentences (for the
development of the formal character of logic, see Scholz [Ge-



259


§71. IS AN INTENSIONAL LOGIC NECESSARY?

sehichte\) It is theoretically possible to establish the logical relations
(consequence-relation, compatibihty, etc ) between two sentences
written m Chinese without understanding their sense, provided
that the syntax of the Chinese language is given (In practice this
IS only possible in the case of the simpler artificially constructed
languages ) The two requirements (i) and (a), which are usually
blended into one, are entirely mdependent of one another Whether
we wish to speak merely of the forms of the language or of the
sense (m some meaning of the word) of the sentences of S^, in
either case an mtensional language may be used, but we can also
use an extensional language for both these purposes The difference
between the extensionality and mtemtonality of a language has no-
thing to do rath the difference between the formal and the material
treatment Now, is it the business of logic to be concerned with the
sense of sentences at all (no matter whether they are given in cx-
tensional or m mtensional languages)? To a certain extent, yes,
namely, m so far as the sense and relations of sense permit of being
formally represented Thus, to the syntax, we have represented
the formal side of the sense of a sentence by means of the term
' content’ , and the formal side of the logical relations between sen*
fences by means of the terms ‘consequence’, ‘compatible’, and
the hke AU the questions which it is desired to treat in the required
logic of meamng are nothmg more than questions of syntax, m the
majority of cases, this is only concealed by the use of the matenal
mode of speech (as is demonstrated by many examples m Part V)
Questions about somethmg which is not formally representable,
such as the conceptual content of certain sentences, or the per-
ceptual content of certam expressions, do not belong to logic at all,
but to psychology AU questions m the field of logic can be for-
mally expressed and are then resolved into syntactical questions
A special logic of meamr^ u superfluous ‘non-formal logic’ is a
contradicUo in adjecto Logic u ^tax

Sometimes the demand for an intensional logic is made m a third
connection it is maintained that hitherto logic has only dealt with
tbrt. vsXeoa/vt. if ’KbA«aa A tk'rvild. alsAs dftsA vi'th, vTi-

tension of concepts But, actually, the newer systems of logic (Frege,
as early as 1893, followed by Rus^and Hdbert) have got far beyond
the stage of development of the mere logic of extension m this sense
Frege himself was the first to define m an exact way the old distmc-
tion between the mtension and the extension of a concept (namely,

17 *



26o past IV. GE^'ERA1, STNTAX

by meins of his distmctioa between • sentcntisl function and its
couiseofstJues). One csn rather maintain the re\‘erse,ih*t modern
logic, in its Utest phase of development, has completely suppressed
extension in favour of intension (cf. the climinanon of classes, § 3S).
This misunderstandins has already been cleared up many times (sec
Russell [Pner. JlfctA.], t, p. 72; Carnap p. sS. Schbh

[Geichzhte], p. 63); it is alscaya Teappeartng, however, amongst
philosophers who are not thoroughly acquainted with modem logic
(and amongst psj'^ologtsts. who, in addition, ccnitise the logical and
the perceptual content of a concept).


(/) RELATIONAL THEORY AND AXIO.MATJCS
§ yia. Relation-u. Theory

In the tArtjjj' c/ reiinonf, the properties of relations are in-
vestigated, particularly the structural properties— that is to
say those which are reiained in isomorphic transformance. A
theory of this kind ts nothing more than the sjutix of many-
termed predicates. IVe have abandoned the uniaJ distioctioQ be-
tween the one-termed predicates and the cla.'«-s)mbols apper-
taining to them, and dcsigruie both class and property by pr'
(see §§ 37, 38). Similarly we no longer differentbte the «*terjned
predicates for b > j from the relational sj-mbols which ha\T hitherto
been correlated with them as syToboIs of cit ension. In this secdoa.
We shall indicate briefly how the most important terms of the
theory of rclauons may be incorporated in the general syntax cf
the predicates.

\Mth regard to the terms used in the theory of relations (such
as ‘sjTnmctricar, 'tfansitivc’, ‘isomorphic’, etc.), it is important
to distinguish between their formulation in the object-language
and their formulation in the syntax-language. By means of this
distinction — the necessitj' of which is usually disregarded— certain
paradoxes in connection with the question of the multiplidry of
the transfinitc cardinal numbers and the possibility of non-
denumcrable aggregates are, as we shall see, clarifled.

\\ e will call an is-termed predicate homogeneous when, from a
sentence constructed from it and n arguments, another sentence
alwaj's arises as a result of any permutation of the arguments. The
majority of the terms of relational theory refer to homogeneous
hvo-termed predicates.



26 i


§ 7 1 0 RELATIONAL THEORY

The relational properties of symmetry, reflexiveness, and so on
are expressed, according to the ordinary method introduced by
Russell, by means of predicates of the second level (or, in Russell’s
o\vn symbolism, by class symbok of the second level) We will
write the definitions m the following form (employing the sym-
bolism of Language 11, but leavmgopcn the question as to whether
the expressions of the zero level are numerical expressions or


designations of objects)

(Fulfilment)- Erf(F)=(3jr)(3y) (i)

(Emptiness) Leer(F)=/««-Erf(F) (a)

(Symmetry)

Sym(F)s[Erf(F).(x)(y)(F(x.y) 3 F(y,x))] ( 3 )

(Asymmetry) As (F) s (x) (y) (F (x.y) D ~ F (> , *)) ( 4 )

(Reflexiveness)

Rea(F)s [Erf(F).MW((F(..:K)vF(y,«)) 5 f (*,,))] ( 5 )

(Total reflexiveness)

Refl«(f)s(Erf(F).(x)(F(.,»))] ( 6 )

(Irreflexivenesj) IiT(f)z(x)(*-F(x,*)) ( 7 )

(Transitivity)

Trans(F)s[Qx)(3y)(3x)(f(x,y).E-(y.x)).

(«)()'Kx)((F(».y).f(y.x))aR(x,x))] ( 8 )

(Intransitivity)

Intr (F) E (x) (y) (=) [(F(x,y). F(y, x))3 ~F(x. x)] ( 9 )


We have altered the usual forms of the definitions (see Russell
[Pnne Math ] Carnap by introducing in the defiruens

of (3) (5). (6). “id ( 8 ) an existential sentence or 'Erf{F)’ as a
conjunction term According to the definitions hitherto given,
transitivity and intransitivity do not exclude one another, and simi-
larly, neither do symmetry and asymmetry, reflexiveness and irre-
flexiveness If, for instance, a relation has no intermediary term
(that is to say, no term which occurs in one pair of the relation as
second term and in another pair as fint term) then it is simul-
taneously both transitive and intransitive (because the implicans in
the definiens of (9) is always false), and for the same reason a null
relation is at the same tune transitive, intransitive, symmetrical,
asymmetncal, reflexive and trreflexive On this account we intro-
duce conditions which require for aymmetnea] reflexive, and tronsi-


Ejfalllhett



262 P.UlTnr. GENERAL SYNTAX

tiAc relations the property of non-emptiness, and further, for a
transitive relation, the occurrence of an infcrinediary term (non-
emptiness of the second power of the relation)- On the basis of our
definition, the two tenns of each of the three pairs exclude one
another. [The term ‘ Erf ’ in (6) can be left out if the individual
domain is non-empty, that is to say, if in the language in question,
‘ (3 *)(*=*)’ demonstrable, as is the case in theordmary languages
of logistics.]

§ yib . Syntactical Terms or Relational Theory

We trill now introduce syntactical terms of relational theory as
opposed to the terms of relational theory of the object-language
which have been defined m the foregoing. The difference bettveen
these ttvo kinds of terms must be veiy carefully noted. Let us tale
as an example the sentence *As(P)’— or, in the word-language:
‘The relation P is an asjTnmetrical rebtion. ’ This sentence — we
will call it Sj—is equipollent to the sentence

In contradistinction to this, we tviUsay that the predicate ‘P* (not
the relation P) is (sjetemically as)-mmetrical or) S-as^enmetrical,
when Sj is not merely true, but 8)'stemica]ly true, I.e. s^alid; and
that * P ' is (logically asjtnmetncal or) L-asjTnmetrical when 6j is
(not merely s-alid but) analytic. In the material mode : The object-
sentence * As (P)’ or ‘P is asjTnmetrical’ c-xpresses the fact that the
relation P does not hold in both directions in any pair; on the
other hand the sj-ntactical sentence *" P’ is S-asjinmetrical ” means
that this fact can be inferred from the transformation rules of the
language-system S (hence, for example, from the natural laws, if
they are formulated as primitive sentences); and the sjmtactical
sentence “‘P’ is L-3S)-iiimetrical *’ means that this is not a
genuinely S}’nlhetic fact, but is already determined by the L-rules
of S, and hence is given in substance by the definition of ' P
We will formubte the definitions indicated here in a somewhat
different manner, so as to avoid the limiting assumption that
universal operators and symbols of proper negation and unplica-
occur in xht ot^eci-kangMge S. The following are our
syntactical definitions. Let pr| be a homogeneous two-termed
predicate. [The definitions can easily be transferred to any
homogeneous ^t\ Sfu*, and Sg*,] Then pr, is called S-noU



§ 7 I& SYNTACTICAL TERMS €JF RELATIONAL THEORY 263

(or L-quU) if always (that is, here and m what follows, for any
closed a^ments, PriC5li»%) contravalid (or contra-

dictory, respectively) pTj is called ^fulfilled (or Ij-fulfilled,
respectively) when a valid (or analytic) sentence of the form
pri(9I,,?l2) exists pti is called S-symmetncal (or L-symmetn-
cal) when ptj is not S null (or X<-null, respectively) and pti (^Ij, 3Ii)
IS always a consequence (or L consequence) of pri(3Ii,2Ij) pti is
called S asymmetncal (or h-asymmitncal, respectively) when
pri(51j,llli) and pri(5Ii,3y are altrays incompatible (or L-incom-
patible) with one another pt^ is called S-reJiexive (or L reflexive)
when pti is not S null (or L-nuU) and pij (Hj, ?Ii) is always a con-
sequence (or L-consequence) of pti^],^ and always a conse-
quence (or L-consequence) of pti(?lt.^i). is called S totally
reflexive (or l^totally reflextve)vihtn pri(2Ii,5Ii) is always valid (or
analytic, respectively), pt| is called S-areflexive (or l^trreflextve)
when pXi(2ti,?Ii) is always contravalid (or contradictory, re-
spectively) ptj is called S irantiltve (or Irtranstlive) when the
two sentences ptiCSi.^) and PtitS*,^) are not always incom-
patible (or L-mcompatibJ«, respectively) with one another, and
when pti(2Ii,?l3) is always a consequence (or L-consequence, re-
spectively) of those two sentences , pti is called S intransitive (or L-
intransitite) when the above mentioned three sentences are always
incompatible (or L-incompatible, respectively) mth one another
In the case of all these terms, corresponding P terms can be
defined , pr^ is called P-nuU when pr^ is S-null but not L-nuU , and
so forth

We will again make clear the difference between the terms of
relational theory of the object language and those of the syntax-
language by means of a juxtaposition

The property of symmetry The property of S-iymmet/y
appertains to certain relations appertams to certam predicates
(namely, to symbob of rela-
tions) (The same holds for
L symmetry )

Thispropeityisexpressedby This property is expressed
the symbol ‘Sym’, or by the by the word ‘S-symmetncal’,
word * s j mmetncal ' , these sym- this word belongs to the syntax-
bob belong to the object-hm- language
guage



PUITIV. GENERAL n’NTAX


264

Assuming appropnate dciinitions for the predicates in a suitable
language S, the foUo^'ing examples hold. The predicate ' brother* is
L-irreflexive, but it is neither S-sj-minetrical nor S-asjTnmetrical.
If it fol!ovi-s from the rules of transfoimaaon of S that, in the district
B, at least one roan has a brother but no man has a sister, then
‘brother in B’ is S-sj-mmettical, but not L-sjtnmetncal, and is
therefore P-sjinmetru^* * Father' is L-irrtflexive, l^asiTEsmemcal
and L-intransitive-

Theorem 7ib.i* (o) If the predicate .‘P* is L-sjuunetrical or
P-s>nimetrtca], then it is also S-sjtnmetrical. (6) If 'P* is S-
sjTnmetrical, then P (not ‘P’) is symmetrical; the converse is not
universally true, (c) Let S be an L-language (which may also be a
descriptive language like I and II); then if 'P* is L-sj-mmctrical in
S, It is also S-sjTometrical ; and conversely, (d) Let S be a logical
language (hence an L-languagc); then if ‘P’ is S-sjTnmctrical or
L-sj-mmetrical in S, P is symmetneal; and converselj'. Corre-
sponding theorerns are true for the remaining terms. For ib #nd
id, it is assumed that the language S contains its otvn sjmtax; $ is
here taken as a oord-Ianguage, m which 'P ts s)-mmetrical’ >s
written for ‘Sj*m(P)’-

It Mould be equall}' possible to express the sjTitaeneal tenns here
defined by means of second level predicates of the chjrel-^larguage^
for example: “P' is S-irreflexive’ by *SIrr(P)‘ and "P* is L-
irrefleiive ' by ‘ Llrr (P) But m * SlixfP) ’and ' Llrr (P) 'P ' woold
be autoroffiout, nhich is not the case in ‘IrrfP)’ (in so far as de*
scripuve arguments are admitted; see p. 237). Those sentences arc
quasi-tyniactical, but ‘ Irr (P) ’ is not (see Example t on p. 234),


§ 71C. Isomorphism

We will define a few more terms of relational theory leading up
to the particularly important term ‘isomorphism’. First we will
give, as before, definitions of symbols of an object -language (with
a sj-mbolism like that of Language II).

(Converse): cnv( 0 (*»y’)s(F(y,x)) (i)

(One-many) :

Un(F) = (*)(>;) (2) [(n».s).F(y.2))D(x=y)] (2)

(One-one): Unun(F)H(Un(F).Un[cnv(f)]) (3)



§7IC ISOMORPHISM


265


(Correlator)

Korr(H, F, G) = (Unun (f/)*(u) [(3 c) (F{u, v)VF{v , «)) =

Q ») (H (k, =:))] . w [Q ,) (G (*, 5.) V G ty, :t)) s Q »)

<^(».3’))]) ' (4)

(Isomorphism)

IsCF,G)sQH)(Koii(If,F,C)) (s)

These defimtions correspond (m a somewhat different formula-
tion) to the usual ones (4) is here formulated for two-termed
predicates, but can easily be transferred to n-termed predicates
for n>2 Just as, earlier, we opposed the terms of relational
theory of the object language (such as ‘Irr’) to correspondmg
syntactical terms (such as •S-irreflexive* and ‘L-irreflexive’),
80 here also we must contrast the term of th; objeeUlanguage
that are defined m (i) to (5) with tyniactual terms that have
previously been either ignored or confused with the former
Let pii be a homogeneous two-termed predicate (the definitions
can easily be transferred to ^r, €fu, or (Ss) PT} is called the
^•eomerse of pi^ if alwa^'s (that is to say, here and m the following,
for any dosed arguments) pct(?li, 5 y is equipollent to pri(?l»,?Ii)
pti is called S-ow mcmy if and ?!, are always synonymous m rela-
tion to {pti(1JIj,?l3),pri(?Iji?l3)) pti is called S-one-one if pti and
the S-converse of pti are S-onc-many Let pti and pr* be homo-
geneous fl termed predicates, then pts is called an S correlator for
prx and pTj if the followuig conditions are fulfilled (i) pTs is
S-one-one, (2) if is a suitable argument for pri then it is also
a suitable argument of the first place for pts, and conversely,
(3) if 13 a suitable argument for pr^ then it is also a suit-
able argument of the second place for pr3, and conversely, (4)
pri(3Ii,3l2, ^)andpr3(S3',^', ,?LOarealwaysequipoIlentm
relation to {ptj(?Ii,?Ii'),pr3(?l3»^0. PtsC^L.^')} homo-

geneous n termed predicates, pti and pr^, are called S-isomorphic
if there is an S-corrclator of ptj and pts For each one of these
terms there is to be defined an analogous L-term and P-term
Theorem 71C.1. Let the language S contain its own syntax
[Here we will take a word-language and will write “P and Q are
isomorphic” (not ‘P’ and ‘Q’) instead of 'Is(P,Q)’] Then
(analogously to Theorem 710 t i and i d) it is true that if ‘P' and



266


p.urr n*. CENisut ststai


‘Q’ are S- (or L-) isomorphic, then P and Q are isomorphic; il
S is a iogical language, then the comerse is also true.

.\n S-correlator for pti and pr* is a predicate of the object-
language. As distinguished from this, ire mean by a
conflation of two homogeneous B-tcrroed predicates, ptj and prj,
a one-one sj-ntactical correlation, Cy which fulSls the following
conditions: (i) if ?Ijis a suitable argument for prj, then >s

a suitable argument for prj; (2) if SIj is a suitable argument
for prt then there is a suitable argument Sfi for pfj such that
PIJ ‘s ?Ij; (3) pri >s alwaj-s equipollent to prj(Oi pij.

Tiro homogeneous B-termed predicates, pfj and pTj,
are called s^mtactlcally isomorphic when there is a sjTitaCtical
correlation for them (that is to saj*, when such a correlation can
be defined in the sj-ntax-language, assuming it to be suffidently
nch).

We will make the difference betireen the concepts of isomor-
phism quite clear by means of a contrasung table ; this is analogous
to the earlier one, but here a third kind of concept, namely, syn*
tactical isomorphism, is introduced.

1. The relation of SWio*
nor^fitm subsists beneeen cer-
tain (homogeneous, two- or
many-t erroedj/redrra/e; (name-
ly, sjTnbols of relations'). (The
same holds for L-Isomorphism.)

This relation is expressed by
the irord ‘S-isomorphic’; this
word belongs to the ^Titax-
tan^ase.

2. The relation of n-ntacb'foJ
likewise subsists

between certain predicates. It
is expressed by the words ‘ sjn*
tactically isomorphic’; these
words belong to the nir/ux-
lan^ayr,

S-isomorphism and syntactical isomorphism are thus both
syntactical concepts which refer to predicates 0/ the object-


relation of tromerpAiirj
subsists beftceen certain (homo-
geneous, nvo- or many-terroed)
relations.


This relation is e.xpres5ed by
the symbol * Js or by the word
'isomorphic'; these ^’mbols
belong to the objeet-lanp/a^c.



;6S TART IV. CC<ER.^L S^'NTAX

maintpin-* that M and U(M) cannot ha\-e the same carding
number. Fracnlel [tWrriucAuB^ca] has given a proof of this
theorem w hich remains \*alid for hts sj-stem S c\ en though it con-
tains the so-called Axiom of Limitation ([v^frnjjrM/rhre], p. 355)-
On the other hand, ho\vcver,wearriveata contrarj’ result as a conse-
quence of the following argument. The Axiom of Limitation means
that in the aggregate-domain which is treated in S — let us call it
n — only those aggregates occur of which the existence is required
by the other axioms. Therefore, only the following aggregates are
existent in B : in the first place, two inilia! aggregates, namely, the
null-aggregate and the denumerably infinite aggregate, Z, re-
quired hy Axiom VII ; and secondly, those aggregates which can
be constructed on the basis of these initial aggregates by applying
an arbitrary but finite number of times certain constructional
procedures. There are only sue kinds of these constructional steps
(namely, the form.ition of the pair-aggregate, of the sum*
aggregate, of the aggregate of sub-aggregates, of the aggregate of
AussoHiUrum;, of the aggregate of selection, and of the aggregt^t
of replacement) Since onlj a denumerable multiplicity of aggre*
ptes can be constructed m this way, there is in B, according to the
Axiom of Limitation, only a denumerable multiplicity of aggre*
ptes, and consequently, at the most, only a denumerable multi*
plifiiy of siib-aggreptes of Z. Therefore U(Z) cannot have «
higher cardinal number than 2 . Actually, on the basis of the t'ro
initial aggreptea and the six constnictional steps, it is easy to
give a method of demimeraiing all the aggrrptes of B, and hence
also of the sub-aggregates of Z, and in this way the sub-aggreptes
of Z can be tinw ocally correlated w ith the elements of Z. TTierefore
\J(Z) and Z lu\c the same cardinal number.

This result appears to contradict Cantor’s theorem; but Mr
rontraJiftioH dj;uppr.irx ns soon as ore differmtiatr hfttroen eqvahty
of cardinal numhers and s^nfacticol etpiahty of cardinal nunhers.
[Since S is a logical language, equality of caniinal numbers and
S-equahty of cardinal numlvrs coincide.] Accorihng to Fraenlel's
definition ((.Ura^rn/rArr]. p. 31^) tv\o (mutually exclusive)
ogRreptes hi and N ime the same cardinal number only if (in B)
there is a transforming aggrepie (i.c. a correlator) Q— that is, an
aggregate of mutually rxelusb c pairs {w, n} w here tn is an clement
of M and n of N such that the paira exhaust M and N. Xow if



§ 7 I</ THE NON-DENUMQlABIi CARDINAL NUMBERS 269

M IS denunierably infinite, a one one correlation of the kind
mentioned before can be effected between the elements of M and
those of U (M), and hence between the elements and the sub-
aggregates of M This correlation, however, is not a correlator
in S but a syntactical correlation In B there is no aggregate Q
which could be a correlator for M and U (M) , that is shown by
Fraenkel’s proof But now Fraenkcl's proof and our own findings
aie no longer m contiadiclion with one another M and U (M),
although they have different cardinal numbert are nevertheless syn
tactically of the same cardinal number

In syntax it is always possible to effect a denumeration of ex-
pressions of any kind (in an anthmetiaed syntax, for mstance, by
means of the senea-numbers of the expressions) Thus m relation
to a fixed syntax-langiiage (which must be presupposed for the
construction of the system S) every aggregate of FrCunkeVs domain
of c^gregates B u syniacticaUy demanerable, two ironsfimte aggre-
gates are alacys syntaettcally of the same cardinal number This is
the element of truth in the criticism brought by the Intuitionists
agamst the concept of the non denumerable aggregates [Poincare
IffSedanken^, pp 108 ff, 134 ff) bases his rejection of the non-
denumerably infinite— subsequently maintained by Brouwer [/«•
tiattoTttstn\ and others— on this nominalistic view, which he him-
self, not very happily, designates as idealistic ] It must, however,
be noted that the syntactical equivalence of all transfinite aggre-
gates of B (from the standpoint of a fixed syntax language) is not
in contradiction with their non-equivalence (within the system S),
and that therefore the distineittm between different transfinite
cardinal numbers mthin a system of the Theory of Aggregates is
justified And indeed, in Fraenkel’s system of axioms, which,
because of the Axiom of Limitation, is, m a broad sense, a con-
structive system, the mequivalence of certam aggregates — for
mstance that of Z and U (Z) — follows from a certam poverty of the
system it does not contain any aggregate which m the given cases
could serve as a correlator In non-constructive axiom-systems —
for mstance, m a system whidj contains no Axiom of Limitation,
and which, on the other hand, operates with existential axioms to
greater extent— the mequivalence, say, of M and U(M) can be
attributed, conversely, to a certain richness of the system U (M)
contains so many element-aggregates that they cannot be wrre



2^0 PART IV. CEIJERAL ST7»TAX

lated in a one-one correspondence with the elements of M. Of
course, this does not mean that sudi a wealth of aggTegatc-<f<>ri^-
tions exists within the s)-stem; obviously the number of aggregate-
designations is denumerable in every system. The richness is only
assumed by means of axioms, and is not demonstrable by designa-
tions (names or descriptions).

Further it must be noted that the difference between the aggre-
gates of the natural numbers, of the real numbers, of the functions
of real numbers, and so on, which Cantor has pointed out and
formulated by attributing to them different cardinal numbers, is
also syntactically representable. This distinction is particularly
significant for the syntactical investigation of a series of languages
each of which is contained m the next as a proper sub-language.
That characteristic of the class of the logical nurncrical functors
which Cantor designates as the non-denumerability of the aggre-
gate of the real numbers is expressed, for instance, in an in-
creasing senes of languages by the fact that every language of the
series, tn addition to the denumerably many such functors of the
prerious languages, can always contain new ones (on this point,
see our earlier remarks on the diagonal method, on Richard's
antinomy, and on the defectiveness of arithmetic; compare
Theorems 6oe.3 snd 6 od.t).

As a result of the distinction between denumerability (in the
8>3tcm under considenuon) and syntactical denoraerabilit}’, the
paradox in connection with the famous Lowcnheim-Skolem
theorem (Skolem [Erfullbarketi]’ cf. Fraenkel [Afm^ettUkre], p-
333) also disappears. This theorem means approximately that for
a non-contradictory axiom-system S of the "rheory of Aggregates
there is alwaj's already a model in a denumerable domain. Such a
model, however, is not coiutnicted by means of terms of S, but by
means of discussions about S, that is to say, by means of syntactical
terms. And the denumerability of the domain whose clcrocnis
constitute the model is not demonstrated by the production of a
correbtor in S, but by the proof of the constnictibility of a syn*
tactical correlation. It is, accordingly, not the denumerability
(in S) of a model which is pnn-cd. but only the sj-ntactica! de-
numerability. Thus the Skolem theorem does not contradict
Cantor’s theorem (or Fracnkel's proof).



§ 7 1 e THE AZIOMA.TIC METHOD


271


§ 7ie. The Axio\utic Method

An axiom system (abbreviation ‘AS*) is usually regarded as a
system of sentences, the so-called axioms, from which other
sentences, the so-caUed theorems or conclusions, may be deduced.
The axioms consist partly of ^mbots whose meaning is assumed
to be known already (for the most part, logical symbob), and partly
of symbols which are introduced for the first time by the AS, the
so called primitive symbob of the AS It is customarily said that
no meaning is presupposed for the latter, but, that the AS — as a
sort of imphat definition — determines their meaning In order to
draw conclusions from the axioms, obviously the rules of forma-
tion and transformation of the language concerned must be known
These rules are usually taatly assumed, but m an exact formubtion
of the AS this taat assumption must be repbced by an esphat
statement Further, it is characteristic of the axiomatic method
that the primitive symbob are, to a certain extent, determined by
the AS only m rebtioa to one another Hence there is sometimes
the possibihty of interpreting the primitive symbob m several
different ways. The statement of a certain mterpretation of the
primitive symbob is designated as the establishment of cone-
btive definitions (see p 78) If it is proved that the axioms are
fulfilled for a certam mterpretation, or at least that their fulfilment
13 not excluded, we say that by this mteipretation a model for the
AS is constructed

Example In drawing up an AS of Geom e tr y , it is usual merely to
state the specifically geometrical axioms In order to render de-
ductions possible, the sentential and fiinctional calculus, together
with elementary arithmetic, must be added.

Usually the AS is formubted in the word language without any
precise statement of the syntactical rules, particularly the rules of m*
ferencc Now there are aev cial quite different possibihties of putting
such an AS mto the exact fonn of a calculus We will state bnefly
the most important methods of formubtion It is desirable to
choose a different terminology for each of the three methods, so
that It may alwa)-s be clear whidi one is the subject of discussion
Therefore we shall speak of “axioms” only m connection with the
first method, of “ primitive sentences” m connection with the



272 PAUT IV. GENERAL S^TJTA*:

second (m accordance with our regular usage in this book), and
of “premisses” m connection with tlte third.

First method: the axioms as sentmliat functions.

For the representation of the AS, a language S with a sentential
and a functional calculus will be taken. {For the examples in the
following, we shall use the sjmboltsm of Language II.) Each of
the k primitive symbols of the AS is represented by a t) (or S);
w e call these 0 the primitive t'onabtes. Each of the m axioms is then
formulated as an Sfu, and, specifically, as an Sfu' if the axiom
contains t different primitive sjTnbols. The same holds of the con-
clusions. In the deductions, however, there is no substitution for
the free primitive vanables. (In the material mode of speech: the
primitive enables do not express universality, but indeterminate-
ncss.) Sfu„ is called a conclusion from the m axioms Sfiij,...
6fUm. if the universal implication-sentence

(Di)...(t.t)[(Sfii,.SK....SfiijDSfu,]

is analytic (or L'-demonst«ble) in S. According to this method,
a model for the AS is to be understood as a series of k substitution-
values 91,, ...Hfj for the primitive variables. If

is wild (or not contrat’alid, or not contpdictory, respectively) In
S, then the model is called a real (or a really possible, or a logically
possible) model. If at least one of the substitution-t*aIues is
descriptive, then the model is called descriptive; othenvise, logical
(or mathematical).

The advantage of this method consists in the fact that by it a
common language may be used for all AS’s, and for all AS’s of the
usual kind a simple language of the usual kind hasung a sentential
and a functional calculus. The primitive variables in this con-
nection are usually “d or p; in the ordinary AS only ®d, *p and *p
occur, and for the most part ’p.

Example: If Ihlbcrt'i AS of Euclidean geometry ([Crwti//. Geom.],
p. 1 ) is presented in accordance with the first method, 8e\cn dif-
ferent primitive variables appear: ’point', 'straight line’, ‘plane’
will each be represented by a •p*;‘hes upon', by a ‘p*; ‘ between *,



§ 7 ie. THE AXIOMATIC METHOD 273

by a and ‘congruence of segments* and ‘congruence of angles’
each by a

On &e fint method, see Carnap {{Eigentltehi], [LogtsttJi^, pp 71 S ,
[AxiomatiX^

Second method the oxiomt (U pnmitvce sentences

The axioms of the AS are formulated as the primitive sentences
of a language Sometimes, m this case, the axioms of a given
AS are the only pmmtiTe sentences of Sj, so that only rules of
inference have to be added But sometimes not only the rules of
inference but also the L-pnmitive sentences of are tacitly
assumed, so that the given axioms must be formulated as additional
primitive sentences of S^ (for the most part descnpave P sentences)
’The conclusions of the AS are the sentences that are valid (or
demonstrable) m Sj The primitive symbols of the AS are here
primitive symbols of Sj, and either they are the only pnmmve
symbols of or they are additional primitive symbols (mostly
descriptive) added to the original logical prunittve symbols of ^
(which IQ the ordinary formulation of the AS are tacitly assumed)
The pruniuve symbols are not $ Hence, the construction of a
model can here not be effected by substitution It is achieved by
means of a translation, 2i, of Sj uto another language S] (usually
a language of saence which has a practical use) In the majority of
cases this will be an expressional translaDon , the statement of the
model consists, as a rule, only of the statement of the Q^-correlates
of the additional primitive qrmbob, the translation of the logical
prumtive symbols being assumed to be established and well
known The model is said to be real (or really possible, or logically
possible) if the class of the Qj^coiielates of the axioms of the AS is
valid (or not contravabd, or not contradictory, respectively) m Sj
If this class is descriptive, the model is called descriptive, if it is
logical, the model is called logical (or mathematical)

Example On a system of geometncal axioms m accordance with
the second method, see §25 Ha “Axiomatic Geometry", anth-
meacal geometry (I) consututes a logical model, physical geometry
(II b) a descriptive model

The Second method affords a greater freedom m mterpretation,
and thus m the construction of models, than the first In the first
method, the domam of the mterpretations of a certam pmmtive
symbol is the domam of the substitution-values of the primitive



274 cener-u. syntax

variable. If, as is usual, it is a case of primitive A’ariables within a
sj'stem of tj’pcs, then the same relations of tj-pes must hold be*
tween the symbols of the model as hold betw een the corresponding
primitive s-ariables. In the second method, the place of substitu-
tion is taken by the far more elastic operation of tiunslation ; here,
for instance, isogenous primithx symbols can have correlates
which arc not isogenous.

Examplet: i. Let Sj, Sj, and Sj be presentations of AS’s of Euclid*
can geometry in accordance with the second method. Let S, take
straight lines as classes of points (see Camap [LogijtiJi]t 5 34) : let S»
take straight lines as relations bem-een points (see Catnap [LogufiA],
§ 35) > and let Sj take straight lines and points as tndinduals (as does
Hilbert [Gnmdl. Grom.]). Three AS’a of this kind, formulated in
accordance with the first method, catmot have a common model. On
the other hand, hy the second method this is possible, in the sense
that Si, St, and S] con all be translated mto the same sub-language
of a logical language, in which * point is interpreted in the usual way
os a tnad of real numbers, a plane as a class of such triads which
satisfy a linear equation, and so on. Thus, by this method, it is easy
to portray formally the retaoonshjp of the three AS’s, which is what
IS meant when it is aaid that they represent ^e same geometry.
3. Let an AS of the Theory of Aggregates be given which okes all
aggregates as individuals (as, for instance, Fraenkel does [ilfnyrt*
Ithre], § 16) but m which only homt^eneous aggregates occur (so
that, for example, as opposed to the AS of Fraenkel, m and [m)
Cannot be elements of the same aggregate). If an AS of such a kind
is presented in accordance with the second method, it can be in-
terpreted as a theory of classes, and, in spite of the equal leivl of the
aggregates, ss a theory of classes of all lex els. *0 and certain ?lt* of
all levels (for instance in Language II) are taken as correlsies of the
;Sgregatc-expressions.

Third method; the axiomj ax premmeu

The AS is represented by means of a (usually indeterminate)
tmtenlial elajt of an assumed language S. The conclusions are
here the L-consequences of this class, and hence the axioms appear
M premisses of derivations (or of consequence-relations). In this, as
in the second method, an interpretation consists of a translation;
and, as in the first method, it is possible to formulate sex eral AS’s
within the same language.

Special and general asiomatict, that is, the theory of certain
individual AS's or of AS’a in general, is nothing more than the
tytilax of the AS s. 'The inx'cstigatioos in axiomatics, which have



275


§ 7if the ahomatic method
been conducted chiefly and mtensirdy by mathematiaana, thus
contain a great number of syntactical discussions and definitions,
many of trhich we have already been able to apply m this outlme
of a general syntax- We have defined «)me terms, m accordance
•with the second method, as properties of languages, and some (in
part, the same ones), m accordance with the third method, as
properties of sentential classes [For instance, the terms * re-
futable*, ‘L-refutahle’, ‘contravahd’, and ‘contradictory’, which
refer to sentential classes, correspond to the terms ‘ contradictory’,
‘L-contiadictory 'mconsistent*, and ‘L-mconsistent’, which
refer to languages ] Com-etsely, it will be possible to make use of
the findmgs and definitions of general syntax for axiomaucs But
we cannot go more fiilly mto this subject in the present work.

Full bibliographical references on the subject of axiomatics up to
the year tgsS are given by Fraenkel [MengenUhre^ § x8 Some new
works on the subject are as follows Hemf/lnom ],LewisandLaog-
ford [;.ogie], and Tarsfci [Methodologu\, [yVidenpruekjfr )



PART V

PHILOSOPHY AND SYNTAX

A, ON THE FORM OF THE SENTENCES
BELONGING TO THE LOGIC OF SCIENCE

§ 72 Philosophy Replaced by the Logic
OF Science

The questions dealt tmh m any theorebcal field — and similarly
the corresponding sentences and assertions — can be roughly
diTided into oijeet’^eshons and logical questions (This differentia
tion has no claim to exactitude, it only serves as a preliminary to
the foUotviog non formal and inexact discussion ) By object*
questions are to be understood those that have to do with the
objects of the domain under considentioo, such as mqumes re*
garding their properties and reUuoos The logical questions, on
the other hand, do not refer directly to the objects, but to sen-
tences, tenns, theories, and so on, which themselves refer to the
objects. (Logical questions may be concerned either with the
meanmg and content of the sentences, terms, etc., or only with the
form of these , of this we shall say more bter ) In a certam sense,
of course, logical questions are also object questions, smcc they
refer to certam objects — namely, to tenns, sentences, and so on —
that IS to say, to objects of logic When, however, we are talking of
a non logical, proper object-domain, the differentiation between
object-questions and logical questions is quite clear For mstance,
m the domam of zoology, the object-questions are concerned with
the properties of animals, the telabons of animals to one another
and to other objects, etc , the logical questions, on the other hand,
are concerned with the sentences of zoology and the logical con-
nections between them, the logical character of the definitions
occurring in that saence, the Ic^cal character of the theories and
hypotheses which may be, or have actually been, advanced, and
so on.

Accordmg to traditional usage, the name ‘phflosophy* serves
aa a collective designation for inquiries of I'cry different kmds



2;S PART V. riiiLOSorinr akd syntax .

Object-questions as ^rell as logical questions are to be found
amongst these inquines. The object-questions are in part con-
cerned trith supposititious objects \chlch are not to be found in the
object-domains of the sciences (for instance, the thing-in-itself,
the absolute, the transcendental, the objective idea, the ultimate
cause of the tvorld, non-being, and such things as s'alues, absolute
nonns, the categorical imperative, and so on); this is especially
the case in that branch of philosophy usually known as mets-
phj-sics. On the other hand, the object-questions of philosophy
are also concerned with tlungs which likewise occur in the em-
pirical sciences (such as mankind, society, language, historj',
economics, nature, space and time, causality, etc.); this is espied*
ally the case in those branches that are called natural philosophy,
the philosophy of history, tlie philosophy of language, and so on.
The logtcal questions occur principally m logic (including applied
logic), and also m the so-called theory of knowledge (or epistemo-
logj’), where they are, however, for the most pan, entangled with
psjchological questions. The problems of ihe so-called philo-
sophical foundations of the various saencts (such as pitj-sics,
biologj', psjchologj', and history) include both object -questions
and logical quesuons.

The logtcal analj-sis of philosophical problems shows them to
vary greatly in ‘character. As regards those object -questions whose
objects do not occur m the exact sciences, cnlical analpls his re-
vealed that they are pseudo-problems. TTie suppiosititious sen-
tences of metaphj-sics, of the philosophy of values, of ethics (in so
far as it is treated as a normative disdplme and not as a psjtho-
sodological inv estigation of facts) arc jiseudo-sentcnccs ; thej' have
no logical content, but arc only expressions of feeling which in
their turn stimulate feelings and volitional tendencies on the part
of the hearer. In the other departments of philosophy the psjxho-
logical questions must first of all be eliminated; these belong to
psychology, which is one of the empirical sdences, and are to be
handled by it with the aid of its empirical methods. [By this, of
course, no veto is put upon the discussion of psychological ques-
tions within the domain of logical investigation; everj'one is at
liberty to combine his questions in the way which seems to him
most fruitful. It is only intended as a warning against the dis-
regard of the dilTerencc betv* een proper logical (or epistemological)



§ 72 PHILOSOPHY RPPIACED BT THE LOGIC OF SCIENCE 279
questions and psychological ones Veiy often the formulation of a
question does not make it dear whetiier it is intended as a psycho-
logical or a logical one, and m this way a great deal of confusion
arises ] The remaining questions, that is, m ordinary terminology,
questions of logic, of the theory of knowledge (or epistemology), of
natural philosophy, of the philosophy of history, etc , are some-
times designated by those who regard metaphysics as unscientific
as questions of scientific philosophy As usually formulated, these
questions are m part logical questions, but in part also object
questions which refer to the objects of the special sciences Philo-
sophical questions, however, according to the view of philosophers,
are supposed to examine such objects as are also mvestigated by
the special sciences from quite a different standpomt, namely,
from the purely philosophical one As opposed to this, we shall
here mamtam that all these remaining philosophical questions are
logical questions Even the supposititious object questions are
logical questions in a misleading guise The supposed peculiarly
philosophical pouit of view from which the objects of science are
to be investigated proves to be illusory, just as, previously, the
supposed pecuharly philosophical realm of objects proper to meta-
physics disappeared under analysts Apart from the questions of
the mdmdual soeoces, only the questions of the logical analysis of
soence, of its sentences, terms, concepts, theories, etc , are left as
genume scientific questions We shall call this complex of ques-
tions the lope of setenee [We shall not here employ the expression
‘theory of science’, if it is to be used at all, it is more appropriate
to the wider domam of questions which, m addition to the logic
of saence, mdudes also the empincal investigation of scientific
activity, such as historical, soaological, and, above all, psycho-
logical mquines ]

According to this new, then, once philosophy is punfied of all
unsaentific elements, only the logic of science remams In the
majority of philosophical mvestigations, however, a sharp dmsion
into saentific and unscientific elements is quite impossible For
this reason we prefer to say the lope of jaenee takes the place of the
tnextneahle tangle of problems schtch tsknonn as philosophy ^Vhether,
on this new, it is desirable to apply the term ‘philosophy’ or
‘saentific philosophy’ to this remainder, is a question of ex-
pedience which cannot be deaded here It must be taken mto



28o part V. PJIILOSOPHT AND SYNTAX

consideration that the word ‘philosophy’ is already heavily
burdened, and that it is largely applied (particularly in the German
language) to speculative metaphysical discussions. The designation
‘theory of knowledge* (or ‘epistemology’) is a more neutral one,
but even this appears not to be quite unobjectionable, since it mis-
leadingly suggests a resemblance between the problems of our
logic of science and the problems of traditional epistemology; the
latter, however, arc always permeated by pseudo-concepts and
pseudo-questions, and frequently in such a tvay that their dis-
entanglement is impossible.

The view that, as soon ns claims to scientific qualifications are
made, all that remains of philosophy is the logic of science, cannot
be established here and will not be assumed in what follows. In
this part of the book we propose to examine the character of the
sentences of the logic of science, and to show that they are syn-
tactical sentences. For anyone who shares tvith us the anti-
metaphysical standpoint it will thereby be shown that all philo-
sophical problems which have any meaning belong to syntax. The
following investigations concerning the logic of science as syntax
are not, however, dependent upon an adherence to this view;
those who do not subscribe (o it can formulate our results simply
as a statement that the problems of that part of philosophy which
b neither metaphysical nor concerned with values and norms arc
syntactical.

Anti-metaphysical views have often been put forward in the past,


ot science) is represented in particular by Wittgenstein and the
Vienna Circle, and has been both established in detail and in-
vestigated in all its consequences by them ; sec Schlick [Metaphytik'],
in’fttde], [Potiiivirmus]i Frank [Kau:algrtet3]\ Hahn

Neurath [H'w. IVeltauff.], [IFrge]; Carnap [^Ttta•
P^yp^ll further bibliographical references are given by Ncurath
(FViw. and in iTrAejMitmr, I, 315 ff, Neurath is definitely

opposed to the continued use of the expressions ‘philosophy’,
icienuric philosophy ‘ natural philosophy ’, * theory of knowledge

The term 'logte of rn'ener’ will 1 m understood by us in a very wide
sense, namely, as meaning the domain of all the questions which are
usually designate as pure and applied logic, as the logical analysis
of the special sciences or of acience as a whole, as epistemology, a»



§72 PHILOSOPHY BEPIACED BT THE LOGIC OF Science 281

problems of foundations, and the like (m so far as these questiona
are free from metaphysics and from all reference to norms, values,
transcendentals, etc) To give a concrete illustration we assign the
following mvesagatjons {with very few exceptions) to the logic of
saence theworks of Russell, Hilbett Brouwer, and their pupils, the
works of the Warsaw logicians, of the Harvard logicians, of Reichen-
bach’s Circle, of the Vienna Circle centring around Schlick the
majority of the wozLa ated m the bibliography of this book (and
others by the same authors) the articles m the journals Erkermtms
and PMosophy of Science, the books m the collections ‘ Schnften
2ur wissenschaftlichen Weltatiffessung'* (edited by Schlick and
Frank), '‘Einheitswissenschaft* (edited by Neuiath), and finally

jV _ _ .Ilf r, ,


I


(geiiLiaij, iij^ u ^AiueiidUiaumunsj, i9<j u \rousu auinuisj, 409 11
(general)


§ 73. The Logic of Science is the Syntax
OF THE Language of Science

In what follows we shall examine the nature of the questions of
the logic of science m the wide sense, including, as already mdi>
cated, the 80*called philosophical problems concerning the founds*
tions of the indindual sciences, and we shall show that these
questions are questions of syntax In order to do this, it must first
be shown that the object~questions which occur m the logic of
saence (for example, questions concerning numbers, thmgs, time
and space, the relations between the psychical and the physical,
etc ) are only pseudo-object questions — i.e questions which, be-
cause of 3 misleading formulation, appear to refer to objects while
actually they refer to sentences, terms, theories, and the like — and
are, accordingly, in reality, logical questions And secondly, it
must be shown that all logical questions are capable of formal
presentation, and can, consequently, be formulated as syntactical
questions. According to the usual view, all logical mvestigation
comprises two parts a formal mquiry which is concerned only
With the order and syntactical kind of the linguistic expressions,
and an mquiry of a matenal character, which has to do not merely
With the formal design but, over and above that, with questions of
meaning and sense. Thus the general opinion is that the formal
problems constitute, at the most, only a small section of the do main



P.UIT V. PHOOSOPHT AXD SYNTAX


of logical problems. As opposed to this, oxir chseussion of general
syntax has already shovm that the formal method, if carried far
enough, embraces all logical problems, c\'en the so-called pro-
blems of content or sense (m so far as these art genuindy logical
and not psychological in character). Accordingly, when we say
that the logic of science is nothing more than the sj^ntax of the
language of saence, we do not mean to suggest that only a certain
number of the problems of what has hitherto been called the logic
of science (as thej’ appear, for example, in the worl.s previously
mentioned) should be regarded as true problems of the logic of
saence. The new we intend to advance here is rather that all
problems of the current logic of science, as soon as they are
exactly formulated, are seen to be svmtactical problems.

It wasWmgenstcm who first exhibited thecJoseconnecticin between
the logic of saence (or "philosophy”, as he calls ii) and sjmtax. In
particular, he made clear the formal nature of lope and enphasited
the fact that the rules and proofs of sj-ntax should hatTs no refer«i«
to the meaning of simbols {{Traetotm). pp. 52, 56. and 164).
Further, he Has shown that the so-called sentences of metiphjria
and of ethics an pteudo-sentcnces. Accordaig to him philosophy is
"cntique of language” (op. or p. 62), its business is “the logieal
clanfieaoon of ideas" (p. 76). of the sentences and concepts of
saence (natural saence), that is. m our terminology, the logic of
saence. ^Vittgenstem’s \new 1$ represented, and has been ftinher
developed, by the Menna Circle, and m this part of the book I owe
a great deal to hts ideas. If I am right, the position here maintained
is in general agreement with his, but goes bcjx>nd it in certain im-
portant respects In nhst follows my view will sometimes be con-
trasted mth his, but this is done only for the sake of greater clarity,
and our agreement on important fundamental questions must not
therefore be otcriooked.

There are two points e^pedally on which the riew here presented
differs from that of Wingenstein, and specifically from his neganie
theses. The first of these theses (op. eil. p. 7S) states : " Propositions
annot represemthe logical form; this mirrors itself in the proposi-
tions. That which mirrors itself in language, language cannot repre-
sent. That which expresses iUrlf m language, roe cannot express by
language.... If two propositions contradict one another, th« w
shown by their structure; simUarly, if one follows from another, etc.
VTiat can be shown eamol be said. ... It w ould be as senseless to
ascribe a formal property to • proposuion as to deny it the formal
property. ’ In other words: ‘Zyiere are no sentences alxiut the forms
of sentences ; there is nocxpressibletyntix. In opposition to this view ,
our construction of s)-ntax has shown that it can be correctly formu-
lated and that sjntactical sentences do exist. It is just as possible to



§ 73 LOGIC OF SCIENCE THE SYNTAX OF LANGUAGE OF SCIENCE 283

construct sentences about the fonns of linguistic expressions, and
therefore about sentences, as it is to consOuct sentences about the
geometrical forms of geometrical structures In the first place there
are the analytic sentences of pure syntax, which can be appbed to the
forms and relations of form of linguistic expressions (analogous to
the analytic sentences of arithmetical geometry which can be ap-
plied to the relations of form of the abstract geometncal structures) ,
and m the second place, the synthetic physical sentences of de-
scriptive syntax which are concerned with the forms of the hnguistic
expressions as physical structures (analogous to the synthetic em-
pirical sentences of physical geometry, se^ § 25) Thus syntax «
exactly formulable tn the same soay as geometry is

Wittgenstem’s second negative thesis states that the logic of
science ("philosophy”) cannot be formulated (For him, this thesis
does not comcide with the first since he does not consider the logic
of science and syntax to be identical , see below ) “ Philosophy is not
a theory, hut an activity A philosophical work consists essentially
of eluadations The result of philosophy is not a number of ‘ philo-
sophical propositions,’ but to make propositions clear’ (p 76)


has climbed out through them, on them overthem (Hemust soto
speak, throw away the ladder, after he has clunbed up on it ) He
must surmount these propostuons, then he sees the world rightly
Whereof one cannot speak, thereof one must be silent” (p 188)
According to this, the investigations of the logic of science contain
no sentences, but merely more or less vague explanations which the
reader must subsequently recognize as pseudo-sentences and
abandon Such an interpretation of the logic of science is certainly
very unsatisfactory [Ramsey first raised objections to Wittgen-
stem’s conception of philosophy as nonsense, but important non-
sense {[Foundations], p 263), and then Neurath, m particular,
([iSozjo/ PAyr],pp 395 f andiPiycAo/J, p 29) definitely rejected it ]
When m what foUows it is shown that the logic of science is syntax.
It is at the same tune shown that the logic of science can be formu-
lated, and formulated not in senseless, if practically mdispensable,
pseudo sentences, but m perfectly correct sentences ’The difference
of opinion here indicated is not merely theoretical, it has an im-
portant influence on the practical form of philosophical mvestiga-
tions Wittgenstem considers that the only difference between the
sentences of the speculative metaphysician and those of his own and
other researches mto the logic of science is that the sentences of the
Vogic ol science — v^mdn^e c^s {IhfuKop'hical e'luciiations — in spite
of their theoretical lack of sense, exert, practically, an important
psychological influence upon the philosophical mvestigator, which
the properly metaphysical sentences do not, or, at least, not m the
same way Thus there is only a difference of degree, and that a very



PART V. PHILOSOPinr AND SYNTAX


284

vague one. The fact that Wittgenstein does not believe in the possi-
bility of the exact fonnulation of the sentences of the logic of sdence
has as its consequence that he does not demand any sdentific
exactitude in his own fonnulations, and that he draws no sharp line
of demarcation between the fonnulations of the logic of sdcnce and
those of metaphysics. In the following discussion we shall see that
translatability into the formal mode of speech — that is, into syn-
tactical sentences — is the criterion which separates the proper sen-
tences of the logic of science from the other philosophical sentences
— we may call them metaphysical. In some of his formulations,
Wittgenstem has clearly overstepped this boundary; this conse-
quence of his belief in the two negative theses is psychologicallv
quite understandable.


syntactical sentences about the language of science; but no new
domam in addition to that of sdence itself is thereby created. The
aentencea of ayniax arc m part sentences of arithmetic, and in part
sentences of physics, and they are only called syntactical because
they are concerned with linguistic constructions, or, more sped*
fically, with their formal structure. Syntax, pure and descriptive, is
nothing more than (he mathematics and physics of language.

Wittgenstem says of the rules of logical syntax (see above) ^t
they must be formulated without any reference to sense or meaning.
According to our view the same thing holds also for the sentences


Sophy “ IS that activity by which the mearjing of propositions is estab-
lished or discovered”; it is a question of “what the propositions
actually ffifon. The content, soul, and spirit of science naturally con-
sist in what is ultimately mtant by its sentences; the philosophical
activity of rendering significant is thus the alpha and omega of all
sdentific knowledge”.

§74. Pseudo-Object-Sentetjces

Wc have already distinguished (in an inexact manner) between
object-sentences and logical sentences. We will now contrast in-
stead (at first also in an inexact manner) the two domains of objtct-
zent^es and syntactical sentences, only those logical sentences
which are concerned with form being here taken into account and
included in the second domain. Now there is an Intermediate field
between these two domains. To this intermediate field we will



§ 74 - PSEUDO-OBJECT-SENTENCES 285

assign the sentences which are foRnulated as though they refer
(either partiaUy or exclusively) to objects, while m reality they
refer to syntactical fonns, and, specifically, to the forms of the
designations of those objects with which th^ appear to deal Thus
these sentences are syntactical sentences in virtue of their con-
tent, though they are disguised as object sentences We will call
them pseudo-d^ect-tenteiues If we attempt to represent m a
formal way the distmction which is here infor mall y and mexactly
mdicated, we shall see that these pseudo-object sentences are
simply quan-tyntaetual sentences of the material mode of speech
(m the sense already formally defined, see § 64)

To this middle territory belong many of the questions and sen
tences relatmg to the mvestigation of what are called philosophical
foundations We will take a simple example Let us suppose that
m a philosophical discussion about the concept of number we
want to point out that there is an essential difference between
numbeis and (physical) things, and thereby to give a warning
against pseudo-questions concerning the place, weight, and so on
of numbers Such a warning will probably be formulated as a
sentence of, say, the foUowmg kind “Five ts not a thing but a
number” (SJ Apparently this sentence expresses a property of
the number fire, like the sentence ” Five is not an even but an odd
number” (SJ In reality, however, ^ is not concerned with the
number five, but with the word ‘five’, this is shown by the formu-
lation 0 , which 13 equipollent to ‘‘‘Five’ is not a thing word
but a number-word.” \\'hUe is a proper object-sentence, Si is
a pseudo-object sentence. Si is a quasi syntactical sentence
(material mode of speech), and Sj is the correlated syntactical
sentence (formal mode of speech)

We have here left out of account those logical sentences which
assert something about the meamng, contenty or seme of sentences
or linguistic expressions of any domam These also are pseudo-
object-sentences Let us consider as an example the following
sentence, Sj “Yesterday’s lecture was about Babylon ” Si ap-
jwars to assert some thing about Babvlon, smee the name ‘ Babylon ’
occurs m It. In reahty, however. Si says nothmg about the town
Babylon, but merely somethmg about yesterday’s lecture and the
word'Babyloa’ Thisiseasily shown by the foUowmg non formal



286


PART V. PHILOSOPHY AND SYNTAX


Babylon it does not matter whether Gj is true or false. Further,
that Gi 13 only a pseudo-object-sentence is clear from the circum-
sunce that Si can be translated into the following sentence of
(descriptive) syntax; “In yesterday’s lecture either the word
' Babylon ’ or an expression synonymous with the word ‘ Babylon ’
occurred ’’ (Sj).

Accordingly, we distinguish three kinds of sentences-

I. Object-sentences z. Pseudo-object- 3. Syntactical

sentences— quz^i- sentences

syntactical sentences
Matersal mode of Formal mode of

speech speech

Examples: "5 is Examples: "Five is Examples: "‘Five’

a prune number"; not a ^mg, but a is not a thmg-word,
" Babylon was a big number"; "Babylon but a number-word" 5
town"; "lions are was treated of m "the word ‘Babylon*
mammals " yesterday's lecture." occurred in ye*-

(" Five u a number- terday’s lecture’’|
word" IS on example ‘“A* —’A’ is a con-
belonging to the su- tradictery sentence."
tonymous mode of
speech.)

The intermediate field of the pscudo-object-sentences, the
boundaries of which have so far been only materially and inexactly
indicated, can also be exactly, and moreover formally, demarcated.
The pscudo-object-sentenccs arc, namely, quasi-syntactical sen-
tences of the material mode of speech. [We can leave the autony-
mous mode of speech out of account here, since there is practically
no danger of a sentence belonging to this mode of speech being
mistalcn for an object -sentence.] The criterion of the material
mode of speech assumes a simpler form when we are concerned
with an object-language S| which o^ntains its own syntax-
language S, as a sub-language. For instance, let Sj be the English
language representing the whole language of science; then the
syntax -language Sj, in which the syntax of Sj is formulated, is a
sub-language of S,. Thu expresses the fact that we regard syntax
not as a special domain outside that of the rest of science but a* a
sub-domain of science as a whole, which forms a single system
(Neurath: Einheitsmssenschaft) having a single language S,.
That a language may contain its own syntax without contradiction



§74 PSETOO-OBJECr-SENTENCES 287

we have already shown E\ cn rf the syntax-language Sj is a sub-
language of Si it IS, of course, both possible and necessary to dis-
tinguish between a sentence Si, of Si (which may also belong to
S,), and a syntactical sentence concerning Si, which belongs to

Sj and therefore also to Si For simphaty’s sake, we will formulate
the criterion of the material mode of speech for the simplest sen-
tential form only (and further, for the sate of brevity and clantj,
we will formulate it for a symbohc sentence) (sec § 64) Let Si be
' P (a) ’ , Si IS called quan-fyniactteal in respect of ‘ a if there exists
a syntactical predicate *Q* such that ‘P(a)’ is equipollent to
*Q(*a’)’ (Sj) and *P{b)’ is equipollent to ‘Q(‘b')’, and corre-
spondingly for every expression isogenous with ‘ a ’ Now ‘ P ’ may
possibly be a syntactical predicate which is equivalent in meaning
to ‘Q’ (this would be shown formally by the fact that ‘P('a’)’
would also he a sentence and moreover a sentence equipollent to
'Q(‘a’)’, and that, further, •p(‘b’)’ would be equipollent to
and correspondingly for every expression isogenous
with 'a'), if this u not the case, we call 3 ^ a sentence of the
material mode of tpeech * Q ’ is called a syntactical predicate corre.
lated to the quasi-syntactical predicate*?', and Si is called a syn-
tactical sentence correlated to the quasi-syntacti^ sentence 6^
In the tranzlahm from the material to the formal mode of tpeech,

u translated tnlo ©j

In order to make it clearer and facilitate its practical apphcation
to the following examples, we will formulate the entenon (still for
the simplest form of sentence) once more, in a less exact, non formal
way (the examples of sentences which come later, especially those
of the logic of saeoce, belong almost entirely to the word-language
m consequence, they are themselves not formubted suffiaently
exactly to make possible the application to them of exact concepts)
Gx IS called a sentence of the material mode of speech if asserts
a property of an object which has, so to speak, parallel to it,
another, and syntactical, property, that is to say, when there is a
syntactical property which belongs to a designation of an object
if, and only if^the original property belongs to the object

It is easy to sec that m the previous example concerning ‘ Baby-
Ion’ this entenoir is fulfilled for the sentence 0^ the syntactical
(m this case the descriptive syntactical) property which is asserted
in Gj of the word 'Babylon’ is parallel to that property which is



jSS PAST T. PHILO50PHT AKt) ST>TAI

iSsertMbSi of thftoirn of BAb)JaQ;for if, indooJy if, Ttsterdir’s
lecture wss oanceioed with i ccmin object, did a d» of
thit object occur in the lecture. The cnterion of the rnateriel n»de
of speech b liewise fulfilled for the Ksieace of the cxxraple
ro n ccT "’" ^ ‘five*; for if, and only if, the propertr expressed

IB gj tli3t of being not a thmc but a number — belongs to some

object (for instance, to the number fire) does the property es-
pre»ed la S*— that of being not a thing-trord but a number-
trord — belong to a desigaatioa of this object (in the example, to
the word ‘fiveO-

§ 75, SDsTences abodt Mewing

In tbW secnon, we shall consider varaous kinds of sentences of
the taatenal mode of speech, espeoaDy those bnds which occur
frequently is philosophiol discussions. On the basis of these js*
resugatiOttS we shall be better able to diacaosc the material mode
of speech in subsequent cases. Further, by this means the whole
character of philosophical problems triD become dearer to us. The
obscunt}* with regiird to this diaractcr 15 chiefly due to the de-
ctpuoa and seU.deception induced by the appheataoa of the
toatenal mode of speech. The disguu« of the material mode of
speech conceals the fact that the so-called problems of phDo*
sophical foundatioas are nothing more ihari questions of the l:^c
of sacnce concerning the sentences and sentential coanecdoas of
the language of saence, and also the further fact that the questions
of the logic of saence are formal — that is to ay, syntacrical —
questions. The true situation is revealed by the translation of the
Sentences of the mateiul mode of speech, which are rtsUy quad*
sj-ntactical sentences, into the correlated syntactical sentences and
thus into the formal mode. We do not mean by tlus that the
material mode of speech should be entirely diiainated. Since it is
in general use and often eauer to understand, 11 may well be re-
tained in its place. But it is a good thing to be conscious of its use,
cc as to avoid the obscurities and pseudo-problems wlucli otbes*
wise easily result from it.

In a sentence c^of the material mode of speech, the illusion that
a genuine object -sentence is present is most easOy dissipated if ;j
belongs in pan to the sjutax-hnguage Sj, but contains at the same



§ 75 SENTENCES ABOUT MEANING 289

tune elements of Sj which do not belong to Sj [Not all sentences
of this kmd are sentences of the material mode of speech For
example, the sentence **The University of Freiburg bears the
inscnption ‘the truth will make you free’” is not a quasi-syn-
tactical sentence but a sunple sentence of descnptive syntax.]
Especially important here are those sentences which express a re-
lation of designation, that is to say, those m which one of the
follcrwmg expressions occurs ‘treats of’, ‘speaks about’, ‘means’,
‘signifies’, ‘names’, ‘is a name for’, ‘designates , and the like
We shall now give a senes of such sentences concemmg meaning,
and, along with them, the coiiebted syntactical sentences The
fiist of these examples has already been discussed [It is, of course,
of no importance whether or not the sentences in the examples are
true ]


Matenal made oj speech
(quasi-syntacQcal
sentences)

10 Yesterdsy’slecturetreated
of Babylon

20 Theword 'daystar’</««i-
nates (or means, or u u name
for) the sun

30 The sentence Si means
(or asserts, or has the content,
or has the meamng) that the
moon IS sphencal

4a The word 'luna' in the
Latin language designates the
moon


5 a The sentence * ’ of the

Chinese language means that the
moon 13 sphencaL


Formal mode of speech
(the correlated syntactical
sentences)

ih In yesterday’s lecture the
word ‘Babylon’ (or a synony-
mous designauon) occurred
zb The word 'daystar* is
synonymous with ‘ sun ’

3 b 5| IS equipollent to the
sentence The moon is sphen-
cal ’

4b There is an equipollent
expressional translation of the
Latm into the English language
in which the word ‘moon’ is the
correlate of the word ‘luna’

Sb There is an equipollent
sentential translation of the
Chinese into the English lan-
guage m which the sentence
‘The moon is sphencal’ is the
correlate of the sentence ‘ ’


The following examples, 6 and 7, show bow the difference be-
tween the meanmg of an exprtssson and the object designated by the
ss^essum can be formally represented [This difference is em-
phasized by the phenomenologists but explained only m a psycho-
logical, not m the logical, sense ]



290 PART V. PinLOSOPKY AND SYNTAX

6a. The expressions ‘merle* 66. ‘Merle’ and ‘blackbird’
and ' blackbird ' have the same are L-synonymous,
meaning {or: mean the same; or:
have the same intentional object).

•ja. ‘Eveningstar’and‘mom- 76. ‘Evening star’ and
ingstar’haveadiflerentmeaningt ‘morning star’ arc not L*syn-
butthcydrngna/ethesameobject. onymous, but P-synon>’mous.

[With respect to a symbolic (P-) language, the above correlates
may also be formulated thus:66. is analytic. 76. ‘9Ii = 9I|’

IS not analytic but P-vatid.]

In the case of sentences the formal representation of the difference
between the fact designated and the meaning is analogous. [The usual
formulations like ‘mean the same' or ‘have the same content' are
ambiguous; m some cases 861s intended, in others 96, and in many
the intention remains obscure.]

8 a. The sentences G^and 86. and 0t are L-equi-

have the same meaning. pollent.

9a. 3i and €| have a dif> 96. 0| and S| are not L-
ferent meaning but they represent equipollent but P-equipollent.
(or: describe) the same fact.

[With respect to a symbolic language; 86. ‘ SjS 6,’ is analytic.
96. ‘ 3|s 3,’ is not analytic but P-valid.)

toa, The sentences of arith- 106. The sentences of arith*
metic Hate {or: express) cetuin metic are composed of numerical
properties of numbers and cer- expressions and one- or many-
tain relations between numbers, termed numerical predicates
combined in such andsucha way.

n o. A particular sentence of 116. A particular sentence of
physics states the condition of a physics consists of a descriptive
spatial point at a given time. predicate and spatio-temporal
co-ordinates os arguments.

The following examples xza, 13 a, and 14 a appear at first to be of
the same kind as lo and 4 a. Actually, however, they demonstrate
parucularly clearly the danger of error which is involved in the use
of the material mode of speech.

This letter ir fl6o«r the 126. In this letter a sentence
son of Mr. Miller. (51^) occurs in which 91, is the

description ‘the son of Mr.
Miller ’.

i*3**’n»^* ”^^^**''’” *** ****“ *3^' There is an equipollent

valdcM designates {oximeam) expressional translation from the
the horse of M. French into the EngUsh Ian-

guagein which ‘the horseof M’is
the correlate of ‘lecheval de M’.

140. The expression *un <16- 146. {Analogous to 136.)

phant bleu means a blue ele-
phant.



291


§ 75 SENTENCE ABOUT R2ANING

Let US assume that Mr IVliller has no son, even m this case the
sentence 12 a may still be true, the letter will then merely be tellmg
a he Now, from the true sentence 12 a, accordmg to the ordinary
logical rules of inference, a false sentence can be derived In order
to mate the derivation more exact, we will use a symbohsm m place
of the word-language Instead of 'this letter’ we will wnte ‘b’,in
stead of ‘ b 13 about a ’ we will wnte * H (b, a) ’ , and mstead of ‘ the son
of a’ we will wnte ‘Son'a’ (descnptianal in Russell’s symbolism,
see §380 Hence for 12a will be wntten ‘H(b Son Miller)’
(Gj) According to a well know’n theorem of logistics (see my
[LoguZiA], § 7 e L 7 2), from a sentence ^r (Srg) in which a descnp-
tion occurs as argument, a sentence is denvable which asserts that
there exists something which has the descnptional property
Accordingly, from S, would be denvable ‘(3*) (Son (x, Miller))
(®i), or, m words “a son of Mr Miller exists ” This, however is a
false sentence Similarly the possibly false sentence “There is a
horse of M ’’ is denvable from 13 a, and the false sentence ‘ there is
a blue elephant’’ from 14a On the other hand, by the usual rules
no false sentences can be denved from the sentences 12^, 13^ and
146 of the formal mode of speech These examples show that the
use of the matenal mode of speech leads to contradictions if the
methods of inference which are correct for other sentences are
thoughtlessly used also m connection with it [It cannot be mam-
tamed that the formulations tza, T30, and 14 a are mcorrect, or that
the use of the matenal mode of apee^ leads necessarily to contra-
dictions , for, after all, the word language is not bound by the rules
of logistics If, therefore, one wishes to admit the matenal mode of


Some sentences contam a relation of meamng which is to some
extent concealed. With sentences of this kind it is not obvious, at
first sight, that they belong to the matenal mode of speech The
most important examples of this arc the sentences which use the
so-called indirect or oblique mode of speech (that is to say, sen-
tences which say something about a spoken, thought, or wntten
sentence, but which do so not by a statement of the ongmal word-
ing but instead by means of a ‘that’, ‘whether’, or other ‘w '
sentence, or of a subordinate sentence without a connective word,
or of an infimtive with ‘ to *) In the foUowmg examples, 15 a and
ibo, iftie formulations 15^ and i6i sliow that the sentences m
which the indirect mode of speech occurs are of the same kmd as
the examples previously discussed, and hence also belong to the
matenal mode of speech



292


PART V. PIIILOSOPHT AND SYNTAX


I. Material mode of speech


I. Sentences in in-
direct speech
150. Charles said
(wrote, thought) Peter
was coming tomorrow
(or: that Peter was
coming tomorrow).

160. Charles said
tehere Peter is.


2. Sentences about
meaning

tsfr. Charlessaid
a sentence which
means that Peter is
coming tomorrow.

t6b. Charlessaid
a sentence which
states where Peter is.


II. Formal mode
of speech

tje, Charles said
the sentence ‘ Peter is
coming tomorrow ’(or :
a sentence of which
this is a consequence).

i6c. Charles said
a sentence of the
form ‘Peter is — ’ in
which a spatial desig*
nation takes the place
of the dash.


The use of the indirect mode of speech is admittedly short and
convenient ; but it contains the same dangers as the other sentences
of the material mode. For instance, sentence 15 a, as contrasted
with sentence ije, gives the false impression that it is concerned
with Peter, while in reality it is only concerned with Charles and
with the word ‘Peter’. When the direct mode of speech is used,
this danger does not occur. For instance, the sentence: "Charles
says ' Peter is coming tomorrow ’ ” does not belong to the material
mode of speech: it is a sentence of descriptive syntax. The direct
mode of speech is the ordinary form used in the word-language for
the formal syntactical mode. (On the construction of the syn-
tactical designation of an expression with the help of inverted
commas, see § 4t.)

The examples so far given sufEce to show that, with certain
formulations in the material mode of speech, there Is the danger of
obscurity or of contradictions. It is true that in such simple cases
as these the danger is easy to avoid. But in less obvious cases of
essentially the same kind, especially in philosophy, the application
of the material mode of speech Im time and again led to incon-
sistencies and confusions.


§ 76. Universal Words

We will call a predicate of which every full sentence is an ana-
lytic sentence a universal predicate, or, if it is a word in the word-
language, a universal word. [Foreverygenusof predicates a uni*



§ 76 UNTVERSAL WORDS 293

versal predicate can easily be de6ned For instance, if pri is a
of any genus whatsoever, we define the universal predicate pr*. of
the same genus, as follows Fr,(i)i)s(pri(Di)V/*-pti(oi)) ] The
investigation of universal words is especially important for the
analysis of philosophical sentences TTiey occur very often in such
sentences both m metaphysics and m the logic of science, and are
for the most part m the material mode of speech In order to
facilitate the practical application of the cntcnon for ‘universal
word ’, let us also formulate it in an informal way A word is called
a universal word if it expresses a property (or relation) which be
longs analytically to all the objects of a genus, any two objects
being assigned to the same genus if their designations belong to the
same syntactical genus Since the rules of syntax of the word-
language are not exactly established, and smce linguistic usage
vanes considerably on just this point of the genenc classification
of words, our examples of umveisal words must always be given
With the reservation that they are valid only for one particular use
of language

Exampltt I ‘Thmg’iaaumversalwordCprovidedthatthedesig-
nations of things consatute a genus) In the word senes 'dog*,
‘animal’, ‘living creature’, ‘thing’, every word is a more compre-
hensive predicate than the previous one, but only the last is a urn-
Versal predicate In the corresponding senes of sentences, ' Caro u
adog’, ‘ IS an animal’, ‘ a living creature’, 'Caro is a thing*, the
content is successively diminished But the final sentence is funda-


designation, the result is not a sentence at all
2 ‘Number’ is a universal word (provided that the numencal
expressions constitute a genus, as for instance m Languages I and
II, as opposed to Russell s language where they form a part of the
class-expressions of the second level) In the senes of predicates,
‘number of the form 2*-H 1 ‘odd number’, ‘number’, only the last
IS a universal predicate In the senes of sentences *7 has the form
*7 IS odd’, ‘7 is a number’, the second is already analytic,
but only the third has the property that every sentence which re-
sults from it if ' 7 ’ is replaced by another 3 again analytic. If ‘ 7 ’
IS replaced by an expression which is not a 3 *10 sentence re-

sults (on the assumptions made at the beginning)

Examples of umtersal tcords ‘thing’, ‘object’, ‘property’,
relation’, ‘fact’, ‘condition’, ‘process’, ‘event’, ‘action’, ‘spatial



PART V. PIULOSOPKY AND SYNTAX


*94

point’, ‘spatial relation', 'space' (sj’stem of spatial points con-
nected by spatial relations), ‘temporal point’, ‘temporal relation’,
‘time’ (system of temporal points connected by temporal rela-
tions); ‘number’, ‘integer’ (m I and II), 'real number’ (in some
systems), ‘function’, ‘aggregate’ (or ‘class’); ‘expression’ (in a
language of pure sjTitax); and many others.

We all use such universal words m our witings in almost every
sentence, especially in the logic of science. That the use of these
words IS necessary is, however, only due to the deficiencies of the
word-languages, i.e. to their inadequate syntactical structure.
Every language can Oe transformed in such a way that universal
words no longer occur in it, and this without any sacrifice either of
expressiveness or conciseness.

We will now dtstmguish ttco mtOibds of employing universal
words (without making an exact and formal differentiation). The
second method involves the material mode of speech, and will be
dealt With later. The first method has to do with genuine object*
sentences. Here a universal word serves to point out the syn-
tactical genus of another expression. In some cases the syntactical
genus of the other expression is already uniiTically determined by
Its form alone; the special indiaiion of it by means of the added
universal word is then only of use in making it more prominent,
as an aid to the comprehension of the reader. In other cases, how-
ever, the addition of the universal word is necessary, since without
it the other expression would be ambiguous. In all these cases of
the first way of using it, the universal word is, so to speak, de-
pendent-, it is an eULnliasy grammatical symbol added to another
expression, something like an index.

Examples: i. "Dy means of the process of cr^’stallizalion. . ..’’
Since crystallixation belongs without any ambiguity to the genus of
the processes, onefnightsimplysayj*' By means of crystallixation....”
ffere the universal word 'process’ only ser>-es ta.point out the genus
to which the Word ‘crysialltzation' belongs. Similarly in the fol-
lowing examples; a. "'nie condition of fatigue.. 3. "The num-
ber file....’’

In the following sentences the universal word is necessary for
univocality. It can be rendered superfluous by the use of a sufiir
(‘ 7 ’ and * 7r ;) or by introducing various explicit expressions in place
of the ambiguous one. 4a. ‘‘The integer 7...." 46. ’"rhe real num-
ber?-'”” S'*- “The condition of friendship....’’ 5A.‘'The relation
of friendship...."



§76 UNIVEBSAI. WORDS 295

In the word language universal words are especially needed as
auxibary symbols for vanabUs, that is, m the formulation of uni-
versal and existential sentences, for the purpose of showing from
which genus the substitution-values are to be taken The word-
language employs as variables words (‘a’, ‘some’, ‘every’, ‘all’,
‘ any ’, and so on) to which no particular genus is correlated as their
realm of values If, as is usual in the symbolic languages, different
kmds of variables were used for the different genera of substitution-
values, the addition of a uruversal word would be superfluous
Accordmgly, the universal word here serves to some extent as an
index to a variable, which mdicates the genus of its substitution-
values


uanumber ” 7^ )” (where**’ is a 5) 8a“Iknowa
thing which " 86 "Q*)( )” (where u a thing-varuble)
9 a “Every numencal property “ ph “(f 0 ( )’’ (where ‘f’ is
a p of which the values are SPt*) 10a “There 1$ a relation ’’
106 “(3F)( )” (where ‘f’ IS a p^

WittgensteintTMCMfuiJp 84says “Sothcvanablename‘**i8the
proper sign of the pseudo-concept object Wherever the word ‘ob-
ject’ (‘thing’, ‘entity’, etc ) is rightly used, it u expressed in logical
symbolism by the vanable name Wherever it is used otherwise,
1 e as a proper concept word, there anse senseless pseudo-proposi-
tions The same holds of the words ‘complex’, ’fact’, ‘function’,
‘number’, etc. 'They all signify formal concepts and are presented m
logical symbolism by variables, not by functions or classes (as Frege
and Russell thought) ExpressionsUke'i is a number’, 'there is only
one number nought’, and all like them are senseless “ Here the
correct view is taken that the umversal words designate formal (m
our terminology syntactical) concepts (or more exactly are not
syntactical but quasi syntactical predicates) and that m the transla-
tion into a symbolic language they are translated mto variables (or,
again more exactly they determme the kmd of variables by which
the words ‘ a * every ’, and so on, are translated , it is only the kmd
of vanables that is determined, and not their design, m the
examples given above, *y* or 's’ can equally well be taken instead
of ‘* ) On the other hand, I do not share Wittgenstein’s opmion
that this method of employing Ae universal words i4 the only ad-
missible one We shall see later that, precisely in the most important
cases, there is another method of use m which the uruversal word is
employed independently ( ‘as a proper concept-word”) There it is
a quesnon of sentences of the material mode of speech which are to
he translated mto syntactical sentences Sentences of this kmd with



PART V, PBILOSOPflTJlXD STTCTAX


296


« unh-ereal word sre held by TIRt^eoJtdn to be nonsecse, beerse
be docs not consider the correct fotmulstjon of syntscticsl sentences
to be possible.


Tf"-! use 0/ tasvmal tDordt pi ^i/atiora in connectioa with oae of
the w. . . interrogatives (‘what*, 'who*, ‘where’, ‘whidi’, etc.)
13 akia to their use in universal and existential sentences. Here
also, in translation into a symbolic language, the univeraal word
determines the choice of the kind of variable. A yes-or-no quo-
tion demands either the affirmation or the denial of a certain sen*
icnce Sj., that is to say, the assertion of either ^ or ''-Si-
[SxarTTple: The question “Is the tablcTound?” requires us to
assert tn answer either: “the table is round" or; "the tablets not
round."] As contrasted with this, aw... question demands in
reference to a certain sentential function the asscrticin of a dosed
full sentence (or sentential framework). In a symbolic question,
the genus of the arguments requested is determined by the kind of
the argument varublea. In the word-languages this genus is in-
dicated by means either of a specific w. . . inierrogative (such u
‘who’, 'where', ‘when*) or of an unspecific w. . . interrogative
(such as ‘ what ‘ which ’) with an auxiliary tmtwnLtl word. Hence
here also the universal word is, ao to speak, an index to a variable.


ExmfUt: t. Suppose I want to ask someone to make an ssiertioo
*be form “Charles was — in Berlin ’’.where atime-detenruastiem
of wfairt I am ignorant but which I wish to leam from the tss^on
u to take the place of the dash. Now the Question must indicate by
some mpsr* ti”* ;i'« — - • , t . , ^


^cstion, the variable whose argument is requested must be bound
by means of a question-operator. e.g. ' (? r) (Charles was t in Berlin)’.]
'‘^Suagethekiodofargumentrequestedismadelcnowa
^er by means of the specific question-word ’when’ r\S'heaw*s
*1 ; Berlin? or jjy means of the universal word 'time’ ot

to an unspecific question-word f'At

what toe t^-as Charles in Berlin?’*).

t® make me an assertion of the form
.f ,t- j r«-'r "'*'^*'’*^*** • relation-word is to take the place

^ or the hke). The symbolic

relational variable 'A*.

hv m<- r^^J"^*^’^”"))’-Iofonnulationinthewofd-Unguagc
‘’f **>* ^ord 'relation’ to



§77 tJNlVEBSAl. WORDS IN MiTERlAL MODE op SPEECH 297


§ 77. Universal Words in the Material
Mode of Speech

la t;e first use of the univeml woid, which we have up to now
been ducussing, it appears as an auxiliaiy symbol determining the
genus of another expression, it was found that, if m place of this
other expression a symbol indicating its own genus was mtroduced,
then the universal word could be dispensed with. As opposed
to this, tn the leeond ute the tamenal toord appe a rs as an inde-
pendent expression, which tn the simplest form occupies the place
ofthe predicate m the sentence m question. Sentences of this kmd
belong to the material mode of speech , For a umversal word is here
a quasi-syntactical predicate, the correlated syntactical predicate
13 that which designates the appertaining optessional genus
[Exan^Ie 'cumber^ is a umversal wonl because it belongs ana-
lytically to all the objects of a genus of objects, namely, that of the
numbers, the correlated syntactical predicate is ‘numerical ex-
pression* (or ‘number-word’), smce tlus appbes to all expressions
which designate a number The sentence “ Five is a number” is a
quasi-syntactical sentence of the material mode of speech , a corre-
lated syntactical sentence is ‘“Five* ts a number-word” ]
Sentences tnth umversal Syntacbcal sentences

teords

(Material mode of speech) (Formal mode of speech)

xya ThemoonxsstAnt^.Sve 176 ‘Moon’ u a tfaing-word
IS not a thing, but a numAer (thing name), ‘five’ is not a
dung-word, but a number-word.

In 170, as contrasted with sentences like "the thing moon ”,

thenumberfiTc ”, the umversal words ‘thing’ and ‘number ’are
independent.

18a ApropertyisnotstAntg j 18A An adjecnve (property-
I word) IS not a tbmg-word

That the formulation i8a is open to objection is shown by the
Rowing consideration 18 a violates the ordinary rule of types
This comes out particularly clearly when an attempt is made to
Jbmulate it symbolically, either by means of ‘(F) (Prop (i^O ~
^umgCF^)’ or by means of *(x)(Prop(*)3 ~Thiag(*))’, m the
* Thing (F^’, and in the second case ‘Prop(x)’, is mcon-
sntent with the rule of types Therefore, if z8a is adimtted as a
sentence (it makes no difference whether ttue or false}, by the usual
syntax of logistics Russell s antinomy can be constructed If this is
to be avoided, special complicated syntactical rules are necessary


PART V. PHILt^OPIIT AND STNT.KX


=95


190. Fnendshtp is 8 rtlatton.

20 a. Fnendship is not a pTX>-
Prrty.


196. 'Fnendship* is a rcls*
lion-word.

2oi. 'Friendship* is not a
property-word.


I9<i corresponds to the sententiil form used by Russell
' ... cRel', the analogous symbolic formulation of 200 would, how-
ever, violate the rule of tj^pes. On the other hand, the correlated
sentences of the formal mode of speech, i9t and 20ft, are, ettn
w ithout any special preliminary adjustments, of the same land and
equally correct. In contrast with the pseudo-object-sentence 190,
a sentence of the form " Fnendship ensues if...”, for instance, is a
genuine object-sentence, and therefore not a sentence of the material
mode of speech.


It is frequently said that the rule of types (even the simple one)
restricts the expressiveness of a language to an inconvenient cirtenl,
and that one is often tempted to use foimuladons which would not
be allowed by it. Such formubtions, howci'er, are often (lie the
examples given) only pseudo-object-scntencts with universal
words. If, m such cases, instead of the object-tenns which one
would like to, but must not,combine,one uses the correlated syn-
tactical terms, the restrictive effect of the rule cf types disappeara.

Independent universal words apj^ar very often in philosophical
aentences, in the logic of science as well as in traditional philo-
sophy. Most of the examples of philosophical sentences whi^ vrill
be gn cn later belong to theraatcrial mode of speech by reason of the
employment of independent universal words.


§ 78. Confusion in PmLosopin' Caused by the
Material Mode of Speech

The fact that, in philosophical writings— cv’en in those which are
free from metaphjsics — obscurities so frequently arise, and that in
pWlosophical discuMions people so often find themselves talking
at cross purposes, is in large part due to the use of the material
instead of the formal mode of speech. The habit of formuUting ir
the material mode of speech causes us, in the first place, to deceive
ourselves about the objeas of our own investigations: pseudo-
object-sentMces mislead us into thinking that we are dealing with
eitn-lmguistic objects such as numbers, things, properties, ex-
penen^, states of affairs, space, time, and so on ; and the fact that,
m reolit)-, it ts a case of language and its connections (such as



§78 CXJKFVSIOS CAUSED BY TOE yiATtSlAL MODE 299

numencal expressions, thing-designations, spatial co-ordinates,
etc ) IS disguised from us by the material mode of speech This fact
only becomes dear by translation into the fonoal mode of speech,
or, m other words, mto syntactical sentences about language and
linguistic egressions

Further, the use of the matenal mode of speech gives nse to
obscurity by employing absolute concepts m place of the syn-
tactical concepts which are relative to language With regard to
every sentence of syntax, and consequently every philosophical
sentence that it is desired to interpret as syntactical, the lan-
guage or iind of language to which it is to be referred must be
stated- If the language of reference is not given, the sentence is
incomplete and ambiguous Usually a syntactical sentence is m-
tended to hold m one of the foUowmg ways

1 for all languages,

2 for all languages of a certain kind,

3 for the current language of science (or of a sub-domam of
saence, such as physics, biology, etc.),

4. for a particular language whose syntactical rules have been
stated beforehand ,

5 for at least one language of a certain kind,

6 for at least one language m general ,

7 for a langu^e (not previously stated) which is proposed as a
language of saence (or of a sub-domam of saence),

8 for a langu^e (not previously stated) whose formulation and
investigation is proposed (apart from the question whether it is to
serve as a language of saence or not)

If the formal syntactical mode of speech is used, then Imguistic
expressions are hemg discussed. This makes it quite dear that the
language intended must be stated In the majonty of cases, how-
ever, even if the language is not expressly named, it will be under-
stood from the context which mterpretation (say, of those just
given) 13 mtended. The use of the matenal mode of speech leads, on
the other hand, to a disregard of the relabmty to language of pktlo-
sophical sentences^ it is responsible for an erroneous conception of
philosophteal sentences as absolute It is especially to be noted that
the statement of a philosophical thesis sometimes (as in mterpreta-
tion 7 or 8) represents not an assertion but a suggestion Any dis-
pute about the truth or falsehood of such a thesis is quite mistaken.



PART V. PinLOSOPHT AND STNTAI


300

a mere emptj* battle of TOrds; xre can at most discuss the utility of
the proposal, or in\-estigate its consequences. But e%-en in cases
where a philosophical thesis presents an assertion, obscurity and
useless controv'ersy are liable to arise through the possibility of
sey’eral interpretations (for instance, x to 6). A few examples may
serve to make this clear. (For the sake of bresity, we shall formu-
late these sample theses in a more elementary manner than would
be done in an actual discussion.)

PhSIosophtcal sfnleneft S}-nia;t{cal senierxei

(Material mode of speech) (Formal mode of speech)

210. Numirrt are classes of aii. Xumerical expressions
classes of thin^rs. are class-expressions of the

second IcstI.

22 a. A’wmim belong to a 226. Numerical expressions
speaal primitive kind of objects, are expressions of the xero-level.

Let us assume that a logidst holds thesis 210. and a formalist
thesis 23 c Then between these tn'o there can be endless fruitless
discussion as to sihich of them is right and stbat numbers tctually
are. The uneertainry disappears as soon as the formal mode of
speech is applied. First of aU. theses si a and ss a should be trans-
lated into St b and 2: {>. Out these sentences are not >Tt complete,
because the statement of the language intended is lac^g. Various
imerpretaaona — such, for instance, as those mentioned preriouily—
are sail possible. Imerpretaoon 3 is obviously not intended- Under


Perhaps, however, the tno dbputants agree that they intend their
theses as proposals in the sense of 7, for instance. In that case, the
question of truth or falsehood cannot be discussed, but only the
question whether this or that form of language is the more ap-
propriate for certain purposes.

* 3 ^. Some refjfionj belong to 236. Some two- (or more-)
the primitise data. termed predicates belong to the

undefined descriptive primitive
. symbols.

^ 34a. ivelarimrareneserprimi- 246.AlJtwo-andmore-tenned
tivc data, they depend upon the predicates are defined on the
properties of their members. basis of the one-termed predi-

cates.

^ ®f theses 23 a and 240, discussion 5 s again fruitless and
deluded unul the disputants pass os’er to the formal mode of



§ 78 COVFtJSIO>I CAUSED BY THE mTEJUAL MODE 3OI

Speech and agree as to which of the interpretations 1 to 8 is intended
for sentences 23 6 and 246

25 o A thing IS a complex of 25 b Every sentence in which
sense>data a thing>designation occurs 13

equipollent to a class of sen-
tences in which no thing-desig-
nations but sense-data designa-
tions occur

26a A thing IS a complex of 26b Every sentence in which
atoms a thing-designation occurs is

equipollent to a sentence m
which space-time co-ordmates
and certain descnpuve functors
(of physics) occur

Suppose that a posiovist mamtams thesis 25 a, and a realist thesis
26a Then an endless dispute will arise over the pseudo-question of
what a thing actually 1$ If we transfer to the formal mode of speech
It 13 in this case possible to reconcile the two theses, even if they are
interpreted m the sense of 3. that is, as assertions about the whole
language of science For the various possibiliues of translating a
thing-sentence into an equipollent sentence are obviously not in-
compatible with one another The nmtrwerty betteeen pontittsm and
rrolism u on tile dispute o&out pseudo-theses which owes its ongtn en-
tirely to the use of the matenal mode of tpeech

Here again we want to emphasize the fact that it does not follow
from the given examples that all sentences of the matenal mode of
speech are necessarily incorrect But they are usually incomplete
Even this docs not prevent their correct use , for in every domam
incomplete, abbreviated modes of speech may frequently be em-
ploy cd with profit But the examples show how important it is m
usmg the matenal mode of speech, especially m philosophical dis-
cussions, to be fully aware of its character, so as to be able to avoid
the dangers inherent in it As soon as, in a discussion, obscunties
and doubts of the kind here descnbed arise, it is advisable to
translate at least the pnnapal thesis involved m the controversy
mto the formal mode of speech, and to render it more precise by
statmg whether it is meant as an assertion or as a suggestion,
and to which language it refers If the exponent of a thesis
refuses to make these statements concermng it, the thesis is m-
complete and therefore ineligible for discussion



PU5T V. PHILOSOPHT AND STTCTAX


§ 79. PiiiLOsoPHiau. Sentences in the I^L\terlxl

AND IN THE FORALU- MODE OF SPEECH

We mil now gi\-e a. series of further esamples of sentences in the
material mode of speech, together with their translations into the
fonnal mode. These arc sentences such as commonly ocoir in
philosophicnl discussions, sometimes in those of the tradinonal
sort, sometimes m in\-esngations which are already expressly
onented in accordance with the logic of science. [For the sale of
brevity, the sentences are, to a cemiri extent, formulated in a
simplified way.] These illustrative sentences (as also those of § 7S)
hare not, for the most part, the simple form of those for whidi we
formulated the criterion of the material mode of speeth in an earlier
section. But they have the general feature which is characteristic
of the material mode of speech; they speak about objects of some
kind, but in such a way that it is posisiblc to construct ccrrelsted
sentences of the fonnal mode of speech which make corresponthng
assertions about the designations of these objects. Since the
original sentence, in most cases, cannot be understood uruvocally, a
particular translation into the formal mode of speech cannot uni«
vocally be given; it cannot even be stated with certainty that the
sentence in question is a pseudo-objcct-sentenee and, hence, a
sentence of the material mode of speech. The translation given
here is accordingly no more than a suggestion and is in no way
binding. It is the task of anyone who wishes to nuTntain the
philosophical thesis in question to interpret it by translating it
into an exact sentence. This Utter may sometimes be a genuine
object-sentence (that is to say, not a quasi-syntactical sentence);
and, in that case, no material mode speech occurs. Otherwise it
must be possible to give the interpretation by means of translation
into a sj-ntaetical sentence. The syntactical sentences of the fol-
lowing examples — like those of the preceding ones — must further
be completed by stating the language which is referred to; from
this statement it can then be seen whether the sentence b an
assertion or a proposal, e.g. a new rule. We hare omitted these
statements in the examples wKdi follow, because as a rule it b
impossible to obtain them ururecaDy from the philosophical sen-
tences of the material mode of spee^ [Here, as in the earlier



§ 79 * PHILOSOPHICAL SENTENCES IN THE TWO MODES 303
examples, it obvioiosly makes no difference to our investigations
whether the illustrative sentences are true or not ]

Philosophical sentences | Syntactical sentences

(Material mode of speech) j (Formal mode of speech)

A Generalities (about things, properties facts, and so on) Here
belong also Examples 7, 9, 17-zo

27a A property of a thing- | zjb A *pt is not a ‘pt
property is not itself a thmg- .
property I

z8a A property cannot pos I zSd There is no pr of a level
sess another property (As op- * higher than the first (As op-
posed to 270 ) I posed to 27 6 )

29<2 The world IS the totality 296 Science is a system of
of facts, not of things j senlcnces, not of names

300 A fact IS a combination ' 30& A sentence is a senes of

of objects (cntiUes, things) symbols

31a If I know an object 316 If the genus of a symbol
then I also know all the possi- is given, then all the possibibues
bilities of Its occurrence m facts of its occurrence m sentences are

also given

32 a Identity is not a relauon 32 b The symbol of identity

between objects is not a descriptive symbol

Sentences 290 to 32 a come from Wittgenstein Similarly many
other sentences of his which at first appear obscure become clear
when translated into the fotmal mode of speech,

33 a This circumstance (or 33 f> This sentence is ana-
fact, process, condition) is logi- lytic, contradictory, not
cally necessary, logically im- contradictory

possible (or inconceivable),
logically possible (or conceiv-
able)

34a This circumstance (or 346 This sentence is valid,

fact, process, condition) IS re^y contravabd, not contra-
(or physically, in accordance valid
with natural laws) necessary,
really impossible, really pos-
sible

35 a The circumstance (or 35 b Sj is an L-consequence

fact, process, condition) Cj is a (or a P-consequence, respec-
logically (pr really) necessary lively) of
condition for the circumstance

C,

33a to 35a are sentences of modality, see § 69



P.<RT V. PIirLOSOPHT AND STNT.Oi


304

36c. A property of an objfct 36^. prj is called an analytic
c IS called an essential (or: »«- (or, if desired: an essential or
temal) property of c, if it is in* internal) predicate in relation
conceivable that c should not to an object-designation ?Ii if
ptossess it (or: if c necessarily pri(Si) is analytic. (Correspond*
possesses it); othenvise it is an ingly for a nvo- or tnore-tenned
inessential (or : erferfH27)property. prcdicate.1
(Correspondingly for a relation.)

The uncertainty of the formulation 360 is shown by the fact that
it leads to obscurities and contradictions. Let us take as the object
c, for example, the father of Charles. According to definition 36 a,
being related to Charles is an csscnnal property of c, since it is in*
concenable that the father of Charles should not be related to
Charles. But being a landowner is not an essential property of the
father of Charles. For, even if he is a landowner, it is conceivable
that he might not be one. On the other hand, being a landowner is
an essential property of the owner of this piece of land. For it is
inconceii'able that the owner of this piece of land should not be a
landowner. Now, however, it happens to be the father of Charles
who is the owner of this piece of land. On the basis of definition
36 a, it has just been proitd that it is both an essential and not an
essential property of this mao to be a landowner. Thus 360 leads
to a contradiction; but 366 does not. because ‘landowner’ is an
analytic predicate in relation to the object *designBtion ' the owner of
this piece of land but it is not an analjtie predicate in relation to
the object-designtuon ‘the father of Oiarles’. Hence the fault of
definition 36 a lies in the fact that it is referred to the one t-hjeet in*
Btesd of to the ohject-eiesipnatims. which msy be different even when
the object is the same.

Thu example shows (as will essfly be confirmed by a closer in*
vesagation) that the numerous discussions and controversies about
external end internal properties and relationx are idle, if, as is usual,
they are based on a defimtion of either the form indicated or one re-*
sembling it, or, at any rate, on one which is formulated in the material
mode of speech. [Such insnstigations are espk."ially to be found in
the work of Anglo-Saxon philosophers, and it was through them
that Wittgenstein, although it is to him ^at we owe the detection of
mmy other pseudo-questions, was himself misled into enquiries of
this nature.] If instead of theusualsort of definition, a def^ition in
the formal mode is gii*cn, then the situation in these commonlr dis-
puted cases becomes unambiguous, and moreover so simple that no
onecananylongerbetemptedtorauephilosophicalproblemaabout it.

B. The te>~ealled philosophy of nvmber; logieel analysis ef arithmetic.

Here belong also Examples 10, 17, at, and 22.

37a. God preated tbe natural I 37ft. The ruturtl-number
numbers (integers); frtetioru | aymbola are primitive tjmbols;



§79 PHILOSOPKICAL SESIIENCES IN THE TW O MODES j05

and real ntunbers, on the other the fractional expressions and
hand, are the Work of man the real number expressions are
(Kronecker ) introduced by definition

38a The natural numbers are 38b The natural number ex-

not given, only an mitial term of pressions are not primitive sym-
the process of counting and the bols (as opposed to 37 b), only
operation of progression frorn ‘0 and *•’ are primitive sym-
one term to the next are given, bols, an St has the form nu or
the other tenm are created pro- St* (Languages I and II )
gresstvely by means of this
operation

39a The mathematical con 396 A pt*j, to which certain

txnvum is a senes of a certain structural properties (density
structure, the tenns of the senes continuity, etc ) are attnbuted m
are the real numhers the axioms, is a primitive sym

hoi The arguments which are
siutable to pti — they are expres-
sions of the zero level— are
called real-number expressions

40a The mathematical con- 406 A pt*i, towhich certain

tiQUum la not composed of structural properties (namely,
atomic elements, but IS awhole thoseofapart-wholerelatioiiofa
which is analysable into ever certain kind) are attnbutedmthe
Airther analysable sub-intervals axioms, isapnmiQvesjTsbo] An
A real number is a senes of in- whose arguments are natural-
tervala contained one inside the number expressions and nhose
other value-expressions are suitable as

arguments to pTi is called a real-
numberexpression [Aso-called
creative sequence of selections is
then represented by an 5 ui , see

p 148 ]

39 a and 40 a present (m a simplified formulation) the antithesis
between tht usual mathtmaUeal conception of the continuum of real
TOmbers, based on the theory of aggregates, and the intuitionwt con-
ception of the conttmsim represent^ by Brouwer and Weyl, which
rejects the former as atomistic 39 b and 40 b may be mterpreted as
suggestions for the construction of two different calculi

C Problems of the so-called gtien or pnmtne data {epistemology,
phenomenology), logical tmalysu of the protocol sentences
Here belong also Examples 23 and 24.

410 The only pnniitivc dbtfl 416 Only two- or more-
are relations between eipen- termed predicates whose argu-
ences ments belong to the genus of the

experience expressions occur as
descriptive primitive symbols



PART V. PHILOSOPHY A>fD SYNTAX


306

420. A temporal scries of
visual fields is given as primitive
data; every visual field is a two-
dimensional system of positions
which are occupied by colours.
(As opposed to 410.)

43a. Thesense-qualities.auch
as colours, smells, etc., belong to
the primitive data.


44a. The fact that the system
of colours arranged according to
similarity (the so-called colour-
pyramid) IS three-dimensional, is
known a pnon (or: is to be ap-
prehended by intuition of es-
sence ; or : IS an internal property
of that arrangement).

4Sa. The colours are not
onginally given as members of
an order, but as individuals; an
empirical relauon of similarity
exists between them, however,
on the basis of which the colours
can be arranged empirically m a
three-dimensional order.


42 fc. A descriptive atomic
sentence consists of a time co-
ordinate, two space co-ordinates
and a colour expression.


43 ft. Symbols of sense-quali-
ties, such as colour-symbols,
smell-symbols, etc., belong to
the descriptive primitive sjm-
bob.

446. A colour-expression
consists of three co-ordinates ; the
values of each co-ordinate form a
aensl order according to syn-
tactical rules; on the basis of
these syntactical rules, therefore,
the colour-expressions consti-
tute a three-dimensional order.

456. The colour expressions
are not compound; they arc
pnmiuve symbols; further, a
symmetrical, reflexive, but not
transitive, pi» to which the
colour-expressions are suitable
as arguments, occurs as a primi-
tive symbol; the theorem of the
three-dimensionality of theordcr
determined by this pr is P-
valid.


The much-disputed philosophical question as to whether the
knowledge of the thrfe-dmenfionaiity oj the eolour-pyraynid is a priori
or mpincal is thus, by reason of the use of the material mode of
speech, incomplete. Tlie answer is dependent upon the form of the
language.


460. Every colour possesses
three components: colour-tone,
saturation, and intensity (or;
colour-tone, white-content, and
black-content).


470. Every colour isataplace.


480. Every tone has a certain
pitch.


466. Every colour-expression
consists of three partial expres-
sions (or : is synonymous with an
expression composed in this
way); one colour-tone expres-
sion, one saturation-expression,
and one intensity-expression.

47fc. A colour-expression is
always accompanied in a sen-
tence by a place-designation.

486. Every tone-expression
contains an expression of pitch.



§79 philosophical SENTENCES IN THE TV.O MODES 307

D The so-ealled natural pfaltaop!^, logical onaJytu of the natural
sciences

Here belong also Examples xi, aSp 26.

49 a Ttffie 13 continuous I 496 The real-number ex-
I pressions are used as tune-
I co-ordinates

See Wittgenstem on this pomt (I Trartfltttf] p 172) “AUproposv-
pons such as the law of cauiapon, the law of contxnuity la nature,
are a pnort mtuipons of the possible forms of the propositions of
saence ” (Instead of"apnontntuiQoasof’'wewouldprefertos3y
“conventions coneemmg’* )

50a Ttmeisone-dimensional, 50^ A time-designapon con-
space IS three-dimensional sists of one co-ordinate , a space-

designation consists of three co-
ordinates

51a Time IS in&ute m both $ib Every posipve or nega-
duections, forwards and back- ove real-number expression
wards can be used as a tune-co-

ordinate

The opposition between the determntsm of classical physics and
the probability determmapoo of quantum physics concerns a syn-
tactical difference in the system of natural laws, thst is, of the P-niles
of the physical language (already formulated or stdl to seek) , this is
shown by the two following examples

520 Every process is um- 526 For every particular
vocally detemuned by its causes physical sentence there is, for

any time-co-ordinate which
has a smaller value than the
time-co-ordinatewhich occurs m
( 5 ,, a class Ri of particular sen-
tences with as time-co-
ordinate, such that is a P-
consequence of

530 The position and velo- 53fr If ®i is a particular sen-
city of a particle is not umvocally tence concerning particles and
but only probably deteimmed by a time-co-ordinate of smaller

a previous constellanon of par- value than that which occurs m
tides Si, then Si is not a P-conse-

quence of a class of such sen-
tences with 9 Ii as time co-
ordinate, "however comprehen-
sive, but only a probability-
consequence of such a class with
a coefficient of probability smal-
ler rl^an I



P.\RT V. PHILOSOPHY AKD SYNTAX


30S


§ 80. Tiie Dangers of tiie Material Mode
OF Speech

If we wish to chancteme the material mode of tfeech by one
general term, we may say, for instarjcc, that it is a tpecial kind of
transposedmode of tfreeh. By a transposed mode of speech we mean
one in w hich, in order to assert something about an object a, some-
thing corresponding is asserted about an object h which stands in a
certain relation to the object a (this does not pretend to be an exact
definition). For example, every metaphor is a transposed mode of
speech ; hut other kinds also occur frequently in ordinarj’ language
— far more frequently than one may at first believe. The use of a
transposed mode of speech can easily lead to obscurities ; but when
sj-stcmatically carried into effect, it is non-contradictor>’.

Examplet of different kinds of transposed mode of speech.
I. An artifiaal example. The term ‘marge* (as a term parallel to
‘ large ') IS introduced by means of the foilowing rule : if a place has
more than lo.ooo inhabitants, then we shall ssy thst the plsce b,
whose name precedes that of o in the alphsbeucal bst of places, is
marge. A rule of this kind can be carried into effect without any
contradiction ; for instance, according to it, the place Berhchingen is
marge, since, in the alphabeucsl list of places, its name is followed
by ' Berlin ' The definition seems absurd, since it makes no dif-
ference to the properties (m the ordinary sense) of a place whether
It IS marge or not. But the same thing holds for the ordinary
matcnal mode of speech also (see below. Example 5), even (as one


tions of a certain kind about him. 3. According to the ordinary use
of language, an action o of a certain person is called legal crime if the
penal law of the country in which that person U\ts places the de-
scription of a kind of action to which a belongs in the list of crimes.
4. According to the ordinary use of language, an action a of a certain
person is called a moral crime if, in the minds of the majority of other


ifutance, of ljab>lon; tee the example in §74) that it has been
treated of in a certain lecture (material mode of speech) if a designa-
tion of the city has occurred in this lecture. For the qualities (in the
ordinary tense) of the city in question, it is notof the least importance
w hether it has the property of haring been treated of in yesterdsy’a
lecture or not. This property is therefore a transposed property.



§ 8o. t)A^GZRS OF THE ^UTERIAL MODE 309

The material mode of speech is a transposed mode of speech
In usmg it, m order to saj somethmg about a word (or a sentence)
we say instead somethmg parallel about the object designated by
the word (or the fact described by the sentence, respectively) The
ongm of a transposed mode of speech can sometimes be explained
psychologically by the fact that the conception of the substituted
object h IS for some reason more vivid and stntmg, stronger m
feehng-tone, than the conception of the original object a This is
the case with the rnatenal mode of speech The image of a word
(for instance, of the word ‘house’) is often much less vivid and
hvely than that of the object which the word designates (m the
example, that of the bouse) Further, the fact, which is perhaps a
consequence of the psychological fact just mentioned, that the
approach and method of syntax have hitherto not been sufBciently
known, and that, m consequence, the majority of the necessary
syntactical terms have not been a part of ordinary language, may
have contributed to the ongm of the mattnal mode of speech For
this reason, instead of saying “The sentence 'a has three books,
h has two books, and a and b together have seven books’ is contra*
dictory”, we say " It is impossible (or mconcetvable) for a to have
three books, b two books, and a and b together seven books”, or
(which has an even stronger resemblance to an object sentence)
''If a has three books, and b two, then a and b together cannot
possibly have seven books ” People arc not accustomed to direct
their attention to the sentence instead of the fact, and it is ap-
parently much more difficult to do so In addition, there is the
circumstance that, m ordinary language, we have no syntactical
expression which is equivalent m meanmg to ‘contradictory’,
while the quasi-syntactical expression ‘impossible’ is ready to
hand

How difficult It IS even for scaentisis to adopt the syntactical pomt


For instance when we of the Vienna Circle cnticize, in accordance
with our ann metaphysical view, certain sentences of metaphysics
(such as “There is a God”) or of metaphysical epistemology (such
as “ The external world is real )wearemterpreted by themajonry
of our opponents as denying diose object sentences and conse-
quently affirming others (such as “‘ITiereis no God” or “The ex-



310 PARTY. PHILOSOPHY AND STTsTAX

temal world is not real etc.). These misundeTatandings are always
occurring in spite of the fact that we have already explained them
many timea (see, for instance, Carnap SchLck

{Ponfirirwj), Carnap and are constantly pointii^

out that we are not talkmg about the (supposititious) facts, but about
the (supposmoous) sentences; m the mode of expression of this
book: the thesis maintained by us ss not an objen-sentence but a
sjntactical sentence.

The sug^tions we have given arc intended only to throw light
upon, and not by any means to answer, the quesDon of the psydio-
logical cxplanahon of transposed modes of speech in general, and
of the material mode in particular. To mvesbgatc it more closely
would be well worth while; but we must leas’e that task to the
psychologists ^ATiat we must here take into account is the fact that
the tnaterul mode of speech is a part of ordinary’ linguistic usage,
and that it will continue to be frequently employed, even by our-
sehes. Therefore it behoves us to pay speoal attention to the
dangers connected with its use.

Most of the ordiQif)' formulations in the matenal mode of
speech depend upon the use of unneml words, (.'wrertaf trerd*
Wfj' easU^ Ucd to ptruio^pnblerux they appear to designate kinds
of objects, and thus make it natural to ask questions concerning the
nature of objects of these kinds. For instance, phQosophers from
antiquity to the present day have associated with the universal
word 'taznber' certain pseudo-problems which have led to the
most abstruse inquiries and controversies. It has been asked, for
example, whether numbers are teal or ideal objects, whether (bey
are extra-mental or only exist in the mind, whether they are the
creation of thought or independent of it, whether they are potential
or actual, whether real or Setitious. The question of the origin of
numbers has been raised, and has been found to be due to a division
of the self, to an original primitive intuition of duality in unity, and
so forth. Similarly, innuincrabJe questions hate been put con-
cerning the natvre of tpcce arj time, not only by speculative meta-
physicians (up to recent times), but also by many philosophers
whose epistemological theses are ostensibly (as with Kant)
oriented in accordance with empirical science. As opposed to all
this, an inquiry which is free from metaphysics and concerned with
the logic of science can only hate as its object the syntax of the
spatio-temporal expressions of the language of science, in the



§8o DANCERS OF TIIE MATERIAL MODE 3II

form, say, of an axiomatics of the space-time system of physics (as,
for instance, the researches of Keidienbach [Axtcmatik]) Further,
mention should be made of the many pseudo-problems concerning
the nature of the phyttcal and the psychical Again, the pseudo-
questions concerning pTt^nttes and relations and with them the
tchole controversy about umversals rests on the misleading use of
universal words All pseudo-questions of this kmd disappear if the
formal instead of the matenal mode of speech is used, that is, if in
the formulation of questions, instead of universal words (such as
‘number’, ‘space’, 'universal*), we employ the corresponding
syntactical words (‘numerical expression’, ‘space-co-ordmate’,
‘predicate’, etc)

We have already met with a number of examples m which the
use of the matenal mode of speech leads to contradictions The
danger of the occurrence of such contradictions is cspeaally great
m the case of languages which are mutually translatable, or, from
the standpoint of one language of science, of two sub-languages
between Ae sentences of which certain relations of equipollencc
(not necessarily of L-equipollence) hold This applies, for in-
stance, to the language of p^chology and the language of physics
If the matenal mode of speech is employed m relation to the psy-
chological language (by the use, for instance, of universal words
like ‘Ae psychical’, ‘psyche’, ‘psychical process’, 'mental pro-
cess’, ‘act*, ‘expencncc’, ‘content of expenence’, 'intentional
object’, and so on), and if, in the same investigation, it is also used
m relation to the physical language (either the everyday language
or the saentific language), hopeless confusion frequently ensues

The danger here indicated has been described by us in detail on
other occasions ([FAyr SpracAe] pp , [[/mty}) Compare also
[PiycAoH p 186, where attenuon isdrawnto the obscunties which
arise from the use of the matenal mode of speech in the sentences
of a psychologist, further, see [Psyehof] p 181 for the ongm of a
pseudo-problem due to the matenal mode of speech The examples
onp 314 under I also belong in part here On the psycho-physical
problem, see p 324

From the earlier examples, whidi could easily be multiplied, it
13 dear that the use of the matenal mode of speech often gives nse
to an obscunty, an ambiguity, which is manifested, for mstance,
m the fact that essentially different translations into the formal
mode of speech are possible In more extreme cases, contradic-



PART V. PHILOSOPHY AND STNT.«


tions also appear. These contra^ctions are, however, frequently
not at all obnous, for the reason that the consequences arc not de-
rived by means of formal rules, but by means of material con-
siderations, m which it is often possible to ai'oid the traps that
one has set oneself by this dubious formulation. Et'en where no
contradictions or ambiguities occur, the use of the material mode
of speech has the disadi-antage of leading easily to self-deception as
regards the object under discussion: one belic\'cs that one is in-
vestigating certain objects and facts, whereas one is, in realitj’,
investigating ihcir designations, i e. words and sentences.


§8i. The Admissibility OF THE Material
Mode of Speech


We have spoken of dangers and not of errors of the material
mode of speech. The natmat mode of tpenk is not in itself erroneous',
it only readily lends itself to wrong use. But if suitable definitions
and rules for the matenal mode of speech are laid down and
sj’stematically applied, no obscunues or contradictions arise.
Suiee, however, the word-language is too irregular and too com-
plicated to be actually comprehended in a s)'stem of rules, one
must guard against the dangers of the matenal mode of speech as
U is ordinarily used in the word-language by keeping in nund the
peculiar character of its sentences. Especially when important
conclusions or philosophical problems are to be based on sentences
of the matenal mode of speech, it is wise to make sure of their
freedom from ambiguity by translating them into the forma] mode.

It is not by any means su^ested that the material mode of speech
should be entirely eliminated. For since it is established in general
use, and is thus more readily understood, and is, moreover, often
shorter and more obvious than the formal mode, its use is fre-
quently expedient. Eien in this bool, and especially in this Part,
the materia] mode of speech has often been employed; here are
some examples:


Material mode of speech
54 a. Philosophical questions
are sometimes concerned with
objects w hich do not occur in the
objea-domain of the empirical


Formal mode of speech
S4b. In philosophical ques-
tions expressions sometimes oc-
cur which do not occur in the
languages of the sdcncts; for



§8l ADMISSIBIUTT OF THE MATERIAL MODE 313

sciences For example the example, the expressions ‘ thing-
thing-m-itself, the transcen- m-itself’, ‘the transcendental’,
dental, and the like (p 378) etc

SS a An object-question is 5S 6 In an object question,
concerned, for instance, with the predicates of the language of
properties of animals, on the zoology (designations of kmds of
other hand, a logical question 13 animals) occur, on the other
concerned with the sentences of hand, m a logical question, de-
zoology (p 278) signations of sentences of the

zoological language occur

560 It 13 just as easy to con 566 It is just as easy to con-
struct sentences about the forms struct sentences m v^hich, as
of linguistic expressions as it is to predicates, syntactical predicates
construct sentences about the occur, and, as arguments syn-
geometncal forms of geometrical tactical designations of expres-
structures (pp 282 f ) sions, as it is to construct sen-

tences in which as predicates,
predicates of the language of
(pure) geometry occur, and, as
arguments, object designations
of the language of geometry

If a sentence of the material mode of speech is given, or, more
generally, a sentence which is not a genume object -sentence, then
the translation into the formal mode of speech need not always be
undertaken, but it must always be possible Translatalnhty into the
formal mode of speech constitutes the touchstone far all philosophical
sentences, or, more generally, for all sentences which do not belong
to the language of any one of the empirical sciences In m-
vestigatmg ttanslatability, the orduiaiy use of language and the
definitions which may have been given by the author must betaken
into cons deration In order to find a translation, we attempt to
use, wherever a universal word occurs (such as * number’ or ‘ pro-
perty’) the correspondmgsyntacucal expression (such as ‘ numeri-
cal expression’ or ‘property-word’, respectively) Sentences
which do not, at least to a certam extent, imivocally determme their
translation are thereby shown to be ambiguous and obscure
Sentences which do not give even a slight mdication to determine
their translation are outside the realm of the language of science
and therefore mcapable of discussion, no matter what depths or
heights of feelmg they may sur Let us give a few warning ex-
amples of such sentences as they occur m the writings of our own
arcle or m those of closely allied authors The majority of readers



PART V. PIIILOSOPIIT AND SlTsTAX


3H

will scarcely, I think, succeed in finding a translation of these into
the formal mode of speech that would satisfactorily represent the
author’s meaning. Even if the author himself is perhaps able to
give such a translation — and in some cases e^'cn this seems doubtful
— his readers will certainly fall into confusion and uncertaintj’.
We shall see that the sentences in which the tvord 'inexpressible’
or something similar occurs arc especially dangerous. In the
examples under heading 1 we find a mylhohsy of the inrspresn'bJe,
in the examples under II a mythoh^' of ht^her things, and in
Sentence 13 both of these.

I. I. There is indeed the inexpressible. 2. The qualities which
appear as content of the stream of consciousness can neitlicr be as*
serted, descnbed, expressed, nor communicated, but can only be
manifested in experience. 3. What can be showw cannot be said.
4. The given experience possesses an utterable structure, but at the
same time :t possesses an ununerable content which is nevertheless
ver>’ well known to us. 5. Human beings must venfy psychological
sentences by their own unutterable expenenee, which is nevertheless
very w ell known to them ; they must examine whether the sentence
in question, the combmaaon of s)'mbols, is isomorphous in
structure) mth their unutterable experience. 6. TTie ununerable
expenenee blue or bmer. . 7 The essence of individualit)' cannot
be represented in w ords, and ismdescnbable, and thereforemesning*
less for saence. 8. Philosophy will mean the unspeakable by clearly
displaying the spcakable. 9. TTie holding (subsistence] of [formal or]
internal properties and relations cannot be asserted by propositions
[sentences]

II. 10. The sense of the world must lie outside the world.

11. How the world is, is completely indifferent to what is higher.

12. If good or bad willing changes the world it can only change the
limits of the world, not the facts. 13. Propositions [sentences] can*
not express anything higher.

Let us suggest a few possibilities of translation which, howe«r,
probably do not correspond to the intentions of the authors. In the
case of Sentence 1 it w ould be necessary to distinguish between two


nations.” j B. “'I'hcrc are unutterable facts”, that ia to say, “There
are facts which are not described by any sentence”; transfalion;
. There arc sentences which arc not sentences.” Concerning 6:
in other words, “The experience designated by the word 'blue*
cannot dwignated by any word”; tramlation: “The experience*
designation blue’ is not an experience*de$ignation.” Sentence 9
means : “ The fact that a property of a certain kind appertains to an



§8l ADMISSIBIUTY OF THE \UTER 1 AL MODE 315

object cannot be asserted by means of a sentence tremslatton
“A sentence m which a property word of a certain kind occurs is
not a sentence ” Sentence 13 means ‘ The higher facts cannot be
expressed by means of sentences’ , tramlatton “The higher sen-
tences are not sentences ”

let It be once more called to mmd that the distmction between
the formal and the material modes of speech does not refer to
genuine object sentences and therefore not to the sentences of the
empmcal sacnces, or to sentences of this kind which occur in the
discussions of the logic of saence (or of philosophy) (See the
three columns, on p 2S6) It is here a question of the sentences of
the proper logic of saence According to the ordinary use of
language it is customary to formulate these partly in the form of
logical sentences and partly in the form of object sentences Our
mvcstigations ha shown that the supposititious object sentences
of the logic of saence arc pseudo-object sentences, or sentences
which apparently speak about objects, hie the real object-sen-
tences, but which in reahty are speaking about the designations of
these objects. This unphes that all the sentences of the logic of
science are logical sentences, that is to say, sentences about lan-
guage and linguistic eTpresstons And our investigauons have
further shown that all these sentences can be formulated in such a
way as to refer not to sense and mcanmg but to the syntacucal form
of the sentences and other expressions — they can all be translated
into the formal mode of speech, or, m other words, into sjTitactical
sentences The logic 0/ jamre is the tynlax of the language of saence


B THE LOGIC OF SCIENCE AS SYNTAX

§ 82. The Physical Language

The logical analysis of physics — as a part of the logic of saence
— ^is the syntax of the phpical language All the so-called epi-
stemological problems concerning physics (m so far as it is not a
question of metaphysical pseudo-problems) arc m part empmcal
questions, the majority of whtdi belong to psychology, and m part
logical questions whui belot^ to syntax. A more exact exposition
of the logical analysis of physid as the sj-ntax of the physical Ian-



316 r.UlTV. PIULOSOPIIY AND SYNTAX

guagc must be left for a special investigation. Here vre shall only

offer a few suggestions totvards it.

The logical analj’sis of physics trill have, in the first place, to
formulate rules of formation for sentences and other kinds of ex*
presswns of the phj'sical language (see § 40). The most important
expressions which occur as arguments are the pomt-eepressions
(designations of a spatio-temporal point, consisting of four rtal-
number expressions, namely, tlirec space-co-ordinates and one
time-co-ordinate) and the domain-expressions (designations of a
limited space-time domain). The physical coefficients of states trt
represented by desenptit-e functors. The descriptisx functors and
predicates can be divided into those having point-expressions and
those hatang domain-expressions as arguments.

The sentences can be classified according to their degree of
generalit)’. We will here only discuss the two extreme kinds of sen-
tences and, for the sake of simpUat)*, only those in which all the
interior arguments arc point- or domain-expressions: the concrete
sentences contain no unrestneted variables; the tares contain no
constants os interior arguments

Either L-rules alone, or L-tules and P-niles, can be laid down as
transformation rules of the physical language. If P-rules are desired,
they mil generally be suted in the form of P-pnmiiive sentences.
In the first place, certain most general laws will be formulated as
P-pnmitivc sentences; we will call these primitive lares. In addi-
tion, descnptivc synthetic sentences of another form— ct’en con-
crete ones — may be stated as P-primilhe sentences. In the ma-
jority of cases, the primitive laws will have the form of a universal
sentence of implication or of equh-alence. The primitive laws and
the other %-alid laws can be either deterministic or lares of
obiUty ; the latter can be fonnulatcd, for instance, trilh the help of
a probability implication. Since the concept of prt^ahility is a t cry
significant one for physics, particularly in view of the latest de-
t elopments, the logical arulysis of physics trill hat e thoroughly to
investigate the syntax of the sentences of probability; and it may
be found possible to estabUsK a con.tiecd'an wttK the concept of
range in the general syntax.

We cannot Ro more fully into the concept of probability here. See
the lectures and discussions ol the Prague Congress (ErfeemilJiii
*. t93o)» further bibliographical references arc given in Erkennlms



3*7


§ 82 THE PHraCM. lASGUACE

II, iSg f,, 1931 , there are also investtgatioiis as yet unpublished, by
Reichenbach, Hempel, and Popper* On the probability implica-
tion, see Reichenbach [H«AncAemfttftA«fr/ogiS]

Syntactical rules will have to be stated concerning the forms
wh-ch the prctocol-mttenetSy by means of which the results of oIn
servation are expressed, may take [On the other hand, it is no t the
task of syntax to determine which semenccs of the established
protocol form are to be actually laid down as protocol-sentences,
for * true ’ and * false ’ are no t syntactical terms , the statement of the
protocol sentences is the affnr of the phy'Siast who is observmg
and innirTTiff protocols ]

A sentence of physics, whether it is a P-pnmitive sentence, some
other vahd sentence, or an mdetcimioate assumption (that is, a
premisswhose consequences are in course of investigation), will be
frr'rd by deducing consequences on the basis of the transfomialioa
rules of the language, until finallv sentences of the form of protocol-
sentences are reached. These will then be compared with the
protocol-sentences which have actually been stated and either con-
firmed or refuted by them. If a sentence which is an L-conse-
quenco of cettim P-pnsutive sentences contradicts a sentence
which has been stated as a protocol-sentence, then some change
must be made in the system. For instance, the P-rules can be
altered tn such a way that those particular primitive sentences are
no longer valid, or the protocol-sentence can be taken as bemg
non-valid, or agam the Lr-niles which have been used m the de-
duccion can also be changed. There are no established rules for the
kmd of change which must be made

Further, it a cot possible to lay down any set rules as to how
new pnmrtiTe laws are to be e^tablisbed on the basis of actually
stated protocol-sentences Onesometcnesspeaksmlhisconnection
of the method of so-called x«<fae<ioB J»ow this designation maybe
retained so long as it is clearly seen that it la not a matter of a regular
method but only one of a pnctical procedure which can be
mvesugated solely m relation to expedience and fruitfulness. That
there can be no rules of mduction is shown bv the fact that the
L-content of a law, by reason of its unrestricted unnersahty,
always goes beyond the L-content of every finite class of protocol-

• {Sole, 193s) These works have meantime appeared, see
Bibliography



3 1 S P.^RT V. PHTLOSOPHT AND STNT «

sentences. Oa the other hssd. exset nJes for deduction can be
Uid down, naraely, the L-mles of the physical lanpiape. Thus the
laws have the character of hyfetkeset m rebtion to the protc»cel-
sentences; sentences of the form of protocol-sentences may be
L-consequences of the laws, but s bw cannot be an b-consequence
of any finite sjTithetic class of protocol-sentences. The laws are cot
inferred from protocol-sentences, but are selected and laid down
on the grounds of the cutting protocol-sentences, which are alwats
being re-examined with thehelpof theextr-cmergincnewprotocol-
sentences. Not only lares, however, but also concrete sentences
are formubted as hypotheses, that « to say, as P-pnmitire sen-
tences — such as a sentence about an unob^ened process by which
certain observed processes can be explained. There is in the strict
sense no refutation (falsificanon) of an hypothesis; for even when
It proves to be L-ineompatible with certain protocol-sentences,
there always ensts the possihiliw of maintAining the hypothesis
and renouncing scknowledgment of the protocol-sentences. Still
less is there tn the stnci sense s complete eonfirmaaon (verifies-
eon) of tn hj'poiheais. When an increasing number of L-conse-
qufiaces of the hjpoihesis agree with the already sdmowledged
protocol-sentences, then the hypothesis is increasingly co.nfirmed ;
there is secordingly only a gradually increasing, but never a final,
confirmation. Further, it is. in general, impossible to test even a
single hypothetical sentence. In the case of a single sentence of
this kind, there are in general no suiuble L-coawjuences of the
form of protocol-sentences; hence for the deducrion of sentences
haling the form of protocol-sentences the remaining hypotheses
must also be used. Thus the test appZTn, cf hottm, not to a sir^le
Alport rrii t-jt to the tehole lyttert cj ph\ na cr a mien of hypotheses
(Duhem, Poinciri).

No rule of the physical language i» definitirt; all rules are bid
down with the reservation that they may be altered as soon as it
seems expedient to do so. This applies not only to the P-rules but
also to the L-rules, bduding those of mathematics. In this re-
spect, there arc only differences in de gr ee; certain rules are more
difficult to renounce than others. [If, however, we assume that
eiTry new protocol-sentence wluch appears within * bnguage U
synthetic, there is this difference between an L-valid, and there-
fore analytic, sentence S, and a P-vaLd sentence Sp namely, that



§82 THE PHYSICAL LANGUAGE 319

such a new protocol-sentence — independently of whether it is
acknowledged as vahd or not— can be, at most, mcompatible with
S, but never with Si In spite of this, it may come about that,
under the mducement of new protocol-sentences, we alter the
language to such an extent that Sj is no longer analytic ]

If a new V-pnmittve sentence Si is stated, but without sufficient
transformation rules by which, from Qi m conjunction with the
other P-pnmitive sentences, sentences of the form of protocol-
sentences could be deduced, then m prinaple Sj cannot be tested,
and 13 therefore useless from the saentiiic point of view If, how-
ever, sentences of the form of protocol-sentences are deducible
from Si m conjunction with the remainder of the P-pnmitive
sentences, but only such as are deducible from the re mainin g
P-prunitive sentences alone, then as a primitive sentence is un-
productive, and saentifically superfluous

A nea descriptive symbol tchsch is |o be introdsutd need not be
reduable by means of a chain of definitions to symbob which
occur in protocol-sentences A symbol of this kind may also be
introduced zsipnmUve rynr&o/ by means of new P-pnmitive sen*
fences If these pnmitive sentences are testable, i e if sentences
of the form of protocol-sentences are deducible fnm them, then
thereby the primitive symbob are reduced to symbob of the
protocol-sentences

Example Let protocol-sentences be the observation sentences of
the usual form. The electric field vector of classical physics is not
definable by means of the symbob which occur m sudi protocol
sentences, it is introduced as a primitive symbol by the Maxwell
equations which are formulated as P primitive sentences There is
no sentence equipollent to such an equation, which contains only
symbob of the protocol-sentences, although, of course, sentences of
protocol form can be deduced from the Maxwell equations m con-
junction with the other primitive sentences of classical physics, m
this way, the Maxwell theory is empincally tested Counter-example
The concept of “entelechy”, employed by the neo-vitalists, must be
rejected as a pseudo-concept It is, however, not a suffiaent justifi-
cation for this rejection to pomtoutthatno definition of that concept
IS given bymeans ofwhitdiit could be reduced to the termsof theobser-
vationsentences , forthesame thing is also trueof a numberof abstract
physical concepts The decisive pomt is rather the fact that no bwa
which can be empirically tested are laid down for that concept

The explanation of a smglc known physical process, the deduc-
tion of an unknown process m the past or m the present, from one



320 PART V. PHILOSOPHY AND SYNTAX

that is known, and the/>«A'ctii;mof a future event, are all operations
of the same logical character. In all three cases it is, namely, a
matter of deducing the concrete sentence which describes the
process from valid laws and other concrete sentences. To explain
a law (m the material mode of speech: a universal fact) means to
deduce it from more general laws.

The construction of the physical ^'stem is not effected in ac-
cordance Kith fixed rules, but by means of eoncen/ionr. These con-
ventions, namely, the rules of formation, the L-rules, and the
P-rules (hypotheses), are, however, not arbitrary. The choice of
them is influenced, in the first place, by certain practical methodo-
logical considerations (for instance, whether they make for sim-
plicity, expedience, and fruitfulness in certain tasks). This is the
case for all conventions, including, for example, definitions. But
in addition the hypotheses can and must be tested by experience,
that is to say, by the protocol-sentences— both those that are
already stated and the new ones that are constantly being added.
Every hypothesis must be compatible with the toul sj’stem of
hypotheses to which the already recognized protocol-sentences
also belong. That hypotheses, in spite of their subordination to
empincal control by means of the protocol-sentences, nevertheless
contain a conventional element is due to the fact that the system
of hypotheses is never univocally determined by empirical material,
however rich it may be.

Let us make brief mention of two theses held by us, upon which,
however, the above view regarding the physical language docs not
depend. The thesis oi physicalitm maintains that the physical lan-
guage is a universal language of science — that is to say, that every
language of any sub-domain of science can be equipollently trans-
lated into the physical language. Prom this it follows that science
is a unitary sptem within which there arc no fundamentally
diverse object-domains, and consequently no gulf, for example,
between natural and psychological sciences. This is the thesis of
the unity of science. We will not examine these theses in greater
detail here. It is easy to see that both are theses of the syntax of
the language of science.

On the view of the physical language here discussed and on the
theses of physicalism and of the unity of science, see Neurath
[Ph^ticaUtm], IPAyrifei/imusl, [S-oxiM. Phyt.'l, [ProtokolU&txe},



§82 THE PHYSICAL LANGUAGE 32 I

[Pj>cAo/], Carnap [PAyr SpraAi\,[Psychol'\,[PTotokolUatze\ In
the discussions of the Vienna Circle, Neurath has been conspicuous
for his early — often inittatoiy — and espeually radical adoption of
new theses For this reason, ^though many of his formulations are
not unobjectionable, he has had a very stimulatirig and fruitful m
fiuence upon its investigations, for instance, m his demand for a
unified language which should not only include the domains of
saence but also the protocol-sentences and the sentences about
sentences, m his emphasis on the fact that all rules of the physical
language depend upon conventional decisions, and that none of its
sentences — not even the protocol sentences — can ever be definitive ,
and, finally, in his rejection of so-called pre~lingtustic elucidations
and of the metaphysics of Wittgenstein It was Neurath who sug-
gested the designations “Physicahsm** and “ Unity of saence —
One of the most important problems of the logical analysis of physics
IS that of the form of the protocol-sentences and of the operation of
testing (problem of vcnficanon), on this point, see also Popper
On the view here expounded the domain of the scientific sentences
u not so restnaed as on the one formerly held by the Vienna Circle
It was originally maintained that every sentence, tn order to be sig-
Oificant, must be eompUuly vtnfiahle (NVittgenstem, Waismann
IWahnchemluhkett] p 229, and Schlick [fCotuo/KoO P 150). every
sentence therefore muse be a molecular sentence formed of concrete
sentences (the so-called elementary sentences) (Wingenstein [7Var-
tatujpp te2,ii8,Camap[/4t((Aau)) On this view therewas no place
for the latss c/ nature amongst the sentences of the language ^ther
these laws had to be deprived of their unrestricted umversahty and
be mterpreted merely as report-sentences, or they were left their
unresbicted umversahty, and regarded not as proper sentences of
the object language, but merely as directions for the construction of
sentences (Ramsey [Foundational PP *37 ^ • Schlick [Kausalitatl
PP 150^ , with references toWittgenstein},8ndhenceBsakmd of syn-
tacbcal rules In accordance with the prmaple of tolerance, we will
not say that a construction of the physical language cotrespondmg
to this earher view is madmtssible , it » equally possible, however, to
construct the language in such a way that theunrestnaedly universal
l*ws are admitted as proper sentences The important difference
between laws and conaete sentences is not obhterated m this
second form of language, but remains in force It is taken into
account m the fact that definitions are framed for both kinds of sen-
tences, and their various syntactical properties are mvesngated The
choice between the two forms of language is to be made on the
grounds of expedience The second form, m which the laws are
treated as equally privileged proper sentences of the object-language,
IS, as It appears, much simpler and better adapted to the ordmary use
of language m the actual sciences *ban the first form A detailed
ennasm of the view according to which laws are not sentences is
given by Popper



PART V. PHILOSOPIIT AND SYNTAX


322

The view here presented allows great freedom in the introduc-
tion of new primitive concepts and new primitive sentences in the
language of physics or of science in general ; yet at the same time it
retains the possibility cj diffamtiating pseudo-coneepu and pseudo-
sentences from real scientific concepts and sentences, and thus of
eliminating the former. [This elimination, howc\*er, b not so
simple as it appeared to be on the basb of the earlier position of
the Vienna Circle, which was in essentials that of Wittgenstein.
On that view it was a question of "the language” in an absolute
sense; it was thought possible to reject both concepts and sentences
if they did not fit into the language.] A newly stated P-primitive
sentence is shown to be a pseudo-sentence if either no sufficient
rules of formation arc given by means of which it can be seen to be
a sentence or no sufficient rules of transformation by means of
which it can, as pretnously indicated, be submitted to an empirical
test. The rules need not be explicitly given; they may also be
tacitly laid down, provided only that they are exhibited in the use
of language. A newly stated descriptive term b shown to be a
pseudo-concept if it b neither reduced to previous terms by means
of a definition, nor introduced by means of P-primitive sentences
that can be tested (see the example and counter-example on
p.3*9)-

Like the individual sentences of the logic of sdence previously
discussed, this presentation of a conception of the logic of science
15 intended only as an example. Its truth b not here in question.
The example is only for the purpose of making it clear that the
logical analj-sis of physics is the syntax of the physical language,
and of further stimulating the formulation, within the domain of
syntax, of views, questions, and investigations concerning the logic
of saence (in the ordinary mode of expression : epblcmology) and
thus making the subject more preebe and more fruitful.


§ 83 . The so-called Foundations of
THE Sciences

Much has been said in recent times about the problems of the
so-called philosophical or logical foundations of the individual
sciences, by which are understood (in our method of designauon)
certain problems of the logic of sdence in relation to the domains



§83 THE SO-CALLED FOUNDATIONS OF THE SCIENCES 323

of the saences Taking the most important examples, we shall
show briefly that these problems are questions of the syntax of the
language of saence

The chief proiZemr ofihe foundations of physics have already been
spoken of m the previous section, and, earher, in Examples 49 to
53 (on p 307) We have seen that the problem of the structure of
tune and space is concerned with the syntax of the space and time
co-ordinates The problem of causahty is concerned with the syn
tactical form of laws, and in particular the controversy regarding
determinism with a certain proper^ of completeness of the system
of physical laws The problem of empirical foundation (problem
of venfication) is an mquiry into the form of the protocol-sentences
and the consequence relations between the physical sentences —
especially the laws — and the protocol sentences The question of
the logical foundations of physical me^urement is the question of
the syntactical form of quantitatite physical sentences (contauung
functors) and of the relations of denvauon between these sen-
tences and the non quantitative sentences (conuining predicates ,
for instance, sentences about pomter-oomadences) Further, such
questions as those concerning the relation between macro- and
micro magnitudes or between macro- and micro laws are to be
formulated as syntactical questions, the eluadation of the concept
of gemdentity also belongs to syntax.

The problems of the foundatttms of biology refer mainly to the con-
nection between biology and the physics of the inorganic, or, more
exactly, to the possibility of tnmslatmg the biological language
mto that sub language Sj of the physical language which contains
the necessary terms for the purpose of descnbmg the morgamc
processes and the necessary laws for the explanation of these pro-
cesses , m other words to the relations between and S, on the
basis ofthetotal language Sywhichcontainsbotbas sub languages
There are, most importantly, two questions which must be dis
tinguished (i) Can the concepts of biology be reduced to those of
the physics of the morgamc? In syntactical form Is every de-
scriptive primitive symbol of Sj synonymous m S, with a symbol
which is definable m Sj ? If thia 13 the case, then there is m relation
to S} an equipollent translation of the L-sub language of mto
that of S] (2) Can the loses of biology be reduced to those of the
physics of the morgamc? In ^ntactical form is every primitive



324 V. PHILOSOPHY AND SYNTAX

law of Sj equipollent in Sj to a law which is N-alid in Sj? If so, then
there is, in relation to Sj, an equipollent translation of Sj (as a
P-language) into Sj. This second question constitutes the scientific
core of the problem of n’/aiion, which howesor, often entangled
with extra-scientific pseudo-problems.

'Tht problems of the foundattons of psychology contain analogues
to those of biology just mentioned, (i) Can the concepts of psycho-
logy be reduced to those of phj-sics in the narrower sense?
(2) Can the has of psychology be reduced to those of physics in
the narrower sense ? (Physicalism answers the first question in the
affirmative, but leaves the second open.) The so-called psycho-
physical problem is usually formulated as a question concerning the
relation of two object -domains: the domain of the psjxhical pro-
cesses and the domam of the parallel phj'sical processes m the
central nervous sj’stem. But this formulation in the material mode
of speech leads vnto a morass of pseudo-problems (for instance :
"Are the parallel processes merely functionally correlated, or are
they connected by a causal relation ? Or » it the same process seen
from two different sides ? "). With the use of the formal mode of
speech it becomes dear that we are here concerned only with the
relation between two sub-languages, namely, the psj’chological and
the phj*sical language; the question is whether two panJlel sen-
tences are alivaj'S, or only in ceitam cases, equipollent with one
mother, and, if so, whether they are L- or P-equipoUent. This im-
portant problem can only be grappled with at all if it is formulated
correctly, namely, as a sj-ntactical problem — whether in the
manner indicated or in some other. In the controixrsy regarding
behaviorism there are two different kinds of question to be dis-
tinguished. The empirical questions which are answered by the
behanoristic investigators on the basis of their observations do
not bdong here; they are object-questions of a speda] sdence- On
the other hand, the fundamental question of behaviorism, which
is sometimes designated as a methodological or an epistemological
problem, is a problem of the logic of sdence. It b often formulated
in the material mode of speech as a pseudo-object-question (e.g.
“Do mental processes exist?", “Is psjxhology concerned only
with ph)’sical behariour?", and so on). If, howeicf, instead of
being formulated in this way it is formulated in the formal mode,
it will be seen that here again the question is one of the redudbility



§ 83 THE SO-CALLED FOUNDATIONS OF THE SCIENCES 325
of the psychological concepts, the fundamental thesis of be
haTionsm is thus closely allied to that of physicahsm

The prdiUms of the fovndatunu of soaolo^ (in the widest sense,
mduding the science of histoiy) are for the most part analogous to
those of biology and psychology

§ 84. The Problem of the Foundation
OF Mathematics

What should a lo^eal foundation of mathematics athseoe^ On this
question there are various views, the fundamental antithesis be-
tween them 13 pamcularly dearly brought out m two doctnnes,
logiasm, which was founded by Frege (1884), and formalmn, re-
presented by Frege’s opponents (The designaftons ‘ logiasm ’ and
‘formalism’ only appeared later) Frege’s opponents maintained
that the logical foundaooa of mathematics is effected by the con-
structioQ of a formal system, a calculus, a system of axioms, which
makes possible the proof of the fonnulae of classical mathematics ,
m this the meaning of the symbols is not to be taken mto con
sideratioQ, the symbols are, so to speak, unphcitly defined by
the primitive sentences of the calculus, the question as to what
numbers actually are — ^which goes beyond the domam of the
calculus — must be rejected. Formalism today represents a view
which IS m essentials the same, but which has been improved upon
m several important pomts, notably by Hilbert. According to this
view, mathematics and logic are constructed together m a common
calculus, the question of fteedom from contradiction is made the
centre of the mvestigations, the formal treatment (the so-called
metamathematics) is earned out more strictly than before As
opposed to tne formalist standpomt, Frege maintained that the
logical foundation of mathematics has the task, not only of setting
up a calculus, but also, and pre-eminently, of giving an account of
the meanmg of mathematical symbols and sentences He tned to
perform this task bv reduemg the symbols of mathematics to the
symbols of logic by means x>f defimtions, and proving the sen-
tences of mathematics "by means of file primitive sentences ol
logic with the help of the logical rules of inference ((Gnoidgerrtae^)
Later Russell and \Vhitehe3d, also representing the standpomt of
logidsm, earned out m an unproi'cd form the construction of



326 TART V. PntLOSOPHT AND STNT-«

nuthenutics on the basis of lo^c ([/;mc. ^WalA.]). We will not go
into certain difficulties with whidi a stnictiwe of this kind is faced
(see Carnap for we are here not so much concerned

with the question whether mathematics can be deri^;ed from lo^c
or must be constructed simultaneously with it, as with the question
whether the construction is to be of a purely formal nature, or
whether the meaning of the symbols must be determined. The
apparently complete antithesis of the opposing snews on this point
can, however, be overcome. The formalist view is right in bolding
that the construction of the system can be effected purely formally,
that is to say, without reference to the meaning of the symbols;
that it is sufficient to lay down rules of transformation, from which
the validity of certain sentences and the consequence relations be-
tween certain sentences follow; and that it is not necessary either
to ask or to answer any questions of a material nature which go
beyond the formal structure. But the task which is thus outlined
is certainly not fulfilled by the construction of a logico-tnathe-
matical calculus alone. For tlus calculus does not contain all the
sentences which contain mathematical symbols and which are
relevant for science, namely those sentences which are concerned
with the applifatim of mathematia, ie. synthetic descriptive sen-
tences with mathematical symbols. For instance, the sentence
*' In this room there are now two people present” cannot be de-
rived from the sentence “Charles and Peter are in this room now
and 1^0 one else ” with the help of the logico-nisthematical calculus
alone, as it is usually constructed by the formalists; but it can be
derived with the help of the logidst system, namely on the basis of
Frege’s definition of 'a*. A logical foundation of mathematics is
only given when a system is built up which enables derivations of
this kind to be made. The system must contain general rules of
formation concerning the occurrence of the mathematical symbols
b synthetic dcscriptJ« sentences also, together with consequence-
rules for such sentences. Only b this way is the application of
mathematics, i.e. calculation whh numbers of empirical objects
and with measures of empirical magnitudes, rendered possible and
sjstcmaiijed. A itnutvre of this kind fulfib, lirtidtanfotisly,
dmonis of both formalism and U^ieisn, For, on the one hand, the
procedure is a purely formal one, and on the other, the meaning
of the mathematical symbols is established and thereby the appli-



§$4 PROBLEM OF THE roUNDATIOV OF mathematics 327
cation of mathematics m actual science is made possible, namely,
by the trwlunon of the mathematical calculus in the total language
The logicist requirement only appears to be m contradiction with
the formalist one , this apparent antithesis arises a 3 a result of the
ordinary formulation m the material mode of speech, namely, ‘ an
interpretation for mathematics must be giTcn in ofder that it may
be apphed to reahty" By translation mto the formal mode of
speech this relation 13 reversed the interpretation of mathematics
IS effected by means of the rules of apphcation The requirement of
logteum 13 then formulated in this way the task of the logical foun
dation of mathematics is not fulfilled by a metamathematies {that «,
by a syntax of mathematics) ed^, but only by a syntax of the total
language, tehich contains both logico-mathemaiual and synthetic
sentences

Whether, m the construction of a system of the kmd desenbed,
only logical symbols m the narrower sense are to be mcluded
amongst the pnautive symbols (as by both Frege and Russell) or
also mathematical symbols (as by Hilbert), and whether only
logical pnmitiTe sentences in the narrower sense are to be taken as
L>pnmitive sentences, or also mathemancal sentences, is not a
question of philosophical significance, but only one of technical
expedience In the construction of Languages 1 and II we have
foUowed Hilbert and selected the second method Inadentuly,
the question is not even accurately formulated , we have m the
general syntax made a formal distmction between logical and
descriptive symbols; but a precise classification of the logical
symbols m our sense into logical symbob m the narrower sense
and mathematical symbob has so far not been given by anyone

The logical analysis of geometry has shown that it is necessary
to distmguish clearly between mathematical and physical geo-
metry The sentences belonging to the two domains, although they
often have the same wordmg m the ordinary use of language, have
a very different logical character Mathematical geometry is a part
of pure mathematics, whether it is constructed aa an axiomatic
system or m the form of analytical geometry The questions of the
foundation of mathematical geometry thus belong to the syntax of
the geometrical axiom-systems, or to the syntax of the systems of
co-ordinates respectively P^swa/geowetTy, on the other hand, is
a part of physics , it arises from a system of mathematical geometry



jaS PARTY. PIHLOSOPHT AND SYNTAX

by means of the construction of the so-called correlative definitions
(see § 25). In the case of the problems of the foundation of physical
geometry, the question is one of the syntax of the geometrical
system as a sub-language of the physical language. The principal
theses, for e.xample, of the empiricist view of geometry: “The
theorems of mathematical geometry arc analytic ", “ The theorems
of physical geometry are synthetic but P-vahd”, are obviously
syntactical sentences.


§ 85. Syntactical Sentences in the Literature
OF THE Special Sciences

In all scientific discussions, object-questions and questions of
the logic of science, i.e. syntactical questions, are bound up with
one another. Even in treatises which have not a so-called epi-
stemological problem or problem of foundation as their subject,
but are concerned \vith specialized scientific questions, a con-
siderable, perhaps even a preponderant, number of the sentences
are syntactical. They speak, for mstance, about certain definitions,
about the sentences of the domain  have been hitherto
accepted, about the statements or derivations of an opponent,
about the compatibility or incompatibility of diiferent assumptions,
and so on.

It is easy to realize that a mathmalical treatise is predominantly
metamathematical, that is to say, that it contains, in addition to
proper mathematical sentences (for instance r "Every even number
is the sum of tvo prime numbers”), syntactical sentences (of such
forms as: "From. ..it follows that...”, “By substitution vc
get. . “We will transform the expression. . and the like).
The same thing is equally true, however, of treatises of empirical
tnence. We will illustrate this by an example from phjsics. In the
following table the first column contains the initial sentences
(abbreviated) of Einstein’s Zur EleKtrodynamik bnetgier KSrper
(1905). The reformulation in the second column is merely for the
purpose of making clear the character of the sentences. In the
third column, the character of the individual sentences or de-
scriptions is stated, and it is shown that the majority of these arc
syntactical.



§85 SYNTACTICAL SENTENCES IVTHE SPECIAL SCrE^CES 329


Sentences from the
original


That Maxwells elec-
tro-dynanucs

lead to asy mm etries m
their application to
bodies in motion
which do not appear to
appertain to the phe
Qomena
IS well known

For example, if one
thinks of reoprocal
causation

Here the obserrable
phenomenon is depen
dent only upon the re«
lative motion of con*
ductor and magnet,
while, according to the
usual new, the case in
which the one body is
m motion must be
stnctly separated from
the case m which the
other is m motion
If, namely, the magnet
moves , then an elec- '
tnc field is the re-
sult,

which proauces an elec-
tric current

But if the magnet does
not move then no
field results,
but on the other hand
an electnvmotive power |
results m the conduc- !


Part^hrase


In the laws which are
consequences of tiie
Maxwell equations
certam asynunetnes are
shown

which do not occur m
the appertaining proto-
col sentences
Contemporary physi-
cucs know that
Example the rccipro
cal causation sentences

Theprotocol sentences
are dependent only up-
on au^ and such sen
fences of the system.

In the ordinary form of '
the system the two
concrete sentences ' ’ '
I and ‘ ’ are not equi-

pollent to each other


If a magnet moves ,
then an electric field
results

If an electric field
anses, a current re-
I suits

I (Analogous )
(Analogous)


Kinds of sentence

(P s =pure syntactical
d s = descriptive syn-
tactical )

p s desenpUon of sen
tences

ps sentence about laws


andaboutprotocol sen-
tences

Historical ds sentence

p 8 desenpuon of sen-
tences

p $ sentence


p s sentence (with de-
scriptions of two sen-
tences)


Object sentence (phy-
sical law)


As before^


As before


As before



F.urr T. J*HILOeOPHT and stntax


330

Sfr.ie^jtsfro^ the

which, however, ...
ca\i5ts . . . eleciric cur~
rents.

Exsmples of a similar
Hnd,

lie the unsuccesiful
attempts to prove a
modoa of the earth re-
lative to the “Ught
medium”.


lead to the supposition
that


...m dectTD^pn^siea
no propercin of the ob-
aervahiephenesnena ...
correspond to the eoo-
eept of absolute rest.


but rather that ... the
aatne eleetro-dynanuc
... laws are valid for all
co-ordinate tjstenu ...

We will taie this sup-
poaidon

(whose content win be
called ia what follows
the ” IVindple of Rela-
tivity”)

aa an hypothesis.


P,sraphratr

(Analcvtous.)


Ai. Steatepces similar
to the previous ones.
As. Such and such
protoeol-scPtences oe-
runins in the history
of physics. By means
of these pmtr>col-sen-
tencessu^and such an
hypothesis ia refuted.
The sentences A sug-
pest the tentative enn-
tmiction of a physical
system S for whi^ the
sentences S are true
(that is to aar> S is a
system of hypoth«ea
which is confinned by
the seote nen * A).

B X. There is 00 term
in the appertainiag
pTOtocdl-senttares (of
the system S) corre-
sponi^np to the term
'absolute rest* in the
sentences of electro-
dynamics.

B a. The ... laws (of the
system S) have the
same form in relation
to all co-ordinate ayi-
tema.


Bs ahal) be called the
’'ftindple of Reis-
tivity”.

B a is stated aa a hypo-
thetical r-nilc.


' I^n;s tj trrie*^
As bcfcirt-


(Loc«!e) pA descr^tia
of sentences.
Historical d-s. descst;
tion of sentences.

P.S. sentence


po. sentence.


P.S. sesteAce.


PA sentence (abea
certain transforma
dons).


p.a. d^nition.


P.S. convention (de;n
tion of 'P-vaL'd in S’



§ 86 thb logic of science is syntax


331


§ 86 The Logic of Science is Syntax

We have attempted to show by a bnef ezammation of the pro-
blems of the logical analysis of physics and of the so called pro
blems of foundation of the different domains—which also belong
to the logic of science — that these are, at bottom, syntactical,
although the ordmary formulation of the problems often disguises
their character Metaphysical philosophy tnes to go beyond the
empirical saentific questions of a domain of science and to ask ques-
tions concemmg the nature of the objects of the do main These
questions we hold to be pseudo questions The non-metaphysical
logic of saence, also, takes a different pomt of view from that of
empirical saence, not, however, because it assumes any meta-
physical transcendency, but because it makes the language forms
themselves the objects of a new investigation On this view, it is
only possible, m any domain of saence, to speak either m or about
the sentences of this domain, and thus only object sentences and
syntactical sentences can be stated

The fact that we differentiate these two kmds of sentences does
not mean that the two investigations must always he kept separate
In the actual practice of saentific research, on the contrary, the
two points of view and the two bnds of sentences are linked with
one another We have seen from the example of a treatise on physics
that mvestigations m the domains of the special sciences contain
many syntactical sentences But it is also true, conversely, that
researches m the logic of saence always contain numerous object
sentences, these sentences arc m part object-sentences of the
domam to which logical analysis is hemg apphed, and m part
sentences concemmg the psychological, soaological, and historical
circumstances under which work is hemg done m that field So
although we can divide the concepts mto logical and descnptive
concepts, and the sentences of simpler form mto sentences of the
logic of saence (that is to say, ^ntactical sentences) and object-
sentences, on the other hand no strict classification of the m-
vestigations themselves and the treatises m which they arc set
forth is possible Treatises m the domam of biology, for mstance,
contam m part biological, and m part syntactical, sentences , there
are only differences of degree, accordmg to which of the two sorts



FART V. PlJlLOSOPinr AST) SYNT.CC


332

of question predominates; and on this basis one may, in practice,
distinguish between specially biological treatises and treatises of
the logic of science. He who wishes to int’cstigate the questions
of the logic of science must, therefore, renounce the proud claims
of a philosophy that sits enthroned above the special sciences,
and must realire that he is working in eiactly the same field as the
scientific specialist, only with a somewhat different emphasis: his
attention is directed more to the logical, formal, sj-ntactical con-
nections. Our thesis that the logic of science is sjmtK must there-
fore not be misunderstood to mean that the task of the logic of
science could be carried out independently of empirical sdcnce
and without regard to its empirical results. The sj'ntactical in-
vestigation of a sj’stem which is already given is indeed a purely
mathematical task But the language of science is not given to us
in a sjutactically established form: whoever desire* to investigate
it must accordingly take into consideration the language which is
used m practice in the special sciences, and only lay down rule*
on the basis of this. In pnnaple, certainly, a proposed new i)!!-
tactical formulation of any particular point of the language of
science >s a convention, i.e. a matter of free choice. But such a
convention can only be useful and productive in practice if it has
regard to the available empirical findings of scientific investigation.
[For instance, in ph)'sics the choice between deterministic laws
and laws of probability, or between Euclidean and non-Euclidean
geometry, although not umvocally determined by empirical
material, is yet made in consideration of this material.] AH work
in the logic of science, all philosophical work, is bound to be un-
productive if it is not done in close co-operation with the special
sciences.

Perhaps we may say that the researches of non-metaphj-sical
philosophy, and especially those of the logic of science of the last
decades, have all, at bottom, been sjTitactical researches, although
unconsciously. This essentia! character of such investigations must
non also be recognized in theory and sj-stcmatically obsen’cd in
practice. Only then will it be possible to replace traditional philo-
sophy by a strict scientific discipline, namely, that of the logic of
sdcnce as the s)-ntax of the language of science. The step from the
morass of subjectivist philosophical problems on to the firm ground
of c-xact syntactical problem* must be taken. Then only shall we



§86 THE LOGIC OF SCIENCE IS SYNTAX 333

have as our subject matter exact terms and theses that can be
clearly apprehended Then only will there be any possibility of
fruitful co-operative work on the part of the vanous investigators
working on the same problems — ^woik fruitful for the mdividual
questions of the logic of science, for the scientific domain which is
being mvestigated, and for science as a whole In this book we
have only created a first working tool m the form of syntactical
terms The use of this mstrument for dealmg with the numerous
and urgent contemporary problems of the logic of science, and the
improvement of it which will follow from its use, demands the co
operation of many mmds



BIBLIOGRAPHY AND INDEX OF AUTHORS

The numbers immediately followinfi an author’s name indicate
the pages of this book on tvhich he is mentioned. The main refer-
ences arc printed in black tj^pe.

The shortened forma of titles which precede them in square
brackets arc those used m citation throughout the book.

• The publications marked with an asterisk have appeared since
the writing of the German ongtnal, and hence are not mentioned
in the test. The tno&t important of these a«: Hdbcrt and Bemaj-s
[Crundl 1934]; Quine (see the author’s renew in

Srkewttnu^ S, I93S, p. 2S5); Tarski [TTahffi.) (cf. Kokos5>T»ska
[fr«iArAflr]).

Exhaustive bibliographies of the literature of logistics and logical
syntax are to be found in: Fraenkel (MfTtgafUhre}; JBrgensen
[TVeatireJj and Lems {i'irrtvj'J.

Ackennann, W.

Zum Hilbertschen Aufbau der reellen Zahlen. Math, Ann. 99,
1938.

Uber die ErfOUbarkeit gewisser Zshlsusdrilcke. MaOt, Am. too,
1938.

See also Hilbert.

Ajdukiewics, K., 167, 176, 1*7.

[5procA«] Sprache und Sinn. Srk. 4. 1934.

D« Weltbild und die BegriiTsappBranir. Erk. 4. 1934-
•Die syntsktische Konnexit3t. Studio Philos. X, 1935.

Ayer, A. J.

^Language, 7>i<tA and Logic. London, 1936.

Bschounn, F., see Camap.

Becker, O., 46, *45. *46, 350, 354,
hfsthematische Existeru. joArb. Phdnom, 1937; also published
separately.

lAfodjliM/en] Zur Logik dw Modslitaten. Jahrb. Pluin<m. It,
1930.

Bchinsnn, H., 49 f., 139, 197, 246.

Bcittigc tur Algebra der Logik, insbeaondere rum Entschei-
dungsproblem. Malh. Ann. 86. rqji.

[Logik] Afaihfmatik und Logik, Leiprig, 1937.
Entschddungsproblem und L^lc der Besichungen. yher. ^fatA.
Fer. 3«. 1938.

Zu den WidcrsprQchen der Logik..,, Jbcr.Matk. lV.4t>, tpjl-
Sind die msthematischen Unetle analytisch oder synihctisch?
Erk. 4, 1934.



Bibliography and index of authors


335


Bemays, P , 96, 97, 173

[Aiissc^etikalk^ AziomaDsche Uatenucbungen des Aussagen-
kaD^ der Prmapia Mathematica Ma 0 i ZS 25,1926
With Schfinfinkel Zum Entschcidungsproblem der mathema-
tischen Logik. Math Aim 99 1928
[Pkthsophu} Die Philosophic der Mathematik und die Hilbcrtsche
Beweistheone. Bl / dt Phtlat 4 1930
See also Hilbert
Black, M

The Nature nf Mathetnaltet London, Z933
Bhimberg, A* E and Feigl, H
Logical Positivism. Joum of Phtloe 28, 1931
Bore], ;.

Lifons tur la Thdane det Foncttom 3rd ed Pans, 1928
(Appendix Discussion between JL Wavre and P L 4 vy on
mtuitiomst logic, repnnted from Revue Mdtapkyt 33, 1926 )
Br^, M , 9
Bndgman, P W

The Logte of Modem Phytta New York, j 927
•A Ph^iast’s Second Reaction to Meng^ehie Smpta Math
2, *934 (cf Fraenkel [Diagonalveyfahren})

Brouwer, L E J , 46S., 148, 161, 222, 269 281, 305 (See also
Inttatvmm )

llntutlpymm} latuiQonism and Foimalism. BuU Amer Math
Soe 20, 1913

IntuiQonutiadie Metigenlehre Jfher Math Vtr 28, 1920
IntuiUonistische Zerlegung mathemabscher GnindbegnBe Jber
Math, Ver 33, 1925

Uber die Bedeutuiig des Satzes vom ausgescblossenen Dntten.
Joum Math 154 1925

Intuidomstische Betrachtungen Gber den Fonnalismus Ber,
Akad Berltn, Phys -math K 2 , 1928
[,SpTache\ Mathematik, Wissensdiaft und Spracbe Monauh.
Math Phyz 36, 1929
BOhler, K , 9

Cantor, G , 1 37 1 , 267 f , 270
Camap, R

[Ai^bati\ Der logtzche Aufbau der Well Berlin (now Meiner,
Leipzig) 1928

[Schetnprobleme'] Schemproileme m der Phzlozopkie Das Fremd-
psyehische und der Realumusstretl Berlin (now Leipzig), 1928
\,Logistt}i\ Abnss der LoguUk (Schr z wiss Weltauff) Vienna,
1929

Die alte und die neue I^ogxk. Erk. X, 1930 (French transl
L’Ancteime el la Noutelle Logique Pans, 1933 )



BIBLIOGRAPHY AND UTOEt OF AUTHORS


336

Carnap, R.

[Axiomatik} Bericht Obcr Dntersuchungtfii rur aJl^cmemen
Axiomaok. Erk. X, 1930.

Die Mathemaiik als Zwcig der Logik- BL/. dl. Philot. 4. 1930 .
[l^ogtstsmus] Die loguisdsche GnmdleB'u’B der Mathemauk.
Erk. 2, 1931.

[Mfta{<kynk] CbcmTndung der Metaphj-sik durch Jogische
Analj-se der Sprache. Erk. 2, 193s. (French transl. : Lc Snenee
el la M/tapkyn^e. Pans, 1934-)

IPkys. SpraAte] Die physikalische Sprache als Dni>*makprBche
der WisscnsdiafL Erk 2, 1932- (English transl. : The Urrity oj
Science. (Psyche Mm.) London, 1934-)

Psychologic in physikahsthcr Sprache. Mit Erwide-
rungen. Erk. 3, 1932.

[ProtokoUsSlee^ (Jber Protokollsauc. Erk. 3, 1932.

On the Character of Philosophic Problems. Phdos. of Screvee, l,
1934-

Loptsche Syntax der Sprache. (Schr. t. wnss. Weltauff.) ^'icnna,
1934. (TTie ongtnal of this book )

Die Aufsabe der rritrmifAn/n/ogi*. (Emhcitsw i$s.) Vienna,
»934' (French transl.. Lt PnMime de la Lapiqiie de Setenet,
Pans, 1935.)

(^flhnomenj Die Anttnoiruen und die Dnvol)st3ndigkeit der
Mathematik. Monatsh. Math. Phyt. 41, 1934
[GOltigkeiiskntenum] Ein GdlogkettsLntenum fdr die SItze der
klassischen Mathematik. Monatsh. Math. Phyt. 43, 1935.

• ni..»....i, f •. » j,n.) London, 1935.

; • • • Erk. S, t93S*

• • • • ;rtrhbse, le, 1935 -

• • ome. Appearing in:

Erk. 6, 1936.

•Testability and Meaning. AppeaririE in: P/ti/ot.of Science, 3, 1936.
Church, A., x6o.

A Set of Postulates for the Foundation of Logic. Arm. of Math.
33- »933;34. 1933.

•The Richard Paradox. Amer. Math. Monthly, 41, 1934.

•An Unsolvoblc Problem of Elementary Number ITieory. etmer,
Joum. Math. 58, 1936.

Ch^^^ltck, L., 9, 313, 345, 246, 249.

Uber die Antinomien der Prinzipien der Mathematik. Math. ZS.
Z4. 1922.

Sur les Fondements de la Logique Modeme. Alti I'. Cemgr.
Intern. Fifoj. (1904), 1925.

Neue Grundlagen der Logik und Mathematik. I, Math. ZS. 30,

1929; II, 34. 1932.

[Nam. Gtvndl] Die nominalistische Grundlegung der Mathe-
nMtik. ;rft. 3. 1933.



BIBIJOCRAPHT AND INDEX OF AUTHORS


337


Chwistek, L.

With W Hetper and J Herzberg Fondements de la M^ta
math&naaquerationelltf BuB Acad Pol,SAr A Math, 1^22
As above Remarques sur la Mitamath^matique rationelle
Loe at
Curry, H B

AnAnalysisofLogicalSubstttution- Amcr Joum Math 51 1929
Grundlagen der kombmatanschcD Logik- Amer Jovm Math
52, 1930

Apparent Variables from the Standpoint of Combinatory Logic.
Aim^of Math.'^ 1933

•Functionality m Combinatory Logic. JVoc Nat Acad Sa 2o

1934-

Dedekmd, R., 137

Was smd und teas solUrt die Zahlent Brunswick, 188S
Dubislav, W 44.

[Analyt'l Vber die sog anatytuchen und synthetischen UrtaU
Berlin, 1926

2 ur kalkOlmasstgen Charaktensierung der Debnibonen. Amt
Pktlos 7, 1928

;l«m«ntarer Nachweis der Widerspruchslosigkeit des Logik-
kalkuls Jaum Math. Idt, 1929
Dia Defimtion Leipzig 3rd ed , (931

the Phlasophu der Mathematik tn ^ Gtgemsart Berlm, 1932
NaturpMlosophu Berlm, 1933
Duhem, P , 3x8.

DOrr, ;

[Letbmz\ Neue Beleuchlung oner Theone van Ldbmz Darmstadt,
1930

Einstein, A., 178, 328
Feigl, H

•The Logical Character of the Pnnaple of Induction. Phtlos of
Set. I, 1934.

•Logical Analysis of the Psycho-Physical I^blem. Phtlos of Sa

1. 1934

See also Blum berg

Fraenkel, A., 97f , i6z, 213, 267 ff, 270, 274, 27s. 335
[Untmuckungerii Untersuchungen Qbn die Grundlagen der
hfengenlehre. Math. ZS 22 1925
Zehn Vorlesungen vber dte GrundUgung der Mengerdehre Leipzig

1927

[MengenlehreJ Emlatimg tn die Mengerdehre Berlin, 3rd ed., 1928
Das Problem des Unendlichen m der neueren Mathemaok.

BLf dt. Philos 4, 1930.

SU


23



338


bibliography and index of authors


Fraenid, A. » , -i

Die hcuugen GegensJtze in der Gnmdlegung der Mathemank.
Erk. I, 1930.

•sur la Notion d’Eiistcnce dans les Math&natiqucs. Erungn.


Math. 34, 1935.

•Sur I’Anome du Choix< Z<®c- nV.

*[Diag<malverfahren] Zum Diagonalvctfshren Cantors. Eund.
Math. 2S, XQ3S.


Frank, Ph., 280 f.

Was bedeuten die gcgenwlrtigcn physikaJisehen Thcorien filr die
allgemeine Erkenntnialebie? Natuncirr. 17, X919; also ii\; Erk.
I, 1930.

IKautalgfseu] Das Kausalgtsetz wnd seine Urensen, (Schr. r. wiss.
Weltauff.) Vienna, 1932.

Frege, G., 44, 49. 99. i34. *43. *44. *s8, *97. 2e»3. *59,

295. 3*5 IT.

BegrifTsschrift. Halle, 1879.

\Grmdlogni\DitGrundlagendeT AriOmetih. Breslau, 18S4. (New
ed. I934-)

[(Snmdgtsetsel Crvndgesetse der Anthnetik. Jena, I, 1S93; II.

1903-

[Zahlen] Cher dtt Zahlen det Hem H. Sekvhm. Jena, 1899.


Gfitachenberger, R.

Symbola. AnfangsgrOndt etntr Ethenntnistkeorie. Karlsruhe, 1920.
Znchen, die FunJamente des Witsem. Eine Absage an die Fhi/o-
tophie. Stuttgan, 1932.

Gentzen, G.

*Die ^Vlde^pruchsC^elhe^t der reinen Zahlentheone. Afath.
Arm. 1X3, 1936.

*Die Widersptuchsfreiheit der Stufenlogik. Afath. ZS.41, t936.
Glivenko, V., 227.

GSdel, K., 28, 55 , 96 f., 99, lOO, 106 f., 129, 130, 131 ff., 134, 139,
160, 173, 189, 197, Z09, 21), 219, 227, 250.

Die VoIlstAndigkeit der Aziome dea logis^en PunktionenkalkOls.
Alonatsh. Alalh. Ph)f. 37, 1930.

[Unmtsekeidbare] Cber formal unentscheidbare Satre der Prin*
apia Mathematica und verwandter Systeme. J. Afonalsh.
Afath. Pk}s. 38, 1931.

IKoi/ckjurum] Venous rtofcs in; Ergein, e. math. Kolloqutums
(K. Menger). Vols. 1-4, *931-33.

Gtelling, K. and Nelson, L., 2ii IT.

Demerkuftgen 2u den Paradoxien von Russell und Durali-Forti.
Abh. d. Fritttchen Sehtde, N.F. 2, *908.



BIBUOGRAPHY AND INDEX OF AUTHORS


339


Hahn, H , 280

[Wus TVeltaifff] Die Bedeutung der wissenschafthchen Wclt-
auffassung:, insbes fOr Madiematik und Physik. Erk X, 1930
Logtk, Mathematik und Naturerkennen (Emheitswiss ) Vienna,

1933

Helmer, O

*AxxomatuehfT Aufbau der Geometne mfomalmerter Darttellung.
Diss , Berlin, 1935
Hempel, C G , 317

*Betirage zvr logucken Analyse des Wakrscheinluhkntsbegnffes
Diss , Berlin, 1934

•Analyse Logique de la Psychologic Rev Synthise, 10,

•Ober den Gebalt von Wahrscheinlichkeitsausaagen Erk 5,

1935

With P Oppenheim *Der Typusbegrsjff tm Ltehte der neuen
Logik Leyden, 1936
Herbrand, J , 53, 134. I73

Retherehes sur la Throne de la DdmoratTatUM Th^ePac Saenees
Pans (Nr star, s^ne A, 1252), 1930 Also m 3 Vaw«* Soe
Saenees Varsovte, Cl in. Nr 33 193©

[Non Contrad} Sur U Non>CoQtradicQon de PAnthnitique
Joum Math 166, 1931

Sur le Problime fondamental de la Logique mathematique
CR Soe Saenees Varsovu, 24, Cl in, 1931
Here, P , 275

[Axiom J Uber AxiomensystcmefiirlTeliebjgeSaejyjteaie Math.
Amt xoi, 1929

Vom Wesen des Logischen .. ;>* a, 1932.

Hetzberg J See Chvnstek.

Helper, W

•Semantiache Anthmetik. CM Soe Saenees Varsovu, 27, CL in

1934.

See also Chwutek.

Heytmg A., 46 fT , 1 66, 203 222, 227, 245, 246, 249 f
[jLogiA] Die forxnalen Regein der mtuitiomstischen Logik. Ber
Akad Berlin, 1930

[Math ] Die fonnalen Regein der intuitionjsmchen Mathematik.
I, n Loc at

[Grvndlegung] Die intuitionisUsche Grundlegung der Mathe-
zaatik. Erk 2, 1931

Anwendung der intuitiomstisdien Logik auf die Definition der
VoUstandigkeit eines KalkQis Intern Math -Kongr ZOnch
1932

*Mathemattsche Grundlagenforschung, IntwUomsmus, Beteets-
theorte (Erg d Math., 111,4.) Berlin, 1934>



BIBLIOGRAPHY AND WOEX OF AtmtORS


340


HHbcrt, D.. 9, 12, 19. 35, 36 , 44 f-, 48. 49 . 79 . 97 . 99 . *04, 128 f.,
140, 147, 158, 160, J 73 . 189, 197. 203. 244. 259. 272 . 274,
281. 32s, 3 * 7 - . ^

{Gnpuil, Geom.l Gnmalagen der Geometne. Leipzig, 1099.
7th edj 1939.

ArioQiatisches Dcnlten. Afath, Ann. ^S, 1918.

Neubegrilndujig der hfsthcmatik. Ahk, Alath, Sem, Htonburg, i.


1922.

IGntndl. 1923] D«e logischen Gmndbgen der Mathematik.
Math. Ann. SS. 1923.

[tJnfndliche} Ubcr das Dnendlichc. Afath. Ann. 95. 1926.

Die Grundlagcn der Mathematik. Mit Bemerkungen Ton Wcyl
und Bemays. Abh. Afath. Sem. Hamburg, 6, 1928,

With Ackemmnn: [Logik] Grwidzilge der theoretUchen Logik,
Berlin, 1928.

Probleme der Gnindlegiing der Mathematik. Moth. Ann. loj,

1930.

[Grundl. 1931) Grundlcgung der elementatrtn Zahlejilchrc.
Math. Ann. 104, 1931.

[Terttuml Beweis del Tertium non datur. Nachr. Gt{. Witt.

GSltingen, malk.-phyt. KJ., 1931.

Wth Bemaya : *[Grundl. 1934J Crvndlagen der Methemattk. I.
Berlin, 1934.

Httne, D., 2S0.

Huntington, E. V.

Sets of Independent Postulates for the Algebra of Logic. 7>o«t.
Amer, Math. Soe. 5, 1904.

A New Set of Postulates for Betweenness, sWth Proof of Complete
Independence. Ttotij. Amer. Math. Soc. 26, 1924.

A New Set of Independent Postulates for the Algebra of Logic,
with Special Reference to Whitehead and Russell's Princi^ia
Mathesnatica. Proc. Nat. Arad. Sei. 18, 1932.

Husserl, E., 49.


Jtskowski, St.

*On the RuUj of Supporitiora in formal Logic, (Studia Logics,
Nr. I.) Warsaw, 1934,
jargensen, J., 258, 335.

[Treatue} A Treatise of Formal Logie. Its Fvolulion and Main
Branches tcith its Relation to Aiathematkt and Philotephy,
3 Fols. Copenhagen, 1931.

iZieU] Uber die Zieic und Probleme der Logistik. Erk. 3, *932.
Kaufmann, Felbt, 46, 51 f„ *39. |6i, 165.

lUne^luhe\ Das UnendlUhe tn der Mathematik und tetne Aus-
sehaltung. Vienna, 1930.

IBer^kungen] Bemerkungen aum Crundlagenstreit in Logik und
Afathetnatik. Erk. 3, 1931.



BIBLIOGRAPHY’ AND INDEX OF AUTHORS 341

Kleene, S C

•A "Ilieory of Positive Integers in Formal Logic Amer Joum
Math, 57, 1935

With Rosser, J B •The Inconsistency of Cemm Formal
Logics Am 0/ 36, 1935
Kokosrynska, M

•[iraXrAeir] tJber den absohiten Wahrheitsbegnff und emige
andere semantiscbe BegnSe. Erk 6 , 1936
Kronecker, L , 305
Kuratotvski, C. See Tarski
Langford, C. H See Lewis
Leibni2, 49

Le&uewski, St, 24, 160

[^euer Slyrtem] Gnindzuge eines neuen Systems der Gnmdlagen
der hlathematik Fund Afath 14, 1929
[OntalogteJ tJber die Gnmdlagen der Ontologie CJ? Soc
•S'nencer Vanovte, 23, Cl iii, 1930
[Defimttmen] (Jbcf De^tionen m der sog Theone der Deduk-
tioa Ihtd 24 Cl III, 1931
L<vy, P See Borel

Lewis, C. 1 , 12, 303, 223 f , 232, 245, 246, 35®. *52. *53 f » *57 f ,
*75. *8i. 335

[Survey] A Sunty of Symbolte Logu Berkeley, 1918
Altemative Systems of Logic. Atonut, 43, 1932
With Langford. C. H [Logu\ Symbohc Logte New York and
London, 1932

•Fxpenence and Aleanmg Pktlos Revxew, 43, 1934
Ldwenhetxn, 270

Lukasiewicz, J , 9, 96, 160, 250, 354
With Tat^ [AussagenkalkSl] Untersuchungen uber den Aus-
sagenkalkul CJ7 Soe Sctenees Fanorte, 23, Cl iii, 1930
[Mekrxserttge] Philosophische Bemerkungen zu mehrwertigen
Systemen des Aussagenkalkuls Loe ett
MacCoU, 254.

MacLane, S

*Abgeldirzte Betaetse sum LogtkkolkaJ Diss GSttingen, 1934
Mannouiy, G

•Die signifischen Gnmdlageo der Mathematik. Erk 4, 1934
Menger, K., 52

Bemerkungen zu Gnmdlagen&agen (especially II Die mengeti-
theorebschen Paradoxien) JSer ATath. Ver 37, 1928
[/nruriOTOenu} Der Intmtionismus Bl f dt Philos 4 1930
Die neue Logik. In Knse urtd Neuaufbau tn den exakten IFwfffl-
sche^ten. Vie nna , 1933



342


BIBLtOCRArRT AND INDEX OF AUTHORS


Mises, R. V., 14^

TTahnchrirdichhit, Statistik vnd ^ahrhfit. (Schr. x. wiss.
Wdtauff.) Mcnna, igsS.

tJber das naturwissenschaflliche WcltbOd der Gtsenwart.
Katvrtziss. 19, 1931.

Morris, C. W.

‘ITje Rdadon of Formal to Instratoontal Logic. In: Essayt in
Piiloifpfy, 1959.

•Tbe Coneopt of Meaning in Pragmatism and Logical PositiTisin.
iVoc. StA IntfrH, Cem^> PMot. (1934). Prague, 1936.

Nagd, E.

•Impressions and Appraisals of Analytic Philosophy in Europe.
Joum. nf Philat. 33, 1936.

Nelson. E. J.. 254, 257.

[ 7 »»rmri«»afl Intensions! Relations. Mind, 39, 1930.

Deducti« Sjatems and the Absoluteness of Logic. Mind, 42,

1033.

On Three Logical Pnndples in Intension. Memst, 43, 1933.
Neumann, J. t,, 96, 98, 139, 147. 166. 173.
lErtomtA.] ZuT Hilberischen Beweistheorie, Math. ZS. sS, 1927-
Die formshstisehe Grundlegung der Miihematil. Erk. 3, I93i.
Neurith, O., 3S0. sSi, 3S3, 286. 320 f.

With others: [ITur. XTrttavff.^ Wmrroehajtlitkf ITV/rji<ffatn«p.
Dft WfenfT Krfis. (Vtstft. d. Vereins Ernst Mach.) Menns,


[ITejfj W'ege der wissenschaftHchen W'eltauffassting. Erk. X|
1930.

Empinsehf Sosiologif. Ofr tnssfnsehaftliehe Gfhalt drr Gtschirhte
undXationaJskcmomie. (Schr. x. wiss. W’eltauff.) Menna, 1931.

IPhytieolirm] Phj'sicalism. The Philosophy of the Mennese
Circle, Momtl, 41, 1931,

IPAjnAn/itwia] Phyaikalismus. ^hVnfia, 50, 1931.

{ 5 orio/. PAjr.] Soziologie im Physikalismus. Erk. 3, jgyr.

[PrttfoAofhatse] ProroVoIlsStte. Erk. 3, 1933.

tPocAof.] Einknimissmsehaft vnd Ptycholosie. fEinheitswiss.)
Vienna, 1933.

•^i^er Phj-slkalisraus und "wirllkhe W'elt". Erk. 4. 1935.

LeDMoppnvnt du CercUde llrnne rt I'Avrnir de rEtnpiriasme
loptjue. Paris, 1933.

Nicod, J.

^ ^ Number of the Primltiee Proposidoru of

Logic. Proe. Camhr. PhiL Soe. 19, 1917.

Ogden. C K. and Richards. I. A,

TIuMrmTv c/ .1 W.,, ^ study of th, hfiunro of Laosuust
vpon ought and of the Stienee t>f Symholitm. London, 1930.

Oppenheim. P. See Hempel.



BIBUOGRAPHY AND INDEX OF AUTHORS


343


Pairy, W T , 254, 257

\KolL'\ Notes m Erg t math Kolloqmumt (ed. by Menger)
Heft 4, 1933

Peaoo, G , 31 f , 44, 97, 99, 144, 158, t66, 2t2
Notatums de Logupte mathdmatujue Turin, 1894
{Fortmdatre] Formulaire de hlathdmatupus Tuiin(i895) 1908
Peirce Ch- S

Collected Papers Ed by Ch Hartshome and P Weiss 5 vols
Cambridge, Mass 1931 fT (Especially Vols 2-4)

PenttiU, A. and Saarmo, U

•Einige giundlegende Tatsachcn der Worttheone Erk 4
1934-

Poincari, H , 46, 161, 269, 318

WisseraeJuft und Hypothese Leipzig (1904), 1914
Wusemehaft tend Methode Leipzig, 1914.

\Gedanker{\ Letzte Gedanken Leipzig, 1913
Popper, K, 3*7, 32*

*Logik der Forsehung Zur Erkermtnutheorse der modemen Natur-
vnssensehaft (Schr 2 wtss Weltauft) Vienna, 1935
Post, ; L., 208

[Introduetum] Introductioa to a General Theory of Elementary
Propositions Amer Joum Math 43, 1921
Presbuiger, M

Cber die VoUstandigkeit euiesgewissen Systems der Anthmeok
Congr Math Warschau (1929), 1930
Qume, W V , 190

•[Syitm] A System ofLogtstu Cambridge, Mass , 1934
*OQtological Remarks on the Propositional Calculus Mind, 43

1934

•Towards a Calculus of Concepts Joum Symbol Ijogie i 1936
•Truth by Conventioa In Phthsophieal Essays for A N
Whitehead, edited by O H I,ce, 1936
•A Theory of Classes Presupposing No Canons of Type Proc
Nat Acad Set 22, 1936

•DefimOon of Substitution Bull Amer Math Soe 42, 1936
•Set-Theoretic Foundations for Logic Jottm Symbol Logic, i,
1936

Ramsey, F P , 50, 86, 114, 211 f , 213, 283, 321

[Foundations'] The Foun^tums of Mathematus, and Other Ixtgual
Essays London, 1931
Reichenbach, H , 78, 281, 311, 317

[Axiomatiki Ajctomatik der relativtstisehen Raum Zett Lehre
Brunswick, 1924

[Pkdosopktel Phxlosophte der Raum^Zett Lehre Berlin, 1928



BIBUOCRAPinr AND INDEX OF AUTHORS


344

Bcidicnbach, H. . „ n i

ITJ'flATj;fcrin/»c^t«ti/off«A] W^hRcheinlfcWceitslogii. Btr. Akaa.

Berlin, 2% 1932. , ^ .

•B'ahrsehcinlkhkeiuUkrt. Bine UnXersuehung Gb^ die loguch/n
tmdtMtherTUJtischen Cfundtasmder Xt'ahttcbeinlichketlsrechnung-
Leyden, 1935'

Richard, J., 3X3. 2l9i 222. *70.

Richards, I. A. See Ogden.

Rosser, J. B.

•A Mathematical Logic wiihont Variables. I. Ann. of ilfatA. 30,
1935. Buke Math, Journ. I, 1935.

See also Kleene.

Rilalow, A., 213.

Der Lasner. Theorie, Getcktchte imd AuflSnaig, Diss. Erlangen,

1910-

Russell, B., 19, 32 , 3S, 44 f., 47 iT., 49 IT., 86, 9Sf., 99, 134 » X 3 *^-»
X 4 Q. t 43 i <44 f-. tsS. t6o, X< 3 . 164 L, 173, X89, I 92 > X 9 $>
S03.lUf., 131, » 44 , 245 f.. 2 M 9 . 253 “ 255 . = 57 f-. * 59 f-.

28 j. * 9 J. 293. 29s. 3 * 5 f-. 327.
t/Vsrtflp/«] The PnnApUt oj il/<j/Ae»nafr«. Cambridge, 1903.
The Theory of Implication. Amer. Jcntm, Math. sS, J 906.


Schlick, M., 51, lor, 280 f., 284, 310, 321.


tnw. 19, 1931.

[RoMfitumui] Positirismus und Realumus. Erk, 3, r932.

Obcrdas Fundament der Erkenntnis. Erk. 4, 1934-
Meaning and Verification. FAiVon Review, 45, 1936.

Scholz, H., 258, 260.

tCewAwhfe) CesehUhte der Logik. Berlin, *931.

«jJh ScltweiQer, H,: *Die sag, Definilionen dureh Ahtiraktioit.
(rorsch. t. Logistik, No. 3.) Leipzig, 1935.

SchBnfinVel, M.

tlber die Daustcme der malhematischen Logik. Math. Ann. 95,

1924.

Sec also Bcmiys.



BIBUOGRAPHT AN1> tNDEX OF AUTHORS


345


SchrSder. E , 44. 158 ^ 1

Vorlettmgen Cber die Algebra der Logik (exable Logik) 3 vols
l*ipzig, 1890-1905
Sheffer, H M

A Set of Five Independent Postulates for Boolean Algebras
Trans Amer Math. Soe 14, 1913
Mutually Prime Postulates BuU Amer Math Soe 22, 19*^

Skolem, Th., 270

[ErfiiUharhetil Logi3ch*kombiaatonsche Untersuchungcn Qber
die Erfullbarkeit oder Beweisbaikeit mathematischer Sitze
Vtdensk. Skr Knstiama, 1920, No 4.

Begrundung der elcmentaren Andimetik durch die relnimerende
Denkweise . Vtdensk Skr Knstiama, 1923. No 6
tJber emige Grundlagenfragen der Mathematik. Skr Norsks
Vtd.-Ahad Oslo 1. Mat Nat Kl 1929, No 4

Taiski, A., 32, 70, 89, 9« f , 160, 167, 172, i?3, i97. 20o» *04. 208,
209. 275-

Sur le Tezine Pmmtive de la Logistique Fund Math 4, 1923
Sur lea Tnith«Fuoctiaas au Sens de MM ^Vhitehead and Russell
Fund. Math 5, 1924

tlber emige ftmdamentale Begnffe der Metamathematik. CJi
See Seienets Varsoute, 23, Cl itl, 1030
Tte.AJ, .... ~ - .. • .* de-

.. : *• •' 30

S laid

AfatA. 17, 1931.

nr*!. f.


\}VuleTspTuchsfT'\ Emige Betrachtungen uber die Begnffe der
o>'VViderspnich9;reiheit und der <u-VoIlstSndigkeit Monatsh
Math Phys 40, 1933

•Emige methodologisdie Untenuchungen uber die Definier-
barkeit der Begnffe Erk 5,1935
•GruadzQge des Systemenkalkuls I Fund Math 25, 1935
•[IFaArA.] Der Wahrheitsbegnff m den fonoalisierten Sprachen
Stud Philos 1 , 1936
See also Lukasiewicz

Vienna Circle, 7^ 44., 280^ 282,, 309^ f See also Carnap^ FetsU
Frank, Gsdel, Hahn, Neuratb, Schbek, Waismann.

Waismann, F , jai

Die Natur des ReduzibOitatsaxiaxiis Monatsh Math Pkys 35
1928



346 BnjUOCRAPOT AND INDEX OF AUTHORS

Waismann, F.

[TVaAricA<Tn/t;A/t«rJ Logische Analyse des Wahnchcmlichkeits-
begriffes. Erk, x, 1930.

•Uber den Begriff der Identit 3 t. ErJc. 6 , 1936.

Waisberg. M.

Uber Axiomensysteme des Aussagenlcalldlls. Jifonalsh, Alath.
Phf- 39 . 1932.

Ein erwdterter KlasscnlndkUl. Afonatih. Math. Ehyt. 40, 1933.
Untcisuchungen Qbcr den PunlcdonenkalkQJ fOr endliche Indi*
vnduenberciche. AfatA. Aim. loS, 1933.

Beitrag rur Metamathematik. Atath. Atm, 109, 1933.

*Beitr 3 ge aum MetasussagenkalkOl. Afonatsh. Math, Phys, 41,

1934.

Warsaw logicians, 9, 160, afii. See also Le;niewski. Lukasiewict,
Tarski.

Wavre, R. See Borel.

Weiss, P., ayS.

The Nature of Systems. (Reprinted from .A/otruf.) jpzS.
Two-Valued Logic, another Approach. Erk. », 1931.

Weyl, H., 46, 99, 148. i8<, 305.

IKontifruiml Das Kontinuum. Leipzig, 1918.

Uber die neuere Gnuidlagenkrise der Mathematik. Afath. ZS.
20, IQSJ.

Die heutige Erkenntnishge in der Mathematik. Sympea. 1, 1925*
(Also published separately.)

Philosophie der Mathematik und Naturtrisscnschaft. Part 1 .
In: fiandb. d, Philos., ed. by Bsumler and Schrflter, hlunieh,
1926. (Also published separately.)

^Vhitchead, A. N., 44, 99, 158.

MeMph>-8ics and Logic of Classes. Atanist, 4z, 1932.

See also Russell.

Wittgenstein, L., 44, 46, 49ff., 51 f., 53, joi, 139, 140, i6x, 186,

trV: *95 f-. 3 ® 3 . 304. 307. 3*2 f-

Logieo-Phltosotdiicus. With introd. by
D. Russell. London, i9zz.

Zermelo, E., 93, jy.

OntmuAunstn ob,r di, Gnadl.c,™ d« JIj*.

Ann. 65, 1908.



INDEX OF SUBJECTS

The namiersTefer to pa^es The most mportant passages are indicated
iy black type

Abbreviatioiis I ^Syntax of I^guage I
II = Syntax of Language II
Gs=Genetal Syntax

A


9, see Expression
a see SymBol
Abbrerotioa, 157 f
Accent, Accented ezpressioB, I 13,
36,11 X32

9fu, see ExpresnoiiolfiaetuM
93 we ExpresstonaJ /rammof^
Aggregate, see Clou
Aggregates, Theory of, II 83, 86,
97f 1 138, t 221, 267
Analytic,! 28,391,43, t[ toof,
jjaf .G i8iff

Antiaoiny, 3,l37f , an ff , aX7 ff ,

231

8 Ic 9> Argumeot, Argumeat*ex<
pressioo. ! 26, II 8t. Syf ,
O 187S


B^sed, 63 68, 72
Behanounsn, 324


Aigumeot, Suitable, G iSS
Anthmetic. I 30!, 59, H 97>
134, G x^, Z05 R , 330 R ,
3<Hf » 3»5ff

Arithmetizatioa, 54 ff . 57, 79
Atomic aestence, II 88
Autofzyinous, 17, 153, 156 f, x6o,

237. 247 f

Autonymous mode of speech, see
Afode

Axiom, see Pnmstwe sentence,
Principle

AnocQ'systns, G »74 f
Axkkbsqc method, system,
^, 2 ^tn
AxM^bcs, 374 f


Boundfv),! 31,66,11 87, G 192
Bradet, 1$, 19


C-, H 100 f , G 171, 17* ff
„ . 17s 182, 183, 185
Ctlculable, 148 f
^culus,4ff,,67, 3j8f
t-hanctenstic, G 202
2*“*37,97. 134ff,i36ff
Wassofespressions.I 37,0 i<
Oosed,! 31,66, G 194
Coextensive, 137, G 187 ;,241
^patible,! 40,11 1X7, G r
Complete, t c. S, ft, G 175, 1.

2 c. language, G 209
Conjunction, 1 i8{,U 89.10
G 202

Conjunctive standard form, 103
Conse«juence. 37, i ^7, 38
_ H MTff.G 168.172.25
'-onsequence, Rules of, 1 37 f
II 98ff,G 171
t^onsequence-class, I 38,0 17
t^equence^^enes, 1 38 f , I


Consiatcfit, G 207, 275
Constant, I j 6,II 84,0 194
Content,! 42, U KO.G nsf
Continuum, 305

Contra-class, sentence G 200,203
Contradiction, 137, 291, 297, 304,
see also Antinomy, and fTin-
tipleoj

Contndictory, I 28, 39 ff, 44J
11 zitf,i28,G 182,207,

275

Contravalid, G 174 275
Convaluahle, II 108
Converse, 264
Co-ordinate, 12

Co-ordinates, Language of, 12, 45
Correlate, 222

Correlated syntactical sentence,

*34. *36

Correbuon, Syntactical, 222
Correbtrve definition, 7, 78, 79
Cotrebtor, G 265



34S


mDES OF StJBJtCTS


D


b, sfe Detcriptive

d-, II: 99. «oi: G: *7off., I74 f-»
182, 1S3, j8j
Definable, 1x4
Defined, G: 172
DtfiratTuitan, definjera, 23
Definite, it; I' 451.; II: 9S.

160 ff., i65ff, i 73:G- 198
Definitioa, 1 : 33 f , 37, 66 ff., 72,
78; II: 88 f.; G: 172, 194
Definition, Expliat, 23, 24; 11 . SSf.

Hegmsive, 23, 68; II. 88, 89
Definition^haic, 24, 71
Definition tn tuu, 24
Definition-schema, 6S
Definition-sentence, II: 71, 72
Demnnstrtble, 1 : 2 8 f., 7$ f. ;

II: 94, 124; G: 171
Denumenhk, 213, 220, 268 ff.
Dependent, G 174
Demible, !• 37l. 39. 7S. H: 94;
G: 171

Dermtion, I 37 f., 33 ff., 39, 75;

II.94f.tG’ 17*


Description, descriprional, 1 : 22 f.,

144 ff.. 154 ff.; G: 193. X95.

2 Q1

Descriptive, t. d. SI, I: 13, 14. ^5.
38; II; 73; G: I77f-. 33«.
231 f.

3. d. I-angtisge, G: X78, 210
3. d. Sjmtas, 7, 53, 5® S.. 79 f-*


*31. «54

Design (Gatali), 25 f., pt. >55
Designation and designated, iS,
153-160

Designition, Syntactical. t54f..ifio
Determinate, ». d. S, Si, H: lot.
115: G: 174
2. d. L^guage, G: 209
Diagonal method, 221
Direct consequence, I: 38; G:
i6Sff., 170 f.

Directly derivable, 27; I: 3*. 745
II: 94; G: I7t

Dtajuncuon, I: igf-i H Sp, io3»
G: 202f.

Double negation, I: 34; >^5


E


Eletnentary sentence, t66; G: 195
Elimination, I: 24 f., 31; II: 89!.;
G: 172

“nuddation", 283, 321
Empty, G: 261
Equality, see iJfntily
Equation, I: 19, 36, G; 243
Equipollent, I: 42; II: 120; G:
176, 1S4, 241

Equipollent translation, G: 226
Equiralence, I: 19; G: 202
Equit-alenee, Symbol of, 1; 16, 19,
49; II: 84: G; 243
Evaluation, II: loS, no


Excluded Middle, see firintifile of
Exclusive content, G: 176
Exiiteace, l4of.

Existential operator, It 21 ; Cs 93,
196 f-

sentence, Unlimited, 47, 163
Expression, 4; 1: 16; G: 167 f.
Expression, Principal, G: 177
ExpressionaJ framevrork, G: 187
Ainction, G: 191
Eztensional. 93, 240 ff.
Extensionality, Primitive sentence.
Axiom, of, see PnnnpU
Thesis of, 139, 245 ff-


f, see FunttoT-vanahU Full expression. Full sentence.

False. 214, 2X7 G: 187

Form, 16, 15s Function, see Expretfianat /., Sm-

Formal, I, 238 f., 281 f. «7iri^/.

Formal mode of speech, see Mode Functional calculus^ I: 35; II : 9®»
Formalism, 300, 325 ff. 10^

ForTnationrtiles.2,4;I:26,62ff- Functor. 13; I: x6, 54f-. 7*5
II: 87ff.:G; i67ff, G:x88

Free (0), I r 21, 66; II : 87; G; X92 Functor- ex pr es sion, II: 84. 87,
«U, see FuTUtor-exprettion G: 18S

|U. see Fimrfor Functor-vanable, II: 84; G: Z95



349


INDEX OF SUBJECTS


©, 13a f

General syntax, *53, X67
Genoa. G 170, 393


G

Geometry, 7, 78 ff, 178, 229,

327

Godue symbols, see SymboU


Heterological, 211, 218
** Higher, The 314 f


H

Homogeneous, 260
Hypothesis, 48, 318 S


1

Identity, Sentence of, see Equabon

Identity, Symbol of, 1 15, xg, 31,
49S.,n 84, G 343 ;f

Impbcation, 1 X9f.32,II 158,

G 202f , 235, 353 S

Impbcation, Strict, see ’Strut"

“Impredicablc”, 138, aix

Impredicanve, 162 ff

In-, srhen not listed see the un-
prefixed leord

Incompatible, I 40, II it?

Incompleteness of srahmeoc, 173,
33xe.

Indefinable, X08, 114, 134, 3x8fi,
33Z

Indefinite, 11, I 45 t, II 99,
113, xdoff, t6sff, 172, G
198

Independent, II ti7,G 174

Indim, 1 X7f , II 86


Junction, see StnlenUal ptru
turn


ft. see CI21S of ceprts t t O H S
I, see Comtotr


I, see Logical

L-,G i8of^ 182S., 362ff , 2650
Luiguage, 1. 4, G 167
Language, Symbobc, 3
language-region, II 88
Language, Science of. 9
Law of nature, 48 52, 81, 148, iSo.

tSy. 307, 3x6ff, 32X
Level, Level number, 11 83 f ,

G xS 6L,26 i

Liar, Antinomy of the, 2x3, 214 f,
2X7L


Indirect mode of speech, see Mode
Individual G x83, 195
Induction Complete, I 321,38
II 92 f, X 2 X
Incomplete, 317
“Inexpressible’ , 282 f , 314
Infinite sententid class, I 37. 39 •
II too

Intensional, 18S, 242, 245 ff
“Internal’ , 304

loterpretatioo, 131 132, 233 239,
3*7

IntuitM&ism, 46 ff , 305
Inverted commas, 18, 155, xsS f
Itrefiexive, 234 f

Irtesoluble, I 38, II 94' *33 ^ •
G 171, 221 f

Isogenous, G 169, 18S, 274
Isolated, G 170
laomorphic, G 224, 265

J

Junction symbols, I i7> tS ; ,
II 93, G 2DX

K

•K’, K-operator, -deseription, I
16. 22 f, 30, II 92. 246


TjT;ii f, Limited operator, I 2 x ,
G xgx

“Logic”, X, 233, 2 s 7 ff . 278 ff
Logic, Intensional, 256, 282
Logic of Modaboes, see Modahtiet
Logical, I 1 21, I 13. M. 25. 38.
73. G X77f

2 1 language, G 178,209!

3 I rules, G tSo f

4 I analysts, 7
Loguasm, 300, 325 ff
Logic of science, see Scieiue



350


INDEX OF SUBJECTS

M


Material mode of speech, see Mode
Mathematics, see Arithmtlic, Num-
ber


Mathematics, Ciassical, 83, 98, 128,
148, 230, 325
Meaning, 189, 288 ff.

“ Meaningless ”, 47, (81), 138. 16*,
163. 283, 3:9, 321, 322
MetaJogic, 9

Metamathematics, 9, 323 ff.
Metaphysics, 7 f., 278 f , 282-284,


309, 320


Modalities, Logic of, G: 237, 246,
250-258, 303

Mode of speech, Auton>Tnous, 23S,
247 ff.

Formal, 239, 280 f., 288 ff.,
299 ff.. 302 ff.

Indirect, 291 f.

Material, 237 f., 239,286,2871!.,
*97 ff-. 302 ff.. 308 ff.

Afodei, O; 272 f.

Molecular sentence, II: 83,
321


31. 11.84

Name, 12 f, 26, 189 f
Name-language, 12, 189
Natural law, see Lm
Negation, I; 19, 20; G. 202 f.
Negation, Double, I - 34. G: 202

Non-contradictoriness. 1 1 •• 124,128 •

G 207ff , 2H

Non-contradictortnes], Proof of
1.8, 134. .19

Non-denumerable, 221, 267 ff.
nu, see Zen tymbol

Null, II-i34f, 0.262 f.


N

Null-content, 176
"Number", 285, 293 f., 295, 300,

304 f.. 31 1

Number, Cardinal, 139, 142 ff.,
326

Real, 11. i47ff.;G: 207, 220,305
Numerical expression, 1: 24,26,72;
11: 87; G: 205
functor, G : 205
predicate, G: 205
symbol (Numeral), 1: 14, 17, 24,
*6, 59, 73; G: 205
variable, I. 17; II: 84; G: 205


Object-language. 4. 160
Object-sentence, 277 f., 284
Open, I: ai, 66, G: iqa
Operable, G' 19a
errand, I; ai;G: 192
Opemor, I: 21. 23; II: 83
G. 191, J93


o

Operator, Deseriptional.22;G: 193
Ltmiied, I: 21 ; G: 191
Sentential, I: 30; 11: 92
Universal, 1: 2t ; II: 93; G: 193,
*96 f.. 231

OstensioR, ostensive definition, 80,
*55


P. see PTedieatt-r.aTiahli
P-. G: i8off, 184 f., 316
Perfect. G: 176
Phenomenology, 289, 305

Physical language, 149 f., ,78, 307
3lSff.. 3*2. 328ff.
t78, 180 f.
aymax, 57, 79 ff.

p”''’®"' *‘°*'t>onal symboU, 12. 45

+ r, Itedicate-expres.ion, IIj s,

87. *34 ff.; 0:188.19,


Predicate-varuble, II: 84; G; 195
Premiss, 27
Prwniss-elass, G: 199
Principal expression, P-symbot,

n_- 9* *^7

Principle of: Austonderung, 98, 268
Complete Induction, I: 32 f.,
38: 11: 92 f., 121
Comprehension, 98, 142
Contra'dicTion, II: 125; G: 203
Double Negation, I: 34; II: 125
ExcludedMiddle, 1: 34,48; 11:125
Cxtensionahty, 11: 92, 98
Infinity, II: 81, 97, 140!.
Limitation, 26S ff.

Reducibility, 86, 98, 142, aia


INDEX OF SUBJECTS


35*


Setecuon, II <izi
Substitution with arguments,
11 92 f 1 125

ProbabiLty, 149, 307, 3x6
Proof, 1 29 33^1 7S, II 94 f ,
G 17I

protocol sentence, see Sentenee
Pseudo-object sentence, see Sm-


Q, see Correiatiott
Qualitative defimtion, 80


R

Range, G 199 f
Realism, 301, 309
Real number, see Mtmier
Reduction, II 103 ;t
-Rftfaeftiw n 10s

Reflexive, 361, 363

Refuuble, I 38,11 G tyi,

T. *’5

Regresaire defimtion, 33, 68 ,

II 88,89

Regular te^eace, 148 {

s

€, see Senunee

f, see Sriientfal vanabU

S , 362 ff

la, see Sententtal tymbol
Schema, i Of pnnutire sentences,

I a9f , 11 91, 96
2 Of proofs and denvatioos,

1 33 f, 11 95 f

Saence.Logicof, 7, 379-384 33iff
Semantics, 9, 249
Senasiology, sanatology, 9
“Sense’ (‘Meaning”), 43, 184,
358 f , 285, 390

Sentence, I i4,35f,73,ll 88,

G 169 f, 332 f

Sentence, Correlated syntactical
*34. 236

Etementary, 166, G *95
Of identity, see Equation
ftimitrve, 29, f 30, 74, II
91 ff , G 171 See Principle
^ocol, 30s, 317 ff , 329 f
Pseudo-object-, 234 2851
Syntactical, 33 f , 284, 286 1
Unlimited existential, 163


I^eudo problem, -sentence, 353.
*78 f , 283 f , 289 ff , 304
309ff.3i3ff.319. 322, 324.

Psycdiolagism, 26, 42. 260, 27^.
389

Psy^ology. psychological language

X51. 3x5. 324

Q

Quasi syntactical, G 234, 23d ff,
256 f , 285 ff
Question, 296


Related, G 169 f
Relation, 260, 262
Relativistic nature of language,

186, 24s. 257. 299. 322

Replacement, ! 36 f , G 169
Resoluble 1 r 6. R, 46, 47.
11 94 >>3. G X7X
3 r lianguage, G 209
Resolution, Method of, 46, 47, i6x
Rules of Inference, 27, 29, 1 32,
II 94


Sentential calculus, 1 30, 34,

11 9tf>96

class, Infinite, see Infimte
frameworL, G 187, 191
fuDcboD, 21, 137, G 191
junction, G 200 f
Operator, see Operator
symbol, 1 84,0 269
variable, n 84 258. G 195
Sequence of numbers 148 f , 305
Senes number, 56 f , 60
Gfu. see Sententicd. function
Sg, see Sentential frameworh
Speech, see Mode
Gl, see Accented expression
' Strict" implication, 203, 232,
237, 251, 2S4ff
Gtu, see System of Levels
Sub-language 179, 225
Substitution r 22, 32, 36f, 73,
11 90fr 96. G 189, 19X f ,
293 f

with arguments, 92 f , 125
Substitution-position, ( place), I
22, G 192



35i


INDEX OF SUBJECTS


Substimti<3n->-»lue, G: 191. *93

Sj-mbol,

Symbolic Isnsuage. 3
S>-iiibols.Equ»l. 15

Gothic. 1 : * 7 f-; ”• Gi

lt>9

Ineompletf, 13S .

Numerieal.seehtfwmfflJ

Pnmim-c. I- 23. 7*. 77: ‘I:*®;

G: !7». 231
PrinapaJ. G *77
Sentential, see S'etilenfia/
Subsidiary, G: *77
SiTnrnetrical, G 361, 263


SjTKKiymoos. *• *•

I 3 o;G:I 76 . *S 4 .s^*

2. a languaffcs. G: 226
Syntactical correlation. 2 **
deaign*t<on, 154 *•, too
Syntax, *. 8
Syntax, Axjomauc. 79

DcsCTiptiw, see DesmptM
General, * 53 . * * 7 . ,
Phj-sical, see

Pure. 6, IS. 56 E, 78

Syntax-language. 4 . 53 . ‘ 53 .
G; 235

Synth«ic,l: 2 S, 4 of-. *o‘. ”5


Tautology, 44. 176; se*

Term-number. 55 . **».

Test (I'erifieaoon), 3*7 *> . 3*3
Tolerance. Principle of, 5 * *•.


3*’

Total content, *79
Transformince. O: SS 3 n
Trtnsformition nJei, s. 4 . *; * 7 * 1 -.
73 . 1 ^ 90 fT ; O . *63 ff-


TVansimt, C: sb*. sbj

Ttanslation, G.

Trarwposed, aee 0/ Sperai

••T*ut’*. ai 4 . ijb, *40
Truth-function, 24®

Ttuth-value tables »o. 20*

Typfi. 11: 84 ff.. 98 . * 37 *..

G »^{-. 249 . S98

Types, Rule of, ***


u


Ub-, aee the unprefiied nofl
Undefined, are Primitive rywWt:
cf

Unity of science, linguige. 286 f..

320 fi


Umeeraal. *« OperaWr, I
Univwaality. a*. 47 ff-. »®3
321

Unit'eraal tvord, 292, 3*6
Unordeied sequence, * 4 “ *•


SJ, see VimabU-fifreitton
r, see rflnablr
Valid, G. 173 f-
Valuation, II' 107, xo$
Value-expreiaion, I: 83
Values, ^urse of, G: 187 f., 24*
Range of, II : 90; C; 19*
Variable, I: | 6 ,ai: 11:84, lB 9 iT.;
G:I 94.*95

Variable, Numerical, aee A'r.


Sentential, see Sx.

Univcraal, *65

Variable-expreaiion, G: I 9 ‘
Variability-number, *98

Variant, Gi *93
Verifieation, aee Tni ,

certn. tc^yitnttioft-otnbcls. 1. w
Vitalism. 3 * 9 . 324

Sf. aee SentmtiaJ funiMn

ot. see 7 a»K-f»o"-f>'«^' G


n’ahifolse, *48, 30J


w


Word-language, 2, 8 , 227 f-


3, aee Ktonerual eapfern’oti
3, tee A’amenrof raruibU
55 f-. 77.

Zero-fj-mbol, 1 : 13 f., 17; G* 205


3 fu, aee P,’umfrieal functor
jpr, see Numerical pr«Araf<


iee CSummcai j-rc— -•

le Sumerieal ty-mboU


SvKTAcnCAi. sraiBOU (uted in connection with Gothic tym^l*'):

34; + . 34: 0 84; ( 5 S). see Suisntuiioni 1 : 0 . see Jieplunnni.




International Library


OP


PSYCHOLOGY, PHILOSOPHY
AND SCIENTIFIC METHOD


Edited by

C K OGDEN. MA

FiUov of Mag^daltnt Collide, Cambnjg!


The latea^aooa] Libmy, of vhicb OTet ooe bucdttd end fifty
Tolumee have aovr been pubtiahed, is both la quality tad quaoaty a
unique aehievemeat in tbs depanmeat of pubbshing Its purpose
IS to give evptessioo, in a coovenient form and at a moderate price
to the fcaiariablc developments wbcb have recently occurred m
Psychology us allied saeoces The older philosophers were
preoccupied by tnetaphysical interests which for the most part have
ceased to attract the younger mvestigators, and their forbidding
tenmnology too often acted as a deterrent for the general reader
The attempt to deal in clear language with current tendencies
whether in England and Amenca or on the Contmeot has met with
a very encoutagmg r ecep t ion, and not only have accepted authorities
been invited to explain the newer theoncs, but it has been found
possible to Include a number of onginal contributions of bgh merit


PuSlisBed by

ROUTLEDGE & KEGAN PAUL LTD

BROADWAV HOUSE 68 74 CARTER LANE, LONDON, E.C4

1958




INTERNATIONAL LIBRARY OF PSYCHOLOGI
PHILOSOPHY AND SCIENTinC METHOD

All prices are net
A. PSYCHOLOGY


CENTRAL AND DESCRIPTIVE

The Mind and its Place io Nature. By C. D. Broad. ;l ISs.

Thought and the Brain. By Henri Pifton. Translated by C. K. Ogde

; 1 .

The Nature of Laughter. By i. C- Gregory. JSs.

The Gestalt Thcor)' and the Problero of Configuration. By Brut
Petermann. Illustrated. ;l 5s.

Principles of Gestalt Psychology. By K. KofTha, D 2s.

Analysis of Perception. By J R. Smythies. ;1 Is.


PERSOMAUrV

The Psychology of Character: wih a Survey of Personality in Gcnen
By A. A. Roback. Bevtsed Edition. ;2 5s.

Problems of Personality: a Volume of Essays in honour of Mortc
Prince. Edited by A. A. Roback. ;t 8s.

Personality. By R. G. Gordon. ;1 3s.


ANALTStS

The Practice and Tlieory of Indhidua] Psychology. By Alfred Adi
;1 5s.

Psychological Types. ByC.G. Jung. Translated with a Foreword 1
H. Godwin Baynes. ;l 12s.

Character and the Hncoiiscioos: a Critical Exposition of the Psjeholo
of Freud and Jung. By J. H. wn der Hoop. ;l.

2



Tte DeTelopment of the Sexual Impulses By R E Money Kyrlc
;1

The Psycholog} of Consciousness By C Daly King Introduction by
W M Marston ;1


LANGUAGE AND SYMBOLISM

The Symbolic Process, and Its Integration in Children By J F
Markey 14s

The Meaning of Meaning a Study of the Influence of Language upon
TTiought and of the Science of Symbolism By C K Ogden and
I A Richards Supplementary Essays by B Malinowski and
F G Crookshank ;1 8s

Principles of Literary Criticism By I A Richards ;1 3s
The Spirit of Language in Cirilization By K Vossler ;1


CHILD PSYCHOLOGY EDUCATION ETC

The Growth of the Mind an Introduction to Child Psychology By
K Koffka Translated by R M Ogden ;1 12s
The Language and Thonght of the Ould By Jean Piaget Preface by
E Oaparede ;1 Is

The Child’s Concepbon of Physical Causality By Jean Piaget ;1 3s
The Child’s Concepbon of the World By Jean Piaget ;1 5s
The Child’s Concepbon of Number By Jean Piaget ;1 5s
Jndgtnent and Reasomng in the Child By Jean Piaget ;1 Is
The Ongm of Intelligence in the ChDd By Jean Piaget ;1 8s
The Child’s Concepbon of Space By Jean Piaget ;2 2s
Educabonal Psychology its Problems and Methods By Charles Fox
Revised Edition ;1 5s

The Mental Development of the Child By Karl Bubler I5s
The Psychology of Intelligence By Jean Piaget 1 8s
The Psychology of Children’s Drawmgs From the First Stroke to the
Coloured Drawing By Helga Eng With many line and coloured
illustrations Second Edition ;! 5s


3



ANIMAL PSYCHOtOOy, BIOLOGY, BTC.


TTie Mentality of Apes, wth an Appendix on the Psychology of Chim-
panzees. By W. Koehler. With nine plates and nineteen figures.
;1 5s.

The Psychology of Animals, in Relation to Human Psychology. By
F. Alverdcs. 15s.

The Social Insects: Their Origin and Evolution. By William Morton
\Vheclcr. With 4S plates. ;1 Ss.

Theoretical Biology. By J. von Ucxkuell. ;1 4s.

Biological Principles. By J. H. Woodger. ;1 IBs.


ANTTIROPOLOQY, SOaOLOQY, RBLIOION, BTC

The Tndiridual and the Community: o Historical Analysis of the
Motivating Factors of Social Conduct. By Wen Kwei Liao. ;1 3$.
Clime and Custom in Sarage Society. By B. Malinowski. With six
plates. 16$.

Sex and Repression la Sarage Society. By B. Malinowski. ;1 Is.
Religious Conrersion. By Same de Sanctis. ;1 5s.


B. pHaosopm’

PhiUsophlcal Studies. By G. E. Moore. ;1 10s.

The Philosophy of “As If”: a System of the Theoretical, Practical,
and Religious Fictions of Mankind. By H. Vaihinccr. Translated
by C, K. Ogden. ;l lOs.

Fire Types of Ethical Theory. By C. D. Broad. ;1 8s.

Speculations: Essays on Humanism and the Philosophy of Art. By
T. E. Hulmc. Edited by Herbert Read. With a frontispiece and
Foreword by Jacob Epstein. ;| Is,

The Mctaphjsical Foundations of Modem Physical Science, with special
reference to Man's Relation to Nature. By E. A. Burtt. ;l 8s.

Bentham’s Theory of Fictions. Edited with an Introduction and
Notes by C. K. Ogden. With three plates. ;1 Is.

4



fdeology and Utopia ao latrodoction to the Sociology of Knowledge
By* Lari Mannheim ;I 8s

Charles Peirce’s Empincism By Justus Buchler ;1 Is
The Philosophy of Peuce Selected Wntmgs Edited by Justus
Buchler ;1 15s

Ethics and the History of Philosophy Selected Essays By C D
Broad ;I 3s

Scnse-Percephon and Matter A Critical Analysis of C D Broad’s
Theory of Perception By Martin E Lean ;1 Is
WTiat is Valoe** An Essay m Philosophical Analysis By Everett W
Hall ;1 5s

Religion, Philosophy and Psychical Research Selected Essays By
C D Broad ;I 5s

The Structure of Metaphysics By Moms Lazerowitz. ;1 5s
Methods and Cntena of Reasoning An inquiry into the structure of
controversy By Rupert Crawshay Williams ;1 12s
Reasons and Faiths By R N Smart In preparation


LOGIC

Tractatus Logico-Philosophicus By L Wittgenstein German teat,
with an English Translation cd regard, and an Introduction by
Bertrand Russell, FRS ;1

Foundations of Geometry and Induction By Jean Nicod With an
Introduction by Bertrand Russell, FRS ;1 3s

The Foundations of Mathematics, and other Logical Essays By F P
Ramsey Edited by R. B Braitbwaite Preface by G E Moore
;1 3s

The Nature of Mathematics a Cntica! Survey By Max Black

Logical Syntax of Language By Rudolf Carnap ;1 lOs

The Conditions of Knowing An Essay Towards a Theory of Knowledge
By Angus Sinclair, OBE,MA,FRSE ;13s

A Treatise on Induction and Probability ByGHvonWnght ;I lOs

Bertrand RasseJJ s Coastmctioa of (be Extemtl World By Charles A
Fntz, Jiinr ;1 3s

Logical Studies By G H von Wnght ;I 8s
5



C. SCIENTinC METHOD

Scientific Thousht: a Philosophical Analysis of some of its Funda*
mental Concepts in the light of Recent Physical De\*clopments.
By C. D. Broad. ;1 12s.

Dynimic Social Research. By John T. Hader and Eduard C. Lindeman.
18s.

The Limits of Science: Outline of Logic and of the Methodology of
the Exact Sciences. By Lcoo Chwistek. Introduction and
Appendix by H. C. Brodie. ;1 12s.

HISTORY, Eta

An Historical Introduction to Modem Psjchologj’. By Gardner
Murphy. With a Supplement by H. KJuver. ;2.

The Historj’ of Materialism and Criticism of its Present Importance.

By F. A. Lange. With an Introduction by Bertrand Russell. ;1 15s.
Phllosoph) of the Unconscious. By E. von Hartmann. ;1 I5s.
Outlines of the Hlstorj’ of Greek PWlosophy. By E. Zeller. New
edition, re*WTitten by Wilhelm Nestle and translated by L. R. Palmer.
;1 3s.

Psjche: the Cult of Souls and the Belief in Immortality among the
Greeks. By Er^rin Rohde. ;1 I5s.

Plato's Theory of Ethics: The Moral Criterion and the Highest Good.
By R. C. Lodge. ;1 12s.

Plato’s Theory of Education. ByR.CLodge, F.R.S. (Canada). ;1 3s.
Plato’s Theory of Art. ByR.CLodge. ;1 5s.

The Philosophy of Plato. ByR.CLodge. ;1 8s.

Plato’s Phaedo. A translation with an Introduction, Notes and
Appendices, by R. S. Bluck. ;1 Is.

Plato’s Theory of Knooledge. The Thcacteius and the Sophist of Plato.
Translated, with a Running Commentary, by F. M. Comford.

Plato’s Cosmology ; The Timaeus of Plato. Translated, with a Running
Commentary, by F. M. Comford. ;1 12s.

Plato and Parmenides. Parmenides’ “ Way of Truth ” and Plato’s
“ Parmenides ". Translated with an Introduction and Running
Commentary, by F. M, Comford, ;I 3s.

Aristotle’s Theory of Contrariety. By John P. Anton. ;1 5s.

6



A LIST OF BOOKS PUBLISHED IN THE LIBRARY BUT AT PRESENT OUT OF PRINT


Analysis of Matter By B Russell

Art of Interrogation By E R Hamilton

Biological Memory By Eugenio Rignano

Chance Lore and Logic By C S Peirce

Child's DiscoTcry of Death By Sylvia Anthony

Colour Blindness By Mary Collins

Colour and Colour Theones By Chnsune Ladd Franklin

Communication By K. Bntton

ComparatiTe Philosophy By P Masson Oursel

Conceotnc Afethod By M Laignel Lavastine

Conflict and Dream By W H R Rivers

Conscious Orientation By J H Van der Hoop

Constitution Types m Delinquency By W A WiUemse

Contributions to Analytical Psychology By C G Jung

Creatire Imagination By June E Downey

Cnme, I^v and Social Science ByJ Michael and M J Adler

Dialectic. By M J Adler

Doctnne of Signatures By Scott Buchanan

Dyoamics of Education By Hilda Taba

Effects of Music. ByMScboco.

Eidetic Imagery By E R Jaenscb

Emotion and lusamty By S Thalbitzer

Emotions of Normal People By W M Marston

Ethical Relatinty By E Westennarck

Examination of Logical PositinsfiL By Julius Weinberg

Growth of Reason By F Lorutier

History of Chinese Pohtical Thought. By Liang Chi Chao

How Animals Find their Way About By E Rabaud

Human Speech. By Sir Richard I^get

Infant Speech. By M M Lewis

Integrahre Ps}choIogy ByW M Marston el a/

Invention and the Unconscious ByJ M Montmasson
Law and the Social Sciences By H Cams
LawsofFeehng ByF Faulhan
Measurement of Emotion By W Whately Smith
Medidne, Magic and Religion By W H Rivers
Menaos on the Mind By I A RKbards
7



Mind and its Bodj . By Charles Fox.

Misuse of Mind. By K. Stephen.

Moral Judgment of the Child. By Jean Piaget.

Nature of Intelligence. By L. L. Thurstonc.

Nature of Learning. By G. Humphrey.

Nature of Life. By E. Rignano.

Neural Basis of Ihought, By G. G. Campion and Sir G. E. Smith.

Neurotic Personality. By R. G. Gordon.

Philosophy of Music, By W. Pole.

Physique and OLiracter. By E. Kretschmer.

Pleasure and Instinct. By A. II. B. Allen.

Political Pluralism. By Rung Chuan Hsiao.

Possibitj. By Scott Buchanan.

Primitive Mind and Nfodem Qfllliatlon. By C. R. Aldnch.

Principles of Experimental Psychology. By H. Pieron,

Problems in Psychopathology. By T. W. Mitchell
Psychology and Ethnology. By W. H. R. Rners.

Psychology and Politics. By W. H. R. Rners.

Psychology of Emotion. By J. T. MacCnrdy.

Psychology of Intelligence and Will. By H. G. Wyatt.

Ps) chology of Men of Genius. By E. Kretschmer.

Psychology ofa Musical Prodigy. By G. Rextsi.

Psychology of Philosophers. By Alexander Herzberg.

Psychology of Reasoning. By E> Rignano.

Psychology of Religious Mysticism. By J. H. Lcuba.

Psychology of Time. By Mary Sturt.

Sciences of Man in the hLaUng. By E. A. Kirkpatrick.

Sclentiflc Method. By A. D. Ritchie.

Social Basis of Consciousness. By T. Burrow.

Social Life in the Animal World. By F. Aherdes.

Social Dfc of Monkeys and Apes. By S. Zuckciman.

Speech Disorders. By S. M. Slinchficld.

Statisdeal Method In Economics and Political Science. By P. Sargan
Florence,

Technique of Controrersy. By B. B. BogoslON-sky.

Telepathy and dalrroyance. By R. Tirehncr.

Theory of LegUbtion. By Jeremy Bcnlham.

Trauma of Birth. By O. Rank.