Logic and Knowledge: Essays 1901-1950

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LOGIC AND KNOWLEDGE

BY BERTRAND RUSSELL

ABC of Relativity Human Society in Ethics and Politics The Impact of Science on Society The Analysis of Matter New Hopes for a Changing World Authority and the Individual Human Knowledge: Its Scope and Limits History of Western Philosophy Logic and Knowledge The Principles of Mathematics Introduction to Mathematical Philosophy The Analysis of Mind Our Knou:ledge of the External World An Outline of Philosophy The Philosophy of Leibniz An Inquiry into Meaning and Truth Principia Mathematica ('leith A. N. Whitehead) Portraits f,-om Memory German Social Democracy Unpopular Essays Legitimacy Versus Industrialism 1814-1848 Unarmed Victory Power In Praise of Idleness Commonsense and Nuclear Warfare Philosophical Essays Why I am Not a Christian The Conquest of Happiness Sceptical Essays Fact and Fiction Mysticism and Logic The Scientific Outlook Marriage and Morals Has Man a Future? Education and the Social Order My Philosophical Deulopment On Education Freedom and Organization, 1814-1914 Political Ideals Principles of Social Reconstruction Prospects of Industrial Civilization Roads to Freedom Practice and Theory of Bolshe'l:ism

Satan in The Suburbs The Amberley Papers Nightmares of Eminent Persons Basic Writings of Bertrand Russell Bertrand Russell's Best

BERTRAND� RUSSELL O.M., F.R.S.

Fellow of Trinity College, Cambridge

Logic and Knowledge ESSAYS 1901-1950

EDITED BY

ROBERT CHARLES MARSH

CAPRICORN BOOKS New York

FIRST PUBLISHED 1956 SECOND IMPRESSION I 964 THIRD IMPkESSIO� 1966 FOURTH IMPRESSION 1968

T his book is copyright under the Berne Convention. Apart from any fair dealing for the purposes of private study, research, criticism or reviev.:, as permitted under the Copy­ right Act 1956, no portion may be reproduced by any process 1.c:ithout icritten permission. En�·uiries should be addressed to tf,e publisher.

Reprinted by arrangement with George Allen and Unwin Ltd. and The Macmillan Company, New York CAPRICORN BOOKS EDITION 197 l

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE ten essays in this volume represent work extending through fifty years in the life of one of the great philosophers of our times. All of these essays are representative, and several can be regarded as among the most important of his writings. None the less, only one of these papers has previously been available in a hard-cover edi­ tion, authorized by Lord Russell, and available through the normal channels of the book trade. Indeed, �most of these papers have previously been available only in libraries with unusually full periodical collections, and this in itself would serve as ample justification for reprinting them in book form. Two collections of essays with partly overlapping contents are all we have had, up to now, to preserve the shorter writings of Russell's most productive decades of work in logic, mathematics, and the theory of knowledge. Nothing included in Philosophical Essays (1910) or Mysticism and Logic (1918) appears here, and an examination of all three books is necessary for a comprehensive view of Russell's papers of the early years of the century. The period which marked the transition to the neutral monism of An Analysis of Mind (1921)-in other words, Russell's philosophical activity (apart from social philosophy) during and immediately after the 1914-18 war-has previously been difficult to study. The appearance here of three papers from those years, none of them previously available in an authorized edition, should serve to fill this troublesome gap in the chronology of Russell's available works. It is the editor's belief that what is ultimately desired is a com­ prehensive edition of Russell's shorter writings arranged on a chronological basis by subject, eliminating only journalistic pieces of limited interest. Such a project is beyond the means of a com­ mercial publisher, in all probability, but it deserves the attention of those interested in preserving in an appropriate form the writings which-for the most part-linked one of our most distin­ guished contemporaries with his audience. Selection has been difficult, and I do not expect everyone to agree with my choices. I have reprinted the three papers of Russell which are starred in Church's Bibliography of Symbolic Logic. These are technical but important. To include them I was forced to omit THE

V

Vl

PREFACE

a group of papers which appeared originally in French in the Revue de M etaphysique et de Morale which are still amo11g the best general discussions of the problems they raise. I regret that I had to make such a choice, but I do not regret the choice that I made. The reader who is not prepared to cope with mathematical logic will, nonetheless, find in certain of the other essays writing as lucid and readable as any of Russell's more popular works. I was introduced to Russell's philosophy by Professor Arthur H. Nethercot at Northwestern University in 1944. In 1951 I took a doctorate at Harvard University with a thesis on Russell's philo­ sophy, and since that year I have had the pleasure, from time to time, of discussing philosophical questions with Lord Russell him­ self. In the production of this book, I alone am responsible for its contents and the views expressed in the introductory remarks which preface each of the essays. Lord Russell has been consulted on all matters relating to the text of the papers, and these, to the best of my knowledge, are here issued in the form which he wishes to be taken as final and definitive. For this aid, and for many other kindnesses, I am greatly in his debt. The first paper has not been reset entirely: the greater part of the symbols are reproduced here by means of photo-engraving from the original edition. For that reason slight variations may be found in the typography of the English text and the symbolic material, since it has not been possible to duplicate exactly the type faces of Peano's Italian printer. There is no instance, however, in which this is likely to introduce an element of ambiguity. In reset­ ting the second paper we have followed the style of the Principia Mathematica rather than the earlier typographical conventions Russell used in the initial publication of this essay. It was Lord Russell's desire that we take advantage of the resetting of the paper to introduce these minor revisions. The dates given for the papers are those of initial publication. In most instances the paper was written in the same year or that immediately preceding publication. The scarcity of some of the papers now made generally available here can be seen from the fact that only one copy of the text of the lectures on logical atomism was known to exist in the whole of Cambridge, and when this disappeared from the University library during the preparation of the book it was necessary for me to borrow the missing material from the library of the University of

PREFACE

Vll

Bristol. I am greatly in debt to Bristol for the consideration shown me in making the original edition of these lectures freely available to me. Because of this kindness, future students of philosophy will be spared the inconvenience of inter-library loans or the necessity of resorting to theft. This collection was planned shortly after I first went to Cam­ bridge in 1953 and was eventually seen through the press during my second period there in 1954-56. I shall always associate Cam­ bridge with a wonderful sense of having achieved freedom from departmentalism, so that I might function, not as a philosopher, or a musician, or an educator, but as a thinker with the right to do whatever he feels to be important. Mr. Walter Beard of George Allen and Unwin undertook super­ vision of the production of the book. He has been obliged to deal with some vexing problems, and his contribution to the volume is not to be underestimated. I am grateful to him for the assistance he has given me, and for the effective but unobtrusive manner in which he attended to difficulties. Trinity College, Cambridge.

ROBERT CHARLES MARSH

CONTENTS page v

PREFACE

1901

The Logic of Relations

I

1905 On Denoting

39

1908 Mathematical Logic as Based on

57

The Theory of Types

1911 On the Relations of Universals and Particulars

103

1914 On the Nature of Acquaintance

125

1918

1 75

The Philosophy of Logical Atomism

1919 On Propositions: what they are and how they mean

283

1924 Logical Atomism

321

1936 On Order in Time

345

1950 Logical Positivism

365

I AM grateful to Mr. Robert Marsh for the industry, patience and accuracy which he has displayed in the following reprints of some of my less well-known writings. In regard to a considerable part of this volume he has undertaken the laborious work of collating versions which differed owing to difficulties arising from war-time censorship. Of many of the articles copies were not easily available and he has had much trouble in finding the material. Mr. Marsh has, in my opinion, shown good judgment in selecting what to reprint and also in his elucidatory introductions. It is not for me to judge whether it is worth while to perpetuate the record of what I thought at various times, but if any historian of bygone lucubra­ tions should wish to study my development he will find this volume both helpful and reliable. BERTRAND RUSSELL

The Logic of Relations In his autobiographical essay MY MENTAL DEVELOPMENT Russell says: 'The most important year in my intellectual life was the year 1900, and the most important event in that year was my visit to the International Congress of Philosophy £n Paris.'• He travelled to Paris with Wh£tehead, his former teacher and then colleague, and they were both struck by the skill shown in the discussion of mathe­ matical and logical problems by Peano and his pupils. Impressed, Russell went home to master Peano's works and particularly his notation. Its influence on the later Russell-Whitehead notation of the PRINCIPIA MATHEMATICA is fairly easily traced. 'THE LOGIC OF RELATIONS was written in I900 and published the following year. It was composed in Peano's notation, although it re­ presents work contemporary with the writing of much of the PRINCIPLES OF MATHEMATICS in which Russell uses an early form of the notation developed fully in the PRINCIPIA MATHEMATICA. Those unfamiliar with Peano' s symbols willfind a concise and admirable discussion of the system in Jorgen Jorgensen's standard rok, A TREATISE OF FORMAL LOGIC, Copenhagen and London, I93 I, Vol. I, p. 176 ff. The Peano notation is actually not difficult to read if one knows that of the PRINCIPIA MATHEMATICA, and the paper is therefore reproduced in its original form. Russell's first publication was in 1895, the year following his first period of residence at Cambridge. It is of no particular interest, although his mathematical investigations during the following four years led to publications that repay examination. However it is with i'his paper that we see clearly the appearance in philosophy of a crea­ tive mind of the first order, and on its publication (in the year in which he was twenty-nine) Russell's eventual position as a 'thinker of reputation' seems to have been assured. Asked what idea in the paper he now feels to be the most important, Russell replied 'my definition of cardinal number' -which here appeared in print for the first time. It was largely on the basis of this paper and• next but one in this collection that Russell was elected a Fellow of the Royal Society in

1908.

•The Philosophy of Bertrand Russell, Evanston and Cambridge, 1944 et seq., p. 12. I

THE LOGIC OF RELATIONS With some Applications to the Theory of Series

This paper first appeared in French in Peano's Rivista di Matematica [Revue de Mathematiques] Vol. VII, pp. u5-48, Turin, 1900-01. The translation is by R.C.M. and has been revised and corrected by Lord Russell.

CONTENTS

page §1

§2 §3 §4

§s

§6

General Theory of Relations Cardinal Numbers Progressions Finite and Infinite Compact Series Fundamental Series in a Compact Series

5 IO

15

23

26

30

of relations, which we find in the works of Peirce and Schroder, is difficult and complicated to so great a degree that it is possible to doubt its utility. Since they lack the distinction be­ tween€ and :>, these authors consider a class as a simple sum of individuals. For that reason, a relation for them is like a sum of pairs of individuals. It follows from this that the fundamental properties of relations are expressed by very long formulae of sum­ mation, the significance of which is not made very evident by the notation. It is, however, the logic of relations which must serve as a foundation for mathematics, since it is always types 'of relations which are considered in symbolic reasoning; that is, we are not THE LOGIC

3

4

LOGIC AND KNOWLEDGE

required to consider such and such particular relation, with the exception of those which are fundamental to logic (like E and o }, but rather relations of a certain type-for example, transitive and asymmetrical relations, or one-one relations. I point out in the present paper that it is possible to simplify the logic of relations enormously by making use of Peano's notation, knowledge of which is assumed in the following text. However it will appear that the logic of Peano is hardly complete without an explicit introduction to relations. Of the basic concepts, take for example the definition of function ( 1 ). The signs xu and ux, which appear on the right in that definition, are not made self-explanatory by the preceding text. The juxtaposition of two letters has not hitherto possessed any meaning other than logical multiplication, which is not in­ volved here. The fact is that the definition of function is not pos­ sible except through knowing a new primitive idea, that of relation. We may observe, for example, the following consequence. From the definition cited, and from P§20 P9 ·4, §22 P2 ·4, §23 P1 ·02, P2 ·o, we derive

This consequence shows that the notation adopted has need of modification. I shall give a more complicated notation from which we cannot derive an equivalent conclusion. I believe, moreover, that the introduction of relations gives occasion for a simplification and generalization of many mathematical theories; and it permits us to give nominal definitions whenever definitions are possible. In the following text, I have adopted some of the sumbols of V Schroder, for example, R, o', 1'. I have not succeeded in making myself conform to the rule of formulation, to put all symbols in a line, and in the case of relations I have had to distinguish between RP and RnP. Otherwise I have adopted all that is given in the logic of Peano, and at the same time the notation Elm suggested by Padoa [Rivista di Matematica, Vol. VI, p. 117]; however I have distinguished eu, where u is a class contained in the range of a relation R, from enu. For that reason the logical product of a class u and a class represented by a Greek letter is always indicated by enu, or 7Tnu etc., and not by eu or ue. [See§ 1, Prop. 1 ·33 ·34 ·35 ·36].

5

T H E L O G I C O F RELAT I O N S

*

§1 . G E N E R A L T H E O R Y O F R E L A T I O N S 1 ·o

Primitive idea : Rel = Relation RE Rel .:) : xRy . = . x has the relation R with y. Re Rel .�. Q= X3l 3:Y3(xRy) I Df Df Re Rel -�- e X3 l 3:Y3(yRx) t

·1

·2 1 · 22

If R is a relation, e can be called the domain of the relation R, that is to say, the class of terms which have that relation with a single term, or with several terms. I always use ( except for the relations which are met with in the formulary) capital letters for relations and the corresponding small Greek letters for the domain of the relations. In definitions ·2 1 ·22 the letter R is assumed to be variable, that is to say, a shall be the domain of a relation A, f3 of a relation B, etc. I regard 3 as a primitive idea of the sort that permits me to put this sign before propositions which are not reducible without its aid to the form XEa.

rel . xee .�.

ex

Di

= y3(xRy)

·3 t

Re

· 32

xee .�. (!X == y3(yRx)

·3 3

Re rel. ue Cls . U�(! .o. eu = y3 !3:U "' x3·(xRy)1 ue = Y3 lxeu .Ox .- xRyt

•34

·a �

U';)e .o. eu = y3 1 3:�3 (yRxH

·36

Re rel

•.j.

·:s

·6

R, R'e

·6t

·7

·7 t

[

Re

.o.

ue = y3Jxeu .ox . yRx t 3:() .=. 3:e

�R .=. ae . : R=R' . =: R'JR' . R'oR

rel .o:. RoR' = xRy .'Jx, y . xR'y

rel .'J. a rel � R'3 (xR'y .=. yRx)

. .

Df Df Df Df

Df

Df Df

Df

Pp

Re rel .�. rel " R'3(xR'y = yRx) e Elm yRx) .-o. · . xR,y -=· yRx : xR1y .:::. yR-x. ·­ Rt,Hi E rel � R'3(xR'y a: xR,y xn._y : �- R1 -Rt ]

·=·

·=·

·a Re rel .o. R = 1 rel " R'3(xR'y . =. yRx) ·s a rel � R3(e = tx . e = ty)

Df Pp

This Pp is of great importance, chiefly in arithmetic. It affirms

6

LOGIC AND KNOWLEDGE

that between any two individuals there is a relation which does not hold for any other pair of individuals. It does not need a hypo­ thesis, since x and y are not subject to any limitation. However one can limit it to the case where x and y are different, since the case where x and y are identical is derived from this by relative multiplication. ·9

[

Re rel .o.

xRy

R=R

·=· 11Rx ·=· :rRy ]

i

·nt R,Se rel . R = S .�.

o.

R = S .=. R = S

(} = ;

·=

xR.y .w. xR..tJ Ra , R,e rel .�.-. x(R 1wR1)y Df Ke Cls'rel .o. w'K R3lxRy .=. :1K " R'3(xR'y)I Dr w'K e rel Pp " !l J · u 6 R1 1 R1e rel .o;. x(RrRJy .=: xRtY . xRJj Df ·0 1 Ke Cls'Rel .:). � ' K = B.3lxRy R'eK .oR·. xR'yl Df ·YR �'K e rP-1 Pp

•93

=

·94

*· t2t· t

·=

. .

R. , R,e rel .o: x R.R, z = :1 y3(xR.y . yR,z) R,R. e rel

Df Pp

It is necessary to distinguish R1nR2 , which signifies the logical product, from R1 R2 , which signifies the relative product. We have R1 " R1 = R1 , but not in general, R1R 1 = R 1 ; we have R1 " R2 = R2nR1 , but not in general R1R2 = R2R1 • For example, grandfather is the relative product of father and /ather or of mother and /ather, but not of father and mother. · t ! Re rel .o. R' Df RR ·t 3 R,Se rel .o. (RS) SR

=

[ x(it§)y ·=· yRSx .:.:::.

=

�ZE

(yRz . aBx} . =. 3:zE (xSz zlty) ·=· zSRy )

Transitive = tr = relnlo(R�R) When one has R�R, one has xRy .yRz.:> .xRz. ·3 Re rel .o:. R' R .=: xRz .=. ay3(xRy . yR.i) ·2

Df

=

If R is a relation which yields a series (which requires that R be transitive and contained in diversity), R2 = R gives the condition for that series to be compact ( uberall dicht) that is to say it contains a term between any two of its terms. (See §5 below.)

7

THE L O G I C O F RELA T I O N S

.=.

· 4 Re rel .�: x-Ry -(xRy) ·� -R e . rel

Df Pp

(-R) = -(R)

I have not found the relative addition of Peirce and Schroder necessary. Here is its definition : Let R and S be relations : their relative sum is a relation P such as Df

xPy .=. · . x-Rz .a � . zSy : e.-Sy .o � xRz xPy

·=·

-3:(-ex " -oy)

·=· -: x(::.R-S)y l

yielding

* 3·

rel pp This Pp says that € is a relation. In this case I have been obliged to abandon the rule of using capitals for relations.

-�

·3

I E€

c = x3)ay3(xey) I e = X3 l3Y3 (ye:J;) I

'4 e 'J e

Df

[e = individual]

Of

( e

( yEe

= Cls-cf\ I

.o. yE Cls .o.

YEO

I

Xef:JI .=. �J.Z3(xez . yez) "51 ey .=. az3(zex • Zelj) • , · x,yeCls . axy ·6 ye Cls'Cls .0. ,.ly = X3(xe'y) ·7 Re rel : xee .'Jx. y3(xRy) = x :'J. R = i [ XE(l .ox : xRy ·=· YEX . ':> . ' , R = E ] ·s u,ve Cls-t /\ .'J. a rel I'\ R3(xRy . =. xeu . yev)

·?>

xe

( Prop 1·8 .o. �rel " P3(,u = n . w = ;)

Pe rel . ,u

= n . w = n .o: :

x w . yEv .-:,;,y. x(EPE)y

. ·.

x-m .v. y-Ev -=:e,y, -l x(EPE),lll

::�.

Prop ]

This proposition proves that, if u, v are two non-null classes, there is a relation which holds between all terms of u and all terms of v, but which does not hold between any other pair of terms. · s t ue Cls - ti\ .�. a rel "' RJ(e = u : xeu •=z • xRu) [ Prop 1 ·8 .o . 3: rel"'P3(n = ,u . ;; = ,u) :�.·. :uu .a:,;, x(EP)u : x-eu .oz . -x(sP)u . o . · . Prop ]

· s! ue Cls-t /\ .::). eu = 1 rel "' R3(e=u : xeu ·=:o • xRu)

Df

8

LOGIC AND KNOWLEDGE

The relation E u is the relation e for the class u alone. It is formed by the relative product of e with the relation which holds only between u and u.

* 4·

Primitive idea : 1 ' = identity This symbol is given in the notation of Schroder. I do not use the symbol = for the identity of individuals, since it has another usage for the equivalence of classes, of propositions, and of rela­ tions. 1

·! •3

l ' e Rel 0 ' = -1 '

Pp Df

o' is diversity. It is a relation as a consequence of Prop

·s t "3

! •33

·u •.&

(1J

1'

(1)

l'

1' o R...e rel . xRy . y 1 ' z .o. xR3' 1 '1 o 1 ' [ Prop 4·33 .a. Prop 1 1'

= l'

[ xl'y

·=· yl ';;c

1 2 1 .a. l ' a 1 '

·.u t " = l '

1 1 1 . Prop ·32 .a. xl 'y

Ill

2 ·5 .

Pp Pp Pp

121

;31 • Prop·32 .a. Prop_ ]

131

[ xl 'y . yl 'y .a. xl '!g :a. l ' :, l ' 1 tlJ , �r:op·34 ,:>. Prop ]

,l J

·n O' = O'

[ Prop·3·4 .::,. Prop J

- �;

R,Pe re l .o: RP a O'

.=. RP o O'

[ RP ;, O' .=: xRy . yPz .az,?1, $• xO'z :

= · -3 (x,y)1(xRy . yPx) =· -3 (x,y)1(yRx . xPg)

*

=: yRx . xPz .as,y,$, yO 'z

:=. RP a O'

)

6· t Nc+ l = Rel " R3 j xRy . xRz- .o.c . yl 'zl · t t l -1-Nc = Rel " R3 1yRx . z Rx .0 . yl 'zl .11 ·2 •3

. . Re

l -1-Nc Re Nc-1- l = l -1- l (Nc-1- l)'"'(l -1-Nc)

=

Df Df Df

Ne-+ 1 is the class of many-one relations. The symbol Ne-+ 1

9

THE LOGIC OF RELATIONS

indicates that, if we have xRy, when x is given, there is only one possible y, but that, when y is given, there is some cardinal number of x's, which satisfies the condition xRy. Similarly, 1 -+ Ne is the class of the converses of many-one relations, and 1 -+ 1 is the class of one-one relations. ·3 1

Nc + l = Rel "' R3(xee �- .,; • "ix e Elm)

·32 l -1-Nc = Rel "' Rs(xeg ·4 l 's 1 -+ 1 [ Prop 4·34·4 .o. Prop ] •:; ·6

-�.:i: •

ex s Elm)

.�. R s 1 -+ 1 Re l -1-Nc .�. RR n 1 ' RE 1 -1- 1

V

U

One does not have RR = 1 ' , since the domain of RR is the same as that of R, which is in general only a part of the domain of 1 ' . [ xRlly .o. a�3(x,ye e z) : Re l +Nc .o. e� e Elm . · .o. Prop ! •7

R,Sel -t- 1 .n. RS e 1 -+ 1

·s

R,Se Nc-t- 1 . ue Cls .

[ xrn . y 1 ' ,xg .o. yea .o:

i

u�e . "eu �a .

l ' ,11-;; .o. xRSz . · .o. °";;(-;u) 'J mi

;rm . xRSz -�- 3 a," .'J3(Xll!J . ySz)

*

6· t

RS = P .�. �iu) = ;u

. 'J .

:1:

yeeu . ZE �eu) :

Ss Nc-r- 1 . R = SS .�. R'�R . R = R

[ xR: ·=· 3 y3(xSy . zSy) .=Rw 3 V3(zSv . 1t:Sv) Se Nc-t-1 . zSy . zSv .o. yl ' v l i l . ) 2 ! . ! 3 1 . o : xR.; . zRw .o . a '!fl(xSy . 1.vSy) . 'J . xRw

·=·

a:Rz

Re rel . Ri�R . R = R

[ �su

f f' o R

·=· a y3(xSy . zSy) ·=· �y3(zSy . xSy) ·=· zR.x : o . R

1 4 l , l 5 1 .o. Prop ] ·2

: 'J .

·=· xee . u= ex :a:

R

;l1 :2: 18: 14: ;61

. �R .�. 3: Nc-1- l "' S3(R = SS)

·=· x,yee . u = ex = eY .o. xRy :a. SS o R R'oR , R = if . 3 R .o: XE(! .'Jz . XE 'i xSu . ySzt

3)

1 2 1 . Hp; l l .o: xR,11 .o. x,yeex .o. xS�x . yS�x .o. xSSy :o. RoSS I l l . 1 3 1 .o. Prop J

11: !21 131

LOGIC AND KNOWLEDGE

IO

P6 · 2 is the converse of P6 · 1 . It affirms that all relations which are transitive, symmetrical, and non-null can be analysed as products of a many-one relation and its converse, and the demonstration gives a way in which we are able to do this, without proving that there are not other ways of doing it. P6 ·2 is presupposed in the definitions by abstraction, and it shows that in general these defini­ tions do not give a single individual but a class, since the class of relations S is not in general an element. For each relation S of this class, and for all terms x of R, there is an individual that the definition by abstraction indicates; but the other relations S of that class do not in general give the same individual. This will be explained better in the applications, for example in the following u

section. Meanwhile, we can always take the class ex, which appears in the demonstration of Prop 6 ·2, as the individual indicated by the definition by abstraction ; thus for example the cardinal num­ ber of a class u will be the class of classes similar to u. § 2 . C A R D I N A L NU M B E R S

V

For the definition of eu see · -1 i

I

Prop 1 · 33 .

sim e r 1 ;i, .=. -(usimi,) . 3:Clsw3 "' (wsimv . w0u) Df

< . . > =/\ . .�.

"! 1 ;1,t 10V = 1-;; V 1;U v=/\ u sim v ·3 u [ Re 1 -+1 . tt=/\ .'J. u'Je . ue = /\ 1 1 1 . v=/\ . Prop 1'1 . 'J . Prop ] Oo = i;/\ tx sim ty [ §1 Prop 1 ·8 . 'J. Prop ] ·6 u sim ix .0: ue Elm

Df

Df

•4

·:s

[u sim

IX

·=· 3: 1 -+l

I'\

R3(uoe . eu

Rd -+1 . uoe . eu = ,x . 'J : Y,ZEU · 6 t u,v e Elm .o. u sim v [ tU Elm . xeu .'J. u sim ,x ve Elm . yev , 'J . v sim 1y l lJ . 121 . Prop 2 · 5 . 'J . Prop ] •7

l o = ,--; "' X3(ue Elm

•

= IX)

. 'J .

yRx . zRx -�- yl 'z

, 'J .

ue Elm 11 1 121

.ou . uSx)

Df

3· t Re rel . uoe . au .o :1 rel "' R'3J e'=u : xR'y .=d!,y. xeu . xRyl

:

Hp . §1 Prop 8·8 . 'J . 3: rel" R•3: e" = u . ? = eu xeu . yeeu - =x,y . xR"yl R "erel : xR "'!/ •.=y,z. XEU • yeeu : Rf'\R ,, = R, :'J : zR'y ·=· XEU . YE eu . xRy ·=· XEU . xRy :;l. Prop ] · t t Re rel . iwe .o. rel " R'3 lxR'y .=. xeu . xRyf 3Elm [ R, ,R.e rel "' R ' alxR'y .=. xm . xRy l .'J: xRtY .=. xeu . xR.'l -=-xR1y :o. Prop]

P · I • 1 1 point out that one can always find a relation the domain of which is a limited portion of that of a given relation, and equiva­ lent to the given relation in that portion.

Df ·!

u,u'e Cls . u sim u' .o. 3 1 -+ 1 "' Ra(� . u'=e)

[ u slm u· . 'J . 3 1 -+1 "' R1(ua� : t : . Prop3·1 .a. Prop ] •3

tt, t'' u' 'V'r- Cls . iwv sim u·.._v' c;.

UV

,eu = u )

= /\ .

u' V'

= /\ .

:ti . u Sim u' .

V

. Sim t,'

-�-

T HE L O G I C OF RE L A T I O N S [ u sim u' . Prop·2 .a. 3 1 ➔1

f\

13

Ra( tt=e . u·=i)

v sim v' . Prop·2 .a. 3 1 ➔1 "' R a(v = e' . tl=e') uv=A, . u ' v'=A. . P= IwR' .a. R=

?WV - �

u\Jv' . Pe 1 -t-l .:,. Prop ]

i�. =/\ :'J. "''Ir, e l -t- 1 ·=·

3 k"R3(xR.'.f) :a. 3: k,.. R3(:z:ee) [ x("''k).lJ R 1 ,Rs£k . R 1 0' R, .a . e,"i>i = /\ :a: 3 kl"\ R3(xee) .az . 1(,,.. Ra(xee) a Elm : a. · . Rek .

3·4 1

R, ,R,

XEe .a: x("''k)y ·=· xRy . · .a. Prop ]

Re l -t- 1 .

e,e e Cls'Ols : xRy .'Jx,y. x sim y :

-

-

x,x'ee. xO'x'.'J x,:z:'. xx' =/\: y,y'ee .yO'y.'J y,y'. ee'=/\ :'J. w'e sim "''e [ xRy ,-;,z,y_ xsim y :a: :tRy .a. 3 l ➔l "' R' 3(x=e' . r-e')

XE() .az . Rz e 1 -+1 "' R'3(x=Q' . ,ex

�') : k= 1 -t-l

I'\

R"a l 3:e "' xa(R''l '&.i ) 1 :

P = 1,i k . Prop 3·4 :a. Pe 1 ➔1 . tr=\J'e . � v'() . a . Prop ]

This proposition gives the foundation of arithmetical addition, in a form which permits the addition of an infinite number with finite or infinite numbers. 3·:s

ue Cls . x,yeu .o. u-ix sim u-ty

( u-,x-,y sim u,-,.-�-,y . ,x si m ,y . Prop 3·3 .a. Prop ] · :s -1 n,1Je Cls . u sim v . xeu . yev . ::) . -zt•tx sim v-ty r Re 1 -t-1 . u=Q . v

g . xRy .a. Prop

111

Re l -t-1 . u:::e . v � . -(xRy) .a . · . 3: "" ze(xRz) : xRz .� . yO'z . · . a . u-,x sim v-,z 121 Prop3·5 .:,. v-,z �im v-,y t3 1 : 1 : . : 2 : . : H : .a. Prop ] · :;! 11,re Cl� . xeu . yev .o: tt sim ·v u-ix sim v-ty

.=.

·t

m,ne'";; .:). m+n = 1;," xgfuSm . vSn . uv=I\ .'Ju,v.

This definition depends on Prop 3 ·3.

Sxl

tMJ ' Df

·2

==

LOGIC AND KNOWLEDGE

k'J-;, .'J. Ik 1a- � p31ue Cls!Cls . u sim k : x,yeu .';)x,y. xy=;=A : � 1-+ 1 "' R3(u-e . � : xeu . �Ry .gx. xSy) :'Ju. ,iu Spl

Df This definition depends on Prop 3 ·41 . It is important to note that this defines the sum of a finite or infinite class of finite or infinite numbers ; but it is necessary for all these numbers to be different, because otherwise, one cannot define them as a class of numbers but only as numbers of classes. For the case where there are some equal numbers in the summation, different considerations are required, and especially multiplication, which I do not develop here in order to avoid excess length. •3

1 aO'Oa

[uS� a .a. ue Elm : vSOa .a. v=f\ : f\-e Elm :a·. Prop]

This proposition proves that for any relation whatsoever of the class s, 1 a differs from oa. ·.i

•,u

xea .'J •. x+la = 1;� y3JuSx . z-eu .'Ju,z. twtz Syl xe;-tOa ,'J·.x-la = 1; � Y3luSx . zeu .'Ju ,z. u-izSyl

Df Df

This definition depends on Prop 3 · 5 1 . 4·tS xe ;-tOa . 'J: x-la l'x

.=. a,l�ro+la

[ 3'-la l ' x . uS x-la . vSx .a. u sim v 1 1 1 . Prop 3·52 .a. · . z-el/. . w-ev .:::): u sim v :,: x-la 1' x x 1 ' x+la ]

·=·

0

[

6

· =· w,z sim wna •·.

111

. .

xe o-tOa .'J: x-- lo O' ro = x O' x+la

·o . Transp .a. Prop ]

Propositions · 5 and ·6 prove that if the number of a class is identical with the number of a class which is obtained by subtract­ ing a term from the given class, then this number is also identical to the number of the class which is obtained by adding a term to the given class, and vice versa. Since we have proved (4 ·3) that I a is different from oa, we have the means of proving that in the class of numbers which obey mathematical induction, starting from oa, two successive numbers are never equal. To develop this subject, it is necessary to examine the theory of progressions, that is, of the series of which the ordinal number is w.

THE LOGIC OF RELATIONS

§3.

*

1.1

PRO GRESSIONS

= Cls "' U3Jal -+ 1"' R3(uoe . eu OU . a u-iu :

w

se Cls . a su-eu . e(su)os . Os . uos l

Df

This is a definition of the ordinal number w, or rather, if one wishes, a definition of the class of denumerable classes. The ordinal numbers are, in effect, classes of series. The class w is the simplest of the classes of infinite series. Since the definition does not presuppose numbers, it will be good to give to this type of series a name which does not imply numbers. Therefore I call this the class of progressions. Here is the definition in words : w is the class of the classes u such that there is a one-one relation R such that u is contained in the domain of R, and that the class of terms to which the different u's have the relation R is contained in u, without being identical with u, and which, if s is any class what­ soever to which belongs at least one of the terms of u to which any u does not have the relation R, and to which belongs all terms of u to which a term of the common portion of u and s has the relation R, then the class u is contained in the class s. Rel-+ 1 .

•t t

e o e . a e-e. o ·

= u3t uoe . eU 'JU

a' u-eu : seCls . a su-eu _- e{su) 'JS .Os . uosl Df we is the class of progressions of which R is the generating relation. W(!

•

Df ·1 2 uew . o . Relu. = 1 -+ 1 n R3(ue we ) ·t 3 Induct .=:. usw . Re Relu . o : se Cls. a su-eu . e (su) 'JS .Os • uos Df ·2 uew . .Re Relu . o . u-eu e Elm [

XE

u-eu .

,'I=

: 1 1 . Induct .a.

Re 1 -+ l . e o e . •3

011

IX V.

eu .a. e(su) as . :[SU-eu

uas .o.

ua tX-.J(!U

ae-e. uewe . o ::

= 1(u-eu)

= =

•3 1 xe,t . o . seqx tea; ·-1 Pe Rel . ,l. P0 11 l '.:r ·ts Pe Rel . ue" . o . pgcqn

-�-

u-eu a,x .a. Prop )

111

Df

= p,, p

Df Df

16

LOGIC AND KNOWLEDGE

P1 ·4 ·5 define by induction the finite powers of relations. This definition is effected by means of the terms of u. In the theory of progressions one cannot do without the powers of relations ; there­ fore, if one wishes to make that theory independent of numbers, it is necessary to define the powers in a manner which does not introduce numbers. The symbols 1 'TT signifies identity in the class TT, and the null relation everywhere else. See §2 Prop 3 · 1 2 . Df

·& lu = 1 e (O�) ·7 Pe l+l. .o. -Plu _:_ P [Plu =1 'nP=P ]

l·s Pe 1+1 . aeu . n . Pa e 1+1 (p()ue 1-+1 Pd -+1 . §lProp 5 ·7 .a: aeu . pa E 1-+1 .a . Paeqa El -t-1 1 1 1 . 12 1 . Induct .�. Prop ] ·st xeu . Ou R xz . � . xl 'z [ xl 'Ou . Ou Rz z .�. zl 'Oii 111 seq :rseqx 111 . : 2 1 . Induct :a. Prop ) Ou Rz x .�. o,. R 121 ·st 1,ew . R'eRelv . xeu . Ov R'zz • o . ztv ( xl'Ou . Oii R'2: z .a. zl '011 .�. uv :11 seqx 0, R'z z . ·uv .a. 011 R' seqz . seq z EV t ] . . 1 .a. Prop Induc I � l 1 1 1 · 9 vew . R'e Reh, . o . � 1+ 1 "" P3j u=n . � : x,yeu . xRy .o x,y. 1 ; x R' ,;;yj

This proposition affirms that two progressions are always two similar series, that is to say that one can find a one-one relation of which the domain is one of the two progressions, and of which the converse relation has the other progression for its domain, and which is such that terms which precede in one series correspond to terms which precede in the other, and vice versa.

=

[ § 1 Prop 1·8 .a. 3: rel "' Pu3(n0 1011 . ;; = iOv) Prop 81 .a: XEU . Ou Rz z ;a. uu Prop 82 ;a: xau .· O, R'z z' .a. z'ev

1 1 1 . 1 21 . 1 31 .a: XEU . iRz P0R'z z ' .a. uu . z'EV . zl'x §1 Prop o · 7 . §3 Prop 1·8 .a. Rz :PoR'z e 1 -+1

111 1 21 181 141

ll> I 1•I . :r»I • Q= rel "' F1:3u "' xa(F = Rz P0 R'z ) I . P = \l'Q .a.·.

Pa 1 -+1 . u::. . r-:';;: x,yeu . xRy .a.1:,,. iixR;i;y .·.a. Prop ]. l·H u' sim u .. � . u'ea,

In this proposition we prove that any class similar to a progression

T H E L O G I C O F REL A T I O N S

i s itself a progression. I f P is the one-one relation between u and u ', and R the generating relation of u, then PRP is the generating relation of u '. ( u'simu .:,. 3 1 ➔1 " P',(wa' . ;;u =u')

Pa 1 ➔1 . - � . ;-u =u' : nu .:>e. z'= i'iz . aeqz'

=

=�-

1_1 1 a(aeqz)

x'Px . xR scqx . seqx P seq z' .:,. z' PRP seq (I/

;21

Hp : 2 : . .PRP =R' .:,. R'r 1 ➔1 . e'u �u• Hp

:a1 . Xo'= ,;o.. ,:,. :co'= ,u�'u'

Hp 1 3 1 . IIE Cls . ,u'-e'u' u . e'(u's) � ,o.·. Ou (PE)s : x(Pe)u's .'J.z • seqx (Pr) u'• 10: • Induct .'J: a;eu .'J.z. �PE)u's Hp 161 . 1 6 1 . Pe 1-t-1 .'J: x'eu' .'J z'. z'u 131 • 1 4 1 . 1 7 1 .'J. Prop ]

*

2.

1

Ra l+l .

e u e we

·1

=

e

'J

e.a

RRR [ Propl ·91 � R' I l l . R'� .'J. Prop ]

(!-e .

ue rop 5·7 .�. (Ria )�eqb a 1-t-i i.1 1 . 1 2 1 . Induct -�- Prop ] 1

121

· n a,b,xeu . o . u " y3Jx(Ra-)b y l e·Elm [ Prop 2·41 . a. Prop ] · 4 3 a,b,xeu . o . x+ab 1 u " y3Jx(Ra )b YI Df Ort +ab ab • -64 Df ·-1 ?S ab l'c· . o . x+ab 1 ' x+c [ lndue;t ] · .& G ab O'c .o . x+ab O' x+c [ Induct ] · -1 1 ab l 'c = x+ab 1' x+c[ Prop 2·45·46 .a. Prop ]

= =

. .

( a+ou = m "' x1(aR011x)=a ] [ Prop 1·81 .J. Prop J

THE LOGIC OF RELATIONS

=

2· S a,beu .o. Ra Rb Ra-f-b [ Ra ROt• = Ra = R11 + 0.. Ra Rb = R11+6 .a. R11 Rseqb = R" Rb R = R-+bR= Reeqf•+b) ,u'"'�l 3 y,(aR b y . yR:c) I a + seqb= m '"' xa(nReeqbz) ,u '"' v: 3u '"' (a,y)a(Ou R11 a . aRb y . y&-r) I : a : . Hp: 2 1 .a. a+scqb IUl"\Z1l 3 Ul"\y1(0u R11 +6y . y&-r) seq(a+b) ; 2 1 . :4: !a. R 11 R b = &11. + b .a. Ra Raeqb = R11 + 1.,qb 1 1 ! • 1 0 ! • Induct .a. Prop ]

=

=

= =

Ill 121 1 31 141 161

=

· H t a,b,xeu .o. (a:+a) +b x+(a+b) [ (x+a) +� = • u'"' u:3 u l"\y1 (:cR a y . yR) z) : = m l"\ Z.1(:cRia Rb z) ; 1 : . Prop 2-r> .�. (x+a)+b = ,u l"\ ZJ(xR11+6z) = x+(atb) ]

11:

=

·ts! a,b,xeu .�. x+a+b (x+a)+b Df · H 3 a,beu .o. a+b b+a 111 Ou +lv = lu = lu +Ou [ Ot, +o., = Ou +ou 121 a+l .. = l u + a . a : seqa+lu =(a+l � ) +lu = (lu +a)+lu 13: . Prop 2·52 .a. seqa+l u = lv-4e(4+ lu ) + tu = l u + seqa : 4 : H p 1 31 . 101 1 21 . 1 4.j . lnduct . a . a+l t. = l u +a 18: • a+b= b-f-a .a. a+seqb a+b+lu = a+lu +b = 1 u +a+b = lu +b+a = b-f-lu +a =seqb-f-a 16 1 : 1 1 . 1 6 1 • Induct .a. Prop ]

=

:a:

[

·6

=

aeu alu

=

.o. alu

=a

IUl"\X3 I Ou (Ra )111 :cl

=

IUl"\:C3 ! Ou Rea :cl

·60 a 0u = 0ua = a ·6 t a,beu .o. a(b+ lu )= ab+a [ a(Ou + lu ) = alu = a = aOu +a a(b-f-lu )= ab+.i .a. a(seqb-f-lu! ) =Jul"\:cal O u (Ra )a�qb-f-luz

= Ou +a = a ] :1!

= IU l"\ Vl 3Ul"\ya(O u Rt»fay . yRo :c) I = •u '"'nl O u Rab-f++ax l = IU l"\VI O u Rcaaeql>-t-a:c I = aseqb+a I2I . Induct .a. Prop -] 1 1 . 12 1 1 ba+ a " 6 t t a,beu .o. (b+iu )a [ (b+lu ) Ou = Ou = bOu + Ou 11 1 (b+l u )a = ba+a .a. (b+lu )(a+lu ) =1U l"\ V I Ou R(6+1u) (ca+l�)x l = IU l"\:caj3y1(0u R(6+1 u)cay . yw,+lu z) j = 1ul"\:c11 3y1(0u R� . yR6+1u :c) I ba+a+b+lu ba+b+a+lu 121 Prop 2·61 . a . ba+b+a+1 u =b(a+1" )+a+lu 1 31 1 2t : ; 3 ! .a: (b+l u )a =ba+a .a. (b+l u )(a+l " ) = b(a+l u ) +a+lu 1 4 1 l l l . j 4 j . Induct .a. Prop ]

=

=

·=

LO G I C AND KNOWLED G E

20

2·6!

a, b, ceu .o. a(b+c)

= ab+ac

111 a(b+Ou ) = ab =ab+aOu a(b+c) = ab+ac .� . a(b+c+lu ) =a(b+c)+a = ab+ac+a 121 Prop 2·61 .a. ab+ac+a =ab+a(c+l.u ) 131 Prop 2 ·53 .a. ab+ac+a =ac+ab+a= ac+a(b+lv )= a(b+lu )+ac 141 ;2; . 1 3 t . 1 4 1 .a : a(b+c) = ab +ac .�. a (b+c+ l u ). = ab+a(c+lu ) .::a(b+lu ) +ac 161 1 1 1 ; 1 5 1 . Induct , � . Prop 1

a,b,ceu .o. (b+c)a = ba+ac

·6.3

[ (b+c)Ou = Ou =bOu +cOu 111 (b+c)a =ba+ca . a. (b+c)(a+l u ) = (b+c)a+b+c =ba+ca+b+c =ba+b+ca+c 121 Prop 2·61 . �= ba+b = b(a+l u ) . ca+c = c(a+lu ) 131 t 21 . 1 31 .a: (b +c) a=ba+ca .a. (b-f-c)(a+l u ) = b(a+l u )+c(a+l u ) 1 4 1 1 1 1 . 1 4 1 . Induct . a . Prop ] · 6.t.

a,beu .o. ab

= ba

[ aOu = Ou =Ou a ab = ba .a. a(b+lu ) =ab+a = ba + a =(b+l u )a I l l • 1 2 1 . Induct .�. Prop ]

f11 121

We have now proved the formal laws of addition and multiplica­ tion : the associative law of addition in P2 ·5 1 , the commutative law of addition in P2 ·53, the distributive law in ·62 and ·63, and the commutative law of multiplication in ·64. The associative law of multiplication results immediately (as for all relational products) from the same law for the logical products. In all that precedes we have never presupposed numbers : the entire theory applies to every progression. Hence follows in a general form all the arith­ metic of finite numbers.

*

3. Re 1+1 . �e . �e-tl- uecne . a,b,ceu .o ::

· t Pe 1+1 . yPz . xPseqaz .o. xPa y [ ::c p■eqa z .�. a W'\ w,(xPa w . wPz) Pd+l .. �. w1(wP1) �Elm 1 2 1 . yP.1 .�. y.w,(wPz) .�. Prop ]

· t t Pe 1+1 • xPy . xPseqa z .o. y Pa z p■eq'J = P*l" = ptv-t,a = ppa I l l .� : zpteqaz . 31Dl(zPw . . wPca z)

=.

�1 1" 121

Il i 121

121 • � , Elm . zPy .�. y,w,(xPw • 10PII z .�. Prop ] · t t xeeu .o. a:-1 u= 1 u1,y1 (sequ=a;)

Df

THE L O G I C O F RE L A T I O N S 3·1 � u " a3(aBzb .w.

( 0. -Rb b

21

b:Bza) Ill

eu

3 " z�(aRs b) . Prop 8·11 .:,. z-1• • " " ll'(seqa Rr b) 1 11 aROlfb .a. bRluaeqa .:,. 3 u " y,(bRr seqa) 181 3 u " za(b&z a) • uu " :i:a(bRs a) .:,. bRaeqsseqa :,. 3 u " y.(bRI seqa) Iii 1 2 : . ; 3 1 . ; 4 1 .a. 3 u " za(a&J b .w. b&z a) .:,. 3 " " y,(seqa Rr b .w. b Rr -aeqa : 1 1 . : 0 1 Induct .:,. Prop ] · ! t Ou e u " X3(aRx b .w. bRz a) .o. al 'b ·H

Ou-e u " X3(a&x b .w. b&za)

•3

a>b .=. � eu " X3(bRx a)

.o. � eU".X3(a&zb .w. b�a) Df

. . � qu

" X3(a�b) ·at ab .w. ab .o. -(ab . a. Prop ] ·o t M' == M To avoid confusions, I have designated by M the relation of be­ ing smaller among the rational numbers. We wish to demonstrate that this relation is equal to its square, which proves that it gives rise to a compact series. In §5 we shall develop the general theory of these series. •9

• 5. R£ 1+1 . �e ·t +a = Op a• ·1

ia = (Ra)

. :1 e-=e . uecue • a,b,c,deu. Df

·t t

.o : :

-a = Op1i• l lnduc, )

Df

T H E L O G I C OF RE L A T I O N S

·a + re = te3J:1 "" ya(x ·a I -tt = xal:1 t� ya(x •31 +u = +u w -u ·.&

q,q'eru .a:. q(D/C)q'

l' +YH l ' -y) I

. = : q= a,',: . q'= IJ/r,

-« e1 1t '"' (;'C,1J,�)3(ri=:nz • c=JJ; :w: 11=-c� . c=JJ;) .�,. a+r1=b ·4 t +d/r, OJ>o/C ·n -d/c Df Op •:i +r u x.� )3: If" (1J, Z)3(;c= +y/.;) .�t -ru ."C3l3 u,�1Jj,Z)3(m- -y/�) t

= = =

=

Df' Df Df Df Df Df Df

+ r u is the class of positive rationals, which are operations on rationals without signs. The classes u, r u, + u, + r u mutually exclude each other : no term of one of these four belongs to any of the three others.

*

§ 4 . F I N I TE A N D I NF I N I TE

=

Cls infln Cls "' u3I 3 U"'X3(u-LX sim MH Cls fin Cls - Cls infin Cls infin = Cls "' u3lxeu .0.11 . u - ix sim u l

t • ·J

=

·t t ·! [ §2 Prop 3·5 .-a: x,ym . -;> . u- ,x sim u- 1y :'J. Prop ] ·! 1 Cls infln Cls '"' u3 I U•tx simu , ( Prop 1 ·2 . xeu .'J. u- ,x sir.i u x-su -�- u-,x u .-a. u-,x simu 1 1 1 . 121 . 'J . Prop ) ·H Cls fin Cls '"' u3 lxeu .o .i: . -(u •tX sim 11.)l ( Prop 1 · 1 ·11 . 'J . Prop ] ·3 ue Cls infin . x-eu .o. u "' tX e Cls infin ( Hp . ym .'J. u-ty "Rimu 1 1 1 . §2 Prop 3·3 .-a. usim w.11x . 'J . Prop ] ·3 1 u "' t.x e Cls fin . x-eu .o. ue Cls fin [ Prop 1 ·3 . Transp .'J. Prop 1 ., ue Cls . 't-ftlll t.x e Cls infln . x-eu .o. ue Cls infi n ti w ix e Cls infin . x-eu . 'J . u "' fX simu ( il I . yeu . §1 Prop 3·51 .�. ti sin: u-,y . 'J . Prop · .i t ue Cls fin . x-eu .o: U\JlX e Cls fin ( Ptop 1 ·4 . Trnnsp -�-. Prop ] ·� 1te Cls . :,:-en .n: ne Cls fin . =. lMX f, Cls fi n [ Prop 1 ·:n ·-H . 'J . Prep 1 ·u /\ e Cls fin l UE Cls infiu . ;> . au ; i) . Prop 1

= = =

Df

Df

;1: :51:

:1 :

:11

L O G I C AND KNOWLEQGE

24

t·& t Elm o Cls fln r ua Elm .� . 3Zl(u= ,z) : Prop 1 ·41 . A• .Cle ftn =�- Prop ·7 ue Cls fin .o. :a:-u 2. S£s .o: :

*

•J

= ;(Cls infln) ; fin = ';;(Cls fin) infin ; fin = xe �tin .=. x+ la e ; fin xe ;fin .=. xO'x + l a

Pp

; infln

·t t ·t ! ·!

o- ;

Df Df [ & Nc-+1 .o. Prop ] [ Prop 1 ·5 .o. Prop l

·t t l Prop 1·22 . §2 Prop 4 ·6 ,-;,. Prop ]

·3 Ra = 1Rel ·3 t Ra e· l+l •31 Ra O O'

. .

+

"' �lxRy = xe � fin • y=a; l a I

[ §2 Prop 3·o2 . Sr Nc+l .o. Prop [ Prop 2·21 .o. Prop ]

· 3 3 (lo = ;; fin [ u e0 • t& .o. ui Cls fin

1 1 1 . Prop 1 ·7 .o. 3-u .o. 3e0 z .o. Prop ]

·3'

1

Df

11(

ea =; fin • tOo

l :u-; fln . t& . a.-O'Oo .o. 3u .o : y,u .o. u-,y s,eo :z: :o. Prop ]

·a� ; fin e Cls infln

*

·1 •1t

3.

Seg . g: : se Cls . Oa es . (lo (s � Induct Prop a·t

=

;; ftn) �s .g. ; fin .:> s

Pp Df

One may, if one wishes, define the finite numbers by mathe­ matical induction, and take as Pp definition 1 · 1 . But I have not succeeded in deducing one of these P's from the other. If one has defined an infinite class by the property of containing a part that is similar to itself, one cannot succeed in demonstrating that the part which one obtains by removing a single individual is similar to the entire class, and this has fatal consequences for the theory of finite numbers. If one then defines an infinite class by the pro­ perty of remaining similar to itself when one adds to it a term which does not belong to it, one excludes the class of all individuals [the universal class] , since one can add nothing to that class. For

T HE L O G I C OF RE LAT I O N S

these reasons I have adopted the definition 1 · 7 and 3 · 1 . 3·1

1 ·1

with the two Pp

; ftn e a,

( Prop 2·3·31 ·33·3-&·8·1 . §3 Prop 1 ·1 . .l. Prop ) •3

·at l

ofin = Ols "' U3 t:1s "' S3(U = oftn ) I o fin = co

Prop 3·2 .o.

o fin o

Df ;1:

a,

Ri 1 ➔1 . u= e . ;tln = ;. P= RRoR .o. u, a>

§2 Prop 1 ·54 .a. ue

o fln

t2:

,.

1 31

1 1 1 . 1 21 . pl: . .l. Prop ]

We have now proved that any class similar to the finite cardinal numbers is a progression, and vice versa. From that we deduce that all the consequences of §3 apply to finite numbers. ·.& ue Ols fin .o. 3 OJf\ V3(u o v) ( '!=/\ .a: vea, .au . uav §2 Prop a•r, . §3 Prop 1 ·91 .a: ueElm . v'eco . ztv .o. v'-1:i: ""' 10 .o. 3: c.o,"\ va(uav) u, Cls fin . vec.o . u-av . ttSx ._ y-eu .-a. tA,JIY , Cls fin . tAJty S x+la . u.v,y a Vvty Prop 3·31 . Prop 2 ·35 . Prop 1 · 1 ·3 . §3 Prop 1 ·91 .-a. w,y ac.o i a: . :4 1 .-a : ue Cls fin . vu, . U'JV . uSx . y-EU .-a. tAJty E Cls fin . ?Ntyeo, . tAJty a V.Jty . "'"''Y S x+la 1 1 1 : 1 2 1 . 151 . In duct :-a. Prop ] ·, t

ue

Ill

12, 1 81 141 IDI

. . y. Prop:)

2·s I veOls • rote . :1�,r • au.iv .o. m, e pi, ·6 v,v·e pu .o: rov' .w. v' o v

","'•pu .

3-v'-v ,i). 3-v'"v; ,:,: :UV .:>;1. 3:v'"'Y• (�Pv) : h (1) • t p u ,:,: :uv ,:>.z . xiv' ::>. v:,v' (9) • (8) .:>. Prop (8) 3:v-v' .:>. "':, t:

[ .

( l) (2)

For the definition of v1r, see § 1 Prop 1 ·34. ·61

v,de pu

·=·

·1 tiTv, ·7t Ts Bel [

.c: vov' .w. v'ov , , , v, v epu . rov . v-=o

§1 Prop 8·82 .;,: ,,, ,,'1pu

(-) ,Rel

· = · vi,;i HV' p

(3)

(1) . ·c2> . (S) . §1 Prop 1 ;98 . Prop

Df (:>) ,Rel

(1)

2·o .:,. (ipua

pu)

I'\ (:>) " (-=) ,Rel

(2) (4)

The P's (2) and (3) of this proof are the Pp, which we should have had to introduce in § 1 , if we had wished to make a complete logic ; ( 2) affirms that the inclusion of a class in a class is a relation, and (3) affirms that the equality of classes is a relation. ·11 ToO' • T'aT T' = T

•73

l

tfrv' .:>. 3v'-11 .:>. 3,rv'-v .:>. 3v' " (:c,y), (:c,y-i" . :t'O'y) (1) . �P.1/ .:>. "T.1rJ: . dr"' (2) (1) . yPz .:,. tfr1ry . nyTv' (1) . (2) . (8) .:>. Prop

·M

1

pu 8 T

[

Prop 1 · 1 .2·71 ·72·73 ;:>. Prop

(1) (8)

1

We have now proved that the class of lower segments is a com­ pact series by reference to T. Similarly it is proved that the class of upper segments is a compact series.

tit

= i, :n::

3. Pe Bel . PoO' . P'-P . lt= u: xeu .o. nxwu»wi'x re pu .o. ro p,e I'\ X3(.xT r.) ( "11 ;, " • r.,pu .:,. Prop ] •1 ·t re e Cls . ID o plt •.J. Trc ptt " xaiz ID " ya(m'ry)I ·H ,,:r =·pu " X3Jye,o .a•. airy I

=

=

For the definition of w -r, see

§1

Prop 1 ·36.

THE LOG I C OF RELATIONS ·a ,c e Cls . w a 1m .a. "' "; a u ,c£ Cls . 1c.> P" • 31c . x_pu •w • 3Uaw'W .o.·• •,

"' ,a e

.p,,

f z, \lw ·=· 3 1r,ul(zac . r.1 JH•) ·=· 31r " 111:3v " g,(3:'PJj,! ,::. :;J "''°"//l(�P11) ·=· .r, Jl(w' ,o) :=. via = :r. Prop ) v.v'e a,p .o: nv o nv' •'-'• nv' o nv

·st

v,v'e .cop".o: iv o-;v· ·""· m,' o iv

*

2. Pe Rel . PoO' .

P1

( §6 Prop 2·61 .a. Prop

= P :-xen"" ; .o ..... nx "" ixw;x = = n� . , :o::

1

,.

n\111 : U

V,V E. e " , e' v' tO?P {1) · -1

Hp (1) . ,t,tl .

l

z1,,_,, " .

-•

z'PI/ • 1l P 1'8qZ' -�- 11• ,, r.'

_. a e' 11' "11•,cseqz- P 11· . 11· P seq seqr) -�= NV' t ' ,· •"

·

(t)

aev· Pr" :

a. MV' P -. teqz' (2) . (3) .:.. ·. uv' .�. : a ,n, .�. seq• 1 n Hp .:.. Oe•• n (4) . (o) • Induct .�. ti:, ,rv .�. Prop 1

(8) (I)

Since this demonstration is a little complicated, I shall repeat it in words. The P affirms that if two progressions v,v ' are such that between any two consecutive terms of v whatsoever one always finds at least a term of v ', then there is not any term of v ' which follows all terms of v. Let x be a term of v, and y a term of v ' between x and seq x. Then the terms of v which do not precede V

x form a progression e xv, and the terms of v ' which do not precede

y form a progression e 11v '. If then x ' is any term whatever of e xv,y ' a term of v ' between x ' and seq x ', one deduces that y ' is a term V

0

of e 11v. Now there is a term y " of e 11v ' which goes between seq x ' and seq seq x ' and such a term must be seq y ' or succeed seq y ' ; thus seq y must precede seq seq x '. It follows then, if z is a v ' which precedes any v, then seq z is also a v ' which precedes any v., But by the hypothesis there is some 'V ' which precedes 'V, therefore the first term of v ' must precede v. We deduce by induction that all terms of v ' precede some terms of v, that is to say, that no term of v ' follows all terms of v. V

V

=

2· 1 Hp Prop t · t .o. nv ml [ '1!1,r-v' .:>. iv',w1(xPy) (1) . t,'OJW .a: :l!Env' .� . zs:irv :a. ,rt,':,,rv z,,rv .a. 3my1(xPy) .a. �v',v1 (xPz) .:,. xi.-v' :a. =rtr.>Jrv' (2) . (3) .a. Prop ]

(1)

(I) (8)

32

L O G I C AND KNOWLEDGE UJ£Cls . wou . av " Z3(uf.>;_z) : xe,.,, .�..: • "dw " Y3(xPy . yPseqx).

•3

w " 'JP(xPy . yPseqx) e Elm : � o �

. · .o. w e wp

This P affirms that if v is a progression in a compact series u, and if w is a class contained within u and succeeding certain terms of v, and if there is a term of w, and one alone, between any two consecutive terms whatsoever of v, and if finally the terms which succeed all terms of v succeed also all terms of w, then w is a progression m u.

:;:c

I Hp . §1Prop1·8 .�: xiv .�ic. 3:l+l " R� ,1 ,==e« .w ,,.,y1(xPy . yPseq:c) x : Ra: = ,l+l l'\ Ra, 1: ,� = e2' , Wl'\yi(XPy . yPseqx) . :>a: = e I XEV . RwERel " R'11aR'b ·=· 3:v " x1(aR« b)I • §3Prop2·16 ::>. Rw, 14-1 (1) RtReh, : XEV .:>«. w «= nt·,wa(xPy . yPseqx) : seqwa: W aeqa, ::>. :cR ,.,toz .�. 1L' tr

if10 Rscqx , seqx R i.,seq ll'a: ::,. U'o;

Wa: Pseq:c . seq:c Pseqw. .:>.

'°• Paeqto.

Rt., R R

=

1!1

sequ.·-'I:

(8)

:c,t,-,0. .:>. Wo P:c .:>. -(�)

(')

(4:) • Hp .:>. tu.i;io., � .:,: yaw .�. �v.l'\ z't(1/Pseqz') Hp (6) . §3Prop. �nl .:>. m:z:11(1/Pseqz') 10>p 1JIW . U"\Z1 ,(yPseq:c') t,' • :c 0 •' .:,. 1/ tor §1 Propo·7 . (1) . . R'

=

=

(2)

=

=

&o R.Rw .:>. R'd+l . to0

(&)

(6)

= 110; w 1

(7) (8)

BICle . w0 rs : :cev . Ws es .:>� eeqwz es ::>: a:(Ru,e)lw .:>z. seq�R.01)110 (9)

(10) (11)

(9) . Hp(9) • Induct .:>: Xev .�, �Rwe) sw Hp (9) . (10) . Rw d+ .:>:- tvs 1W .� . 1Vs II (IJ

(7) • (8) • (11) .:>. Prop ]

•.&

·?S

l

0

= I\ . nv = nw) [ Prop2·8 .:>. Prop av; " Z3(nz = nv) .o. vi " Z3(nz = nv) sElm

!ila>p "w3(m.o

nz = nv . ..rPz' .a. nz'· n; = "" . z'Pz .:>. ,,_,,_

6

a�zs(nz

= m, = nv

(1) (2)

(1) . (2) ,:>. Prop )

= nv) .o. l'v = mi "Z3(nz· = nv)

• 6f -W8a>F. aWnf\Z3�Z�'W) 0

.o.

l,w=7Wnt\J3(;;.=:m»)

Dt

Df

l'v and l ' w, as J ust defined, are true limits, while "A'v and "A' v of the section preceding are either limits or maxima or minima. Since l'v belongs to the class VTT, l'v cannot belong to the class v, which, moreover, by definition, has no maximum. Likewise I ' w

33

T H E L O G IC O F RELAT I O N S

does not belong to the class w, which has no mm1mum. If a wP or a wP possesses a limit, it is only possible to have one, but it is possible not to have any at all. In the derived classes, 'TTOJ, V

U

oJ'TT, S� z, "'no,

Prop 8·1 ·2 . :ce";; ,g, a .Q)P" vl(:c.s _,;) ( 1) . (2) . § r, Prop 1 ·8 .�. Prop ]

,g, :c-1

(1) (2)

"''na>

= ;;-n . "';;weEim ·5 xn,.; .o. "'' ron = n-'i . "' cmreElm

·, t :i� .o. "'' iw [ :CIR·; ,g:

VICD P ,01', �V;r

�; . Prop 3·11 ·31

n-J; .o.

,g,

:g. :af'\'a>;r

3a, P"Vf(XUV) .0.,rn"''°"' (2)

= I\ .01 m>n .o. 1•liw = A ·G

*

0) . (2) �- Prop )

Dem 8·4 N• (2) .�. Prop I

"'no,

.o. "'am -= I\ ;�" .o. "'run ==

·7 no;

· 7 .t

(1)

A

-I PSRE,\l . P�' • p p . � : t»Btt .o:i: . �i��a, • 1 xT,y = x,yenru . a;ay . xy 1-

= . . xT,y .=. x,ye;w . (CJY • x- = y

•1t · 1 i -xTa11 .=. x,yewn . xoy . ro- = y

. .

= tt .�::

Df Df Df

· t 3 xT,JJ = x,yero; . X'JY . a- = y Df · � xe:nw .o. � wT, "'Z'3(l'z a,) [ Prop 8 ·1 .g: y1 1ya1,u . y1 Py1 ,gy1 ,y1 . 3na, f'\ m,(ny1T1m . mT1ny1) (1 ) ( 1 ) . Prop 2· 1 ·3 . viwp . x = nv : zav ,gz, tbi. tno, . n:T L 1l·% . ·n·�_ T,.n(�r.q ?): 11•1a�1(yl' w� ) I . · .o. 1(;1CUT1 .l'ro ::t, 1 1V

=

=

=

THE

L O G I C OF

RELAT IONS

35

This P proves that any term of 1rw ( that is to say, all lower seg­ ments of u) is the upper limit of a progression of the terms of 1rw. If v is a progression in u, x a variable term of v, 1TV is the limit of the segments 1rx ; but this fact does not suffice for the demonstra­ tion of 4 ·2, since one has no reason to believe that 1rx always be­ longs to the class 1rw, that is to say, that if x is a u, x is the upper limit of a progression in u. • .i t XF, am .o. a wT1 "' Z3(1,z

= a,)

This P follows from Prop 3 · 1 1 as 4 ·2 follows from 3 · 1 . ·H

'!3 ·t.& ·! :s •i6 ·u

xe nw .o. xe wn .o. xe nw .o. xe c1J; .n. xe nw .o. xe w; .o.

-

--

a wT1 3'3(1'.: = x) a wt; " Z3(1,.z x) a wT1 Z3(l,z x) awf. " z3(1 z x) amT. "' 33(1, z X) awf. "' $3(1,z x) "'

"'

= = = = =

The demonstration of Prop ·22 to ·27 is similar to that of Prop 2 . There are other propositions of the same form which we do not know how to demonstrate, and which do not always appear to be true. This is the proposition �·e ,rw . •n. awfa " .;3(1.� a·) • 3 .zemT, .o. l'.i e :,co ( :r,: .:». �Tscq.r .,l . 3 l""IJl(.rT:111 . :r.11Tsl•q.r) \1) Prop 2·3 : zu .:> ..r 1·z llt"!Jl(J.:tTJJ • •-r/JT.i-t••t-1· J 1 , . = u·,: :offl'l'1( ,rt •,. .r 1 : . · .a· c:10, p . l 1 z = :rr .:). Prop l .31 .:e wf1 .o. 1,:e co:i ·H :e coT, .o. l ;e :rco

=

•33

-3, · 3?S ·3G ·:n

�e coi\ .o. l'ze w:r ze roT1 .o. l'ie ;;°' ze wf1 .o. l�e wn $£ mT• .o. l'.ze ;co ze wi\ .o. lFB con

Here also there are eight other propositions of similar form which do not always appear to be true. Such is the proposition ze coT, .o. l'.re ncu

LOGIC AND KNOWLED GE

Note to §6. We can now summarize the principal results of §6. A compact series (a ) is a series which has a term between any two of its terms. One such series is defined by a transitive relation P, which implies diversity, and which is such that P 2 = P. If xPy, one may say that x precedes y. If there areJerms outside the series considered which have the relation P or P with other terms, one can always fi� another relation, equivalent to P in the series being considered, and ·such that all the terms which hold that relation or its converse belong to the series being considered. (§5 Prop 1 ·6.) In consequence, it is more simple, and no less general, to take as the type of a compact series the entire domain of a suitable relation and its converse. Let u be one such series, P its generating relation. A progression in u is a series of type w, contained in u, and such that one always has xP seq x, if x is a term of the progression. We call wP the class of progressions in u. At the same time, wP is the class of regressions, that is to say, of series of type w, for which xP seq x. One may construct a wP and a wP, of which all terms may be found between any two terms whatsoever of u. Every class v contained in u defines four classes in u : ( 1 ) TTV, which contains all terms such that there is a v which succeeds them ; V

(2) TTV, which contains all terms such that there is a v which precedes them ; (3)

V7T,

(4)

V7T,

V

which contains all terms which precede any term of v ; ,

which contains all terms which succeed any term of v.

If v is a progression, ( 1) and (4) alone are important ; for a regression (2) and (3) alone are important. If v is a progression, any term of u belongs to ( 1) or (4), and ( 1) does not have a last term ; but one cannot know (in the general case) if (4) has a first term or not. One has some similar remarks if v is a regression. One now advances the theory of segments, which constitutes a

37 generalization of the theory of real numbers. One has four classes of segments : T H E L O G I C OF R E L A T I O N S

( 1 ) The class TTw, which is formed of all the classes v is any wP whatsoever ; V

(2) The class TTw, which is formed of all classes

V

'TTV,

'TTV,

where

where v is

V

any wP whatsoever ; (3) The class wTT, which is formed of all the classes VTT, where v u

is any wP whatsoever ;

V

V

(4) The class wTT, which is formed of all the classes VTT, when v is any wP whatsoever. Each of these four classes is a , of which the generating relation is derived from logical inclusion. Any term of WTT is the product of u and the negation of the corresponding term of TTw ; and the same for TTW and wTT. The classes TTW and wTT may have common terms ; for example, if u is the class of rational numbers, and v is a progression in u which does not have a rational limit, v ' a regression which determines the same section (in the sense of Dedekind). If u is a series which satisfies Dedekind's postulate of continuity, TTw and wTT have not common terms ; for then there is a third term in all classes which belong to the class wTT, and in no class of TTW. 0 U In each of these four classes, TTw, TTw, w1T, wTT, one can construct a progression or a regression, which will always have a limit be­ longing to one of the four classes, but not always to the class which contains the same progression or regression. Further, any term of each of these four classes is the limit of certain progressions, or of certain regression, but not necessarily ( so far as appears) of both ; and the terms of the specified progressions or regressions do not have to belong to the same class as the term which is their limit. These results are summarized below : Any term of 'TT W is the limit of a progression in 1rw and a progression in w7r

'' ,, ,, D

,, ,, ,, ,, " ,,

V

'TT W W'TT' V

W'TT'

,, ,, ,, ,, ,, ,,

,, ,, ,,

''

' ' , , , , regression

,, ,, ''

,,

,, ,, ,,

V

'TT W 'TT W V

'TT W

,, ,, ,, ,, '' ,,

'' ,, ,,

,, ,, ''

V

W'Tr W7T

V

W'TT

LOGIC AND KNOWLEDGE

All progressions in Tr w o r in

,,

,,

, , regressions

,,

,,

,,

V 1T W

' ' 1T W 11;1

' ' 1T W

,, ,, ,, " ,, ,,

w1r

V W1T

have a limit in 1r w

, , ,, ,, ,, ,, ,,

W1T

u

W7T

,, ,, ,, ,, ,, ,,

� 1T W

W1T V

W1T

Thus : Trw is identical to the class of limits of progressions in V TT W

W1T V WTT

,, ,,

,, '' ,,

,, ,, ,, ,, ,, ,,

'' ,, '' '' ,, ,,

,, ,, ,,

,,

,,

, , regressions

,,

,,

,, ,, ,,

1rw

or

W1T

V

V

1T W

, , W7T

1T W

" W1T

1T W

" W1T

u

u

We have not succeeded in proving that each of these four classes is a completely perfect series, but each of them is perfect either to the right or the left, that is to say, either for regressions or for progressions. The logical sum of 'TTW and

V

w'TT,

or of 'TTW and

is a perfect series, but in general that series will not be compact. For if there exists in u a progression v and a regression v ' having the same limit in u (which is known to be possible), then 1rv and v' 1r will be consecutive in the series 'TTW v W7T for v' 1r contains only a single term which does not belong to 'ITV, namely the common limit. Therefore 'TTW u w'TT is not in general a continuous series. We have not succeeded in proving that any progression or re­ gression in u has a limit, because we do not know an example of a compact series of which no term is not a principal element (in the language of Cantor). We have not been able to prove that there are terms of 1rw which are limits of regressions, etc. One knows after Cantor how to prove all these theorems if u is a denumerable series [Rivista di Matematica, Vol. V, pp. 1 29-62] . We do not develop this subject, since it has been referred to previously by Cantor. In §6 we have only wished to deduce the results which are valid for all compact series, without introducing other conditions. w'TT

On Denoting The volume of MIND for 1 905 appears superficially to be an out­ dated collection of the sort of papers that usually fill journals issued by and /or academic men. One would assume from it that the conflict between idealists and pragmatists over the nature of truth was the most important thing in the world. Embedded in this context of philo­ sophic war/are and dwarfed by the seventy-eight page disquisition on 'Pragmatism v. Absolutism' that precedes it is a fourteen-page paper by Russell which he has called his finest philosophical essay. The editor of MIND, Professor G. F. Stout, regarded it as unusual and unconventional, but none the less had the sound judgmen t to print it . How many of his readers understood it remains an open question. ON DENOTING is a milestone in the del.Jelopment of contemporary philosophy, revealing once more Russell's inventiveness and striking originality in thought. Ironically it contains a minor error. G. E. Moore has pointed out that Russell's 'shortest statement' at the close of the paper is faulty because of the ambiguity of the verb 'to write'. 'Scott is the author of WAVERLEY' does not, therefore, have the same meaning as 'Scott wrote WAVERLEY', since Scot t ( like blind Milton) may be the author of the work without being the person who literally wrote it /or the first time. Russell has accepted this correction 'with equanimity'.• The right to feel patronizing about this slip is reserved by law to those who have done as much for philosophy as Russell and Moore. The fuller development of these ideas is the well-known theory of descriptions, the full statement of which was to come five years later with the publication of the first volume of the PRINCIPIA MATHE­ MATICA. • The Philosophy of Bertrand Russell, Evanston and Cambridge, 1 944 et seq., p. 690. Moore's we ll-known essay is on p. 1 77 ff. of the same volume.

39

ON DENOTING a 'denoting phrase' I mean a phrase such as any one of the following : a man, some man, any man, every man, all men, the present King of England, the present King of France, the centre of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. We may distinguish three cases : ( 1 ) A phrase may be denoting, and yet not denote anything ; e.g., 'the present King of France'. ( 2) A phrase may denote one definite object ; e.g., 'the present King of England' denotes a certain man. (3) A phrase may denote ambiguously ; e.g., 'a man' denotes not many men, but an ambiguous man. The interpretation of such phrases is a matter of considerable difficulty ; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain. The subject of denoting is of very great importance, not only in logic and mathematics, but also in theory of knowledge. For ex­ ample, we know that the centre of mass of the solar system at a definite instant is some definite point, and we can affirm a number of propositions about it ; but we have no immediate acquaintance with this point, which is only known to us by description. The distinction between acquaintance and knowledge about is the dis­ tinction between the things we have presentations of, and the things we only reach by means of denoting phrases. It often happens that we know that a certain phrase denotes unambigu­ ously, although we have no acquaintance with what it denotes ; this occurs in the above case of the centre of mass. In perception we have acquaintance with the objects of perception, and in thought we have acquaintance with objects of a more abstract logical character ; BY

41

LOG IC AND KNOWLEDGE

but we do not necessarily have acquaintance with the objects denoted by phrases composed of words with whose meanings we are acquainted. To take a very important instance : there seems no reason to believe that we are ever acquainted with other people's minds, seeing that these are not directly perceived ; hence what we know about them is obtained through denoting. All thinking has to start from acquaintance ; but it succeeds in thinking about many things with which we have no acquaintance. The course of my argument will be as follows. I shall begin by stating the theory I intend to advocate ;• I shall then discuss the theories of Frege and Meinong, showing why neither of them satisfies me ; then I shall give the grounds in favour of my theory ; and finally I shall briefly indicate the philosophical consequences of my theory. My theory, briefly, is as follows. I take the notion of the variable as fundamental ; I use ' C (x)' to mean a propositiont in which x is a constituent, where x, the variable, is essentially and wholly un­ determined. Then we can consider the two notions 'C (x) is always true' and 'C (x) is sometimes true't. Then everything and nothing and something (which are the most primitive of denoting phrases) are to be interpreted as follows : C ( everything) means ' C (x) is always true' ; C (nothing) means ' " C (x) is false" is always true' ; C (something) means 'It is false that "C (x) is false" is always true'.§ Here the notion 'C (x) is always true' is taken as ultimate and indefinable, and the others are defined by means of it. Everything, nothing, and something are not assumed to have any meaning in isolation, but a meaning is assigned to every proposition in which they occur. This is the principle of the theory of denoting I wish * I have discussed this subject in Principles of Mathematics, Chap. V, and § 476. The theory there advocated is very nearly the same as Frege's, and is quite different from the theory to be advocated in what follows. t More exactly, a propositional function. t The second of these can be defined by means of the first, if we take it to mean, ' It is not true that "C (x) is false" is always true'. § I shall sometimes use, instead of this complicated phrase, the phrase ' C (x) is not always false', or 'C (x) is sometimes true' , supposed defined to mean the same as the complicated phrase.

ON DENOTING

43

to advocate : that denoting phrases never have any meaning in themselves, but that every proposition in whose verbal expression they occur has a meaning. The difficulties concerning denoting are, I believe, all the result of a wrong analysis of propositions whose verbal expressions contain denoting phrases. The proper analysis, if I am not mistaken, may be further set forth as follows. Suppose now we wish to interpret the proposition, ' I met a man'. It this is true, I met some definite man ; but that is not what I affirm. What I affirm is, according to the theory I advocate : ' "I met x, and x is human" is not always false'. Generally, defining the class of men as the class of objects having the predicate human, we say that :

'C (a man)' means ' "C (x) and x is human" is not always false'. This leaves 'a man', by itself, wholly destitute of meaning, but gives a meaning to every proposition in whose verbal expression 'a man' occurs. Consider next the proposition 'all men are mortal'. This pro­ position• is really hypothetical and states that if anything is a man, it is mortal. That is, it states that if x is a man, x is mortal, what­ ever x may be. Hence, substituting 'x is human' for 'x is a man', we find : 'All men are mortal' means ' "If x is human, x is mortal" is always true' . This is what is expressed in symbolic logic by saying that 'all men are mortal' means ' "x is human" implies "x is mortal" for all values of x'. More generally, we say :

'C (all men)' means ' "If x is human, then C (x) is true" is always true' . Similarly 'C (no men)' means ' "If x is human, then C (x) is false" is always true' . 'C ( some men)' will mean the same as 'C (a man)', t and • As has been ably argued in Mr. Bradley's Logic, Book I , Chap. I I . t Psychologically 'C (a man)' has a suggestion of only one, and 'C (some men)' has a suggestion of more than one ; but we may neglect these suggestions in a preliminary sketch,

44

LOGIC AND KNOWLEDGE

'C (a man)' means 'It is false that " C (x) and x is human" 1s always false'. 'C (every man)' will mean the same as 'C (all men)'.

It remains to interpret phrases containing the. These are by far the most interesting and difficult of denoting phrases. Take as an instance 'the father of Charles II was executed'. This asserts that there was an x who was the father of Charles II and was executed. Now the, when it is strictly used, involves uniqueness ; we do, it is true, speak of 'the son of So-and-so' even when So-and-so has several sons, but it would be more correct to say 'a son of So-and-so'. Thus for our purposes we take the as involving uniqueness. Thus when we say 'x was the father of Charles II' we not only assert that x had a certain relation to Charles II, but also that nothing else had this relation. The relation in question, without the assumption of uniqueness, and without any denoting phrases, is expressed by 'x begat Charles II'. To get an equivalent of 'x was the father of Charles II', we must add, ' If y is other than x, y did not beget Charles II', or, what is equivalent, ' If y begat Charles II, y is identical with x'. Hence 'x is the father of Charles II' becomes : 'x begat Charles II ; and "if y begat Charles II, y is identical with x" is always true of y'. Thus 'the father of Charles II was executed' becomes : 'It is not always false of x that x begat Charles II and that x was executed and that "if y begat Charles II, y is identical with x" is always true of y'. This may seem a somewhat incredible interpretation ; but I am not at present giving reasons, I am merely stating the theory. To interpret ' C (the father of Charles II)', where C stands for any statement about him, we have only to substitute C (x) for 'x was executed' in the above. Observe that, according to the above interpretation, whatever statement C may be, 'C (the father of Charles II)' implies : ' It is not always false of x that "if y begat Charles I I, y 1s identical with x" is always true of y',

ON DENOTING

45

which is what is expressed in common language by 'Charles II had one father and no more'. Consequently if this condition fails, every proposition of the form ' C (the father of Charles II)' is false. Thus e.g. every proposition of the form 'C (the present King of France)' is false. This is a great advantage in the present theory. I shall show later that it is not contrary to the law of contradiction, as might be at first supposed. The above gives a reduction of all propositions in which denoting phrases occur to forms in which no such phrases occur. Why it is imperative to effect such a reduction, the subsequent discussion will endeavour to show. The evidence for the above theory is derived from the difficulties which seem unavoidable if we regard denoting phrases as standing for genuine constituents of the propositions in whose verbal ex­ pressions they occur. Of the possible theories which admit such constituents the simplest is that of Meinong. * This theory regards any grammatically correct denoting phrase as standing for an object. Thus 'the present King of France', 'the round square', etc., are supposed to be genuine objects. It is admitted that such objects do not subsist, but nevertheless they are supposed · to be objects. This is in itself a difficult view ; but the chief objection is that such objects, admittedly, are apt to infringe the law of con­ tradiction. It is contended, for example, that the existent present King of France exists, and also does not exist ; that the round square is round, and also not round, etc. But this is intolerable; and if any theory can be found to avoid this result, it is surely to be preferred. The above breach of the law of contradiction is avoided by Frege's theory. He distinguishes, in a denoting phrase, two ele­ ments, which we may call the meaning and the denotation.t Thus 'the centre of mass of the solar system at the beginning of the twentieth century' is highly complex in meaning, but its denotation is a certain point, which is simple. The solar system, the twentieth century, etc., .are constituents of the meaning ; but the denotation • See Untersuchungen zur Gegenstandstheorie und Psychologie (Leipzig, 1 904) the first three articles (by Meinong, Ameseder and Mally respec­ tively). , See his 'Ueber Sinn und Bedeutung , Zeitschrift fur Phil. und Phil. Kritik, Vol. 1 00.

t

LOGIC AND KNOWLED GE

has no constituents at all.• One advantage of this distinction is that it shows why it is often worth while to assert identity. If we say 'Scott is the author of Waverley', we assert an identity of denotation with a difference of meaning. I shall, however, not repeat the grounds in favour of this theory, as I have urged its claims elsewhere (loc. cit.), and am now concerned to dispute those claims. One of the first difficulties that confront us, when we adopt the view that denoting phrases express a meaning and denote a denota­ tion, f concerns the cases in which the denotation appears to be absent. If we say 'the King of England is bald', that is, it would seem, not a statement about the complex meaning 'the King of England', but about the actual man denoted by the meaning. But now consider 'the King of France is bald'. By parity of form, this also ought to be about the denotation of the phrase ' the King of France'. But this phrase, though it has a meaning provided 'the King of England' has a meaning, has certainly no denotation, at least in any obvious sense. Hence one would suppose that 'the King of France is bald' ought to be nonsense ; but it is not non­ sense, since it is plainly false. Or again consider such a proposition as the following : 'If u is a class which has only one member, then that one member is a member of u', or, as we may state it, 'If u is a unit class, the u is a u'. This proposition ought to be always true, since the conclusion is true whenever the hypothesis is true. But 'the u' is a denoting phrase, and it is the denotation, not the mean­ ing, that is said to be a u. Now if u is not a unit class, 'the u' seems to denote nothing ; hence our proposition would seem to become nonsense as soon as u is not a unit class. Now it is plain that such propositions do not become non• Frege distinguishes the two elements of meaning and denotation everywhere, and not only in complex denoting phrases. Thus it is the meanings of the constituents of a denoting complex that enter into its meaning, not their denotation. In the proposition ' Mont Blanc is over 1 ,ooo metres high', it is, according to him, the meaning of 'Mont Blanc', not the actual mountain, that is a constituent of the meaning of the pro­ position. In this theory, we shall say that the denoting phrase expresses a mean­ ing ; and we shall say both of the phrase and of the meaning that they denote a denotation. In the other theory, which I advocate, there is no meaning, and onl y sometimes a denotation,

t

ON DENOTING

47

sense merely because their hypotheses are false. The King in The Tempest might say, 'If Ferdinand is not drowned, Ferdinand is my only son'. Now 'my only son' is a denoting phrase, which, on the face of it, has a denotation when, and only when, I have exactly one son. But the above statement would nevertheless have remained true if Ferdinand had been in fact drowned. Thus we must either provide a denotation in cases in which it is at first sight absent, or we must abandon th� view that the denotation is what is concerned in propositions which contain denoting phrases. The latter is the course that I advocate. The former course may be taken, as by Meinong, by admitting objects which do not sub­ sist, and denying that they obey the law of contradiction ; this, however, is to be avoided if possible. Another way of taking the same course ( so far as our present alternative is concerned) is adopted by Frege, who provides by definition some purely con­ ventional denotation for the cases in which otherwise there would be none. Thus 'the King of France', is to denote the null-class ; 'the only son of Mr. So-and-so' (who has a fine family of ten), is to denote the class of all his sons ; and so on. But this procedure, though it may not lead to actual logical error, is plainly artificial, and does not give an exact analysis of the matter. Thus if we allow that denoting phrases, in general, have the two sides of meaning and denotation, the cases where there seems to be no denotation cause difficulties both on the assumption that there really is a denotation and on the assumption that there really is none. A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science. I shall therefore state three puzzles which a theory as to denoting ought to be able to solve ; and I shall show later that my theory solves them. ( 1 ) If a is identical with b, whatever is true of the one is true of the other, and either may be substituted for the other in any proposi­ tion without altering the truth or falsehood of that proposition. Now George IV wished to know whether Scott was the author of Waverley ; and in fact Scott was the author of Waverley. Hence we may substitute Scott for the author of ' Waverley ', and thereby prove that George IV wished to know whether Scott was Scott.

L O G I C AND K N O W L ED G E

Yet an interest in the law of identity can hardly be attributed to the first gentleman of Europe. (2) By the law of excluded middle, either 'A is B' or 'A is not B' must be true. Hence either 'the present King of France is bald' or 'the present King of France is not bald' must be true. Yet if we enumerated the things that are bald, and then th� things that are not bald, we should not find the present King of France in either list. Hegelians, who love a synthesis, will probably conclude that he wears a wig. (3) Consider the proposition 'A differs from B'. If this is true, there is a difference between A and B, which fact may be expressed in the form 'the difference between A and B subsists'. But if it is false that A differs from B, then there is no difference between A and B, which fact may be expressed in the form 'the difference between A and B does not subsist'. But how can a non-entity be the subject of a proposition ? 'I think, therefore I am' is no more evi­ dent than 'I am the subject of a proposition, therefore I am', provided 'I am' is taken to assert subsistence or being,• not exist­ ence. Hence, it would appear, it must always be self-contradictory to deny the being of anything ; but we have seen, in connexion with Meinong, that to admit being also sometimes leads to contradictions. Thus if A and B do not differ, to suppose either that there is, or that there is not, such an object as 'the difference between A and B' seems equally impossible. The relation of the meaning to the denotation involves certain rather curious difficulties, which seem in themselves sufficient to prove that the theory which leads to such difficulties must be wrong. When we wish to speak about the meaning of a denoting phrase, as opposed to its denotation, the natural mode of doing so is by inverted commas. Thus we say : The centre of mass of the solar system is a point, not a denoting complex ; 'The centre of mass of the solar system' is a denoting complex, not a point. Or again, The first line of Gray's Elegy states a proposition. • I use these as synonyms.

49 'The first line of Gray's Elegy' does not state a proposition. Thus taking any denoting phrase, say C, we wish to consider the relation between C and 'C', where the difference of the two is of the kind exemplified in the above two instances. We say, to begin with, that when C occurs it is the denotation that we are speaking about ; but when 'C' occurs, it is the meaning. Now the relation of meaning and denotation is not merely linguis­ tic through the phrase : there must be a logical relation involved, which we express by saying that the meaning denotes the denota­ tion. But the difficulty which confronts us is that we cannot suc­ ceed in both preserving the connexion of meaning and denotation and preventing them from being one and the same ; also that the meaning cannot be got at except by means of denoting phrases. This happens as follows. The one phrase C was to have both meaning and denotation. But if we speak of 'the meaning of C', that gives us the meaning (if any) of the denotation. 'The meaning of the first line of Gray's Elegy' is the same as 'The meaning of ''The curfew tolls the knell of parting day'',' and is not the same as 'The meaning of ''the first line of Gray's Elegy''.' Thus in order to get the meaning we want, we must speak not of 'the meaning of C', but of 'the meaning of "C" , ' which is the same as 'C' by itself. Similarly 'the denotation of C' does not mean the denotation we want, but means something which, if it denotes at all, denotes what is denoted by the denota­ tion we want. For example, let 'C' be 'the denoting complex occurring in the second of the above instances'. Then ON DENOTING

C = 'the first line of Gray's Elegy', and the denotation of C = The curfew tolls the knell of parting day. But what we meant to have as the denotation was 'the first line of Gray's Elegy'. Thus we have failed to get what we wanted. The difficulty in speaking of the meaning of a denoting complex may be stated thus : The moment we put the complex in a pro­ position, the proposition is about the denotation ; and if we make a proposition in which the subject is 'the meaning of C', then the subject is the meaning (if any) of the denotation, which was not intended. This leads us to say that, when we distinguish meaning and"denotation, we must be dealing with the meaning : the mean­ ing has denotation and is a complex, and there is not something

50

L O G I C AND KNOW LEDGE

other than the meaning, which can be called the complex, and be said to have both meaning and denotation. The right phrase, on the view in question, is that some meanings have denotations. But this only makes our difficulty in speaking of meanings more evident. For suppose C is our complex ; then we are to say that C is the meaning of the complex. Nevertheless, whenever C occurs without inverted commas, what is said is not true of the meaning, but only of the denotation, as when we say : The centre of mass of the solar system is a point. Thus to speak of C itself, i.e., to make a proposition about the meaning, our subject must not be C, but something which denotes C. Thus 'C', which is what we use when we want to speak of the meaning, must be not the meaning, but something which denotes the meaning. And C must not be a constituent of this complex ( as it is of 'the meaning of C') ; for if C occurs in the complex, it will be its denotation, not its meaning, that will occur, and there is no backward road from denotations to meanings, because every object can be denoted by an infinite number of different denoting phrases. Thus it would seem that 'C' and C are different entities, such that 'C' denotes C ; but this cannot be an explanation, because the relation of 'C' to C remains wholly mysterious ; and where are we to find the denoting complex 'C' which is to denote C ? Moreover, when C occurs in a proposition, it is not only the denotation that occurs ( as we shall see in the next paragraph) ; yet, on the view in question, C is only the denotation, the meaning being wholly rele­ gated to 'C'. This is an inextricable tangle, and seems to prove that the whole distinction of meaning and denotation has been wrongly conceived. That the meaning is relevant when a denoting phrase occurs in a proposition is formally proved by the puzzle about the author of Waverley. The proposition ' Scott was the author of Waverley' has a property not possessed by ' Scott was Scott', namely the property that George IV wished to know whether it was true. Thus the two are not identical propositions ; hence the meaning of 'the author of Waverley' must be relevant as well as the denota­ tion, if we adhere to the point of view to which this distinction belongs. Yet, as we have just seen, so long as we adhere to this point of view, we are compelled to hold that only the denotation

O N DE N O T I N G

51

can be relevant. Thus the point of view m question must be abandoned. It remains to show how all the puzzles we have been considering are solved by the theory explained at the beginning of this article. According to the view which I advocate, a denoting phrase is essentially part of a sentence, and does not, like most single words, have any significance on its own account. If I say 'Scott was a man', that is a statement of the form 'x was a man', and it has 'Scott' for its subject. But if I say 'the author of Waverley was a man', that is not a statement of the form 'x was a man', and does not have 'the author of Waverley ' for its subject. Abbreviating the statement made at the beginning of this article, we may put, in place of 'the author of Waverley was a man', the following : 'One and only one entity wrote Waverley, and that one was a man'. (This is not so strictly what is meant as what was said earlier ; but it is easier to follow.) And speaking generally, suppose we wish to say that the author of Waverley had the property (x, y)' will mean 'cf, (x, y) is always true', and so on. Then the distinction between the assertion of all values and the assertion of any is the distinction between ( 1 ) asserting (x) . "y,

whence by ( 1 2),

r : . (X) • . 3 ! X'y . X'y C a'X . X'y C fJ. That is to say, every member of fJ has predecessors which are /J 's. Thus in fJ there are infinite descending series. These series are all

contained between the beginning of x and the first stopping-point of events existing when x begins. There must therefore be no lower limit to their duration. They are all contemporary with the whole of µ, i.e. -+ In fact, Further,

-+

fJ Cp'S"µ . -+

v -+

u -+

p' S "µ = µ u fl = S'x - P"(S'x - P" S'x) . -+

p'S"(µ u /J) = µ .

Thus µ is too small a class to form an instant, and µ u f3 is too large. -+ For if we have a Cp'S"a, but the two are not equal, we need to -+ enlarge a and therefore diminish (or at least not increase) p'S"a. -+ -+ in order to arrive at a = p' S"a ; but if p' S"a C a and the two are not equal, we need to diminish a and therefore increase ( or at least not -+ diminish) p'S"a in order to secure equality. These considerations may be illustrated by considering the class A, where A = µ u (/3 - P"{J). We have -+ -+ p'S"fl = µ ; hence µ Cp'S"(fl - P"{J). Also -+ -+ -+ p' S"A = p' S" µ n p' S" (/1 - P"{J), -+ = (µ u fJ) n p'S" ({J - P"{J), -+ = µ u {/J n p 'S" (/3 - P"/J)}.

35S

ON ORDER IN TIME

T hus w e shall have A E I n if

_.

/3 - P"/3 = f3 n p' S" (/3 - P"{J).

_.

Now

{J - P"/3 C p 'S" ({J - P"{J).

Therefore if A E In we require

_.

p 'S" (/3 - P"/3) C - P"{J,

1.e.

P"{J C P" ({J - P"{J) V P" ({J - P"/3), P"{J C P" (/3 - P"/3).

1.e.

This is a condition exactly analogous ( except for the substitution of P for P) to the previous one, viz. v

_.

u

_.

u _.

P"S'x C P" (S'x - P"S'x), and exactly similar considerations apply to it. If we put

y = P"{J - P" (/3 - P"{J), and if y is not null, it contains infinite ascending series of events all contained in a stretch between the end of P" ({J - P"{J) and the end of P"/3. A few general considerations may help to make these possibilities clear. Let the duration of x be represented by XX'. Then the

X

B

A

X'

existence of fJ implies the existence of a stretch XA such that no event existing at X stops before A, but, given any event which stops in XA, it not only begins in XA, but has predecessors which begin and end in XA, and these have others, and so on ad infinitum. The class fl consists of events which begin in XA, wherever they may stop. Now suppose that there is a stretch BA such that every event that begins in BA also stops in BA. All such events are included in P"fJ. fl- P"/3 will be events which begin after X and stop at or after A. By hypothesis, none of these begins in BA. Thus all the events that begin in BA are members of y.

L O G IC A N D K N O W L E D G E

The hypothesis concerning X was : No event existing at X stops before A. The hypothesis concerning A is : No event existing at A begins after B. If both hypotheses are verified, the duration XA has no instant at the beginning or at the end. We can, of course, define continually smaller regions, by pro­ longing the above procedure ; but there is no reason to suppose that they approach a point as their limit. From the above it appears that there must be instants if there is a minimum to duration, i.e. if an infinite series of non-overlap­ ping events must ultimately reach any given region ; or, conversely, if a P-series consisting entirely of contemporaries of a given event must be finite. This may be expressed as follows : -+ -+ (R,x) : R C: P . C'R C S'x . :) · H ! B'R. This hypothesis ensures that every event has a first instant. If, for -+ -+ V B'R, we substitute B'R, we ensure that every event has a last instant. Thus if, among the contemporaries of x, there are no infinite series of non-overlapping events, x has a first and a last instant. -+ u -+ This matter may be put as follows : If S'x - P"S'x is to be an instant, we require, as we have seen, Put Then

bSx . xS I Ph . :) . (Ha) . aSx . - (xS I Pa) . aPb. -+ -+ u a = S'x n P'b . a E a - P"a.

-+ -+ -+ -+ -+ -+ -+ -+ -+ a E S'x n P'h . P'a C - S'x u - P'b . P'a C P'b . P'a C P"S'x. Therefore -+ -+ P'a C - S'x,

and therefore - (xS I Pa). Thus 3 ! a - P' 'a gives the desired result. This follows if - (xS I P2b) ; and so it does if - (xS I pnh), where n is any finite integer. This leads to the same hypothesis as the above for the existence of instants. The event x will have a first instant if there is an event a which stops just when x begins. This happens if -+ -+ aSx . P"S'a C P'x,

3 57 i.e. if a overlaps x, but nothing that ends sooner does so. In this case, -+ V -+ S'x - P"S'x E In. For -+ -+ -+ -+ P"S'a C P'x . ::> . P'a C P'x . ::> . - (xS I Pa). Therefore -+ -+ a E S'x - P"S'x. Again, xS I Pz . ::> . (3 y) . xSy .yPz. O N O RDER I N T I ME

V

But by hypothesis Therefore Again,

yPz . zSa . ::> .yPx. xSy .yPz . ::> . - (zSa).

- (xS I Pa) . ::> :

r--.1

(3 y) . xSy .yPa :

::> : xSy .yPz . ::> . - (zPa).

Hence

xSy .yPz . ::> . aPz. Hence +u -+ Z E P"S'x . ::> . z E P'a. But -+ -+ a E S'x - P" S'x. Therefore v -+ v -+ v -+ Z £ P"S'x . ::> . z E P" (S'x - P"S'x). Therefore V -+ -+ S'x - P"S'x E In. The hypothesis -+ -+ aSx . P"S'a C P'x is equivalent to aSx . - (aS I P l Sx). For -+ -+ V -+ -+ P"S'a C P'x . ::> . P"S'a C - S'x . ::> . - (aS I P I Sx). V

And since

-+ -+ -+ -+ +-+ P"S'a C - P'x, P"S'a C - S'x . ::> . P"S'a C P'x.

Thus

-+ -+ V P"S'a C P'x . =!! . - (aS I P I Sx) .

= . - (xS I P J Sa).

LOGIC AND KNOWLEDGE

Thus a sufficient condition for the existence of instants is

H!s

.!...

s I P I s.

If this hypothesis is satisfied, there are two events a, x such that -+ -+ u -+ +-

aSx . P"S'a C P'x . P"S'x C P'a,

and the last instant of a is the first of x. If S C: S I P I S, then, putting Q = P I S and R = S I P,

s e: s I Q e: s I Q e: s I Qn, 2

where n is any finite integer, and

S C: R I S C: R2 j S C: Rn I S.

S C: S I Qn states that, if two events a and b overlap, there are n events of which the first is a contemporary of a, the second ends later than the first, the third than the second, etc., and b ends later than the nth. S G Rn I S, similarly, states that, if two events a and b overlap, there are n events of which the first begins later than a, the second than the first, etc., and the nth is contemporary with b. If either of these does not happen, one of the events a and b has a first instant and the other a last. The above sufficient condition may be stated in the form -+ -+ -+ +(3.a,x) : aSx : P"S'a C P'x . v . P"S'a C P'x. Note that aS I P I Sx • = . a begins before the end of x, V

Thus and

aS I P I Sx • = . a ends after the beginning of x.

a (S .!... S I P l S) x . = . a begins at the end of x,

a (S .!... S I P I S) x . = . a ends at the beginning of x.

Observe that, putting we have

Q=pI

s . R = s I P,

-+ = . a e-+S'x - R"S'x, -+ -+ a (S £ S I P l S)x . = . a e S'x - Q"S'x.

a (S .!... S I P l S)x . V

V

ON ORDER IN TIME

3 59

We have seen that, if

-+ u u -+ a (S ..!_ s 1 P l S)x, S'x - P"S'x e In. In this case, -+ v -+ -+ -+ S'x - P"S'x = S'x n S'a, for -+ u u -+ a ..!-. S I P I Sx . S'a C - P" S'x. Therefore -+ -+ -+ v -+ S'a n S':" C S'x - P"S'x. Again, aSx . - (xS I Pz) . :, . - (aPz), and zPa . :, . zPx, so that xSz . :, . - (zPa). Hence xSz . - (xS I Pz) . :, . zSa, 1.e. -+ v -+ -+ -+ S'x - P" S'x C S'x n S'a. Hence -+ v -+ -+ -+ S'x - P" S'x = S'x n S'a.

=.

Let µ be an instant for which 3 ! µ - R"µ . 3 ! µ - Q"µ, where R = S I P . Q = P I S. That is to say, there is an event a which is a member of µ, and begins not later than any other member of µ ; and there is an event b which is a member of µ and ends not earlier than other member. b

a

Thus our hypothesis is Now

Thus 1.e.

a e µ • b E µ : y E µ • :, 11 • - (aRy) . - (yQb).

= : aSz . - (yQb) . = : zSb . - (aRy) .

=>z .

- (zPy),

=>z .

- (yPz).

y£µ . �Y : aSz . zSb . =>• • zSy, -+ -+ -+ z e S'a n S'b . :> :y e µ . => v . zSy : ) : z ep'S"µ : :, : z E µ ,

3 6o

L O G I C AND KNOWLEDGE

1.e.

-+ -+ S' a n S'b C µ,

i.e.

-+ -+ S'a n S'b = µ.

In this case,

aRc . ::) : c - £µ : :) : - (cSa) . v . - (cSb), aRc . - (cSa) . :) . aPc. But - (aQb). Therefore - (cSb). Thus

aRc . ::) . - (cSb),

i.e.

Thus, putting S I P I S = H, we have

µ £ In . 3 ! µ - R"µ . 3: ! µ - Q"µ . -+

- (aS I P I Sb). -+

-+

V

::, . (3:a,b) . µ = S'a n S'b . a (S ..:.. H ..:.. H) b.

-+

-+

-+

Conversely, assume S'a n S'b £ In. Put µ = S'a n S'b. Therefore

µ E In :) : d£µ .

= . µ C -+S'd. =.

-+ -+ -+ -+ -+ -+ -+ S'a n S'b £ In . ::, : d£ S'a n S'b . S'a n S'b C S'd. -+ +Suppose that dQb. Then 3 ! S'b n P'd. Now -+ +C £ S'b n P'd . dSa . ::, . aRc . ::, . - (cPa), cSb . bQa . :) . cHa . ::) . - (aPc). Thus

-+ +c E S'b n P'd . dSa • bQa . :) • cSa • :) • c £ µ.

Therefore

+bQa • dQb • dSa • ::, • 3: ! µ n P'd . ::, • d- £ µ, -+ V bQa . ::, • Q'b C - µ . ::, • b £ - Q"µ.

whence Similarly Now

Thus "-I

bRa • ::, • a£ - R"µ. +- +- (bQa) . . P'b C P'a, -+ -+ - (bRa) . . P'b C P'a. -+ -+ -+ (bQa) . - (bRa) . S'a C S'b, whence S'a £ In.

= = =.

3 61

O N O R D ER I N T I M E

Similarly -+ -+ "' (aQb) . "' (aRb) . S'b C S'a , whence S'b E In. Hence -+ ➔ -+ -+ -+ -+ S'a E In . v . S'b E In . v . 3: ! S'a - S'b . 3: ! S'b - S'a. Suppose that -+ -+ -+ -+

=.

J E S'a - S'b . bPy . z E S'b - S'a . zPa.

Then

bQa . bRa.

Thus -+ -+

-+

-+

S'a, S'b "' E In . S'a n S'b E In . :> : a E - R"µ

Hence -+

V

-+

-+

U

. b E - Q"µ . v • b E - R"µ . a E - Q"µ. -+

-+

S'a , S'b E - In . µ = S'a n S'b . :> :

-+

'-'

V

S'a n S'b E In . = : a E - R"µ . b E - Q"µ . v . b E - R"µ . a E - Q"µ, and in this case we have "' (aS I P I Sb) . "' (bS I P I Sa) ,

i.e. one stops as the other begins. It might be thought that, if there is not a first instant when an event exists, there will be a last instant when it does not. But the conditions for the two are independent. -+ The terms which overlap with x and begin before it are S'x -+ n S"P'x. The terms which end just before x are a(aPx : aP I Sb . :>b . "' bPx) , i.e. -+ -+

P'x - P' 'S"P'x.

Put

-+

-+

-+

-+

"A = S'x n S"P'x . µ -= P'x - P"S"P'x . R = S I P.

Therefore

-+

-+

-+

-+

-+

"A v µ = (S'x n S"P'x) v (P'x n S"P'x - P"S"P'x) -+

and

-+

-+

-+

= S"P'x - P"S"P'x = R'x - P"R'x, -+

-+

-+

-+

"A = S'x n R'x • µ = P'x - P"R'x.

-+ Obviously "A v µ C p 'S'' ("A v µ). We need further V

- >..- µ C P"µ V P"µ V P""A V P""A. U

L O G I C AND KNO W L E D G E

++- -+ u .... - R'x =p'P"P'x Cp'P"µ C P"µ if g: ! µ. Thus it remains to prove We have

-+ P"R'x C P" (,\ v µ). -+ -+ -+ -+ R'x = ,\ v (R'x n P'x) = ,\ v P'x.

Thus we have to prove -+ P"P'x C P" (,\ v µ). Now +- -+ -+ .... P"P'x C P",\ . : aP2x . ::::> a • a ! P'a n R'x n S'x. +- -+ +- -+ Since PC:. R, we have g: ! R'a n R'x. Put a = R'a n R'x. Assume h E a - R"a. Then

=

aRb . bRx : bRc . ::::> c . - (aRc . cRx). But R2 C:. R. Therefore

aRb . bRx : bRc . :::>c . - (cRx), i.e. aRb . bR .!.. R2x. -+ . Now bR .!.. R2x . :::> . bSx, unless S'x E In. For b .!.. R2x . :::> : b S I P I Sw . :::>10 • - (wPx). Therefore bS I P I Sb . :::> . - (bPx). Also b� . :::> . - (xPb). Hence But

Hence

bS I P I Sb • :::> : bR .!.. R2x • :> • bSx.

-+ -+ -+ u -+ - (bS I P l Sb) . :::> : S'b n P"S'b = A . S'b n P"S'b = A : -+ -+ -+ -+ -+ -+ :> : S'b C p'S"S'b : :::> : S'b =p'S"S'b : -+ :> : S'b E In.

S'b - E In • :::> : bR ...!.. R2x • :::> • bSx.

Further, bR .!.. R2x . aP2x . :::> • aPb. For

(3:c) . aPc . cPx, . ·. bSa . :> • bRc . ::::> • bR2x, . ·. - (bSa), • ·. aPb. Hence -+ -+ aP1x • bR .:. R2x • :> • b £ S'x n R'x • aPb.

ON ORDER I N T I M E

Thus

-+ -+ -+ -+ 3 ! R'x - R"R'x . :> : P "P 'x C P "">., . v • S'x E In.

Hence -+ -+ -+ -+ 3 ! P 'x - P "R'x . 3 ! R'x - R"R'x . -+ -+ -+ :> : R'x - P "R'x £ In . v . S'x E In. Now -+ -+ -+ -+ -+ -+ P 'x C P "R'x . :> : S"P 'x C S"P "R'x . :> • R'x C R"R'x. Thus

-+ -+ -+ -+ 3 ! R'x - R"R'x . :> • 3 ! P 'x - P "R'x.

Hence

-+ -+ -+ -+ -+ 3 ! R'x - R"R'x • :> : R'x - P "R'x £ In . v • S'x E In. -+ -+ The condition 3 ! R'x - R"R'x is sufficient, but we have not shown it to be necessary ; it means that there is a term which begins just before x. Imagining a Dedekind section, x has perhaps no first moment, but the term in question begins at the end of the time before x. To sum up : When the whole class of events can be well ordered, and also when methods exist of constructing certain kinds of well­ ordered series of events, the existence of instants can be proved. But in the absence of such possibilities, I do not know of any way of proving the existence of instants anywhere if it is possible that all the events existing at the beginning of some event ( or at the end) continue during a period when others begin and cease (or have previously existed during such a period). A sufficient condition for the existence of instants is

ii ! s � s I P I s.

Another is, putting M = S � P I S, S I P C: M I P, which ensures that every event has a first instant ; while, putting N = S � P I S, the condition u

u

S j P C: N I P

ensures that every event has a last instant.

Logical Positivism Few philosophers have altered the thought of their own times as powerfully and directly as Russell, and this has sometimes placed him in the anomalous position of bei.ng simultaneously a classic from the antiquity prior to the 1 9 1 4-1 8 war and an active and controversial contemporary. His great achievement, the PRINCIPIA MATHEMATICA was written (with Whitehead) while Russell was, by conventional standards, a young man; he was only forty-one at the time of the appearance of its final volume in 1 9 1 3 . Borrowing from an old joke, we have been obliged in the past three decades to distinguish between the great philosopher Bertrand Russell, whom we all respect, and Mr. B•rtr•nd R'""ss•ll, with whom we frequently disagree. The whole character of modern British philosophy (and that por­ tion of recent American philosophy which shows its influence) goes back to the revolutionary consequences of the thought of Russell and Moore at Cambridge in the nineties. The concept of philosophy, its problems and methods, developed by Russell and Moore was provided wi.th a rigorous procedure by the formulation, by Russell and White­ head, of a new logic of greater power and scope than any known previously. The life-cycles of philosophers are interesting. There are those, such as Hume, who do their greatest work as young men and turn, in later years, to other things; while others, such as Kant, are undistinguished in their youth and only in middle life reveal the unquestionable marks of genius. As Russell grew older his thoughts and writings introduce rhemes quite removed from those on which he centred his attention in the early years of the century, and many of the ideas he advanced in the years be/ore 1 9 20 were, in the long run, developed by others than himself. The essay on logical positivism reprinted here from the REVUE DES MATHEMATIQUES is therefore of considerable interest, since it surveys the chief philosophic tendencies of this century from the van­ tage point of one who, perhaps more than anyone else, was resfJonsible for them all. Portions of this paper were included in the text of HUMAN KNOW­ LEDGE : ITS SCOPE AND LIMITS ( 1 948). This is the first republication of the full text.

LOGICAL POSITIVISM is a name for a method, not for a certain kind of result. A philosopher is a logical positivist if he holds that there is no special way of knowing that is peculiar to philosophy, but that questions of fact can only be decided by the empirical methods of science, while questions that can be decided without appeal to experience are either mathematical or linguistic. Many members of the school would describe their position briefly as a determination to reject 'metaphysics', but 'metaphysics' is such a vague term that this description has no precise meaning. I should prefer to say that questions of fact cannot be decided without appeal to observation. For example : seventeenth century Contin­ ental philosophers would contend that the soul must be immortal because the soul is a substance and substances are indestructible. A logical positivist would reject this argument, but would not necessarily hold that the soul is mortal, since he might think that psychical research gives empirical evidence of survival. This, however, does not distinguish the logical positivist from earlier empiricists. What is distinctive is attention to mathematics and logic, and emphasis upon linguistic aspects of traditional philosophical problems. British empiricists, from Locke to John Stuart Mill, were very little influenced by mathematics, and had even a certain hostility to the outlook engendered by mathematics. On the other hand, Continental philosophers, down to Kant, re­ garded mathematics as the pattern to which other knowledge ought to approximate, and thought that pure mathematics, or a not dis­ similar type of reasoning, could give knowledge as to the actual world. The logical positivists, though they are as much interested in and influenced by mathematics as Leibniz or Kant, are complete empiricists, and are enabled to combine mathematics with empiri­ cism by a new interpretation of mathematical propositions. It was 'LOGICAL POSITIVIS M'

367

LOGIC AND KNOWLEDGE 3 68 work on the foundations of mathematics and on mathematical logic that gave the technical basis for the school, and without some understanding of this basis it is impossible to do justice to the grounds for their opinions. Mathematics, from Pythagoras onward, was mixed up with mysticism. Plato's eternal world was inspired by mathematics. Aristotle, though more empirical and more biological than Plato, still thought the capacity for doing sums so remarkable that the arithmetical part of the soul must be immortal. In modern times, Spinoza took geometry for his model, and hoped to deduce the nature of the universe from self-evident axioms, and Leibniz thought that all questions could be decided by reasoning, as appears in his account of what could be achieved by the Characteristica Universalis : 'If we had it, we should be able to reason in meta­ physics and morals in much the same way as in geometry and analysis',• and again : 'If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to their slates, and to say to each other (with a friend as witness, if they liked) : Let us calculate' .t Kant's theory of knowledge cannot be disentangled from his be­ lief that mathematical propositions are both synthetic and a priori. My own work on the principles of mathematics was to me, at first, mainly interesting as a refutation of the view that mathematical propositions assert more than can be justified by deductive logic. Hegel (especially in his Greater Logic) made a quite different use of mathematics. The great men of the seventeenth and eighteenth centuries were so much impressed by the results of their new methods that they did not trouble to examine their founda­ tions. Although their arguments were fallacious, a special Provid­ ence saw to it that their conclusions were more or less true. Hegel fastened upon the obscurities in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses. It is interesting that some of his worst absurdities in this field were repeated by Engels in the Anti­ Duhring, and that, in consequence, if you live in the Soviet Union

• Leibnizens gesammelte Werke (Pertz and Gerhardt's edition, Vol. VII, p. 2 1 . t Ibid., p . 200.

LO GICAL P O S I T I V I SM

and take account of what has been done on the principles of mathe­ matics during the last one hundred years, you run a grave risk of being liquidated. Let us enumerate a few of the errors that infected mathematics in the time of Hegel. There was no defensible definition of irra­ tional numbers, and consequently no ground for the Cartesian assumption that the position of any point in space could be de­ fined by three numerical co-ordinates. There was no definition of continuity, and no known method of dealing with the paradoxes of infinite number. The accepted proofs of fundamental proposi­ tions in the differential and integral calculus were all fallacious, and were supposed, not only by Leibniz, but by many later mathe­ maticians, to demand the admission of actual infinitesimals. As regards geometry, it was supposed that the truth of the Euclidean system could be known without any· appeal to observation. The resulting puzzles were all cleared up during the nineteenth century, not by heroic philosophical doctrines such as that of Kant or that of Hegel, but by patient attention to detail. The first step was Lobatchevsky's non-Euclidean geometry, which showed that only empirical observation can decide whether Euclidean geometry is true of actual space, and that geometry as a part of pure mathe­ matics throws no more light upon actual space than the multiplica­ tion table throws on the population of an actual town. The next step was Weierstra&s's work on the differential and integral calculus, which eliminated infinitesimals and substituted limits. Then came Georg Cantor's definition of continuity and arithmetic of infinite numbers. And last came Frege's definition of cardinal numbers, and his proof that arithmetic requires no concepts and no premises that are not required in deductive logic. It seems strange that, although numbers had been used for many thousands of years, no one could define either 'number' or any particular number until Frege did so in 1884. And what is perhaps even more strange is that no one noticed what Frege had achieved until I read him eighteen years later. The definition of the number I had great importance in clearing up metaphysical confusions. 'One' is a predicate of certain classes, e.g., 'satellite of the earth' ; but when a class has only one member, it is nonsense (in the strict sense) to say that that member is one, unless the unit class was a class of classes, in which case it is

L O G I C A N D KN O W L E D G E

generally false. E.g. you may say : 'In such-and-such a Parliament there is only one political party', but the party is not one, unless it has only one member. More generally, if I say 'there are three men in the room', the correct statement is 'the class of men in the room is a triad'. This may seem a trivial matter, but it is amazing how much bad metaphysics it refutes. For example : the scholastics used to say 'One and Being are convertible terms'. It now appears that 'one' is a predicate of concepts, not of the things to which the concepts are applicable ; the predicate 'one' applies to 'satellite of the earth' but not to the moon. And for other reasons 'being' applies only to certain de­ scriptions, never to what they describe. These distinctions, it will be found, put an end to many arguments of metaphysicians from Parmenides and Plato to the present day. This development in the pri'nciples of mathematics suggested that philosophical puzzles are to be solved by patience and clear think­ ing, the result being, in very many cases, that the original question is shown to be nonsensical. Logical positivism arose largely out of this suggestion. Carnap maintained at one time that all philoso­ phical problems arise from errors in syntax, and that, when these errors are corrected, the problems either disappear or are obviously not soluble by argument. I do not think he would still maintain quite so extreme a position, but there can be no doubt that correct logical syntax has an importance which was not formerly recog­ nized, and which logical positivists have rightly emphasized. Wittgenstein's Tractatus Logico-Philosophicus, published shortly after the First World War, laid great emphasis upon syntax, and provided a stimulus which helped in the formation of the 'Wiener Kreis', where logical positivism first took the form of a definite school. The Wiener Kreis and the admirable periodical Erkenntnis did excellent work, until they were ended by Hitler and the Anschluss. It was by the Vienna school that the hierarchy of lan­ guages was developed, a doctrine which I had briefly suggested as a way of escaping from Wittgenstein's rather peculiar syntactical mysticism. He had maintained that the/orm of a sentence can only be shown,, not stated. The apprehension of /orm, in his doctrine, was something that was ineffable in the strict sense, and only pos­ sible in virtue of some kind of mystical insight. This point of view was very alien to the spirit of logical positivism. It was therefore

LO GICAL POSITIVISM

37 1

natural that the Wiener Kreis, admitting Wittgenstein's problem, should seek other ways of solving it. It appeared that, given any language, it must have a certain in­ completeness, in the sense that there are things to be said about the language which cannot be said in the language. This is con­ nected with the paradoxes-the liar, the class of classes that are not members of themselves, etc. These paradoxes had appeared to me to demand a hierarchy of 'logical types' for their solution, and the doctrine of a hierarchy of languages belongs to the same order of ideas. For example, if I say 'all sentences in the language L are either true or false', this is not itself a sentence in the language L. It is possible, as Carnap has shown, to construct a language in which many things about the language can be said, but never all the things that might be said : some of them will always belong to the metalanguage'. For example, there is mathematics, but how­ ever 'mathematics' may be defined, there will be statements about mathematics which will belong to 'metamathematics', and must be excluded from mathematics on pain of contradiction. There has been a vast technical development of logic, logical syntax, and semantics. In this subject, Carnap has done the most work. Tarski's Der Begriff der Wahrheit in den formalisierten Sprachen is a very important book, and if compared with the attempts of philosophers in the past to define 'truth' it shows the increase of power derived from a wholly modern technique. Not that difficulties are at an end. A new set of puzzles has resulted from the work of Godel, especially his article Vber formal un­ entschei.dbare Satze der Principi,a Mathematica und verwandter Systeme ( 193 1 ), in which he proved that in any formal system it is possible to construct sentences of which the truth or falsehood cannot be decided within the system. Here again we are faced with the essential necessity of a heirarchy, extending upwards ad infi­ nitum, and logically incapable of completion. This whole subject has become so technical, and so capable of quasi-mathematical definiteness, that it can hardly be regarded as belonging to philosophy as formerly understood. True, it solves what were philosophical problems, but so did Newton in writing on what he still called 'natural philosophy'. But we do not now regard planetary theory as part of philosophy, and I think that on the same ground much of the recent work on logic, syntax, and

L O G I C AND K N O W L ED GE 372 semantics should b e regarded as definite knowledge, not philoso­ phical speculation. So far, I have been dealing with topics that arise out of the con­ sideration of mathematics and logic. I come now to what logical positivism has to say about empirical knowledge, and here I find myself, on some important points, no longer in agreement with most members of the school. There are here two not wholly disconnected questions. The one is as to scientific as opposed to deductive inference ; the other is as to what is meant by the 'significance' of a sentence. The question of scientific inference is one which has been acute ever since the time of Hume. It has been common to include under 'induction' all such inferences as would be considered valid in science but are not justified by the rules of deduction. I think myself that induction has less importance in this problem than is generally thought. What is clear, and generally admitted, is : ( I) That scientific as opposed to deductive inference can only make the conclusion probable ; ( 2) That it cannot even do this except by assuming postulates, or a postulate, for which there is, and can be, no empirical evidence. This is an awkward conclusion for an empiricist, but it seems to be unescapable. I shall not in this article deal further with this problem, but shall instead examine the doctrine that 'significance' and 'knowledge' are both confined to experience. Some modern empiricists-in particular, the majority of logical positivists-have, in my opinion, misconceived the relation of knowledge to experience. This has arisen, if I am not mistaken, from two errors : first, an inadequate analysis of the concept 'experience', and second, a mistake as to what is involved in the belief that some assigned property belongs to some (undetermined) subject. Two specific problems arise, one as regards significance, the other as regards knowledge of what are called 'existence pro­ positions' i.e. propositions of the form 'something has this pro­ perty'. It is maintained, on the one hand, that a statement is not 'significant' unless there is some known method of verifying it ; on the other hand, that we cannot know 'something has this property' unless we can mention a specific subject that has the property. I wish to give reasons for rejecting both these opm1ons.

LOGICAL POSITIVISM

3 73

Before examining the abstract logic of these two problems, let us consider them, for a moment, from a common-sense point of view. To begin with verification : There are some who maintain that, if atomic warfare is not checked, it may lead to the extermination of life on this planet. I am not concerned to maintain that this opinion is true, but only that it is significant. It is, however, one which cannot be verified, for who would be left to verify it if life were extinct ? Only Berkeley's God, whom, I am sure, logical positivists would not wish to invoke. Going backwards instead of forwards, we all believe that there was a time before there was life on the earth. Those who regard verifiability as necessary to signific­ ance do not mean to deny such possibilities, but in order to admit them they are compelled to define 'verifiability' somewhat loosely. Sometimes a proposition is regarded as 'verifiable' if there is any empirical evidence in its favour. That is to say, 'all A is B' is 'verifiable' if we know of one A that is B and do not know of one that is not B. This view, however, leads to logical absurdities. Suppose there is no single member of A concerning which we know whether it is a B, but there is an object x, not a member of A, which we know to be a B. Let A ' be the class consisting of the class A together with the object x. Then 'all A ' is B' is verifiable in terms of the definition. Since this impiies 'all A is B', it follows that 'all A is B' is verifiable if there is, anywhere, a single object known to be a B. Consider now a generalization of a different sort, such as we may wish to make in connexion with the doctrine of natural kinds. The generalizations I have in mind are those of the form : 'all predicates of the class A are true of the object B'. Applying the same definition of 'verifiability', this is 'verifiable' if some, or at least one, of the predicates of the class A is empiric­ ally known to be true of B. If this is not the case, let P be some predicate known to be true of B, and let A ' be the class consisting of the class A together with P. Then 'all predicates of the class A ' are true of B' is verifiable, and so, therefore, is 'all predicates of the class A are true of B'. From these two processes it follows that, if anything is known to have any predicate, all generalizations are 'verifiable'. This consequence was not intended, and shows that the above wide

374

L O G I C AND KN O W L E D G E

definition of 'verifiability' is useless. But unless we allow some such wide definition, we cannot escape from paradoxes. Let us next consider propositions containing the word 'some', or an equivalent, e.g. 'some men are black', or 'some quadrupeds have no tails'. As a rule, such propositions are known by means of instances. If I am asked 'how do you know that some quadrupeds have no tails ?' I may reply 'because I once had a Manx cat, and it had no tail'. The view which I wish to combat maintains that this is the only way of knowing such propositions. This view has been maintained by Brouwer in mathematics, and is maintained by some other philosophers in regard to empirical objects. The paradoxes resulting from this opinion are very similar to those resulting from the above doctrine as to verifiability. Take such a proposition as 'rain sometimes falls in places where there is no one to see it'. No sane person would deny this, but it is impossible to mention a raindrop that has never been noticed. To deny that we know that there are occurrences not observed by any one is incompatible with common sense, but is necessary if we never know such propositions as 'there are A's' except when we can men­ tion A's that we have observed. Can any one seriously maintain that the planet Neptune or the Antarctic Continent did not exist until it was discovered ? Again only a Berkelian God will enable us to escape from paradoxes. Or again : we all believe that there is iron in the interior of the earth, but we cannot give instances beyond the depth of the deepest mine. Adherents of the doctrine that I am combating interpret such facts hypothetically. They say that the statement 'there is undis­ covered iron' is an abbreviation, and that the full statement should be : 'if I did certain things, I should discover iron'. Suppose, for the sake of precision, we take the statement 'there is iron more than 1 ,ooo kilometres below the surface of the earth'. It is unlikely that anybody will ever find this iron, and, in any case, how can it be known what a person would find ? Only by knowing what is there to be found. A hypothetical of which the hypothesis will probably always be false tells us nothing. Or consider : 'there was once a world without life'. This cannot mean : 'if I had been alive then, I should have seen that nothing was alive'. Let us now consider the above two doctrines more formally, from a strictly logical point of view.

LOGICAL POSITIVISM

3 15

A. M E AN I N G A N D V E R I F I C A T I O N

There is a theory that the meaning of a proposition consists in its method of verification. It follows (a) that what cannot be verified or falsified is meaningless, (b) that two propositions verified by the same occurrences have the same meaning. I reject both, and I do not think that those who advocate them have fully realized their implications. First : practically all the advocates of the above view regard verification as a social matter. This means that they take up the problem at a late stage, and are unaware of its earlier stages. Other people's observations are not data for me. The hypothesis that nothing exists except what I perceive and remember is for me identical, in all its verifiable consequences, with the hypothesis that there are other people who also perceive and remember. If we are to believe in the existence of these other people-as we must do if we are to admit testimony-we must reject the identification of meaning with verification. 'Verification' is often defined very loosely. The only strict mean­ ing of verification is the following : A proposition asserting a finite number of future occurrences is 'verified' when all these occur­ rences have taken place, and are, at some moment, perceived or remembered by some one person. But this is not the sense in which the word is usually employed. It is customary to say that a general proposition is 'verified' when all those of its consequences which it has been possible to test have been found to be true. It is always assumed that, in that case, probably the consequences which have not been tested are also true. But this is not the point with which I am concerned at present. The point with which I am concerned at the moment is the theory that two propositions whose verified consequences are identical have the same significance. I say 'veri­ fied', not 'verifiable' ; for we cannot know, until the last man perishes, whether the 'verifiable' consequences are identical. Take, e.g. 'all men are mortal'. It may be that on 9 February, 1991, an immortal man will be born. The presently verifiable consequences of 'all men are mortal' are the same as those of 'all men born before the time t are mortal, but not all those born later', where t is any time not more than a century before the present. If we insist upon using the word 'verifiable' rather than

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'verified', we cannot know that a proposition is verifiable, since this would involve knowledge of an indefinitely long future. In fact, that a proposition is verifiable is itself not verifiable. This is be­ cause to state that all the future consequences of a general pro­ position are true is itself a general proposition of which the in­ stances cannot be enumerated, and no general proposition can be established on purely empirical evidence except one applying to a list of particulars all of which have been observed. E.g. I may say 'the inhabitants of such-and-such a village are Mr. and Mrs. A, Mr. and Mrs. B, etc., and their families, all of whom are known to me personally ; and all of them are Welsh'. * But when I cannot enumerate the members of a class, I cannot, on purely empirical grounds, justify any generalization about its members except what follows analytically from its definition. There is however still a point to be made in favour of the verifiers. They contend that there is a distinction between two kinds of cases. In one, we have two propositions whose consequences hitherto have been indistinguishable, but whose future conse­ sequences may diverge ; e.g. 'all men are mortal' and 'all men born before A.D. 2000 are mortal'. In the other, we have two proposi­ tions whose observable consequences can never diverge ; this is especially the case with metaphysical hypotheses. The hypothesis that the starry heavens exist at all times, and the hypothesis that they only exist when I see them, are exactly identical in all those of their consequences that I can test. It is specially in such cases that meaning is identified with verification, and that, therefore, the two hypotheses are said to have the same significance. And it is this that I am specially concerned to deny. Perhaps the most obvious case is other people's minds. The hypothesis that there are other people, having thoughts and feel­ ings more or less like my·own, does not have the same significance as the hypothesis that other people are only parts of my dreams, and yet the verifiable consequences of the two hypotheses are identical. We all feel love and hate, sympathy and antipathy, ad­ miration and contempt, for what we believe to be real people. The emotional consequences of this belief are very different from those of solipsism, though the verifiable consequences are not. I • Such general enwnerative statements involve many difficulties, but I will ignore them.

3 77 should say that two beliefs whose emotional consequences differ have substantially distinct significations. But this is a practical argument. I should go further, and say, as a matter of pure theory, that you cannot, without incurring an endless regress, seek the significance of a proposition in its conse­ quences, which must be other propositions. We cannot explain what is the significance of a belief, or what makes it true or false, without bringing in the concept 'fact', and when this is brought in the part played by verification is seen to be subsidiary and deriva­ tive. LOGICAL POSITIVISM

B. I N F E RE N T I A L E X I S T E N C E - P RO P O S I T I O N S

A form of words containing an undetermined variable-for instance, 'x is a man'-is called a 'propositional function' if, when a value is assigned to the variable, the form of words becomes a proposition. Thus 'x is a man' is neither true nor false, but if for 'x' I put 'Mr. Jones' I get a true proposition, and if I put 'Mrs. Jones' I get a false one. Besides giving a value to 'x', there are two other ways of obtain­ ing a proposition from a propositional function. One is to say that the propositions obtained by giving values to 'x' are all true ; the other is to say that at least one of them is true. If 'f (x )' is the func­ tion in question, we will call the first of these 'f (x) always' and the second 'f (x) sometimes' (where it is understood that 'sometimes' means 'at least once'). If 'f (x)' is 'x is not a man or x is mortal', we can assert 'f (x) always' ; if 'f (x)' is 'x is a man', we can assert 'f (x) sometimes', which is what we should commonly express by saying 'there are men'. If 'f (x)' is ' I met x and x is a man', 'f (x) sometimes' is ' I met at least one man'. We call 'f (x) sometimes' an 'existence-proposition' because it says that something having the property f (x) 'exists'. For instance, if you wanted to say 'unicorns exist', you would first have to define 'x is a unicorn' and then assert that there are values of x for which this is true. In ordinary language, the words 'some', 'a', and 'the' (in the singular) indicate existence-propositions. There is one obvious way in which we get to know existence­ propositions, and that is by means of instances. If I know '/ (a)', where a is some known object, I can infer 'f (x) sometimes'. The

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question I wish to discuss is whether this is the only way i n which such propositions can come to be known. I wish to maintain that it is not. In deductive logic, there are only two ways in which existence­ propositions can be proved. One is the above, when '/ (x) some­ times' is deduced from 'f (a)' ; the other is when one existence­ proposition is deduced from another, for instance 'there are bipeds' from 'there are featherless bipeds'. What other methods are pos­ sible in non-deductive inference ? Induction, when valid, gives another method. Suppose there are two classes A and B and a relation R, such that, in a number of observed instances, we have (writing 'a R b' for 'a has the rela­ tion R to b') a1 is an A. b1 is a B. a1 R b1

a2 is an A. b2 is a B. a2 R b2 a,,,, is an A. b,,,, is a B. an R bn and suppose we have no contrary instances. Then in all observed instances, if a is an A, there is a B to which a has the relation R. If the case is one to which induction applies, we infer that probably every member of A has the relation R to some member of B. Con­ sequently, if an+l is the next observed member of A, we infer as probable : 'there is a member of B to which a,,,,+1 has the relation R'. We infer this, in fact, in many cases in which we cannot adduce any particular member of B such as we have inferred. We all believe that probably Napoleon III had a father, although no one has ever known who he was. Not even a solipsist, if he allows him­ self any views as to his own future, can escape from this sort of induction. Suppose, for instance, that our solipsist suffers from intermittent sciatica, which comes on every evening ; he may say, on inductive grounds, 'probably I shall be suffering pain at 9 p.m. tonight'. This is an inference to the existence of something tran­ scending his present experience. 'But,' you may say, 'it does not transcend his future experience.' If the inference is valid it does not ; but the question is : 'how is he to know now that the inference is probably valid ?'. The whole practical utility of scientific infer­ ence consists in giving grounds for anticipating the future ; when

3 79 the future has come and has verified the inference, memory has re­ placed inference, which is no longer needed. We must, therefore, find grounds for trusting the inference be/ore it is verified. And I defy the world to find any such grounds for trusting inferences which will be verified, which are not equally grounds for trusting certain inferences which will be neither verified nor falsified, such as the inference to Napoleon Ill's father. We are faced with the question : in what circumstances is induc­ tion valid ? It is futile to say : 'Induction is valid when it infers something which subsequent experience will verify'. This is futile, because it would confine induction to cases in which it is useless. We must have reasons, in advance of experience, for expecting something, and exactly similar reasons may lead us to believe in something that we cannot experience, for example, the thoughts and feelings of other people. The plain fact is that much too much fuss is made about 'experience'. Experience is needed for ostensive definition, and therefore for all understanding of the meanings of words. But the proposition 'Mr. A had a father' is completely intelligible even if I have no idea who Mr. A's father was. If Mr. B was in fact Mr. A's father, 'Mr. B' is not a constituent of the statement 'Mr. A had a father', or, indeed, of any statement containing the words 'Mr. A's father' but not containing the name 'Mr. B'. Similarly I may understand 'there was a winged horse' although there never was one, because the statement means that, putting 'fx' for 'x has wings and is a horse', I assert 'fx sometimes'. It must be understood that 'x' is not a constituent of 'fx sometimes' or of 'fx always'. In fact, 'x' means nothing. That is why beginners find it so hard to make out what it means. When I infer something not experienced-whether I shall or shall not experience it hereafter-I am never inferring something that I can name, but only the truth of an existence-proposition. If induction is ever valid, it is possible to know existence-proposi­ tions without knowing any particular instance of their truth. Sup­ pose, for instance, that A is a class of which we have experienced some members, and we infer that a member of A will occur. We have only to substitute 'future members of A' for 'members of A' to make our inference apply to a class of which we cannot mention any instance. LOGICAL P O S I T I V ISM

LOGIC AND KNOWLEDGE 3 80 I incline to think that valid inductions, and, generally, inferences going beyond my personal past and present experience, always depend upon causation, sometimes supplemented by analogy. But, in the present article, I wished only to remove certain a priori objections to a certain kind of inference-objections which, though a priori, are urged by those who imagine themselves able to dis­ pense with the a priori altogether. There is, I think, a danger that logical positivism may develop a new kind of scholasticism, and may, by being unduly linguistic, forget the relation to fact that makes a statement true. I will give one illustration of what I should regard as scholasticism in the bad sense. Carnap, and others of the same school, have very rightly pointed out the confusions that arise when we do not distinguish adequately between using a word and naming it. In ordinary cases this danger does not arise. We say : Socrates was a man, 'Socrates' is a word of eight letters. We can go on to say : 'Socrates' is the name of a man, but ' "Socrates" ' is the name of a word. The usual method of designating a word by putting it in inverted commas is useful, presupposing that we know what a word is. But difficulties arise when we carry on the same process as regards sentences or propositions. If the words 'To-day is Tuesday' occur without quotes, the sentence is used, not named. But when the sentence occurs in quotes, what am I naming ? Do I include sentences, in other languages, which have the same meaning, e.g. 'aujourd'hui est mardi' ? Conversely, suppose that in the Hottentot language the noise 'to-day is Tuesday' means ' I like cheese'. Clearly this should not be included in what I designate by the phrase in quotes. We must say : a sentence in quotes designates the class of those utterances, in no matter what language, that have the significance that I attach to the sentence when I use it : we must include the Frenchman's phonetically dissimilar remark, and exclude the Hottentot's phonetically similar remark. It follows that we cannot tell what is designated by a sentence in quotes without first in­ vestigating what is meant by saying that two utterances have the same significance. At any rate, we cannot do so if the sentences, when used, are to be possible values of the propositional variables used in logic, e.g. when we say 'if p implies q, then not-q implies not-p'. This shows that what is meant when a set of words is put in quotes is not such a simple matter as one would sometimes

L O GI CAL POSITIVISM

suppose from the works of some logical positivists. I n this way, there is danger of a technique which conceals problems instead of helping to solve them. Absorption in language sometimes leads to a neglect of the con­ nexion of language with non-linguistic facts, although it is this connexion which gives meaning to words and significance to sentences. No one can understand the word 'cheese' unless he has a non-linguistic acquaintance with cheese. The problem of mean­ ing and significance requires much that is psychological, and demands some understanding of pre-linguistic mental processes. Most logical positivists fight shy of psychology, and therefore have little to say about meaning or significance. This makes them, in my opinion, somewhat narrow, and not capable of producing an all-round philosophy. They have, however, the great merit that their method allows them to tackle problems one by one, and that they are not obliged, as philosophers used to be, to produce a com­ plete theory of the universe on all occasions. Their procedure, in fact, is more analogous to that of science than to that of traditional philosophy. In this respect I am wholly at one with them. I value their rigour, exactness, and attention to detail, and speaking broadly, I am more hopeful of results by methods such as theirs than by any that philosophers have employed in the past. What can be ascertained, can be ascertained by methods such as theirs ; what cannot be ascertained by such methods we must be content not to know. There is one matter of great philosophic importance in which a careful analysis of scientific inference and logical syntax leads-if I am not mistaken-to a conclusion which is unwelcome to me and (I believe) to almost all logical positivists. This conclusion is, that uncompromising empiricism is untenable. From a finite number of observations no general proposition can be inferred to be even probable unless we postulate some general principle of inference which cannot be established empirically. So far, there is agreement among logical positivists. But as to what is to be done in conse­ quence there is no agreement. Some hold that truth does not con­ sist in confonnity with fact, but only in coherence with other propositions already accepted for some undefined reason. Others, like Reichenbach, favour a posit which is a mere act of will and is admitted to be not intellectually justified. Yet others make attempts

LOGIC AND KNOWLEDGE

-to my mind futile-to dispense with general propositions. For my part, I assume that science is broadly.speaking true, and arrive at the necessary postulates by analysis. But against the thorough­ going sceptic I can advance no argument except that I do not be­ lieve him to be sincere.