Wittgenstein's Notes on Logic

Wittgenstein’s Notes on Logic

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Wittgenstein’s Notes on Logic

Michael Potter

3

3 Great Clarendon Street, Oxford OX DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Michael Potter 2008 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2008 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organizations. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by the author Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk ISBN 978–0–19–921583–6 10 9 8 7 6 5 4 3 2 1

One often makes a remark and only sees later how true it is. Wittgenstein, 10 October 1914

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Preface I wrote the first draft of this book while I was a Senior Research Fellow in the Department of Philosophy at Stirling University, funded by the AHRC. The friendly welcome I received there, in a department which I came to see as an oasis of academic seriousness and respect, contributed very markedly to my well being and hence to the successful completion of the project. Such conducive research environments are rare; I suspect that university administrators who recognize them when they arise, and cherish them as they should, are even rarer. A happy feature of the AHRC project was a series of workshops and a concluding conference on the Tractatus and its history, at various of which I tried out some of the ideas expressed here. I have pleasant memories of the constructive atmosphere at these meetings, and am grateful for all the feedback I received, even when it was no more articulate than a raised eyebrow. Something similar goes for a series of seminars on the first draft of the book which I conducted in Cambridge in the Lent Term, 2006. I am sure the final version is better as a result. In the archival research for the book I have been assisted by librarians at the Houghton Library, Harvard University; the Russell Archive, McMaster University; and the Manuscripts Room, Cambridge University Library. I am grateful to Blackwell Publishing for permission to reprint the Notes on Logic as an appendix. In compiling the version printed here I was grateful for earlier editorial work by Michael Biggs. Will Crouch compiled the index for me. Nicholas Griffin, Stephen Read, David Cardwell, and Brian McGuinness have responded helpfully to requests for information. I have also benefited from detailed comments on the first draft by some unusually generous colleagues. I am very grateful to Ian Proops, Peter Sullivan, and two anonymous readers for Oxford University Press, not least because each of them has saved me from a number of embarrassing displays of my own ignorance and stupidity. Wittgenstein scholars who read this book will, of course, look first for their own names in the index. Most will, I fear, come away disappointed. If I had footnoted everyone I have read, and explained in full what I thought of their

views, the book would have been twice as long and twice as late, but not, I suspect, twice as useful. My gratitude for what I have gained from their work is no less sincere for remaining unitemized. To Brian McGuinness, however, I owe a particular debt which it would be churlish not to single out: his editorial work, most notably on Wittgenstein’s letters, has saved me an enormous amount of time; and his writings on Wittgenstein display a combination of historical accuracy and philosophical acuity which I can only dream of emulating. It was Peter Sullivan who arranged the AHRC project in the first place and hence created the circumstances in which it was possible for me to write this book. I must have discussed almost every idea in it with him at some point, and many were no doubt originally his. Since neither of us now knows which these are, and some of those which are certainly his he now prefers to deny, this preface is the only feasible place to record that debt. MDP

Contents Contents

vii

Introduction

1

1 Finding a problem

6

1.1 1.2 1.3 1.4 1.5

Early life Manchester The Principles of Mathematics Logicism Russell’s paradox

2 First steps 2.1 2.2 2.3

Cambridge On denoting Sense-data

6 7 10 13 15

18 18 20 23

3 Matter

26

3.1 3.2 3.3 3.4 3.5

26 27 30 33 36

The project On matter Dawes Hicks The relation of sense-data to physics The atomistic assumption

4 Analysis 4.1 4.2 4.3

Inference or construction? Wittgenstein’s conception Practicalities

39 39 43 45

viii

Contents

5 The fundamental thought 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Why logic? Logical constants as incomplete symbols There are no logical constants There are no real variables Logic as a special science Logic as contentless The fundamental thought

6 The symbolic turn 6.1 6.2 6.3 6.4

Propositions The rejection of psychologism The reliability of language Conflicting conceptions

7 Simplicity 7.1 7.2 7.3 7.4

Realism Solipsism Idealism Reconciliation

49 49 51 52 54 57 58 61

63 63 64 66 68

70 70 73 74 76

8 Unity

78

8.1 8.2 8.3

78 80 82

The copula There cannot be different types of things The theory of types is superfluous

9 Fregean propositions 9.1 9.2 9.3 9.4

Frege’s notion of assertion Propositions are not names of truth-values Whose influence? Propositions as articulate

10 Assertion 10.1 10.2 10.3 10.4

The judgment stroke as force indicator Asserted and unasserted propositions Assertion as psychological Psychology

11 Complex and fact 11.1 A world of facts, not of things 11.2 Influences 11.3 Russell on facts

86 86 88 90 91

94 94 95 98 99

102 102 105 106

Contents

ix

12 Forms

109

12.1 12.2 12.3 12.4 12.5

109 111 113 114 116

Form as name Form as function The form of a fact The unity of the proposition The symbolic turn again

13 Russell’s theory of judgment 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

The original multiple relation theory A problem for the original theory Russell’s revised theory Wittgenstein’s further objection Acquaintance Another formulation The fate of the multiple relation theory Other accounts

14 Meaning 14.1 14.2 14.3 14.4

Russell’s lectures on logical atomism Propositions are not names of their meanings Meanings as facts The demise of propositional meaning

15 Metaphysics 15.1 15.2 15.3 15.4 15.5

Disjunctive facts Negative facts Summing facts General facts Logical data

16 Sense 16.1 16.2 16.3 16.4

Semantic value The semantic value of a form The compass-needle analogy Grain

17 Truth-functions 17.1 17.2 17.3 17.4

Using primitive signs Truth-tables Truth-diagrams Comparison

118 118 120 122 124 125 126 128 129

132 133 135 137 140

142 142 143 145 147 149

151 151 153 155 156

158 158 160 160 163

x

Contents

18 Truth-operations

165

18.1 The problem 18.2 The solution 18.3 Duality

165 166 168

19 Molecular propositions 19.1 19.2 19.3 19.4

Terminology Which fact? Poles The inputs

20 Generality 20.1 Variables as classes of propositions 20.2 Notation 20.3 Undecidability

21 Resolving the paradoxes 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8

Russell’s theory of types Wittgenstein’s vicious circle principle Types as classes of propositions Types and molecular propositions Types and generality Uniting generality and truth-functions The general form of proposition Unsayability

22 Typical ambiguity 22.1 Typical ambiguity 22.2 Independent indefinables 22.3 Whitehead

23 Identity 23.1 Russell’s definition 23.2 Eliminating identity 23.3 The notational problem

24 Sign and symbol 24.1 24.2 24.3 24.4 24.5

Seeing through to the symbol Same sign, different symbol Same symbol, different sign Symbol in terms of sign The symbol vanishes

170 170 171 173 174

177 177 180 181

184 184 186 187 189 190 191 192 193

196 196 199 200

204 204 206 207

209 209 210 212 214 216

Contents 25 Wittgenstein’s theory of judgment 25.1 25.2 25.3 25.4

Russell’s later views The theory of judgment in the Notes Wittgenstein’s later theory of judgment Ramsey

26 The picture theory 26.1 26.2 26.3 26.4 26.5

Coincidence of structure The picturing analogy Truth The identity theory Possibility

27 Tractarian objects 27.1 27.2 27.3 27.4

Relations as objects Widening the scope Facts in the Tractatus Confusion?

28 Philosophy 28.1 28.2 28.3 28.4

Metaphysics Psychology Epistemology Value

xi 218 218 219 220 222

224 224 226 227 229 231

232 232 233 236 237

241 241 243 244 245

29 Themes

249

29.1 29.2 29.3 29.4 29.5 29.6 29.7 29.8

249 250 253 254 255 259 260 261

Working methods Characteristics What if Fundamental thoughts Influences on Wittgenstein Influence on Russell Influence on Frege Conclusion

A History of the text A.1 A.2 A.3 A.4 A.5 A.6

Narrative The manuscripts Russell’s labelling The Costello version Wittgenstein’s dissertation Conclusion

263 263 265 268 271 274 274

xii

Contents

B The Notes on Logic

276

The Birmingham Notes The Cambridge Notes Textual notes The Costello version

276 284 290 292

Citations Notes on Logic Tractatus

297 297 297

Index

299

Bibliography

305

Primary sources Secondary sources

305 306

Introduction In 1911 Wittgenstein arrived in Cambridge to study philosophy with Russell. In 1913, just before he left to spend a year communing with his soul in Norway, he produced the Notes on Logic, a sort of summary of the conclusions he had reached during his time in Cambridge, and his first philosophical work. My intention in what follows is to engage in a study of that period and this work. Most of Wittgenstein’s surviving pre-Tractarian writings were published, in stages, some time ago (in 1957, 1961, and 1979). Since then it has been a familiar method of all serious exegetes of the Tractatus to mine these writings for remarks to support their interpretations. One reason they do this is simply because they can: although Wittgenstein had his prewar Cambridge notebooks destroyed, and two of his later notebooks are probably missing, what remains (the Notes on Logic already mentioned, the Notes dictated to Moore, three surviving wartime notebooks, and a handful of letters) is a body of evidence of a scale nicely poised to intrigue but not intimidate the diligent scholar. But it is also because they must: some of the remarks in the Tractatus are so obscure that only by relying on the earlier notebooks for support do we have any hope of divining their meaning. Although I hope in this book to contribute to the same project of Tractarian exegesis, I aim to do so by a somewhat different method. Instead of studying the Tractatus, and drawing on Wittgenstein’s earlier writings only when they contribute to understanding it, I shall here be focusing on the 1913 Notes on Logic, treating them if not quite as a terminus in Wittgenstein’s work then at least as worthy of study in their own right. There are several benefits to be derived from this approach. One is that it gives us a better chance of understanding Wittgenstein’s own reasons for some of the views he held. This is probably a good reason to study the early works of almost any major philosopher, but it is especially so with Wittgenstein, whose own arguments for his views are so often too compressed to be comprehensible without understanding the context in which he formulated them. An obvious example is that commentators have presented a remarkably large number of Wittgenstein’s logical claims as consequences of his picture theory. But not all



Introduction

of that theory is present in the Notes. In particular, the analogy with pictures which gives the theory its name came to him almost a year after he compiled them. Understanding the parts of the theory which were already in place by then puts us in a much better place to understand what further work he thought there was for the picturing analogy to contribute. Another benefit has to do with Wittgenstein’s method of working, by which remarks were first written down, then compiled and rearranged (almost endlessly in the case of some of his later work). These rearrangements sometimes gave the remarks, in the resulting juxtapositions, resonances which they did not originally have. However allusive these resonances may be, however possible it is that Wittgenstein may sometimes have been struck by them and used his final arrangement of the text to invite them, the fact remains that it is likely to be worthwhile to study the remarks in their original setting. But I also hope here to recommend, by exemplification, an approach to the study of the Tractatus that, if not actually denying the boundary between biography and philosophy, at least regards the interaction between them as potentially fruitful. I do not apologize for introducing biographical observations into what is primarily a philosophical study; and the reason is that I have found it often helps, in judging which interpretation of Wittgenstein’s remarks is plausible, to have a sense of how he thought and how he worked. If we are to gain the maximum insight from his work, we need to understand, certainly, what motivated him to address the problem he did in the way that he did. But the exegetical task of understanding him aright can at times seem harder with Wittgenstein than with some other philosophers. (Russell is an obvious example.) One thing that makes it easier, though, is the striking unity, if not in Wittgenstein’s thought then in his method of thought. Almost all of his ideas are, in a certain sense, simple. Once we have grasped the sort of simplicity that is in question, it can then become a useful measure by which to assess our interpretations in the future. And it is here that an understanding of the man is relevant. Wittgenstein wrote the Tractatus during the First World War, of course, but it had its birth in the two years he spent working in Cambridge with Russell between 1911 and 1913. He compiled the Notes on Logic at the very end of that period, as a summary for Russell (and perhaps, to an extent we shall discuss later, for himself) of the work he had accomplished. The destruction of his notebooks, mentioned earlier, makes the Notes almost our only guide to the work he had been doing in Cambridge. By studying them, therefore, we can hope to discover which of his ideas Wittgenstein owed to this period and which to the very different circumstances in which he worked later, first in Norway and then on active service during the war. We can also hope to lay bare some of the influences which helped to form his views. The acknowledgment Wittgenstein made in the preface to the Trac-

Introduction  tatus to ‘the great works of Frege and the writings of my friend Bertrand Russell’ is endlessly quoted. So, too, is a diary entry from 1931 in which he listed Frege and Russell (along with others) as people whose ideas he had used in what he called his ‘work of clarification’.1 But there is some gap between acknowledging an influence and determining what that influence was; and in any case we are by no means compelled to assume that Wittgenstein was conscious of, and chose to acknowledge, all the influences that shaped his thinking. Once again, though, the benefits will not just be biographical: knowing where he got his ideas from is often a useful tool for understanding what those ideas really were. Wittgenstein’s writings have been worked over so thoroughly in the half century since his death that the lack until now of any book-length study of the Notes on Logic is rather remarkable in itself. This is especially so when one pays attention to the significance of their timing, just highlighted. One reason for this neglect may be that the Notes are, even by Wittgenstein’s gnomic standards, hard to understand on a first reading. If they were, to some extent at least, compiled only for Russell’s benefit (or for Wittgenstein’s own), then that is of course part of the explanation. But another part of the explanation lies in the rather complicated circumstances of their composition, which have furnished us with a text (or rather a series of texts) whose convoluted and repetitive structure has compounded the difficulties in comprehension. An important aspect of this book, therefore, will be to disentangle these texts in order to leave the way to philosophical understanding of Wittgenstein’s intentions much clearer. That historical detective work will be the subject of Appendix A, and the Notes on Logic themselves are reprinted with a critical apparatus as Appendix B. I use Bn and Cn to mean the nth paragraphs of the Birmingham and Cambridge versions of the Notes respectively. (Decimal numbers unqualified are, of course, references to the Tractatus.) The bulk of the book itself is taken up with exegesis—not, certainly, of every sentence of the Notes, but at least of what I take to be their central claims. The aim will be to show that, once the problematic structure of the surviving text is disentangled, the Notes are a much more coherent and substantial work than has hitherto been recognized. The reader will no doubt notice, however, that the Notes on Logic are not, to begin with, mentioned very often. The reason for this lies in the fact that I have attempted here to describe not only the contents of these Notes, narrowly conceived, but the whole of Wittgenstein’s period working with Russell in Cambridge. Just what part of that period Wittgenstein saw the Notes as summarizing is now hard to determine—I shall discuss this question further in §11.3—but even if he viewed them as a report on all of his discoveries up to that point which he thought worth preserving, it would be natural for many 1

CV, 19.



Introduction

more of those discoveries to have occurred to him in the second year of study than in the first: that first year was, after all, when he came to Cambridge as, in effect, a self-taught philosophical novice. In relation to his first year of study, therefore, our evidence concerning what Wittgenstein was thinking is much more conjectural: there is little hard evidence apart from a couple of letters to Russell. The aim of the first half dozen chapters, therefore, will be to make those conjectures that seem possible concerning the work Wittgenstein did before what is in the Notes on Logic. In practice, this will involve us to a considerable extent at first in examining Russell’s work during this period. Such an examination would be appropriate in any case, since it constitutes the context in which Wittgenstein was working. But the closeness and complexity of the working relationship between Wittgenstein and Russell makes the latter’s work during this period of more than usual importance. It is clear that for much of this period they saw themselves as collaborators in a common project. So Russell’s writings can give us significant clues to what Wittgenstein’s own views were. In my discussions of Wittgenstein’s and Russell’s work I shall generally adopt their logical notation. In particular, the reader needs to be familiar with the following: ∼p pvq p q p⊃q ( x) φ ( x) ( x) φ ( x)

not- p p or q p and q If p then q (in Russell’s dubious idiolect: ‘ p implies q’) For all x, φ ( x) (Russell: ‘φ ( x) is always true’) For some x, φ ( x) (Russell: ‘φ ( x) is sometimes true’)

E

Russell usually used dots rather than brackets to indicate scope, but readers not at home with this convention can probably let the sense carry them through. It is also worth emphasizing at the outset that Wittgenstein did not show much sign of wanting Quine’s distinction between use and mention. One of the emerging themes of the book will be what Wittgenstein owed to Frege and what to Russell, but one thing which a casual inspection of the Notes on Logic tells us is that he was not inclined while he was compiling them towards Frege’s pedantry in the use of quotation marks: for Wittgenstein they are sometimes a naming device, but sometimes no more than a form of parenthesis. And Wittgenstein’s propositional letters ‘ p’, ‘q’, etc. are sometimes schematic, sometimes not. For instance, he plainly intended ‘ p’ is true if and only if p as a schema to stand for ‘Snow is white’ is true if and only if snow is white

Introduction  and similar propositions. I shall take that as a general licence, in expounding Wittgenstein’s thought, to be no prissier than he was about the use–mention distinction, except occasionally when it seems to matter.

Chapter 

Finding a problem What led Wittgenstein to study philosophy with Russell in Cambridge? The narrative of Wittgenstein’s life before 1911 is well summarized in the available biographies,1 so I shall confine myself in this chapter to picking out a few points that deserve emphasis in the current context. . Early life Wittgenstein’s father, a steel magnate, was one of the richest men in Austria. He was not an aristocrat, but Wittgenstein evidently acquired in his youth many of the attitudes of the rich. (One example, perhaps, is the tendency he had in the early part of his life to dabble in various fields, a tendency which displays a sort of enthusiastic amateurism sometimes to be observed in the independently wealthy.) Between the ages of 14 and 17 Wittgenstein was educated at the Oberrealschule in Linz. This was, notoriously, the same school as Hitler, but they overlapped for only one year (1903–4), during which Hitler was in class IIIA and Wittgenstein in class V, and there is no reason to think that Hitler influenced Wittgenstein’s philosophy any more than that Wittgenstein influenced Hitler’s anti-Semitism. Wittgenstein’s Jewishness is in any case not a prominent theme in accounts of his early life: all four of his grandparents were baptized and, although he was no doubt aware of his Jewish roots, I know of no reason to think that he became at all self-conscious about them until much later. (Some of the remarks his acquaintances made about him when he came to Cambridge display casual amusement at the oddity of foreigners, but I have come across none that allude to his Jewishness.)2 In 1906, when he was seventeen, Wittgenstein went to Germany to spend three semesters studying engineering at the Technische Hochschule in Charlottenburg (a suburb of Berlin). Stories from his later life attest to his fascination with how things work, his capacity for spatial reasoning, and his ability to mend quite complicated pieces of machinery. Although this may have been to some extent a natural talent, the training at the Technische Hochschule, which emphasized the practical over the abstract, was specifically designed to foster 1 McGuinness, Young Ludwig; Monk, The Duty of Genius. idea of Jewishness’.

2

Cf. McGuinness, ‘Wittgenstein and the

Manchester



it. On the other hand, the mathematical component of the training was rather limited: higher mathematics (i.e. differential and integral calculus and analytic geometry) in the first year of the course, and mechanics in both years. In addition there were what the course timetable3 describes as descriptive geometry and graphical statics, but these will have been practical courses, more technical drawing than anything that we would nowadays regard as mathematics. This point bears emphasis, if only because some writers on Wittgenstein’s philosophy have overestimated his mathematical knowledge—assuming, perhaps, that his training as an engineer included a substantial mathematical education. Evidence that Wittgenstein made some sort of effort to extend his mathematical knowledge has survived in his copy of the German translation of Lamb’s Hydrodynamics, which he presumably bought in Berlin around this time. What is curious about the volume is the markings Wittgenstein made in the margins—not so much the fact that they occur only in the first four of the twelve chapters of the book (although that does perhaps suggest that his interest was that of a dilettante rather than a serious student), but the nature of the comments themselves. Apart from a few corrections of obvious misprints in the text, Wittgenstein’s marginalia are almost all rewordings that he seems to have regarded as stylistic improvements.4 One might, of course, see this as an early sign of Wittgenstein’s later deliberate interest in language and preciseness of expression, but there is something else about them too: the impression one has is almost that Wittgenstein was not really interested in the mathematics at all; or, if he was, one would like to have been able to explain to him that this was not the right way to go about studying a mathematical text. . Manchester In 1908, after a short period constructing kites for meteorological research, Wittgenstein became a research student at Manchester. The university he was joining was one of the leading scientific research institutions in the world. It seems that he originally hoped to work with Rutherford, the professor of physics, who had just been awarded the Nobel prize; Chadwick (later to win the Nobel prize for discovering the neutron) was Rutherford’s assistant; Geiger and Marsden were performing their famous experiments on the scattering of alpha particles; De Hevesy, who joined the department two years later, would receive the Nobel prize for his work on the use of isotopes as tracers. The chemistry department, unquestionably the finest in Britain, contained organic chemists such as Perkin, Haworth (who first synthesized vitamin C) and Robinson. In the event, though, Wittgenstein ended up working 3 See Hamilton, ‘Wittgenstein and the mind’s eye’. Wittgenstein’s copy of Lamb’s Hydrodynamics’.

4

Spielt and McGuinness, ‘Marginalia in



Finding a problem

not with Rutherford but at the engineering laboratory, whose head, newly appointed in 1908, was Petavel, inventor of a device for measuring variations in pressure caused by exploding gases and later director of the National Physical Laboratory. Manchester’s strength was not confined entirely to the experimental sciences. In the mathematics department were Littlewood, one of the best mathematical analysts in the world, and Lamb. The presence in Manchester of Lamb, whose book Wittgenstein had been so idiosyncratically studying, may well have been one of the things that attracted Wittgenstein there. When he arrived, Wittgenstein lost no time in approaching Lamb with questions about some equations he had devised. As he related it to his sister, Lamb will try to solve the equations that I came up with and which I showed him. He said he didn’t know for certain whether they are altogether solvable with today’s methods and so I am eagerly awaiting the outcome of his attempts.5

Perhaps this seemed (or was intended to seem) impressive to Wittgenstein’s sister: he had come up with some equations which one of the foremost applied mathematicians of the day did not know how to solve! But of course what every applied mathematician knows is that devising equations one cannot solve is easy. The hard part is to model a system in such a way that the resulting equations are soluble. His meeting with Lamb evidently mattered greatly to Wittgenstein at the time: his letter to his sister makes plain his extreme state of nervous tension throughout the day on which he made it. As things turned out, however, Lamb did not play the central role in his life that Wittgenstein seems to have hoped for; and it is hard to avoid the suspicion that what Wittgenstein was reporting was really a polite brush-off from a professor confronted with a somewhat eccentric student and some rather curious equations. In his first year at Manchester Wittgenstein at least began to attend Littlewood’s lectures on mathematical analysis, but we do not know how long he carried on. What we do know is that early in his time at Manchester Wittgenstein became interested in the philosophy of mathematics and, after three years at Manchester, decided to go to Cambridge to study with Russell. Quite how this interest in the philosophy of mathematics arose is something of a mystery, however. The account offered by McGuinness6 would be hard to improve on, both as a summary of what is known and as a caution against speculation that goes beyond it. Some commentators have presented Wittgenstein’s interest in the philosophy of mathematics as flowing naturally from the mathematics he was engaged in, but I am sceptical about this since, as we have just seen, he had not 5

To Hermione Wittgenstein, Oct. 1908.

6

Young Ludwig, ch. 3.

Manchester



Wittgenstein’s patented jet rotor

really done very much mathematics by this stage in his life. It has been common, perhaps in an attempt to make this account more plausible, to present Wittgenstein’s research in Manchester as having a strong mathematical component; and it is true that the reminiscences of Eccles, Wittgenstein’s friend during his time in Manchester and himself an engineer, refer to a theoretical aspect to Wittgenstein’s work, but the only evidence we have of his engineering work, a patent he filed in 1910, leaves little trace of it. The patent involved the idea of mounting a jet nozzle on each end of the rotor blade of a propeller. It is a curious mixture. There are certainly elements in the design that are original and farsighted: powered flight was still new in 1910, and the idea of using any sort of jet engine to power an aeroplane, although not itself original, had yet to reach the engineering mainstream. And the idea of placing sources of propulsion at the tips of a propellor, although again not original, was eventually used successfully thirty years later by another Austrian, Friedrich von Doblhoff, to construct a helicopter with no need of a tail rotor. On the other hand, Wittgenstein’s implementation of the idea does not address the practical difficulties involved in turning it into a workable engine. One set of difficulties is created by Wittgenstein’s idea of mounting combustion chambers on the end of rotating propeller shafts. The propeller blades would have to be very strong in order to withstand the stress generated by the very high moments of the rotating combustion chambers. This would (at least with the materials then available) require the blades to be heavy, fur-



Finding a problem

ther increasing the forces involved. Nor does the patent address the difficulties involved in supplying fuel and oxygen to a combustion chamber on the end of an arm rotating at high speed. A further flaw is that his design has four separate combustion chambers: it would be difficult to control them, either independently or together, and any difference in thrust between the engines would put further strain on the propeller blades and make the assembly unstable. Wittgenstein described his idea in the patent as applying indifferently to ‘aeroplanes, helicopters, dirigible balloons, or other forms of aerial machines’, but the practical problems would probably be less serious in a helicopter, since the rotation rate of the rotor is typically lower than in an aeroplane propellor and the blades are much longer. The point of labouring these design issues here is not so much to suggest that Wittgenstein was a poor engineer as to cast doubt on the common representation of him as a skilled mathematician. One could quite quickly estimate the moment generated by a combustion chamber rotating at high speed, and yet Wittgenstein’s patent application takes no account of this. Indeed, it is very hard to believe that he made any calculations before he submitted it. The design shows ingenuity and imagination, it is true, but it would have needed a lot more work before it could become a practical engineering project. . The Principles of Mathematics The plain, if somewhat unsatisfying, fact is that we do not really know what first led Wittgenstein to take an interest in the philosophy of mathematics. What we do know is that Wittgenstein’s interest led him to Russell’s Principles of Mathematics. How this came about is described in a much later reminiscence of Rush Rhees. Wittgenstein himself told me that while he was working in the Engineering Laboratory, he and two others doing research there began to meet for one evening each week to discuss questions about mathematics, or ‘the foundations of mathematics’. . . At one of these meetings Wittgenstein said he wished there were a book devoted to these questions, and one of the others said, ‘Oh there is, a book called The Principles of Mathematics, by Russell: it came out a few years ago.’ Wittgenstein told me that this was the first he had heard of Russell: and that this was what led him to write to Russell and to ask if he might come and see him. I believe it was from The Principles of Mathematics that Wittgenstein learned of Frege.7

Wittgenstein may, for all we know, have read philosophy before this—he is reported8 to have read Schopenhauer’s World as Will and Idea in his youth, and as a consequence to have adopted for a time a version of epistemological idealism—but Russell’s Principles is the first philosophical work whose influence 7

Recollections of Wittgenstein, 213–14.

8

See von Wright, ‘Biographical sketch’, 5.

The Principles of Mathematics



on him we can trace directly. (A copy which Wittgenstein bought in October 1912 has survived9 but must surely be a duplicate or replacement for one he had already bought in Manchester.) To modern readers (of whom there are not as many as one might expect, given its place in the history of the subject) Russell’s Principles comes across as a transitional work: it contains extended passages which we can recognize as analytical philosophy in quite the modern sense, but these are juxtaposed to passages written in a style that strikes us as wholly antiquated, introducing bizarrely elaborate classifications for no apparent reason that develop into an architectonic of almost Kantian complexity. Whatever its faults, though, its influence on Wittgenstein is unquestionable. (Tradition has it that he continued to think highly of the book much later in his life.) It will therefore be in place for us to explain here some of the philosophical background from which it arose. Much of that background was supplied not by Russell but by Moore, from whom, on fundamental questions of philosophy, Russell said that he had derived all the chief features of his position.10 Russell’s later recollection was, more specifically, that the movement of which this book was part (a movement which led him to reject the neo-Hegelian idealism espoused by Bradley which was then popular in Britain) was born in conversations between him and Moore in 1898.11 The first publications to exhibit this movement are Moore’s articles on ‘The nature of judgment’ and ‘The refutation of idealism’, the central claim of which is that by conceiving of propositions as objective complex entities independent of any knowing mind we can resist the temptations of idealism. But if the overall shape of the project is clear, the details are not. The targets of Moore’s criticism are broadly spread: although it is Bradley’s post-Hegelian denial that absolute truth is ever attainable which is the principal target, at times Berkeley’s view that esse est percipi or Kant’s view that the relations the objects of experience bear to one another are supplied by the mind are also attacked. Moore’s conception of a proposition is embodied in two central doctrines. The first is that the entities of which a proposition is composed (which he called ‘concepts’) are themselves the items the proposition is about. Propositions are the objects of judgment, and the concepts that make up the proposition are therefore part of what we judge, but the view is nonetheless realist because this is ‘no definition of them’; ‘it is indifferent to their nature’, he says, ‘whether anyone thinks them or not’.12 Concepts are, that is to say, objective entities, and a proposition consists of such entities somehow related so as to form a complex. Moore opposed this to Bradley’s view that when I have an idea of something, that thing is itself part of the idea. This opposition is 9

See Hide, ‘Wittgenstein’s books at the Bertrand Russell archive and the influence of scientific literature on Wittgenstein’s early philosophy’. 10 Principles, xviii. 11 MPD, 54. 12 ‘The refutation of idealism’, 4.



Finding a problem

plainly not exhaustive of the possibilities, but once he had disposed (no doubt rightly) of Bradley’s view, Moore seems to have seen no need of an argument for his own. Moore slid, that is to say, from conceiving of the components of a proposition as objective (which holds true, for instance, of Frege’s senses) to concluding that they are the very same as the things the proposition is about (which does not). Nonetheless, the doctrine is central to the refutation of idealism as Moore conceived of it. Once it is definitely recognized that the proposition is to denote not a belief (in the psychological sense), it seems plain that it differs in no respect from the reality to which it is supposed merely to correspond, i.e. the truth that I exist differs in no respect from the corresponding reality my existence.13

It follows, Moore held, that truth does not depend upon any relation between ideas and reality, nor even between concepts and reality, but is an inherent property of the whole formed by certain concepts. . . The ultimate elements of everything that is are concepts.14

The lacuna in Moore’s argument is significant for our present purposes because his conclusion—that a proposition must, if it is to be independent of the mind, contain parts of the external world—is one that Russell embraced wholeheartedly. Moreover, Russell did not, any more than Moore, consider at this stage any alternative resembling Frege’s notion of the sense of a name. By the time Russell did come across Frege’s conception, he seems to have been too deeply embedded in his own to be able to engage with it. When he confirmed in response to Frege’s query that ‘in spite of all its snowfields Mont Blanc itself is a component part of what is actually asserted in the proposition “Mont Blanc is 4000 metres high” ’, he offered as his reason that ‘if we do not admit this, we get the conclusion that we know nothing at all about Mont Blanc’,15 but did nothing to explain why this should follow. This disagreement between Frege and Russell is often expressed in terms of names rather than sentences. For Russell the part of the proposition that corresponds to the proper name ‘Mont Blanc’ is the mountain itself; for Frege it is not the mountain but the sense of the name. Russell’s was, that is to say, what is sometimes called a one-step, Frege’s a two-step semantic theory. Moore’s second central doctrine was that there are no internal relations between concepts—no relations between concepts that are part of the nature of the concepts related. What it is for a proposition to be true is just for the concepts it is composed of to be externally related to each other in a certain way. Once again, it is easy to see what the target is. Bradley had held that all relations are internal, and had concluded as a result that since in particular knowledge must be conceived of as an internal relation between the knower 13

‘Truth’. 14 Moore to BR, 11 Sep. 1898 (Griffin, Russell’s Idealist Apprenticeship, 300). Frege, 12 Dec. 1904.

15

To

Logicism  and the proposition known, the simple act of coming to know the proposition will turn it into something different from what it was. No truth, according to Bradley, is wholly true; truth is ‘subject always to degree’.16 There is room to doubt whether Moore meant by ‘internal’ the same as Bradley. And even if Moore was right to reject Bradley’s extreme conclusion that nothing is ever wholly true, it is much less clear why Moore should have said that there are no internal relations between concepts at all: as in the case of the first doctrine, he seems to have been oblivious of the need for an argument. Nonetheless, once again Moore’s view was shared by Russell, who as early as 1899 confidently asserted that ‘all relations are external’.17 The shadow cast by these two doctrines, that names refer directly to their objects without the mediacy of sense, and that there are no internal relations, is long. For they were both not only adopted by Russell but maintained, in a certain sense, by Wittgenstein in the Tractatus. Wittgenstein did not maintain that propositions contain the parts of the world they are about, but he did side with Russell against Frege in rejecting18 the notion that names have sense as well as reference. And his doctrine19 of the logical independence of elementary propositions can be thought of as a reexpression in Tractarian terms of Moore’s denial of internal relations between objects. Moreover, it is notable that the Tractatus contains hardly any argument in support of either claim. Indeed neither of them is discussed in Wittgenstein’s surviving pre-Tractarian writings. It might seem perverse, therefore, for me to stress these two views in a book about the Notes on Logic, in which they do not occur. My ground for mentioning them nonetheless is that I think the most plausible explanation for Wittgenstein’s failure to discuss them is that he never saw any reason to question them; and indeed they became so embedded in his conception that, like Russell, he found it hard to see the need for argument.

. Logicism The doctrine that there are no internal relations between concepts runs into an obvious difficulty in the case of identity statements. If the identity ‘a = a’ expresses anything about a—a relation between a and itself—it seems clear that this must be internal. So if there are no internal relations, we are forced to conclude that it does not express anything at all. This is perhaps not so bad in itself, but we shall need to say something about the identity ‘Hesperus = Phosphorus’, which, apparently at least, expresses genuine astronomical information. And a lot more will have to be said about arithmetic, in which apparently informative identity statements (such as ‘7 + 5 = 12’) play such a central role. 16

Appearance and Reality, 321.

17

CP, II, 143.

18

3.3.

19

5.134.

Finding a problem



The work in which this was first attempted was Russell’s Principles. What Russell added to Moore’s conception of propositions in order to account for arithmetic (and indeed for mathematics more generally) was the notion of a denoting concept. A denoting concept is what one might call an ‘aboutness shifter’:20 its task is to enable a proposition to be about something else that is not itself part of the proposition. On the view that Russell had derived from Moore, let us recall, the proposition expressed by the sentence ‘I met John’ contains me, John, and the universal meeting. The proposition expressed by ‘I met a man’ will similarly have to contain me, meeting, and a third element expressed by the phrase ‘a man’. But what is this third element? It cannot be any particular man, since it is just the same proposition whichever man it was that I actually met. We seem forced by this to hold that the third element is a concept that is somehow related to whatever man I might have met; but this concept, if a constituent of the proposition, is not one of the things the proposition is about. The proposition is not about a man: this is a concept which does not walk the streets, but lives in the shadowy limbo of the logic-books. What I met was a thing, not a concept, an actual man with a tailor and a bank-account or a public-house and a drunken wife.21

Yet there must be some connection between the man with the bank-account and the propositional component in question. In the Principles Russell calls the propositional component a denoting concept—elsewhere sometimes a denoting complex—and the relation it has to the man that of denoting. ‘A concept denotes when, if it occurs in a proposition, the proposition is not about the concept but about a term connected in a certain peculiar way with the concept.’22 Russell seized on denoting as a central element in the epistemology of mathematics. The concept all numbers, though not itself infinitely complex, yet denotes an infinitely complex object. This is the inmost secret of our power to deal with infinity. An infinitely complex concept, though there may be such, can certainly not be manipulated by the human intelligence; but infinite collections, owing to the notion of denoting, can be manipulated without introducing any concepts of infinite complexity.23

A proposition about all numbers therefore does not itself contain all numbers; rather it contains a concept which denotes all numbers. The concept is finite, and hence capable of being grasped by our finite intelligence, even though what the concept denotes is infinite. In the Principles denoting concepts thus act as the bridge between what we are capable of grasping directly and what we are not; they enable a proposition to be about something (in this case the class of natural numbers) which is in a certain sense out of our reach. 20

Makin, ‘Making sense of “On denoting”’.

21

Principles, §56.

22

Ibid.

23

Principles, §72.

Russell’s paradox  Merely to invoke the notion of denoting is not, plainly, to explain mathematics. A second element in the development of Russell’s views occurred in 1900 when he attended the International Congress of Mathematicians in Paris and learned for the first time of the work of Peano, which demonstrated the expressive power of symbolic logic in expressing mathematics. The claim Russell made in the Principles was that all the propositions of mathematics could be reexpressed in the vocabulary of classes and would thereby turn out to be truths of logic. Peano’s work focused largely on the task of expressing the theorems of mathematics: he was much less concerned with the issue of how to prove them. However, a theory which aimed to achieve this had been developed by Frege, first in his Begriffsschrift and then in more detail in the Grundgesetze. Russell had been given a copy of the Begriffsschrift by Ward (one of the philosophers at Trinity), but had not read it.24 He became acquainted with the first volume of the Grundgesetze around the beginning of 1901, but wrote the Principles in ignorance of most of Frege’s writings. Only when the main text was complete did he make a study of them and add an appendix summarizing and criticizing them.25

. Russell’s paradox Shortly before this, however, Russell had discovered a problem not just for Frege but for any prospect of a logicist reduction of mathematics. The problem was that we cannot unproblematically assume that for every propositional function φ there is an extensional entity, the class of φ s, corresponding to it. To see why not, consider the class K of all those classes which do not belong to themselves. Then for every class x it is the case that x belongs to K if and only if x does not belong to itself. In particular, then, K belongs to K if and only if K does not belong to K. This is a contradiction, and the argument that leads to it is known as Russell’s paradox. We shall come in a later chapter to the elaborate theory Russell eventually devised to get round this difficulty. In the Principles he did no more than sketch the outline of a ‘theory of types’ that might resolve the matter, leaving the details as a matter requiring further work. The totality of all logical objects, or of all propositions, involves, it would seem, a fundamental logical difficulty. What the complete solution of the difficulty may be, I have not succeeded in discovering; but as it affects the very foundations of reasoning, I earnestly commend the study of it to the attention of all students of logic.26 24

Russell, Autobiography, 65. 25 See Linsky, ‘Russell’s notes on Frege for Appendix A of The Principles of Mathematics’. 26 Principles, §500.



Finding a problem

Perhaps it was natural that Wittgenstein would be intrigued by this problem and would take up Russell’s recommendation to attempt a solution. The first evidence we have of Wittgenstein working on any philosophical problem dates from April 1909, when a friend of Russell called Philip Jourdain made the following note in his correspondence book. Russell said that the views I gave in a reply to Wittgenstein (who had ‘solved’ Russell’s contradiction) agree with his own. . . In certain cases (e.g., Burali-Forti’s case, Russell’s ‘class’, . . . Epimenides’ remark) we get what seem to be meaningless limiting cases of statements which are not meaningless.27

Jourdain was perhaps a natural person to approach, as he had already published on the topic,28 but there is also something characteristic of Wittgenstein —his blend of confidence and diffidence—in the fact that he did not write to Russell himself but to someone he may well have known was in contact with him. Wittgenstein’s letter to Jourdain has not survived: all we have is Jourdain’s description just quoted of the views of his own which he offered in reply. It is hardly likely, though, that Wittgenstein, a self-taught novice, had come up with a ‘solution’ to the paradoxes of any interest or subtlety; and Jourdain, whose correspondence does not elsewhere display much grip on the concept of tact, will no doubt have explained his error to him with clinical directness. Wittgenstein was thin-skinned at the best of times, and a brush-off from Jourdain might on its own be enough to account for the fact that it was another two years before he felt able to approach Russell in person. However, he may equally have been influenced by his father, who (at least on Wittgenstein’s presentation of the matter) was ‘disappointed in all his other sons’ and ‘very anxious this one should do something respectable like engineering and not waste his time over such nonsense as philosophy’.29 His elder sister’s later recollections leave us in no doubt about the strength of the conflict he felt during this period. Reflection on philosophical problems suddenly became such an obsession with him, and took hold of him so completely against his will, that he suffered terribly, feeling torn between conflicting vocations. . . It shook his whole being. . . During this time Ludwig was in a constant, indescribable, almost pathological state of agitation.30

The issue that so piqued Wittgenstein’s interest was unquestionably important. As soon as it was discovered, Russell’s paradox became the central problem in the philosophy of mathematics, a position it held at least until the publication of Gödel’s incompleteness theorems in 1931. The attraction for Wittgenstein, a young man of ambition and talent searching for a field in which to make his mark, is therefore easy to understand: the problem was 27 Jourdain’s correspondence book, 20 Apr. 1909, quoted in Grattan-Guinness, Dear Russell —Dear Jourdain, 114. 28 E.g. ‘On the question of the existence of transfinite numbers’. 29 BR to OM, 7 Mar. 1912. 30 Quoted in Rhees, Recollections of Wittgenstein, 2.

Russell’s paradox  recent, simple, and intriguing; and a satisfying solution to it would be certain to bring its author attention. Not only was this apparently the first philosophical problem Wittgenstein worked on seriously, but the desire to solve it was probably what drew him to Cambridge, and therefore to philosophy as a career rather than a hobby (to the extent that a man of means like Wittgenstein recognized that distinction). At any rate, it was the philosophy of mathematics rather than philosophical logic that he stated to Russell as his interest when he arrived at Cambridge;31 and there is ample evidence (in the diaries of his Cambridge friend David Pinsent, for example) that he continued throughout the two years he spent there to regard solving the paradoxes as one of his principal ambitions. 31

BR to OM, 18 Oct. 1911.

Chapter 

First steps Whatever philosophical study Wittgenstein undertook in Manchester, it was plainly not something that could be described as a training and, although earlier influences on Wittgenstein are no doubt relevant at various points, any study of the genesis of the Tractatus naturally begins in earnest with his arrival in Cambridge to study under Russell. (Later, indeed, he told Ramsey1 that the book had taken seven years to write, thus implicitly identifying his arrival in Cambridge as the point at which his work on the Tractatus really began.) Wittgenstein once advised his friend Eccles of the importance of going to work in really first-class places.2 In his choice of Cambridge (more particularly, Trinity) in 1911, as of Manchester in 1908, he certainly took his own advice. Even if it was not the centre of the universe that some of its members were (and are) inclined to suppose it, a college that boasted Russell, Moore, McTaggart, and Hardy—Whitehead had only recently resigned his Cambridge post and gone to work in London—was plainly the best place Wittgenstein could have chosen to pursue an interest in the philosophy of mathematics. Wittgenstein’s decision to approach Russell as a possible supervisor also conforms to a pattern. In 1908 he had sought out Lamb, whose book on Hydrodynamics he had been studying; now it was the author of the Principles he went to see. Russell’s description3 of their first meeting, in October 1911, suggests that Wittgenstein was in as extreme a state of nervousness as he had been when he met Lamb. This time, though, the outcome would be vastly more fruitful. . Cambridge During that first Michaelmas Term Wittgenstein’s attendance at Russell’s lectures on the foundations of mathematics was evidently something of a trial subscription. He was still officially a research student at Manchester, had not matriculated as a member of Cambridge University, and presumably attended the lectures only on Russell’s sufferance, staying in rented accommodation in 1

Ramsey to his mother, 20 Sep. 1923.

2

[July 1912].

3

To OM, 18 Oct. 1911.

Cambridge  the town while he did so. Moore was also in the audience for Russell’s lectures and we have his lecture notes, which give the clear impression that the logical background Russell assumed in his audience was not great. There is hardly a logical symbol in Moore’s notes and what is covered is mainly very elementary material on cardinal and ordinal numbers. At the end of the Michaelmas Term Wittgenstein asked whether Russell thought he had a future in philosophy. Russell replied that he should write something over Christmas and Russell would give an opinion once he had read it. At some point in this decision-making process Wittgenstein also visited Frege to ask his advice, and it may be that this meeting took place over the Christmas vacation at the end of 1911, since Russell’s later recollection4 was that when he arrived in Cambridge Wittgenstein did not yet know Frege personally; another possibility is that Russell was mistaken and the meeting took place the previous summer. At any rate, Frege’s advice, whenever exactly it was given, was that Wittgenstein should study with Russell; and Russell’s, when he had read Wittgenstein’s vacation essay, was the same. Shortly thereafter Wittgenstein did matriculate formally as a student at Cambridge. In his second term at Cambridge Wittgenstein attended the continuation of Russell’s course on the foundations of mathematics, as well as a course offered by Moore. The title of Moore’s course was ‘Psychology’, but this hardly conveys the content to the modern reader. Moore had devoted much of the previous term to a mind-numbingly careful discussion of how to define psychology, and the remainder to a critique of Russell’s multiple relation theory of judgment; in this second term he spent much of his time on what is nowadays called the philosophy of mind. Psychology as an empirical science was rarely mentioned. Russell in his lectures did at least now get as far as discussing the paradoxes, the subject which had fired Wittgenstein’s interest, but once again the technical level was very low. Whatever understanding of the theory of types Wittgenstein did acquire was at any rate not obtained from Russell’s lectures, which were pitched at a level that no doubt matched the technical capabilities of most of his audience well enough but could hardly be described as taxing. Quite soon Wittgenstein began to have frequent meetings with Russell to discuss philosophy outside of lectures. Russell’s letters to Ottoline Morrell from this period occasionally mention Wittgenstein’s views, but once we allow for the licence Russell no doubt took in turning them into amusing stories for his lover, the views he reports—for example, that nothing empirical is knowable, and hence, notoriously, that he could not be certain there was not a rhinoceros in the room5 —give us only the murkiest sense of the problems Wittgenstein was working on. What the letters do convey vividly is the demands which he quite soon began to make on Russell’s time and patience, 4

CP, XI, 178.

5

2 Nov. 1911.



First steps

demands which Russell treated for the most part with an amused tolerance that from the perspective of the modern academic seems quite remarkable. A partial explanation is provided by the nature of Russell’s post at Trinity: being neither a university lectureship nor a college fellowship, it carried no administrative responsibilities and a strictly limited teaching load (three lectures a week). One should allow for this, no doubt, and also for the sense one has that in a university such as Cambridge these were altogether more spacious times. After a while, though, another factor began to play a central role in their relationship. There is ample evidence of the veneration which Wittgenstein quite soon began to feel for Russell. One of the most touching incidents was when Wittgenstein described hearing Beethoven’s Choral Symphony with Russell as one of the great moments in his life:6 it was not just the music, we are to understand, that made the moment great. Perhaps by then Wittgenstein saw Russell as a surrogate father (his own, to whom he was evidently not close, having died in January 1912); Russell certainly began to see Wittgenstein as a surrogate son and said as much to Ottoline.7 . On denoting It is plain that during the first of his two years in Cambridge Wittgenstein came to see himself as engaged in a joint project with Russell: his letters refer to ‘our’ problems8 and ‘our’ theory.9 However, the direct evidence of Wittgenstein’s own work during that year is scant. To get a sense of what their joint project was it will therefore be necessary for us in the next few chapters to devote a considerable amount of space to Russell’s own work. We shall start by examining a fundamental change that had come about in his position since he wrote the Principles. As we noted in §1.5, what he offered in that book was no more than a gesture towards a solution to the paradoxes. It was his desire to improve on this gesture that led him to focus his attention on those denoting concepts (such as, most famously, that of the present King of France) which do not denote anything. When he wrote the Principles, he had of course been aware that according to his theory there would have to be such concepts; but now he was convinced of the need to gain a better understanding of how they function, since the paradoxes seem to show that ‘the class of all classes that do not belong to themselves’ is a denoting concept which does not denote anything. He had said in the Principles that a proposition in which a denoting concept occurs ‘is not about the concept but about a term connected in a certain peculiar way with the concept’.10 If the concept does not denote anything, 6

BR to OM, 9 June 1913. 7 21 Aug. 1912. Dec. 1912. 10 Principles, 53.

8

16 Aug. 1912; [Summer 1912] (CL, no. 6).

9

26

On denoting  the term in question does not exist (in Russell’s terminology it does not ‘have being’), and so the way in which it is connected with the concept will perhaps be peculiar enough. But the moment of revelation for Russell came when he saw that the relationship is peculiar even when the term does have being. For if there is a relationship between the concept and the thing it denotes, there will be a true proposition expressing that relationship, and this true proposition will be about the concept. Yet a denoting concept, let us recall, is defined as one whose job is to occur in a proposition but to point at something else which the proposition is about. So how can any proposition be about the denoting concept itself? What sort of entity should occur in a proposition in order for the proposition to be about, say, the denoting concept expressed by the phrase ‘the first line of Gray’s Elegy’? Not, certainly, the denoting concept itself, since if it is doing its aboutness-shifting job properly, it will ensure that the proposition ends up being not about the concept but about what it denotes, i.e. about the sentence ‘The curfew tolls the knell of parting day’. Nor, clearly, is it any use to put in the proposition the denoting concept ‘the meaning of the first line of Gray’s Elegy’, since that would make the proposition be about the meaning of the sentence ‘The curfew tolls the knell of parting day’, which again is not what we want. And other attempts, Russell argued, run into similar troubles. There is something in the argument just sketched that is apt to puzzle the reader on first acquaintance. It is supposed to show that there can be no informative proposition about the concept expressed by the phrase ‘the first line of Gray’s Elegy’. Yet this last sentence seems to express a proposition that is about just this concept. Russell insists that it is not what he wants. Why? At this point he introduces a further constraint. The relationship between a concept and its denotation (if any) is not, he says, ‘linguistic through the phrase’. He asserts this without argument, as if it is obvious, but his reason for holding it depends on his Platonistic conception of logic, according to which logical relations, such as that of denotation, are independent of language. Concepts exist, Russell believed, whether or not we choose to devise means to express them in language; so the relationship between the concept and its denotation exists independent of language, and hence so does the proposition expressing it. Consequently, any sentence in which a linguistic item (such as the phrase ‘the first line of Gray’s Elegy’) is mentioned cannot be what we are after, since the proposition it expresses will be (partially) about language whereas the proposition we are trying to express would, if it existed, be independent of language. The argument we have just described (which is known as the Gray’s Elegy argument because of the example he used to make the point) led Russell to reject the theory of denoting he had put forward in the Principles. What he replaced it with was an account according to which the true structure of the



First steps

proposition a sentence expresses is to be revealed by translating it into the predicate calculus with identity. The sentence ‘I met a man’, for instance, might be translated as ( x)(M x Rax), where ‘M x’ means that x is a man, ‘R xy’ means that x met y and ‘a’ denotes me. In the analysed sentence ‘a man’ is no longer a denoting phrase but occurs only as part of the predicate ‘is a man’: the work of denoting is done by the notation of quantifier and variable. And, as undergraduates learn in their elementary logic course, ‘The present King of France is bald’ can be translated as ( x)(K x (y)(K y ⊃ x = y) B x), where ‘B x’ means that x is bald and ‘K x’ means that x is a king of France. Once again, the denoting phrase (in this case ‘the present King of France’) has disappeared in the translation, to be replaced by quantified variables. What was significant about this method of translation was that it showed how the grammatical form of a sentence might differ from the logical form of the proposition the sentence expresses. Thus in the standard example, ‘The present King of France is bald’, the sentence has a subject, ‘The present King of France’, which does not correspond to any constituent of the proposition it expresses. The theory thus avoids the need to appeal to a shadowy realm of non-existent objects (often called ‘Meinongian’ although this is a little unfair to Meinong) to explain the meaning of the sentence. This is a general method of considerable power. Wherever in philosophy we come across linguistic items which appear to refer to entities which are in some way problematic, the possibility now arises that the terms in question may be what Russell soon called ‘incomplete symbols’, that is to say expressions which have no meaning on their own but which are such that any sentence in which the expression occurs can be translated into another in which it does not. By this means we eliminate reference to the problematic entities without rendering meaningless the sentences which apparently refer to them. It is not known when Wittgenstein first came across the theory of descriptions of ‘On denoting’, but it is certain that he adopted it wholeheartedly and probable that he did so quite early. As he somewhat patronizingly told Russell, ‘your theory of descriptions is quite CERTAINLY correct’.11 And Jourdain was surely inspired by Wittgenstein when, in a letter to Frege written after he had come to know Wittgenstein personally at Cambridge, he referred to ‘what seems to be a fact, namely, that Russell has shown that propositions can be analysed into a form which only assumes that a name has a “Bedeutung”, and not a “Sinn” ’.12 Moreover, it was not just the theory of descriptions that Wittgenstein agreed with: his letters to Russell show that he was soon experimenting with the method of incomplete symbols as an approach to analysing compound propositions. Some of the consequences of Wittgenstein’s acceptance of this method will emerge shortly. E

E

11

[Nov. or Dec. 1913] (CL, no. 32).

12

15 Jan. 1914.

Sense-data  . Sense-data Although Russell intended his logic to form a foundation for mathematics, he did not, of course, conceive of that as its only role. He also hoped to use it to ground an account of our knowledge of the external world. In order for his theory of descriptions to be of use in explaining that knowledge, however, he had to forge a link between logic and epistemology. What he assumed was that the direct logical relation between a name in my idiolect and the object it denotes mirrors a similarly direct epistemological relation (which he called acquaintance) between me and the object. With this assumption in place, he hoped that his theory would enable him to determine what the objects of acquaintance are. To understand how it would do this, we need to recall how it dealt with non-referring expressions. Russell analysed ‘The present King of France does not exist’ as ∼( x)(K x (y)(K y ⊃ x = y)). (In words: it is not the case that there is exactly one present king of France.) The key step occurred when he decided to apply an analysis of the same form in every case where we say that something does not exist. If we say that Homer did not exist, for instance, we should be taken to mean that no one person wrote both the Odyssey and the Iliad. The apparent predication of a property of existence is replaced by an existential quantifier, with the result, Russell thought, that we avoid the difficulties involved in supposing there to be a person, Homer, with the awkward property of non-existence. ‘Homer’ is thus for Russell an example of a term that is grammatically a proper name, but not logically so, since the correct logical analysis of ‘Homer does not exist’ reveals ‘Homer’ to be really a definite description in disguise. And in the same sort of way ‘Sherlock Holmes does not exist’ might be analysed by replacing ‘Sherlock Holmes’ with a definite description such as ‘the detective who lived at 221b Baker Street’. Russell used the term ‘logically proper name’ for any proper name which functions as such not just grammatically but logically—for any name, that is to say, which logical analysis does not reveal to be really a disguised definite description. But in ordinary language logically proper names are the exception rather than the rule. For it is not just words for spurious classical poets and fictional detectives that turn out to be disguised descriptions. The eliminative doctrine applies in any case where I can say intelligibly, even if falsely, that someone does not exist. Since I can wonder whether Plato existed, ‘Plato exists’ must express a non-trivial proposition fit to be the object of my wonderment. So ‘Plato’ is (at least in my idiolect) a disguised definite description. The same applies to anything whatever of whose existence I can coherently entertain a doubt: the term referring to it must on this view be a disguised definite description. It follows that a term ‘A’ in my language can be a logically proper name only if the sentence ‘A does not exist’ is not merely false but E



First steps

absurd: the object A must be something of whose existence I am so certain that I cannot intelligibly doubt it. This is plainly a very demanding criterion: even tables, chairs, and pens do not fulfil it since they might be holograms, tricks of the light, or hallucinations. The only things in the empirical realm that do fulfil the criterion are what Moore and Russell called ‘sense-data’, by which they meant whatever the things are that are immediately given to us in experience. (The word was coined in 1882 by Josiah Royce and popularized by William James,13 but it was Moore14 who first used it with the meaning just given.) Even if the green table on the other side of the room were an illusion, the patch of green at the centre of my visual field when I look in that direction would certainly exist. It was part of Russell’s view to maintain that if I say something about the table (that it is oblong, for example), the proposition that I express does not contain the table itself but may contain various sensedata, such as the green patch just mentioned. And of course I have other senses too: whatever is presented immediately and indubitably to these other senses—sounds, pains, feelings of roughness beneath the finger, or bitterness on the tongue—will according to Russell and Moore be sense-data too. What, then, is the relation between me and the green patch in my visual field? It might reasonably be called a sort of experience, but it is distinguished from other cases of what we normally call experience by its indubitability. If I think I am experiencing a blow on the back of the head, I may be mistaken: perhaps the pain has some other cause. But I cannot be mistaken that I am in pain. (An amputee who describes a pain in a non-existent leg is mistaken about the source of the pain, not about the fact of it.) This special sort of experience is what Russell called acquaintance. Acquaintance is thus a relation which, when it holds, provides an immediate and indubitable link between the mental world of the subject and something else. Moore used the term ‘direct apprehension’ for the same relation. In the case of my experience of the contemporary external world, then, the object of acquaintance is a sense-datum. Russell often talked of acquaintance as a relation between me and the sense-datum, but he regarded this as a loose way of talking and strictly speaking took the object that is related to the sensedatum by acquaintance to be not me myself but my awareness of the sensedatum, which he called a sensation. Acquaintance thus acts, according to this view, as a bridge between the mind and the world, relating an event within my mind (the sensation) to something non-mental (the sense-datum). In an act of memory, too, acquaintance links the subject to something non-mental: the difference is only that in this case the object of the act is not represented as being simultaneous with the act itself. By taking items of experience as building blocks in this way Russell showed 13 See Milkov, ‘The History of Russell’s concepts “sense-data” and “(knowledge by) acquaintance” ’. 14 ‘The subject matter of psychology’, 57.

Sense-data  evident sympathy with a central strand of empiricism, but he was very far from being an empiricist, since he did not think that they are the only constituents of propositions or that all our knowledge is derivable from them. There are, he thought, other cases in which the object of acquaintance is not a sensedatum, one of which is that of acquaintance with universals. He thus felt able to maintain a liberal ontology of universals such as love or meeting, which he thought were constituents of propositions such as ‘John met Mary and fell in love with her’. Universals, he somewhat over-exuberantly claimed, are ‘unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of systems, and all who love perfection more than life’.15 Russell’s theory has a curious side-effect, however. Recall his argument for the identification of the simple entities as those things whose existence it would be incoherent to doubt. That argument was that if A is a simple entity, then the sentence ‘I doubt whether A exists’ cannot be intelligible, since if it were intelligible, the Russellian analysis would reveal ‘A’ to be not a logically proper name but a disguised description. But we can evidently run an exactly analogous argument in the case of the sentence ‘it is possible that A does not exist’: if this is intelligible, the Russellian analysis will reveal ‘A’ to be a disguised description once more. But if we simply use this argument to place a further constraint on the simples, Russell’s theory seems to collapse under the weight, since we now need the simples to be entities whose existence is not only indubitable but necessary, and common sense suggests that even sense-data do not fulfil this criterion: I may be sure that there is a patch of green in the centre of my visual field, but can I not also represent to myself the possibility that there might not have been this patch (if, for instance, I had painted the wall a different colour)? The only way out if there were to be any simples in the world at all was for Russell to say that despite appearances to the contrary I cannot in fact represent the possibility of there not having been precisely this patch of green. Kripke, famously, rejected Russell’s analysis of ‘A does not exist’, but only at the price of accepting that proper names need not refer to anything and hence leaving ‘A exists’ unanalysed. Russell, on the other hand, took our ability to represent A’s non-existence as evidence that we conceive of A as complex. He held, therefore, that the notions of possibility and necessity are properly applicable not to propositions but only to propositional functions. If talking of propositions as possible was to be legitimate at all, it would have to be explained as a way of saying something not about how the world could have been but about how it actually is.16 15

Problems, 57.

16

Cf. CP, VI, 194–5.

Chapter 

Matter When Wittgenstein arrived in Cambridge, Russell himself was at a transitional point in his work. The text of Principia Mathematica had been delivered to the publisher in 1909 (although correcting the proofs occupied him intermittently until February 1913).1 He wrote his ‘shilling shocker’, The Problems of Philosophy, in the summer of 1911 just before Wittgenstein’s arrival. Now he was ready to embark on a new project, and the subject he chose was the problem of matter. . The project Even if the world contains such things as sense-data, they are plainly not the same as the objects ordinary language speaks of, nor are they the same as the particles (electrons, protons, neutrons, or whatever) which physics takes as fundamental. In 1911, when he wrote The Problems of Philosophy, Russell’s view had been that sense-data are the only items in the external world with which we are directly acquainted, and hence of whose existence we may be quite certain. He noted, however, that the inference from them to the existence of matter constituting an independent external world ‘does not lead to any difficulties, but on the contrary tends to simplify and systematize our account of our experiences’. For that reason, he thought, ‘there seems no good reason for rejecting it’.2 If I look away or leave the room and come back in, or if I shut my eyes and then open them, the various sense-data I experience have a regularity which is most easily explained by positing a table from which they are all derived. Or, to put the point in a rather less homely way, the simplest physical theory that correctly predicts the data of sense is one which quantifies over the matter of physics and not just over sense-data. This, Russell believed, gives us reason to adopt the hypothesis that matter—‘that thing if any wh[ich] corresponds to sense-data and is independent of perception’3 —exists. Russell now made what I shall call the strong constructional conjecture—the conjecture, namely, that it might prove ‘possible, by a logical construction, to define in terms of sense-data alone some object having the properties which 1

To OM, 16 Jan., 22 Feb. 1913.

2

Problems, 11.

3

Russell’s lectures, 1912.

On matter  dynamics assigns to matter’.4 A theory of perception might be expected to provide an account of sense-data in terms of matter: it is because certain pieces of matter are configured in a certain way that I can be acquainted with a certain sense-datum. Russell’s project was, if possible, to do the converse— to provide, that is to say, an account of matter in terms of sense-data—and hence to provide an explanation of how physics can be based on the sensedatum theory. Wittgenstein cannot have played any part in Russell’s formulation of his project, since Russell discussed the problem, if not the solution, in his Cambridge lectures as early as November 1911, when his view of Wittgenstein was still that ‘it is really rather a waste of time talking with him’.5 Russell himself attributed his interest in the problem at least partly to Whitehead,6 who was working on a related project himself. The constructional conjecture and its variants soon became a central preoccupation for Russell, and in 1912 he twice delivered versions of a paper on the subject. By then, however, the relationship between him and Wittgenstein had changed markedly. Just before he began writing his paper on matter, Russell reported that he had ‘got a number of new technical ideas from him’, which he thought ‘quite sound and important’.7 So it is natural to look in this paper for signs of Wittgenstein’s influence. What we shall see is that the paper does indeed contain such signs, not all of them constructive. Perhaps it is not quite accidental, then, that Russell never published this paper, and that the account he did eventually publish, ‘The relation of sense-data to physics’, was not written until the beginning of 1914, when Wittgenstein was in Norway. . On matter Russell’s conjecture that matter can be constructed out of sense-data need not in itself be taken to indicate dissatisfaction with his earlier view that the former can be inferred from the latter: a logical construction would, after all, be preferable to an inductive inference. However, in his paper ‘On matter’ he did in fact express dissatisfaction with the inference. He now rejected the argument from theoretical simplicity mentioned earlier, saying that we have no reason to suppose that the universe is simple, and hence no reason a priori to prefer simple explanations to complicated ones. Perhaps this is Wittgenstein’s voice we hear, since elsewhere in his writing Russell did not hesitate to prefer simple explanations to complicated ones. Moreover, it is notable that Russell’s tone in the passage is uncharacteristically uncertain. Indeed, this hesitancy pervades the whole paper. Having made his constructional conjecture Russell discussed hypothetically what its consequences 4

CP, VI, 512.

5

To OM, 16 Nov. 1911.

6

To OM, 29 Dec. 1912.

7

To OM, 23 Apr. 1912.



Matter

would be, but then in a brief passage offered an argument to show that the conjecture is in fact false. Having reached this conclusion, he then went back and made a first set of revisions, probably before he delivered the paper at Cardiff, altering the passage about the consequences of the strong constructional conjecture from a real to an unreal conditional. What Russell had only just realized was that if his constructional conjecture was to have any hope of being correct, it would have to be weakened so that the constructional base included not only sense-data but things of the same kind as sense-data that are not in fact sensed by anyone. Let us call this the weak constructional conjecture: that the matter of the physicist consists of ‘collections of constituents of the nature of sense-data, some actually perceived, some not’.8 Russell’s views were shifting even as he wrote the paper. In the first draft he had written, ‘There is, in fact, reason to suppose that what I immediately see when I look at an object does not exist when I am not looking at it.’9 Now he deleted this and said instead, It is possible that the arguments against naive realism could be met by a theory which regards a piece of matter as consisting entirely of constituents of the nature of sensedata, by including everything that could be a sense-datum to any possible observer.10

Here, then, we have the first sign that Russell now countenanced, albeit hesitantly, a new kind of inference, not from sense-data to physical objects, but from sense-data to what he now began to call ‘qualities’—things of the same nature as sense-data. Wittgenstein read the beginning and end of Russell’s paper, and thought they were excellent.11 His view, apparently, was that physics could be reinterpreted directly in terms of sense-data, so that the assumption of matter would drop out as unnecessary. ‘If there is no matter,’ he told Russell, ‘physics and astronomy, and all the other sciences could still be interpreted so as to be true.’12 Presumably, therefore, one of the things he approved of was Russell’s statement (which he later deleted) of just this view near the end of the paper. It seems probable that all physical science, in so far as it is verifiable, could be interpreted in terms of sense-data alone, and might therefore retain whatever truth it can be known to possess even if there were no such thing as matter; and if this is the case, it follows that we cannot know either the existence or the non-existence of matter.13

But Wittgenstein’s initial approval for Russell’s paper did not last once he had read the whole of it.14 Russell’s letters speak of minor revisions to the paper in preparation for reading it in Cardiff, which are probably the minor revisions to the manuscript to which I have just referred. If so, the version which Wittgenstein read a few days later already concluded that the strong 8

CP, VI, 95. 9 CP, VI, 512. 10 CP, VI, 85–6. 11 BR to OM, 22 May 1912. Apr. 1912. 13 CP, VI, 516. 14 BR to OM, 26 May 1912.

12

BR to OM, 23

On matter  constructional conjecture was as a matter of fact false. It seems likely, though, that Wittgenstein’s main disagreement lay deeper. What he objected to was the conception of metaphysics that lay behind any attempt such as Russell’s to construct matter out of things of the same kind as sense-data. Wittgenstein, that is to say, had a conception of analysis that is genuinely eliminative in a sense in which Russell’s was not. We shall discuss the difference between their two conceptions of analysis in the next chapter. The conjecture that the focus of their disagreement lay in this difference receives some circumstantial support from the second, more substantial round of revisions to the paper which Russell probably made in October 1912, in preparation for reading the paper again, this time to the Moral Sciences Club in Cambridge with Wittgenstein in the audience. It was then, I conjecture, that Russell deleted a long passage from the middle of the paper, including the discussion of the metaphysics of constructed entities just mentioned. The passage Russell inserted in the revised version contained a more developed and clearer version of the argument that his strong constructional conjecture was false. This new version was based on the observation that, although the predictions against which physics is tested may be translated into claims about the configuration of sense-data, the theory used to make the predictions contains parameters which are not determined by the sense-data alone. The example Russell uses to make the point is that of the stars and planets. If we refer them to polar coordinates with the earth as origin, their angular coordinates will be observable, but the radius vector will always be a pure inference. . . Thus the radius vector introduces an unobservable distinction between two cases which give the same sense-data. It is introduced, of course, because it enables us to state simple laws for the motions of the heavenly bodies; but what is called the verification of these laws applies only to the angular coordinates, since the radius vector is not amenable to observation. . . We may so choose the coordinates of any physical system as that some of them shall have a one-one correspondence with sense-data, while others shall be entirely independent of sense-data, i.e. different values of them will not correspond to different sense-data. Those that correspond with sense-data may be called verifiable coordinates, while those that do not may be called hypothetical coordinates. Thus in the case of the planet, its angular coordinates are verifiable, while its distance from the earth is hypothetical.15

A distant star appears to me as a point of light in a certain position in the night sky: the distance from me to the object is not part of the sense-datum but is an hypothesis introduced by the astronomer so as to simplify the laws used to predict where in future similar sense-data will appear to me. The inference Russell draws is that the relation of the momentary state of the world to our sense-data is many–one, not one–one: there are various ways the world could 15

CP, VI, 87–8.



Matter

be which would give rise to just the sense-data that we in fact experience. So any construction of matter in terms of sense-data will have to contain extra hypothetical parameters whose purpose can only be to simplify the formulation of physics. Russell’s attempts at constructing matter are not the only things in the paper Wittgenstein might have disagreed with, however. At one point, for example, Russell speculates on whether there might be non-logical a priori principles which would license the principle of induction. In support of this possibility he cites an example: if A is earlier than B, and B is earlier than C, then C is not earlier than A. If the world were periodic, so that after a certain interval everything became exactly as it had been at an earlier time, we should still regard the earlier occurrence as numerically different from the later occurrence; and so far as I can see, we should do this in consequence of our a priori certainty that time cannot be circular.16

That some non-logical truths might be knowable a priori is of course something Wittgenstein denied in the Tractatus,17 and it would not be at all surprising if his opposition to it had arisen as early as 1912. Even in its revised form, though, Russell’s paper remains curiously hesitant. As late as the day of the Moral Sciences Club meeting at which he was due to read it, he wrote to Ottoline, ‘I don’t know yet whether to say there is matter or there isn’t.’18 The claim that physics can probably be interpreted in terms of things of the same kind as sense-data remains, but Russell does not expand on his earlier sketch of how the reinterpretation is to be carried out.

. Dawes Hicks The position Russell had now argued himself into—in part, surely, at Wittgenstein’s urging—was undoubtedly complicated. He had rejected the inference from sense-data to matter; but he had also shown that his original conjecture that we can construct matter out of sense-data is false, and conjectured instead that we can construct matter out of what he called ‘qualities’, i.e. things of the same kind as sense-data which may not actually have been sensed by anyone. But what licenses the new inference that is now required from sensedata to qualities? It is far from obvious that it is any easier to establish than the previous inference directly from sense-data to matter. To see why Russell thought the new inference preferable, let us turn to the concerns expressed by George Dawes Hicks in his 1912 critique of The Problems of Philosophy.19 Russell’s correspondence with Ottoline Morrell has preserved for us Wittgenstein’s reaction to Dawes Hicks’ article. 16

CP, VI, 514.

17

2.225.

18

25 Oct. 1912.

19

‘The nature of sense-data’.

Dawes Hicks  This morning I made him read two pages on sense-data by a muddle-headed person named Dawes Hicks, but the muddle made him quite ill. He declaimed for a long time and I thought he would murder me!20

But even if Dawes Hicks was muddle-headed, the issue he raised is close to Russell’s central difficulty: if sense-data are objects distinct from all the familiar objects of everyday life, how is knowledge about the latter possible? Russell’s theory bears on the surface an obvious risk of falling foul of a problem that bedevilled early modern philosophy: if physical objects are unknowable (as, for instance, real essences are supposed to be for Locke), then it is hard to see what sense there is in positing their existence nonetheless; Humean scepticism beckons. In the 1912 paper on matter that we have been discussing Russell acknowledged the point explicitly. ‘Even if we could know that sense-data “correspond” to some reality independent of perception, we could know nothing whatever as to the intrinsic nature of this reality,’21 he said. Now, however, he denied this. To avoid the Humean danger, Russell had to insist that sensedata are just as real as physical objects. Among objects, he was forced to say, the sense-data are merely the ones that are as a matter of fact directly known; other objects, such as tables and chairs, are not known, but this does not, he now said, make them unknowable in principle. ‘I do not know of any reason why the mind should be “disqualified” from knowing the physical thing; the question is one of fact, do we know the physical thing or do we not?’22 If, as Russell insisted, sense-data are physical, it is quite difficult to work out quite what they are. We plainly cannot (at least in general) treat the sensedatum as a property of the object perceived, since in some cases (such as mirages and other similar illusions) there is in fact no object to be perceived. Russell was therefore surely right to insist, in his reply to Dawes Hicks, that the sense-datum is a distinct object and not a mere property of the object perceived.23 One approach might be to think of the sense-datum theory as a contribution to the psychology of perception. (Sometimes, indeed, one has the impression that Russell and Moore were trying to do psychology without the bother of actually doing any experiments.) Consider for simplicity, as Russell and Moore most often did, the case of vision. It would then be natural to think of the sense-datum as some sort of physical occurrence within the visual apparatus. In that case one can ask at which point in the sensory process it is supposed to be located. If we think of the sense-datum as constituted by the physical configuration of receptors in the eyeball, we explain mirages well enough but struggle to explain hallucinations and dreams, which do not involve the eye at all but are generated within the brain itself. The further up the chain of vision we push the sense-datum, though, the closer the theory comes 20

5 Sep. 1912.

21

CP, VI, 93.

22

CP, VI, 186.

23

CP, VI, 187.



Matter

to treating us as brains in a vat. The sense-datum becomes simply the brain state that is correlated with the mental act of sensation, so that the relation of acquaintance is reduced to the mere bridging role of Descartes’ pineal gland. It begins to seem not only wildly over-optimistic to hope that the theory might be the foundation of an account of our knowledge of the external world, but straightforwardly mistaken. The propositions we utter, which we take to be about the table, turn out on analysis really to be about sense-data. As long as these were interpreted as properties of the table—its solidity, roughness, colour, etc.—that did not seem too implausible; but if they are really brain states, the proposition simply misses its intended target completely. If sense-data are not properties of the physical object, however, the question arises whether they exist when no one is looking at them. In The Problems of Philosophy Russell had been happy to accept that even if they are not mental, they ‘must be, in part at least, “in” the mind, in the sense that their existence would not continue if there were no seeing or hearing or touching or smelling or tasting’.24 The greenness I perceive when I look at the wall no longer exists, in other words, when I look away. Why Russell thought this is less clear. He mentions Berkeley, but with only partial approval. ‘His contention,’ Russell says, ‘was almost certainly valid, even if some of his arguments were not so.’25 But Russell does not develop the point, except to say it depends on ‘various reasons of detail’,26 and the reader is left uncertain what the valid argument might be which establishes that the sense-datum depends for its existence (rather than merely for the fact that it is a datum to sense) on the act in which it is sensed. The obvious way of avoiding even the strictly limited idealism that this concession to Berkeley amounts to is to accept that sense-data may exist when they are not being experienced, and that there may be things of the same sort as sense-data that no one ever has experienced or ever will experience. On this view, which Russell now adopted, something becomes a sense-datum when it is apprehended, just as a man becomes a husband when he marries; but in neither case is the object itself brought into existence (or changed intrinsically) as a result. But although Russell now accepted that it is possible for qualities to persist after they cease to be sense-data, he continued to doubt for ‘empirical reasons of detail’27 whether they do. Once again, though, he did not trouble to explain what these reasons of detail are. We can now see that the advantage of positing qualities rather than matter lies in the similarity of nature between the entities that are inferred and those that are experienced directly. In the Problems Russell had maintained that we infer the existence of matter from the existence of sense-data. His new view was that we infer the existence of unsensed qualities from the existence of 24

Problems, 20.

25

Ibid.

26

Problems, 21.

27

CP, VI, 186.

The relation of sense-data to physics  sensed qualities (i.e. sense-data). So, although he could not see how to dispense with inferred entities entirely, now at least the inferred entities had what he claimed was the virtue that they were all of the same kind as the entities of which I have direct acquaintance. An obvious source for the notion that this is an advantage was Hertz, who expressly recommended that in a physical theory the inferred entities (which he called concealed masses) should be of the same nature as the observed masses. Russell had of course read Hertz, but so had Wittgenstein, and it would be no surprise if Russell’s adoption of this methodological principle owed something to his prompting.

. The relation of sense-data to physics Russell continued to think about his project on matter until March 1913, when he put it to one side in order to work on a book on epistemology. He had reached stalemate because his conception of qualities—things of the same nature as sense-data, including those that are not in fact the data of sense—was not sufficiently rich to be able to generate matter from them. Wittgenstein’s scepticism about the possibility of carrying out the project remained implacable. ‘I was very interested to hear your views about matter,’ he told Russell in January 1913, ‘although I cannot imagine your way of working from sensedata forward.’ Russell did eventually return to the question of matter, and part of the impetus for him to do so came in October 1913, when he received the manuscript of an article Whitehead had just written on space. He and Wittgenstein read it together.28 This article contained Whitehead’s attempt to show how physical space could be constructed out of perceptual space. The significance of the idea does not seem to have struck Russell immediately, however. It was not until January 1914 that he came back from a holiday in Switzerland and dictated most of ‘The relation of sense-data to physics’ to a secretary.29 What he did in this paper was to construct matter out of what he now called sensibilia.30 A sensibile was to be, just as a quality had been, an entity of the same kind as a sense-datum, whether or not it is in fact sensed by anyone.31 As we shall see in a moment, however, his conception of what kind this is had evolved significantly. Russell’s central idea in the paper was to identify a physical object with the class of all the appearances it presents from various perspectives, and to define the matter of the thing to be the limit of these appearances as the distance from the object diminishes: matter, roughly, is what an object would look like from all angles if you got very close to it. If there is to be matter in 28

BR to OM, 2 Oct. 1913. 29 See Blackwell, ‘Our knowledge of “Our knowledge” ’. VIII, 7. 31 Moore, in ‘The status of sense-data’, preferred the English coinage ‘sensibles’.

30

CP,



Matter

places where no one is looking, as physics (not to say common sense) requires, we evidently have to assume the existence of unsensed sensibilia. In October 1912, as we have seen, he had an argument to show that matter cannot be constructed out of sense-data, because there are in physics hypothetical coordinates which do not correspond to sense-data: the example he used was the distance from us to a distant star, which is not part of the sense-datum since all that is present to the senses is the direction in which the star appears to us. In 1914 he overcame this difficulty by the brashly simple device of supposing that distance is part of the sense-datum. According to the new theory, therefore, a sensibile is something like a three-dimensional vector (i.e. it has both length and direction), and its origin (the location of the observer) is itself a position in three-dimensional space. Russell’s conception of the sort of thing that a sense-datum is had thus become significantly richer. Perhaps this is why he introduced the new word ‘sensibile’ for them, rather than continuing with the word ‘quality’ that he had used before. In February 1913 he had spoken of the need for a ‘fundamental novelty as to the nature of sensation’.32 Now he had found (or invented) it. If we succeeded in showing that tables and chairs are logical constructions out of sensibilia, that would show that the assumption that they exist has no more explanatory power than the assumption that sensibilia exist, since in any explanation involving a table we could simply substitute the logical construction of the table out of sensibilia. It would not on its own show that tables do not exist, however. It would not, for instance, rule out the possibility that sensibilia could also be constructed out of tables and chairs, in which case some further principle would be required to settle which should be eliminated in favour of the other: Occam’s razor on its own would not be enough. What made Russell try to eliminate matter in favour of sensibilia and not the other way round was an epistemological rather than a metaphysical consideration, namely that we can have direct acquaintance with sensibilia but not with matter. We should pause here to consider in more detail Russell’s enriched conception of sense-data as involving distance as well as direction. Humans do, it is true, have the ability to judge distance with respect to vision and hearing. There are two mechanisms for judging visual distance. The first, which uses stereoscopic vision to compare the views of a body as observed by the two eyes, depends on the fact that the objects we normally observe have an internal structure, and it is therefore highly fallible when applied to a uniform patch of colour. The second mechanism, which uses the focusing of the lens in the eye, is also easily misled in the case of uniform patches and is in any case much less accurate than the first. In practice, our ability to judge distance is highly influenced by contextual factors, as is vividly demonstrated by a whole host of trompe l’œil illusions. It may be added that both methods are useless 32

To OM, 23 Feb. 1913.

The relation of sense-data to physics  at large distances from the object, although this is of little import for Russell’s account since he identifies matter with the limit of the appearances as distance diminishes. What is more troubling, however, is that the methods break down at small distances too. When an object is less than 10 cm from the eye, we have difficulty focusing on it at all. Similar remarks apply to sound: there is no analogue of lens-focusing, and our stereoscopic sense of distance is routinely fooled every time we put on a pair of headphones. The possibility of optical and auditory illusions is not itself a difficulty for Russell’s account, of course, but the ease with which we can be fooled in our judgments of distance—and know that we are being fooled—adds a layer of implausibility to his decision to treat the distance to an object as part of what is indubitable in experience. It has the curious result, for instance, that the discussion one might read in a hi–fi magazine of the breadth and depth of the sound stage created by a hi–fi system becomes literally true. In relation to touch, of course, the kinaesthetic sense more plausibly provides us with a means of judging relative position, so that the distance as well as the direction from us of the object touched is plausibly part of what we apprehend. It is notable, therefore, that Russell was inclined to downplay the significance of touch, perhaps because it is of rather limited use in astronomy. Even by laying claim to an enriched conception according to which distance is part of the sense-datum, however, Russell could not derive his conception of matter from sense-data alone. Although he continued to claim, as ‘a probable inference from empirically ascertained causal laws’,33 that sensedata do not persist after they cease to be data, he was forced also to posit the existence, as sensibilia, of those appearances an object would have from a certain perspective even if there is no being with sense organs positioned so as to experience these sensibilia as sense-data. This assumption was a practical necessity if he was to identify the physical object with its appearances, but Russell was plainly uncomfortable with it and hoped that ‘the part played by unperceived “sensibilia” could be indefinitely diminished, probably by invoking the history of a “thing” to eke out the inferences derivable from its momentary appearance’.34 However, the notion of an unsensed sensibile is very puzzling. In the case where the sensibile is sensed, Russell is willing to grant its physiological subjectivity, i.e. causal dependence on the sense-organs, nerves, and brain. The appearance which a thing presents to us is causally dependent upon these, in exactly the same way as it is dependent upon intervening fog or smoke or coloured glass.35

What, though, of the case where there are no sense-organs positioned so as to sense the appearance of the thing from the relevant perspective? We might 33

CP, VIII, 9.

34

CP, VIII, 26.

35

CP, VIII, 7.



Matter

accept, on grounds of continuity, that in such a case the putative sense-datum would exist if sense-organs were appropriately positioned, but it seems very strange to say that it does exist. Russell affected not to recognize this strangeness. We have not the means of ascertaining how things appear from places not surrounded by brain and nerves and sense-organs, because we cannot leave the body; but continuity makes it not unreasonable to suppose that they present some appearance at such places.36

The ontology Russell conjures up is thus extravagantly rich. For every object and every position there exists a distinct thing which is the appearance that object would have if viewed by us from that position. Moreover, if appearances are, as Russell says, dependent on the sense-organs, nerves, and brain, presumably there will be for each such object and position various different appearances corresponding to different sorts of sensory apparatus. Russell claimed explicitly that if per impossibile there were a complete human body with no mind inside it, all those sensibilia would exist, in relation to that body, which would be sense-data if there were a mind in the body. What the mind adds to sensibilia, in fact, is merely awareness: everything else is physical or physiological.37

It is hard to see, in that case, why we should not say the same about cameras. But the appearance an object presents to infra-red film, for instance, may be very different from what it presents to ordinary film. Russell was in grave danger of populating the world with (among very many others) such physical entities as the appearances things present to an ASA 800 film exposed at aperture f8. Yet he continued to say that there is no reason to think a sense-datum persists after I have experienced it. If I stare fixedly at the brown stain on the wall for four seconds, therefore, the sense-datum I perceive persists, presumably, for just four seconds. What, then, of the case in which I do not look at the wall? The stain still presents, Russell claims, an (unsensed) sensibile to the perspective which in the other case I occupied. How long does this sensibile persist? No very plausible answer to this question presents itself. . The atomistic assumption In the period we have been discussing, then, Russell offered several different accounts of how physics can be derived from sense-data, or entities of the same kind as sense-data. All are hopeless, but for different reasons; and with accounts as varied as these we need not expect their failures to have a 36

CP, VIII, 7–8.

37

CP, VIII, 8.

The atomistic assumption  common source. However, there does seem to be one key move in Russell’s thinking that is worth isolating. It occurred when he identified sense-data as the simplest items we may attend to. When I speak of a ‘sense-datum’, I do not mean the whole of what is given in sense at one time. I mean rather such a part of the whole as might be singled out by attention: particular patches of colour, particular noises, and so on.38

Russell then supposed without argument that all our knowledge of the world derives from such items. ‘Sense-data at the times when they are data,’ he said, ‘are all that we directly and primitively know of the external world.’39 But it is very far from clear that this is true. Russell never seriously considered the possibility that the act of singling out an item of experience by attention somehow reconfigures it. It seems quite possible that there may be knowledge of the world that we acquire directly but not by means of any item we are capable of singling out by attention. Or, to put it another way, there might be some sorts of experience we have which are such that attending to their constituents is inherently distorting. In the case of an object in motion, for instance, attention might have the effect of freezing the object and hence eliminating the motion. A series of static images on a cinema film can give us the illusion of motion, but Russell was much too swift to think that this shows the cinema to be ‘a better metaphysician than common sense’,40 since it certainly does not follow from this that motion can be reconstructed out of those components of our experience of it that we are capable of attending to. This, one feels, is one of the points in Russell’s work at which the dangers of armchair psychology are most acute. The central point to note for our purposes, though, is how Russell changes his conception of what the basic entities are in response to an epistemological requirement. He took it as given, that is to say, that we do know many facts about the external world. Having also made the atomistic assumption that there are objects in the world with which we are directly acquainted in perception, his project was then to explain the former in terms of the latter. In order to do so, he was prepared to enrich to the point of implausibility the structure that he supposed the immediate objects of perception to possess. Logical atomism is perhaps the issue on which Wittgenstein’s debt to Russell is most apparent; the point of difference comes in their different responses to the epistemological problem just noted. While Russell was pulled in one direction or another by the technical constraints of his project, Wittgenstein was not. Russell’s common-sense instinct led him in the Problems (1911) to treat the existence of matter as a ‘probable inference’; in ‘On matter’ (1912) he attempted to construct matter out of sense-data, and concluded, hesitantly, that it could not be done, but conjectured instead that it can be constructed 38

CP, VIII, 6.

39

Ibid.

40

CP, VIII, 77.



Matter

out of things of the same nature as sense-data; in ‘The relation of sense-data to physics’ (1914) he enriched his conception of this nature in such a way as to make the construction feasible, but with the same goal as before of providing a philosophical licence for our common-sense belief in the existence of matter. Wittgenstein, on the other hand, seems to have been resolute throughout in resisting the need to provide this belief with a grounding of any kind. What is striking about the few scattered remarks of his about Russell’s project that have been preserved from this period is their consistency: they show no hint that he felt any temptation to pursue Russell’s route towards an enriched conception of the internal structure of sense-data. And that is surely because he never felt the tension which had drawn Russell down this path.

Chapter 

Analysis Whatever sympathy Wittgenstein may have had with the basic presumptions of Russell’s project, we saw in the last chapter that he was sceptical of its goal of constructing matter out of sense-data or things of that kind. He thought that physics could be explained without proceeding via a logical construction of matter. What this exposes is a fundamental difference between the two men in their attitude to constructed entities and hence in their conception of the method of analysis.

. Inference or construction? The advantage of Russell’s method of analysis as he advanced it up to 1912 was that it explained how we can understand the false sentence ‘The present King of France is bald’ without supposing there to be a shadowy, non-existent King of France, because analysis reveals that the proposition the sentence expresses does not contain any component corresponding to the denoting phrase ‘the present King of France’. An exactly similar analysis shows that the present President of France is not part of the proposition expressed by the sentence ‘The present President of France is bald’. But although we can understand this sentence without being required to suppose that France has a president, it will not be true unless she does. The analysis reduces dramatically the ontological demands of understanding, because the proposition does not contain the President as a component, but it leaves the demands of truth unchanged. A proposition contains as components only entities with which I am acquainted, but its truth may require the existence of various other entities (in this case, the present President of France). Then, as we have seen, Russell’s conception changed radically. In 1912 he made, and then in 1914 he purported to establish, the weak constructional conjecture, that matter can be defined as a logical construction out of things of the same kind as sense-data. Bolstered, perhaps, by his success in this endeavour, Russell now began to recommend the method of construction more generally: he adopted, that is to say, the methodological slogan that ‘wherever



Analysis

possible, logical constructions are to be substituted for inferred entities’.1 The idea was not Russell’s: he attributed ‘the suggestion and the stimulus for its application entirely to [his] friend and collaborator Dr Whitehead’.2 Nor was the idea itself new:3 within mathematics, it had already been current for some years, and had been exploited throughout Principia. Rather than assuming the existence of the real numbers, for instance, we devise a logical construction which we can show to have the properties that we believe the real numbers to have. Having done this, we then use the logical construction in place of the real numbers. What was new was only the application of this method in the case where the entities to be replaced by logical constructions are not numbers but people. If any assemblage of matter (such as the President of France) can be represented as a logical construction out of sensibilia, the truth of the proposition that the President of France is bald will not require the existence of an inferred physical entity, a flesh-and-blood president, but only of various classes of sensibilia. Instead of a host of disparate empirical inferences to the existence of tables, chairs, electrons, quarks, or interest rates, we make a single logical inference to the existence of the classes required to make the axiom of reducibility true. For a time, indeed, Russell thought that the success of the hypothesis of reducibility in explaining empirical phenomena constituted a justification for assuming it. This has the virtue of unification, no doubt, but one might wonder whether it also exhibits the advantages of theft over honest toil. Where does that leave common sense? It does not in itself show that ordinary objects do not really exist alongside their logically constructed surrogates. This is, of course, an issue that Russell had already faced in the mathematical case. If we construct logical objects with all the mathematical properties of the real numbers, say, do the real numbers themselves, as we previously conceived of them, drop away as unnecessary? In the first draft of ‘On matter’ in May 1912 Russell explicitly contrasted two attitudes one could take to constructed entities. If it is possible, by a logical construction, to define in terms of sense-data alone some object having the properties which dynamics assigns to matter, the cautious theorizer will adopt such a definition as the basis of his dynamics. In so doing, he insure[s] the certainty that he is not assuming the existence of something which perhaps does not exist, and he obeys one of the greatest principles of all philosophizing, I mean Occam’s razor: ‘Entia non multiplicanda praeter necessitatem.’ His task as logician, mathematician, and cautious man of science is then completed, and he may leave to the philosopher the task of inquiring whether, by means of some a priori principle, he is able to give to the symbols another interpretation in which they convey knowledge concerning things not given in sense. We, as philosophers, must pursue this inquiry, 1

CP, VIII, 11. 2 CP, VIII, 12. physical objects’.

3

See Miah, ‘The emergence of Russell’s logical construction of

Inference or construction?  knowing that mathematics and science can no longer help us, and that our conclusion, if affirmative, is forever incapable of empirical verification.4

I mentioned in the last chapter that there were parts of ‘On matter’ which Wittgenstein disagreed with. This intriguing passage, I shall suggest, may well be one of them. For here Russell somewhat curiously treats Occam’s razor as a methodological principle applicable only by the ‘cautious man of science’ and not by the metaphysician. If by the method of logical construction we succeeded in eliminating mention of some kind of entity (quarks, perhaps) from our scientific discourse, that would show that science has no need of such entities; and these entities would, if they exist, be unknowable. Russell, however, leaves open the possibility that the metaphysician would nonetheless wish to assert that they exist. In relation to such entities he thus recommends agnosticism rather than outright scepticism. When he revised the paper on matter, he dropped the passage just quoted, but his sympathy for agnosticism did not die. In his 1918 lectures, for instance, he considered the possibility that my desk is a logical construction out of its appearances. If you can get on without assuming the metaphysical and constant desk, you have a smaller risk of error than you had before. You would not necessarily have a smaller risk of error if you were tied down to denying the metaphysical desk. That is the advantage of Occam’s Razor, that it diminishes your risk of error. Considered in that way you may say that the whole of our problem [of logical constructions] belongs rather to science than to philosophy.5

Russell’s way of drawing his distinction between the attitudes of the scientist and the metaphysician is curious, because it is the opposite of what we observe in practice: it tends to be metaphysicians who doubt the existence of tables and chairs; men of science, however cautious, show no such inclination. And even on his own terms Russell’s attitude of agnosticism towards the table is surely a little puzzling. According to his Principle of Acquaintance, all the entities that occur in the propositions we understand are ones with which we are directly acquainted. It is far from clear how on this view we can even understand the hypothesis that a real, metaphysical table exists over and above the construction out of sense-data that Russell recommends to the cautious man of science. Wittgenstein would no doubt have applauded the idea of drawing a distinction between the cautious man of science and the metaphysician. Later he was often concerned to distinguish philosophy sharply from the natural sciences, but the concern is already visible in the Notes on Logic, where he remarked that ‘the word “philosophy” ought always to designate something over or under but not beside, the natural sciences’.6 He would, however, have been much 4

CP, VI, 512–3.

5

CP, VIII, 243.

6

B67.



Analysis

more likely to draw the distinction between the man of science and the metaphysician the other way round from Russell. One of the methodological principles that is prominent in his later philosophy is that the philosopher should not attempt to change the practice of other subjects (mathematics, science, etc.): they should carry on just as before. His inclination was therefore to endorse the fact that scientists talk about such things as electrons and protons, while allowing nonetheless that the philosopher might have a very different view of what is going on. The issue arises not only in relation to the problem of matter, of course, but in all the cases in which we apply the method of incomplete symbols which Russell developed after ‘On denoting’. Wittgenstein shows no trace of Russell’s desire to say that any of the eliminated entities might ‘really’ exist— behind our backs, as it were. Indeed, what little evidence we have of his early philosophical views suggests that this habit of ontological parsimony was something he acquired very early. He seems to have arrived at Cambridge quite at ease with sparse ontologies—Russell described him as ‘the only man I have ever met with a real bias for philosophical scepticism’7 —and in their disagreements it was always Russell who was more inclined to feel the draw of common-sense realism: it was Wittgenstein who had made him ‘more of a sceptic’.8 Wittgenstein’s already mentioned refusal to accept that anything empirical is knowable might, it is true, be the sort of naive scepticism one often meets in philosophical novices, but it might also, more narrowly, be the inept expression of a refusal to infer from the immediate objects of experience the existence of anything else. At base, the disagreement between Wittgenstein and Russell has to do with our understanding of the variable. As we noted in §1.4, Russell had originally hoped that his notion of a denoting concept would somehow explain our ability to talk about objects we are not directly acquainted with. As Moore was quick to notice, however, his theory of descriptions did not eliminate denoting concepts completely, but only reduced them all to one, namely the variable; the problem of explaining how we talk about objects with which we are not acquainted remained, but the bridging role which Russell had previously assigned to denoting concepts now fell wholly on the variable. You say ‘all the constituents of propositions we apprehend are entities with which we have immediate acquaintance’. Have we, then, acquaintance with the variable? and what sort of entity is it?9

Russell’s response was to admit the problem Moore had pointed out. I admit that the question you raise about the variable is puzzling, as are all questions about it. The view I usually incline to is that we have immediate acquaintance with 7

To OM, 2 May 1912.

8

To OM, 20 May 1912.

9

To BR, 23 Oct. 1905.

Wittgenstein’s conception  the variable, but it is not an entity. . . I only profess to reduce the problem of denoting to the problem of the variable.10

Russell’s problem was to explain, without merely lapsing into formalism, how we succeed in being acquainted with the variable even though its domain includes objects with which we are not acquainted. Russell’s shift from the method of inference to the method of construction changed the nature of the epistemological gap the variable had to bridge, but it did not explain how to bridge it. . Wittgenstein’s conception Wittgenstein’s response, characteristically, was not to solve Russell’s problem directly but to adopt a different conception of analysis which obviated the need for a solution. Russell’s programme aimed to construct out of sensibilia logical proxies for the ordinary objects of the world such as the President of France. If it had succeeded, it would have permitted us, in principle at least, to reinterpret sentences mentioning the President as being about his proxy, and therefore in the end, if we carry through the analysis in full, about various sensibilia. Russell’s programme, that is to say, is one way of trying to reinterpret talk about the world as talk about sensibilia; but it is only one way. In his way of proceeding, matter is replaced in the first stage of the analysis with a proxy consisting of various classes of sensibilia. Since classes are in turn according to Russell logical fictions, to be eliminated in the second stage of analysis, there is therefore a sense in which matter can be said to have been eliminated completely. But because the first stage replaces matter with what appears to be a logical surrogate, the analysis invites—indeed seems designed to encourage— the speculation that there is matter nonetheless. Russell’s discussion does not show any clear awareness of, or interest in, the possibility of a more radical reinterpretation in which matter is eliminated in such a way that no apparent proxy for it is provided. Wittgenstein, on the other hand, was much readier than Russell to say not only that there is no present President of France but that no single entity does duty for him either. What logical analysis tells us is that the present President of France is a complex made up out of components related in a certain manner, but Wittgenstein denied that the complex is a further entity over and above the components. When we say that France has a president, we should be interpreted as saying that there are some components related in a certain way, a way which ordinary language speaks of as constituting a president, not that any one object on its own is President of France. Wittgenstein’s conception of analysis is therefore more radical than Russell’s. The fundamental idea 10

To Moore, 25 Oct. 1905.



Analysis

of his eliminative programme is that no proposition is genuinely about a complex: the apparent complex will always disappear on analysis to be replaced by a statement about its components. Every proposition which seems to be about a complex can be analysed into a proposition about its constituents and the proposition which describes the complex perfectly.11

This latter proposition, Wittgenstein explains in a gloss, is ‘that proposition which is equivalent to saying the complex exists’.12 In order to illustrate the sort of analysis Wittgenstein had in mind, consider another example. My computer is complex. It is in fact very complex, but let us for current purposes assume (as I myself generally do when part of it does not work and needs replacing) that its immediate components—main unit, screen, keyboard, and mouse—are its ultimate constituents. In that case the statement that my computer is on the desk can be analysed as the conjunction of the statement that each of these constituents is on the desk and the statement that they are connected together so as to form a computer rather than just a set of parts awaiting assembly. In this analysis we do not require a quantifier ranging over complexes: the analysis is genuinely reductive in a sense in which Russell’s was not. Propositions which appear to be about matter turn out on analysis to be about simples, and the variable is no longer required to act as a bridge between the parts of the world with which I am acquainted and those with which I am not. To express the point more generally we need to invent a notation: let us write [ p] for the complex, if there is one, consisting of the objects referred to in the proposition p, combined in such a way that p. So Caesar’s death, for instance, is just [Caesar died]. And in one of Wittgenstein’s own examples a broom may be thought of as a complex consisting of the brush and the broomstick: in the notation just introduced the broom is the complex [the broomstick is attached to the brush]. On the other hand, there is no such complex as [Charles I died in bed], since Charles I died on the scaffold. Wittgenstein’s idea, then, was that a proposition φ ([ p]) apparently about the complex [ p] should be analysed as the conjunction of a proposition φ ! (a1 , . . . , an ) about the constituents a1 , . . . , an of the complex and the proposition p which describes the complex, i.e. which ‘is equivalent to saying the complex exists’.13 In the case of a relational proposition aRb, for instance, Wittgenstein’s proposal was that we analyse φ ([aRb]) ≡ φ ! (a,b) aRb, where φ ! is a propositional function appropriately related to the original function φ . 11

C25. 12 Ibid. 13 At B17 Wittgenstein describes the analysis as a ‘logical sum’, but this is plainly just a slip: he meant logical product.

Practicalities  Notice, though, that the apparently innocuous issue of deriving from φ the ‘appropriately related’ function φ ! masks a considerable degree of complexity. In the case of the first example considered earlier, of course, the answer is simple: my computer is on the desk just when its components are, so φ ! is simply the conjunction of the relevant instances of φ . Similarly, in Wittgenstein’s own example, ‘My broom is in the corner’ could perhaps be replaced by ‘The broomstick is there, and so is the brush, and the broomstick is fixed in the brush’.14 But even a casual consideration of examples such as ‘The army surrounded the castle’ and ‘My computer weighs 30 kg’ suggests that the link is not always nearly as simple as this. In the Notebooks Wittgenstein tried repeatedly to ignore this issue, apparently proposing that we should always define φ ([aRb]) to mean the same as φ (a) φ (b) aRb.15 This is perhaps one instance (among many) of the remarkable insouciance which he manifested throughout the period we are studying towards the details of analysis. Although he was willing to concede that how the analysis is to be done is ‘an important question’, nonetheless, ‘its answer is not unconditionally necessary for the construction of logic’.16 Not until June 1915 did he finally face up to the difficulty, conceding that a proposition such as ‘The watch is lying on the table’ obscures a large (and highly contextsensitive, not to say indefinite) degree of complexity. Yet the concession was only temporary: the finished text of the Tractatus betrays puzzlingly little trace of it.

. Practicalities Wittgenstein seems to have been confident that physics can be interpreted so as to be about sense-data but plainly had not the least idea how the reinterpretation was to be carried out. What we see emerging here is a clear difference in conception between him and Russell. According to Wittgenstein, logic is never concerned with the details of which resources we in practice need in order to represent the world around us; rather is it concerned with those structural features which must be present in those resources simply in order that it should be possible for them to represent a world at all. That is not to say that he had no interest in identifying the simple entities that analysis would reveal as the ultimate constituents of the world. This is a recurring theme in the Notebooks, where there is, for instance, the lengthy discussion of the notion of simplicity which I referred to a moment ago. The absence of firm evidence that he was interested in the question while at Cambridge need show no more than that he did not yet see any reason to doubt 14

PI, §60.

15

5 Sep. 1914.

16

B17.



Analysis

the account favoured (with minor variations) by both Russell and Moore at the time. One of the striking features of the Tractatus, then, is Wittgenstein’s lack of curiosity about how the reduction of various sorts of discourse to talk about simples is actually to be carried through. The discussion of simplicity in June 1915 resonates powerfully with Wittgenstein’s later philosophy, but not with the Tractatus, where the obvious examples which create problems for his conception of analysis (such as ‘The army surrounded the castle’) are quite absent. Contrast this with Russell’s recommendation that 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.17

Examples of Wittgenstein’s failure to test his theory by its ‘capacity for dealing with puzzles’ abound in the Tractatus. For instance, he treated the incompatibility of ‘This is red’ and ‘This is green’ as showing that redness and greenness cannot be simple and require further analysis,18 but there is no sign that he spent much (or indeed any) time thinking out how that analysis might go. And he acquired this casual attitude to the practicalities of analysis early, remarking to Russell in 1912 that matter is a ‘trivial problem’.19 What is harder, perhaps, is to see why Wittgenstein took this attitude. As we shall see in the next chapter, he had from very early a conception of logic as being of ‘a totally different kind than any other science’,20 and would therefore no doubt have disliked the analogy Russell drew between logical puzzles and scientific experiments. He thought, that is to say, that nothing in logic can have a status even analogous to that of an experiment. But something more than the observation that logic is different from physics is plainly needed if we are to show that logical theories cannot be confirmed or refuted by logical puzzles. We should recognize the possibility of a non-philosophical explanation for Wittgenstein’s behaviour. The interest he began to take in the practicalities of analysis when he returned to philosophy at the end of the 1920s was strongly influenced by his interaction with members of the Vienna Circle, who took a decidedly scientific approach to philosophy and certainly saw themselves as undertaking a project of analysis by which the discourse of natural science could be represented in a formal logical language. It was probably under pressure from them that Wittgenstein came to see the genuine difficulty which the incompatibility of ‘This is red’ and ‘This is green’ poses for his theory.21 Russell, at the time when Wittgenstein worked with him, was engaged in the beginnings of a project that was similar in many ways to the Vienna Circle’s. 17 CP, IV, 420. 18 6.3751. remarks on logical form’.

19

BR to OM, 30 Apr. 1912.

20

To BR, 22 June 1912.

21

Cf. ‘Some

Practicalities  So was Whitehead. Yet their effect on Wittgenstein seems to have been the opposite of the Vienna Circle’s fifteen years later: he simply left the technical work to them. The explanation cannot be merely that they were both so much better at it than Wittgenstein, although this is no doubt true. This is a matter of symbolic facility and technical ingenuity: Russell and Whitehead were both highly competent mathematicians, which Wittgenstein was not. Rather is it connected with the desire, or even need, which Wittgenstein evidently had at this early stage in his life to find a field in which he could make a distinctive contribution. Some of the things Wittgenstein did—his jet engine patent, experiments on the psychology of rhythm—have in retrospect a certain resemblance to the dabblings of a wealthy dilettante. But they also betray signs of his ambition, a desire to find a new field in which he could make his mark. When he arrived in Manchester, there were no doubt other available engineering projects less speculative (and probably, in the short term, more fruitful) than trying to develop a jet engine; but because avionics was a new science, any success he might have had in it would have had that much more impact. And in the same way one of the things that attracted him to experimental psychology was surely that it too was an unploughed field: the Cambridge laboratory in which Wittgenstein did his experiments was only just being set up at the time. The need to find a creative equilibrium with Wittgenstein is a recurring theme in Russell’s letters to Ottoline, and he often seems to have conceived of this as much in terms of marking out his own territory as of negotiating a productive means of collaboration. Wittgenstein no doubt felt the point at least as keenly. His efforts to respond to it, and hence find a contribution that was distinctively his own, perhaps encouraged him in what became one of his most characteristic philosophical techniques, that of seeing past a piece of formalism so as to produce a criticism that is independent of its details. Something of this technique we have perhaps already seen in embryo in his marginalia to Lamb’s Hydrodynamics, but it is at its most intense in his application of it to Principia Mathematica. Wittgenstein certainly studied Principia. We know this, because when Russell tried to sell Wittgenstein’s copy of Volume I (presumably left behind in Cambridge when he departed for Norway and therefore part of the job lot Russell bought from him after the war), it ‘still had decorations by Wittgenstein’.22 Until Wittgenstein’s copy resurfaces, it can only be a matter of speculation whether the decorations have the somewhat eccentric character of Wittgenstein’s earlier annotations of his copy of the Hydrodynamics, or whether they indicate a serious engagement with the formal technicalities of the logic Whitehead and Russell developed. In the meantime, though, it is surely worth drawing attention to the curiously hesitant quality of Wittgenstein’s criticisms of Principia: again and again he gestures towards 22

BR to Ogden, 23 June 1922.



Analysis

errors in Russell’s presentation but stops just short of saying exactly what they are or, more importantly, quite what ought to be done about them. Perhaps the most startling fact to note in relation to Wittgenstein’s conception of analysis, though, is how early he came upon it. Both the idea itself and his insouciant attitude to the details of its implementation seem already to have been in place by April 1912, when he told Russell—without, surely, any idea how it could actually be done—that ‘if there is no matter, . . . physics and astronomy, and all the other sciences could still be interpreted so as to be true’.23 23

To OM, 23 Apr. 1912.

Chapter 

The fundamental thought Later, not just in the period leading up to the Tractatus but throughout his philosophical career, Wittgenstein was an inveterate keeper of Tagebücher— journals—in which he recorded day-by-day his philosophical thoughts, and there is no reason to think that his method of working was any different while he was at Cambridge. It may well be that he was carrying on a habit he had learned as an engineering student: it bears an obvious similarity to the practice scientists have of keeping lab books. However, the notebooks devoted to Wittgenstein’s Cambridge period have not survived: he left them in storage in Cambridge when he left for Norway in 1913, and after the war asked Russell to have them destroyed.1 So the only evidence we have from Wittgenstein’s own pen of his philosophical progress consists of letters he wrote to Russell, usually during vacations when one or other of them was away from Cambridge. Our aim in the next few chapters will be to find out what these letters can tell us about the development of his thought. . Why logic? When he arrived at Cambridge in October 1911, Wittgenstein told Russell that he wanted to work on the philosophy of mathematics. Yet the surviving letters make it clear that from quite early on the focus of his work was the philosophy not of mathematics but of logic: he was trying to construct a theory of the symbolism, by which he meant an account of the structure of propositions which would both characterize the notion of logical truth and explain its nature. With remarkable stability of purpose, moreover, this remained the focus of his work until at least the end of 1915: the book which he was then beginning to compile was originally, it seems, to be called Der Satz (‘The Proposition’).2 But this was not really a change of focus from Wittgenstein’s original intention to work on the philosophy of mathematics. By the time he arrived in Cambridge Whitehead and Russell had published volume I of Principia Mathematica, the book in which they aimed to make good Russell’s claim in the 1

1 Nov. 1919.

2

Bartley, Wittgenstein, 45.



The fundamental thought

Principles that mathematics is part of logic. Quite soon, it seems, Wittgenstein decided to subject the philosophical foundations of Principia to a detailed critique. One impetus to this will no doubt have been his meetings with Frege, whose views on Principia may be fairly gauged from the letter he wrote to Jourdain in 1914. This expounds over several pages Russell’s confusions over the use of the word ‘variable’, but nowhere in the letter do Frege’s criticisms advance beyond the first fifteen pages of the Introduction. (His first draft of the letter makes a brief comment on the explanation of the ramified hierarchy on pages 52–4.) Frege had never been the most sympathetic reader of other philosophers’ work, but by this stage in his life he evidently found it hard to turn to the second page of a book until he had corrected all the errors of expression on the first. It is easy to see how a conversation with him might have given Wittgenstein the idea of writing an extended critique of the Introduction to Principia, with the aim of putting the logic presented there on a firm philosophical foundation. It is clear, too, that Russell himself soon accorded with Wittgenstein’s project of criticizing and revising the early pages of Principia: from April 1912 onwards he repeatedly spoke of abandoning technical philosophy to Wittgenstein;3 in February 1913 he remarked, ‘Wittgenstein has persuaded me that the early parts of Principia Math[ematic]a are very inexact, but fortunately it is his business to put them right, not mine’;4 and by August Pinsent was writing in his diary, ‘It is probable that the first volume of the “Principia” will have to be rewritten, and Wittgenstein may write himself the first eleven chapters. That is a splendid triumph for him!’5 The first eleven chapters constitute precisely the parts of Principia that deal with logic proper, Section A on ‘The theory of deduction’ and Section B on ‘Theory of apparent variables’: it was not proposed, it seems, that Wittgenstein should revise Section C on ‘classes and relations’. Russell recognized quite early that Wittgenstein’s approach involved above all a concern with the most fundamental philosophical issues. ‘He doesn’t want to prove this or that, but to find out how things really are.’6 It probably took him some time, however, to realize just how true this was. When he invited Wittgenstein to rewrite the first two sections of Principia, the Tractatus was presumably not quite what he had in mind: when he eventually realized that Wittgenstein was not going to make the kind of revisions he envisaged, he had to make the revisions by writing an Introduction to the second edition himself. Throughout, Wittgenstein’s approach to the task evidently shared something with Frege’s, mentioned earlier. His concern was, as Frege’s had been, overwhelmingly with the very earliest parts of Principia. There is very little evi3

To OM, 30 Apr., 1 June, 24 July, 4 Sep., 14 Oct. 1912, 22 Jan., 7 Feb., 23 Feb. 1913. OM, 23 Feb. 1913. 5 29 Aug. 1913. 6 To OM, 8 Mar. 1912.

4

To

Logical constants as incomplete symbols  dence requiring us to suppose that he ever got much beyond the Introduction, and the vast majority of his comments are directed at issues that crop up in the first twenty pages of that. Moreover, the issues which concerned him were generally those that one might notice if one focused on Russell’s manner of expression rather than on the general drift of his exposition. To the mathematician’s eye the most singular move in the whole Introduction, which cries out for more justification than Russell could give it, is surely the Axiom of Reducibility. Yet we have no evidence that Wittgenstein was concerned with this problematic axiom until the summer of 1913, and even then he seems at first to have recognized only dimly what makes it controversial. . Logical constants as incomplete symbols Suppose for a moment that we start, as Wittgenstein evidently did, with Russell’s conception of propositions as complexes. If p and q are two such complexes, what is there in the world that would make the conjunction ‘ p and q’ or the disjunction ‘ p or q’ true? Russell’s earlier view had been that these may be explained as more elaborate complexes: ‘ p and q’, for instance, would be a complex consisting of the complex p, the complex q and the universal conjunction. In his lectures during February 1912, however, we find Russell saying (according to Moore’s notes), ‘logical constants are not the sort of constants wh[ich] can be substituted: e.g. or, not, true, 0, 1, 2 etc. All of these are incomplete symbols (I think, but am not sure).’ We know that Wittgenstein liked to read Russell’s lectures before he gave them, in order (presumably) to get his objections in early.7 So it may be that a trace of Wittgenstein’s influence is to be found in what Russell had to say here, but whether in the suggestion or in the hesitation it is hard to be sure. There are two parts to Russell’s suggestion. The first part, that occurrences of logical constants in a proposition cannot be replaced by variables, is something he had long held, at any rate in application to constants such as ‘or’ and ‘not’: he stated it explicitly in the Principles, for example. Moreover, it is something Wittgenstein himself later held: he stated it in the Notes dictated to Moore;8 and in the case of numbers it is a view he expounded in the Tractatus and probably held by the summer of 1914, if not before. As we saw in the last chapter, the second part of Russell’s suggestion, that logical constants are incomplete symbols, need not on Russell’s lips have committed him to denying that there are such things as logical constants. It would, on the other hand, have committed him to what we would nowadays call a research programme, namely to provide contextual rewriting rules for all the contexts in which logical constants can meaningfully occur. And there is evi7

BR to OM, 7 May 1912.

8

¶47.



The fundamental thought

dence that Wittgenstein may have been engaged during the summer of 1912 in a variant of such a programme. In one letter he suggests to Russell that universal generalizations might be incomplete symbols: Will you think that I have gone mad if I make the following suggestion?: The sign ‘( x) φ ( x)’ is not a complete symbol but has meaning only in an inference of the kind: from ! φ x ⊃x ψ x φ (a) follows ψ a. Or more general: from ! ( x) φ x #0 (a) follows φ (a). I am—of course—most uncertain about the matter but something of the sort might really be true.9

The caution with which Wittgenstein presented his ideas here presumably has something to do with the distance he was now going beyond what Russell had suggested in his lecture. The idea then had been that the logical constants are incomplete symbols. In the current case that would amount to holding that any sentence involving a universal quantifier should be rewritten in such a way that the universal quantifier disappears. But now Wittgenstein was flirting with something much more like the idea that the sense of a logical constant is given by the inference rules that govern its application. In another letter10 around this time Wittgenstein introduced rather similar speculations concerning the meaning of disjunctive propositions, but once again he offered them only hesitantly and we have no evidence that he developed them very far (nor any reason to think that he would have been well advised to do so). The proposals Wittgenstein makes in these letters are thus experimental at best. Moreover, it is hard for us now to fill out his sketches in anything like a satisfying manner. This is at least partly because Wittgenstein himself soon abandoned them (in many cases wisely), with the result that his later, much more detailed writings contain no further clues as to what he had in mind. . There are no logical constants Let us turn now to an earlier letter Wittgenstein wrote to Russell. It dates from early in the summer of 1912, when Wittgenstein had not yet left Cambridge for the long vacation, and the philosophical remarks it contains are the earliest from Wittgenstein’s pen that we have. In contrast to the letters from later that summer, this one contains thoughts that turned out to be of fundamental importance for his future work: what chance has preserved for us as his first philosophical remarks are among his most significant. Logic is still in the melting pot but one thing gets more and more obvious to me: The propositions of logic contain ONLY apparent variables and whatever may turn out to be the proper explanation of apparent variables, its consequences must be that there are NO logical constants. Logic must turn out to be a totally different kind than any other science.11 9

1 July 1912.

10

[Summer 1912] (CL, no. 4).

11

22 June 1912.

There are no logical constants  Wittgenstein’s most immediate target in saying that there are no logical constants was of course Russell, whose conception entailed that items of distinctively logical vocabulary such as ‘not’ and ‘or’ refer to entities. But the remark also opposes him to Frege, whose inventory of logical objects included natural numbers, real numbers and truth-values, all of which he reduced in the Grundgesetze to a single kind of logical objects which he called value-ranges. If Wittgenstein had by this time adopted the view described in the last chapter, that there is nothing in the world to which an incomplete symbol refers, his pronouncement that there are no logical constants might be read at first sight as no more than a redescription of the research programme he had embarked on, apparently at Russell’s instigation, of providing elimination rules for logical constants. Even if so, however, it is plain that the motivations of the two men were different. For Russell the idea that logical constants are incomplete symbols was a technical conjecture to be tested: the test is whether we can find appropriate rewriting rules that allow us to eliminate the logical constants from all the contexts in which they occur meaningfully. For Wittgenstein, on the other hand, the reason to think that there are no logical constants is not a technical one but springs from his conception of the nature of logic. The completion of the research programme would therefore at best provide confirmation of what he already thought he knew. This is characteristic of a difference in philosophical method which runs right through their work: Russell, it is clear, enjoyed technical challenges for their own sake; but more than that, he thought that technical work could lead to philosophical conclusions. Wittgenstein, by contrast, was inclined to indulge in technical projects only in order to fill the gaps in his system which his philosophical speculations had already identified, and usually with the greatest reluctance. Russell once confessed it to be one of his dreams ‘to found a great school of mathematically-minded philosophers’. Wittgenstein, he thought, was ‘exactly my dream’.12 But it is hard to believe that Wittgenstein was ever quite what Russell was hoping for: his work was never really mathematically minded in the way that Russell meant. It is worth noting, too, that in Russell’s usage the phrase ‘logical constant’ was of wide range: he applied it, for instance, to what he called propositional forms, such as the general form of all subject-predicate propositions or the general form of all relational propositions.13 No doubt Wittgenstein intended his remark to apply to all such supposed entities as well. His usage did later waver a little: in the wartime Notebooks he was willing to grant that there is one ‘logical constant’, namely the general form of proposition.14 However, it would take a lot of work to represent this as a significant retreat from his 1912 view, since (as Wittgenstein’s use of scare-quotes hints) whatever the sense is in 12

To OM, 29 Dec. 1912.

13

Cf. McGuinness, Approaches, 104.

14

NB, 5 May 1915.



The fundamental thought

which the general form of proposition can be thought of as a logical constant, it is at some distance from the uses Russell and Frege made of the term. On the other hand, perhaps the exact scope Wittgenstein meant the term to have is not especially important, because he does not seem to have supposed that he had a general argument that there are no logical constants. Thus in the Notes on Logic we find not the general claim that there are no logical constants but only particular instances of it: in various places he targeted Russell’s conception of ‘not’ and ‘or’ and of variables, and Frege’s conception of truthvalues, and he offered piecemeal arguments against these different targets. He did not think it sufficient simply to appeal to the general claim. Perhaps the reason is that the general claim had for him the character more of a guiding methodological principle—he later called it his ‘fundamental thought’—than of a firm conclusion for which he could offer a persuasive argument.

. There are no real variables On Russell’s understanding variables, too, were (somewhat confusingly) logical constants. This is because, at least when he wrote the Principles and for some time thereafter, he tended to use the word ‘variable’ to refer not to the signs ‘ x’, ‘y’, etc. that are used in his logical notation, but to what these signs express. Even in Principia Russell still often speaks this way: for instance, he talks of a logical assertion as concerning a variable x in the same way that a non-logical assertion concerns Socrates or Plato.15 So when Wittgenstein said in his letter to Russell that the propositions of logic do not contain real variables, this may be read as a particular case of the general claim just discussed: if there are no logical constants, there are no real variables in the sense of the Principles, and hence no propositions that are about them. It is worth observing, though, that (in conformity with the pattern we noted) Wittgenstein does not present his expulsion of real variables from logic as a consequence of the general claim. Rather he holds out the hope that a correct account of the variable might somehow have the general claim as a consequence. If we are to follow Wittgenstein’s thought processes here, we should start by considering his claim that the propositions of logic contain only apparent variables; or, as we should say nowadays, that any variables in them may only occur bound and not free. Wittgenstein’s complaint therefore has something to do with Whitehead and Russell’s practice of stating theorems in such a way that they involve real variables as well as apparent variables. ‘There is,’ as he later put it, ‘no proposition which is expressed by “ x = x”, for “ x” has no 15

PM, I, 93.

There are no real variables  signification; but there is a proposition “( x) x = x” and propositions such as “Socrates = Socrates” etc.’16 But if this were Wittgenstein’s only point, it would surely not go at all deep, since the practice just mentioned seems to be only a harmless abbreviating convention. After all, very many theorems in logic and mathematics have the logical form of universal generalizations. So it would be a convenience to adopt the convention that if we assert a formula involving an apparent variable ‘# φ ( x)’, we should be taken to mean to assert the universal generalization ‘# ( x) φ ( x)’. All the convention does, it seems, is to make the formula shorter and hence somewhat easier to read. That leaving off initial universal quantifiers is best thought of as no more than a convenient abbreviating convention is indeed a rather obvious point, and not one that originates with Wittgenstein: the insistence that whatever can be asserted cannot contain a free variable is ubiquitous in Frege, for instance. What is capable of truth or falsity is a complete thought, he says, and what expresses a complete thought is a sentence. If we replace an occurrence of a name in a sentence with a variable, the resulting formula does not express a thought. So although in the Begriffsschrift 17 Frege adopted the abbreviating practice just mentioned, he made it clear that it was only an abbreviation, and therefore harmless. (Indeed he adopted at the same time, presumably to emphasize his awareness that it is only an abbreviating device, the further convention that variables whose binding is supplied only implicitly by the abbreviating device should be written with Latin letters, whereas he always used German letters for ordinary bound variables.) On the face of it, then, Wittgenstein was merely advocating a piece of elementary logical hygiene which he had learnt from Frege; a piece of hygiene, moreover, with which Whitehead and Russell would have agreed readily enough. They stated quite clearly, after all, that ‘ “ x is hurt” really makes no assertion at all, till we have settled who x is’.18 They made use of the notion of the assertion of a propositional function in Principia, therefore, only because they found it technically convenient to do so. But it is convenient, and the hygiene involved in eliminating it, although in a sense elementary, is by no means trivial. It is far from obvious, that is to say, how to amend the formal presentation of logic in Principia so as to eliminate the need to assert propositional functions rather than just propositions. This is because in Principia the notion of asserting a propositional function, as when we write ‘# φ ( x)’, is taken not as an abbreviation for ‘# ( x) φ ( x)’, as in effect it is by Frege, but as a distinct primitive idea. Because of this, various primitive propositions are then needed to govern the transitions between the two sorts of assertion. For example, they had to adopt the rule, ‘In any assertion 16

B13.

17

§11.

18

PM, I, 14.



The fundamental thought

containing a real variable, this real variable may be turned into an apparent variable of which all possible values are asserted to satisfy the function in question.’19 The result of all this is that it is rather hard to determine from a casual inspection how easy it would be to recast ∗9 of Principia so as to remove the need to take assertion of a propositional function as a primitive notion. The complexity of the task may be judged, indeed, from Russell’s introduction to the 1925 second edition of Principia, where he attempted to complete it by offering a new ∗8 to replace ∗9 in which assertion of a propositional function was no longer taken as a primitive. Note, moreover, that in his letter to Russell, Wittgenstein did not make the general claim that propositions contain only apparent and not real variables (even though that is certainly true), but restricted himself to the special case of the propositions of logic. This is at first sight odd since, as we have already observed, there is an obvious argument leading directly to the conclusion that no proposition can contain a free variable. (An expression containing a free variable can only be a formula, not a sentence. And a formula cannot express a proposition; only a sentence can do that.) This suggests rather strongly that the general point was at this stage a peripheral issue, and that what really concerned Wittgenstein was a specific point about the nature of logic. One reason Wittgenstein limited himself to the claim that the propositions of logic do not contain real variables is no doubt that what motivated him to make it was a conception of logic as a subject without a subject matter. On Russell’s conception a proposition containing a sign for a real variable says something about that variable. But according to Wittgenstein the propositions of logic are not about anything; and that is why they cannot contain real variables. Russell’s use of real variables, which ought to have been (as it was for Frege) no more than a convenient abbreviation, had on Wittgenstein’s diagnosis misled Russell into imagining that logic has a subject matter of which these variables form a part. ‘By employment of variables instead of the generality-sign, it comes to seem as if logic dealt with things which have been deprived of all properties except thing-hood.’20 But there is another point Wittgenstein is making here. When he says that the propositions of logic contain only apparent variables, he is ruling out not only real variables but constants too. The reason is, once again, his conception of logic as a subject without a subject matter. He does not here state the converse, that the propositions of logic are the only true propositions which do not contain any constants, but he may for a time have believed it. At any rate, it was not until he was in Norway that he developed a criterion—tautologousness—to distinguish between logical truths and those that are wholly general but true only by accident. 19

PM, ∗9.13.

20

B71.

Logic as a special science  . Logic as a special science According to Wittgenstein, the reason that the propositions of logic are not about anything is that ‘logic must turn out to be a totally different kind than any other science’. He explained the point in a little more detail in the Notes, where he said that a reason against the existence of logical constants ‘is the generality of logic: logic cannot treat a special set of things’.21 His view contrasts starkly with that of Russell, for whom the world has a physical aspect, which is the business of physics, and a logical aspect, which is the business of logic. In order for Russell’s view of logic to make sense, there has to be something, a subject matter, that logic may be said to be about. Over the decade since the Principles Russell had been forced, in his various responses to the paradoxes, to revise several times what he thought this subject matter was, but he never seems to have given up the view that it has a subject matter of some kind. Even the conjecture he made in his 1912 lectures that the logical constants may be incomplete symbols is not really a denial of this view. He hoped, perhaps, that all the contexts in which ‘∼’ and ‘v’ occur meaningfully could be analysed in such a way that these logical constants do not occur in the analysis. But in that case other notions with some title to be regarded as logical would presumably occur in the analysans. These other notions would then turn out to be the real logical constants. There is no suggestion that there might turn out to be no logical constants at all. Wittgenstein’s view, on the other hand, is just that: when language is correctly analysed, it must turn out that there is nothing in the world corresponding to any of the logical words in our vocabulary. His reason was that Russell’s assumption of the existence of a logical realm existing somehow alongside the physical one rendered it inexplicable why the laws obeyed by the former should be applicable to the latter. His alternative explanation, discernible already (at least in outline) in his letter of June 1912, is that logic applies in reasoning about anything whatever just because it has no subject matter. Wittgenstein said in the Notes both that philosophy does not exist alongside the natural sciences22 and that ‘philosophy consists of logic and metaphysics: logic is its basis’.23 It follows in particular that logic cannot exist alongside the natural sciences either. One way of expressing this would be to press the question of how Russell’s notion of acquaintance could apply in the case of logic—to ask the question how it would be possible to apply logic if there were such a thing as logical experience. ‘We could say: if there would be a logic, even if there were no world, how then could there be a logic, since there is a world.’24 This way of making the point is of a later date, and reflects the fact that by then Wittgenstein had a developed account to offer of the nature of logic: logic is an expression of the structure of the world, and so 21

C9.

22

B67.

23

B61.

24

5.5521.



The fundamental thought

if there were no world, there would be no logic. But the underlying point remains the one he had made in 1912, namely that Russell’s conception makes the propositions of logic into substantive facts about a logical realm which exists independent of the physical world (and would exist even if there were no world); far from explaining the universality of logic, Wittgenstein thought, this makes it utterly mysterious. . Logic as contentless Logic ‘cannot treat a special set of things’, because if it did it would not be wholly general. The generality of logic that Wittgenstein is appealing to here stands in contrast to the other sciences. We might think of it as the business of scientists to formulate laws of general applicability, but in saying this we assume implicitly that the range of applicability of these laws will not be completely general. The laws of supply and demand may apply quite generally to economic transactions between people, but they are of no use whatever if we are studying the flow of water on a river bed. The laws of fluid dynamics will be relevant in this case, but they in turn are of little use in studying the motion of the planets. If we think of logic as a science, however, we are immediately struck by its special character: it is applicable in reasoning about anything whatever. There seems to be something radically incoherent about supposing that there could be anything not subject to the laws of logic. In previous chapters of this book it has almost exclusively been Russell’s influence on Wittgenstein that we have been discussing. From now on, however, it will be increasingly necessary to take into account the influence, in writing and in person, of Frege. I have already mentioned a visit Wittgenstein made to Frege in 1911. The only other visits that we know of were at Christmas 1912 and Christmas 1913, and lasted a few days at most. Although the personal contact between them was thus very limited, the effect Frege’s thinking had on Wittgenstein was nonetheless profound. The letter we are now discussing already shows signs of his influence, for the very question that now engaged Wittgenstein—how to explain the nature of logic—was one that had preoccupied him. The laws of logic, Frege said, have a special title not possessed by the laws of geometry and physics because ‘they are the most general laws, which prescribe universally the way in which one ought to think if one is to think at all’.25 Notice how Frege here advances beyond the thought just adumbrated that logic is the maximally general science—the thought, that is to say, which might be captured by saying that its propositions contain only apparent variables—in order to make the stronger claim that it represents a normative constraint on 25

Gg, I, xv.

Logic as contentless  thought itself. The generality of logic, he might have said, is not accidental generality.26 And when he came to offer an explanation for logic’s universality, his thought took a correspondingly unexpected turn. Geometry and physics have truth as their goal, he said, but only logic has truth as its subject matter. Logic is concerned with the predicate ‘true’ in a quite special way, namely in a way analogous to that in which physics has to do with the predicates ‘heavy’ and ‘warm’ or chemistry with the predicates ‘acid’ and ‘alkaline’.27

For Frege, the point of conceiving logic in this manner was that it allowed him to resist psychologism: the laws of logic are not laws of thought but laws of truth. I understand by ‘laws of logic’ not psychological laws of takings-to-be-true, but laws of truth. If it is true that I am writing this in my chamber on the 13th of July 1893, while the wind howls out-of-doors, then it remains true even if all men should subsequently take it to be false. If being true is thus independent of being acknowledged by somebody or other, then the laws of truth are not psychological laws: they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace.28

When Wittgenstein found himself in a prisoner of war camp at the end of the war, he quoted passages of Grundgesetze from memory to another prisoner in order to convince him of the glories of its preface.29 It is hard not to imagine that this was one of the passages he recited; hard, too, to disagree with his assessment of it. When Frege tells us that logic is concerned with the predicate ‘true’ as physics is with the predicate ‘heavy’, we may read him as saying that logic is about truth just as physics is about heaviness. But this should give us pause when we unite it with his further claim that truth is not a property as heaviness is. The peculiarity of truth, that is to say, is not so much that it ‘is obviously something so primitive and simple that it is not possible to reduce it to anything simpler’, but rather that it makes no specific contribution to the senses of sentences in which it occurs. ‘What distinguishes it from all other predicates is that predicating it is always included in predicating anything whatever.’30 The special character of logic therefore consists for Frege in the peculiarly evanescent quality of its subject matter. The word ‘true’ is not an adjective in the ordinary sense. . . The sense of the word ‘true’ is such that it does not make any essential contribution to the thought. If I assert ‘it is true that sea-water is salty’, I assert the same thing as if I assert ‘sea-water is salty’.31

The quotation is from a note in Frege’s Nachlaß of uncertain date headed ‘My basic logical insights’, but Wittgenstein could have found the same idea in Frege’s published writings if he read them carefully enough. 26 Cf. 6.1232. 251.

27

PW, 128.

28

Gg, I, xvi.

29

McGuinness, Young Ludwig, 270.

30

PW, 129.

31

PW,



The fundamental thought

One can, indeed, say: ‘The thought, that 5 is a prime number, is true.’ But closer examination shows that nothing more has been said than in the simple sentence ‘5 is a prime number’. The truth claim arises in each case from the form of the declarative sentence, and when the latter lacks its usual force, e.g. in the mouth of an actor upon stage, even the sentence ‘The thought that 5 is a prime number is true’ contains only a thought, and indeed the same thought as the simple ‘5 is a prime number’.32

Truth’s peculiarity, in other words, is that it is not properly to be conceived of as a property at all.33 But if logic is about truth and truth is in this sense redundant, it is a short step to conclude that logic is not about anything. Wittgenstein’s conclusion that logic is contentless thus derives from two tenets central to Frege’s thinking. But however short a step it is from Frege’s views to this conclusion, it is one that Frege himself never took. Why not? The short answer is that he did not take the redundancy just mentioned to show that the word ‘true’ has no sense. Early and late, he always shied away from saying that truth is redundant and emphasized instead not its redundancy but its indefinability. ‘What true is, is indefinable’,34 he wrote in an early note of his views on logic; and in 1918 he still maintained that truth is ‘sui generis and indefinable’,35 before wondering ‘whether it can be called a property in the ordinary sense at all’.36 But he always stopped short of saying that there is no such property. This short answer, though correct, immediately invites the further question why Frege resisted at just this point. After all, if we grant that the sense of ‘It is true that sea-water is salty’ is just the same as the sense of ‘Sea-water is salty’, this might tempt us (as Frege himself granted) ‘to think that the word “true” has no sense at all’. Frege’s response, however, was to resist this temptation because in that case a sentence in which ‘true’ occurred as a predicate would have no sense either. All one can say is: the word ‘true’ has a sense that contributes nothing to the sense of the whole sentence in which it occurs as a predicate.37

Rather than treating the absence of the word ‘true’ from the second sentence as showing that it contributes nothing to the sense of the first, therefore, he concluded instead that the sense of the word somehow lurks in the locality of every sentence, implicated in, but not quite contributing to, its sense. The position to which Frege was driven in order to avoid admitting the redundancy of truth is unquestionably puzzling: it is hard to see how a word can have a sense which makes no contribution to the sense of sentences in which it occurs. But if Frege’s stance is so very implausible, there must have been something driving him to adopt it. Perhaps, then, part of his reason for resisting the redundancy of truth was that he saw just where it would lead. 32 36

‘Über Sinn und Bedeutung’, 34. 33 Cf. PW, 129 and 233. ‘Der Gedanke’, 61. 37 PW, 251–2.

34

PW, 174.

35

‘Der Gedanke’, 60.

The fundamental thought  Since inventing the Begriffsschrift in 1879 Frege had been keen to emphasize his logical system’s fruitfulness, its ability to ‘bring forth judgments that at first sight appear to be possible only on the basis of some intuition’.38 In contrast to the old logic, which only takes out of the box what has just been put into it, the polyadic logic of the Begriffsschrift yields a content sufficiently rich that mathematics can be founded on it. If that content rests ultimately on nothing but the concept of truth, then that concept, even though sui generis and indefinable, cannot, Frege thought, be wholly redundant. . The fundamental thought I mentioned earlier that Wittgenstein called his insight that there are no logical constants his Grundgedanke, his ‘fundamental thought’. The description comes from the Notebooks,39 but Wittgenstein was sufficiently impressed by it to think it worth repeating in the Tractatus itself.40 Why did he regard it as fundamental? Some commentators, understanding ‘fundamental thought’ to mean something like ‘central claim’, have found the remark puzzling. And indeed it is not completely straightforward to present the claim that logical constants do not represent as the argumentative lynchpin of the Tractatus. But the rather lowly place (4.0312) Wittgenstein gives the remark in the numbering system of the Tractatus suggests that this is not quite what he meant. If instead we interpret Wittgenstein’s remark rather more autobiographically, as meaning that his insight concerning logical constants was what started him on the journey away from Russell’s conception of logic towards the account given in the Tractatus, it is much less puzzling. Indeed the remainder of this book will provide ample confirmation of the central role Wittgenstein’s fundamental thought played in the process which led him to the Notes on Logic, and thence to the Tractatus. What might be taken at least as a central motivation of the Tractatus is perhaps not so much the claim that logical constants do not represent, but rather the reason he offers for thinking that this should be so, namely that logic must turn out to be a totally different kind than any other science. Russell’s conception put logic alongside the natural sciences and hence rendered its special nature inexplicable. Wittgenstein’s, on the other hand, offered the hope of dissolving the epistemological problem because if logic is contentless there is nothing there to know. The centrality of this thought in the Tractatus is to be explained by the use to which Wittgenstein puts it in subverting the transcendental idealism of Kant’s Critique. By the time of the Tractatus, that is to say, Wittgenstein saw the emptiness of logic as obviating Kant’s need for a transcendental deduction of the 38

Bs, §23.

39

NB, 25 Dec. 1914.

40

4.0312.



The fundamental thought

categories. No deduction is needed because there are no categories to require validation in their application to the world. And without a transcendental deduction there is no pressure towards transcendental idealism either. But there is nothing to suggest that he understood that until 1916. What is striking is how the view that logic is empty is fully present in 1912, although the later application of it is not. And it is plain that this emptiness is something which, for Wittgenstein, flowed from his fundamental thought. But the flow was autobiographical at least as much as it was strictly logical.

Chapter 

The symbolic turn One of the most profound themes in the Tractatus is its exploration of the harmony between the structure of the world and the structure of the symbols we use to talk about it. This theme began to emerge in Wittgenstein’s writing while he was at Cambridge. The ‘symbolic turn’ in his thought is in its way as significant a moment as the fundamental thought we discussed in the last chapter. What led him to it was his emerging concern, already mentioned, with understanding the nature of propositions. . Propositions The account of propositions which Moore proposed in 1898 and Russell advocated in the Principles was that a proposition is a complex which may have among its components parts of the actual world. John, for instance, is on their account a constituent of the proposition that John loves Mary. ‘That’s right, John himself, right there, trapped in a proposition.’1 Not all the components of propositions are parts of the empirical world, of course: loving, for instance, is not. And, as we saw in §2.3, even John was soon ousted from the proposition on account of not being simple enough. But what replaced him, whether sense-data or sensibilia, were in turn conceived of as parts of the world. Central to Wittgenstein’s symbolic turn was his abandonment of this conception. Wittgenstein conceived of propositions instead as symbols having names of objects, not the objects themselves, as constituents. To say that a proposition has symbols as components rather than the parts of the world those symbols refer to is so far to disagree with Russell only on a point of terminology, namely how to use the word ‘proposition’. What is significant is not the change of terminology but the change of focus it betokens. Wittgenstein regarded the study of propositions, understood now in his sense and not Russell’s, as the goal of his work. But what exactly was his sense of ‘proposition’? Saying that a proposition is a symbol is not yet to identify it. What the symbolic turn amounts to is the recommendation that in identifying a proposition we should pay attention to those features of it that are relevant 1

Kaplan, ‘Dthat’, 13.



The symbolic turn

to its ability to say what it does. This, admittedly, is still programmatic— it will be some time before we are in a position to ascertain which are the features Wittgenstein regarded as relevant—but even at this programmatic stage we can recognize here a distinctive turn in Wittgenstein’s thought away from Russell’s. Certainly the decision to focus on symbols entails that we should treat names as types, not tokens. Wittgenstein himself made this point in his characteristically oblique manner. ‘It is to be remembered that names are not things, but classes: “A” is the same letter as “A”. This has the most important consequences for every symbolic language.’2 If names (and other words) are types, not tokens, a sentence will then also be a type. But it will not be what Wittgenstein meant by a proposition. This is because the proposition consists only of those features of the sentence that symbolize, i.e. those that are essential to the sentence’s ability to say what it does. That certainly entails that a proposition is a type, not a token, but it entails quite a lot more, since a sentence of ordinary language may have features that are not relevant to what it expresses, and any such irrelevant features will have to be ignored in identifying the proposition involved. We cannot now reconstruct just when Wittgenstein decided to take this course. The first sign we have of it is in a letter to Russell of January 1913, where he appealed to the idea that a thing just is ‘whatever can be symbolized by a simple proper name’. We shall be discussing various aspects of this remarkable letter in the course of the next three chapters, but the point to focus on now is the casual manner in which Wittgenstein moves here from talk about ‘things’ to talk about ‘whatever can be symbolized by a simple proper name’. Wittgenstein wrote the letter only a few weeks after one of his visits to Frege, and it is natural to wonder, as we explore the contours of the symbolic turn for which it is the earliest written evidence, what role this visit might have played in his decision to take it. . The rejection of psychologism To approach this question, it will be helpful to compare the symbolic turn which I am attributing to Wittgenstein with what has become known as the linguistic turn. The moment when Frege took the linguistic turn—the moment, indeed, when it was born—occurred in the Grundlagen when, having stated the context principle, that ‘only in the context of a proposition do words mean anything’,3 he then used this principle to transform the Kantian question, ‘How are numbers given to us?’ into the linguistic question, ‘How are number words used?’ Frege explicitly advertised one of the purposes of the 2

B27; cf. 3.203.

3

§62.

The rejection of psychologism  context principle as being to resist psychologism. If we ignore this principle, he said, we are ‘almost forced to take as the meanings of words mental pictures or acts of the individual mind’,4 and hence, he thought, to descend into psychologism. The linguistic turn, in Frege’s hands at least, was thus a component part of his war on psychologism. Already this is enough to open up some distance between Frege’s linguistic turn and Wittgenstein’s symbolic one, since for Wittgenstein psychologism was never the target. This is not because he had any more sympathy with it than Frege did, but because he regarded this as a battle already won. He had probably never read Lotze as Frege had, and he therefore had no need to object to that author’s treatment of thoughts as distinctive segments of our stream of ideas. Nor, presumably, had he read the psychologistic work of Erdmann which incited Frege to such lengthy vituperation.5 In 1913 Wittgenstein wrote a scathing review6 of a not very distinguished textbook by a logician called Coffey, and this may indeed be the only book belonging to the earlier tradition in logic that he ever read. (Later he claimed not even to know what a logic textbook contains,7 and one wonders whether he had actually read this one right through: all the examples he mentioned in his book review are from the first fifty pages of Volume I.) One of the most prominent features of Wittgenstein’s symbolic turn is the implicit assumption that we have no direct access to the world and must therefore deduce its features from features of the symbols we use to represent it. This does not, certainly, amount to a denial of metaphysics—logical positivism avant la lettre—but it does amount to a consciousness of its limits. In this respect Wittgenstein’s attitude bears an obvious similarity to Frege’s, but for Frege it was thought that should be regarded as being out of reach of direct philosophical discussion: we analyse the structure of language as a means to analyse the structure of thought. We have to accept that thinking, for us humans at least, involves language essentially. As Frege put it, ‘We think in words. . . Without symbols we would scarcely raise ourselves to conceptual thought.’8 For Wittgenstein, on the other hand, it is the world that is in a certain sense out of reach: we are compelled to approach features of the world by means of features of the symbols we use to represent it. Notice, though, that whether either project, Frege’s or Wittgenstein’s, has any hope of success depends on what degree of fit there is between language and whatever language expresses, whether that is in the first instance taken to be thoughts, as by Frege, or the world, as by Wittgenstein. Notice, too, that even if we assume that there is such a fit, that does not commit us to a view about the direction of fit. For all that has been said so far, it remained open to Wittgenstein to say that the world might owe its shape to language, 4

Gl, Intro. 5 Gg, I, xv–xvii. 6 ‘Review of Coffey, Science of Logic’. 7 Parak, ‘Wittgenstein in Monte Cassino’, 149. 8 ‘Über die wissenschaftliche Berechtigung einer Begriffsschrift’.



The symbolic turn

so that what it is for an object to be simple would be for it to be the reference of a simple symbol; or to say that language owes its shape to the world, since a symbol would not be capable of expressing anything about the world if it did not conform to the requirements which the world imposes. The first of these two views we might label a sort of idealism, the second a sort of realism. Nothing Wittgenstein said in the Notes committed him to either view, or even to seeing them as alternatives to be arbitrated upon. One of the most intriguing features of the Tractatus is that it resists any attempt to impose one direction rather than the other (probably because by then Wittgenstein had come to regard the very notion of a direction of fit between language and world as misconceived). . The reliability of language If the linguistic turn is to have any prospect of success, we have to assume that there is some correspondence, even if not a perfect one, between the structure of language and the structure of what it represents. This assumption is an important strand in Frege’s thought. He observed in a letter to Russell, for example, that ‘to the decomposition of the sentence there corresponds a decomposition of the thought’.9 Presumably, though, Frege should not be taken as claiming that the sentences of ordinary language are already structured so as to reveal with perfect clarity the thoughts they express. His purpose must have been only to point to a certain sort of linguistic analysis, the decomposition of a sentence into logically relevant components, as capable of revealing the structure of the underlying thought. Even so, it remains that if we have to rely on language as a guide to the structure of what language represents, we are in an obvious difficulty because of the unreliability of language in this regard. This, too, is a recurring theme in Frege, whose desire to expel psychologism from logic encouraged in him a suspicion of ordinary language. It cannot be the task of logic to investigate language and determine what is contained in a linguistic expression. Someone who wants to learn logic from language is like an adult who wants to learn how to think from a child. When men created language, they were at the stage of childish pictorial thinking. Languages are not made so as to match logic’s ruler.10

This, incidentally, is a strand of thought that can also be found in Russell: he, too, protested against the confusion that results from ‘the notion that words occur in propositions, which in turn is due to the notion that propositions are essentially mental and are to be identified with cognitions’.11 But now there is an evident tension in the position. To avoid psychologism we must recognize both that all our thoughts are expressed in words and 9

28 July 1902.

10

To Husserl, 30 Oct. to 1 Nov. 1906.

11

Principles, §51.

The reliability of language  that ordinary language constantly misleads us as to the true structure of those thoughts. In principle, one way to resolve this tension would presumably be to adopt a language in which sentences do correspond to thoughts, but those which express the same thought, and which are therefore for logical purposes equipollent, need not be distinguished. Only now that logical analysis proper has become possible can the logical elements be recognized, and we can see the clearing in the forest. All that would be needed would be a single standard sentence for each system of equipollent sentences, and any thought could be communicated by such a standard sentence.12

Presumably Frege was here alluding to what he would later call a logically perfect language.13 Perhaps he thought at first that his Begriffsschrift came close to being such a language. He recognized, of course, that his notation was incomplete: he noted, for example, that it would have to be extended if we wanted to use it to talk about geometry. And he granted, too, that it might ‘fail to reproduce ideas in a pure form’;14 but he does not seem to have meant by this that his Begriffsschrift might inevitably distort the thoughts it expressed. Indeed, he claimed that ‘we can restrict the discrepancies to those that are unavoidable and harmless’.15 But, after Russell’s discovery of the contradictions and his own failure to repair the damage they caused to his formal system, Frege seems increasingly to have thought of logical perfection as an unachievable aspiration rather than an ambitious but in principle attainable goal. If our language were logically more perfect, we would perhaps have no further need of logic, or we might read it off from language. But we are far from being in such a position. Work in logic just is, to a large extent, a struggle with the logical defects of language, and yet language remains for us an indispensable tool.16

Frege seems here to be on the verge of saying that language might be inevitably distorting, and the goal of a logically perfect language in principle unattainable. That is not to say, though, that Frege ever gave up on the idea that the route to thought is via language. As Dummett succinctly puts it, ‘Language may be a distorting mirror: but it is the only mirror we have.’17 It is worth noting how close what Frege says here is in spirit to one of the central themes of the Tractatus. Wittgenstein displayed much the same hesitancy as Frege concerning the significance of the features of sentences in ordinary language. Sometimes he claimed that the presence of some feature in an ordinary language sentence is an indication that the feature must be playing some symbolizing role. He thought, for instance, that in ‘not- p’ the occurrence of p is necessary. Ordinary language would not contain the whole propositions if it did not need them: However, e.g., ‘not- p’ may be explained, there must always be a meaning given to the question ‘what is denied?’18 12 To Husserl, 30 Oct. to 1 Nov. 1906. 6. 18 B21.

13

PW, 256.

14

Bs, Preface.

15

Ibid.

16

PW, 252.

17

OAP,



The symbolic turn

Elsewhere, though, Wittgenstein insisted that ‘distrust of grammar’—by which he presumably meant the grammar of ordinary language—‘is the first requisite for philosophizing’.19 And he specifically warned against thinking that we can read off the structure of the proposition from the structure of the ordinary language sentence. ‘Ordinary language conceals the structure of the proposition: in it, relations look like predicates, predicates like names, etc.’20 In ordinary language, that is to say, sentences sometimes contain features that do not play a symbolizing role: ‘Language disguises the thought.’21 It is the task of the philosopher to identify and eliminate (or at any rate see beyond) these features. So Wittgenstein’s criterion, according to which a thing is whatever can be represented by a simple symbol, is not to be applied straightforwardly to ordinary language. Or perhaps we would be better to say that it cannot be applied to ordinary language, because ordinary language does not contain any simple symbols. For this reason it is a constant theme of Wittgenstein’s work from now on to seek the level at which the structure of the proposition—the structure which does correspond precisely to the simplicity of the parts of the world being represented—is revealed. At once this makes it clear how distant Wittgenstein’s notion of a symbol is from that of a sign as used in linguistics: anyone who imagined that the structure of the world might be revealed simply by the linguistic analysis of sentences, in any ordinary sense of that word, would plainly be beyond the reach of reason. Wittgenstein’s propositions are not, as Moore’s were, indistinguishable from the worldly facts that make them true; but nor are they sentences in the sense familiar from linguistics; they occupy a third level between these two, where the structure of the world is revealed but how things stand in it is not. . Conflicting conceptions Whatever the similarities, then, the symbolic turn in Wittgenstein’s thought which we have been discussing is not Frege’s. Frege had a clear conception of complexity in language, and of the idea that it is logic’s role to explain and reflect that complexity, but he did not have a corresponding conception that simplicity in language might be a mirror or proxy for simplicity in the world. The reason is that his conception of the relationship between language and the world did not allow for it. For him, every expression in language relates to the world only mediately via its sense, and his notion of sense, if it was to do the work he wanted it to do, could not leave room for the sort of simplicity Wittgenstein appealed to. Consider, for instance, an object picked out by what would have counted for Wittgenstein as a simple proper name (a demonstrative perhaps). Even 19

B63.

20

B69.

21

4.002.

Conflicting conceptions  if Frege counted this name as simple, it was central to his conception of an object to allow that there could be other names referring to the same object, and these other names need not be simple. Accordingly, whatever complexity parts of the world possess cannot on Frege’s conception of sense be straightforwardly read off from the complexity of the terms we use to refer to them. There could for him be no direct connection between language and the world, and therefore no direct inference in either direction between the simplicity of an expression in language and the simplicity of the object it refers to. Perhaps, then, Wittgenstein’s conception is best thought of as a synthesis of two influences, Frege’s and Russell’s. His conception of what it is for objects to be simple is one which, as we saw in chapter 4, he obtained from Russell: they are what resist elimination by Russell’s method of definite descriptions (and, more generally, the method of incomplete symbols of which it is an instance). What is not present in Russell, but pervades the writings of Frege, is a recognition of the importance of studying the distinctive role of language in representing the world. (Russell, on his own admission,22 did not become sensitized to the value of studying language, rather than using it as a medium for the study of thought, until 1918.) It is only by putting these two strands together that we obtain the idea, embedded in Wittgenstein’s letter to Russell of January 1913 and from then on a fixed point in his thinking, that a thing is ‘whatever can be symbolized by a simple proper name’. In chapter 3 we noted the distance that Wittgenstein seems to have maintained from Russell’s oscillations concerning the nature of sense-data. Russell treated the relation of acquaintance as fundamental, and therefore hoped to identify the simple objects of experience by means of a sort of informed introspection. For Wittgenstein, on the other hand, the simple objects are whatever is represented by the simple symbols. This conception need not in itself contradict Russell’s, since for all that has been said so far a ‘symbol’ might be, or be correlated with, a mental event such as the sensation of a sense-datum. That partly explains, perhaps, why there is no suggestion that Wittgenstein actively opposed Russell’s view of sense-data while he was at Cambridge. But once Wittgenstein began to fill in more of his conception of what a symbol is, his conception of the simples inevitably drifted away from Russell’s. All the indications are that this happened while Wittgenstein was in Norway during the Autumn of 1913. That was when he settled on the Tractarian conception of propositions as expressing possibilities, and of objects as the constant elements on which these possibilities hinge. Hence his comment in a letter to Russell that ‘the individual primitive signs . . . are not at all the ones you thought’.23 22

MPD, 98.

23

[Nov. or Dec. 1913] (CL, no. 32).

Chapter 

Simplicity Wittgenstein took the symbolic turn, I have suggested, when he used it as a criterion for the simplicity of an object in the world that the object should be symbolized by a simple name. We now need to explore what the idea of simplicity that is at work here amounts to. We shall do this by comparing Wittgenstein’s account with Frege’s. Both authors offered explanations of how names contribute to the meaning of sentences in which they occur, but where Frege distinguished three distinct items—the referent of the name, its sense, and the idea I associate with it—Wittgenstein had only one, the object which is the name’s referent. One way of gaining some understanding of Wittgenstein’s conception, then, is to ask what Wittgenstein’s reasons were for collapsing the distinctions Frege had drawn.

. Realism I said in §1.3 that Russell had a semantic theory for names according to which they refer directly, whereas Frege had a theory according to which names (and singular terms generally) refer only mediately via their senses: the primary purpose of the notion of sense was to allow for different names of the same object to make different contributions to the thoughts expressed by sentences in which they occur. A quick way of putting Wittgenstein’s understanding of the semantic role of names would (for the time being, at least) be to say that in this matter he sided with Russell’s theory against Frege’s. When he came to this view is uncertain: it is implicit in Wittgenstein’s letter of January 1913, and the way in which the letter alludes to this conception suggests that it was by then already an accepted part of his thinking. But there is no hint either there or in the Notes of his reason for adopting it. As we noted in §2.2, one of Jourdain’s questions to Frege in January 1914 was whether he accepted Russell’s demonstration that the distinction between sense and reference is unnecessary, but if it is Wittgenstein’s view he is there reporting, Jourdain leaves us tantalizingly short of a hint as to his reason for holding it. Another natural place to look for enlightenment on this dispute would be

Realism  the writings of Russell and Frege themselves, but what makes their disagreement disappointing as a moment in the history of philosophy is that neither of them ever really engaged with the other’s arguments. For instance, I know of no evidence that Frege ever actually read ‘On denoting’, and his response to Jourdain’s enquiry does not take up the implicit invitation to address Russell’s views directly. Frege’s famous argument for the notion of sense on the basis of the non-trivial informational content of the identity ‘Hesperus = Phosphorus’ is hardly persuasive on its own, since it is so plausible to think of ‘Hesperus’ and ‘Phosphorus’ as disguised definite descriptions, for which Russell’s analysis provides a rather convincing explanation of the informativeness of identity statements. If we accept that analysis, an application of Occam’s razor then allows us to remove the senses of the terms ‘Hesperus’ and ‘Phosphorus’ from our ontology. The underlying point which Frege’s example is intended to highlight, of course, is that the names ‘Hesperus’ and ‘Phosphorus’ have a perspectival character: they both refer to the same object, namely the planet Venus, but they do so in different ways, under different modes of presentation. In order for Frege’s argument to be persuasive against Russell, he would have to offer a reason to think not merely that some singular terms have this perspectival component but that all do—that none simply refers to an object raw and unmediated. The Gray’s Elegy passage, discussed in §2.2, offers what Russell claimed was an argument against Frege’s notion of sense. However, he originally formulated that argument as an attack not on Frege but on his own earlier theory of denoting. If a denoting concept and the object it denotes are at the same level, as Russell had believed, it is indeed reasonable to ask about the propositions expressing the relationships between them. But against Frege the argument shows at most that there is a regress: a further step would be required to show that the regress is vicious. So if we want to find Russell’s real objection to Frege’s two-step semantic theory, we must look elsewhere. Russell did not examine Frege’s writings in any detail until 1902, by which time the conception of propositions as containing parts of the world, which he adopted during conversations with Moore in 1898, was probably too embedded in his thinking for him to question it. But what had led them to adopt this conception was a desire to embrace realism, a desire which they had expressed by way of an identity theory of truth: what it is for a belief of mine to be true is for it to be the case. If we add to this the view that what is the case, a fact, can be a fact about the world only if it contains parts of the world, we arrive at the conclusion that the object of my belief must contain parts of the world too. Frege, who also adhered to just such an identity theory, had therefore, in consistency, to deny this last premise



Simplicity

and conceive of facts as belonging to the realm of sense, not to the world itself. ‘A fact,’ on this account, ‘is a thought that is true.’1 For Russell, on the other hand, facts are as worldly as it gets, so it is easy to see how it would have seemed to him as if Frege’s account fell fatally short of the world. I believe that in spite of all its snowfields Mont Blanc itself is a component part of what is actually asserted in the proposition ‘Mont Blanc is more than 4000 metres high’. We do not assert the thought, for this is a private psychological matter: we assert the object of the thought, and this is, to my mind, a certain complex (an objective proposition, one might say) in which Mont Blanc is itself a component part. If we do not admit this, then we get the conclusion that we know nothing at all about Mont Blanc.2

Quite soon Russell adopted views which were incompatible with the identity theory, at least in this naive form, and eventually he himself realized this and abandoned it. (So, incidentally, did Moore.)3 But Russell’s response was to change the complexes to be analysed from propositions to judgments, not to remove the objects from the analysis. So his old rejection of idealism survived, recast only slightly to say that objects with which I am acquainted occur as components in my judgments, rather than, as before, in the propositions that I judge. Frege’s notion of sense continued, that is to say, to seem to him to erect a needless veil between us and the world. But to anyone outside the argumentative loops of Russell’s theorizing, this way of expressing a commitment to realism is apt to seem like no more than an unmotivated metaphysical dogma. What we need is an independent argument for the idea that realism requires a one-step semantic theory. In the Tractatus the so-called ‘argument for substance’4 is supposed to fulfil this role. And if Frege’s notion of sense is seen as mediating between language and the world in such a way as to contradict this requirement, then we reach Wittgenstein’s desired conclusion that names do not have sense but refer directly to their objects. However, the argument for substance, at least in the form in which it occurs in the Tractatus, probably dates from some time after Wittgenstein had left Cambridge: the most natural way of explaining it makes it depend on assumptions which, though part of the Tractarian system, Wittgenstein probably did not work out until he was in Norway. So if we are to discern in the argument for substance any clue to what originally led Wittgenstein to adopt Russell’s one-step semantic theory, we must try to extract its essence by disengaging it from its Tractarian framework. What underlies the argument, surely, is the naive thought that sometimes language must just latch directly onto the world since, if it did not, the process of representation by which we succeed in expressing ourselves would never be able to get started: each attempt at saying something would depend for its meaningfulness on the meaningfulness of something else. In order to have some 1

‘Der Gedanke’, 74.

2

To Frege, 12 Dec. 1904.

3

SMPP, 308.

4

2.021–2.0212.

Solipsism  intuitive plausibility, this simple thought need not be turned into a specific proposal involving Russell’s theory of descriptions and his distinction between internal and external negation. Instead, we can present it at a much simpler level as an expression of the realist’s instinct that language’s success in referring to things in the world is ultimately due to the fact that we knock against them from time to time. If we think of the argument in this light, its key premise is the claim, already stressed in the Notes, that ‘propositions have a sense which is independent of their truth or falsity’,5 and therefore ‘we must be able to understand a proposition without knowing whether it is true or false’.6 If we press that view back to its base, we obtain the conclusion that there must be propositions whose sense we can understand even if we know nothing at all about how things stand in the world. In other words, the realist’s instinct (familiar enough from the more recent philosophical literature in favour of semantic externalism) is that sometimes in language we do not say, but simply presuppose, that there is a world for our words to refer to.

. Solipsism That, or something very like it, is the realist’s argument for identifying the reference of a name with its sense. In arguing for his notion of sense Frege generally preferred to emphasize the distinction between the sense of a name and the idea associated with it in my mind. The reason he repeatedly offered for this distinction was that it was required in order to explain communication. When I tell you something you did not know, there is a single thing we both grasp, namely a certain thought. That thought, and by extension the senses which are its component parts, cannot be ideas, since these, being mental entities, are intrinsically private to the mind. Neither of us has the thought, but both of us grasp it. For Russell, on the other hand, it was not really part of the task he was engaged in to explain communication. On his view, indeed, the fact that we communicate at all emerges as a kind of miracle. For the sense-data experienced by me are not the same as those experienced by anyone else. Even if you are in the room with me, the angle at which you look at the table, and hence the exact sense-data you obtain from it, will be different. As a consequence, the logically proper names in your idiolect never mean quite the same as those in mine.7 The only entities that are common to the propositions that you express and the ones I express are universals. Now the propositions of mathematics and logic, Russell thought, have no components that are not universals; so they can be grasped by more than one person. Ordinary propositions about the world, however, always contain, when completely 5

B9.

6

C5.

7

Russell, CP, VIII, 174.



Simplicity

analysed, simple objects of acquaintance, whether sense-data or ideas. Since, as a matter of fact if not of necessity, no two people have ever been acquainted with exactly the same object, it follows that no such proposition has ever been grasped by more than one person. This does not absolutely rule out the possibility that we genuinely communicate with each other; but if we somehow succeed in doing so, the proposition is not in any straightforward sense the vehicle of communication. It is not, as it is in Frege’s account, an entity which embodies what it is that is communicated. A fundamental difference between Frege’s theory of reference and Russell’s, therefore, is that Frege aimed to explain how communication is possible, whereas Russell treated communication only as an observed fact from which it might be possible to draw conclusions (for instance, about the existence of other minds). Wittgenstein, on the other hand, was quite happy to circumvent the issue entirely by tolerating solipsism. ‘He admits,’ Russell reported, ‘that if there is no matter then no one exists but himself.’8 And if we embrace solipsism, the problem which led Frege to distinguish ideas from senses, and Russell to make communication a kind of miracle, simply dissolves. Wittgenstein’s response to this disagreement between Russell and Frege is characteristic in several respects: it is characteristic that he should have been unwaveringly determined to accept the consequences of his views, however unpalatable or frankly implausible they might have been; characteristic that he did not feel the draw of Russell’s robust (if somewhat selective) common sense; characteristic, too, that he should adopt a view one of whose effects was not so much to argue against an opposing view but rather to undermine it by reconfiguring the project to which it was intended to contribute.

. Idealism In the last two sections I have traced out the connections between realism and the identification of sense and reference, and between solipsism and the identification of idea with sense. I want now to examine the consequences of identifying idea with reference. In particular, I want to mention a particular feature that ideas are normally taken to have, namely that they do not have unexpected aspects. If an idea is presented to me in a certain way, it is natural to think of that as the only way it could be presented to me: if it were presented in a different way, it would just be a different idea. Indeed talking of how an idea is presented to me is already misleading. The transparency of ideas to their bearers consists in this, that ideas cannot be distinguished from how they are presented: an idea just is its presentation. 8

To OM, 23 Apr. 1912.

Idealism  Now Russell certainly did not wish to identify a sense-datum with a mental entity such as my idea of it. Yet various readers of his work thought that he did, and as a result he was required to insist indignantly that he had been misunderstood.9 The issue on which Russell was so often misunderstood no doubt has a connection with the discussion of solipsism in the last section: part of the reason for the misunderstanding was that no two people have ever experienced precisely the same sense-datum. In order to counter the inference that sense-data are private, and hence mental, it was necessary for Russell to claim that this is only an empirical fact about sense-data and not part of their nature. But another component in the explanation for the misunderstanding is that on Russell’s conception sense-data do have the feature I have just alluded to, namely that there is no gap between how they seem and how they are. If they are objects, we might say, they are so only in a thin sense which renders them transparent to us. We may contrast them in this respect with the ordinary objects of common-sense experience, which are opaque: they may have various aspects to them, sometimes unknown and unexpected ones. It is helpful here to consider Frege’s response when Jourdain asked him whether there was really any need for a distinction between sense and reference. Frege asked Jourdain (or would have done if he had sent the letter he drafted) to imagine two explorers who come upon the same mountain, but see it from different perspectives in different valleys. If one gives this mountain the name ‘Afla’, the other the name ‘Ateb’, it is then a genuine discovery, which we require the notion of sense to explain, that these are names of the same thing. In the form in which Frege tells the story,10 it does no more than to make once more the point we discussed in the last section, that we need the notion of sense if we are to explain communication. But it is easy to see that just the same issue arises if we alter the story so that there is only one explorer, not two: the explorer travels to both valleys in turn and sees the same mountain from different sides without realizing it. In its new form, the point the story makes cannot any longer be one about communication, but must rather be about our conception of objects. When the explorer learns that Ateb is the same mountain as Afla, this no doubt counts as a discovery, but not one that overturns anything he formerly believed or that requires him to revise his language. It causes him, we might say, to add some information to his map of the world, but not to delete anything. What this transposition of Frege’s example demonstrates is that although Frege repeatedly offered the need to explain communication as a motivation for the notion of sense, it is not the only one. His notion of sense also seeks to capture the realist’s belief that objects are, in the sense just mentioned, opaque and not transparent: the way in which they are presented to us need not 9

CP, VIII, 88.

10

PMC, 80.



Simplicity

exhaust what there is to them. Now on Russell’s account the opaque objects are to be constructed logically out of the transparent ones. One way of putting the realist’s point would be to doubt whether it is even possible to perform such a construction. How can logic on its own, using transparent objects as its only building blocks, deliver opaque ones? Where is the opacity supposed to come from? Another way of putting something like the same point would be to note that Russell’s elaborate 1914 construction of matter from sensibilia still seems hopelessly far away from an explanation of how matter has mass, for example, when sensibilia (one assumes) do not. Of course, to press this concern is so far only to insist that objects may have unknown aspects, not to allow that they have unknowable ones. Nonetheless, the two thoughts tend in the same direction. They are both ways (the second of course much stronger than the first) of responding to realism’s sense of the world as something to be explored, not constructed. Anyone who supposes that the surface of the world is the whole of it runs, the realist thinks, a constant risk of unpleasant surprises. The difficulty this realist conception of the world poses for atomism, whether Russell’s or Wittgenstein’s, is that their method of analysis equates transparency with logical simplicity and is thus at a loss to explain how it can be that our current conception of the world already makes room for it to have a complexity to which we are not yet privy.

. Reconciliation My aim in this chapter has been to draw out some connections between Wittgenstein’s understandings of realism, solipsism, and idealism. In explaining the contribution a name makes to determining the sense of any sentence in which it occurs, Frege distinguished between three elements, the idea I associate with the name, the sense of the name, and its reference. Wittgenstein’s conception of objects certainly does not coincide neatly with any of the three, but what I have been suggesting here is that it shares something with each. Depending on which we emphasize, his view will seem to be characterizable as idealism, solipsism, or realism. Of course, something like this thought is present in the Tractatus,11 but the route by which Wittgenstein reached it there is one he did not travel until 1916. First he connected solipsism with realism. Here we can see that solipsism coincides with pure realism, if it is strictly thought out. The I of solipsism shrinks to an extensionless point and what remains is the reality coordinate with it.12

Then he connected idealism with solipsism as well. 11

Cf. 5.64.

12

NB, 2 Sep. 1916.

Reconciliation  This is the way I have travelled. Idealism singles men out from the world as unique, solipsism singles me alone out, and at last I see that I too belong with the rest of the world, and so on the one side nothing is left over, and on the other side, as unique, the world. In this way idealism leads to realism if it is strictly thought out.13

Wittgenstein’s route to the identification is therefore intimately connected with his later understanding of the metaphysical subject, an understanding of which not a glimmer is to be seen in the Notes. That would have to wait not only for the removal of the empirical subject from the analysis of judgment, which he arrived at in Norway, but, more importantly, for the emergence of transcendental reflections as a theme in his philosophical thinking during the summer of 1916. It is striking, though, how short a distance there is still to be travelled. If Wittgenstein was to reconcile the Russellian conception of simplicity with a Fregean conception of thought, he had little choice but to collapse the traditional distinction between realism, idealism, and solipsism; and once he had done that, he could do no other than to reduce the self to an extensionless point, leaving only the reality coordinate with it. 13

NB, 15 Oct. 1916.

Chapter 

Unity Merely to take the symbolic turn—to conceive of a proposition as symbolizing what it expresses, rather than being identical with it—is not yet to go very far towards uncovering the structure of propositions. And in his letters to Russell during 1912 Wittgenstein was still operating with a Russellian conception of that structure. A proposition, that is to say, he still thought of as a sort of complex. Two questions preoccupied him. What are the components of the complex? And what is the manner of combination by which these components unite to form a proposition?

. The copula It is a familiar requirement of English grammar that every declarative sentence should contain a verb. We have already noted Russell’s fitful tendency to regard features of the sentences of our language merely as proxies for the features of the propositions they express. In accordance with this tendency, therefore, he made the obvious move of projecting this linguistic requirement onto the complex which the declarative sentence expresses and insisting that that complex must contain a corresponding element: this essential ingredient of a proposition he also, for want of a better word, called a ‘verb’. But Russell was also impressed by the devices which languages such as English have for converting verbs or verbal phrases into verbal nouns. We can derive hitting from hits, dying from dies, mortality from is mortal. Russell therefore held that verbs have a ‘curious twofold use’: they can occur in propositions as verbs or as verbal nouns, but although the mode of occurrence is different, he claimed that the entity that occurs is the same in the two cases. His argument to this effect1 is based on the assumption that if the verb really were different from the verbal noun, we could make a statement to that effect: we could say, for instance, ‘the verb is mortal differs from the noun mortality’. But in that statement is mortal occurs as a noun, not as a verb. So the supposition is self-refuting. 1

Principles, §49.

The copula  This is not a very good argument. Just the same problem occurs if we try to state the contrary: in the statement ‘the verb is mortal is the same entity as the noun mortality’ the verb is mortal occurs as a noun, not a verb. But let us leave that to one side and for the moment accept Russell’s conclusion— accept, that is to say, that the proposition ‘Socrates is mortal’, in which is mortal occurs as a verb, is equivalent to the proposition ‘Mortality belongs to Socrates’, in which it occurs as a verbal noun. Since the mode of occurrence of mortal is different in the two cases, Russell was compelled to admit that they are different propositions. Nonetheless, they are logically equivalent, and it is therefore hard to see why we should not always substitute the latter for the former. In that case, though, it is a natural further thought that if a verb can sometimes be replaced by a verbal noun, then it always can. There will still (presumably) need to be a verb to bind the disparate elements of the complex into a proposition, but it will now be only a bare verb without any specific meaning, i.e. what grammarians traditionally called a copula. This, then, is the sort of analysis that Wittgenstein was working with during 1912. The number of elements in the analysis will vary according to the type of proposition that is in question, and so there will have to be different copulae for the different types. An atomic subject-predicate proposition φ a is analysed as #1 (a,φ ): the subject a and the predicate φ are here bound together by the copula #1 . The proposition ‘Socrates is mortal’, for example, contains ‘Socrates’ and ‘mortality’, bound together by #1 . This copula is thus the form which all subject-predicate propositions have in common. Similarly, an atomic relational proposition aRb is analysed as #2 (a,R,b): here a, b, and R are bound together by a different copula #2 . Wittgenstein also countenanced as a degenerate case the one-place copula #0 , enabling him to analyse ‘a exists’ as #0 (a). We shall look later at how Wittgenstein proposed to analyse these copulae. In the meantime let us focus on the idea that we can convert the apparent verb of the proposition into a noun and hence regard mortality as an object. This is evidently problematic. If we can turn verbs into nouns at will, for instance, one might wonder what objection there could be to also turning the copula is into the verbal noun being and hence dissolving the proposition entirely. One might also wonder what conception Wittgenstein had of the pure copulae #n that he thought was consistent with his fundamental thought of June 1912 that there are no logical constants. Given these difficulties, it is perhaps no great surprise that Wittgenstein quite soon abandoned the idea of explaining propositional unity by means of the copula. What is suggestive, however, is the timing of the abandonment, which occurred just after Wittgenstein had visited Frege during the Christmas vacation at the end of 1912. In chapter 6 I mentioned a letter Wittgenstein wrote to Russell soon after this visit. It provides, I said, the first clear ev-



Unity

idence of the symbolic turn in Wittgenstein’s thought. In the same letter, however, we also find Wittgenstein reporting that he had abandoned the idea that what supplies unity to a proposition is a copula distinct from its meaningcontributing components. I have changed my view on ‘atomic’ complexes: I now think that qualities, relations (like love) etc. are all copulae! That means I for instance analyse a subject-predicate proposition, say, ‘Socrates is human’ into ‘Socrates’ and ‘something is human’, (which I think is not complex). The reason for this is a very fundamental one: I think that there cannot be different Types of things! In other words whatever can be symbolized by a simple proper name must belong to one type. And further: every theory of types must be rendered superfluous by a proper theory of symbolism. For instance if I analyse the proposition Socrates is mortal into Socrates, mortality and ( x,y)#1 ( x,y) I want a theory of types to tell me that ‘mortality is Socrates’ is nonsensical, because if I treat ‘mortality’ as a proper name (as I did) there is nothing to prevent me to make the substitution the wrong way round. But if I analyse (as I do now) into Socrates and ( x)φ x it becomes impossible to substitute the wrong way round because the two symbols are now of a different kind themselves.2 E

E

The essence of Wittgenstein’s new proposal, then, was to be that ‘Socrates is mortal’ is analysed not into two names, ‘Socrates’ and ‘mortality’ connected by a copula, but into one name ‘Socrates’ and a second component of a kind which he soon began to call a form; there was no need to appeal to a separate copula #1 to explain the unity of the proposition because that function was taken over by the form.

. There cannot be different types of things Wittgenstein gives two reasons for rejecting the copula. The first is that there cannot be different types of things. It is obvious straightaway, of course, that if this is right, it will put paid to any theory that treats Socrates and mortality as things, for it then entails that they are of the same type, and hence that ‘mortality is Socrates’ is just as grammatical as ‘Socrates is mortal’. It is clear enough, too, how a fundamental grammatical distinction such as the one Wittgenstein offers might suffice to explain the ungrammaticality of ‘mortality is Socrates’. But that does not prove Wittgenstein’s claim, since it does not demonstrate that there are not other cases of ungrammaticality for which a distinction of type among things is necessary. So why did Wittgenstein think that there cannot be different types of things? In the letter he gives no reason for the claim (which in any case he later abandoned). However, it is certainly a view that is to be found in Frege. When he was considering how to deal with Russell’s paradox—one of the most obvious 2

To BR, 16 Jan. 1913.

There cannot be different types of things  reasons that might have pressed him to recognize different types of thing— Frege briefly entertained the proposal that classes might be of a different kind from other objects. (He suggested calling them ‘improper’ objects.) But he immediately insisted that there would have nonetheless to be some functions which could have both kinds of objects as arguments. At least the relation of equality (identity) would be a function of this sort. (An attempt might be made to escape this by assuming a special sort of equality for improper objects. But that is certainly ruled out. Identity is a relation given to us in such a specific form that it is inconceivable that various kinds of it should occur.)3

But Frege seems here to be doing little more than restating in different language his claim that all objects are of the same kind. If there are indeed different kinds of objects, then there will be different kinds of identity relation that can hold between them: Frege regards this as ‘inconceivable’, but he does not explain why. We may suspect, then, that his argument from the universality of the relation of identity cannot really be what motivated his view. After all, he was quite content to accept a distinction of type between objects, firstlevel concepts, second-level concepts, and so on. Moreover, concepts are for Frege objective entities, and there must therefore be a relation between them analogous to that of identity between objects. This seems to be only terminologically different from saying that there are different kinds of identity, one kind that may hold between objects, another between first-level concepts, and so on. His real reason for thinking that there cannot be different kinds of object seems rather to be that any difference of type between entities must stem from a fundamental difference of grammar. Russell’s paradox could on its own at most indicate the existence of such a difference; it could not constitute it. This is thus an instance in which Frege’s deep instincts, if not his logical practices, were in harmony with what I have called Wittgenstein’s symbolic turn. Frege’s view that there is only one kind of object is of course equivalent to the claim that the universal quantifier is genuinely universal. It was his deep-seated adherence to this doctrine that was responsible for his inability to deal with the effect of Russell’s paradox. It is interesting to compare Frege’s response to this difficulty with Russell’s. Frege’s argument that there is only one relation of identity would hardly have swayed Russell, who would simply have accepted that functions such as identity are typically ambiguous: we talk as if there is one relation of identity that applies to objects of all types, but strictly there is a whole family of such relations, one for each type. (For Russell identity is a defined relation, but that does not affect the point at issue here.) The danger inherent in the device of typical ambiguity, though, is that we do not take it seriously. Taking the type distinctions seriously would involve using 3

Gg, II, 254.



Unity

for each type i a distinct equality sign ‘=i ’. Moreover, it would be important, if we did this, to recognize that here ‘=i ’ is a complete symbol, not a function of the subscript i. But Whitehead and Russell were not so careful. Throughout Principia they did not use subscripts to indicate differences of type. So the same sign ‘=’ is used for equality in every type. But, as Wittgenstein would shortly point out, it can never express the common characteristic of two objects that we designate them by the same name but by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where then would be the common element in the designations? Nevertheless, one is always tempted, in a difficulty, to take refuge in different ways of designation.4

Although he here makes the point about names, it evidently applies also to expressions for relations. We must not pretend, by using the same sign in different cases, that we have expressed any similarity in what the sign expresses in each case. . The theory of types is superfluous Let us turn now to Wittgenstein’s second reason for rejecting the idea that propositions consist of names related by a copula, namely that the theory of types is superfluous. When Wittgenstein introduces this second reason with the word ‘further’, what he means is presumably that this is to be understood as an additional reason for the same conclusion, since if the first reason is correct, so that all names are of the same type, no theory of symbolism will be able to prevent us from swapping ‘Socrates’ and ‘mortality’ in the old analysis so as to obtain the nonsensical proposition that mortality is Socrates. As with his first reason, though, Wittgenstein does not in the letter to Russell supply an argument for his claim. What he does is only to offer an example of how, by distinguishing between names and forms, we can explain without any appeal to the theory of types why ‘Socrates is mortal’ makes sense but ‘mortality is Socrates’ does not. Wittgenstein’s account here is evidently little more than a rewording of what he would have gleaned from Frege. Frege’s recommendation was that we should analyse ‘Socrates is mortal’ not into ‘Socrates’ and ‘mortality’ but into ‘Socrates’ and ‘ x is mortal’. The proposition is what results if we substitute the first of these expressions into the argument place in the second. If, on the other hand, we try to substitute the second expression (‘ x is mortal’) into the first (‘Socrates’), what we get is not so much nonsense as perplexity: since ‘Socrates’ has no argument place, there is simply no sort of substituting that can be done in it. Of course, the more examples we supply in which our 4

B3, cf. 3.322.

The theory of types is superfluous  ‘proper theory of symbolism’ explains away apparently problematic combinations of signs, the more superfluous the theory of types will seem, but such individual cases fall some way short of explaining why the theory of types must be superfluous. The notion that a correct theory of symbolism will render the theory of types superfluous is in fact one that recurs throughout Wittgenstein’s thought from this point on. On the face of it, though, one might wonder what the contrast is that Wittgenstein is drawing. After all, one might think that a theory of types just is a proper theory of symbolism, in which case it would be mystifying how the latter could render the former superfluous. Such, evidently, was Russell’s view when he wrote to Wittgenstein just after he read the Tractatus for the first time. The theory of types, in my view, is a theory of correct symbolism: a simple symbol must not be used to express anything complex: more generally, a symbol must have the same structure as its meaning.5

Even after all this time, though, Russell had still not quite got things right. That’s exactly what one can’t say. You cannot prescribe to a symbol what it may be used to express. All that a symbol can express, it may express. This is a short answer but it is true!6

It is hard to see what Wittgenstein could have found to object to in Russell’s apparently innocent remark. One can only presume that in Russell’s (surely innocuous) use of the word ‘must’ he discerned an expression of the view he was concerned vehemently to oppose, namely that the theory of types is something that can be superimposed on language after it is operational. The case of ordinary language exemplifies this point quite well. The words we use can be categorized into grammatical types such as noun, verb, adjective, adverb, etc. But if I understand a word, then it already belongs to a grammatical category; and if I define a new word, the definition must be such as to determine its category. I cannot later decree the word’s grammatical type when I already know what it means. It is incoherent to conceive of grammar as something that can be imposed on words when they already have a sense, since grammar is no more than a systematization of how words contribute towards the senses of the sentences that they form. There has been a tendency for commentators to misunderstand this, especially in the formulation Wittgenstein dictated to Moore. ‘A THEORY of types is impossible. It tries to say something about the types, when you can only talk about the symbols.’7 Wittgenstein was not denying that propositional functions fall into a hierarchy of levels—something he evidently maintained in the Tractatus and would in any case have found it hard coherently to deny. Nor 5

13 Aug. 1919.

6

LW to BR, 19 Aug. 1919.

7

NdM, ¶18.



Unity

was he denying—as would, once again, be hard—that it is possible to draw up, post facto, rules describing which combinations of signs make sense and which do not. Such a set of rules could no doubt be called a theory of types. What is a mistake, according to Wittgenstein, is to regard such a theory as determining, rather than merely describing, which combinations make sense. The names ‘Socrates’ and ‘Plato’ are similar: they are both names. But whatever they have in common must not be introduced before ‘Socrates’ and ‘Plato’ are introduced. The same applies to a subject-predicate form, etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e., types are not indefinables.8

Wittgenstein’s point is that it makes no sense to introduce a new grammatical category into language in advance of having any words to put into it. Grammatical categories such as ‘proper name’ or ‘verb’ are not buckets into which we cast words before they have meanings; they are descriptions of resemblances between the meanings which various existing words possess. We introduce the words ‘Socrates’ and ‘Plato’, and the way we introduce them makes it the case that they are both proper names. ‘Proper name’ is our grammatical term to express what it is that such words have in common; but what they have in common they owe to the manner in which they were given meaning, not the other way round. Wittgenstein is here drawing on an analogy between the indefinables of a logical system and the unknowns of a physical system. Applied mathematicians talk about the number of degrees of freedom in a system, meaning by this the number of unknowns in the equations which may be determined independently of each other. If we have a set of equations in five unknowns with four degrees of freedom, for instance, we can choose values for any four of the unknowns freely, but the value of the fifth will then be fixed by the choices we have made for the other four. When we set up a formal system, Wittgenstein thought it important to ensure that there are no more indefinables than there are degrees of freedom in the system. This idea is something he evidently obtained from Hertz, who remarked that we have accumulated around the terms ‘force’ and ‘electricity’ more relations than can be completely reconciled amongst themselves. We have an obscure feeling of this and want to have things cleared up. Our confused wish finds expression in the confused question as to the nature of force and electricity. But the answer which we want is not really an answer to this question. It is not by finding out more and fresh relations and consequences that it can be answered; but by removing the contradictions existing between those already known, and thus perhaps reducing their number. When these painful contradictions are removed, the question as to the nature of force will not have been answered, but our minds, no longer vexed, will cease to ask illegitimate questions.9 8

C28.

9

Prinzipien der Mechanik, Introduction.

The theory of types is superfluous  The methodological principle which Hertz offers in this passage is that philosophical perplexity often arises because our system of beliefs has more indefinables than degrees of freedom. It is a principle that remained central to Wittgenstein’s thinking long afterwards. In 1939, for instance, he quoted the passage from Hertz at a meeting of the Moral Sciences Club and said, as the minutes report, ‘that he must confess that this passage seemed to him to sum up philosophy’.10 Expressed in these terms, Wittgenstein’s point was that if, after introducing the names ‘Socrates’ and ‘Plato’, we regarded ‘proper name’ as a further indefinable, the system would have more indefinables than degrees of freedom. He concluded that differences of type between linguistic items can arise only through differences in their grammatical roles. This is to be distinguished from Wittgenstein’s further view, expressed in the June 1912 letter but later rejected, that there cannot be different types of things. The latter will follow only if name is assumed to be a single grammatical category. As I have said, it is tempting to detect Frege’s influence in Wittgenstein’s adoption of this last assumption. The reason that Frege held name to be a single category was that he believed the only possible source for such differentiations to be his distinction between saturated and unsaturated expressions. For him, therefore, all names, being saturated, belong to the same category. But Wittgenstein does not offer any argument to explain why we should follow Frege in this belief. 10

Quoted in Klagge and Nordmann, Public and Private Occasions, 379.

Chapter 

Fregean propositions One of the central doctrines of Frege’s logical theory from 1890 onwards was that a proposition is a name of a truth-value. The main task of this chapter will be to analyse the argument Wittgenstein offers in the Notes to show that this doctrine was misconceived. His argument turns on the use Frege made of the sign ‘#’ as a symbol to represent assertion. . Frege’s notion of assertion It is worth emphasizing straightaway that although it was Frege who devised the turnstile sign ‘#’, his use of it to signify assertion is not at all the same as its widespread modern use to signify logical derivability. This is partly because the modern usage gives a sense to occurrences of the sign with sentences both before and after it, as, for instance, when we say that if A # B, then C # D. In Frege’s usage, by contrast, the turnstile cannot be embedded in this manner, since it is intended to signal the act of assertion, not to be part of what is asserted. (For that reason it is incorrect to read ‘# A’ as ‘A is asserted’: the former, unlike the latter, does not say that A is being asserted but actually asserts it.) And even in the simple usage with a single sentence # A,

the modern understanding of the turnstile is different, since it is taken to assert not merely that A is true but that it is provable by logical means. In practice, of course, Frege asserted sentences in Grundgesetze only when he believed them to be provable, but that is merely because Grundgesetze is a work of logic. It would be quite correct, in Frege’s symbolism, to use the turnstile to assert anything that is true, whether or not the ground of its truth is logical. When Frege originally introduced the turnstile in his Begriffsschrift, he had not yet formulated his view of sentences as names of truth-values. In the

Frege’s notion of assertion  Begriffsschrift he conceived of the turnstile as a compound sign: the function of the horizontal part was to ‘combine the signs that follow it into a totality’,1 so as to make them capable of being judged; the vertical part of the sign, which he called the ‘judgment stroke’, then expressed the assertoric force of the judgment. Frege’s explanation does not really clarify what the purpose of the horizontal stroke is, but suggests that he saw it as a sort of copula, conferring propositional unity on the signs that follow it. If so, then by the time of the Notes Wittgenstein had come to regard it as unnecessary: what follows it must already possess the requisite unity. More to the point, though, Frege himself must soon have come to see that it is unnecessary. In any case this account of the turnstile plainly could not survive Frege’s adoption around 1890 of his mature semantic theory, according to which every sentence is a name of one or other of the two truth-values, which he called ‘the True’ and ‘the False’. From then on he continued to use the sign ‘#’, and to represent it as compound, but the parts were now different. The horizontal part of the sign no longer served as a sort of copula; instead it had the rather curious function of converting names which are not already sentences into names of the False; names that are already sentences it left untouched. Since ‘2’ is a name of a number, not of a truth-value, for instance, ‘−2’ is in Frege’s post-1890 notation a name of the False. But of course, no one would in any normal circumstances think of writing that. (Frege himself did, but only as an intentionally outlandish example to demonstrate how he intended his notation to operate.) Frege now used the vertical part of the turnstile, the judgment stroke, to indicate that the thought indicated by the name following it was being uttered assertorically. Of course, he did not place the turnstile in front of anything other than a name of a truth-value: to do so would always be an error. So the horizontal part of it is in practice eliminable: if we already have before us a name of a truth-value, adding the horizontal in front of it simply produces a slightly more elaborate name for the same truth-value. Since the horizontal acts in this way as a kind of identity function, adding two horizontals has the same effect as adding one: perhaps Frege intended the way the signs merge into each other visually to mimic this. For present purposes, then, let us concentrate on the judgment stroke. If sentences are names of truth-values, some device is needed merely to cope with the grammatical difference between uttering a name and expressing a proposition. When I assert a name of the True, I speak truly; when I assert a name of the False, I speak falsely. The effect of the judgment stroke is to convert a name of a truth-value into the assertion that what that name refers to is the True. Now ordinary language contains a device which fulfils the 1

Bs, §2.



Fregean propositions

grammatical role of converting a name of a sentence into something meet to be asserted, namely the phrase ‘is true’: by writing ‘Snow is white’ is true I express the thought that snow is white; and if, more generally, ‘A’ names any proposition, by writing A is true I express something not expressed by simply writing the name ‘A’. Frege’s judgment stroke can be thought of as combining two roles, then: first it converts such a name into something of the right form to be asserted, just as the ‘is true’ operator does; then it asserts that. . Propositions are not names of truth-values It is plausible enough that ‘Snow is white’ is true succeeds in expressing the thought that snow is white; but it is much less plausible that anything analogous holds for other sorts of names. If I assert Gödel’s theorem is true, I no doubt commit myself to the truth of Gödel’s theorem, but have I expressed it? In an obvious sense I have not. The example in which we use the quotation-name of a sentence is special just because there is in this case a means of semantic descent more direct than adding ‘is true’, namely to delete the quotation marks. In essence, Wittgenstein’s objection to Frege’s doctrine that sentences are already names even before we add quotation marks to them is that it deprived Frege of the resources to explain why expressions like ‘Gödel’s theorem’, which name propositions, can only ever be parasitic on sentences, which express them. In order to formulate his objection, Wittgenstein in effect decomposes the judgment stroke into two components in the manner I sketched in the last section. He then points out a difficulty for the first of the two components, the operator ‘is true’ which converts a name into an assertoric sentence. Wittgenstein expresses his objection to this operator by means of an analogy which may well have been inspired by one of Frege’s own. In Grundgesetze2 Frege had objected, against one formalist account of arithmetic, that it was like defining signs to be white if they belong to white objects. This is illegitimate because it 2

§59.

Propositions are not names of truth-values  only makes sense if we already know what the word ‘white’ means, in which case we are not free to define afresh what it is for a sign to be white. To make the point vivid, Frege noted that if the definition were in good order, we could make a black patch on a sheet of paper white by the simple device of agreeing to use it as a sign for a white sheet of paper. In the Notes Wittgenstein adapts this example in order to turn it against Frege himself. The focus of the example is still a black patch on a sheet of paper, but the question at stake is now what is involved in saying that a particular point on the sheet is white or black. In Frege’s theory, if I merely utter a sentence A, I am only setting up an assumption for consideration. If I want to say that the sentence is true, I must utter it with assertoric force: we would normally represent that by writing ‘A is true’ (or, in his formal notation, by writing ‘# A’). In something like the same way, if I indicate a point on the sheet of paper and name it P, I am not yet saying anything. If I say ‘P is black’, on the other hand, I have made an assertion which may be true or false. But, as Wittgenstein points out, the analogy goes wrong at just this point. I can indicate a point of the paper, without knowing what white and black are; but to a proposition without sense nothing corresponds, for it does not designate a thing (truth-value), whose properties might be called ‘false’ or ‘true’.3

There are, Wittgenstein is saying, two stages involved in saying that a particular point on a sheet of paper is black: the first, naming the point, does not involve knowledge of the difference between white and black; the second, predicating blackness of the point indicated, does. And if Frege’s account were right, there would similarly be two stages involved in asserting something: first we would construct a name of a truth-value; then we would predicate truth of it. It should therefore be possible, by analogy with the case of the sheet of paper, for the first stage, that of constructing a name which expresses what we wish to assert, to be done by someone who does not know what truth and falsity are. And that is ridiculous. In order to express a thought, I have to realize that thoughts aim at truth. It is incoherent to imagine someone coming to an understanding of language as a device for picking out one or other of the truth-values while still ignorant of what the point of this practice is. In contrast to the case of naming a point on a sheet of paper, there is nothing analogous to blind pointing by which I could, as it were, succeed in expressing a thought by accident. Frege’s mistake, in other words, was that he had left himself no resources with which to restrict our freedom of action in constructing names of truth-values in such a way that their usefulness in making assertions is built into them. 3

B10; cf. 4.063.

Fregean propositions



. Whose influence? Wittgenstein’s argument against Frege has attracted much scholarly attention. Ever since Anscombe,4 it has been a commonplace for commentators to note Russell’s influence on the way the argument is formulated. In one place Frege notes in passing that the proposition A on its own is an expression for a truthvalue but does not make any assertion; if we did not make this distinction, he says, we could not express something that is a mere assumption (Annahme), ‘the putting of a case without a simultaneous judgment as to whether or not it obtains’.5 Russell, when he summarized Frege’s views in an appendix to the Principles, saw in his use of the word Annahme an echo of Meinong’s,6 and incorrectly supposed that Frege was using it in something recognizably like Meinong’s way, as a technical term for any use of a sentence not prefixed by an assertion-sign. Anscombe is quite right to note that when Wittgenstein criticizes Frege in the Notes, he does indeed use the word ‘Annahme’ in this Meinong-like sense. And it is quite possible, of course, that Wittgenstein picked up this usage from Russell. There is surely nothing strange in this: it is, after all, easy to imagine that if Wittgenstein and Russell ever discussed Frege’s views, as presumably they did, they might have resorted to Russell’s preferred terminology in doing so. But that is not the only route by which Wittgenstein might have picked up the usage. He might equally have got it from Moore, who discussed Meinong’s idea of an Annahme in his lectures during Lent Term 1912. The point to stress, though, is that the fact that Wittgenstein used terminology in ways then current in Cambridge is not a sufficient reason to take the further step of supposing that Wittgenstein saw Frege’s works through the lens of Russell’s summary of them. After all, if we are to discuss Frege’s conception of propositions, it is convenient to have a word for the occurrence of a proposition unasserted, and for that purpose ‘Annahme’ is surely as good a word as any, even if it is not really Frege’s. Wittgenstein himself, at any rate, evidently thought so, since he continued to use the word in this sense through to the Investigations.7 But to use a word that is not Frege’s is not yet to show that one misunderstands him. Russell’s discussion8 of Frege’s conception of sentences as names of truth-values is riddled with confusions and misunderstandings; Wittgenstein’s, as I have outlined it above, is not. That is in itself enough to cast doubt on the idea that Wittgenstein approached Frege’s writings via Russell: if he had, one would surely expect his criticisms of them to have been equally confused. I think, though, that we can go further. Not only does Wittgenstein’s criticism of Frege show an informed understanding of Frege’s own views, rather than with Russell’s summary of them, but it is itself based on a profound en4

IWT, 105–6.

5

Funktion und Begriff, 21.

6

Über Annahmen.

7

PI, §22.

8

Principles, §477.

Propositions as articulate  gagement with Frege’s conception of logic. Wittgenstein’s argument against Frege depends on a certain understanding of the relationship between thought and truth, and hence between propositions and truth. It is essential to a thought that it should be capable of being true, and essential to a proposition that it should express a thought. Two things follow from this: first, truth is not a property of thoughts as redness is a property of flowers; and second, it is essential to a proposition that it should have a structure that enables it to express something capable of being true. Wittgenstein’s argument that propositions are not names of truth-values is merely an application of these two principles. I noted earlier that the example Wittgenstein uses to make his point is based on one Frege himself uses against a kind of formalism, but of course that is at best evidence only of superficial borrowing, not of deep influence. What does make Wittgenstein’s argument profoundly Fregean is that the two guiding principles it relies on are ones Frege himself throughout his working life made central to his conception of logic. The first, that truth is not to be thought of as a property, we have already discussed. As for the second, that it is essential to a proposition that it should express something capable of being true or untrue, this is explicit in a pre-Begriffsschrift sketch9 of Frege’s logical doctrines. A criterion for whether a mode of connection constitutes a thought is that it makes sense to ask whether it is true or untrue. Associations of ideas are neither true nor untrue. . . The expression in language for a thought is a sentence. We also speak in an extended sense of the truth of a sentence. A sentence can be true or untrue only if it is an expression for a thought.10

Here, then, we have an instance of Wittgenstein’s tendency to think, in Kienzler’s nice phrase, ‘with Frege against Frege’.11 We have met several examples in earlier chapters where Wittgenstein exploited Fregean principles, but what makes his employment of them here especially notable is that he directs them, to devastating effect, against Frege himself. What he displays is an inconsistency in Frege’s own views, and he does so in a manner which makes it all the more surprising that Frege did not spot it himself. . Propositions as articulate Here is another, quicker way of seeing what is wrong with Frege’s idea that sentences are names of truth-values. According to that doctrine, I speak truly whenever I assert a name of the True. In particular, since ‘the True’ is a name of the True, I speak truly when I assert ‘the True’. But what am I then asserting? If I represent my assertion as ascribing a property to an object, then by saying that the True is true I seem indeed to have said something—trivial, admittedly, but correct. However, if I insist, as Frege did, that there is no such 9

See Dummett, FOP, 77.

10

PW, 174.

11

Wittgensteins Wende, 227.



Fregean propositions

property as truth for me to be ascribing here, and no sense to which the phrase ‘is true’ corresponds, that explanation cannot be right. At this point, though, it becomes difficult to see what else I can be saying if not that. When I assert ‘the True’, it seems that I am performing the incoherent feat of telling the truth without there being any particular truth I am telling. On this way of putting the point, therefore, what Frege had failed to do by treating propositions as names of truth-values was to build into them the requirement that they should be complex. ‘Propositions,’ Wittgenstein said, ‘can never be indefinables, for they are always complex.’12 The names of the two truth-values, ‘the True’ and ‘the False’, contravene this dictum. It is important to understand clearly why a proposition cannot be simple. Consider, for example, two notations for postal addresses in a town: in the first system, each address consists of a number followed by a street name; in the second, every house has a name of its own. It is not hard to see which notation a postman would prefer. The compositional structure of the atomic proposition is, we might think, grounded only in this point about economy of expression. That, however, is not Wittgenstein’s point at all. To see that it mistakes his intentions, it is enough to note that the degree of economy effected by the device under discussion, although sometimes substantial, is always finite. It is not a device which enables us by learning a finite number of items to express infinitely many different senses. It can therefore only make a difference to the relative feasibility of the notation, not to its absolute possibility. But this issue of relative feasibility was far from Wittgenstein’s concerns. Later in his life he came to address seriously the issue of what it amounts to to say that something is possible even though it would be quite impracticable for any human to do it, but throughout his early philosophy there is a consistent tendency for him to regard issues of feasibility almost as missing the point. One sometimes suspects that he thought the gap between possibility and feasibility was just what servants were there to bridge: he would not have regarded the views of the postman as relevant. Wittgenstein’s point was rather that a proposition is essentially articulate. The reason that the notation with street names is preferable to the one where each house has its own name is not that there are fewer signs in it, but rather that the first notation is capable of telling the postman where each house is: the composition of the address corresponds to the actual arrangement of the houses. Wittgenstein makes the obvious point that the single-word Latin sentence ‘ambulo’ (which means ‘I walk’) is not a counterexample to this, because the existence of other related words such as ‘ambulas’ and ‘ambulat’ with the same root and different terminations shows it to be really hiding the articulate structure ambul|o within the single-word form.13 However, there is another 12

B64.

13

Ibid.

Propositions as articulate  fairly obvious kind of apparent counterexample to the thesis that propositions are essentially articulate that needs to be considered. ‘How is it possible,’ Wittgenstein asked, for ‘kilo’ in a code to mean ‘I’m all right’? Here surely a simple sign does assert something and is used to give information to others.—For can’t the word ‘kilo’, with that meaning, be true or false?14

To answer his question we have to recall our earlier discussion of the symbolic turn. The task of a sentence of ordinary language is to bring to mind a symbolizing fact. But, as was stressed before, not everything in the sentence is relevant to this task. We are used, when we read English sentences, to ignoring a host of irrelevant features in discerning what symbolizes in them. What Wittgenstein’s example shows is that sometimes in ordinary language the opposite is true. Some signs have in certain respects less complexity than the facts that we read into them. The most obvious examples of this are, like Wittgenstein’s in the above quotation, abbreviations; and in such cases the natural thought is that the abbreviation simply inherits the complexity of the expression it abbreviates. The obvious hypothesis would be that abbreviations are the only such cases, but there is nothing in Wittgenstein’s writing to show that he was committed to this hypothesis, at any rate if we limit ourselves to explicit abbreviations. The irony is that the articulacy of propositions is once again something Wittgenstein could have learnt from Frege. For it was above all Frege who stressed that thoughts are essentially complex. He found it ‘extraordinary’, for example, ‘that some linguists have recently viewed a “sentence-word”, a word expressing a whole judgment, as the primitive form of speech and ascribe no independent existence to the roots, as mere abstractions’.15 Yet Wittgenstein’s accusation is, in effect, that if sentences were names of truth-values, it would be impossible to explain what was extraordinary about this view. What is wrong about Frege’s assimilation of sentences to names is that it offends against a principle which Frege himself held dear. 14

NB, 4 Oct. 1914.

15

PW, 17.

Chapter 

Assertion . The judgment stroke as force indicator As we noted in the last chapter, the judgment stroke is not interchangeable with the ‘is true’ operator. For a sentence of the form ‘A is true’ may, like any assertoric sentence, be used for purposes other than simple assertion. It may, for instance, be used to form compound sentences, such as If A is true, then I am a Dutchman or A is not true. Frege’s judgment stroke, on the other hand, cannot be embedded in this manner, because it combines two roles: it both converts a name of a truth-value into a sentence and indicates the assertoric force of the utterance, which the phrase ‘is true’ cannot do on its own. Nor does this point to some weakness of the word ‘true’: it is just as wrong to read ‘# A’ as ‘A is asserted’ as it is to read it ‘A is true’, since, once again, ‘A is asserted’ can be embedded in compounds but ‘# A’ cannot. What this shows is that even if we accept Wittgenstein’s recommendation to absorb the verb ‘is true’ into the proposition, there is still a function left for Frege’s turnstile to perform, namely that of indicating assertoric force. Natural language, as Frege was fond of pointing out, has no sign for assertion, relying instead on a variety of means (especially context) to indicate when an assertion is being made. Frege regarded this as a defect of natural language and introduced the turnstile in order to correct it, but it is far from clear that he was right to do so: as we shall see shortly, the turnstile has been responsible for considerable confusion. At the very least, its grammar takes some getting used to. If a text contains the assertion ‘# p’, then what the author asserts is that p; it is an easy slip to disquote and represent him, incoherently, as asserting that # p. Because it is an indicator of force, and has no semantic content, the turnstile must not be thought to modify what p expresses. I may assert, doubt, suppose, or entertain the bold conjecture, that p; and in each case it is the same proposition p to which I adopt an attitude. As Wittgenstein ob-

Asserted and unasserted propositions  served, ‘ “#” . . . belongs as little to the proposition as (say) the number of the proposition.’1 In an appendix to the second volume of Grundgesetze Frege discussed various ways of avoiding the paradox which Russell had recently pointed out to him. Because of their tentative nature Frege did not prefix his proposals with the assertion-sign. Geach has called this ‘hypocritical’,2 but surely it is worse than that. Once Frege had adopted the convention that a sentence of the Begriffsschrift must be prefixed with an assertion sign if it is to count as an assertion, simply writing a sentence without such a sign was not merely hypocritical but ungrammatical. He could, of course, have introduced a further sign to mark the force of tentative assertion, but even tentative assertion is a kind of assertion and therefore a linguistic act. There is perhaps even something in Wittgenstein’s much later suggestion3 that the assertion-sign should be thought of as a punctuation mark. There was a time when books on English usage insisted that the exclamation mark should be used only to terminate an exclamation or command, and not, as is nowadays common, as a sort of generalized emoticon. If the stricter convention were followed universally, we could equally consider using the full stop only to terminate assertions. But then there might also be a point in having a sign to mark the start of an assertion, to fulfil the function, like cinema ads, of reassuring us that we have not arrived halfway through. In ordinary written English, of course, capital letters usually fulfil this role well enough, but in logic, where we cannot capitalize willy-nilly, we might feel the need of another convention, such as Frege’s. Whatever the practical merits of such a proposal, however, the real point Wittgenstein was making, both early and late, was of course a negative one: the assertion sign must at any rate not be thought of as more than a punctuation mark. This is not to doubt its role as a mark of force, but only to proscribe any further role for it. . Asserted and unasserted propositions Wittgenstein’s recommendation to treat the assertion sign as a punctuation mark thus has some purchase as a correction to Frege, but it applies with even greater urgency to Russell, who had struggled to explain the difference in role between occurrences of a proposition asserted and unasserted in logical arguments. Russell shared Frege’s anti-psychologistic view of logic. Consequently, he thought that expelling psychology from logic required him to treat the turnstile, when it occurs in logical inferences, as logical and not psychological. 1

B32.

2

‘Saying and showing in Frege and Wittgenstein’, 63.

3

PI, §22.



Assertion

Moreover, since he did not in the Principles have Frege’s distinction between content and force, he supposed there that the difference in role between ‘ p’ and ‘# p’ must reflect a difference between the entities these expressions refer to. He therefore treated ‘ p’ and ‘# p’ as distinct entities, which he called the unasserted and asserted proposition respectively. But since the conception of assertion in question here was logical, not psychological, an asserted proposition was for him just the same thing as a true one. ‘It is plain,’ he said, that true and false propositions alike are entities of a kind, but that true propositions have a quality not belonging to false ones, a quality which, in a non-psychological sense, may be called being asserted.4

But what is the ‘quality’ which distinguishes true propositions from false ones? ‘There are,’ he had to admit, grave difficulties in forming a consistent theory on this point, for if assertion in any way changed a proposition, no proposition which can possibly in any context be unasserted could be true, since when asserted it would become a different proposition. But this is plainly false; for in ‘ p implies q,’ p and q are not asserted, and yet they may be true.5

He could have added, indeed, that the view would also require him to abandon Moore’s simple identity theory of truth. When I believe something, the object of my belief cannot be the asserted proposition, both because I can believe falsely and because if it were, I could hope to discern its truth by introspection on the object of my belief. But if what I believe is the unasserted proposition, then in the case when my belief is true, what is true, the asserted proposition, is not the same as what I believe. Yet it was Russell himself who insisted in a letter to Frege that whether I judge a proposition or only imagine it, ‘the object is the same in both cases. . . The judgment stroke therefore means a different way of being directed towards an object.’6 In truth, Russell had got himself into a tangle, as is almost visible on the page. Assertion does not seem to be a constituent of an asserted proposition, although it is, in some sense, contained in an asserted proposition. If p is a proposition, ‘ p’s truth’ is a concept which has being even if p is false, and thus ‘ p’s truth’ is not the same as p asserted. Thus no concept can be found which is equivalent to p asserted. Yet assertion is not a term to which p, when asserted, has an external relation; for any such relation would need to be itself asserted in order to yield what we want. Also a difficulty arises owing to the apparent fact, which may however be doubted, that an asserted proposition can never be part of another proposition: thus, if this be a fact, where any statement is made about p asserted, it is not really about p asserted, but only about the assertion of p.7

If the view that asserted propositions are different entities from unasserted ones leads to so many difficulties, one might expect Russell to have abandoned 4

Principles, §38.

5

Ibid.

6

24 May 1903.

7

Principles, §478.

Asserted and unasserted propositions  it rapidly, but he did not. Instead he adopted the rather desperate strategy (which he often used when he found himself in a hole) of reaffirming the view while granting that he had no answer to the difficulties it leads to. ‘Leaving this problem to logic,’ he neatly advised, ‘we must insist that there is a difference of some kind between an asserted and an unasserted proposition.’8 That was in the Principles. Was he any clearer by the time of Principia? There he probably did not intend to allow the turnstile to embed in larger propositional contexts, although he contrived to create an appearance to the contrary by adopting9 a convention according to which ‘# p, # p ⊃ q, so # q’, for instance, is abbreviated to ‘# p ⊃ # q’. Nonetheless, he did still speak of the asserted proposition as a different entity from the unasserted proposition.10 For Russell, then, a proposition is somehow different when it occurs asserted from when it occurs unasserted. Wittgenstein rejects this doctrine very firmly in the Notes: in the Birmingham Notes he points out that ‘the assertion sign is logically quite without significance’;11 and in the Cambridge version he tells Russell that ‘there are only unasserted propositions’.12 He does not offer an argument for this. No doubt, the argument we offered in the last chapter could be adapted to meet the case: if, as that argument aims to show, a proposition, even an unasserted one, ‘must already contain the verb’,13 it already contains everything that is needed to determine what it expresses, i.e. what is said to be the case when the proposition occurs asserted. But Wittgenstein probably did not formulate the argument with that in mind: after all, he was explicit that its target was Frege, not Russell, and he did not allude to it again when he came to make his point against Russell. In truth, though, he did not need to: the view could easily be attacked on other grounds, not least those Russell himself offered in the Principles. One thing that follows from Wittgenstein’s view, of course, is the hopelessness of trying to disentangle truth from assertion. ‘Could we not express ourselves,’ Wittgenstein asks, by means of false propositions just as well as hitherto with true ones, so long as we know that they are meant falsely? No! For a proposition is then true when it is as we assert in this proposition; and accordingly if by ‘q’ we mean ‘not-q’, and it is as we mean to assert, then in the new interpretation ‘q’ is actually true and not false.14

It is incoherent, that is to say, to suppose that we could lie all the time: lying is parasitic on the practice of telling the truth. This point is again a Fregean one, since it was Frege who stressed the curiously intimate relationship between truth and assertion. The word ‘true’, he said, ‘seems to make the impossible possible: it allows what corresponds to the assertoric force to assume the form of a contribution to the thought’.15 8

Principles, §38.

9

PM, I, 104–5.

10

PM, I, 92.

11

B32.

12

C40.

13

B10.

14

B9.

15

PW, 252.



Assertion

. Assertion as psychological Russell’s misbegotten doctrine of asserted propositions arose from two views of his: first that the turnstile has, as we might now say, semantic content, rather than being merely an indicator of force; second, that what it indicates is something logical, not psychological. That the first of these views is wrong is something Frege had been clear about long before he met Wittgenstein. On the second, however, Frege’s views were not perhaps as clear as some16 have held. What the judgment stroke indicates is for Frege an act, the mental act of judging a thought to be true. There is a strand in Fregean commentary17 that presents him as holding that we cannot judge falsely. That is uncharitable: the few passages in which he seems to speak that way—most notably an unsent draft of a letter—are surely best passed over as no more than loose phrasing.18 But if judgment, for Frege, is a mental act, hence both psychological and fallible, what should he then say about assertion? Judgment and assertion are not the same, as he had long realized. ‘When we inwardly recognize that a thought is true, we are making a judgment; when we communicate this recognition, we are making an assertion.’19 For Frege, the realm of psychology is the mental, and what is characteristic of the mental is that it is private: it is a recurring theme of Frege’s writings that my ideas are intrinsically mine and hence unavailable to anyone else. So if judgment is psychological and assertion is the public manifestation of judgment, should we suppose that assertion is in some way psychological too? Pain, we may agree, is private and hence psychological. Crying out plainly is not: although often a sign of pain, it may be feigned, by actors and malingerers, for their own ends. But crying out in pain has a psychological component, which it inherits from the pain: an actor cries out, but does not cry out in pain. Is assertion in this respect like crying out or like crying out in pain? In order that I should assert something, is it required that I should judge it? There are indications that Frege himself was tempted to think of assertion as entailing judgment. It is consistent with such a view, for instance, that he should have been so exercised by the case of actors on stage, who utter sentences which they do not, in propria persona, intend their hearers to believe. He held that because they are not really judging as true the sentences they utter, the actors are not asserting anything either.20 To some extent, no doubt, ‘assertion’ is a philosopher’s term of art; and to that extent we are free to legislate its use as we please. Dummett is right to maintain, nonetheless, that there are clear advantages to thinking of assertion as wholly non-psychological. Frege’s view misses the essentially public nature 16 E.g. Goldfarb, ‘Wittgenstein’s understanding of Frege: The pre-Tractarian evidence’. 17 E.g. Anscombe, IWT, 114. 18 See Kremer, ‘Judgment and truth in Frege’. 19 PW, 139. 20 ‘Über Sinn und Bedeutung’, 34–5.

Psychology  of assertion. Precisely because we can indicate it linguistically, what is essential to assertion is the expression of it, not whether we mean it in our hearts. If we had in natural language an explicit device like Frege’s judgment stroke to indicate assertion, then we would expect actors on stage to use it with full vigour: anything less could scarcely count as a convincing performance. It thus seems right to say, as Dummett21 does, that the actor is doing more than asserting, not less. Similar remarks may be made about liars and conmen. It may indeed, as Davidson22 claims, be hard to specify the linguistic conventions governing when something is to count as assertion; but that is not sufficient reason to deny that such conventions exist, so that if something I say conforms to them, then it is an assertion. For I can be convicted of error if it turns out to be wrong, just as I can be blamed for failing to keep an insincere promise. It follows from this that assertion (unlike judgment) is not in Frege’s sense psychological, since it is not private.

. Psychology At first sight, this conclusion—that assertion is not psychological—is in flat disagreement with Wittgenstein’s claim that it is. ‘Assertion,’ he claims in the Notes, ‘is merely psychological.’23 However, this is a straightforward contradiction only if Wittgenstein’s use of the word ‘psychology’ is the same as Frege’s. What should give us pause is that Wittgenstein describes the correlation of names with their meanings as psychological.24 What Wittgenstein is describing here is the process by which we turn linguistic items into symbols: they acquire membership of the various grammatical categories by virtue of being given symbolizing roles in this way. This is what we might nowadays call linguistics: Wittgenstein makes no distinction between it and psychology. The point here is one we discussed in chapter 7. By refusing to distinguish between the subjective and the intersubjective Wittgenstein collapsed the distinction between ideas and senses, and hence that between psychology and linguistics. When he said that attaching a meaning to a name is a matter for psychology, he intended to distinguish psychology not from linguistics but from logic. Wittgenstein says, moreover, that epistemology is the philosophy of psychology.25 Epistemology studies the process by which linguistic items acquire meanings, and thereby symbolic roles. This process is not the concern of logic,26 which studies these symbolic roles themselves. For Frege the distinction between ideas and senses was fundamentally one between what is private and what is public, between the subjective and the intersubjective. Wittgenstein denied this distinction. 21

FPL, 311.

22

‘Moods and performances’.

23

C40.

24

B43.

25

B62.

26

B24.



Assertion

I mentioned earlier that Wittgenstein does not seem to have been troubled about whether his views were solipsistic. This is not quite the same as saying that he was a solipsist, but at any rate it is clear that he did not, as Frege did, intend the word ‘psychology’ to demarcate the mental as a private sphere in contrast to the public sphere of language. For, as we have just noted, he was explicit that the correlation of name and meaning is psychological,27 and that is a matter of linguistic convention if anything is. For this reason we can, I think, agree with Wittgenstein’s claim that ‘assertion is psychological’, by which he meant only that it is not a matter for logic, without disagreeing with the conclusion we reached earlier that assertion is not in Frege’s sense psychological, since it is not private. This question of whether assertion should in any sense be regarded as psychological was for Frege by no means a peripheral one. Long before he met Wittgenstein, he had noted the curious connection between the use of the word ‘true’ and the assertoric form of the sentence. If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3 make 5. In the same way I assert that it is true that my idea of Cologne Cathedral agrees with reality, if I assert that it agrees with reality. Therefore it is really by using the form of an assertoric sentence that we assert truth, and to do this we do not need the word ‘true’. Indeed we can say that even where we use the form of expression ‘it is true that . . . ’ the essential thing is really the assertoric form of the sentence.28

There is in Frege’s Nachlaß a document headed ‘The following may be of some use as a key to the understanding of my results’. Perhaps it is not too fanciful to wonder whether Frege wrote this document in preparation for, or as a response to, one of his meetings with Wittgenstein. Here Frege repeats his view, which he had stated elsewhere, that the word ‘true’ indicates the essence of logic. But then he shifts his ground. The thing that indicates most clearly the essence of logic is the assertoric force with which the sentence is uttered. But no word, or part of a sentence, corresponds to this; the same series of words may be uttered with assertoric force at one time, and not at another. In language assertoric force is bound up with the predicate.29

I suggested in §5.6 that Wittgenstein’s fundamental thought could be seen as the combination of two Fregean principles, that the subject matter of logic is truth and that truth is redundant. It seems now as if, in order to resist this conclusion, Frege was prepared to say that the subject matter of logic is assertion. It is plain that if he were indeed to say that, and not give up on his virulent anti-psychologism, he had to be clear that assertion is not psychological in his sense of the word. I have suggested that he may not have been as clear about this as he should have been. There would, on the other hand, have 27

B43.

28

PW, 129.

29

PW, 252.

Psychology  been no strong reason for Frege to resist the idea that part, at least, of logic is psychological in Wittgenstein’s sense, since this only means that it takes cognisance of the realm of sense. Frege was committed to the idea that it is the richness of this third realm that explains the richness of arithmetic. He approved, for instance, of Dedekind’s proof that there are infinitely many things in the third realm,30 a proof that is central to Dedekind’s way of showing that arithmetic is part of logic. It follows that Frege’s conception of logic, like Dedekind’s, has a component that is in this sense (but not in Frege’s own) psychological. Wittgenstein, on the other hand, would have insisted that logic is not psychological even in his own wider sense: he used the words ‘logical’ and ‘psychological’ in what he plainly intended as contrastive senses. So Wittgenstein, in addition to having a wider conception of psychology than Frege, also had a narrower conception of logic. It is therefore plausible to hear Wittgenstein’s voice in the background when Jourdain asked Frege, ‘whether you now regard assertion (#) as merely psychological’.31 If Frege was, as I have suggested,32 in some confusion over the matter, perhaps it is telling that both his draft reply (never sent) and his actual reply avoid answering Jourdain’s question outright. 30

PW, 136. 31 15 Jan. 1914. pre-Tractarian evidence’.

32

Contra Goldfarb, ‘Wittgenstein’s understanding of Frege: The

Chapter 

Complex and fact The distinction between complex and fact is easily stated. A complex is an arrangement of things; that they are arranged in this way is a fact. A complex is thus something we can refer to by means of a description; we specify a fact, on the other hand, by means of a ‘that’ clause. This distinction is one of the most important in Wittgenstein’s philosophy, and it pervades the Notes on Logic. . A world of facts, not of things In chapter 4 I discussed Wittgenstein’s analysis of a proposition about a complex into ‘a proposition about its constituents and the proposition which describes the complex perfectly’. His idea, we saw there, was to analyse φ ([aRb]) ≡ φ ! (a,b) aRb.

where φ ! is a propositional function related appropriately to φ . In the case of a proposition which says only that a certain complex exists, φ ! is vacuous, so that the analysis reduces to something like the following: [aRb] exists ≡ aRb.

(1)

So if we take Wittgenstein’s eliminativism seriously, the existence of the complex referred to on the left of this biconditional is reduced to the fact expressed on the right. At the last stage of analysis, that is to say, we obtain a world of facts, not of complexes. What recommended Wittgenstein’s eliminative conception of analysis over Russell’s might be described as a ruthless application of Occam’s razor; or, less charitably, as the triumph of ontological scepticism over common sense. But what reason is there for wielding the razor in this direction? Consider the more general equivalence [ p] exists ≡ p, (2) and leave to one side whether it could hold in full generality. The question we need to address is whether, in those cases in which it is true, there is any reason

A world of facts, not of things  to treat one side rather than the other as more fundamental. Note first that it is problematic to try to privilege the left-hand side of the equivalence over the right in all such cases, since that would require us to assume that exactly one of the two complexes [ p] and [∼ p] exists, without providing any explanation for this curious coincidence. The view would thus seem to load onto the world the whole responsibility for logic. Whenever a complex goes out of existence, another complex, its logical contradictory, has to come magically into existence in its stead. This makes the connection between p and not- p, according to which not- p is true just in case p is false, a kind of miracle, a gift donated to logicians by a generous world. But when we restrict ourselves to the very simplest case, where p is an atomic proposition, it is not so clear that this objection still applies: one might perhaps think that it is a criterion of the atomicity of p that p and not- p should be related in this manner. (I shall discuss this thought further in §15.2.) Is there any other reason, in this atomic case, to privilege the right-hand side of (2) over the left? Russell, at any rate, thought not. In 1910 he outlined a theory according to which what makes an atomic proposition true or false is the existence or non-existence respectively of the corresponding complex. But in fact his conception oscillated uncertainly. We have already come across his willingness to turn verbs casually into verbal nouns. He often displayed a similarly relaxed attitude to the process of nominalization which turns the fact that Caesar died into Caesar’s death: one moment he gave as an example the complex ‘a-in-the-relation-R-to-b’;1 the next he used the fact ‘Socrates is mortal’.2 On occasion, he referred to complexes by means of that-clauses, and he observed that the grammar of such clauses gives them no complete meaning on their own. ‘We feel that the phrase “that so-and-so” is essentially incomplete, and only acquires full significance when words are added so as to express a judgment.’3 But, not having a clear grasp of the distinction, he regarded this only as an argument (and even then not a decisive one) against the existence of propositions, not as a way of arbitrating between complex and fact. Whatever Russell’s uncertainty, it is clear that he conceived of the denizens of his world as much more complex-like than fact-like. However, the difficulty with this conception of the world as consisting ultimately of complexes arises when we consider our most basic judgments about it. Russell used the term perception for the experience of an atomic complex which grounds our indubitable judgment that the complex exists. If I perceive a red circle, for instance, that may, according to Russell, ground my judgment that the red circle exists. But as Wittgenstein put the point much later, ‘To say that a red circle 1

CP, VI, 10.

2

CP, VI, 12.

3

CP, VI, 119.



Complex and fact

is composed of redness and circularity, or is a complex with these component parts, is a misuse of these words, and is misleading.’4 One way of explaining what Wittgenstein was referring to here is by noting a fundamental difference in structure between complex and fact. How much structure Russell supposed a complex to carry is somewhat uncertain: perhaps the truth is that he supposed it to carry whatever structure it needed in order to serve the task at hand. At any rate, a complex cannot be simply the mereological sum of its components, since that will exist whenever the components do.5 We may disagree about the precise ingredients of a salad Niçoise (whether it should contain eggs, for example), but whatever those ingredients are we will surely agree that they do not constitute a salad until they are put in the same bowl; I would be a poor host if I simply laid them out for my guests to assemble themselves. Ordinary language sometimes wavers. If I dismantle my computer in the hope of fixing it, I may prefer to say that my computer is in pieces rather than that it no longer exists. But at any rate many of the objects of everyday life are, like salads, complexes made up of parts related in certain ways. That the parts are related in the requisite manner is a fact about the complex. We may say that the complex exemplifies the fact. What is important—or at any rate one thing that is important—is that on this understanding of the distinction the correspondence between complexes and facts is one–many. One complex, that is to say, may exemplify several different facts. The salad, for instance, exemplifies the fact that the bowl contains the requisite ingredients (tuna, anchovies, etc.); but it exemplifies, too, the fact that the dressing coats the other ingredients evenly. The red circle in Wittgenstein’s example exemplifies the fact that it is red; but it also exemplifies the fact that it has a certain radius; and, too, that it occurs in a certain position in my field of vision. It is misleading to say that a red circle is a complex consisting of redness and circularity because it privileges a particular fact about it, namely that it is a red circle. But complexes do not speak: they are what they are, and do not present any particular facts as salient. Complexes, therefore, cannot be what ground propositional thought, because they do not have the right kind of structure to do so. Only facts can do that. We might wonder once again, though, whether the argument we have offered applies in the atomic case. Perhaps there are complexes of a kind that only exhibit a single fact. For such complexes the issue of salience would not arise. But that is not the point. The lack of a one–one correspondence between complexes and facts is only a symptom of the difference, not the underlying reason to eliminate the former in favour of the latter. 4

PG, 200.

5

Cf. Russell, CP, VI, 355.

Influences  . Influences When he discussed the distinction between complex and fact in the 1930s, Wittgenstein was quite clear where he had obtained the insight just mentioned, that a red circle is not a complex consisting of redness and circularity: ‘Frege was aware of this and told me.’6 Although the distinction is not something Frege emphasized in his published writings before 1913, he did make the following observation in his 1918 article, ‘Der Gedanke’. That the sun has risen is not an object which emits rays that reach my eyes, it is not a visible thing like the sun itself. That the sun has risen is seen to be true on the basis of sense-impressions. But being true is not a material, perceptible property.7

This shows Frege to have been aware of the distinction between complex and fact, but he cannot8 here have been responding to the Tractatus: he sent the final version of ‘Der Gedanke’ to be published a couple of weeks before he first saw the Tractatus. For Frege, though, facts never quite played the dominant ontological role that they did for Wittgenstein. In ‘Der Gedanke’ he remarks that ‘a fact is a thought that is true’, but he does not go on to make any great play with the notion. The offhand way he mentions it suggests, indeed, that his meaning is almost deflationary: a fact is merely a thought that is true. Wittgenstein, on the other hand, famously chose to make facts salient at the beginning of the Tractatus. If he himself was not quite so emphatic in the 1913 Notes, Russell was explicit, presumably on his behalf, in the first of his logic lectures at Harvard the following year. ‘What we analyse,’ Russell said then, ‘is a fact, not a thing. A complete description of the real world is not given by enumerating the things in it.’ Although Frege’s influence would be sufficient to explain the importance Wittgenstein attached to the distinction between complex and fact, he could also have found it in a continental tradition stemming from Brentano. Stumpf,9 for instance, noted in 1907 that Brentano had in his lectures thirty years earlier distinguished between what he called the specific and the presentational content of a judgment. He went on to note that the former is what is expressed linguistically in a ‘that’-clause and to propose the word Sachverhalt as a technical term for it. Others in the same tradition such as Reinach10 and Marbe11 also discussed the distinction between complex and fact in various guises. Of the two aspects to the distinction that we have highlighted—the difference of identity conditions and the difference of nature—it is the latter that most of these authors emphasize. Only with Reinach12 do we begin to see a stress on the idea that a variety of facts (that this rose is red, that this rose forms the 6

7 8 9 PG, 200. p. 61. Contra Sullivan, Formal concepts, 110. ‘Erscheinung und psychische Funktionen’. 10 ‘Zur Theorie des negativen Urteils’. 11 ‘Beiträge zur Logik und ihrem Grenzwissenschaften’. 12 Ibid. ch. I, §7



Complex and fact

substrate of redness, that redness is inherent in the rose) might be exemplified in a single complex. We do not know which of these authors Wittgenstein read, but it is surely quite likely that he read some of them: if not, it is at least a curious coincidence that he later used Sachverhalt in something rather like Reinach’s sense and that he chose to illustrate his discussion13 with the very same example (‘This rose is red’) as Reinach and Marbe. Stumpf is also someone Wittgenstein is very likely to have known about. Moore was reading him in the summer of 1912, and Wittgenstein would in any case have been likely to come across the name in the course of his experiments on rhythm at the Psychological Laboratory. McTaggart, who was also at Trinity and first met Wittgenstein in October 1912,14 is also known to have taken an extensive interest in Stumpf’s work, and his own writings are notably free of the confusion in question.15

. Russell on facts Whatever the influences were that helped Wittgenstein towards the distinction between complex and fact, it is at any rate clear that Russell was not one of them. It remains to ask, though, whether there was any influence in the other direction, from Wittgenstein to Russell. If we are to answer that question, we need to say a few words first about the dating of the Notes. I shall explain in Appendix A my reasons for thinking that the three sections of what I there call the Birmingham Notes are compilations of remarks from three distinct notebooks in which Wittgenstein had been writing his thoughts day by day during various parts of his time at Cambridge. If this is right, then it has some bearing on the subject of the current chapter to note that the distinction between complex and fact is exploited in all three sections of the Birmingham Notes; yet the letter of January 1913 does not betray a glimmer of it. This suggests quite strongly that the Birmingham Notes do not contain very much material drawn from before January 1913. On the other hand, I do not think that Wittgenstein could have started writing the notebooks from which the Birmingham Notes are derived much later than January 1913. By saying this I contradict the popular view that the Notes are in their entirety a report of Wittgenstein’s work in the summer of 1913 (or, even more impressively, just during his three-week holiday in Norway with Pinsent). This view does admittedly originate in a remark by Russell himself, who seems to have been under the impression that much of the work that Wittgenstein was reporting to him had been done in August and September16 but, although I cannot refute the notion conclusively, the sheer number 13

B7. 14 BR to OM, 29 Oct. 1912. 15 Geach, ‘Saying and showing in Frege and Wittgenstein’, 67. 16 To Lucy Donnelly, 19 Oct. 1913.

Russell on facts  of substantial philosophical ideas the Notes contain surely makes it wildly unlikely. But it has contributed to the more general impression occasionally voiced that Wittgenstein worked in extremely concentrated spurts before collapsing exhausted. Another strand in the same thread, now disproved, used to hold that Wittgenstein wrote the whole Tractatus in the summer of 1918: the more mundane truth is that he had by then been working on the book itself (as opposed to the notebooks from which much of it was derived) for almost three years. Indeed, there is every reason to think that his working methods were not at all impulsive, but patient and methodical. One reason the myth took hold was no doubt that it resonates with something in our conception of the genius as someone possessed by an external force. It is not sufficient that Mozart wrote wonderful music: we express his genius by emphasizing the astonishing speed with which he wrote it. In the case of the Notes, though, there is another, much more prosaic reason why some have been tempted to suppose that they were written on holiday in Norway, namely that many of the ideas contained in them were evidently new to Russell when Wittgenstein explained them to him in October 1913. If, as I am claiming, some of them may date from as early as the previous January, why did Russell not know them already? The conclusion we must draw, I believe, is that this is a sign of a marked change in the working relationship between the two men. Russell alluded to something of this in his letters to Ottoline, where he confessed that he no longer told Wittgenstein about his own ideas, because he ‘treats infant theories with a ferocity which they can only endure when they are grown up’.17 Perhaps Wittgenstein was now in his turn becoming selective about which of his ideas he shared with Russell. If so, then it may be an instance of this emerging intellectual gap that in May 1913 (when, if the dating I have been urging is right, Wittgenstein may already have been using his new conception of facts for several months) Russell could still write: It may be questioned whether a complex is or is not the same as a ‘fact’, where a ‘fact’ may be described as what there is when a judgment is true, but not when it is false. (I do not suggest that this is a proper definition of a ‘fact’, but merely that it serves to point out what sort of object is in question.) However this may be, there is certainly a one–one correspondence of complexes and facts, and for our present purposes we shall assume that they are identical.18

By asserting that the correspondence between complex and fact is one–one, Russell demonstrated his continuing failure to adopt Wittgenstein’s conception: he was still using the word ‘fact’ for what Wittgenstein now called a complex, namely what there is when a judgment is true, whereas Wittgenstein used it for what is the case if the judgment is true. Indeed the passage 17

24 Apr. 1913.

18

ToK, 79–80.



Complex and fact

just quoted reads rather as if Wittgenstein had already pressed on Russell a distinction between complex and fact, but Russell had not yet really grasped what it was. One reason Russell did not immediately grasp the distinction was no doubt that he had no need of it in order to make the point about structure. That is to say, he could, if he wished, construct structured entities having the unique parsing into constituents that was the main structural feature of Wittgensteinian facts. One point of interpretation is perhaps worth noting here. Since Russell was slow to grasp Wittgenstein’s distinction between complex and fact, and since the distinction is never explicitly explained in the Notes, one might wonder how reliable a guide Russell’s translation of these notes is to Wittgenstein’s intentions. Can we be sure, in other words, that ‘fact’ here really does mean fact and not complex? I think that we can. Both words occur in the Cambridge notes, dictated in English; both occur, too, in the Birmingham notes as translated by Russell. Moreover, the usage throughout is consistent with the distinction in the form in which Wittgenstein later made it explicit. This would be an odd coincidence unless Russell was consistently translating Tatsache as ‘fact’ and Komplex as ‘complex’. It is worth mentioning, too, that however slow on the uptake he may have been, Russell did eventually embrace Wittgenstein’s view, at least to the extent of adopting his terminology: his logic lectures at Harvard in 1914, which began by saying that what we analyse is a fact, not a thing, refer repeatedly thereafter to facts, not complexes; in the Lowell Lectures, meanwhile, he defined a fact as ‘that a certain thing has a certain quality, or that certain things have a certain relation’;19 and again in 1918, still before he had read the Tractatus, he used the language of facts in claiming, for instance, that he did not believe in complex entities of the kind for which we habitually use proper names.20 19

OKEW, 51.

20

CP, VIII, 170.

Chapter 

Forms At the centre of Wittgenstein’s project was the task of explaining the unity of the proposition. We have already noted how in his letter of January 1913 he began to move away from an analysis of propositions as complexes united by a copula—an analysis which, if not actually Russell’s, was certainly Russellian in spirit and therefore wholly merited Wittgenstein’s description of it as our theory of symbolism. In the letter he adopted the Fregean idea that the unity is to be explained not by a pure copula but by one of the meaning-contributing components of the proposition. However, the way in which he implemented this Fregean idea was still Russellian. As a result, it is hardly a position that could have satisfied him for long. . Form as name As we saw in chapter 8, Wittgenstein proposed in his January 1913 letter to analyse ‘Socrates is human’ into two parts, of such different kinds that it is impossible to reassemble them the wrong way round: the first is a name and the second what we have been calling a form. I contrasted this proposal with Wittgenstein’s earlier view that the proposition contains two names ‘Socrates’ and ‘humanity’, combined by means of what he called a copula. What I did not discuss then was what the kind of entity which is responsible for the unity of the proposition (whether it is a form or a copula) is supposed to be. What remained constant between the old theory and the new was precisely his conception of the sort of thing this entity is. In the old theory he analysed the copula of a subject-predicate proposition as ( x,y)#1 ( x,y); or, in words, as ‘Something is predicated of something’. In the new theory he analysed the form of ‘Socrates is human’ as ( x)φ x; or, in words, ‘Something is human’. And similarly he now analysed the form of aRb as ( x,y)R( x,y).1 I suggested in §8.1 that Frege’s influence can be detected in Wittgenstein’s move from copula to form. The nature of the form was to be sufficient in itself to explain why ‘Socrates is mortal’ makes sense but ‘Mortality is Socrates’ does not. This was, I suggested, Wittgenstein’s attempt to implement Frege’s E

E

E

1

LW to BR, 16 Jan. 1913.



Forms

distinction between saturated and unsaturated expressions. But by maintaining his old analysis of the form, Wittgenstein failed to implement his Fregean insight convincingly. He was still struggling, it seems, to shoehorn the Fregean distinction into what was still at base a Russellian conception of propositions as complexes. Wittgenstein was no doubt right to insist in the letter that the component of an atomic proposition which plays the role of the unsaturated expression must be simple, since if it is not, we shall then have to analyse it and a regress quickly threatens. But his proposal that this simple component is ‘something is human’ is hard to take seriously, since this is so obviously complex. Indeed, one might well wonder whether the idea was really Wittgenstein’s at all. Since this part of the account—the idea that the verb of the proposition should be analysed as existential—is present in both the old theory Wittgenstein was rejecting and the new one he was now recommending, it evidently predates the letter. Moreover, although it is surely an odd idea, Wittgenstein does not pause to explain or justify it, which suggests strongly that Russell was already familiar with it. And indeed when Russell, writing his Theory of Knowledge manuscript a few months later, needed something to play the role of the form of a relation, he used just the same device.2 If it was part of what Wittgenstein was still inclined to call ‘our theory of symbolism’3 —part, that is to say, of a collaborative enterprise that he and Russell were conducting jointly—it is surely quite plausible that it was Russell’s idea in the first place. By the time Russell used the device in his Theory of Knowledge manuscript, however, Wittgenstein had probably already abandoned it. The passage in which we see him doing so occurs early in MS1 of the Birmingham Notes. It is easy to suppose that only such symbols are complexes as contain names of objects, and that accordingly ‘( x,φ )φ x’ or ‘( x,R,y) xRy’ must be simple. It is then natural to call the first of these the name of a form, the second the name of a relation. But in that case what is the meaning of (e.g.) ‘∼( x,y) xRy’? Can we put ‘not’ before a name?4 E

E

E

The answer Wittgenstein intended us to give to this last rhetorical question is of course ‘no’. The inference he drew, therefore, was that ‘( x,R,y) xRy’ is not simple and hence not a component of ‘aRb’. This argument against attributing the unity of the proposition to a component of existential form is surely convincing. It may therefore seem surprising to discover him returning to the very same point in the wartime Notebooks. E

I thought that the possibility of the truth of the proposition φ a was tied up with the fact ( x,φ ) φ x. But it is impossible to see why φ a should only be possible if there is another proposition of the same form. φ a surely does not need any precedent. (For suppose that there existed only the two elementary propositions ‘φ a’ and ‘ψ a’; and that ‘φ a’ were false: Why should this proposition only make sense if ‘ψ a’ is true?)5 E

2

ToK, 114.

3

To BR, 26 Dec. 1912.

4

B5.

5

NB, 23 Oct. 1914.

Form as function  There is no need, though, to draw from this the conclusion that Wittgenstein had become dissatisfied with his earlier argument against treating the form of a proposition as existential. It is one of the features of his way of working that he quite often returned some time later to issues that he had apparently settled. Sometimes this seems to have been because he had come to doubt his earlier arguments; sometimes, perhaps, it was just because he had forgotten what his earlier arguments were; quite often, though, he knew what they were but wanted to think the matter through again in any case. I mention this here because interpreters of Wittgenstein have sometimes used his later discussions of an issue to argue that he cannot have been clear about the issue earlier. There will no doubt be occasions when that is so, but it is a harder point to prove against Wittgenstein than against many other philosophers, because his method of working seems so often to have involved him in thinking through a problem again and again, reaching the same conclusion each time by a slightly different route. He once enjoined himself, ‘Don’t worry about what you have already written. Just keep on beginning to think afresh as if nothing at all had happened yet.’6 And the Notebooks provide ample evidence that he obeyed his own injunction.

. Form as function If we reject the idea that the third component of ‘aRb’ is ‘( x,y) xRy’, that does not yet point us to what else it might be. The next step was for Wittgenstein to see that the answer must be radically different: the third component cannot just be another name. Propositions, he said, ‘cannot consist of names alone; they cannot be classes of names’.7 Unusually, he repeated this point later in the Birmingham Notes, this time adding helpfully, ‘This is easily shown.’8 And indeed it is easily shown. For a class of names offers no prospect of supplying the resources to explain the unity of the proposition. Wittgenstein also thought this observation worth including in the Tractatus.9 When we read it in this context, though, it is apt to engender puzzlement. Our immediate response to being told that a proposition is not a class of names is to reply, ‘Yes, but whoever said it was?’ Not Frege, certainly: for him a proposition is a name, not a class of names. Russell’s position is a little more complicated, since by 1913 he held that propositions are incomplete symbols. But this does not make them classes of names; and before he abandoned propositions, he had consistently held that any proposition must contain at least one verb. The answer to this puzzle is surely that it was Wittgenstein himself who had, albeit temporarily, held in effect that a proposition is a class of names. E

6

NB, 15 Nov. 1914.

7

B1.

8

B47.

9

3.142.



Forms

Abandoning this view led him to require that the third element in ‘aRb’, the element he called its form, must be something radically different from a name. But if it is not a name, what is this third component of ‘aRb’? A casual account of Frege’s view might be that the third component is the incomplete expression ‘ xRy’. But this is casual. The expression ‘ xRy’ is plainly not a component of ‘aRb’ in any ordinary sense of the word ‘component’, if only for the trivial reason that the letters ‘ x’ and ‘y’ occur in the former but not in the latter. Frege’s rather less casual answer was to think of the incomplete element in the expression of a proposition as what Geach has called a linguistic function, i.e. a function taking linguistic expressions as arguments and having a linguistic expression as its value. That, at any rate, is surely10 the natural way to read Frege’s explanation of the notion of a function in the Begriffsschrift. If in an expression . . . a simple or complex sign occurs in one or more places and if we regard that sign as replaceable in all or some of these occurrences by something else (but everywhere by the same thing), then we call the part that remains invariant in the expression a function, and the replaceable part the argument of the function.11

Thus, for example, the sentence ‘Socrates is mortal’ is the result of applying to ‘Socrates’ the function which for any name α as argument has !α is mortal" as its value; and ‘aRb’ is the result of applying to ‘a’ and ‘b’ the function whose value for the arguments α and β is !αRβ ".12 The difficulty with this, however, is that we may doubt whether it is any longer something that can appropriately be called an analysis of the ‘structure’ of the proposition. That an object is a value of a function for some argument is not generally thought of as contributing to an analysis of the object. (To observe that 4 is the cube root of 64 does not contribute to an analysis of the number 4, for example.) This has led Dummett to claim that ‘to talk of expressions and their structure, we need the notions of part and whole, not those of function and value’.13 The problem with the function–argument analysis, that is to say, is that given only the value of a function for some argument, it is not possible to recover the function or the argument. For this reason it is inappropriate to regard either the argument or the function as a constituent or part of the value, since we naturally suppose that anything is uniquely analysable into its ultimate constituents, and that the parts of a thing may be discerned by scrutiny of it.14

Scrutiny of the proposition ‘Socrates is mortal’ cannot reveal the function whose value for the argument ‘Socrates’ it is, because there is any number of such functions. So if the purpose of analysis is to reveal the constituents of a thing and the manner of their combination, Frege’s application of the notion of a function cannot be relevant to the analysis of the proposition. 10

Contra Baker and Hacker, Logical Excavations, 172n. 11 Frege, Bs, §9. functional model of sentential complexity’. 13 ‘An unsuccessful dig’, 396.

12 14

See Sullivan, ‘The Ibid.

The form of a fact  . The form of a fact Wittgenstein’s breakthrough came when he saw that he could respect the essence of Frege’s point by saying that what symbolizes in ‘aRb’ is not the sign ‘R’ on its own but the fact that it occurs between the names ‘a’ and ‘b’. From this point on, therefore, he maintained that a proposition is a fact, not a complex. When it was applied at the level of symbolism, the distinction between fact and complex thus allowed him to refine the Fregean insight that what makes a proposition expressive is the structure it has. Frege, let us recall, had conceived of the expressions for concepts and relations as what he called unsaturated, i.e. as containing argument places which have to be completed with names of objects in order to form a complete sentence. What corresponds in Wittgenstein’s conception to a saturated expression is what he in 1913 calls a ‘name’; what corresponds to an unsaturated expression is what he calls a ‘form’. What is expressive is not the complex consisting of the names and form but the fact that they are combined in a certain manner. Not: ‘The complex sign “aRb” ’ says that a stands in the relation R to b; but that ‘a’ stands in a certain relation to ‘b’ says that aRb.15

Or, as he glossed it in the Cambridge Notes, In aRb it is not the complex that symbolises but the fact that the symbol a stands in a certain relation to the symbol b. Thus facts are symbolised by facts, or more correctly: that a certain thing is the case in the symbol says that a certain thing is the case in the world.16

In a relational proposition involving the names ‘a’ and ‘b’, what expresses the relation is not one identifiable component of the proposition on its own; rather is it the fact that the names ‘a’ and ‘b’ are related in some particular way. In the case of a proposition written ‘aRb’, therefore, the sign ‘R’ does not express anything on its own; it does so only when combined with the names ‘a’ and ‘b’ in a certain way. What is expressive is not the complex consisting of the three signs, but a fact about this complex, namely that in it the sign ‘R’ occurs with something to the left of it and something else to the right of it. The sign ‘R’ functions only as a label to distinguish this relationship between the names ‘a’ and ‘b’ from other possible relationships (such as the one exemplified in ‘aLb’, for instance). Borrowing our terminology from Long,17 who in turn borrowed it from the Tractatus,18 let us call signs used as labels in this fashion indices. The correlation between an index and what it labels is a matter of convention. Indeed, the fact that there is an index at all is conventional. What is required in order to express that aRb is only that the names ‘a’ and ‘b’ should stand in a certain relationship. In the present case, of course, the relationship is, as it happens, that of having the index ‘R’ standing between them, but there is 15 18

B57; cf. 3.1432. 5.02.

16

C44.

17

‘Are predicates and relational expressions incomplete?’, 93.



Forms

nothing essential about this. It would be perfectly possible to have a language in which some relations are expressed not with the aid of indices but rather by other devices such as spatial relationships between the signs. This last sort of notation is indeed sometimes used for functions in mathematics: what does the symbolic work in the exponential expression xy , for instance, is not that ‘ x’ and ‘y’ have an index standing between them but that they are spatially related in a certain way. In cases where there is an index, of course, mathematicians adopt the convention of using the index to refer to the function, and for just that reason they find cases where there is no index, such as that of the exponential function, awkward. They are reduced to the device of setting exp( x,y) = xy , so that they have an index ‘exp’ which they can then use to refer to the function. Wittgenstein’s conception is that the form of a proposition is the symbolizing relationship between the names in the proposition which makes it the case that the proposition says what it does. The point of this conception is in a sense the mirror image of the point of conceiving of the world as made up of facts, not complexes. In order to make judgments about the world, what we must perceive are facts, not complexes; and the symbols that express those judgments, likewise, must be facts, not complexes.

. The unity of the proposition This is Wittgenstein’s response to the problem of the unity of the proposition. Russell’s theory had been that a proposition is a sort of complex entity containing (at least in some cases) the things it is about. But this conception faced a difficulty when it came to explaining what it is that makes it the sort of complex that can be true or false. The elements of a complex do not become a proposition on their own, so we are naturally tempted to posit something else—propositionally expressive glue, so to speak—which combines the elements in the complex in such a way as to make them expressive. But this is only to postpone the problem, not to solve it, since we are now owed a corresponding explanation of the expressiveness of a new proposition, namely the one which says that the original complex is expressive. This is known as ‘Bradley’s regress’, and versions of it dogged Russell’s various attempts to explain the expressive unity of the proposition. Frege, on the other hand, avoided the difficulty of Bradley’s regress by supposing that what makes a proposition expressive is the unsaturatedness of the predicate or relation symbol it contains. He analysed the sentence ‘aRb’, for example, as formed by substituting the names ‘a’ and ‘b’ into the argument places in the expression ‘ xRy’. Writing the expression for a relation as ‘ xRy’ rather than just ‘R’ is a way of reminding us that it is unsaturated, requiring the addition of two names if it is to form a complete expression (a sentence).

The unity of the proposition  What the expression ‘ xRy’ represents, therefore, is a certain sort of function— a function which with the names ‘a’ and ‘b’ as arguments yields the sentence ‘aRb’ as its value. Frege did not need to posit a further element, the ‘copula’ of traditional logic, gluing ‘a’, ‘b’, and ‘R’ together to form ‘aRb’, because the relational expression ‘ xRy’ already contains, through its unsaturatedness, the requisite glue. But how does this explain the expressiveness of ‘aRb’? Frege’s view was that the unsaturated relational expression ‘ xRy’ has corresponding to it something which, although not an object, is nevertheless objective: he called it a relation. The relation possesses an unsaturatedness analogous to that of the relation symbol. So if it is determined what relation ‘ xRy’ expresses and what objects ‘a’ and ‘b’ refer to, no further resources are needed to determine the thought expressed by the sentence ‘aRb’. This cannot yet amount to an explanation of the unity of the proposition, however, because unsaturatedness is a feature of function-signs such as ‘ x + y’ just as much as of relational expressions such as ‘ xRy’. Wittgenstein’s conception of propositions as facts seeks to capture what is right about Frege’s conception of unsaturatedness, while also explaining Frege’s other insight, blurred by Frege’s faulty assimilation of sentences to names, as to the essentially articulate nature of propositions. Wittgenstein’s idea is not only that the verb in a sentence has an essentially different nature from the subject, but that this different nature is such that it on its own explains the unity of the proposition. The two kinds of components of a proposition are fundamentally different, as Frege’s two kinds—saturated and unsaturated expressions—had been. The proposition ‘aRb’, for example, contains three components. Two of them are the names ‘a’ and ‘b’. The third component is not the index ‘R’ but the form of the fact which constitutes the proposition. ‘There is no thing which is the form of a proposition.’19 In Fregean terms, the unsaturatedness of what ‘ x is human’ expresses suffices to ensure that when it is combined with ‘Socrates’, what results is a proposition: no glue is needed. But the form of the proposition is not a kind of function, because what results when names are inserted into its argument-places is not a complex name, as it would be in the case of a function, but a symbolizing fact. Russell himself was certainly aware of the problem of propositional unity which Wittgenstein was addressing. He put the point succinctly in the notes he made in October 1912 for a projected article, ‘What is logic?’ In a complex, there must be something, which we may call the form, which is not a constituent, but the way the constituents are put together. If we made this a constituent, it would have to be somehow related to the other constituents, and the way in which it was related would really be the form; hence an endless regress. Thus the form is not a constituent.20 19

B48.

20

CP, VI, 55.



Forms

If the form is not a constituent, what is it? To say, as Russell had done, that it is a component of the proposition but not a constituent, is so far only to label the problem, not to solve it. But if the idea that a proposition is not a complex but a fact is Wittgenstein’s solution to the problem, Russell did not immediately recognize it as such. He soon abandoned ‘What is logic?’, telling Ottoline that he would leave this sort of thing to Wittgenstein; and in the Theory of Knowledge manuscript, as we have noted already, he indicated that he was vaguely aware of a distinction between complex and fact, but evidently did not think it terribly important. Presumably it took the Notes on Logic to make him recognize that the distinction might play a role in solving Bradley’s problem. ‘Chiefly through the work of an Austrian pupil of mine,’ he told Bradley, ‘I seem now to see answers about unities; but the subject is so difficult and fundamental that I still hesitate.’21 . The symbolic turn again The idea that we can infer features of the structure of reality from the structure of the symbols we use to represent it is what I have been calling the symbolic turn. From this we obtain the idea that the structure of an elementary proposition corresponds precisely to the structure of what it expresses. Every proposition which says something indefinable about a thing is a subject–predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate proposition, and so on. An indefinable simple symbol can only be a name, and therefore we can know, by the symbol of an atomic proposition, whether it is a subject-predicate proposition.22

Notice, though, that Wittgenstein claims this precise correspondence only for the case of elementary propositions (propositions which ‘say something indefinable’). The reason is presumably that his conception of the relationship between symbolism and the world allowed him to infer from the simplicity of a name to the simplicity of the object it refers to, but not yet in the opposite direction. If, for instance, there is just one thing on the table, ‘the thing on the table’ is a complex name of something simple. For a similar reason, although on this conception whatever a simple name refers to is an object, we need a further argument if we are to hold also that no name, simple or complex, can refer to something unsaturated. For Wittgenstein, though, this is not something to be argued for so much as a guiding philosophical principle: it relies on the harmony between the symbolism and what it expresses that I have called his symbolic turn. In this instance, though, 21

30 Jan. 1914 (RA1.710.047583H).

22

B77.

The symbolic turn again  it is also something that Frege maintained: on his view, whatever refers to something unsaturated is itself unsaturated. Expressed in Russell’s language, this amounts to saying that noun and verb are intrinsically different grammatical categories. The unsaturated expression ‘ x is mortal’ and the saturated expression ‘mortality’ cannot, contrary to what Russell had claimed, refer to the same thing. When Wittgenstein claims that there is ‘no name which is the name of a form’,23 that is just the analogue in his terminology of Frege’s claim that no saturated expression can refer to something unsaturated. Wittgenstein’s claim that ‘no name is the name of a form’ is thus related to Frege’s discussion of the concept horse. The phrase ‘the concept horse’ is saturated and therefore what it refers to is saturated. But concepts are unsaturated. The conclusion Frege drew was, notoriously, that despite appearances what the phrase ‘the concept horse’ refers to is an object, not a concept. Frege was not as troubled by this conclusion as he might have been. ‘Here,’ he was content to say, ‘we are confronted with an awkwardness of language’24 which puts ‘a quite peculiar obstacle in the way of understanding’. By a kind of necessity of language, my expressions, taken literally, sometimes miss my thought; I mention an object, when what I intend is a concept. I fully realize that in such cases I was relying upon a reader who would be ready to meet me halfway—who does not begrudge a pinch of salt.25

Wittgenstein—plainly a man to begrudge a pinch of philosophical salt whenever he could—was not the sort of reader Frege could rely on. In the latter parts of the Birmingham Notes we see him gradually enlarging the range of expressions which he recognizes as missing their targets in this way. In the Tractatus, of course, this is expanded into a central theme, but in the Notes the point is made piecemeal. What is lacking at this stage, that is to say, is any sign that he had yet begun to think through in any generality the consequences of the idea that there are attempts at expressing things which inevitably misfire. 23

B48.

24

‘Über Begriff und Gegenstand’, 196.

25

Ibid. 204.

Chapter 

Russell’s theory of judgment In June 1913 Wittgenstein’s objection to Russell’s multiple relation theory of judgment led Russell to give up writing his book on Theory of knowledge. It is one of the most famous incidents in their interaction, and has sometimes been presented as a turning point in their relationship, when the roles of master and pupil were reversed. The truth, perhaps, is more complicated. Russell’s attitude to Wittgenstein’s views had already become one of deference some months earlier: in February 1913, for instance, Pinsent noted that Russell ‘acquiesced in what he said without a murmur’;1 and as early as June 1912 Russell told Ottoline, ‘I feel he will do the work I should do, and do it better. He starts fresh at a point which I only reached when my intellectual spring was nearly exhausted.’2 If one insists on identifying a moment when the son’s battle to wrench psychological control from the father was won, it was probably not in June 1913 but the previous October: for that was when Russell abandoned his project of writing an article with the title ‘What is logic?’, telling Ottoline that Wittgenstein’s criticisms disturbed him profoundly3 and that he would in future leave this sort of thing to Wittgenstein.4 Nor was Wittgenstein’s objection responsible on its own for Russell’s turn away from philosophy: that, as we shall see later, was brought about by an accumulation of factors, not one catalytic incident. But the objection has piqued the interest of philosophers for reasons that are not simply biographical. Offering an exegesis of Wittgenstein’s objection seems to have become a sort of rite of passage for scholars of early analytic philosophy.5 Perhaps this is because the clues that are tantalizingly scattered through the surviving documents have been thought to stop just short of telling us quite what Wittgenstein’s objection was. . The original multiple relation theory We have seen that when he wrote the Principles Russell thought of propositions as complexes which actually contain parts of the physical world. ‘A 1 4 Feb. 1913. 2 1 June 1912. 3 13 Oct. 1912. 4 14 Oct. 1912. 5 E.g. Griffin, ‘Wittgenstein’s criticism of Russell’s theory of judgment’; Wahl, ‘Bertrand Russell’s theory of knowledge’; Landini, ‘A new interpretation of Russell’s multiple-relation theory of judgment’; Weiss, ‘On the demise of Russell’s multiple relations theory of judgement’; Stevens, ‘Re-examining Russell’s paralysis’.

The original multiple relation theory  proposition, unless it happens to be linguistic, does not itself contain words: it contains the entities indicated by words.’6 So the snow-covered peak itself is a constituent of the proposition that Mont Blanc is the tallest mountain in Europe. In a case such as this, in which the proposition is true, it is hard to see any difference between the proposition, as Russell conceived of it, and the fact. We have, in other words, an identity theory of truth according to which a true proposition just is the fact that makes it true. But in that case, what about false propositions? The proposition that Charles I died in bed cannot be identified with the fact of his doing so, because there is no such fact. So we might try instead to posit a complex made up from Charles I, dying, and his bed: this complex—let us call it a ‘fiction’—is like a fact except that it happens not to exist. Truth and falsity are thus properties that objective propositions may have: true propositions are facts; false ones are fictions. ‘When a proposition happens to be true,’ Russell said in the Principles, ‘it has a further quality, over and above that which it shares with false propositions.’7 But what is this ‘further quality’? Or, to put it another way, what is the difference between fact and fiction? In the Principles Russell says merely that this question ‘belongs no more to the principles of mathematics than to the principles of everything else’8 and can therefore be left to the logicians. In truth, though, the question he was there so patently dodging was a fundamental difficulty for his 1903 conception of propositions, as he gradually came to realize. The key insight came with ‘On denoting’, when Russell realized that the apparent references to mysterious fictional entities that abound in ordinary language are (sometimes, at least) a by-product of misleading grammatical forms that can be eliminated on analysis. As a result of this analysis we are freed from supposing that the King of France somehow has his being in a shadowy realm of non-existent entities. Applied to the present question, that insight amounts to this: it might be that the term apparently denoting the proposition vanishes from the correct analysis of statements attributing propositional attitudes. If so, we shall be freed from assuming the subsistence of objective falsehoods just as we were freed by the earlier analysis from assuming the subsistence of a non-existent king of France. Yet if this was the key insight, Russell was slow to realize it. Even after the revelation that came to him in the summer of 1905, Russell did not straightaway apply the device of incomplete symbols to propositions so as to dismiss false propositions from his ontology. In 1906 he was willing to concede only that ‘the view which denies objective falsehoods is, on the face of it, more plausible’, and still thought that ‘the difficulties in its way are formidable’.9 It was not until 1910 that he rejected objective falsehoods unequivocally, on the 6

Principles, §51.

7

Ibid. §52.

8

Ibid.

9

‘On the nature of truth’, 249.



Russell’s theory of judgment

ground that positing them ‘leaves the difference between truth and falsehood quite inexplicable’.10 From now on Russell tried to analyse a propositional attitude ascription as relating the bearer of the attitude not to the erstwhile proposition but to its components. Judgment is not a dual relation of the mind to a single Objective, but a multiple relation of the mind to the various other terms with which the judgment is concerned. Thus if I judge that A loves B, that is not a relation of me to ‘A’s love for B’, but a relation between me and A and love and B.11

Of course, Russell is not here proposing an analysis of the proposition ‘I judge that A loves B’, for any such analysis would only push the problem back one stage. Rather is he proposing to analyse only the fact, if it is a fact, that I make the judgment; the problem of false propositions does not arise for such a fact, because in the case in which I make no judgment there is no fact and hence nothing to analyse. The central feature of Russell’s theory is that the proposition that is judged disappears from the analysis of judgment. The ascriptions of other propositional attitudes (belief, disbelief, understanding, etc.) will be analysed similarly but with different relations in place of judging. It does not follow, though, that propositions disappear entirely: for theoretical purposes it is possible to reconstruct them as long as we accept that they have a rather different form from what we had previously supposed. Russell could now offer as a surrogate for the proposition aRb the fact that ( x,φ )φ ( x,a,R,b): what it says is that someone bears some propositional attitude relation to the terms a, R, and b.12 Propositions therefore become subjective in the sense that if no one bears any attitude to them at all, they do not exist even in this extenuated sense. Russell believed that his theory solved the problem of false propositions: it explained how I can judge something even though what I judge is false, because in judging that aRb I do not bear any relation to the proposition (which has a wholly different structure) but only to its constituents a, R, and b. E

. A problem for the original theory During the early summer of 1913 Russell was writing a manuscript on the Theory of Knowledge in which the multiple relation theory of judgment was to play a central role. Epistemology had for some time held a central place in Russell’s conception of philosophy. ‘There is one great question’, he had told Ottoline in 1911. ‘Can human beings know anything, and if so, what and how?’13 Nonetheless, the very fact that he was writing about epistemology at quite this point probably owes something to Wittgenstein’s presence at his shoulder. He 10

CP, VI, 119.

11

CP, VI, 122.

12

Cf. ToK, 115.

13

13 Dec. 1911.

A problem for the original theory  had recently put aside two incomplete projects, the work on matter which we discussed in chapter 3 and the essay on the nature of logic which he had decided to leave to Wittgenstein. He even delayed telling Wittgenstein about the epistemology book until he was eighty pages into writing it,14 apparently because he was afraid of his criticisms. When he did eventually tell Wittgenstein, his fear proved well founded. Shortly before he got to the part of the book which was to deal with the multiple relation theory, Wittgenstein offered him an objection to it. Wittgenstein came to see me last night with a refutation of the theory of judgment which I used to hold. He was right, but I think the correction required is not very serious. I shall have to make up my mind within a week, as I shall soon reach judgment.15

And when Russell came to the section on judgment about five days later, he did indeed reject the 1910 theory which I described in the last section (the theory which he ‘used to hold’). The reason he gave was as follows. In an actual complex, the general form is not presupposed; but when we are concerned with a proposition which may be false, and where, therefore, the actual complex is not given, we have only, as it were, the ‘idea’ or ‘suggestion’ of the terms being united in such a complex; and this, evidently, requires that the general form of the merely supposed complex should be given. More simply, in order to understand ‘A and B are similar’, we must know what is supposed to be done with A and B and similarity, i.e. what it is for two terms to have a relation; that is, we must understand the form of the complex which must exist if the proposition is true.16

It is natural to wonder whether Russell is here reproducing Wittgenstein’s objection. The language he uses in reporting it is admittedly not Wittgensteinian: he expresses the point as an epistemological one concerning our understanding of the complex rather than, as Wittgenstein would surely have done, as a grammatical point about the words with which the judgment is expressed. Perhaps, though, Russell’s mode of expression owes something to his present purpose, which is to direct us towards his new theory (which, as we shall see shortly, Wittgenstein liked no better) rather than to convince us of the error of his old one. It is true, too, that Russell does not explicitly attribute the objection to Wittgenstein, but this need not in itself be especially significant. In some of his later work Russell was punctilious about crediting Wittgenstein whenever he reported his ideas in print, but only a few days after writing the passage just quoted we find him commenting to Ottoline on ‘the difficulty of not stealing his ideas—there is really more merit in raising a good problem than in solving it’.17 It is tempting to hear in this remark the voice of bad conscience, a recognition by Russell that he had just been guilty of stealing something and should therefore resolve to avoid that danger in the future. The most striking 14

To OM, 14 May 1913.

15

BR to OM, 21 May 1913.

16

ToK, 116.

17

1 June 1913.



Russell’s theory of judgment

feature of Russell’s account, though, is its curiously hesitant tone. Having put the point several times in different ways, Russell concludes lamely, ‘I do not know how to make this point more evident, and I must therefore leave it to the reader’s inspection, in hopes that he will arrive at the same conclusion.’18 This inclines me to think that the objection Russell raises against his 1910 theory is at base Wittgenstein’s. Russell represents the objection as being one that applies only in the case where the judgment is false—where, in other words, there is no complex consisting of the relata combined as the proposition says—but the problem is recognizably derived from the one that in chapter 8 we saw Wittgenstein using against the theory that aRb should be analysed as #2 (a,R,b). In order to understand ‘A and B are similar’, we must know what is supposed to be done with A, B, and similarity. We must know, Russell might have said, what are the types of A, B, and similarity. In the case where A and B are indeed similar, Russell supposed that we can read these types off from the fact of their similarity when we perceive the complex; what he now conceded was that when there is no such fact, we cannot. Wittgenstein’s point was thus that the types of A, B, and similarity ought somehow to be deducible from the fact that I judge that A and B are similar, whether or not they are similar. As he put it in the Notes on Logic, ‘Every right theory of judgment must make it impossible for me to judge that this table penholders the book. Russell’s theory does not satisfy this requirement.’19 Russell’s theory did not satisfy the requirement because in it all the components of what is judged, including the verb, occurred as terms. Russell therefore did not have the resources to distinguish, in what is judged, between terms (such as ‘similarity’) which derive from verbs and those (such as ‘penholders’) which do not. . Russell’s revised theory What Russell did in the Theory of Knowledge manuscript was to amend his theory so that the judging relation has a further argument place in which the form of what is judged occurs. If A judges that Socrates is mortal, that is to be analysed as J(A, #1 , Socrates, mortality), where #1 is the subject-predicate form. If A judges that John loves Mary, that judgment is to be analysed as J(A, #2 , John, Mary, loving), where #2 is the form of relational judgments. And so on. Now if the form of the proposition was to occur as a term in the judging relation, Russell needed to make a specific proposal about what entity that 18

ToK, 116.

19

B33.

Russell’s revised theory  is. The proposal he made was in essence the one that Wittgenstein had been using in January 1913. Wittgenstein had then taken the form of a subjectpredicate proposition ‘Socrates is mortal’ to be ‘something is mortal’. Since Russell was treating mortality as a second constituent, the same device led him to take the form as ‘someone has some property’. And in the same way, the form of a relational proposition such as ‘John loves Mary’ would now be ‘someone bears some relation to someone’. I have presented Russell’s 1913 theory as a response to the objection Wittgenstein presents in the Birmingham Notes. Since we cannot be sure when Wittgenstein formulated that objection, the question arises whether we can be sure that Wittgenstein intended it to apply primarily to the 1910 theory rather than to the 1913 modification. One reason to think he did is that he included a variant of the same objection in the Tractatus. ‘The correct explanation of the form of the proposition “A judges p” must show that it is impossible to judge a nonsense. (Russell’s theory does not satisfy this condition.)’20 In this formulation, at least, Wittgenstein must have intended the objection to apply primarily to Russell’s 1910 theory. For Russell had given up his 1913 theory without publishing it, and Wittgenstein knew this. (Russell famously told him that the objection had paralysed him.) It would be a little odd, even by his somewhat idiosyncratic standards, if Wittgenstein had chosen to include in the Tractatus an objection to a theory of Russell’s that no one except him had seen and which at the time he was composing the Tractatus he believed Russell to have long ago abandoned. The lectures on logic which Russell gave at Harvard in 1914 confirm this. Russell there presented his ‘old theory of judgment’, analysing the fact that S judges that x has R to y as a relation between S , x, R, and y, before commenting, in what is plainly an allusion to Wittgenstein’s criticism, that ‘if R was a thing you could substitute another thing (z) for it, and if you do, the j[udgment] is meaningless’.21 If Wittgenstein’s criticism was directed at the version of the theory in which the form of what is judged does not occur as a term, it does not follow, of course, that Russell’s original purpose in adding the form was to circumvent this difficulty. It may be, for instance, that Russell had already decided to include the form for reasons connected with the so-called ‘direction problem’ and his correspondence with Stout. Nonetheless, he evidently thought at first that including the form would meet Wittgenstein’s objection. And it is fairly easy to see why he might have thought this: he hoped that the form of what is judged could have encoded in it the typing information that is needed to rule out nonsense; the form of all relational propositions, for instance, would in effect stipulate that the middle term must be a verbal noun (such as ‘similarity’), not an ordinary noun (such as ‘penholders’). 20

5.5422.

21

Eliot’s notes, 9 Apr. 1914.



Russell’s theory of judgment

. Wittgenstein’s further objection A couple of days after Russell had written the section of his manuscript expounding the revised theory, the following incident took place. Wittgenstein came to see me—we were both cross from the heat—I showed him a crucial part of what I have been writing. He said it was all wrong, not realizing the difficulties—that he had tried my view and knew it wouldn’t work. I couldn’t understand his objection—in fact he was very inarticulate—but I feel in my bones that he must be right, and that he has seen something I have missed. If I could see it too I shouldn’t mind, but as it is, it is worrying, and has rather destroyed the pleasure in my writing.22

There can be little doubt that what Russell showed to Wittgenstein was somehow related to the new version of the theory of judgment. But what was wrong now? The first issue that arises in answering this is whether it was the general proposal of treating the form as a term in the judging relation that Wittgenstein objected to or Russell’s specific proposal as to the identity of that form. We have seen already that the specific proposal was indeed one that Wittgenstein had tried and rejected, because it would have the absurd consequence that we can put a negation sign before a name. And perhaps some of his inarticulate comments were indeed directed at explaining this difficulty. However, I think we can rule out the idea that this was the whole of Wittgenstein’s objection. If it had been, after all, he could simply have suggested to Russell that he replace his faulty conception of the form of the proposition with Wittgenstein’s own. It seems much more likely, therefore, that Wittgenstein’s objection was to any proposal according to which the form of the proposition, however it is conceived, occurs as a term in the judgment relation. If that is right, then the objection will have been in essence the one he had already offered, namely that a theory of judgment should make it impossible to judge that the table penholders the book. For Wittgenstein had indeed tried and rejected something very like the view Russell now proposed. If I analyse the proposition Socrates is mortal into Socrates, Mortality and ( x,y)#1 ( x,y) I want a theory of types to tell me that ‘mortality is Socrates’ is nonsensical, because if I treat ‘Mortality’ as a proper name (as I did) there is nothing to prevent me to make the substitution the wrong way round.23 E

Wittgenstein thought, that is to say, that the presence of the form as a further term in the judging relation did not meet his objection: it still did not make it impossible to judge nonsense unless we simply add a stipulation according to which ‘mortality’ is of a different type from ‘Socrates’. Russell ends the chapter of the manuscript on the understanding of propositions by drawing a map showing various connections between the elements 22

BR to OM, 27 May 1913.

23

To BR, 16 Jan. 1913.

Acquaintance  of the judging relation. Perhaps it is of significance, therefore, that when Russell tried some years later to recollect Wittgenstein’s objection, he expressed it by saying that you cannot make what I should call a map-in-space of a belief. . . You cannot get in space any occurrence which is logically of the same form as belief. . . The discovery of this fact is due to Mr Wittgenstein.24

The point is that the map at once shows the futility of supposing that we can add the form of what is judged as a term in the relation. By drawing a map—a spatial complex of a certain sort—Russell showed that he conceived of the fact that I judge something as having the same structure as this spatial complex. But the entities that are related in such a complex are things, not forms. Any attempt to include the form as a term in such a relation will inevitably fail: by making it a term of the relation we turn it into a name and therefore make it refer, if it refers at all, not to the form of the putative proposition but to something else. Judgment cannot be what Wittgenstein calls ‘a relation in the ordinary sense’, because a relation in the ordinary sense has objects as its constituents, whereas one of the constituents of a judgment is the verb of the proposition judged, which is not an object.

. Acquaintance It is futile, then, to include the form as a term in the judgment if judging is a relation in the ordinary sense, i.e. one with the same structure as a map. But for all we have said so far, the possibility remains open that judging may be a higher-order relation in which the form does genuinely occur as a form. Wittgenstein needed a further argument to eliminate this possibility. There is no thing which is the form of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell’s theory of judgment.25

The phrasing is at first sight puzzling, since in Russell’s theory judgment is not a relation that sometimes holds between things and sometimes between forms and things. Which relation is he referring to, then? The most likely candidate is the relation of acquaintance. Russell held that if I am to judge something, I must be acquainted with all the terms of the judging relation. So in the case where I judge that Socrates is mortal, I must be acquainted with Socrates, with mortality, and with the subject-predicate form. If the latter is to occur in this relation as a form, it cannot be the same sort of acquaintance 24

CP, VIII, 198–9.

25

B48.



Russell’s theory of judgment

as in the other cases. So Russell would have to accept that acquaintance is not a univocal notion. Perhaps this ‘goes against’ Russell’s theory, but it does not yet refute it. Why should Russell not simply have granted that our acquaintance with forms is a different sort of relation from our acquaintance with objects? The difficulty with this is one that Russell himself pointed out, in a section of his manuscript which he wrote only a couple of days after the incident reported in the last section. How can an object be at once simple and a ‘fact’, in the sense in which a ‘fact’ is opposed to a simple particular and is the sort of object whose reality makes a proposition true? Why, if pure forms are simple, is it so obviously inappropriate to give them simple proper names, such as John and Peter?26

These are good questions, and Russell could not answer them. He therefore resorted to a familiar kind of bluster. ‘These logical questions can no doubt be answered,’ he wrote, ‘but for our purposes the epistemological questions are more pressing.’ A few days later Russell told Ottoline that although ‘in all likelihood they are just’, Wittgenstein’s criticisms ‘have to do with problems I want to leave to him’.27 If he meant by this that the problems he had in mind belonged to logic rather than to epistemology, then, as we have just seen, the problem of acquaintance with forms was his own prime candidate. And if he used this as an excuse to carry on advancing a view whose difficulties had already been pointed out to him, then perhaps his behaviour merited the damning assessment that he himself offered a few weeks later: ‘It is the first time in my life that I have failed in honesty over work.’28 . Another formulation For the time being Russell soldiered on, even though Wittgenstein’s criticisms had ‘destroyed the pleasure’29 in his writing and made the work ‘a task rather than a joy’.30 A couple of weeks later, however, Wittgenstein finally provided Russell with the succinct formulation of his objection that had hitherto eluded him. I can now express my objection to your theory of judgement exactly: I believe it is obvious that, from the proposition ‘A judges that (say) a is in a relation R to b’, if correctly analysed, the proposition ‘aRb v ∼aRb’ must follow directly without the use of any other premiss. This condition is not fulfilled by your theory.31

To some extent, the difference in expression between this and the formulation in the Tractatus—that the correct explanation ‘must show that it is impossible to judge a nonsense’—is cosmetic: Wittgenstein is here choosing to put 26

ToK, 130. 27 28 May 1913. 28 To OM, 20 June 1913. 1 June 1913. 31 [June 1913] (CL, no. 14).

29

To OM, 27 May 1913.

30

To OM,

Another formulation  the point in a somewhat more formal manner that gestures towards Russell’s modes of expression. In the Principles it is taken to be a characterization of what it is for p to be a proposition that p ⊃ p should be true, but in Principia v and ∼ are taken as primitive, so that p ⊃ p is an abbreviation for p v ∼ p. To require that from ‘A judges that p’ should follow p v ∼ p is therefore to require that p should be a proposition, i.e. that it should not be nonsense. Wittgenstein’s reformulation of the point is curious nonetheless. If he was simply making again the point that a theory of judgment must show that it is impossible to judge nonsense, this is a very odd way to make it. It is plausible, therefore, that Wittgenstein is here responding not to the theory we have been considering, but rather to some new variant Russell had proposed, probably during their argument a fortnight earlier, as a way out of his predicament. The explanation may be contained in a set of notes32 Russell probably made around this time. (The first page of these notes is written on the back of a rejected sheet from the main manuscript which Russell wrote around the time of Wittgenstein’s visit on 26 May, so the notes themselves may well date from very shortly after that.) In the notes Russell explores a variant multiple relation theory which is evidently an attempt to get round Wittgenstein’s objection. Judgment, he says, will ‘still be a multiple relation, but its terms will not be the same as in my old theory’.33 Russell suggests that what is involved in judgment is not the positive fact that aRb or the negative fact that ∼aRb, but rather what he calls the ‘neutral fact’ which obtains when either of these facts obtains. It is surely very natural to represent this neutral fact as aRb v ∼aRb. Russell conjectures that the neutral fact is a constituent of whichever of the positive and the negative fact actually occurs. ‘There will only be a neutral fact,’ he observes, ‘when the objects are of the right types.’34 That is to say, the neutral fact that aRb v ∼aRb is equivalent to saying that aRb makes sense. So if Wittgenstein’s requirement that it should be impossible to judge nonsense is to be directed against this new idea of Russell’s, it is natural to reexpress it as the requirement that whenever A judges that aRb, the neutral fact, i.e. aRb v ∼aRb, must follow. Even in Russell’s notes, though, we see him recognizing that to regard judgment as a relation to neutral facts ‘entails the old difficulties; it seems not intimate enough. . . It can’t be quite right.’35 Wittgenstein’s letter was an attempt to explain succinctly why. To explain the point more fully, let us write #n (a0 , . . . , an ) for the result of assembling a0 , . . . , an in accordance with the form #n (so that, for instance, #2 (a,R,b) is just aRb). Wittgenstein insists that from J(A, #2 ,a,R,b) it should follow that aRb is a proposition, i.e. that aRb v ∼aRb; or, in greater generality, from J(A, #n , a0 , . . . , an ), 32

CP, VIII, App. B.1.

33

CP, VII, 197.

34

CP, VII, 199.

35

Ibid.



Russell’s theory of judgment

it should follow that #n (a0 , . . . , an ) is a proposition, i.e. that #n (a0 , . . . , an ) v ∼#n (a0 , . . . , an ).

Russell—the Russell of Principia at any rate—had a simple method for dealing with technical lacunae of this sort: if in doubt, just add another primitive proposition. Thus if in the present case we add the axiom J(A, #n , a0 , . . . , an ) ⊃ (#n (a0 , . . . , an ) v ∼#n (a0 , . . . , an )), the required conclusion will follow at once. But this could hardly have satisfied Wittgenstein, since adding this axiom does not show, but only asserts, that it is impossible to judge nonsense. Including the form in the judging relation as an extra term #n does not help one jot with this. Perhaps that is why Wittgenstein places a stronger demand on the theory in his letter than in the Notes. The conclusion that we cannot judge nonsense, he now insisted, ‘must follow directly without the use of any other premiss’. In the notebook entry he included in the Birmingham Notes he had required only that the theory should ‘make it impossible’ to judge nonsense. What he was now demanding was that the theory should, as he put it in the Tractatus, show (and not merely assert) that it is impossible to judge nonsense. All told, then, the exact formulation of his objection that Wittgenstein offered in his famous letter is probably not so much his central argument against Russell as a sort of intermediate skirmish, a response to a somewhat desperate rearguard action by Russell to shore up the defences of his theory by his favoured manoeuvre of adding a primitive proposition to it. . The fate of the multiple relation theory When he received the letter containing the succinct expression of Wittgenstein’s objection, Russell had in fact already stopped work on the book, but only temporarily: the day after receiving it, he still said that he intended to start again in September. If Wittgenstein wanted to get Russell to give up completely, he would have to make one last attack. The principal purpose of Wittgenstein’s letter was to make arrangements for Russell to have lunch with his mother, who was visiting London at the time. Perhaps Wittgenstein used this lunch as the opportunity to make his attack; perhaps this time he succeeded. At any rate it is noticeable that before the lunch Russell speaks only of feeling that Wittgenstein must be right but not being able to see why, whereas the following day we find him writing to Ottoline of a recent and profound change in his understanding of the effects of Wittgenstein’s criticisms of his work. ‘It was very difficult to be honest about it,’ he told her, ‘as it makes a large part of the book I meant to write impossible for years to come

Other accounts  probably.’36 Now, at last, Russell acknowledged the strength of Wittgenstein’s objection, which until then he had felt rather than understood. ‘Yesterday,’ he rather over-dramatically added, ‘I felt ready for suicide.’ By 1916 Russell had become convinced that Wittgenstein’s criticism had been ‘an incident of first rate importance in my life’.37 At the time, though, he seems to have bounced back quickly enough. By the beginning of 1914, indeed, he could tell Ottoline: ‘I don’t think my brain has been more clear, or more willing to do all I ask of it, any time since 1900.’38 And just in case she felt any temptation to doubt his own assessment of his intellectual stature, Russell was in the happy position of having independent evidence he could adduce for it. Whitehead . . . has been telling me that my mind has improved greatly in the last two or three years, that in fact it has risen to an altogether higher class. He says . . . if my present work develops as it promises, it will put me among the few great philosophers.39

‘Although this is so agreeable,’ Russell continued modestly, ‘I think it is true.’ . Other accounts As I mentioned at the beginning of this chapter, there have been many attempts by commentators to explain what Wittgenstein’s objection amounted to—too many for me to give them all detailed attention here. One which is particularly worth noting, though, is Griffin’s.40 Wittgenstein’s objection turns out, he says, ‘to be an argument of the most extraordinary subtlety. . . What Wittgenstein was able to show was that Russell’s multiple relation theory of judgment was inconsistent with the theory of types, the lynchpin of Principia.’41 The difficulty, according to Griffin, is that to ensure that an elementary judgment such as aRb makes sense, Russell would have to appeal to further judgments (such as the judgment that a and b are suitable arguments for a firstorder relation) which according to the theory of types are of higher order than the original judgment whose sensicality he was trying to establish. Griffin’s account depends on quite detailed features of Russell’s theory of types. However, it is just this feature that makes it unlikely to be correct. I have already noted that it is striking how little engagement Wittgenstein’s writings from this period display with the formal parts of Principia. Wittgenstein did address some of the particularities of the formal system of Principia, but some of the comments he made about them are inept in the extreme. It seems intrinsically more likely that Wittgenstein’s objection would have had a philosophical rather than a narrowly technical basis. When Griffin says that 36

20 June 1913. 37 To OM, 4 Mar. 1916. 38 8 Jan. 1914. 39 To OM, 23 Feb. 1914. multiple relation theory of judgment’. 41 The Private Years, 461.

40

‘Russell’s



Russell’s theory of judgment

the argument he attributes to Wittgenstein is ‘of the most extraordinary subtlety’, he could just as well have said ‘very elaborate’. And that tells against it too. One of the features of Wittgenstein’s arguments is that they are rarely elaborate; most of them, indeed, are simple—infuriatingly so, when one has struggled to work out what they are. Perhaps that is why I find it difficult to regard Griffin’s argument, ingenious though it is, as truly Wittgensteinian. If the account I have offered here is correct, on the other hand, Wittgenstein’s criticism was indeed fundamentally simple. If I say ‘A judges that p’, I do not, of course, myself judge that p, but I do, in the course of saying it, have to express what it is that A is judging (namely that p). If I am to do that, I must use the verb of p as a verb. Russell’s theory, in any of its variants, forces the verb of p to occur as a term in the judging relation, and in such a position it cannot function as a verb. Even if I am right about what Wittgenstein’s objection was, that does still leave open the possibility that Griffin might be right about why the objection paralysed Russell. The evidence does not support even this concession, however. The notes that have survived of Russell’s lectures on epistemology at Harvard the following year quote him as expressing just what I have represented as Wittgenstein’s objection. If I believe that Jones hates Smith, the occurrence of ‘hate’ in this fact is, he says, ‘not substantive’. Although it is a single fact, nonetheless it ‘contains two verbs’ and ‘both must occur as verbs’. This, he concludes, ‘constitutes [the] oddity of propositional thought’. And when Russell lectured on the same matter four years later, his recollection of Wittgenstein’s criticism had not changed. ‘When A believes that B loves C,’ he said, you have to have a verb in the place where ‘loves’ occurs. You cannot put a substantive in its place. Therefore it is clear that the subordinate verb (i.e. the verb other than believing) is functioning as a verb, and seems to be relating two terms, but as a matter of fact does not when a judgment happens to be false. That is what constitutes the puzzle about the nature of belief.42

What first led Wittgenstein to realize that Russell’s theory was susceptible to this difficulty? Perhaps it is not wholly coincidental that Moore spent some time criticizing Russell’s theory in his Cambridge lectures during Wittgenstein’s first year there. While granting that a false judgment could not be a relation of the person making the judgment to a single non-existent entity, Moore complained that it could not be a relation to the proposition’s components either. It is totally impossible that belief should consist in having a relation to something which simply isn’t; but it seems equally impossible that it should consist solely in having relation to something which indubitably is. 42

CP, VIII, 198.

Other accounts  Moore makes it clear that he objects to Russell’s multiple relation theory as much as to the theory it replaced. ‘To both forms of solution which [Russell] suggests there is fundamental difficulty, that whole J[udgment] solely consists in my having relation to object which undoubtedly is.’ What both theories fail to explain is that ‘To believe that I am in Cambridge, is same thing as to believe that prop[osition] that I am in Cambridge is true, or that there is a fact corresponding to prop[osition].’ In his lecture notes Moore never quite expresses the objection satisfactorily: he struggles repeatedly, in fact, to bring it to a clear formulation. False belief, he observes at one point, consists ‘somehow in taking for a fact, something which isn’t a fact’. That, surely, is the essence of Wittgenstein’s objection to Russell’s original multiple relation theory, and hence, as we have seen, of his later objections too.

Chapter 

Meaning Russell was troubled, as we have seen, by the difficulty of explaining false propositions. He had begun with an account which identified true propositions with facts, but this left false (or ‘unasserted’) propositions as a puzzling kind of sham, like facts except for their non-existence. The great advantage of the multiple relation theory was that according to it true judgments have facts (or, as Russell was just as happy to say, complexes) corresponding to them, whereas false judgments have nothing at all. Rejecting this theory left Wittgenstein with the problem that had originally motivated it. His elegant answer was that what it is in the world that makes p true, if it is true, is the same as what makes ∼ p false; and conversely, what makes p false, if it is false, is just what in that case makes ∼ p true. This allowed him to satisfy the desire to ensure that every proposition, true or false, should have some part of the actual world responsible for its truth or falsity. This part of the world he called its meaning: the meaning of ∼ p is the same as the meaning of p,1 but they point at it, as it were, from opposite directions. If propositions are to have meaning at all, the advantage of saying that p and ∼ p have the same meaning is that it avoids the need to commit ourselves to the fictional facts that had troubled Russell. But it comes at a price. We must give up any pretence of the delightful simplicity of Moore’s original identity theory. Given that what makes p true if it is true is the same as what makes ∼ p true if it is true, it plainly cannot more than half the time be the case that what makes a proposition true is itself. It is something of a misnomer, therefore, to call the meaning of a proposition its truth-maker: if the proposition is true, its meaning is indeed what makes it true; but if it is false, its meaning is what makes it false. A proposition’s meaning might therefore more accurately (if less elegantly) be called its true-or-false-maker. And although the ‘meaning’ is the worldly correlate of the proposition, it is not in any recognizable sense what the proposition means. So the word ‘meaning’ is a rather awkward one to use in this context. Russell at any rate thought so: when he was explaining Wittgenstein’s theory, he preferred to call the true-or-false-making entity the ‘objective reference’ of the proposition.2 Even Wittgenstein himself later expressed unease with his own terminology.3 1

B37.

2

CP, VIII, 297.

3

NB, 2 Nov. 1914.

Russell’s lectures on logical atomism  . Russell’s lectures on logical atomism Before we discuss the details of Wittgenstein’s notion of meaning, something needs to be said about Russell’s 1918 lectures on logical atomism, in order to explain why they are an important source for the interpretation of the Notes on Logic. For this we need to continue the narrative some way beyond the point when the Notes were compiled. Russell was due to spend the period from March to June 1914 at Harvard delivering the Lowell lectures. After Wittgenstein had departed for Norway, therefore, Russell spent the autumn writing these lectures, later published as Our Knowledge of the External World. Although Russell mentioned Wittgenstein in the preface to this book, his influence on it is detectable much more in what it does not discuss than in what it does. Only in chapter 2, which describes Wittgenstein’s conception of atomic facts, and of logic as the study of the forms of facts, does Russell make significant use of his ‘vitally important discoveries’.4 The Lowell Lectures were intended for a general audience, but while he was at Harvard Russell also gave two more specialized lecture courses, one on epistemology and one on logic (with a graduate student called Costello as his teaching assistant); it was in preparation for the latter course that he went to the labour of translating and arranging Wittgenstein’s Notes in February 1914. Russell did not publish the logic lectures himself, but much of their content is preserved for us in the notes of T. S. Eliot, a philosophy student at Harvard who soon achieved fame for other reasons. During the Autumn of 1913 Russell and Wittgenstein exchanged letters frequently, but shortly before Russell was due to leave for Harvard there was a rift and philosophical discussions between them almost ceased. After he returned from Harvard, Russell read the notes Wittgenstein dictated to Moore, but he did not understand them and Moore, he claimed,5 could not help him. (Moore indignantly responded6 that he had not helped because Russell had not asked.) At this point the war intervened. Straightaway philosophical communication between Russell and Wittgenstein became impossible (and, after a year or so, non-philosophical communication too). From then on, moreover, Russell did very little by way of original philosophy. One reason for this was no doubt the war: it energized him politically and led him to devote much of his time to campaigning against it. Not only that, but his motivation to work in philosophy deserted him. The reasons for this, as is often the case in such matters, were complex. The war certainly (and understandably) played its part in making philosophy seem pointless to him. For example, his reaction to a reminder about a lecture he had agreed to give in Manchester early in the war was to ask, ‘Are there really people in Manchester interested in philosophy still? I am not.’7 A year or so later, though, he felt more inclined to attribute his lack of motivation to Wittgenstein. 4

OKEW, vii.

5

To LW, 21 June 1919.

6

Moore’s diary.

7

To Samuel Alexander, 17 Oct. 1914.



Meaning

I had to produce lectures for America, but I took a metaphysical subject although I was and am convinced that all fundamental work in philosophy is logical. My reason was that Wittgenstein persuaded me that what wanted doing in logic was too difficult for me. So there was no really vital satisfaction of my philosophical impulse in that work, and philosophy lost its hold on me. That was due to Wittgenstein more than to the war.8

He now spoke of the war not as drawing him away from philosophy, but rather as filling the gap his discouragement with it had left by supplying him with the ‘new and less difficult ambition’ of political activism. And years later, in part of his autobiography believed to have been written around 1931, Russell had yet another explanation: the effect of writing Principia, he now thought, had been to induce in him a long-term intellectual exhaustion which made it impossible for him to work in technical philosophy again. In 1917, however, Russell did return to philosophy after a break of three years, giving two courses of popular lectures. Each course consisted of eight lectures at weekly intervals: the first course, on the philosophy of mathematics, began in October 1917; the second, on logical atomism, in January 1918. Russell’s prosecution (for making statements ‘likely to prejudice His Majesty’s relations with the United States of America’) came to court while the second course was still in progress, and shortly afterwards he served a six-month prison sentence (reduced to five for good behaviour), during which he converted the first lecture course into a book, his Introduction to Mathematical Philosophy. Meanwhile, the course on logical atomism was published in the form of verbatim transcripts of the lectures. In the Introduction to Mathematical Philosophy Russell mentions Wittgenstein only in a footnote9 (attributing to him a recognition of the importance of the notion of tautology). Elsewhere in the book Wittgenstein’s influence is detectable only very sparingly. The logical atomism lectures, on the other hand, are according to Russell ‘very largely concerned with explaining certain ideas which I learnt from my friend and former pupil, Ludwig Wittgenstein’, who ‘originally supplied many of the theories contained in them’.10 Even a casual inspection of the lectures confirms that Russell should here be taken at his word. Large parts of them consist simply of exposition, in language suitable for a non-technical audience, of ideas from the Notes. In some instances the reasons he offers for the claims he makes are plainly Wittgenstein’s. In others he repeats things Wittgenstein says in the Notes, but seems unable to recollect Wittgenstein’s reasons for saying them and quickly changes the subject. Hardly anywhere does he directly contradict Wittgenstein’s views as he then believed them to be. It might be thought odd that Russell agreed to publish the lectures if they were very largely a report of Wittgenstein’s views. Two obvious reasons sug8

To OM, 4 Mar. 1916.

9

p. 205.

10

CP, VIII, 160.

Propositions are not names of their meanings  gest themselves. One is that his lectureship at Trinity had not been renewed, and he needed the money. The other is that, as he noted in a preface to the lectures, he did not at that point know whether Wittgenstein was alive or dead, and therefore could not judge how soon there might be any other opportunity for Wittgenstein’s ideas to reach the wider audience he thought they deserved. He had no way of knowing that the Tractatus was by then nearing its final form. Further evidence of the extent to which Russell had by now turned away from philosophical logic is provided by the first paper he published on his return to philosophy, ‘On propositions: what they are and how they mean’. In this paper, written in February 1919, Russell began by offering what was in effect a summary of Wittgenstein’s views about propositions in the Notes on Logic, but then he devoted the remainder to considerations of psychology, which he had been reading up in prison. In 1913 he had told Ottoline that he felt inclined to leave logic to Wittgenstein: here is evidence that this is just what he did. For our current purposes, however, Russell’s misfortunes are our fortune. Because he had not been active in the field for so long, Russell’s views seem truer to Wittgenstein’s than might otherwise be the case. In the lectures, especially, he was more interested in recollecting what Wittgenstein’s views had been than in developing his own. This is relevant to us here, because there had been, as we have noted, very little philosophical contact between them since Wittgenstein’s departure for Norway in October 1913. The views of Wittgenstein’s which Russell was recollecting are therefore overwhelmingly those of the Notes on Logic. For this reason Russell’s lectures represent a significant additional source for the interpretation of these notes. . Propositions are not names of their meanings ‘Especially false,’ Wittgenstein said, is the statement “propositions are names of complexes”.’11 Especially false, because it combines two errors, that of conceiving of the meaning of the proposition as a complex rather than a fact, and that of conceiving of the relationship between the two on the model of the relationship between a name and its meaning. The main target at this point is Russell: we shall focus first on his view that a proposition names its meaning, and postpone to the next section the question of whether the meaning is a complex or a fact. We need at this point to distinguish two questions. As we saw in §2.2, Russell had adopted the view that what look like names often are in fact disguised definite descriptions. Those names which resist this treatment and turn out on analysis to be genuinely names Russell calls logically proper names. Russell would have 11

B4.



Meaning

needed no convincing that propositions are not simple names: if they were, we would on his view have direct acquaintance with what they refer to, and would therefore know whether they are true or false whenever we understood them. But there remains the question, not settled by this quick argument, whether propositions are names in the wider sense that includes disguised descriptions. Wittgenstein’s reason for saying that they are not is neatly expressed by Russell in his 1918 lectures (lectures which, as we have just noted, consist largely of exposition of Wittgenstein’s views). It is perfectly evident as soon as you think of it, that a proposition is not a name for a fact, from the mere circumstance that there are two propositions corresponding to each fact. Suppose it is a fact that Socrates is dead. You have two propositions: ‘Socrates is dead’ and ‘Socrates is not dead’. And those two propositions corresponding to the same fact, there is one fact in the world which makes one true and one false. That is not accidental, and illustrates how the relation of proposition to fact is a totally different one from the relation of name to the thing named. For each fact there are two propositions, one true and one false, and there is nothing in the nature of the symbol to show us which is the true one and which is the false one. If there were, you could ascertain the truth about the world by examining propositions without looking round you.12

The argument that Russell is expounding here depends on two of Wittgenstein’s doctrines which we have met already. First there is the doctrine with which this chapter began, that the feature of the world that makes a proposition true or false is the same as the feature that makes its negation false or true. Then there is the doctrine, which we encountered in connection with the argument for substance, that we can understand a proposition independent of knowing whether it is true or false. Expressed less epistemically, what is relevant here is the observation—central to Russell and Moore’s 1898 revolt against idealism—that when a feature of the world makes a proposition true or false, the proposition itself is not altered thereby. It follows from this that truth and falsity are not internal to a proposition but are different relationships that may hold between a proposition and the relevant feature of the world. But if a proposition were a name, there would be only one such relationship, namely that of reference. Therefore propositions are not names of those features of the world that make them true or false. According to Wittgenstein, then, propositions are distinguished from names by having two possible relations to their meanings. He called this feature of propositions their bipolarity. This is a promising proposal for avoiding Russell’s problem of fictional complexes, and we shall do what we can to develop the details of it in the coming pages. Notice, though, that neither Wittgenstein himself nor Russell on his behalf offered an argument for it. What recommends it, apart from whatever virtues the detailed theory will turn out to possess, is 12

CP, VIII, 167–8.

Meanings as facts  only the intuitive thought that what makes a proposition true if it is true is the very same as what makes its negation false. If that thought should lose its appeal, the proposal will be left without a priori support.

. Meanings as facts Wittgenstein’s idea, we have said, was that with each proposition there is associated some part of the world which is its meaning. But which part? We discussed in §11.1 his view of facts as conceptually prior to (because closer to the possibility of their representation than) complexes. It is to be expected, therefore, that he would conceive of the meaning of the proposition that Caesar died as being not the complex consisting of his death but the fact of it. And so he did. The meaning of a true atomic proposition he called a positive fact. What makes an atomic proposition true is thus that something is the case, not that something exists. Wittgenstein also had to suppose that there are negative facts to act as the meanings of false atomic propositions. If an atomic proposition is false, there is no positive fact corresponding to it, but we are not required to suppose that something else magically springs into existence. The non-occurrence of the positive fact is itself already the negative fact. If A loves B, it may be said, that is a good substantial fact; while if A does not love B, that merely expresses the absence of a fact composed of A and loving and B, and by no means involves the actual existence of a negative fact. But the absence of a fact is itself a negative fact; it is the fact there is not such a fact as A loving B. Thus, we cannot escape from negative facts in this way.13

The logical book-keeping is thus done on this account not by a mysterious feature of the world itself but simply by the grammar of factuality. Straightaway, though, we should recognize that Wittgenstein’s proposal about meaning has embedded in it all the metaphysical problems that come with a commitment to negative facts—problems which will be a recurring theme here. In the Notes Wittgenstein is not at all explicit about what other facts, apart from positive and negative atomic facts, his notion of propositional meaning requires; nor does he give a worked out account of which facts are the meanings of which propositions. So let us try to fill in the gaps he has left. Some cases are easy enough. The meaning of not- p is the same as the meaning of p. If ‘ p and q’ is true, its meaning is just the sum of the meaning of p and the meaning of q. (We shall say a little more in the next chapter about what ‘sum’ means here.) The problematic case is to say what the meaning is of ‘ p or q’ if it is true (or, dually, what the meaning is of ‘ p and q’ if it is false). What Wittgenstein says about the meaning of ‘ p or q’ is as follows: 13

Russell, CP, VIII, 280.



Meaning

Among the facts which make ‘ p or q’ true, there are some which make ‘ p and q’ true; but the class which makes ‘ p or q’ true is different from the class which makes ‘ p and q’ true; and only this is what matters. For we introduce this class, as it were, when we introduce ab-functions.14

This remark does not yield up its meaning instantly. Wittgenstein says that ‘the meaning is the fact’ (singular),15 yet he seems now to be suggesting that what makes a disjunctive proposition true is not a single fact but a whole class of them. On the face of it, this makes no sense unless we understand him here to be using ‘fact’ to mean ‘possible fact’. We might think, then, that when he speaks of the class of facts which make a proposition true, he means not those that do make it true, but rather the class of possible facts any of which will, if it is indeed a fact, be a candidate to be the meaning of the proposition in question. On this reading, the meaning of a true proposition would in general be the sum of all those possible facts belonging to this class that are actually facts. But it unlikely that this can really be what Wittgenstein meant. The context, let us recall, was Russell’s failed project of explaining the semantics of propositional thought without appealing to non-existent entities such as the death in bed of Charles I. And that context surely makes it improbable that Wittgenstein would really have intended to refer to what might nowadays be called a ‘trans-world class’, i.e. a class some of whose members are merely possible and not actual. A clue to what Wittgenstein intended is contained in the Cambridge version of the Notes, where he remarks that in the explanation just given we have been ‘talking of all p’s and all q’s’. What he intends to vary, in other words, seems to be not the possible world referred to but the propositions involved. However, if this is what he means, what he says in the Birmingham Notes is obviously wrong. For any fact whatever makes ‘ p or q’ true for some values of p and q; and equally any fact whatever makes ‘ p and q’ true for some p and q. So the class of facts that make ‘ p or q’ true for some values of p and q is exactly the same as the class of facts that make ‘ p and q’ true for some values of p and q. Perhaps this is why the version of the remark that Wittgenstein included in the Cambridge Notes omits the passage about the two classes being different. But Wittgenstein has here got himself in a tangle quite unnecessarily. What he probably meant to say was that although for some values of p and q what makes ‘ p or q’ true is the same as what makes ‘ p and q’ true, for some other values what makes them true is different. In other words, he was merely alluding to the obvious point that if ‘ p or q’ is true, that may be because p is true, because q is true, or because p and q are both true. And the natural thing to say, therefore, is that what makes it true, if it is true, is whichever of these is indeed the case. More precisely, if ‘ p or q’ is true, its meaning is the sum of the meanings of whichever of p and q are true. 14

B50.

15

C39.

Meanings as facts  We do not need to make any additional stipulation about the contrary case in which ‘ p or q’ is false, since in that case the meaning is fixed by our earlier prescriptions: if ‘ p or q’ is false, ‘not- p and not-q’ is true, so its meaning is the sum of the meanings of not- p and not-q, i.e. the sum of the meanings of p and of q; hence this is also the meaning of ‘ p or q’. And dually we do not need to say anything else about the meaning of ‘ p and q’ when that is false, since from what we have already said it will be the sum of the meanings of whichever of p and q are false. We may summarize the account in the following table, in which we write ‘M( p)’ for the meaning of ‘ p’ and ‘+’ for the operation of summing facts. M(not p) = M( p) p

q

T T F F

T F T F

M( p or q) M( p)+M(q) M( p) M(q) M( p)+M(q)

M( p and q) M( p)+M(q) M(q) M( p) M( p)+M(q)

It is easy to extend this account to quantifiers. If ( x f x is true, its meaning is the sum of the meanings of all its instances; if it is false, its meaning is the sum of the meanings of all its false instances. Dually, if ( x) f x is true, its meaning is the sum of the meanings of all its true instances; if it is false, its meaning is the sum of the meanings of all its instances. The theory just outlined does not always give especially intuitive results. It is easy to check, for instance, that according to the theory M( p v ∼ p) = M( p), whatever the truth-value of p. If M( p) &= M(q), therefore, M( p v ∼ p) &= M(q v ∼q). This is at the very least uncomfortable because it has the consequence that different logical truths will have different meanings: the meaning of ‘ p or not- p’ will generally be different from the meaning of ‘q or not-q’. Wittgenstein did not at this stage have anything very developed to say about logical truths: that would come in Norway. But he does at least seem to have realized that whatever motive we have for identifying something in the world that makes a proposition true or false does not apply to logical truths, and therefore to have made them a sort of exception. But if it is clear that Wittgenstein already made logical truths an exception, it is less clear what sort of exception. In the Cambridge Notes he appears to tell us that they do not have a meaning at all.16 But if we take this claim seriously, it causes severe difficulties for the theory, because logical truths may be components of propositions that do have meanings. Consider, for example, ( p v ∼ p) q: presumably we want it to have a meaning, even though on the view under consideration its component p v ∼ p has no meaning. The amendments we E

16

C36.



Meaning

shall have to make to our theory to meet this difficulty quickly rob it of any appealing simplicity it might have had. In fact, though, Wittgenstein probably did not intend to say that p v ∼ p has no meaning. The corresponding passage in the Birmingham Notes says not that p v ∼ p is meaningless but that it is senseless.17 The conclusion to draw is probably that here we have no more than a difficulty of translation: when Wittgenstein said in English that a proposition ‘has meaning’, what he meant (confusingly) was that it has sense. Of course, this still leaves us to explain how ( p v ∼ p) q, for instance, can have sense even though part of it, p v ∼ p, does not; but we shall postpone that problem until we discuss Wittgenstein’s notion of sense in chapter 16. (We shall see there, in fact, that according to Wittgenstein ( p v ∼ p) q has the same sense as q.) Nonetheless, even if we do not read Wittgenstein as saying that logical truths have no meaning, that is not to say that his theory is at all palatable, since it leads us out of one difficulty straight into another. Consider the case where p and q are true propositions expressing different atomic facts. Then according to the theory just offered, M(( p v ∼ p) q) = M( p v ∼ p) + M(q) = M( p) + M(q) &= M(q). So this is a case where two propositions have the same sense but different meanings. It is hard to believe that Wittgenstein would have regarded this as satisfactory, or even coherent, but it is equally hard to see how to avoid it. . The demise of propositional meaning This is perhaps the point at which to comment on the sketchiness of Wittgenstein’s theory of meaning. The reconstruction of that theory just offered is hardly elaborate, but in the Notes Wittgenstein does not state it fully—the parts he does state he seems to bungle—and it is hard to tell in how much detail he ever bothered to work it out. This is an instance of a pattern in Wittgenstein’s work which we have noted already. Unlike Russell, he never let technical considerations drive his philosophical views, and he filled in the details of technical proposals only reluctantly. A cynic might say, I suppose, that he did not work the theory out in detail because he already had an inkling that it did not work, and there is probably some truth in this. We shall later come across at least one other case in which Wittgenstein’s reluctance to work out the details of a theory, despite Russell’s pestering, is plausibly a symptom that he subconsciously knew it did not work. (If so, this would not make him unusual, of course: many of us can report comparable experiences.) 17

B40.

The demise of propositional meaning  In this case, though, there is a further reason for the sketchiness of the account. Wittgenstein never spelt out the details, even in the Tractatus, because by the time he wrote it he had all but given up the idea that propositions have meaning at all. The Tractatus does, it is true, contain the claim that ‘the propositions “ p” and “∼ p” have opposite senses, but to them corresponds one and the same reality’.18 Since by this time Wittgenstein used the word ‘reality’ to mean any sum of positive or negative facts,19 he was thus still asserting that p and ∼ p correspond to the same sum of positive and negative facts, i.e. that they have what in the Notes he called the same meaning. But the word ‘bipolarity’ does not occur in the Tractatus and the claim that p and ∼ p correspond in different ways to the same fact is by now no more than an idle wheel, one of a number of remarks from his early notebooks which he included in the finished work perhaps more for nostalgic reasons than because they were relevant to the exposition of his current views. What he seems by then to have abandoned is the idea that it is important to identify just which bit of reality a proposition corresponds to. While propositions still have a sense, in the Tractatus their meaning has faded into the background—not quite denied, but no longer doing any significant work in the account. One way of putting this point stresses the overriding importance of elementary propositions in Wittgenstein’s conception. The problems that make Wittgenstein’s theory of meaning collapse occur only when we consider the meanings of molecular propositions, so nothing prevents us from continuing to say, if we wish, that elementary propositions and their negations have meanings (which will always be positive or negative facts). Another way of putting the reason for the demise of propositional meaning might be that by the time of the Tractatus Wittgenstein no longer adhered to the correspondence theory of truth which the notion of meaning entails. Once he had adopted the picture theory, that is to say, it became less important to single out which part of the world it is that is responsible for making a proposition true or false. If we know the sense of a proposition, we know what it says. If it is true, things are as it says they are. Or, we might also say, what makes it true or false is how things stand. Once we have explained, through our account of sense, how things have to stand for the proposition to be true, what further point is served by seeking to identify a particular part of the world as especially responsible for bringing this about? 18

4.0621.

19

2.06.

Chapter 

Metaphysics . Disjunctive facts In order to make sense of Wittgenstein’s remarks we found it necessary in the last chapter to appeal to what one might be tempted to think of as conjunctive facts—the fact, for instance, that this rose is red but that one is not. In order to provide meanings for generalizations, we had to extend this to allow for facts made up of infinitely many component facts. At no point, though, did we find ourselves having to appeal to disjunctive facts—the fact, for instance, that at least one of the roses in my garden is red. Of course the fact that Wittgenstein’s account did not appeal to disjunctive facts does not in itself show that he thought there are no such things. But if there were, they would make the account significantly simpler: the natural thing to say would then be that the meaning of ‘ p or q’, if it is true, is simply the disjunctive fact that p or q. That he did not adopt this much simpler account is what shows that he cannot have believed in them. What reason might he have had for this? One argument against disjunctive facts that has become popular was advanced by Fine.1 Suppose that the fact that p or q is thought of as having p and q as constituents. In that case the occurrence of the fact that p or q entails, it is argued, the occurrence of both the constituent facts that p and that q. This is merely an instance of the general principle that the existence of a compound entity entails the existence of its constituents. It is hard, though, to regard this argument as completely persuasive, because it depends on the premise that we should regard p and q as components, in the relevant sense, of the fact that p or q. It would presumably be open to the defender of disjunctive facts to claim that there is a manner of composition, of which they are an instance, for which the existence of the compound does not entail the existence of the components. Another, perhaps rather more Wittgensteinian, argument against disjunctive facts is that even if there is a manner of combination such as has just been countenanced, the fact that p or q will have to differ from the fact that p and q, and the difference can only be that one contains or where the other contains 1

Cf. Fine, ‘Facts’, 67.

Negative facts  and. But the existence of such logical ties as or and and stands in direct contradiction to Wittgenstein’s ‘fundamental thought’ that the logical constants do not represent. This kind of argument is limited, however: it shows at most that if we wish to countenance disjunctive facts, we need a radically different conception of their structure, not that no such conception is possible. After all, exactly analogous arguments apply to negative facts, which, we have seen, Wittgenstein explicitly accepted in the Notes. Perhaps a more plausible explanation for Wittgenstein’s reluctance to countenance disjunctive facts is offered by a combination of Occam’s razor and a conception of the world as determinate. It cannot be a fact that p or q unless one or other of p and q is a fact. So whichever of these is a fact is part of the world. What need is there to posit other, less definite facts? It is surely a rather natural thought that devices such as disjunction are linguistic constructs designed to represent our ignorance about the ways of the world, not part of the world itself. . Negative facts What of negative facts? Russell later said that ‘the absence of a fact is itself a negative fact’.2 If this was supposed to be a report of Wittgenstein’s view on the matter, then Russell did not get it quite right: what he should have said is that the absence of a positive atomic fact is itself a negative fact. Wittgenstein did not think that the absence of a non-atomic fact is a fact. For, as we have just seen, Wittgenstein rejected disjunctive facts. So the absence of the fact that p and q, for instance, cannot itself be a fact, since if it were, it would just be the disjunctive fact that not- p or not-q. From now on, then, let us focus on the atomic case. Even if a positive atomic fact represents the existence of something, it is not clear that the corresponding negative fact has to be thought of as the existence of something else. In the Notes, Wittgenstein suggests that the manner in which the absence of something can determine reality is like the blind spot in the visual field. The comparison of language and reality is like that of retinal image and visual image: to the blind spot nothing in the visual field seems to correspond, and thereby the boundaries of the blind spot determine the visual image—as true negations of atomic propositions determine reality.3

Wittgenstein was presumably trying by the use of this analogy to make it plausible that reality might be characterizable as much by the absence of something as by its presence. Let us now consider what can be said about the structure of a negative fact. It cannot consist of the positive fact together with another element called 2

CP, VIII, 280.

3

B11.



Metaphysics

negation, for reasons that parallel the ones offered earlier against disjunctive facts: there is no such ‘thing’ as negation, and in the case in question the positive fact does not exist. A more plausible solution, perhaps, would be to roll up the negation into the verb of the fact, so that if the positive fact is that aRb, the negative one is that a∼Rb. Of course, this is not a proposal we can use as an analysis of complex facts: we cannot analyse the fact which makes ‘the present King of France is bald’ false as being the same as the fact that the present King of France is non-bald. But that is no objection in the present case, since here it is only the facts corresponding to atomic propositions that are in question, and ‘the present King of France is bald’ is not atomic. Indeed a quick way to see that it is not atomic is simply to note the difference in sense between ‘The present King of France is non-bald’ and ‘It is not the case that the present King of France is bald’. The current proposal, that negation should be rolled up into the verb of the proposition, was considered and rejected by William Demos in his article objecting to negative facts.4 However, his reason for rejecting it is barely more than a restatement of his denial that there are negative facts. It is also, more significantly, the treatment of negative facts that Russell offers in the first section of his 1919 paper ‘On propositions’. In other respects this section of the paper consists largely of an exposition of ideas from the Notes on Logic: Russell notes that it ‘contains nothing essentially novel’, and claims, a little inaccurately, that he has ‘defended its doctrines elsewhere’.5 This supports the hypothesis that his account of negative facts was Wittgenstein’s. If so, it is an account with which Wittgenstein himself later became uneasy. In the Notebooks he repeatedly mentions his concern with the structure of the negative fact and its relationship to the positive. ‘It is the dualism, positive and negative facts, that gives me no peace. For there cannot be such a dualism. But how to escape from it?’6 Even when it is limited to the atomic case, the proposal is indeed far from unproblematic. One of the things that makes facts more plausible truthmakers for atomic propositions than complexes, I suggested in §11.1, is the implausibility of leaving the logical book-keeping to the world to sort out; but now it seems as if we are reinstating that problem by supposing that the primitive ways in which objects can combine to form facts come in pairs: for every simple way R in which objects can be related there is another way ∼R, with the convenient property that in every case exactly one of the two relations holds. We might try to get out of this difficulty by viewing the convenient property just mentioned as a criterion for the simplicity of R. In other words, the feature 4

‘A discussion of a certain type of negative proposition’.

5

CP, VIII, 278.

6

NB, 25 Nov. 1914.

Summing facts  we noted earlier in the case of ‘the present King of France is bald’—that it does not have a well-defined negation—is representative of something more general: what it is for a proposition to be atomic is for there to be two mutually exclusive possible facts, one positive and one negative, such that whichever of them obtains is its meaning. Nonetheless, this hardly solves the difficulty. Once the forms of facts are aligned in contradictory pairs, R and ∼R, we can perhaps regard it as a criterion of atomicity that aRb and a∼Rb should be mutually exclusive and exhaustive possibilities. But what reason is there to think that forms can be paired off in this way at all? In his 1919 presentation of the theory, Russell described the difference between R and ∼R as ‘ultimate and irreducible’,7 which was his favourite way of labelling a problem rather than solving it. It may in any case be doubted whether the proposal now under discussion drives a sufficiently large wedge between negative facts, which Wittgenstein admitted, and disjunctive facts, which he rejected. For if the idea of rolling up the logical tie into the verb is valid in the case of negation, it could in principle be extended, albeit less tidily, to other logical constants such as disjunction. These difficulties suggest, perhaps, that a more radical approach is required. If we really want to save negative facts, perhaps we should reject the temptation to conceive of their structure in such a way that the difficulty we have been discussing even arises. . Summing facts No fact, Wittgenstein believed, can contain a logical tie. So even what I earlier called ‘conjunctive’ facts are not really conjunctive—do not combine their constituent facts by means of a logical tie of conjunction. In that case, though, we need an explanation of how there can be any complex facts; any facts, that is to say, other than the positive and negative ones to which atomic propositions correspond. The way I put the matter earlier was by appealing to facts that are sums of other facts: the facts we are tempted to call ‘conjunctive’ should more properly be thought of as something like mereological sums of their components. One might express the view, then, by saying that a mereological sum of facts is itself a fact. That, at any rate, is how Russell seems to have conceived of non-atomic facts. In his Introduction to the Tractatus he used the language of part and whole, distinguishing8 between facts which contain parts which are facts and facts which contain no such parts. Mereological sums are sometimes held to be metaphysically innocent. According to this view, speaking of the sum of some things is just another way of 7

CP, VIII, 280.

8

CP, IX, 104.



Metaphysics

speaking of those things. The most vigorous proponent of this view was David Lewis. To be sure, if we accept mereology, we are committed to the existence of all manner of mereological fusions. But given a prior commitment to cats, say, a commitment to cat-fusions is not a further commitment. The fusion is nothing over and above the cats that compose it. It just is them. They just are it. Take them together or take them separately, the cats are the same portion of Reality either way. Commit yourself to their existence all together or one at a time, it’s the same commitment either way. If you draw up an inventory of Reality according to your scheme of things, it would be double counting to list the cats and then also list their fusion. In general, if you are already committed to some things, you incur no further commitment when you affirm the existence of their fusion. The new commitment is redundant, given the old one.9

In Russellian terms a mereological sum is an incomplete symbol which could be eliminated on analysis if our underlying logic was plural. But Wittgenstein’s logic was not plural, and he displayed no sign of having been tempted in that direction. Perhaps, therefore, assuming the existence of complex facts is, strictly speaking, an inflation of his ontology beyond the positive and negative facts that compose them; but if we take David Lewis’s line, we do not need to think of it as an important inflation. This is presumably what Wittgenstein meant when he noted that ‘whatever corresponds in reality to compound propositions must not be more than what corresponds to their several atomic propositions’. What corresponds to p q if it is true, for instance, is the sum of the fact that p and the fact that q; but that is really no more (or at any rate not much more) than those two facts considered together. According to Wittgenstein, then, there are complex facts which are sums of positive and negative fact. In the Birmingham Notes (but, for reasons we shall come to shortly, not in the Cambridge ones) Wittgenstein made the further claim that there is, in a certain sense, no more to the world than the positive and negative facts. ‘If we formed all possible atomic propositions, the world would be completely described if we declared the truth or falsehood of each.’10 Anything we can say about a sum of facts can easily be reworded as a statement about the constituent facts. So to describe the world it is not necessary to mention such composite facts at all. One point worth noting about this conception of the world is that it is in a certain sense flat. Each positive or negative fact that there is has an internal structure: it is composed of a form and a certain number of objects. But there is no other structure to the world, except in the thin sense that atomic facts agglomerate into more complex facts as mereological sums. What there is not is any sort of structure to facts that would allow for the notion that they might have alternative atomic decompositions. 9

Parts of Classes, 81–2.

10

B36; cf. 4.26.

General facts  . General facts Much of the metaphysical picture outlined in the Notes Russell accepted in his 1918 lectures without demur. The one point on which he disagreed, however, came when Wittgenstein insisted that the positive and negative facts are all there is to the world. General facts, Russell believed, are a counterexample to this claim. ‘You cannot,’ Russell said, ‘ever arrive at a general fact by inference from particular facts, however numerous.’11 There must, he insisted, be at least one fact over and above all the particular facts, namely the fact that they are all of them. It is perfectly clear, I think, that when you have enumerated all the atomic facts in the world, it is a further fact about the world that those are all the atomic facts about the world, and this is just as much an objective fact about the world as any of them are.12

Russell was here repeating something he had said in the Lowell Lectures four years earlier. It is easy to see that general propositions, such as ‘all men are mortal’, cannot be known by inference from atomic facts alone. If we could know each individual man, and know that he was mortal, that would not enable us to know that all men are mortal, unless we knew that those were all the men there are, which is a general proposition.13

Now this looks superficially like, but is not the same as, the claim from the Introduction to the Tractatus that ‘the world is fully described if all atomic facts are known, together with the fact that these are all of them’. The confusing similarity between the two claims is the result of a terminological shift: in the Tractatus atomic facts (Sachverhalte) are always positive, whereas in the 1914 and 1918 lectures Russell was following Wittgenstein’s prewar usage according to which atomic facts may be positive or negative. Suppose for a moment that a, b, and c are the objects and f x and gx the forms, so that there are six possible positive facts that may obtain, namely fa, f b, fc, ga, gb, gc. If I tell you that among these possible facts fa, f b, fc, and gb (say) do actually obtain, I do not describe the world completely: to do that I would have to add the information that these are the only ones that obtain; or equivalently I would have to say that ga and gc do not obtain. The point Russell is making in the Introduction is that a complete description of the world cannot (except in the unusual case in which all the atomic facts obtain) be wholly positive; it will have to contain the negative information that no other of the atomic facts actually obtain. But in the 1918 lectures Russell’s point is different. Suppose now that in addition to saying that fa, f b, fc, and gb obtain, I also tell you that ga and gc do not. By the lights of the Tractatus I have now described the world completely. But Russell objects to this on the ground that the information supplied 11

CP, VIII, 206.

12

CP, VIII, 207.

13

OKEW, 55–6.



Metaphysics

does not license the inference to ( x) f x. For that inference to be valid I have to know that there are no objects other than a, b, and c, no forms other than f x and gx. The point, in other words, is not now about which of the possibilities are actualized, but about what the possibilities are. Russell’s insistence on general facts is striking partly because it is one of the very few points in his 1918 lectures on logical atomism on which Russell disagrees with the Notes on Logic at all, but also, more pointedly, because Russell does not give any indication that he is disagreeing with them. What is the explanation for this? By the time of the Tractatus, of course, Wittgenstein had a developed view which allowed him to reject Russell’s point. To admit it as a further fact about the world that just these atomic facts are possible would be tantamount to granting that there could have been a different range of possible atomic facts. But that would be to say that it is possible for the possibilities to be different from what they are, and hence to appeal to a second-order notion of possibility, which is precisely what Wittgenstein’s univocal account of possibility14 denies. In the Notes, on the other hand, the notion of possibility is not even discussed, and it is hard to see how, with the resources Wittgenstein then had at his disposal, he could resist Russell’s argument. It seems likely, indeed, that the point simply had not occurred to him. This explains an otherwise puzzling difference between the Birmingham and Cambridge versions of the Notes. In the Birmingham version Wittgenstein insisted, ‘If we formed all possible atomic propositions, the world would be completely described if we declared the truth or falsehood of each.’15 I am inclined to think that Wittgenstein repeated this claim to Russell in Cambridge, but that Russell then offered the objection just outlined. Without any response to this objection Wittgenstein was forced to withdraw the general claim. Instead he restricted himself to the more limited observation that ‘whatever corresponds in reality to compound propositions must not be more than what corresponds to their several atomic propositions’,16 stumblingly repeated a little later: ‘Molecular propositions contain nothing beyond what is contained in their atoms; they add no material information above that contained in their atoms.’17 By a ‘molecular proposition’ Wittgenstein presumably meant a truth-function of elementary propositions, so what Russell’s objection now reduced him to asserting was no longer a substantial metaphysical claim, but rather a complete triviality. If in his conversations with Russell Wittgenstein withdrew the general claim pending the considerations in the Tractatus which allowed him to reinstate it, that would be at least a partial explanation for Russell’s confident denial of it in lectures in which he otherwise followed Wittgenstein’s line so slavishly. When he denied it in 1918, Russell was indeed still reporting what he still had no reason to doubt was Wittgenstein’s view. 14

6.375.

15

B36.

16

C2.

17

C10.

Logical data  If my conjecture is right, it is also remarkable for what it implies about the working relationship between Wittgenstein and Russell. If, as seems to be the case, the question whether generalization introduces further facts had simply not been raised between the two men before October 1913, that can hardly fail to highlight their lack of genuine collaboration over the previous few months.

. Logical data The issue just discussed is important for epistemology, at least if we think, as Russell did, that ‘all empirical evidence is of particular truths’. For in that case ‘if there is any knowledge of general truths at all, there must be some knowledge of general truths which is independent of empirical evidence, i.e. does not depend upon the data of sense’.18 As Russell went on to note, this result ‘is important, since it affords a refutation of the older empiricists’.19 He concluded that ‘there is general knowledge not derived from sense, and that some of this knowledge is not obtained by inference but is primitive’. He therefore owed an account, which he did not supply, of how it is possible to have primitive knowledge of facts not derived from sense. Russell called these facts ‘logical data’, but he might just as well (if more provocatively) have called them synthetic a priori truths. It was precisely this notion of logical data that Wittgenstein wanted to resist. He may have been temporarily nonplussed by Russell’s apparent counterexample of the fact that the atomic facts are all there are, but it remained his ambition to find an explanation which allowed him to deny this. Part of his reason for this was epistemological: he wanted to deny that there could be ‘logical experience’, since such a thing would contradict his fundamental insight that logic ‘must turn out to be a totally different kind than any other science’.20 But the point is also metaphysical: if there are general facts not reducible to atomic facts, as Russell maintained and Wittgenstein wished to find a principled reason to deny, then logical atomism, in the form in which Wittgenstein then understood it, is false. Wittgenstein’s atomism, that is to say, amounted at the time of the Notes to the view that all there is to the world is atomic facts; or, more precisely, all facts are just sums of (positive and negative) atomic facts. In particular, therefore, there is no room in this conception for any such thing as a logical fact. In light of this, it is reasonable to ask in what sense Russell understood himself to be a logical atomist despite his willingness to allow that not all facts are reducible to the atomic facts. The phrase itself owes nothing to Wittgenstein: Russell described his own view as ‘un atomisme logique’21 during a discussion in Paris in March 1911 before the two men had met. But it was never as good 18

OKEW, 56.

19

Ibid.

20

To BR, 22 June 1912.

21

CP, VI, 412.



Metaphysics

a description of Russell’s view as it was of Wittgenstein’s. The ontology of the Tractatus famously22 consists of facts, not of things. Wittgenstein no doubt meant various things by this, but one of the things he meant was that a conception of the world will not count as truly atomistic unless it holds that all facts are reducible to atomic facts. Russell was not an atomist in this sense: for him atomism was primarily a doctrine about things, not about facts. Even if we accept that all facts—in particular, general facts—are reducible to (i.e. nothing more than sums of) positive and negative atomic facts, there remains the question whether there are any logical facts. Wittgenstein denied that there are (and in the Lowell Lectures,23 presumably under his influence, Russell agreed). But is there anything other than a prejudice against the synthetic a priori that counts as a reason against them? Wittgenstein’s atomism does at least entail that if there are any logical facts, the atomic facts that they reduce to will have to be logical facts too. So the question reduces to whether an atomic fact could be logical. Wittgenstein’s ‘fundamental thought’ already rules out the possibility that there could be logical objects. So an atomic logical fact would have to have non-logical objects as its constituents. What, then, would distinguish it from a non-logical fact? Nothing, presumably, except its necessity. But that—that it is necessarily so—would then be a further fact. In addition to the atomic fact that p, in other words, there would also be the fact that # p. But this leaves the entailment # p ⊃ p unexplained, unless we suppose that it is a further fact, thus inviting an obvious infinite regress. The only escape route consistent with Wittgenstein’s understanding of atomism, therefore, is to conclude that there are no logical facts at all. The concepts of necessity and possibility are conspicuously absent from the Notes. The reason, I assume, is that they await the Tractarian conception of a proposition as expressive of a possibility as to how things might be. That conception is quite absent from the Notes—it is probably one of the things he arrived at in Norway—but once it is in place, the notions of necessity and possibility quickly follow. It is tempting, perhaps, to say that this makes modality in the Tractatus a matter of language, not of the world: we express how things are by means of a contrast with how else they might have been, and hence we fashion our conception of possibility. But if we think this through, what we come to realize is the flimsiness of the attempt to pin on the Tractatus a contrast between language and world at all. For the possibilities of combination are already present in the atomic facts; or, at any rate, in their being all the facts. And there seems no reason to place that on the side of language rather than of the world. 22

1.1.

23

OKEW, 53.

Chapter 

Sense A proposition has a meaning, which is the fact that makes it true or false. But it was central to Wittgenstein’s view that we can understand a proposition without knowing which of these two possibilities holds.1 The meaning is not something we come to know simply by virtue of understanding the proposition, since it depends also on how things stand in the world. So there must be a second ingredient in what a proposition expresses, something we can grasp in advance of finding out what its meaning is. This second ingredient Wittgenstein called the sense of the proposition. It consists in the conditions under which the proposition is true and the conditions under which it is false. What we know when we understand a proposition is this: We know what is the case if the proposition is true, and what is the case if it is false. But we do not know (necessarily) whether it is true or false.2

By saying that a proposition has both sense and meaning, Wittgenstein was of course quite consciously mimicking Frege, but although he kept the same terminology, and some of the same motivation, there are important differences in their usage. We have already seen that for Wittgenstein the meaning of the proposition is the fact that makes it true or false, not (as for Frege) its truth or falsity itself. And although the sense is still (as it was for Frege) what I grasp when I understand the proposition, Wittgenstein’s conception of it is based, as we shall see shortly, on a rather different understanding from Frege’s of the contributions made to it by the proposition’s components. . Semantic value A proposition can have various structures, subject-predicate, relational, etc. Wittgenstein focused solely on the relational case, not because he thought all propositions have this form—he was explicit that they do not—but purely for expositional simplicity. So let us follow him in this, and focus for convenience on a relational proposition ‘aRb’. Here there are three components, the two names ‘a’ and ‘b’, and the form ‘ xRy’. The contribution an expression makes 1 C5; B42; cf. 4.024. 2 C6. The version of this remark in the Birmingham Notes (B41) omits, presumably in error, the requirement that we should know what is the case if the proposition is false.



Sense

to the sense of propositions in which it occurs is its semantic value. Wittgenstein held that the sense of the proposition is determined by the semantic values of its components. To understand the sense of ‘aRb’, therefore, it suffices to grasp the semantic values of its three components.3 But what are these semantic values? Giving the sense of a proposition consists, we have said, in determining the conditions under which it is true and the conditions under which it is false. Wittgenstein starts by considering the idea that to understand the proposition p it is sufficient to be able to fill in the blank in the schema ‘ p’ is true if and only if . . . in such a way that what results is true. But, as he notes, this is too weak a constraint. It is incorrect to say: we understand the proposition p when we know that ‘ “ p” is true’ ≡ p; for this would naturally always be the case if accidentally the propositions to right and left of the symbol ‘≡’ were both true or both false.4

It is accidentally the case, for instance, that ‘Snow is white’ is true if and only if grass is green. We need a tighter constraint on the schema if we are to rule out cases like this. In order to obtain such a constraint, Wittgenstein suggested, we need to introduce more structure into the T-schema, i.e. an extra variable, and insist on a formal equivalence with respect to that variable. On the face of it the most obvious such variable would be the possible world in which the proposition is to be true. Our criterion for understanding the proposition p would then be that we could fill in the gap in the scheme For every possible world W, ‘ p’ is true-in-W if and only if in W . . . in such a way as to make it true. The completion For every possible world W, ‘Snow is white’ is true-in-W if and only if in W grass is green is now ruled out, because there are possible worlds in which snow is white but (because of a genetic mutation, perhaps) grass is red. I have already suggested in my discussion of Wittgenstein’s account of meaning that he was reluctant to appeal to possible worlds in his semantics. His technique on that occasion was to resort to actual-world quantification to achieve the restrictions he needed. It is therefore no great surprise that in his account of sense he was not attracted by the appeal to possible worlds just sketched, but instead restricted the schema by an actual-world quantification. 3

B41.

4

B44.

The semantic value of a form  What is needed, he says, ‘is bound up with the introduction of the form of p’.5 And again, ‘What is wanted is the formal equivalence with respect to the forms of the proposition, i.e. all the general indefinables involved.’6 In the case of our relational proposition ‘aRb’, for instance, quantifying over the name positions gives us ( x,y)(‘a’ means x ‘b’ means y

⊃

‘aRb’ is true ≡ x has R to y),

where ‘a’ and ‘b’ are being used, like ‘R’, schematically. One thing we can deduce from this straightaway is what our earlier discussion of Wittgenstein’s conception of simplicity already led us to expect, namely that the semantic value of a name is its meaning. The object-level quantification in the schema ensures that different names which refer to the same object make the same contribution to the sense of propositions in which they occur, and so names do not have Fregean sense. . The semantic value of a form What remains, though, is to work out the semantic value of the form ‘ xRy’. Wittgenstein’s account of this is repeated in both the Birmingham and Cambridge versions of the Notes.7 What is striking about this account is the care he took to proceed via the form, so as to reach the sense of the proposition only derivatively. He was not content, for instance, simply to say that ‘aRb’ is true if a has R to b, and false otherwise. His reason for taking the more elaborate route is that only by going via the form of the proposition could he ensure the connection he wanted between the senses of ‘aRb’ and ‘cRd’: the relationship that ‘aRb’ expresses as holding between a and b must be the same as the relationship that ‘cRd’ expresses as holding between c and d. That is the constraint the theory had to meet, and it is clear Wittgenstein understood it. He emphasized, for instance, the need to explain ‘propositions such as ( x,y) xRy and similar ones’ in such a way as to show that they ‘obviously have in common with aRb what cRd has in common with aRb’.8 As he could have learnt from Russell, who had alluded to the point in the Principles,9 the dependency of a proposition on the names occurring in it is not merely that of a function on its arguments. Given his reluctance to countenance possible-world talk, it is no surprise to find Wittgenstein insisting on an extensional understanding of forms. E

Let us consider symbols of the form ‘ xRy’; to these correspond primarily pairs of objects, of which one has the name ‘ x’, the other the name ‘y’. The x’s and y’s stand in various relations to each other, among others the relation R holds between some, but not between others.10 5

B44.

6

C40.

7

B43; C38.

8

B51.

9

Principles, ch. 7.

10

B43.



Sense

In other words, what it involves to specify the semantic value of the form ‘ xRy’ is primarily to determine which pairs of objects are related by the relation R and which are not. But it is easy to see that this is not sufficient to determine which form we intend. To simplify the exposition, let us consider for a moment the monadic case. The parallel proposal in this case would be that we can fix the semantic value of a propositional function simply by determining which objects fall under it and which do not. Consider a world with three objects a, b, and c, and two atomic forms f and g; suppose that the positive facts that occur are fa, f b, and ga. I might try to determine a propositional function by specifying that its extension is to be the class {a,b} (so that a and b fall under it but c does not). But this plainly does not specify the function uniquely: it does not fix whether it is f x or f x v gx, for instance. The point is, of course, that although it is true in the world just described that these two forms are equivalent—i.e. the same objects satisfy them—the truth-conditions of propositions involving them are not equivalent: the sense of fa is not the same as the sense of fa v ga. So it is not possible to fix the semantic value of a form solely by determining its extension: we need to make our specification more fine-grained. Wittgenstein’s proposal for achieving this finer grain is that we have to correlate the proposition with facts. He therefore introduces the idea that facts may be gleichsinnig with, or entgegengesetzt to, a proposition. Russell translates this in the Birmingham Notes by talking of facts as ‘of like sense’ with, or ‘of opposite sense’ to, the proposition.11 In the Cambridge Notes Wittgenstein talks instead of the proposition as ‘true to the fact’ or ‘false to the fact’.12 (Notice, incidentally, that not all facts can be categorized as being of like or of opposite sense to a proposition. The fact that the cup is warmer than the saucer, for instance, is neither of like nor of opposite sense to the proposition that the cup is sitting on the saucer: they stand in quite different dimensions of variation.) Now that we have the notion of correlation with facts in place, we can return to the problem that led us to invoke it, namely that of determining the semantic value of a form. This, we can now say, involves specifying which facts are of like sense with the form and which of contrary sense. In the example considered earlier, where the positive facts are fa, f b, and ga (so the negative facts are ∼ f c, ∼gb, and ∼gc), we could fix the form f x by specifying that fa and f b are of like sense to it and ∼ f c is of opposite sense; to fix on the form f x v gx, we would specify that fa, f b, and ga are of like sense, ∼ f c, ∼gb, and ∼gc of opposite sense. Wittgenstein says elsewhere13 that a form is like a line dividing a plane. Presumably the analogy he intends is that the form divides a certain group of facts (the plane) into two classes. In the case we are considering here, f x is like a line which has fa and f b on one side of it, ∼ f c on the other. Notice that 11

Ibid.

12

C38.

13

B23.

The compass-needle analogy  f x v ∼ f x now emerges as a special case because all the facts are of like sense with it: it does not divide a plane. Similarly, f x ∼ f x does not divide a plane, since it is of opposite sense to all the relevant facts.

. The compass-needle analogy One of the challenges an account of sense faces is how to deal with the counterfactual element. It is worth pausing to note how Wittgenstein’s account deals with this challenge. Grasping a proposition is not simply a matter of seeing the fact that actually makes it true or false, but rather of understanding what would make it true or false however things turn out to be. One would like to say here, ‘which facts would make it true or false’. But we cannot say that, because facts are actual: there are no such things as merely possible facts. Our first temptation, that is to say, is to explain the sense of the proposition as a categorization of possible facts as being of like or opposite sense to it, but we are forced to recognize that that explanation is not available to us. This is yet again the problem of false propositions which had bedevilled Russell. Wittgenstein’s account of meaning had elegantly allowed him to provide a meaning for the false proposition that Charles I died in bed. The reason that the proposition is false is precisely that the fact that is its meaning is incompatible with its sense. So far, so good. However, it is also part of understanding the proposition to know that if Charles I had died in bed, that—which would then have been a fact—would have been of like sense with the proposition. But as things actually stand, it is not a fact; it is nothing at all. So instead Wittgenstein suggests that as part of our account of the sense of the proposition ‘Charles I died in bed’ we point to the fact that Charles I did not die in bed, and specify that this fact is of contrary sense to the proposition. In an attempt to make this account plausible, Wittgenstein exploits another meaning of the word ‘sense’, as applying to the direction of an arrow. For Frege this would have been no more than an unintended pun; but for Wittgenstein it is the basis of an extended analogy. The sense of a proposition is akin to the direction of an arrow: p and ∼ p point to the same fact, i.e. they have the same meaning; but they point to it in opposite directions, i.e. they have opposite senses. It is hard to avoid the thought that Wittgenstein has in mind some sort of analogy with electromagnetism. Perhaps the proposition is to be thought of as akin to the needle of a compass placed next to a coil of wire. When current passes through the wire, the compass needle aligns itself with the magnetic field created; if the direction of the current is reversed, the compass needle turns to point in the opposite direction. Wittgenstein’s use of the word ‘pole’ thus echoes its use to refer to the ends of a magnet, or the terminals of a battery. Reversing the direction of current flow by swapping



Sense

the terminals corresponds to negation. Of course, variants of this analogy are possible too: I might hold the compass needle fixed in my hand, in which case a magnetic field in the opposite sense to the needle will not make it rotate but will nonetheless generate a force, which I will be able to detect. As with any analogy, not every aspect of the story is relevant, and there is room for doubt as to which aspects Wittgenstein wished especially to draw on. One aspect that surely attracted him, though, is that the compass needle is disposed to react in a certain way to the presence of a magnetic field whether or not the field is actually present. Wittgenstein presumably hoped to draw on this analogy to make plausible the idea that a proposition provides the means to categorize facts as being of like or contrary sense even though some of the facts to be categorized are not actual. A proposition is to be thought of, that is to say, as carrying within it the disposition to be of like or contrary sense to the facts, however the facts turn out to be. . Grain It follows from Wittgenstein’s account that two propositions can have different senses only if there is a difference in the circumstances that would make them true. But this does not straightaway determine how fine-grained the notion is. Russell, for instance, was quite happy with the idea that there are logical facts, and therefore could have maintained that different logical truths have different senses. Wittgenstein reached the contrary view that all logical truths have the same sense by way of his rejection of logical facts. Sense is therefore much less fine-grained for Wittgenstein than for Russell. However, it is also much less fine-grained for Wittgenstein than for Frege, who in Grundgesetze held that logically equivalent propositions may differ in sense. According to Dummett, this alone is enough to show that sense is not [for Wittgenstein] correlative to understanding, since one may, in the ordinary meaning of ‘understand’, understand two sentences without realizing that they are equivalent.14

But this is to locate the difference between Frege and Wittgenstein at the wrong place. There may indeed be a meaning of ‘understand’ on which what Dummett says is right, but that is not Wittgenstein’s meaning. On his usage, we do not fully understand a proposition until we grasp what it says about the world (or, in the case of a tautology, that it says nothing). And that is surely a usage that is to be found in ordinary language too. In the metaphor Wittgenstein later introduced, we do not understand the proposition until we picture what it says. So on Wittgenstein’s use of the word ‘understanding’, sense is correlative to understanding. 14

FPL, 680.

Grain  For all we have said so far, though, this is no more than a difference of terminology. To see why it is more than that, we must return to the contrast between Wittgenstein’s and Frege’s conceptions of logic. Both started from the idea that the sense of a proposition is what it says. We might be tempted to qualify this by adding that the sense is what the proposition says about the world, but for Wittgenstein (though not for Frege) that is otiose, since there are no logical objects and hence there is nothing else against which the world can be set—nothing else for the proposition to be about but the world. Now logical truths all say the same thing, which means, as just noted, that they say the same thing about the world. But what they say about the world is nothing: they do not rule any atomic facts in or out, since if they did so, that would make those atomic facts logical, contrary to Wittgenstein’s view that there are no such things as logical facts. It follows that if logic is assessed only with respect to what it says, it is trivial. The only space left for Wittgenstein to find anything non-trivial in logic is not in what it says but in how it says it. This is where the true disagreement between Wittgenstein and Frege is to be found. For Frege (and indeed for Russell) the fruitfulness of logic was proof that it has content; for Wittgenstein the only explanation for logic’s special character was that it has none. This left Wittgenstein with an emerging problem over what to say about the logicist project that Frege and Russell had been conducting. Part of the reason for their conviction that logic is non-trivial was their belief that logic entails substantial parts of mathematics: since they took it as evident that mathematics is not trivial, it followed that they could not admit logic to be trivial either. If Wittgenstein wished to maintain nonetheless that logic is trivial, he had either to insist, implausibly, that mathematics is trivial too or to provide a quite different, non-logicist, explanation of its role in reasoning. Perhaps it is no accident that just when Frege began to think his logicist programme was a failure, he also came to accept that logically equivalent propositions have the same sense.15 By the time Wittgenstein knew him, therefore, he would have been willing to grant that the sense of any logical truth is trivial, and hence that the explanation for the complexity of mathematics is to be found elsewhere. 15

To Husserl, 9 Dec. 1906.

Chapter 

Truth-functions

A truth-function is any function taking a finite list of truth-values as arguments and producing a truth-value as its result. In the Notes, to Russell’s exasperation, Wittgenstein chose to call them ‘ab-functions’. I shall mention his reasons for this later, but in the meantime I shall stick to the more standard terminology. In this chapter I want to describe the notations Wittgenstein used at various times for describing a truth-function. Since the eventual aim will be to discuss the relationship between a logical connective and the corresponding truth-function, it will be sensible to distinguish notationally between the two. I shall therefore, for the moment at least, use ‘not’, ‘or’, ‘and’, etc. for the connectives, and reserve ‘∼’, ‘v’, ‘.’, ‘⊃’, ‘≡’, ‘|’ for the various truth-functions.

. Using primitive signs One sort of notation for expressing truth-functions is simply to make an arbitrary choice of sign to stand for each one. But the number of n-ary truthn functions is 22 , so even in the binary case there are sixteen that need to be symbolized. It turns out, though, that all the truth-functions can be obtained by composition from a small subset of them. So all we need to do is to choose a symbol for each of these and then express others as compounded out of them. If such a notation is to be adequate to express all truth-functions, of course, those for which we have primitive symbols must be chosen with care. The truth-functions ∼ and v are jointly adequate, for example; so are ∼ and ⊃; but ∼ and ≡ are not. This point is not mathematically sophisticated, and no doubt many of those who worked on aspects of the propositional calculus in the nineteenth century were aware of it. Frege, for instance, presented a formalization of the propositional calculus in the Begriffsschrift in which he chose ∼ and ⊃ as the primitive functions, no doubt conscious that they are truth-functionally adequate, whereas Russell chose ∼ and v. There are two truth-functions each of which is expressively adequate on its own. They are the functions p | q and p † q defined as follows:

Using primitive signs  p

q

T T F F

T F T F

p|q

F T T T

p

q

T T F F

T F T F

p†q

T T T F

These two truth-functions are known as ‘Sheffer stroke functions’ after their discoverer. Confusingly, there is no agreement which of the two to denote by the stroke sign ‘|’. In the table above I have followed Wittgenstein’s usage according to which p | q expresses ‘not- p or not-q’; the dual function, which I have written here p † q, expresses the joint denial of p and q. The paper in which Sheffer showed that each of these functions on its own is adequate to express all truth-functions1 appeared late in 1913, but Russell was sent a copy in April and there is ample evidence in the Notes that Wittgenstein knew its content. He uses Sheffer’s notation, for example,2 and he states Sheffer’s result that | is adequate to express all truth-functions.3 I mentioned in §8.3 the idea Wittgenstein got from Hertz that the indefinables in a system should be independent from one another. In the Notes Wittgenstein applied that idea to the truth-functions in three ways, none of them very convincing. First, he claimed that ∼ and v are not independent: if by this he meant that one can be defined in terms of the other, then the claim is trivially false; if not, it is wholly unclear to me what else he could have meant. Second, Wittgenstein thought that if a system can be presented using one indefinable, then presenting it using two must contravene Hertz’s principle. This is true for vectors in space, of course—different bases for the same space always have the same number of vectors in them—but it is not true for axiom systems, as ∼ and v show. Wittgenstein’s attachment to this faulty reasoning seems to have made him unduly impressed by the fact that Sheffer’s stroke function | is on its own adequate to express all the truth-functions. It may indeed be more elegant to present logic using one primitive sign rather than two, but it does not have any ultimate theoretical privilege. Third, Wittgenstein thought that the existence of alternative sets of indefinables shows that ‘these are not the right indefinables’.4 It is hard to see why he might have believed this, since not even the corresponding claim about vectors is true—any non-trivial vector space has various different bases—nor is it true for his favoured Sheffer stroke |, since the dual stroke function † is also adequate. Russell reminded him of this in 1919, suspecting as a consequence that duality ‘persisted covertly in your system’,5 but Wittgenstein brushed the point away with what looks very like bluster. ‘This doesn’t matter! . . . All is said in my book about it and I feel unable to write it again. Try to understand it till we meet.’6 1 2

‘A set of five independent postulates for Boolean algebras, with application to logical constants’. B31. 3 B30. 4 B18. 5 13 Aug. 1919. 6 19 Aug. 1919.



Truth-functions

. Truth-tables One obvious objection to the notations we have just been considering is that because they are chosen more or less arbitrarily, they do not wear the truthfunctions they stand for on their faces. An alternative approach is to devise a sign for the truth-function which explicitly identifies the truth-table it corresponds to. Examples of truth-tables in Wittgenstein’s handwriting have been preserved on the back of one sheet of some of Russell’s notes for his work on matter. So truth-tables were among the things they discussed at their meetings (most likely some time after Russell had received Sheffer’s paper). Whether representing a truth-function by means of a table was Russell’s idea or Wittgenstein’s is hard to say. (If I had to guess, I would opt for Russell.) But even if these are the earliest known examples, they have no great significance in themselves. The idea of notating a function on a finite domain by means of a table of arguments and values is an obvious one, and had been used frequently by nineteenth-century authors with whom Russell would have been familiar. All that he and Wittgenstein were doing was to apply this idea to the particular case where the domain in question consists of the two truth-values. The significant step for logic is rather that of identifying this case as an important one to study. A truth-function, as we are using the term here, is a function in the mathematical sense: an n-ary truth-function takes a list of n truth-values as arguments and delivers a single truth-value as its value. Because the values delivered by truth-functions always belong to the same set as the arguments, it is possible to compose truth-functions with one another to form further truthfunctions. The criterion of identity for truth-functions is the extensional one familiar from mathematics: functions count as identical if they deliver the same value as output whenever they are given the same arguments as input. What this amounts to when expressed in terms of truth-tables is that all that matters in settling the identity of the truth-function a truth-table represents is the main column, i.e. the one corresponding to the final result. A slightly more compact alternative to truth-tables, used in the Tractatus itself,7 is to agree once and for all a convention as to the order in which the lines of the truth-table occur, so that a listing of the last column could then suffice by itself to represent the whole truth-table. The truth-function p | q, for instance, might then be written as (F T T T)( p, q). . Truth-diagrams Although, as we have just seen, Russell and Wittgenstein were both familiar with the idea of truth-tables by October 1913, there are none in the Notes. 7

4.442.

Truth-diagrams  Instead Wittgenstein made use of another notation for representing truthfunctions which, unlike the truth-table, is undoubtedly of his own devising. It works as follows. First write ‘T’ and ‘F’ (what Wittgenstein called the ‘inner poles’) next to letters representing each argument place of the function. Then write one ‘T’ and one ‘F’ on the diagram (what Wittgenstein called the ‘outer poles’). Now add a series of lines connecting each outer pole with the sets of inner poles which, when used as arguments for the function, produce that pole as value. (Wittgenstein used the letters ‘a’ and ‘b’ instead of ‘T’ and ‘F’, but his reasons for doing so need not detain us at the moment.) Here is an example. T

                                       

.. .... ... ... ... .. .. .. .. .. .. .. .. .... .... ... . .. .. .. ... ... .. .. .. .... .. . . ... . .. ... ... .. .

T q F

                                                    

T p F

! F

The outer ‘T’ is connected to the pairing of the two inner ‘T’ poles; the outer ‘F’ is connected to the other three pairs. So this diagram represents the same function as the following truth-table. p

q

T T F F

T F T F

T F F F

In other words, it represents the conjunction function p q. In the same way the following example represents the function which takes the value T just in case its arguments are both T or both F, i.e. the function p ≡ q. F..

T q F

                                             

T p F

                        

.. . .. .. ..

                        

. .... ... ... ... ...

# # " # "

T

There is evidently no significance in the particular spatial arrangement chosen for the diagram: all that matters is which outer poles are connected to which sets of inner poles. For instance, we could write the diagram for ∼ p as follows.



Truth-functions T $ $ T p F $ $ F

But it would amount to the same to write this linearly as F − T p F − T, or indeed as T − F p T − F. In the same way the diagram for p is just T p F, or indeed F p T. As Wittgenstein said in a letter to Russell, ‘The proposition p has two poles and it does not matter a hang where they stand.’8 It is also permitted, of course, for the same argument to occur more than once in a truth-function. The diagram for p ≡ p, for instance, is obtained from the diagram for p ≡ q above by substituting p for q. F .. .. T p F

                                             

T p F

                        

... . .. .. ..

                         

.... ... ... ... ...

# # " # "

T

Now p ≡ p is an example of what Wittgenstein would very soon after he had written the Notes begin to call a tautologous function, i.e. a function which takes the value T for all combinations of arguments. The fact that the function is tautologous can be read off from the truth-diagram once we have learnt to ignore impossible links—connections between an outer pole and opposite poles of the same atomic proposition. In the diagram for p ≡ p just given, for instance, there are two impossible links from the outer F-pole to both T and F poles of p. If we ignore these impossible links, we observe that the only remaining links are to the outer T-pole, and so the function is tautologous. A recurring theme of Wittgenstein’s letters from Norway is Russell’s evident struggle to understand his truth-diagram method. Certainly what Wittgenstein says in the Notes themselves could hardly count as a sufficient explanation on its own. And his response to Russell’s requests for clarification might be thought to lack a certain sympathy. It distresses me that you did not understand the rule dealing with signs in my last letter because it bores me BEYOND WORDS to explain it. If you thought about it for a bit you could discover it for yourself! . . . I beg you to think about these matters for yourself: it is INTOLERABLE for me to repeat a written explanation which even the first time I gave only with the utmost repugnance.9

(It was at about this time10 that he told Moore he thought Russell was past it.) Truth-diagrams can be composed in a manner that replicates the effect of composing the truth-functions they represent. It cannot have helped Russell 8

LW to BR, [Nov.] 1913 (CL, no. 28).

9

[Nov. or Dec. 1913] (CL, no. 32).

10

19 Nov. 1913.

Comparison  in his efforts to understand the method that Wittgenstein did not give him any examples of compound diagrams in the Notes. He would probably have found it helpful to be told, for instance, that we can compose the diagrams for p q and p ≡ q to obtain the following diagram for p ≡ ( p q).

                                                                                                    

F........ ........ . % ......................... % ....... ....... %..... ... .. % . . ... T p F .... %T p F .... ... . . . ... % . . .. .... % ................... .... ..... ..........

T

T q F

                                                                                                                 

.. ... . ... .. .. .... .. .. .. .. .. .... .... ... . .. .. .. .. .. .. .. .. .. ... .. .. .... . . ... ... .... .. ... . .

! F

... ... ............. .... ................. .............................

T

We noted earlier that all that matters in settling the identity of a truthfunction is the output for various possible inputs, and that this is represented in the corresponding truth-table by means of the main column. When the truthfunction is represented by means of one of Wittgenstein’s truth-diagrams, the analogous point is that all that matters is which sets of inner poles are connected to which of the outer poles: intermediate poles drop out of the picture as irrelevant. Or, as Wittgenstein put it, ‘In this notation all that matters is the correlation of the outside poles to the poles of the atomic propositions.’11 And, he ought to have added, we need as before to ignore impossible links. When we do that, we see, for instance, that the diagram for p ≡ ( p q) just given is equivalent to the following diagram for p ⊃ q.

                                                     

F .. .. .. ... ... ... ... ... ... ... ...

T q F

                                      

T p F

& & T

As we shall see in chapter 20, Wittgenstein came to hope (wrongly) that the truth-diagram notation could be extended so as to represent quantifiers as well as truth-functions. But that idea emerges only at the very end of the Cambridge version of the Notes; in the Birmingham Notes the method is used only for truth-functions. . Comparison Some time after the Notes (perhaps when he realized that it does not provide a general method of the kind he sought for symbolizing quantified sentences) Wittgenstein put the diagram method aside. It makes no appearance in the 11

C31.



Truth-functions

wartime notebooks or in the Prototractatus. But he did not forget it completely: an account of the method is one of the few things he added to the Tractatus typescript while he was a prisoner of war after the 1918 armistice. In the account he gives there12 he offers as an illustration the diagram for p ⊃ q above. He also works through an example to show how truth-diagrams can be used to decide whether a proposition is a tautology. The case he discusses is the obviously tautologous proposition ∼(q ∼q), whose truth-diagram he pieces together from the diagrams for conjunction and negation shown earlier. F

                                                  

T

T $ $ T q F $ $ F

... ... ... .... ... .. .. .. ... .. .... ... ... ... .. .. .. ... ... ... .

                                                      

T q F

# # #

F

T We can see from the diagram that ∼(q ∼q) is a tautology because the only link between the outer F pole and the inner poles is impossible. In the Tractatus Wittgenstein bravely described this as an ‘intuitive method’13 for recognizing a tautology. It is worth comparing with the nowadays more familiar method of showing that ∼(q ∼q) is a tautology by means of the following truth-table. q

T F

∼(q ∼q) T FF T FT

Theoretically, of course, there is nothing to choose between truth-diagrams and truth-tables, since the two methods are obviously equivalent. In practice, on the other hand, truth-diagrams are at a significant disadvantage to truth-tables: even in simple cases they are hardly faster to write, and in more complicated cases the diagram tends quickly to become a tangled mess. 12

6.1203.

13

Ibid.

Chapter 

Truth-operations From a single proposition p we form its negation not- p. From two propositions p and q we form various further propositions, e.g. p or q, p and q, etc.; similarly (though less often) with three or more propositions. For each such method of forming propositions it is common to speak, as we frequently did in the last chapter, of a certain truth-function as corresponding to it. We speak, for instance, of ‘the truth-table for negation’ or ‘the truth-table for the Sheffer stroke’. On the one hand, then, there is a function which takes senses of propositions as inputs and returns the sense of a proposition as output. (Later, most notably in the Tractatus itself,1 Wittgenstein called such functions operations, and although this is not the term he used in the Notes, it will be convenient to adopt it here.) On the other hand, there is a truth-function, i.e. a function taking some finite number of truth-values as arguments and giving a single truth-value as value, which is representable by a truth-table or truth-diagram. What I want to discuss in this chapter is the relationship between the two. . The problem For Frege, of course, this issue hardly arises, since he took sentences to be names of truth-values and could therefore simply combine them using signs for the truth-functions. He could regard the expression ‘ p v q’, for instance, as naming the output which results if we apply the truth-function v with the truth-values named by ‘ p’ and ‘q’ as inputs. On his account, it seems, the truth-operation just is the truth-function. For Wittgenstein, though, this beguilingly simple account was not available, since he rejected the conception of sentences as names of truth-values on which it depends. There is thus a mismatch in type between sentences and the sorts of items that can fill the argument places of a truth-function. And even if that mismatch could be corrected by some procedure that converted sentences into names, there would remain a further, more serious mismatch at the output level: the output of a truth-function is a truth-value, and the point of Wittgenstein’s objection to Frege’s account of sentences as names of 1

5.23.



Truth-operations

truth-values was precisely that there is no general route back from there to the proposition expressing what it is that has that truth-value. Simply putting an assertion sign in front of it, or tagging ‘is true’ on the end, will not do the trick. In other words, if it is a mistake to think of ‘ p’ as a name of a truth-value, it must equally be a mistake to regard ‘∼ p’ as such.2 What is uncontroversial, though, is that some operations, those we might call ‘truth-operations’, give rise to truth-functions. Examples are the operations of disjunction, conjunction, and negation. We have seen, on the other hand, that such truth-operations cannot be regarded as simply being the same as the corresponding truth-functions, or as obtained by composing them with some suitable connecting devices. For all we have said so far, then, one might entertain a doubt as to whether the truth-operation is obtainable from the truth-function at all. One reason for doubt would be if the relationship between a truth-operation and its corresponding truth-function were not one–one. If, for instance, we individuated operations in such a way that p | q counted as different from ∼ p v ∼q, then unless some reason could be given for privileging one of these operations over the other, neither of them would be obtainable from the corresponding truth-function. That is not the point at issue here, however. Wittgenstein individuated operations in just such a way as to ensure that the correspondence is one– one. Moreover, he evidently did think that it is possible to derive the truthoperation from the truth-function. He mentioned with evident approval, for example, Frege’s truth-functional explanations of ‘not p’ and ‘if p then q’.3

. The solution How, then, do we obtain the sense of the output of a truth-operation from the sense of the input propositions? Let us consider as an example the case of the connective ‘or’. In order to make the connection between the truth-function and the truth-operation, we first define

% T if ‘ p’ is true Val( p) = F if ‘ p’ is false Our task is now to give a sense to expressions of the form ‘ p or q’ in such a way that we always have Val( p or q) = Val( p) v Val(q). 2

Cf. 4.431.

3

B22.

The solution  In order for this equation to hold, we must have ‘ p or q’ is true ≡ Val( p or q) = T ≡ Val( p) v Val(q) = T

≡ Val( p) = T or Val(q) = T ≡ ‘ p’ is true or ‘q’ is true

We can use this equivalence to derive an account of the sense of any proposition in which ‘or’ occurs. For instance, ‘aRb or cS d’ is true

≡ ≡

‘aRb’ is true or ‘cS d’ is true the facts about the meanings of ‘a’ and ‘b’ accord with the sense of ‘ xRy’ or the facts about the meanings of ‘c’ and ‘d’ accord with the sense of ‘ xS y’.

This account derives from the truth-function since it is this that tells us the condition for propositions with the structure ‘aRb or cS d’ to be true. The key step in this account of disjunction is the requirement that Val( p or q) = Val( p) v Val(q). It is this that enables us to derive the equivalence ‘ p or q’ is true ≡ ‘ p’ is true or ‘q’ is true, and hence gives us a way of deriving the sense of ‘ p or q’ from the senses of p and q. Can this be generalized to any truth-function? If f is an n-ary truthfunction, can we introduce a corresponding n-ary truth-operation f ! by means of the specification that for any propositions p1 , p2 , . . . , pn Val( f ! ( p1 , p2 , . . . , pn )) = f (Val( p1 ), Val( p2 ), . . . , Val( pn ))? What our discussion has shown is that simply writing down this equation is not itself enough to secure what we want. What we need to show is that the equation suffices to give a sense to f ! ( p1 , p2 , . . . , pn ) whenever p1 , p2 , . . . , pn have senses. There is of course no great problem about doing this, but the procedure has a feature that is worth noting. If we mimic the reasoning we gave earlier for the case of disjunction, we obtain easily enough ‘ f ! ( p1 , p2 , . . . , pn )’ is true ≡ Val( f ! ( p1 , p2 , . . . , pn )) = T ≡ f (Val( p1 ), Val( p2 ), . . . , Val( pn )) = T ≡ ...



Truth-operations

But what do we write next? We want to convert this last line into a condition depending on which of ‘ p1 ’, ‘ p2 ’, . . . , ‘ pn ’ are true. We could do that, of course, if we simply assumed the availability (‘in the metalanguage’, as we would say nowadays) of the truth-operation f ! whose sense we are trying to secure. If we do not assume that, however, we have to inspect the function f in order to determine just what the condition we want is. In other words, the method for conversion, although it depends uniformly on f , does not depend schematically on it. This last point is of some significance since it shows that the way that ‘ p’ and ‘q’ occur in ‘ p or q’ is fundamentally different from the way that ‘or’ does. This difference is what makes it permissible to quantify over the former positions in the proposition but not over the latter. . Duality One further point is worth noting about the relationship between truth-functions and truth-operations. What we have been discussing is a means by which we can, for any truth-function f , find an n-ary truth-operation f ! such that Val( f ! ( p1 , p2 , . . . , pn )) = f (Val( p1 ), Val( p2 ), . . . , Val( pn )), where Val is a function defined so that % T if ‘ p’ is true Val( p) = F if ‘ p’ is false. What we should note is that the identity of f ! depends directly on the choice of the function Val. If, for instance, we used instead the function % T if ‘ p’ is false Val∗ ( p) = F if ‘ p’ is true then we could certainly by an analogous procedure find an n-ary truth-operation f !! such that Val∗ ( f !! ( p1 , p2 , . . . , pn )) = f (Val∗ ( p1 ), Val∗ ( p2 ), . . . , Val∗ ( pn )), but it would not be the same operation as f ! . Since Val∗ ( p) = Val(∼ p), it follows that ‘∼ f !! ( p1 , p2 , . . . , pn )’ is true ≡ ‘ f !! ( p1 , p2 , . . . , pn )’ is false ≡ Val∗ ( f !! ( p1 , p2 , . . . , pn )) = T ≡ ≡

f (Val∗ ( p1 ), . . . , Val∗ ( pn )) = T f (Val(∼ p1 ), . . . , Val(∼ pn )) = T

≡ ‘ f ! (∼ p1 , ∼ p2 , . . . , ∼ pn )’ is true.

Duality  So the relationship between f ! and f !! is that ∼ f !! ( p1 , p2 , . . . , pn ) has the same sense as f ! (∼ p1 , ∼ p2 , . . . , ∼ pn ). In other words, f !! is the same as the operation ( f ∗ )! derived from the dual f ∗ of the truth-function f (i.e. the truthfunction whose truth-table is obtained from that of f by swapping all the occurrences of T and F). The point to note is that whether we associate the truth-function f with f ! or with f !! depends entirely on whether we decide to adopt Val or Val∗ . Each of these functions establishes a link between the concept of truth and the objects T and F, but it is a matter of convention which way round we establish that link. Of course, to any English speaker our choice of the letters ‘T’ and ‘F’ as names of the two objects indicates which convention we intend to adopt: once we have made that choice, to adopt the opposite convention would be wilfully confusing. Nonetheless, it is worth stressing that the other convention, although notationally unhelpful, would not be in any deep way mistaken. Our choice of name is not a property of the object itself, and no object has any essential connection with truth (or, for that matter, with falsity).

Chapter 

Molecular propositions As we saw in chapter 9, it is for Wittgenstein not the complex that does the symbolizing in a proposition, but rather a particular fact about the complex, namely that fact about it which ensures that the proposition expresses the sense that it does. It is this fact—the symbolizing fact—that we see in the complex when we read it as expressing the proposition. The question therefore becomes urgent of trying to say what this symbolizing fact is. It is not the identity of the sentences that is in question now. The sentence ‘ p or q’, for instance, is readily enough described as a complex consisting of the sentence ‘ p’ followed by the word ‘or’ and then the sentence ‘q’. What we want to do now is to identify what the fact is that we see exemplified in this complex when we read it as expressing a proposition. . Terminology In the Cambridge Notes Wittgenstein did not explicitly define what he meant by a ‘molecular’ proposition, but it is clear from the text that he meant any proposition obtainable from elementary propositions by the application of a finite number of truth-operations.1 Confusion is possible, however, because Russell’s own usage around this time was different. What Wittgenstein called elementary propositions Russell preferred to call ‘atomic’; the propositions Wittgenstein called molecular Russell called ‘elementary’; and Russell used ‘molecular’ to mean non-atomic. There are traces, both in the Cambridge Notes and in the correspondence they conducted thereafter, of a running battle Wittgenstein and Russell were fighting over terminology. Russell did not object to new words for new concepts, but he did (reasonably enough, perhaps) put pressure on Wittgenstein to keep to old words for old concepts—to use established terminology where possible. ‘What you call ab-functions are what the Principia calls “truth-functions”,’ he observed at one point. ‘I don’t see why you shouldn’t stick to the name “truth-functions”.’2 And in the Cambridge Notes he briefly managed to get Wittgenstein to call his elementary propositions by Russell’s term ‘atomic’.3 1

C11; C14; C31.

2

CL, no. 29.

3

C31; C33.

Which fact?  This terminological battle resulted in a certain amount of confusion, not just for the modern reader but also, it seems, for Russell. In his translation of the Birmingham Notes4 he used the term ‘complex proposition’ for what Wittgenstein was in English calling a ‘molecular proposition’. (The word Russell was translating was probably ‘zusammengesetzt’.) And at one point he seems to have forgotten that he had encouraged Wittgenstein to call Elementarsätze ‘atomic propositions’, translating the term instead as ‘simple propositions’.5 Wittgenstein’s letters to Russell during the autumn of 1913 show recurring evidence that Russell was still struggling to grasp his terminology: in one of them, for instance, we find Wittgenstein helpfully glossing ‘zusammengesetzte Sätze’ by using Russell’s term ‘elementary propositions’.6 Thus Wittgenstein’s claim that ‘every possible complex proposition is a simple ab-function of simple propositions’7 might have been clearer if Russell had translated it, ‘Every possible molecular proposition is a simple truth-function of elementary propositions.’ And Wittgenstein was here only reminding us of what he meant by a molecular proposition. There is no trace of Wittgenstein’s later claim in the Tractatus that all propositions are (possibly infinite) truth-functions of elementary propositions, and indeed various remarks in the Notes actively exclude it. . Which fact? In chapter 18 I explained a uniform procedure whereby each truth-function f gives rise to a corresponding truth-operation f ! . If we have a notation for describing truth-functions, we can convert it into a notation for the corresponding truth-operations. So each of the methods of representing truthfunctions that we described in chapter 17 corresponds to a method of representing molecular propositions. I mentioned earlier Wittgenstein’s hope that he might be able one day to find a ‘perfect notation’ in which there is a one– one correspondence between propositional signs and propositions. One of the attractions he saw in the truth-diagram notation is that although it is not quite such a notation, at least it is closer to being one than Russell’s or Frege’s is. For if a proposition is written in truth-diagram notation, it is at any rate possible by a sort of selective inattention to detail—by shutting one eye, so to speak— to see in the complex the relevant (i.e. symbolizing) fact. The diagram for ∼∼ p, for instance, is T–F–T p F–T–F. It is in this case a relatively easy matter to ignore the intermediate detail so as to see here the same symbolizing fact as in T p F, and to conclude from this that ∼∼ p = p. But even once we have devised an appropriate notation for expressing molecular propositions, how, in general, are we to discern the symbolizing fact 4

B31.

5

Ibid.

6

[Nov. or Dec. 1913] (CL, no. 32).

7

B31.



Molecular propositions

in it? Let us consider first the apparently trivial case where the truth-function in question is the identity function. The truth-table method, for instance, gives us the rather laborious expression p

T F

T F

for the proposition p. The abbreviated truth-table method gives us the rather more manageable expression (T,F)( p). Wittgenstein’s truth-diagram method gives us T p F, or, if we follow his notational eccentricity of using ‘a’ and ‘b’ to indicate the two poles, a p b. But for each of these we can ask the same question: what is it in the expression that symbolizes? In the case of the last mentioned notation, Wittgenstein himself offered an explanation. What symbolizes in the expression a p b, he said, is that ‘a is on the left of p and b on the right of p’. The puzzle that this explanation generates, however, is to explain the manner in which ‘ p’ occurs in here. If ‘ p’ is a complex, e.g. the sentence expressing the proposition, then according to Wittgenstein’s own doctrines it does not yet symbolize anything, since what symbolizes in a proposition is a fact, not a complex. What does symbolize is the fact that the pole letters ‘a’ and ‘b’ occur on either side of the complex. So the effect of Wittgenstein’s device of attaching ‘a’ and ‘b’ to p is just the same as that of writing ‘ “ p” is true’, i.e. it turns the complex into something expressing a proposition. But of course Wittgenstein knew quite well that this is incoherent. ‘The verb of a proposition is not “is true” or “is false”, but what is true must already contain the verb.’8 The alternative is to say that ‘ p’ occurs in ‘a p b’ not as a complex but as a fact. But now Wittgenstein’s explanation of what symbolizes in ‘a p b’, namely that ‘a’ is on the left of p and b on the right of p, makes no sense whatever. A fact is not the sort of thing that can stand in a spatial relation to anything else. If Wittgenstein was right to say that ‘what is true must already contain the verb’, he would have to say that ‘ p’ already symbolizes the proposition. The most Wittgenstein could hope to achieve by writing ‘a p b’ would in that case be to symbolize the same proposition in a rather more complicated manner, in just the same way as writing ‘It is true that p’ does. But that would be to render ‘a’ and ‘b’ wholly redundant. The only alternative, it seems, would be to say that ‘ p’ on its own does not express a proposition but is some sort of incomplete symbol. This, of course, looks worryingly similar to Russell’s misbegotten distinction between asserted and unasserted occurrences of the proposition, a distinction which in §10.2 we presented Wittgenstein as rejecting. 8

B10; cf. 4.063.

Poles  . Poles One thing which the three methods of representing truth-functions considered in chapter 17 have in common is that as we have described them so far, they all use the letters ‘T’ and ‘F’ in a uniform manner. But what manner is that? When we introduced them as devices for notating truth-functions, the answer was clear: truth-functions are functions whose arguments and values are objects for which we use the two letters ‘T’ and ‘F’ as names, and the role of these letters in the notation corresponds directly to their use as proxies for these objects. But when the signs are intended to refer not to truth-functions but to truth-operations, that direct connection drops away. As we noted in §18.3, the connection of the objects we have called ‘T’ and ‘F’ with the concept of truth is one we have to establish by convention. The role of the letters ‘T’ and ‘F’ is now only that of a mnemonic to remind us which of the two conventions we have chosen. In chapter 17 I talked as if Wittgenstein used ‘T’ and ‘F’ (or ‘W’ and ‘F’ in German) as signs for the two poles of a proposition. As noted in the last section, though, he did not: he used the letters ‘a’ and ‘b’ instead (and correspondingly called operations that can be represented by means of truth-diagrams ‘ab-functions’). I omitted mention of this before in order to stress that how he chose to label the two poles is a distinct issue from the relative merits of truthdiagrams and truth-tables. Whatever point Wittgenstein wanted to make by using the letters ‘a’ and ‘b’ he could have made just as well by substituting them for ‘T’ and ‘F’ in truth-tables (and, I suppose, calling them ‘poles’ rather than ‘truth-values’). This use of ‘a’ and ‘b’ for the two poles seems to have been a matter of some importance to Wittgenstein. The text of the Notes bears witness to another battle he fought on this point with Russell, who found it hard enough to work out from Wittgenstein’s explanations how truth-diagrams were supposed to work without the additional difficulty of having to remember which letter stood for truth. In the dictation made in his presence Wittgenstein briefly (at Russell’s insistence, presumably) used ‘W’ and ‘F’ instead,9 but a couple of paragraphs later he reverted to ‘a’ and ‘b’.10 Moreover, Wittgenstein maintained the practice of using ‘a’ and ‘b’ in the notes he dictated to Moore in Norway and continued to call the truth-functions ab-functions in his first two wartime notebooks.11 Why, then, was Wittgenstein so intransigent? What was wrong with Russell’s suggestion of using ‘W’ and ‘F’ instead of ‘a’ and ‘b’? One reason that has sometimes been suggested is that Wittgenstein wanted to point up the duality between truth and falsity already noted: by swapping the roles of ‘a’ and ‘b’ we 9

C31.

10

C33.

11

E.g. NB, 17 Dec. 1914.



Molecular propositions

transform a proposition into its dual. Now Wittgenstein was plainly aware of this duality—he discussed it explicitly in the Notes dictated to Moore—and there is little doubt that he was intrigued by it: the bipolarity of the proposition is, after all, an attempt to express this symmetry between a proposition and its negation. However, the idea of swapping the roles of ‘a’ and ‘b’ over can hardly be the whole explanation for Wittgenstein’s insistence on using these letters, since it is as straightforward to make this point using ‘W’ and ‘F’ as it is using ‘a’ and ‘b’. More plausible, I think, is the concern that using ‘W’ and ‘F’ might reify truth and falsity: it might encourage us to think that truth and falsity are logical objects.12 In the Notes dictated to Moore Wittgenstein emphasized the idea that ‘a’ and ‘b’ are not names by describing them as ‘scratches’. What Wittgenstein says in the Notes is that ‘a proposition has two poles, corresponding to the case of its truth and the case of its falsehood’.13 He seems to have seen the role of ‘a’ and ‘b’ as being not to name truth and falsity, but somehow to make visible the fact-like nature of what it is that makes p symbolize. They are there only to serve as hooks against which p can stand in a fact-like relation. Indeed, it is hard to avoid the thought that Wittgenstein intended ‘a p b’ to remind us of ‘aRb’: he wanted to suggest that the symbol ‘ p’ on its own is incomplete in something like the way that ‘R’ is. How he could possibly have made this idea consistent with other things he believed can only be a matter of speculation, however. Wittgenstein had abandoned his use of the ab-notation by the time of the Tractatus: even when he resurrected truth-diagrams there, he used the conventional labelling of the poles with ‘W’ and ‘F’. The reason is presumably that in the Tractatus truth-diagrams are presented only as an ‘intuitive method’ which enables us to ‘recognize a tautology as such’,14 not as a notation for expressing propositions. By that time he had abandoned (for reasons to be explained in §20.3) the idea that a truth-diagram is somehow a transparent medium for conveying the proposition it expresses. Perhaps, too, he had realized the futility of attempting to make p more fact-like by putting scratches on either side of it.

. The inputs Rather similar considerations apply when we ask how propositions occur in compound propositions. Suppose first that the occurrences of ‘ p’ in ‘∼ p’ and of p and q in ‘ p ⊃ q’ function as names: they do not name the propositions p and q themselves, perhaps, but one might suppose that they name complexes related in some way to p and q. Some such view as this is widespread in Russell’s writings of the period. In Principia, for example, he reads ∼ p as 12

Cf. 4.441.

13

C13.

14

6.1203.

The inputs  ‘the negation of p’ and p ⊃ q as ‘ p implies q’—locutions which require the letters to be replaced by names rather than sentences if they are to make sense. Wittgenstein rejects the view. Indeed he takes it to be obvious (even15 to the ‘plain man’) that the logical connectives do not express relations. He does not expand on why this is obvious, however. One reason he might have had in mind is simply that what we put alongside the logical connectives are sentences, not names. If we think of ∼ p as expressing a property of p, and p ⊃ q a relation between p and q, then the substituands for the letters must be names, as Russell’s way of speaking assumes. So what Wittgenstein may have taken to be obvious is that what is negated in ∼ p is exactly what is expressed in p. ‘In not- p, p is exactly the same as if it stands alone; this point is absolutely fundamental.’16 The reason this is ‘obvious to the plain man’ is that it is what ordinary language already suggests. Ordinary language would not contain the whole propositions if it did not need them: However, e.g., ‘not- p’ may be explained, there must always be a meaning given to the question ‘what is denied?’17

In other words, the inputs to the process, as well as the output, must be propositions and hence (symbolizing) facts. Wittgenstein also tells us that ‘propositions, owing to sense, cannot have properties or relations’.18 If we put these two claims together, we obtain the result that ‘∼’ is not a property and ‘⊃’ is not a relation. On the other hand, Wittgenstein’s argument for this conclusion is not all that persuasive, since it seems to depend rather too much on features of language that might turn out not to be essential. Ordinary language, after all, certainly contains many features it does not need. A fact, being a sum of component facts (positive or negative) has no internal structure other than that of the components. Applied in the particular case of a symbolizing fact (i.e. a proposition) this amounts to saying that a fully perspicuous expression of such a fact, an expression which fully captures its basic structure, cannot have any need of a notion of alternative decomposition. But consider now a sign for a molecular proposition such as ‘ p ⊃ q’. The symbolizing fact expressed by this sign is, like any fact, uniquely decomposable into atomic parts. But the very same symbolizing fact is expressed by the sign ‘∼ p v q’, or by ‘∼( p ∼q)’. Since the same fact may be expressed in these various different ways, none of them can be a true expression of the structure of the symbolizing fact they have in common. In the Tractatus Wittgenstein tried to express this last point by emphasizing the significance of the fact that our notation for expressing molecular propositions makes essential use of brackets. 15

B18.

16

C40.

17

B21.

18

C15.



Molecular propositions

The apparently unimportant fact that the apparent relations like v and ⊃ need brackets —unlike real relations—is of great importance. The use of brackets with these apparent primitive signs shows that these are not the real primitive signs; and nobody of course would believe that the brackets have meaning by themselves.19

We now know, of course, that Wittgenstein was wrong about the particular point he was making here: Polish notation provides the same expressive resources as the methods we discussed earlier, but does not involve any brackets. And even if we put that issue aside, there is plainly something fumblingly wrong about what Wittgenstein says. His thought was presumably that the genuine primitives are in some sense independent of each other and hence do not interact, but he mislocated this point by aligning it with the distinct issue of unique decomposition, which in the usual notation is what brackets are designed to ensure. Yet this does not prevent there being something right in the general territory of Wittgenstein’s remark. There is a distinction to be drawn between a conception of complexity which leaves room for alternative decompositions of the same complex and one that does not. Wittgenstein was wrong to think that the use of brackets is the characteristic mark of the first of these kinds of complexity, but he was right to stress the importance of the distinction. What is significant here is the contrast between language and the world. As we noted in §15.3, Wittgenstein’s conception of the world was in a certain sense flat: it can be completely described simply by listing all the positive and negative facts; neither brackets nor anything else of comparable complexity is required. The fact that language is in this respect different is due to the wider range of possibilities—in particular, disjunctive possibilities—that it aims to express. This case exemplifies a more general feature of Wittgenstein’s way of thinking. There are several points in the Notes where he seems to have been confused about technical matters in logic, or where subsequent developments have shown suppositions he made to be wrong. It is curious how often the philosophical views he was making are not refuted by exposing the technical error in the pieces of logic he used to make them. And this suggests rather strongly that the technicalities were never for him the real reasons for holding the views in question. 19

5.461.

Chapter 

Generality One of the most obvious gaps in the account of logic we have presented so far is that it says nothing about how quantification operates. This seems to be an issue Wittgenstein had only recently begun to address when he compiled the Notes: it is notable how little he says there about it, and some of what he does say is contained only in the Cambridge Notes, not the Birmingham ones. Evidently the same steps have to be gone through as in the molecular case. First there is the question of a perspicuous notation for expressing quantified propositions. Then we need an account of how the sense of such propositions depends on the senses of their components. And finally there is the question of what it is in the chosen notation that symbolizes.

. Variables as classes of propositions If in a proposition we replace all the occurrences of the name ‘a’ with a variable ‘ x’, we obtain a propositional function. We must not think of ‘ x’ here as performing anything like the role of a variable name, however, for there is no such thing. Perhaps this is the reason why Wittgenstein preferred to call it not a variable but a prototype. But what is a propositional function? We saw in chapter 5 that Wittgenstein objected, for impeccable logical reasons, to the primitive idea of the assertion of a propositional function which Whitehead and Russell introduced in Principia, ∗9. But he could not simply banish propositional functions from logic completely: they are too useful. What he needed to do was to provide an explanation for those uses of them that cannot be eliminated, most notably their use within quantified expressions such as ( x) φ ( x) or ( x) φ ( x). One of the central ideas of the Tractatus is that we should break down our explanation of how a quantified expression such as ( x) φ ( x) or ( x) φ ( x) symbolizes into two stages: first, the propositional function φ ( x) determines a class of propositions; then an operator is applied to this class to express that all or some of the propositions in the class are true. This idea is already visible in the Notes. ‘What is essential in a correct apparent-variable notation is this: (1) it must mention a type of propositions; (2) it must show which components of E

E



Generality

a proposition of this type are constants.’1 By the time it reached the Tractatus this remark had been refined into the observation that ‘that which is peculiar to the “symbolism of generality” is firstly, that it refers to a logical prototype, and secondly, that it makes constants prominent’.2 Wittgenstein thus recommended that the propositional function φ ( x) should be thought of simply as a means of specifying a class of propositions, namely all those propositions we obtain if we replace the variable x in the function with any legitimate substituand. ‘If we change a constituent a of a proposition φ (a) into a variable, then there is a class pˆ {( x) φ ( x) = p}.’3 In much the same way, we could alternatively keep the name ‘a’ constant and replace the remainder of the proposition with a variable. The resulting propositional function would specify a different class, the class of all propositions about a. The conventional way of thinking about generality, repeatedly encouraged by Russell, focuses on the variable ‘ x’, which is thought of as having as its possible substituands a class of objects. Wittgenstein focuses instead on the whole symbol ‘φ ( x)’, which he takes as having as its range a class of propositions. It would therefore be natural, I suppose, to see a link between Wittgenstein’s account of the variable and Frege’s context principle, that ‘only in the context of a proposition do words mean anything’,4 since the context principle may be thought of as motivating just such a shift of focus from objects to propositions. But although the similarity is suggestive, I do not think it is especially deep. Wittgenstein’s treatment of the variable shares with the context principle the methodological tendency to treat the proposition as primary, but it does not contribute, as Frege intended the context principle to do, to the goal of avoiding the temptation ‘to take as the meanings of words mental pictures or acts of the individual mind’.5 Although Wittgenstein repeated the context principle in the Tractatus,6 no doubt in conscious homage to Frege, it did not play for him there the role it played for Frege in the Grundlagen. What is striking, indeed, is how little the Grundlagen seems to have influenced Wittgenstein compared with Frege’s other writings. For instance, even when Wittgenstein denies the central claim (or, more accurately, presumption) of the Grundlagen, that numbers are self-subsistent objects, he never so much as gestures towards Frege as a possible target. The deeper resonance of Wittgenstein’s explanation of propositional functions is rather with Frege’s treatment of variables in the Begriffsschrift. There Frege takes the proposition as primary not in order to avoid psychologism but in order to avoid the temptation to think of variables as analogous to names. Russell, by contrast, although he had long shared Frege’s understanding of propositional functions as obtainable from a proposition by replacing a constant with a variable, frequently succumbed to the temptation to regard the E

1

C49.

2

5.522.

3

B19.

4

Gl, §62.

5

Gl, Preface.

6

3.3.

Variables as classes of propositions  variable as a kind of ambiguous name, and hence the propositional function as ambiguously denoting the various propositions that are its instances.7 Wittgenstein’s idea in a sense reverses the conventional way of conceiving of variables: replacing a constant with a variable serves to make prominent not what has been removed from the proposition but what remains. The idea that the role of a propositional function is to determine a class of propositions is hinted at occasionally by Russell. In the Introduction to Principia, for instance, he said that ‘a function is what ambiguously denotes some one of a certain totality, namely the values of the function’.8 But Russell did not suggest that the totality of values is to replace the function. He had tried various devices (most notably his substitutional theory of 1908) to eliminate propositional functions from his ontology, but never this one. There is surely something characteristic about how Wittgenstein uses the device: the idea itself is very simple, and on its own it is hardly deep; yet it brings about a radical reconfiguration of our way of looking at the notion. Russell had in Principia conceived of the variable as specifying a class of things; now we are asked to conceive of it instead as specifying a class of propositions. This reconceptualization of quantifier–variable notation has two effects. One is that by regarding propositional functions only as notations for picking out classes of propositions it removes any temptation to think of them as a kind of entity. Higher-order quantification is no longer in any sense quantification ‘over’ propositional functions. On Wittgenstein’s understanding, a quantified expression which apparently quantifies over propositional functions does not really do so: a propositional function ‘can only occur in a proposition through its values’.9 On Russell’s account the expression (φ ) φ !a, for instance, had been explained as asserting that all the predicative functions φ ! xˆ apply to a; whereas on Wittgenstein’s it is now the logical product of all the propositions of the form φ !a. The second consequence Wittgenstein’s presentation has is that it removes the possibility that the variable could somehow enable a general proposition to include within its range cases which no individual proposition expresses. The variable had been the last remnant of Russell’s notion of a denoting concept— a concept whose presence in a proposition was supposed to enable the proposition to be about entities with which we are not acquainted. Now, finally, this last remnant was discarded. One effect of thus discarding the variable we have already seen: it renders incoherent Russell’s pre-1912 project of inferring the existence of objects of one kind from our acquaintance with objects of another. For that project depended on our supposed acquaintance with a variable which ranges over objects with which we are not acquainted. However, this introduces a tension into Wittgenstein’s position. The notational device of quantification enables me to express a logical product without 7

E.g. PM, I, 39.

8

Ibid.

9

PM, I, xiv.



Generality

having to write down all the instances, but are the instances contained in the symbol? It seems that they must be, since if I do not grasp the instances, I have not fully grasped what the proposition says. To someone who recognizes the symbol in the sign, the proposition ( x) f x ⊃ f a must be completely trivial; but it can be trivial, it seems, only to someone who recognizes ‘ fa’ as already occurring in the symbol ‘( x) f x’. And that, as Frege remarked, is wildly implausible. If I utter a sentence with the grammatical subject ‘all men’, I do not wish to say something about some Central African chief wholly unknown to me. It is thus utterly false that I am in any way designating this chief when I use the word ‘man’, or that this chief belongs in any way whatsoever to what the word ‘man’ means. It is likewise equally false that in such a sentence many judgments are put together by means of the common name.10

As an explanation of our power to grasp generalizations Wittgenstein’s appeal to the process by which we convert signs into symbols is thus every bit as mysterious as Russell’s appeal to the variable. The consequences of this tension in Wittgenstein’s position for his understanding of the theory of types will occupy us in the next chapter. . Notation Suppose that we accept Wittgenstein’s idea of decomposing a notation for quantification into two stages: in the first, a symbol picks out a class of propositions; in the second, an operator is applied to that class. Plainly this is so far only programmatic: we need a more detailed account of both parts of the process. Since in the Notes Wittgenstein wanted the device of truth-diagrams to play the central role in his explanation of symbolism, what he needed to do was to extend that method to quantified expressions. The remarks about this issue at the end of the Summary are among the few things in the Cambridge Notes that are not simply reexpressions of, or glosses on, what is in the Birmingham version. Perhaps, therefore, the ab-notation for generality is one of the things he had been working on during his holiday in Norway with Pinsent, and was therefore not contained in the notebooks from which the Birmingham Notes were drawn. If that is the case, however, one has to say that he had not made much progress with it. All he had to offer Russell on the subject11 was the following: a−( x)−aφ xb−( x)−b a−( x)−aφ xb−( x)−b

E

( x)φ x : ( x)φ x :

E

E

for for

10 ‘Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra der Logik’, 454. 11 C43.

Undecidability  This is plainly so half-baked that one suspects he had not really thought about the matter at all and simply made something up on the spot when Russell asked him about it. In one of his letters from Norway Wittgenstein was willing to concede that the rule for ab poles ‘applies first of all only for what you called elementary propositions’, (i.e. what Wittgenstein called molecular propositions—truthfunctions of atomic propositions). But he insisted that ‘it is easy to see that it must also apply to all others’.12 Not unreasonably, however, Russell claimed not to understand, and in his next letter Wittgenstein was reduced to patent bluster. ‘I beg you to think about these matters for yourself: it is INTOLERABLE for me to repeat a written explanation which even the first time I gave only with the utmost repugnance.’13 The fact that Wittgenstein hoped to extend the method of truth-diagrams so that it would encompass a notation for quantification explains, incidentally, his otherwise mystifying response to another of Russell’s observations in the same letter. ‘What you call ab-functions are what the Principia calls “truth-functions”. I don’t see why you shouldn’t stick to the name “truthfunctions”.’14 Wittgenstein’s reply to this was, ‘Whether ab-functions and your truth-functions are the same cannot yet be decided.’ The explanation for the reply can only be that Wittgenstein was now using ‘ab-functions’ to mean whatever functions can be expressed using the method of truth-diagrams. If he had succeeded in extending this method as he hoped, ab-functions, in the sense in which Wittgenstein was now using the term, would not have coincided with Russellian truth-functions but would have included those expressible using quantifier-variable notation. It should be stressed, though, that this was a change of terminology on Wittgenstein’s part. In relation to the Notes on Logic, Russell’s observation is perfectly correct: Wittgenstein does indeed use ‘ab-function’ there as a synonym for what the Principia calls ‘truth-functions’. . Undecidability The desperation of Wittgenstein’s attempts to avoid the question Russell was pressing is patent. As is so often the case where instances of psychological avoidance are concerned, there was a good reason for Wittgenstein’s bluster: there cannot be a notation for quantification which in every case makes manifest which proposition is being symbolized. There might, that is to say, have been some hope that Wittgenstein could extend the truth-diagram notation to cover some cases of generality, but it was forlorn for him to hope that he could extend it to cover all such cases. The reason is the Church– Turing undecidability theorem, which tells us that once we have a notation to 12

[Nov. 1913] (CL, no. 30).

13

[Nov. or Dec. 1913] (CL, no. 32).

14

25 Oct. 1913.



Generality

express multiple generality, it is not mechanically decidable which sentences are logically valid. It was precisely the purpose of the truth-diagram method to make it immediately apparent which proposition a particular diagram expresses; and, in particular, to make it immediately apparent which diagrams express logically valid sentences and which do not. So any adequate notation for generality will inevitably leave us some way short of this goal. Moreover, what the mechanical undecidability of first-order logic reveals is something more than merely the failure of Wittgenstein’s favoured notation. It reveals also a curious feature of Wittgenstein’s conception of propositions as symbols fully revealing what they express. It is perfectly possible to be confronted with a logically valid sentence in some logical notation (Russell’s, for instance) whose validity we have not yet grasped. Until we do grasp this, we have not yet recognized the symbolizing fact that is being presented to us, and hence do not understand the proposition being expressed. I take it, for instance, that most of us cannot instantly spot whether ( x)( f x ⊃ gx) ⊃ ∼( x)( f x ∼gx)

(1)

E

is logically valid. Yet if we were asked whether we understand it, we would be inclined to treat that as a question only about our grasp of Russell’s symbolism. On Wittgenstein’s account that is incorrect: in order to understand the proposition we have to understand what it says, and that includes understanding whether it is logically valid or not. On this view the question I might set in a logic exam asking whether (1) is logically valid is really a question about signs: what I am testing is which proposition the sign (1) expresses, not whether that proposition is valid. What I want to find out is thus in a certain sense inherently linguistic. If I try to ask the question within the language, what I am asking is trivial to anyone who understands the question. Of course, that is not to say that I could not achieve the aims of the logic exam by asking the question within the language: by answering it correctly the student could demonstrate precisely the understanding which I want to test. Nonetheless, it would remain the case that what I am testing is not the same as what I am asking. (In just the same sense the examiner in a French oral asks questions to test not whether the candidates know the answers but whether they understand the questions.) The underlying point is that there is a difference in complexity between two tasks, that of recognizing the sign and that of understanding the symbol. What the undecidability result shows is that in the case of polyadic quantified logic that difference is stark: the first task is mechanically decidable, in the case of any logically adequate formal language, whereas the second is not. In the simpler case of propositional logic, on the other hand, there remains a difference of complexity, but it is only one of degree: both tasks are mechanically

Undecidability  decidable, but the second is more complex than the first. The mechanical undecidability of polyadic logic is what makes it the case that there cannot be a wholly transparent notation which reveals in every case the logical structure of what it expresses: if there were, logic exams would be easier than they are.

Chapter 

Resolving the paradoxes Our discussion of Wittgenstein’s account of quantification in the last chapter left unaddressed what sort of theory of types it commits us to. To answer this question we need to look in more detail at the motivation for believing in logical types at all. That motivation derives from Russell’s paradox, the problem which had originally attracted Wittgenstein’s notice back in 1909.

. Russell’s theory of types The paradox which Russell discovered in 1902 concerns classes: it is contradictory to suppose that there is a class of all those classes which do not belong to themselves. Russell’s solution to this paradox made use of the method of incomplete symbols which he discovered in 1905. In his theory terms apparently referring to classes are incomplete symbols which disappear on analysis. The solution was to be that any sentence in which the term ‘the class of all classes which do not belong to themselves’ occurs would resist rewriting according to the translation rules and would therefore turn out not to express a proposition at all. This solution does not just drop out all by itself, however. It is easy enough to formulate rewriting rules for eliminating class terms (so that, for instance, a proposition that appears to be about the class of all men turns out really to be about the property of manhood), but if that is all we do, we simply transfer the focus of attention to the corresponding paradox for propositional functions, which involves the function which applies to just those propositional functions which do not apply to themselves. In order to avoid such paradoxes as this, Russell found it necessary to stratify propositional functions into types. The primary method of stratification is to categorize a propositional function according to the types of its real variables. If this is the only method of stratification we adopt, the theory of types is said to be ‘simple’. The distinctive feature of a simple theory of types is that since the only principle of classification is based on the types of the real variables, and propositions do not contain real variables, all propositions belong to a single type. As Russell was aware, the simple theory of types suffices to eliminate his original paradox. When we attempt to define the class of all classes that do not belong to

Russell’s theory of types  themselves, we are stymied. But there are other paradoxes which the simple theory does not solve. One of the best known is Grelling’s:1 if we define the adjective ‘heterological’ by the prescription that it applies to just those adjectives which do not apply to themselves, then the word ‘heterological’ itself is heterological if and only if it is not. So far, this is a paradox in ordinary language: the problem Russell had to address is that it can also be formalized in a simple theory of types. Let us write ‘Het x’ for ( φ )( x means φ zˆ ∼φ x). E

Now ‘Het’ means Het zˆ, and so Het(‘Het’) ≡ ( φ )(‘Het’ means φ zˆ ∼φ (‘Het’)) E

≡ ∼ Het(‘Het’),

since ‘Het’ cannot mean two different things. In order to solve paradoxes such as this, Russell adopted a ‘ramified’ theory of types, so called because propositional functions (and hence propositions) are categorized not only according to the types of the real variables they contain but also according to the apparent variables. The derivation of a contradiction from the definition of Het is stymied because the variable φ in this definition is permitted to range only over propositional functions of a single order. We can, for instance, define a function by ( φ )( x means φ !ˆz ∼φ ! x), E

where the variable φ ranges only over predicative functions, but the function thus defined is not predicative, since it contains a quantification over predicative functions. So in the ramified theory, unlike the simple theory, the function does not fall in the range of the variable φ occurring in its definition. This solves Grelling’s paradox. The philosophical principle which Russell adduced in support of the theory of types was what he called the vicious circle principle: whatever involves all of a collection cannot be one of the collection. But as a philosophical basis for the theory this principle has two weaknesses. First, Russell does not really explain why it should be true. In order for it to seem problematic that something should belong to a collection which it involves we must presumably conceive of the notion of involvement as contributing in some way to the constitution of the entity in question, so as to make the consequent circularity vicious. After all, no one thinks that the tallest man in the room is somehow 1

Grelling and Nelson, ‘Bemerkungen zu den Paradoxien von Russell und Burali-Forti’.



Resolving the paradoxes

compromised by the fact that he may be singled out in terms of a collection to which he belongs. The manner in which we here pick him out, although impredicative, is harmlessly so, because it is not an internal property of him. By formulating the vicious circle principle quite generally Russell divorces it from the surroundings in which it might most naturally be justified. Second, which theory of types the vicious circle principle justifies, simple or ramified, depends on how we understand the notion of involvement in question. If every propositional function involves the real variables in it, that grounds classifying them into levels; if every propositional function also involves the apparent variables occurring in it, that justifies ramifying the hierarchy so as to distinguish propositional functions of different orders. Why should we say that the propositional function φ x involves its instance φ a, and not conversely that φ a involves φ x? Russell says that this is ‘sufficiently obvious, in any particular case’, because the proposition ‘Socrates is human’, for instance, ‘can be perfectly apprehended without regarding it as a value of the function “ x is human” ’. As Gödel later observed,2 Russell’s statements of the vicious circle principle waver between ‘involves’, ‘is definable only in terms of’, and ‘presupposes’. If we hoped that it would arbitrate between the simple and ramified theories, we should first have to settle which kind of involvement is relevant. The fact that Russell was so vague about this central issue suggests that perhaps he did not really envisage it as playing so decisive a role. What was driving his decision to adopt the ramified theory was really the need to avoid Grelling’s paradox, not the vicious circle principle. . Wittgenstein’s vicious circle principle Wittgenstein seems to have thought up several solutions to Russell’s paradox at various times. We have already noted that he wrote to Jourdain with one while he was at Manchester in 1909. Then in 1912 Pinsent wrote in his diary that Wittgenstein had explained to him a new solution he has discovered to a problem {in the most fundamental Symbolic logic} which was puzzling him greatly in Iceland, and to which he made a somewhat makeshift solution then. His latest is quite different and covers more ground, and if sound should revolutionize lots of Symbolic logic: R[ussell], he says, thinks it sound, but says nobody will understand it: I think I comprehend it myself however (!). If W[ittgenstein]’s solution works, he will be the first to solve a problem which has puzzled R[ussell] and Frege for some years: it is the most masterly and convincing solution too.3

If, as one assumes, the ‘problem which has puzzled Russell and Frege for some years’ is that of finding a philosophically defensible solution to Russell’s paradox, it is perhaps a shame that Pinsent did not record Wittgenstein’s ‘most 2

CW, II, 121.

3

25 Oct. 1912.

Types as classes of propositions  masterly and convincing’ solution for posterity. Whatever it was, though, perhaps it did not satisfy Wittgenstein for very long: the following summer we find him reporting from Norway that he was once more ‘thinking about the beastly theory of types’.4 The principle Wittgenstein eventually settled on as guiding the resolution of the paradoxes was that ‘no proposition can say anything about itself, because the symbol of the proposition cannot be contained in itself; this must be the basis of the theory of logical types’.5 If giving an account of the theory of types was really Wittgenstein’s main goal while he was at Cambridge, then this principle is in one sense a poor showing for two years’ work. For Russell, as we have just seen, had already tried to justify his own solution to the paradoxes by appealing to the vicious circle principle, that whatever involves all of a collection cannot be one of the collection. What Wittgenstein was now advocating was not much more than a transposition of Russell’s principle into the language of symbolism that he had by then adopted. It is, that is to say, what Russell’s vicious circle principle becomes when viewed through the lens of Wittgenstein’s symbolic turn. A proposition involves all those propositions which it says something about, and hence by the vicious circle principle cannot say anything about itself. On the other hand, although Wittgenstein’s vicious circle principle looks very similar to Russell’s, the background it relies on is different. Wittgenstein’s phrasing makes his solution to the paradoxes dependent upon his theory of symbolism, since whether a putative proposition attempts what is impossible, namely to say something about itself, is to be settled by examining its symbol. This might at least be thought to overcome the first weakness which we noted earlier in Russell’s vicious circle principle: Wittgenstein, by giving it a symbolic turn, could motivate it by appealing to the non-circularity of understanding. On the other hand, Wittgenstein’s vicious circle principle on its own is surely programmatic in just the same manner as Russell’s. On its own, that is to say, Wittgenstein’s principle does not settle whether the theory of types should be simple or ramified. That will be settled only once we have a developed account of when one proposition ‘says something about’ another; or, equivalently, an account of propositional symbols that allows us to determine when one contains another. . Types as classes of propositions We saw in the last chapter that Wittgenstein conceived of quantification in two stages: first, the formation of a symbol to pick out a class of propositions; second, the application to that class of an appropriate operation. The only method for achieving the first stage of the process that he countenanced in 4

To BR, 5 Sep. 1913.

5

B76; cf. 3.332.



Resolving the paradoxes

the Notes was by means of a propositional function, conceived of as obtained from a proposition by replacing some component with a variable. Suppose now that we continue the process of replacing parts of the proposition with variables as far as it can be taken. We shall then obtain a symbol every component of which is a variable. This will pick out a class of propositions which will be maximal, in the sense that the classes picked out by the propositional functions referred to earlier will be subclasses of it. Wittgenstein called this maximal class of propositions a logical type.6 A logical type may thus be thought of as a kind of limiting case of a class of propositions picked out by a propositional function: the class picked out by a propositional function will always be a subclass of some logical type. So to determine what sorts of quantification are possible we need to determine what the logical types of propositions are. Suppose, for instance, that ‘Socrates is mortal’ is an atomic proposition. What is its logical type? If we replace ‘is mortal’ with a variable, we obtain a class of propositions φ (Socrates). But prima facie it seems as if there might be various such classes, depending on the range of the variable φ : there is the narrow range consisting only of those propositions ascribing qualities (i.e. simple properties) to Socrates; but there might be broader ranges containing propositions that ascribe increasingly complex properties to Socrates, such as that of being mortal if Plato is wise or, more complex still, that of having all the qualities of a great philosopher. And, as Ramsey famously noted,7 the same ambiguity between narrow and broad ranges is present in the other part of the proposition. If we replace ‘Socrates’ with a variable, we obtain a class of propositions ‘ x is mortal’: this is normally taken by logicians to be the narrow range consisting of the propositions obtained by replacing ‘ x’ with a simple proper name; but we should consider the possibility that there may also be broader ranges which include propositions where we replace it with words such as ‘someone’ or ‘anyone’. More generally, the logical type of ‘Socrates is mortal’ is the class of propositions we obtain if we replace both parts of the proposition with variables so as to obtain a class picked out by the symbol ‘φ ( x)’. To determine which class this is we need to settle the range of both these variables. If both ranges are taken as narrowly as possible, we obtain the elementary subject-predicate propositions, so the logical type certainly contains all of these. The question we have to settle is whether it contains any other propositions. According to Wittgenstein this ‘is not dependent upon any convention, but only upon the nature of the symbol “φ ( x)” ’.8 We saw earlier that Wittgenstein thought Russell had attempted to superimpose type restrictions by stipulation on symbols whose sense already determined their types; and here Wittgenstein explicitly advances the alternative conception. Moreover, Wittgenstein is specific about how this is to be settled. In the 6

B19.

7

FoM, 125.

8

Ibid.

Types and molecular propositions  Notes he says that his vicious circle principle ‘must be the basis of the theory of logical types’;9 in the Tractatus he says, more explicitly, that it is the ‘whole’ of the theory of types.10 What this means, I take it, is that the question we are now considering, of the logical type of the proposition that Socrates is mortal, is to be settled solely by reference to Wittgenstein’s vicious circle principle: this is the only constraint capable of restricting the range of values of a variable. Anything else could only be an artificial constraint—an illegitimate attempt to restrict by stipulation the range of applicability of a symbol. But to be explicit about the means of solving a problem is not yet to solve it. Whether Wittgenstein’s vicious circle principle leads to a simple or a ramified theory of types still depends on how the complexity of a propositional symbol is to be understood.

. Types and molecular propositions Consider first the case of a molecular proposition. We are agreed that ‘Socrates is mortal’ and ‘Plato is human’ are of the same logical type, but what about ‘Socrates is not mortal’ or ‘Socrates is mortal unless Plato is human’? To settle whether these are of the same type as ‘Socrates is mortal’, we need to ask what symbols they contain. Does ∼ p contain the symbol p? If it does, and if similarly ∼∼ p contains ∼ p, then (assuming, as is plausible, that containment is transitive) ∼∼ p contains p. But, as we have already noted, ∼∼ p is the same proposition as p. So p contains itself, which directly contradicts Wittgenstein’s principle. The conclusion, then, is that the symbolizing fact in ∼ p does not contain the symbol p; in the same way the symbolizing fact in p v q does not contain the symbols p and q; and similarly for all the other truth-functions. So on Wittgenstein’s conception of what is relevant to symbols the way is clear for the application of truth-functions not to alter the logical type of a proposition. The answer to the question we posed at the beginning of this section is therefore that ‘Socrates is not mortal’ and ‘Socrates is mortal unless Plato is human’ are of the same logical type as ‘Socrates is mortal’. This is what led Wittgenstein to emphasize, for example, that the function p | q is merely a mechanical instrument for constructing all possible symbols of ab-functions. The symbols arising by repeated application of the symbol ‘|’ do not contain the symbol ‘ p | q’.11

The stroke ‘|’ contributes to the structure of signs for molecular propositions without contributing correspondingly to the structure of the symbols these signs express. 9

B77.

10

3.332.

11

B31.



Resolving the paradoxes

However clear Wittgenstein may have been about this, however, it is in tension with things he said elsewhere. For instance, he stressed the intuition that ordinary language would not contain the whole propositions if it did not need them: However, e.g., ‘not- p’ may be explained, there must always be a meaning given to the question ‘what is denied?’12

This intuition surely encourages us to think that the symbolizing fact in ∼ p does contain the symbolizing fact in p as a component. And if we need any further encouragement, it is supplied in the Cambridge Notes. ‘In not- p, p is exactly the same as if it stands alone; this point is absolutely fundamental.’13 Wittgenstein’s conception of ∼∼ p as the same symbol as p thus conflicts with his intuition that in ∼ p it is p that is being denied. If he was to resolve this tension, he needed to explain our conviction that in ∼ p it is p that is being denied in such a way that we are not required to see the latter symbol as contained in the former. Chapter 19 was our attempt to extract from the Notes whatever clues they contain towards the resolution of this tension. Our lack of progress there may be taken to indicate how much work still remained for Wittgenstein to do on this question.

. Types and generality Let us take it as granted, however problematically, that truth-functions do not raise the type of a proposition. What about the corresponding question for quantification? The Birmingham Notes give no clue what Wittgenstein’s answer to this was. Perhaps this was what he was contemplating as he sat beside a fjord in Norway thinking about the beastly theory of types: if so, it was presumably the product of this contemplation that he offered Russell, almost as an afterthought, at the very end of the Cambridge Notes. Take (φ ) φ ! x. Then if we describe the kind of symbols, for which ‘φ ! x’ stands and which, by the above, is enough to determine the type, then automatically ‘(φ ) φ ! x’ cannot be fitted by this description, because it CONTAINS ‘φ ! x’ and the description is to describe 14 ALL that symbolises in symbols of the φ ! kind.

What this explanation makes clear is that Wittgenstein intended his account to justify the ramified theory of types of Principia, not the simple theory. He here considers the propositional function (φ ) φ ! x, in which the variable φ ranges over all the predicative functions of x. Could this function itself be predicative? Could it, that is to say, be among the functions over which the variable ranges? Wittgenstein’s answer is that it could not, because otherwise any explanation of how it symbolizes would inevitably be circular. 12

B21.

13

C40.

14

C51.

Uniting generality and truth-functions  Thus Wittgenstein conceived of a variable as containing the symbols for all the propositions which are instances of it. The broadest of the ranges we canvassed earlier for the variable φ (Socrates) is therefore barred to us. The proposition ‘Socrates has all the qualities of a great philosopher’ is not of the same type as ‘Socrates is mortal’, because mortality is one of the qualities which great philosophers possess: the symbol of the former proposition contains the symbol of the latter. The difficulty with this, though, is that it seems to go against Wittgenstein’s decision to regard the role of a proposition as being to express a certain sense. That decision, let us recall, led him to regard two symbols as the same if they make the same contribution to determining the sense of the proposition. Wittgenstein had not yet worked out in detail his account of the sense of propositions involving quantification, but what he had already said about molecular propositions (that p = not-not- p, for instance) shows that he regarded logically equivalent propositions as having the same sense and hence as being the same symbol. The problem this causes is that it is possible for two propositions of different types to be logically equivalent. But the same symbol cannot belong to two different types. So Wittgenstein’s account of sense drove him towards a simple theory of types, while his understanding of a propositional variable as involving the symbols for the propositions in its range directed him towards a ramified theory. . Uniting generality and truth-functions What we have seen is that at the time of the Notes there must have been a fundamental difference in Wittgenstein’s understanding of how molecular and generalized propositions depend on their components: a molecular proposition does not contain symbols for its components, whereas a generalized proposition does. This was the reason why he thought that molecular propositions belong to the same type as their components but generalized propositions do not. Later, though, Wittgenstein united the two methods of forming propositions by adopting a generalization of Sheffer’s stroke. In the Tractatus he introduced a single operation N which could be applied to any class of propositions so as to express the joint denial of all the propositions in the class.15 So if (ξ ) is a symbol picking out a class of propositions, N(ξ ) expresses the proposition which is true if and only if all the propositions in the class are false. If (ξ ) picks out the class expressed by the propositional function f x, for instance, then N(ξ ) = ( x) ∼ f x.

But Wittgenstein now widened his conception of the kind of symbols that could be used to pick out a class of propositions: in addition to propositional 15

5.502.



Resolving the paradoxes

functions, he allowed finite lists of propositions. This enabled him to treat molecular propositions as particular cases of the same method. Thus N( p) = ∼ p

N( p,q) = ∼ p ∼q.

Now that the symbols for molecular propositions were the result of the same general process of symbolization as general propositions, it became even less plausible than before to think that the theory of types should discriminate between them. To think that, we would have to hold that the symbol ‘ f x’ picks out the class of propositions { fa, f b, fc, . . . } in such a way as to involve the symbols for all the instances fa, f b, etc., but that the symbol ‘{ p, q}’ does not in the same manner involve the propositions p and q. And why would one think that? But even if we accepted this as a reason to hold that molecular and general propositions are in the same logical boat, and hence that Russell’s ramified hierarchy is untenable, that would not in itself determine which direction to go: towards a simple theory of types in which neither the formation of molecular nor of general propositions raises the type of a proposition; or towards an ultra-ramified theory in which both do. To settle that, we would have to decide whether the symbol for a class of propositions involves the symbols for the propositions in the class, in whatever sense of ‘involves’ is relevant to the vicious circle principle. But the vicious circle principle is, as we noted earlier, only programmatic. So more would still need to be said about the structure of symbols. . The general form of proposition Just when Wittgenstein invented the N-operator is a question on which the written record is curiously silent: the device is absent from the Notes dictated to Moore as well as from the surviving wartime notebooks, but it occurs on the first page of the Prototractatus. Whenever it was that Wittgenstein adopted the N-operator, however, the tension implicit in his adherence to a ramified theory of types, already severe, surely now became intolerable; yet it took Wittgenstein a remarkably long time to resolve this tension. When he began to compile the Prototractatus in late 1915 or early 1916 he still seems to have adhered to a ramified theory.16 Matters were then made still worse by the central role that came to be played in Wittgenstein’s conception by the general form of proposition. The culmination of the Tractatus, as Wittgenstein originally conceived it, was to be a statement of the general form of proposition. Propositions, he said, form the range of the propositional variable [ p0 , ξ , N(ξ )]. 16

Cf. PTLP, 4.432.

Unsayability  What matters for current purposes is not the particular form of this variable but merely its existence. If there is a general form of proposition (whether this or anything else), then an expression for this general form is a variable which has as its range all propositions, in which case we are forced to have a simple theory of types. And Wittgenstein’s philosophical instincts pulled him strongly towards a belief in such a variable. This had its roots in Wittgenstein’s much earlier insight, present already in his letter to Russell of June 1912, that there cannot be any such thing as a logical experience. If the most general form of proposition could not be given, then there would have to come a moment where we suddenly had a new experience, so to speak a logical one. That is, of course, impossible.17

Thus he insisted in November 1916 that it must be possible to erect the general form of proposition, because the possible forms of proposition must be a priori. Because the possible forms of proposition are a priori, the general form of proposition exists.18

And yet he evidently had not yet seen his way through to the simple theory, for in the next sentence he seems still to admit the ramified theory at least as an open possibility, but to encourage us to ignore it for the purposes of the present discussion. In this connection it does not matter at all whether the given fundamental operations, through which all propositions are supposed to arise, change the logical level of propositions, or whether they remain on the same logical level.19

And as late as January 1917, at the very end of the surviving Notebooks, Wittgenstein continued to feel an unresolved tension between these two competing conceptions. In the sense in which there is a hierarchy of propositions there is, of course, also a hierarchy of truths and of negations, etc. But in the sense in which there are, in the most general sense, such things as propositions, there is only one truth and one negation.

In one sense, he was saying, a ramified theory is correct; but in another sense a simple theory is. . Unsayability Wittgenstein did eventually resolve the tension, and he did so in favour of the simple theory of types: in the published version of the Tractatus almost all the remaining traces of the ramified theory are at last removed, and propositions are no longer presented as belonging to a hierarchy of types. (The principal exception is Wittgenstein’s lingering mistrust of the ancestral,20 which is hard 17

NB, 9 July 1916.

18

NB, 21 Nov. 1916.

19

NB, 21 Nov. 1916.

20

4.1273.



Resolving the paradoxes

to motivate in the context of a simple theory of types.) But the final decision to opt for the simple theory must have occurred rather late, since by the beginning of 1917, when he still thought there was in a sense a hierarchy of propositions, almost seventy pages of the Prototractatus were already filled in. Why did it take him so long? As we have seen, his philosophical instincts— the unity of the self, the emptiness of logic, and his conception of propositional sense—pushed him unequivocally towards a simple theory. Only one thing pushed him in the other direction—the need to solve the paradoxes. He realized, no doubt, that a simple theory suffices to solve Russell’s original paradox of the class of all classes which do not belong to themselves. But Russell had been driven towards a ramified theory by the need also to solve paradoxes such as Grelling’s. If Wittgenstein was to adopt a simple theory, he therefore needed some reason other than ramification to prevent the formation of the propositional function ( φ )( x means φ zˆ ∼φ x), E

which is responsible for generating Grelling’s paradox. The reason Wittgenstein settled on was his doctrine of unsayability, which disallows the use within his system of semantic concepts such as meaning. The meaning of a symbol is shown by its use in propositions, but we cannot say in language what the symbol means. In particular, therefore, we cannot define a propositional function Het (and even if we could, we could not say that ‘Het’ means Het). This suffices to block Grelling’s paradox (and others like it) without the need to ramify types. Now it is true that this solution was not really available to him at the time of the Notes, since only the earliest traces of the distinction between saying and showing are visible there. But unsayability—in particular, the unsayability of semantic relations such as that of meaning—is something Wittgenstein made a great deal of in the notes he dictated to Moore in Norway the following year. So he did have by then all the components he needed to resolve the tension we have been discussing. Yet it seems to have taken him three years from then to see how the resolution should go. At the very least this shows rather vividly that his reasons for adopting the doctrine of unsayability must have been very distant from the technical considerations we have been discussing here. Its ability to contribute to a resolution of the paradoxes can only have been a welcome consequence of the doctrine, not its original motivation. Even so, the slowness of Wittgenstein’s adoption of a simple theory of types remains one of the central puzzles in our understanding of the genesis of the Tractatus. What it demonstrates above all is once again the extent to which Wittgenstein’s logical insights were independent of formal considerations. He had a conception of the sort of thing a perspicuous notation would be, and

Unsayability  then he set about finding it. And in the same way he had a conception of what the general form of proposition amounts to before he saw how this translated into the formal system. It was philosophical, not technical, considerations that motivated these conceptions. Although Wittgenstein’s discussion of types inevitably suggests that Principia is his intended target, we have noted already how little trace of that book there is in his discussion. It is not merely that the Notes do not discuss the details of the theory of types that is presented there: they bear remarkably little sign that their author had even got beyond the Introduction. Wittgenstein made routine use of the logical signs Whitehead and Russell employed in Principia, but of course he would not have had to read the book in order to acquire his knowledge of them. Only once, at the very end of the Cambridge Notes,21 did he make use of a notation (a variable ranging over predicative propositional functions) that is distinctively type-theoretic. And if, as is curiously plausible, he delayed until the summer of 1913 any serious consideration of the parts of Principia which deal with quantification, there is no sign that his engagement with them became deeper during the war years. (By 1922 his recollection of Principia was so shaky that he could no longer remember22 whether the sign used there for the existential quantifier was ‘ ’ or ‘E’.) Although it was a desire to solve the paradoxes in the philosophy of mathematics that put him on his course towards the Tractatus, his method of achieving that goal was, to the last, strikingly indirect. E

21

C51.

22

To Ogden, 10 May 1922.

Chapter 

Typical ambiguity In §5.4 we saw how Wittgenstein thought Russell had been led into error by his practice of using formulae containing real variables. To correct this mistake, it would be necessary to rewrite Principia so that any use of such formulae is either eliminated or explained. In the elementary parts of the book, where it is simply a treatment of first-order predicate calculus that is in question, eliminating the problematic occurrences is feasible (although, given the eccentricities of the treatment, not trivial). Very soon after that, however, Whitehead and Russell use formulae containing real variables in a different manner that aroused new suspicions in Wittgenstein, namely as a device for expressing various propositions simultaneously at different levels in the typetheoretic hierarchy.

. Typical ambiguity The device of typical ambiguity is a response to a difficulty which arises once we accept that the paradoxes prevent us from quantifying over absolutely everything. We find ourselves nonetheless wanting to make assertions that apply to each level in the type-theoretic hierarchy. We might wish to say, for instance, that everything, at whatever level in the hierarchy, is self-identical. If we say that, however, we quantify over all the levels in the hierarchy, which is precisely what the theory of types renders illegitimate. The solution Whitehead and Russell adopted was to make use of the device of typical ambiguity. In order, for instance, to regard ‘# φ ( x)’ as expressing something determinate, we need to know what sort of variable x is; we need to know, that is to say, which type it is supposed to range over. Until that is determined, what we have written remains ambiguous as to type. Their convention was that by writing ‘# φ ( x)’ we should be taken as ambiguously asserting any such determination of the formula. Nowadays we would formulate a typically ambiguous generalization metalinguistically by means of a schema which only becomes a sentence expressing a determinate proposition once the schematic letters in the schema are

Typical ambiguity  replaced by specific symbols of the object language. It is striking, therefore, that in the Notes Wittgenstein uses the word ‘scheme’ in what is recognizably this modern sense. Those symbols which are called propositions in which ‘variables occur’ are in reality not propositions at all, but only schemes of propositions, which only become propositions when we replace the variables by constants. There is no proposition which is expressed by ‘ x = x’, for ‘ x’ has no signification; but there is a proposition ‘( x) x = x’ and propositions such as ‘Socrates = Socrates’ etc.1

One central case in which Whitehead and Russell made use of the device of typical ambiguity was the axiom of reducibility. Wittgenstein complained about this in a letter to Russell probably dating from the summer of 1913. Your axiom of reducibility is ! ( f ) φ x ≡ x f ! x; E

now is this not all nonsense as this proposition has only then a meaning if we can turn the φ into an apparent variable. For if we cannot do so no general laws can ever follow from your axiom. The whole axiom seems to me at present a mere juggling trick. Do let me know if there is more in it. The axiom as you have put it is only a schema and the real P[rimitive] p[roposition] ought to be ! (φ ) ( f ) φ x ≡x f ! x, E

and where would be the use of that?2

But Wittgenstein had mislocated the source of the problem. Whether a variable is typically ambiguous is nothing to do with whether it is real or apparent. What makes this hard for the reader of Principia to see is that their notation omits any visible indication that a variable has been disambiguated. Modern treatments of the theory of types standardly indicate which type a variable is supposed to range over by means of a subscript. If the theory of types is simple, so that there is only one dimension of variation, the subscript need only be a single numeral; if the theory is ramified, as that in Principia is, we need a more complicated notation, but the principle is the same. What it amounts to to disambiguate a schema of Principia is to decorate all the variables that occur in it with imaginary type-subscripts indicating the types they range over. But when we do this, we must decorate the real variables as well as the apparent ones. So in Principia ‘( x) x = x’ is typically ambiguous just as ‘ x = x’ is: in both cases the letter ‘ x’, which looks as if it is expressing a variable, does not really do so until it is decorated with a subscript determining the logical type it ranges over. If we adopted a notation which insisted on attaching 1

B13.

2

CL, no. 20.



Typical ambiguity

type-subscripts to all variables, the point would emerge more clearly. The law of self-identity, for example, would in such a language be written ( xσ ) xσ = xσ , where σ is schematic; if we replaced σ with a specific type-subscript in the object language, the resulting proposition would express the self-identity of all the entities in the particular type over which the corresponding disambiguated variable ranged. What Wittgenstein had failed to see, then, is that this point applies to apparent variables just as much as to real ones. So what makes the axiom of reducibility ‘only a schema’ is not really, as Wittgenstein evidently supposed, the unbound occurrence of ‘φ ’. The formula which Wittgenstein offers as the ‘real primitive proposition’ is just as much a schema as Russell’s version. It may indeed be reasonable to ask (as Wittgenstein does) what the use of the axiom of reducibility is, but that question has nothing to do with whether we can turn the φ in it into an apparent variable. The relevance of real variables to the point Wittgenstein was fumbling for was only that Russell’s willingness to use formulae containing real variables to assert ambiguously all the instances of the formula in a single type made him less critical than he should have been of extending this device so as to assert ambiguously instances belonging to different types. The first kind of ambiguous assertion, although strictly incorrect, is innocuous, because by prefixing the formula with a universal quantifier we can turn the real variable into an apparent one. The second kind is not innocuous, because no corresponding manoeuvre is available to us. There is also another sense in which Wittgenstein’s criticism of the axiom of reducibility is inept. Wittgenstein’s objection is that the ‘axiom’ is really just a schema awaiting determination of the types in order to obtain a proposition usable in inference. But in that case it is curious that he had not spotted the problem already in ∗9, where we find primitive propositions such as ‘# φ x ⊃ ( z) φ z’; this is a schema in which φ awaits the determination of its type, just like the axiom of reducibility. It was not until he was in Norway that Wittgenstein was able to give a more plausible reason to think that reducibility is a ‘mere juggling trick’. The objection was not that it was a schema, but that even if it is true, it is only accidentally true.3 And to be in a position to formulate this objection he needed to be clear about the distinction between truth and tautology, and that, above all, is what he achieved in Norway. But even if it is understandable that he could not formulate his objection to the axiom of reducibility until later, it is surely notable nonetheless that the letter quoted earlier gives the impression of being the first time he had thought much, if at all, about it. E

3

LW to BR, [Nov. or Dec. 1913] (CL, no. 32).

Independent indefinables  . Independent indefinables I mentioned in §8.3 Wittgenstein’s tendency to see Russell’s theory of types as an attempt to impose artificial restrictions on symbols whose grammar has already been fixed by the meanings we have assigned to them. We see this tendency at work in many of Wittgenstein’s comments about Principia. Every proposition belongs to a logical type. Something Wittgenstein was repeatedly concerned to stress, however, was that types should not be thought of as containers into which propositions can be sorted. To see the point Wittgenstein was trying to make we need to take a step back and consider his views on the issue of how language gets its meaning. Once more, the starting point is Fregean. One of the central concerns of his Grundgesetze had been to ensure that what he set up there was not a purely formal language. In order that logic can be applied, it is necessary that we invest the primitive terms occurring in it with meaning. Wittgenstein called the primitive terms indefinables. These, he said ‘must be independent of each other’. The reason was that he wanted—to exploit the analogy from mechanics which we mentioned in chapter 8—a system in which the number of indefinables is equal to the number of degrees of freedom. If an indefinable is introduced, it must be introduced in all combinations in which it can occur. . . In short for the introduction of indefinable symbols and combinations of symbols the same holds, mutatis mutandis, that Frege has said for the introduction of symbols by definition.4

This is why Wittgenstein held that words such as ‘individual’ or ‘particular’ must not be taken to be primitive ideas in logic. ‘It is easy to suppose,’ he observed, ‘that “individual”, “particular”, “complex” etc. are primitive ideas of logic. Russell e.g. says “individual” and “matrix” are “primitive ideas”.’5 Wittgenstein went on straightaway to offer a diagnosis of Russell’s error. It is, he said, to be explained by the fact that, by employment of variables instead of the generalitysign, it comes to seem as if logic dealt with things which have been deprived of all properties except thing-hood, and with propositions deprived of all properties except complexity. We forget that the indefinables of symbols [Urbilder von Zeichen] only occur under the generality-sign, never outside it.6

The fact that Russell added the German ‘Urbilder von Zeichen’ suggests that he was uncertain about the translation, and indeed it might more happily be translated here as ‘prototypes of signs’. What Wittgenstein is referring to is simply Russell’s variables and pure forms. Wittgenstein takes Russell’s device of ambiguous assertion to have misled him into thinking that logic has a subject matter, namely the abstract forms of propositions and their constituents. 4

B51.

5

B71.

6

B71.



Typical ambiguity

Part of the point here is that the type of a proposition is intrinsic to it: if it belonged to a different type, it would be a different proposition. But this is only part of the point. It is intrinsic to the table that it is made of wood: if it were not, it would be a different table. In this case, however, it is perfectly intelligible, even if false, to say that the table is made of metal. In the case of the proposition, however, it does not even make sense to say that it belongs to a certain type. To see why, suppose for a moment that it does make sense to say that a proposition belongs to a certain type. Then it cannot also make sense to say the same of a proposition of some other type. This is because if one proposition can be substituted for another salva sensu then they are of the same type. (That, after all, is just the point of categorizing propositions according to type.) But if it does not make sense to say of a proposition that it has any type other than the one it in fact has, it cannot make sense to say that it does not have that type (since that is tantamount to saying that it has some other type). And in that case it does not even make sense to say that the proposition does have the type it in fact has, since if it did make sense, the negation of that claim would make sense too. The conclusion we must draw from this is that logical types are not really properties of propositions at all.7 This conclusion, that types are not properties, is significant partly because it points to a fundamental difficulty with the theory of types, namely that it seems impossible to state it without violating its own constraints. But it is significant also because it is one of the first signs in Wittgenstein’s work (perhaps, indeed, the very first) of the theme of unsayability that would play such a large role in the Tractatus. We cannot say which type a proposition belongs to, and yet, we feel, the proposition surely does belong to it. We seem to be gesturing frustratedly towards a truth that is just out of our expressive reach. . Whitehead Typical ambiguity does pose a genuine problem, even if Wittgenstein did not accurately pinpoint its source. The problem is that the device of typical ambiguity creates an illusion of unity where, if we take the theory seriously, there is none. If we assert a typically ambiguous formula φ ( x), we are to be taken as asserting an infinite number of formulae of this form, in each of which the variable x is disambiguated to range over a particular type. But each of these infinitely many formulae obtained by disambiguating the typically ambiguous formula φ ( x) is distinct: there is nothing except an accident of notation to unite them. A remark early in the Notes, although perhaps not originally directed at the problem of typical ambiguity, is certainly applicable to it. 7

B16.

Whitehead  It can never express the common characteristic of two objects that we designate them by the same name but by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where then would be the common element in the designations? Nevertheless one is always tempted, in a difficulty, to take refuge in different ways of designation.8

Wittgenstein was not the first person to have been suspicious of the device of typical ambiguity, however. Whitehead himself expressed the point to Russell in January 1911. I vaguely conjecture that some careless sentence of mine has led you to believe that I object to limiting the type of a variable. This is a gross libel. So far from that, my view is that our symbols remain mere unmeaning forms until the types of all the variables are determined.9

A couple of days later he put the point even more clearly. ‘According to me until all ambiguities are definitely settled there is simply a sequence of meaningless shapes.’10 In the same letter Whitehead also appealed to a distinction between what we now call the object language and the metalanguage— between what we prove within a formal theory and what we say about it from without. We do talk in summaries, etc. of all types. Certainly we do,—and the words have a very complicated meaning, as do most notions of common thought. . . In the case of addition it has been worth while defining the complication in the S[ymbolic] T[heory]; in the case of ‘all types’ it is not, chiefly because the complication is much greater. Hence any essential use in the S[ymbolic] T[heory] of ‘all types’ is forbidden to us.11

According to the theory of types it is illegitimate to quantify within our formal language over all the levels in the type-theoretic hierarchy. Nonetheless, we may do so informally in ‘summaries’ (i.e. the pieces of explanatory text placed at the beginning of each section of Principia). There is an evident difficulty with this idea. What forced Whitehead and Russell to introduce the complexities of the theory of types was the paradoxes, which showed that there is something profoundly mistaken about the attempt to quantify over everything. If that is so, then it remains mistaken whether the attempt is made formally or informally. A paradox is a paradox whether it is written down in a formal language or not. What Whitehead was advocating thus seems to be once again an instance of the sort of chickening out from the consequences of a theory which we saw with Frege and the concept horse, and which Wittgenstein so obviously disapproved of. Whitehead did at least recognize that he and Russell needed to say more about this issue than they had in the text of Principia so far, even if he perhaps did not know what more that was. ‘This correspondence has convinced me 8

B3; cf. 3.322.

9

27 Jan. [1911] (RA1 710.057409).

10

29 Jan. 1911 (RA1 710.057414).

11

Ibid.



Typical ambiguity

that we must sketch somewhere the talk the sort of thing ‘all types’ (in summaries) means.’12 Whitehead also recognized that if the typical ambiguities in schemas were to be removed, there had to be a procedure for determining their types, and this procedure would itself have to be typically ambiguous, thus leading to the danger of an infinite regress. We have . . . rules for the assigning of meaning to the symbols so as to produce a prop[osition]. There are really a succession of sets of rules one set for each type as it is defined. All the sets have a formal identity of procedure in relation to the various symbols. To say that a prop[osition] holds for any type is to say,—If after any acts of definition of types, the corresponding sets of formal rules of interpretation be applied to a sequence of symbols, we can then rightly assert the proposition or propositional function which is then obtained. Thus the P[rimitive] I[dea] ‘act of definition of a type’ is used—but not in the S[ymbolic] T[heory]. In the S[ymbolic] T[heory] we prove for the one or two types we do define, and outside the S[ymbolic] T[heory] we say that whoever works away at defining more types will find the corresponding prop[osition]s still true. . . . What I want to see (and we get no further forward till we have it) is a full discussion by you of the legitimacy of ambiguity of type under the assert[ion] sign.13

Russell’s reply (now lost) was evidently to the effect that we can escape the difficulty by explaining that when we assert a formula containing an ambiguous variable, this is to be considered true if it holds however the variable is disambiguated. Whitehead was not reassured: What do you mean by ‘considered true’? This is the point of the whole question—the law of excluded middle cries aloud for vengeance. The truth is that under this phrase you hide the fact that you are really dealing with ‘all types’.14

The end result of this correspondence was that Whitehead wrote a ‘prefatory note’ which appears at the beginning of Volume II of Principia. (The remarks in this note do not apply just to Volume II, incidentally: it was bound in there simply because by this time Volume I had already appeared.) If typical ambiguity was one of the issues Wittgenstein was thinking about during the summer of 1913, it is to be expected that he should have wanted to talk about it to Whitehead, who evidently grasped the issue more clearly than Russell. Wittgenstein at any rate took considerable trouble to go and visit Whitehead in August 1913,15 and then again before leaving for Norway in October:16 one suspects that these were not merely courtesy calls. The issue of typical ambiguity is a prime candidate for the application of the distinction between sign and symbol which we shall be discussing in chapter 24, and indeed Whitehead’s way of expressing his concerns about it in his 12 16

Ibid. 13 Ibid. 14 To BR, 31 Jan. 1911 (RA1 710.057458). LW to BR, 17 [Oct.] 1913.

15

McGuinness, Young Ludwig, 179.

Whitehead  letters to Russell is strongly suggestive of Wittgenstein’s later way of conceiving of that distinction. A ‘proposition’ of Principia which contains a typically ambiguous variable is, strictly speaking, merely a sign: it becomes a symbol only when the ambiguity is resolved. But there is nothing in the sentence on the page that tells us how to do that. We resolve the ambiguity by reading the sentence as saying something, i.e. by disambiguating all the variables that occur in it, and when we do that, the resulting proposition says something about the entities in some particular type. More generally, then, a sign is, as Whitehead said in the particular case of a typically ambiguous formula, ‘simply a sequence of meaningless shapes’ until it is read as a particular symbol. What Whitehead’s discussion makes salient but is quite absent from the Notes, however, is the distinction between what can be said in the symbolic theory and what can only be said outside it—or, as we would now say, between object language and metalanguage. Once we attend to that distinction, however, it is then a short step to recognizing the danger that the paradoxes which the theory of types was designed to solve will simply recur in the metalanguage.

Chapter 

Identity Whatever its virtues, the account of logic that Wittgenstein offers us in the Notes is, as he plainly knew himself, incomplete. One of the issues that he knew still required further thought was how to represent the relation of identity. . Russell’s definition It was evidently Wittgenstein’s ambition at the time when he compiled the Notes to extend his method of truth-diagrams so that it would provide a way of symbolizing all the logical notations of Principia. But on the face of it there was no need for him to address the question of identity, because in Principia it is not one of the primitive notations. In Principia identity is a defined relation: x = y =Df (φ ) φ ! x ≡ φ !y.

Two things are thus said to be equal, according to this definition, just in case they have all their predicative properties in common. By October 1913, however, we know that Wittgenstein had formulated an objection to the definition: he reported the objection to Moore,1 but for some reason did not include it in the Notes. Nor, unfortunately, did Moore record it, so we are left to guess just what it was. We have noted that by about this time Wittgenstein had begun to suspect that the axiom of reducibility is no more than a ‘juggling trick’. It would be natural in that case to suspect Russell’s definition of identity in turn. For Russell needed reducibility in order to obtain the indiscernibility of identicals in full generality. Without this axiom, he could not rule out the possibility that there are objects which are equal according to the definition (because they share their predicative properties) but differ as to their higher-order properties. But although Wittgenstein could perhaps have objected to Russell’s definition on this ground, I doubt if he did. It is more likely that his objection is the one that appears in the Tractatus. The objection there is not that the definition does not entail the indiscernibility of identicals, but rather that it depends unjustifiably on the identity of indiscernibles. 1

Moore’s diary, 3 Oct. 1913.

Russell’s definition  Russell’s definition of ‘=’ won’t do; because according to it one cannot say that two objects have all their properties in common. (Even if this proposition is never true, it is nevertheless significant.)2

To see just what Wittgenstein’s point was, let us start by noting Russell’s observation in 1913 that it is erroneous to regard [the identity of a sense-datum] as constituted by its predicates, because a precisely similar sense-datum may exist in another place. (By ‘precisely similar’ I mean ‘having the same predicates’.)3

But even if two sense-data cannot be distinguished by means of their predicates, Russell did not think they would be indistinguishable: he thought they could be distinguished by means of their relational properties. For if a and b differ in relation to other entities with respect to the relation R, for instance, so that there is an object c such that aRc &≡ bRc, then by applying Russell’s definition of identity with the propositional function xˆRc as φ we obtain a &= b. Wittgenstein was not disagreeing with this. His point was that even if as a matter of fact a and b are distinguishable, they might not be. We can understand what it means to say of two distinct objects a and b that they have all their predicative properties in common: the sentence (φ ) φ !a ≡ φ !b is significant even if it is false. The conclusion Wittgenstein drew from this was that the relation Russell had defined was not identity. How, though, could Wittgenstein be so sure that it is significant to say that two objects have all their properties in common? In the Tractatus, this follows from the doctrine that elementary propositions are logically independent of one another, and that doctrine in turn depends on Wittgenstein’s hostility to the Kantian synthetic a priori, a hostility which may well be of early date. On the other hand, Wittgenstein need not have formulated the logical independence of elementary propositions in full generality in order to believe that it is significant to say two objects have all their properties in common. He had ample motivation for that view independent of his developing views about possibility. For otherwise the fact that they do not have all their properties in common would be a further fact about them not reducible to the atomic facts. And we have noted already his conception of atomism as involving the claim that there can be no such fact. One of the oddities of this story, of course, is that there is no record of Wittgenstein reporting his objection against Russell’s definition to Russell himself. Perhaps he knew what Russell’s reaction would be. After all, Russell thought that there are general facts which are not reducible to the atomic facts—there is, for instance, the fact that there are no other atomic facts—and we noted earlier the evidence in the Cambridge Notes to suggest that when Russell put this point to him, Wittgenstein had nothing to say in reply. Why, in that case, 2

5.5302.

3

CP, VI, 94.



Identity

should there not also be further facts not reducible to atomic facts, such as the fact that two objects do not ever have all their properties in common? Whatever other of Russell’s philosophical views Wittgenstein had succeeded in persuading him to give up, Russell did not waver in his belief that there is an a priori structure to which the atomic facts are subject. After the war, when Russell eventually learned of Wittgenstein’s objection to his definition from the typescript of the Tractatus, he was initially persuaded: it was, he said, ‘a destructive criticism from which there seems no escape’.4 Quite soon, though, he came to doubt it. However, his doubts were, as so often with Russell, largely regressive. Without the definition of identity, he thought, mathematics would be lamed. This was hardly the sort of objection to impress Wittgenstein: so much the worse, he would have said, for mathematics. Russell’s only non-regressive defence of his definition was epistemological. ‘I should maintain,’ he said, ‘that to say that a and b are two means that they have different properties.’5 By now, though, Russell was not engaging with Wittgenstein’s system as a whole, but assessing the plausibility of his claims piecemeal. If I have recognized a and b as distinct entities, Russell insisted, they must ipso facto have different properties. Quite so, but Wittgenstein did not deny this. His point was not that there are two different objects which share all their properties, but only that it is intelligible to say that there are. The relation Russell had defined did indeed coincide with identity in the actual world, but Wittgenstein thought that in other worlds it need not. If something is possible, it does not follow that it is possible for it to be actual. . Eliminating identity Even if the objection Wittgenstein reported to Moore before departing for Norway was the one just described, the most that it can achieve is to show that Russell’s definition is incorrect, because the relation it defines might not be identity. What it cannot do in itself is to show that there is no such relation as identity at all. It leaves open, for instance, the possibility that identity is simply a primitive relation which may hold or fail to hold between objects. What rules this out is rather Wittgenstein’s conception of the metaphysics of facts. For each proposition, let us recall, there is on his account a fact which is its meaning. Suppose that identity is a primitive relation, so that sentences of the form a = b are atomic. So in the case when a = b, the fact that a = b is a positive fact—the very same, indeed, as the fact that a = a. Now if a positive fact does not occur, there is a negative fact which occurs in its stead. So in the case when a &= b, the negative fact in question ought to be the fact that a &= a. But there cannot be any such fact. 4

CP, IX, 107.

5

CP, X, 108.

The notational problem  To put the point in other words, the bipolarity of the proposition requires that the negation of any elementary proposition should represent a possible situation. But if simple things are identical, then they are just the same thing, so to represent that they are not identical would be to represent that a single thing is not itself. But there is no room in Wittgenstein’s conception for such a fact, and there is therefore no relation of identity. But this has a consequence for his conception of simple objects. It was part of his conception of simplicity that it is unintelligible to say of two distinct simple objects that they are identical. His explanation of any informative identity statement would therefore always be that the things which are asserted in it to be equal are complex. When exactly Wittgenstein arrived at this conception of objects as the constant elements around which possibilities turn, and hence of identity as not properly a relation, is not clear: what little he says about identity in the Notes dictated to Moore does not show that he had fully embraced such a conception by the time he dictated them in April 1914. Yet he must by then have been tantalizingly close to just this conception, since it seems to be firmly in place right at the beginning of the wartime notebooks, where Wittgenstein observes that to say of two things that they are identical means nothing.6 One of the characteristic methods we have seen Wittgenstein applying to technical problems that got in his way is to dissolve them rather than to solve them—to reconfigure the territory in such a way that the alleged problem disappears. Something like this is what Wittgenstein did in the case of identity, when he arrived at the conclusion that there is, properly speaking, no such relation. What Wittgenstein did in order to reach this conclusion was to adopt a fundamentally modal conception of sense according to which what varies between possible worlds is how the objects are configured, not which objects there are. Only now, perhaps, might we describe Wittgenstein’s view as genuinely logical atomism. . The notational problem If we agree that there is no genuine relation of identity, we might hope that there should be no need for a notation for identity (since in that case there is nothing in the world for such a notation to express). And in its occurrences to link proper names we can indeed eliminate the need for the sign of identity by the device of adopting a language in which one object never has two different names. But that does not absolve us completely from the need to have a notation for identity. For Wittgenstein’s conception depends on Russell’s theory of descriptions, according to which f (the g) is to be analysed as ( x) f x (y) gy ≡ x = y, E

6

NB, 5 Sep. 1914.



Identity

which still makes use of the sign of identity. Wittgenstein therefore needed an account of the notation of identity as it occurs linking variables in sentences such as this. Wittgenstein’s letters throughout the autumn of 1913 document his continuing failure to make progress with this problem. Identity is the very Devil and immensely important; very much more so than I thought. It hangs—like everything else—directly together with the most fundamental questions, especially with the questions concerning the occurrence of the same argument in different places of a function. I have all sorts of ideas for a solution of the problem but could not yet arrive at anything definite. However, I don’t lose courage and go on thinking.7

Shortly afterwards, Wittgenstein believed that ‘the ab-notation’ (by which he meant his method of truth-diagrams) provided a general test of whether a proposition was a logical truth. But in the case of identity he had to admit that this was still no more than a hope. ‘The ab-Notation for identity is not yet clear enough to show this clearly but it is obvious that such a Notation can be made up.’8 A little while later, he repeated his belief that logical propositions are those whose truth or falsehood is, in a perspicuous notation, visible in the sign itself. Nonetheless, he had to admit that identity still defeated him. I have not yet succeeded in finding a notation for identity that satisfies this condition; but I have NO doubt that it must be possible to find such a notation. . . I find identity, as I say, still far from clear. So I will deal with that another time.9

Wittgenstein did eventually solve the problem: in the Tractatus he adopted10 a notational convention according to which the ranges of nested variables are to be understood exclusively. For example, the sentence ‘( x,y) xRy’, which according to Russell says that there exist things related by R, on Wittgenstein’s reading says that there exist distinct things related by R. Similarly, ‘( x,y) xRy’ is now to be read as saying that anything is related by R to anything else. If we adopt this way of reading the quantifiers, the Russellian analysis of f (the g) can be expressed without any use of the identity sign as E

( x) f x gx ∼( x,y) gx gy. E

E

It seems to have taken Wittgenstein some time to hit on this notation. Not until November 1914 did he mention it in his notebook (although it is hard to tell from the entry whether he had just invented the device or was reminding himself of something he had devised earlier).11 Yet the core idea is surely already present in the Notes, where he observes that it is ‘useless to replace φ (a,a) by φ (a,b) a = b’.12 7

To BR, 29 Oct. 1913. 8 To BR, [Nov. 1913] (CL, no. 32). 5.532. 11 NB, 29 Nov. 1914. 12 B29.

10

9

[Nov. or Dec. 1913] (CL, no 32).

Chapter 

Sign and symbol One of the themes of this book has been Wittgenstein’s understanding of the notion of a symbol. This notion was placed at the very centre of his conception of logic as soon as he took the symbolic turn described in chapter 6. By characterizing a simple object as what is expressed by a simple symbol, he committed himself to a conception of symbols quite different from the signs that routinely confront us on the page. In this he was to some extent following Russell, whose conception of simplicity committed him to the view that hardly anything which counts grammatically as a proper name is logically such. In other respects, however, Wittgenstein’s conception of symbols owes much more to Frege’s notion of sense. In this chapter we shall explore this conception, both in the form that is implicit in the Notes and in Wittgenstein’s development of it in the Tractatus. . Seeing through to the symbol ‘In regard to notation,’ Wittgenstein reminded us, ‘it is important to note that not every feature of a symbol symbolizes.’1 The way I put it in the discussion of Wittgenstein’s symbolic turn was that recognizing the symbol is a matter of seeing through the irrelevant details to the feature of the complex that does the symbolic work. At that point in the discussion this was an approximate stipulation, since it left open quite what the work is that we are trying to do. Recent chapters have made the issue more determinate. The sense of a proposition is what we have to grasp in order to be said to understand it. So the symbolic work the proposition has to do is just that of expressing this sense. The symbolizing fact in the proposition is therefore the fact which ensures that it expresses the sense it does. The symbolizing fact is what we see exemplified in the complex when we understand the proposition. So it must determine the sense: we could not have two different senses symbolized by the same fact, because there would be nothing else to determine which sense was meant. On the other hand, the symbolizing fact does no more than to determine the sense. So propositions 1

C14.



Sign and symbol

that express the same sense must have the same symbolizing fact, and hence must be the same. In summary, propositions are the same if and only if they express the same sense. That, at least, is what Wittgenstein evidently held. It is why he says, for instance, that ‘ “not-not- p” is the same as “ p” ’.2 It is a doctrine which gives rise to some difficulties, however. To explore these difficulties it will be convenient to make use of the distinction Wittgenstein draws in the Tractatus3 between a propositional sign and a proposition. Both are facts, but the identity criteria are drawn differently in the two cases. A propositional sign is what I shall call a signifying fact: it is the fact that various syntactic units are arranged in the sentence in a certain manner. In the sentence ‘John met Mary’, for instance, the propositional sign is the fact that the word ‘John’ is followed by the word ‘met’, which is followed in turn by the word ‘Mary’. The proposition, on the other hand, is a symbolizing fact: it is what the signifying fact becomes when we read it as saying something, namely that John met Mary. This distinction between propositional sign and proposition, between signifying and symbolizing fact, is in the Tractatus an instance of a more general distinction between sign and symbol. A symbol is what a sign becomes when we read it (or a sentence in which it occurs) as saying something. Wittgenstein did not himself draw this terminological distinction between ‘sign’ and ‘symbol’ until he was composing the early part of the Prototractatus around the end of 1915. Although both these words (as well as their cognates, ‘signify’ and ‘symbolize’) occur in the Notes, there is no reason to think that this amounts to any more than stylistic variation. This naturally invites the question whether the terminological distinctions are relevant to a discussion of the Notes at all, given the anachronism that Wittgenstein had not yet formulated it when he wrote them. I hope, though, that enough has been said in earlier chapters about Wittgenstein’s concern for symbols to establish that the anachronism is only terminological: if he had not settled on his later terminology, this was not because he had not yet drawn the distinction, but rather because he had hardly any need at that stage to talk about signs at all. . Same sign, different symbol Symbols are what signs become when we invest them with meaning. When we read the words on the page, we turn them into the living expression of a situation; mere signs, on the other hand, do not yet say anything about the world. One might be tempted to think, perhaps, that what Wittgenstein meant by a sign was a mere pattern of words on the page. But in that case my talk of the ‘signifying fact’ would seem puzzling: a pattern is a complex, not a 2

C14.

3

3.12.

Same sign, different symbol  fact. Yet my usage here follows Wittgenstein’s own: in the Tractatus he states explicitly (three times) that a propositional sign is a fact.4 The question that arises, therefore, is which fact. If we focus on the sign rather than the symbol, which features of the pattern of words before us on the page should we attend to? I have forgotten the Greek I learnt at school so comprehensively that I can no longer even recognize which are the nouns and which the verbs; all that has remained is the residual ability to read the characters. So when I am shown a piece of classical Greek and read the words on the page without the least understanding of their grammar or their sense, am I identifying the signifying facts or not? According to the Tractatus, ‘The propositional sign consists in the fact that its elements, the words, are combined in it in a definite way.’5 Moreover, it is clear from the examples Wittgenstein offers that instances of different grammatical categories can, sometimes at least, count as the same word. At any rate, this will have to be the case if the sign–symbol distinction is to be of any use in explaining the device of typical ambiguity discussed in chapter 22. For the point there was that variables of different types are genuinely of different grammatical categories: to think of them as forming a single category would just be to collapse the distinctions on which the theory of types depends. Wittgenstein’s answer therefore appears to be that the grammatical category of a word belongs not to the sign but only to the symbol, and hence that my almost total ignorance of Greek is no bar to identifying its signs. Once we take this view, though, it is far from clear what stable notion of signifying fact is available. The point of regarding propositional signs as facts, not complexes, was to give them a grammar that put them on course to have a representational function, but the grammar of signs now seems too plastic to do any such representational work. When Wittgenstein introduced the terminological distinction between propositional sign and proposition, it was, superficially at least, because he wanted to talk about cases where the same propositional sign is used to express two different propositions or, more generally, where two different symbols can be read into the same sign. This is supposed to occur most notably in the case of puns. The issue is not quite as clear as it seems on the surface, however. In ordinary language there are cases in which the same sentence is used to express two different propositions (i.e. the same complex exemplifies two different symbolizing facts), but it is worth noting that there are two ways in which this can happen. In some cases the right explanation for an ambiguity is that there are two different propositional signs exemplified in the same sentence. For instance, we might plausibly explain the ambiguity of the complex ‘not p or q’ by describing two different facts: first, that ‘not’ occurs in front of ‘ p or q’; second, 4

3.14, 3.143.

5

3.14.



Sign and symbol

that ‘or’ occurs between ‘not p’ and ‘q’. It seems right to describe both of these as facts at the level of signs rather than symbols because they are straightforwardly syntactic. There are, in other words, two different propositional signs exemplified in the complex ‘not p or q’. In order to avoid this ambiguity, of course, all that is required is to add appropriate brackets to the sentence in order to disambiguate it. Other examples require other devices (such as conventions to disambiguate variable clashes in predicate logic). We might call a notation ‘uniquely parsable’ if no sentence exemplifies more than one signifying fact. On the other hand, a slightly different case arises when an ambiguity cannot be explained by a difference of compositional fact residing in the same complex; in this other sort of case the explanation of the ambiguity is that one sign is used to name two different objects, e.g. two different people called ‘John’ either of whom we might be referring to in the sentence ‘John met Mary’. To regard this as a difference of propositional sign we need to treat the two uses of the word ‘John’ as different signs, not just different symbols. Let us call a language ‘uniquely readable’ if no component of it is ambiguous as to the contribution it makes to determining the sense of a sentence in which it occurs. What is clear, though, is that if we are to avoid linguistic confusion, we need to eliminate both sorts of ambiguity. Let us therefore call a language ‘logically adequate’ if it is both uniquely parsable and uniquely readable. In that case no sentence in the language will exemplify more than one signifying fact, and no signifying fact will allow more than one symbolizing fact to be read in it. I mentioned earlier that Wittgenstein wavered in the Notes over whether the features of ordinary language should be taken as reliable indicators of logical structure. In the Tractatus Wittgenstein famously said that ‘all propositions of our colloquial language are actually, just as they are, logically completely in order’,6 but presumably he was by now placing considerable weight on his use of ‘propositions’ rather than ‘propositional signs’. He was suggesting not that colloquial language is logically adequate in the sense just adumbrated, but rather that if we do succeed by the use of colloquial language in expressing a proposition, then that proposition is logically completely in order. Making sense is not, on this view, something we can do by halves. . Same symbol, different sign Perhaps it is debatable whether Wittgenstein should have insisted that a difference of symbol entails a difference of sign. What is uncontroversial, however, is that two different propositional signs may be used to express the same proposition, even in a logically adequate language. In Russell’s notation, for 6

5.5563.

Same symbol, different sign  instance, ∼q, ∼∼∼q, ∼∼∼∼∼q, etc. are all signs for the same proposition. Even specifying all the signs in this notation which express the same proposition as ∼q is a non-trivial task. It is not just q preceded by an odd number of instances of ‘∼’, since we have to include other signs such as ‘∼q v ∼q’, or indeed ‘∼q ( p v ∼ p)’. The difficulty of specifying the ‘definite rule’ for generating all such signs is one to which Ramsey drew attention in his critical notice of the Tractatus. ‘It may,’ he observed, ‘be doubted whether it is possible to formulate this rule as it seems to presuppose the whole of symbolic logic.’7 And Ramsey was (as so often) right: to settle whether p v ∼ p is the same proposition as p, for instance, is just to settle whether p is a tautology, and so the former question cannot be any easier to solve than the latter. I mentioned in chapter 6 Frege’s quest for a ‘logically perfect language’. With the Wittgensteinian terminology now available to us we can express logical perfection as the demand that there should be a one–one correspondence between signs and symbols, so that we never have two different propositional signs expressing the same proposition, as happens with ∼q and ∼∼∼q in Russell’s notation. It was plainly Wittgenstein’s hope at the time of the Notes to be able to find such a notation. It is not only ordinary language that is not perfect; neither are the tidier languages of the Begriffsschrift and Principia. This is something Wittgenstein warned us to be conscious of, since otherwise there is a danger that we shall be misled by inessential features of our symbolism. If p = not-not- p etc., this shows that the traditional method of symbolism is wrong, since it allows a plurality of symbols with the same sense; and thence it follows that, in analysing such propositions, we must not be guided by Russell’s method of symbolizing.8

If we can find a notation in which not-not- p is represented by the same symbol as p, Wittgenstein believed, then that must be the right notation and Russell’s must be wrong. The very possibility of Frege’s explanations of ‘not- p’ and ‘if p then q’, from which it follows that ‘not-not- p’ denotes the same as p, makes it probable that there is some method of designation in which ‘not-not- p’ corresponds to the same symbol as ‘ p’. But if this method of designation suffices for logic, it must be the right one.9

On the other hand, if a perfect notation is impossible, as Frege eventually came close to realizing, then of course these pronouncements will have to be softened somewhat. The content of ‘wrong’ in the first quotation will have to be understood as no more than that the traditional method of symbolism is misleading if taken as a guide to the underlying complexity of what it expresses. A symbolism will be more accurate as a guide to this the more nearly it approaches the goal of perfection. 7

FoM, 278.

8

B26.

9

B22.



Sign and symbol

As we saw in chapter 20, a symbolism capable of expressing polyadic quantified logic transparently is impossible. Eventually, Wittgenstein himself seems to have realized this, or at any rate to have given up on the ambition. But it was plainly his goal at the time of the Notes. There is therefore some justice in Russell’s later tetchiness on the subject. At one time, Wittgenstein agreed with me in thinking that a logical language would be useful in philosophy, and I attributed this view to him in the introduction which I wrote to his Tractatus Logico-Philosophicus. Unfortunately, by this time, he had not only abandoned the view, but had apparently forgotten that he ever held it. What I said about it therefore appeared to him as a misrepresentation. His followers ever since have vehemently rejected the suggestion that a logical language could possibly be useful.10

. Symbol in terms of sign There is a natural temptation to explain the symbolizing fact in terms of the various signifying facts that can be used to express it. The account Wittgenstein offered in the Tractatus is as follows. That which denies in ‘∼ p’ is . . . not ‘∼’, but that which all signs of this notation, which deny p, have in common. Hence the common rule according to which ‘∼ p’, ‘∼∼∼ p’, ‘∼ p v ∼ p’, ‘∼ p ∼ p’, etc. etc. (to infinity) are constructed. And this which is common to them all mirrors denial.11

It is worth noting carefully what Wittgenstein does not say in this passage. He does not say that the symbolizing fact in an expression for the negation of p is that it is ‘∼ p’ or ‘∼∼∼ p’ or ‘∼ p v ∼ p’ or ‘∼ p ∼ p’, or any of the others in a range of signs which can be generated according to a certain rule. If he continued to believe at the time of the Tractatus, as he had when he wrote the Notes, that there are no disjunctive facts, it would evidently have been necessary for him to avoid such an account, since according to it the symbolizing fact would be disjunctive. Nor does Wittgenstein say that what symbolizes is the rule according to which all the various signs for the negation of p are constructed. At the level of phenomenology, at any rate, this would not be a plausible view to hold. As Ramsey noted, even in the case of Russell’s notation it may be doubted whether it is possible to formulate the rule at all; and it seems wildly implausible to ascribe to me even an implicit grasp of such a rule merely on the basis of my ability to read signs such as ‘∼ p’ and ‘∼∼∼ p’ reliably. Wittgenstein’s use of the word ‘hence’ in referring to the putative ‘common rule’ suggests that he wished to avoid such an ascription. For each proposition there is a rule codifying what is common to all the signs expressing it. Such rules ‘are equivalent to 10

MPD, 165.

11

5.512.

Symbol in terms of sign  the symbols and in them their sense is mirrored’,12 but it does not follow from this (or at any rate not directly) that anyone who has the capacity to recognize the symbol in any sign constructed according to the rule must have an implicit grasp of the rule. Wittgenstein apparently wanted us to think of symbolizing facts not as disjunctions of signifying facts or as rules for generating such facts but as occurring at an altogether different level—in, to borrow Frege’s later usage, a different realm. Symbols are not worldly, but nor are they of anything like the same nature as signs. One way to think of this might be by analogy with looking at a distant object through a patterned window: when we focus on the distant object, the pattern on the window becomes invisible. In much the same way, recognizing the symbolizing fact is a matter of adjusting our logical focal length so that the sign by means of which the proposition is expressed becomes invisible. Something very like this is indeed a familiar experience in reading. It is the difference, perhaps, between the experience I have of reading my native language and that of reading one in which I am not fluent: in the former, especially if the content of what I am reading interests me, I hardly notice which words are actually being used; in the latter I am conscious of each sentence as I laboriously translate it word by word. In my earlier sketch of their respective theories I followed the orthodoxy of describing Russell’s as a one-step, Frege’s as a two-step semantic theory. Which, then, was Wittgenstein’s? As I remarked in §7.1, he did not need the notion of the sense of a proper name for the role which Frege originally claimed for it of explaining the different contributions two names of the same object may make to the thoughts expressed by sentences in which they occur. Even in a non-perfect language in which there are two signs for the same object, Wittgenstein held that how they symbolize will be identical: the semantic value of a name is the object it refers to, and so there will be only one symbol to be discerned in the two signs. If he had succeeded in the project of devising a notation that at least approximated to logical perfection, he might then have felt able to limit the sign–symbol distinction to the criterial role of defining what it is for a language to be logically perfect. It would then have been possible to say that, for logically perfect languages at least, his semantic theory counted as one-step in Russell’s sense. But, as we have seen, Wittgenstein eventually gave up the attempt to provide an explicit language in which the symbols are transparently mirrored in the signs used to express them. This left the symbol with the role of mediating the transition from sign to object, not, as in Frege’s case, to make good a conception of objects as multifaceted, but nonetheless for a purpose that is still recognizably that of Frege’s senses, namely that of enabling language to represent a world whose complexity it cannot hope to match. Frege and Witt12

5.514.



Sign and symbol

genstein both bridge the gap between language and world in two steps, with thoughts as the intermediate point, but Wittgenstein’s picture theory, by insisting on the transparency of the second step, makes the first, from language to thought, that much more mysterious. Frege’s third realm of sense has seemed mysterious to many of his readers: twentieth-century philosophy is littered with attempts to eliminate or demystify it. Perhaps the Notes are one of the first such attempts. If so, this is a respect in which the Tractatus is not a continuation of the project sketched in the Notes. The Tractatus not only does not eliminate the mediating role of symbols but it in a certain sense identifies that role with the mystical. This is of course a large theme, and one that is quite absent from the Notes. What is significant to note here is only that Wittgenstein’s abandonment of the goal of a logically perfect language, although not a prerequisite for there to be the mystical, undoubtedly contributes to its mysteriousness. Recognizing the symbol in the sign is a process which the doctrine of unsayability tells us cannot be put into words: the impossibility of a logically perfect language gives us a clue to its impenetrability. . The symbol vanishes The job of a proposition, according to the Notes, is to divide facts into two classes, those that are of like sense, and those of opposite sense. Wittgenstein invoked an analogy with a line dividing a plane, but tautologies and contradictions were to be treated as special cases: the proposition p v ∼ p, for instance, is of like sense with the facts, whatever they are; and in the same way p ∼ p is of contrary sense to the facts. Wittgenstein’s response to this was to say that here the analogy with a line dividing a plane breaks down: for any line in the plane, there are points on each side of it. Wittgenstein concluded that p v ∼ p is senseless.13 Whether we say this, or instead say that p v ∼ p has a trivial sense, is surely in itself just a matter of terminology. But Wittgenstein evidently intended what he said to have a consequence for our understanding of the proposition. By saying that p v ∼ p has no sense, he intended to suggest that there is nothing to be expressed, and hence no symbolizing fact to be discerned in the propositional sign. Later, the analogy he drew between propositions and pictures would provide him with a reason for holding this view. If a proposition is a picture of how things would have to be for it to be true, then a logical truth is not a proposition because there is nothing to picture. Saying that this rose is either red or not red puts no constraint on the rose or anything else. To picture how the world has to be for this to be the case is not to picture anything. 13

Cf. §14.3.

The symbol vanishes  If we try to assemble pictorial elements so as to picture a situation in which the rose is either red or not red, we stumble: what results is not a trivial picture, but no picture at all. What is notable here, though, is how Wittgenstein seems, as on other occasions, to have adopted a view before he had much reason to hold it. This is characteristic of Wittgenstein’s method of theory formation, which put a much greater premium on suggestive analogies than on reasons. Until we have formulated the picturing analogy, it is not really clear why we should say that p v ∼ p is senseless, rather than that its sense is trivial. And even when that analogy is in place, it is still only an analogy: it does not in itself dictate what the senselessness of a tautology amounts to. In these remarks in the Notes about the vacuity of the propositions of logic Wittgenstein was plainly only a small step away from the characterization of them as tautologies which he developed in Norway shortly afterwards. But he was also an equally small step away from an obvious problem for the corresponding conception of the symbolism. If in the case of the sentence ‘This rose is either red or not red’ there is no symbol to be discerned, then the same is true of the sentence ‘This rose is both red and not red’. Both, according to Wittgenstein, say nothing. Yet they somehow contrive to say it in diametrically opposite ways. The duality between positive and negative facts, which gave him no peace,14 here has its echo in the case where no facts are in question. 14

NB, 25 Nov. 1914.

Chapter 

Wittgenstein’s theory of judgment In chapter 13, I described how Wittgenstein tore Russell’s multiple relation theory of judgment to shreds. That discussion left one salient piece of unfinished business, however. The nub of Wittgenstein’s complaint against Russell’s analysis, let us recall, was that in it the verb of the proposition believed did not occur as a verb. If I say that A believes that p, I must express what it is that A believes, even though I do not myself assert it. So p must occur in the analysis as a fact, not as a complex. But even if we are now clear what was wrong with Russell’s theory, it still remains to be determined what should be put in its place. . Russell’s later views Although it does not belong to the principal thread of our narrative, it is perhaps worth noting first that Wittgenstein’s criticisms did not lead Russell altogether to give up hope of resurrecting some version of the multiple relation theory. When he reprinted in 1917 one of the articles in which he mentioned it, he added in a footnote, ‘I have been persuaded by Mr Wittgenstein that this theory is somewhat unduly simple, but the modification which I believe it to require does not affect the above argument.’1 Unfortunately Russell did not go on to explain the modification he had in mind. Some time later, though, he was a little more explicit. ‘Belief will really have to have different logical forms according to the nature of what is believed.’2 What he had in mind was presumably that he could escape from the difficulty by positing different kinds of judging—a judging relation J1 appropriate for judging subject-predicate propositions, another judging relation J2 appropriate for judging relational propositions, and so on. As he noted, ‘The apparent sameness of believing in different cases is more or less illusory.’ And of course he had been quite happy to take the analogous course with the concept of truth as a way of resolving the liar paradox: in Principia he held that there is not just one concept of truth but a whole hierarchy of them. So it would not have seemed to him to be too large a step to say similarly that there 1

CP, VI, 154.

2

CP, VIII, 199.

The theory of judgment in the Notes



is a whole hierarchy of judging relations. (At any rate, it is surely a lot more plausible to say this about judgment than about truth.) A moment’s thought shows, however, that the hierarchy of judging relations will have to be of considerable complexity. For eventually Russell would have to deal with the cases where we judge not only atomic propositions but more complex ones built up from them. For instance, he was in danger of needing a special relation just for judging ‘ p or q’ where p is a subject-predicate proposition and q is relational. One of his students, Dorothy Wrinch, published an attempt to extend the multiple relation theory to molecular propositions in 1919, but it was (as she herself admitted in her article) both very tentative and very complicated.3 Russell himself made no attempt to develop the idea, and it seems very likely that he regarded this as one of the matters he would leave to Wittgenstein. . The theory of judgment in the Notes In the Notes, as we have seen, Wittgenstein was in the grip of the idea that what makes the proposition p expressive is its relationship to its two poles a and b. So when he wanted a proposition to occur as a fact, not a complex, one way he liked to indicate this was to draw in its poles explicitly. His way of representing the idea that in ‘A believes that p’ the proposition p is expressed was therefore to say that it consists in a relationship not between A and the proposition p but between A and the poles of p. ‘This is obviously not,’ Wittgenstein says with some understatement, ‘a relation in the ordinary sense.’4 Indeed not, because the poles a and b are not objects in the ordinary sense. In fact, they are not objects at all. So it is puzzling what relationship I am supposed to enter into when I believe something. If I conceive of it as a relationship between me and the proposition p, it seems that p will inevitably occur in the relationship as a complex, not as a fact. On the other hand, it cannot in any useful sense be thought of as a relationship between me and the poles of the proposition, because there are no such things. So it is hard to avoid the thought that the diagram A a

p

b

which Wittgenstein offered in the Notes to represent his analysis of ‘A believes that p’ is in its way just as desperate as the map of belief Russell had drawn in his Theory of Knowledge manuscript. Wittgenstein was plainly struggling. All he could offer was the rather vague suggestion that ‘the epistemological questions concerning the nature of judgment and belief cannot be solved without a correct apprehension of the form 3

‘On the nature of judgement’.

4

C41.



Wittgenstein’s theory of judgment

of the proposition’.5 As it turned out, though, he was looking in the wrong place. His error lay not in the bottom half of the diagram (the understanding of the proposition p), but the top half (the occurrence of the subject A as a term in the relation). As long as A occurs in our analysis of the judgment, we will have no alternative but to conceive of the judgment as consisting in a relation between A and what is judged; yet we cannot relate A to the symbolizing fact p, precisely because it is a fact. The only entity in the vicinity for A to be related to is a complex, not a fact, and hence not what we want. Wittgenstein’s error was thus that he had placed incompatible demands on his theory. The proposition had to occur in the diagram as a proposition in order that the judgment should express what is judged; but the subject A could not occur in the judgment as a further term outside what is judged and related to it, because ‘propositions, owing to sense, cannot have predicates and relations’.6

. Wittgenstein’s later theory of judgment We can trace out a route (perhaps even Wittgenstein’s own route) to his eventual theory of judgment if we start again from the theory of Russell’s that he rejected. Let us consider, for simplicity’s sake, a relational proposition ‘aRb’. As we saw in chapter 13, Russell wanted to analyse this as #2 (a,R,b), whereas Wittgenstein had rejected this idea. But the question of how far into the proposition the form reaches is not what is at issue at the moment. So let us here simply side with Wittgenstein and represent the symbolizing fact in ‘aRb’ as having two constituents ‘a’ and ‘b’ combined together according to a certain form. And let us, borrowing a notational device from Geach,7 refer to this form as §R. In that case Russell’s theory, adapted to recognize Wittgenstein’s analysis of the proposition, would have analysed A’s judgment that aRb as J(A,§R,a,b), i.e. it would have included as a term the form of what is judged. But that, as we have seen, is precisely what Wittgenstein objected to—that §R occurred as a term and not as a form. So let us respond to this objection by removing the form. We cannot let it disappear entirely, though: judging that a has R to b is not the same as judging that a has S to b, and that difference has to be represented in the analysis. So let us indicate the difference by subscripting the judgment relation correspondingly, thus obtaining J§R (A,a,b). 5

B55.

6

C15.

7

Mental Acts, 52.

Wittgenstein’s later theory of judgment  Thus far, we have not gone beyond anything available to Wittgenstein at the time of the Notes. But now there is one further step to be taken, and it relates to the term in the relation that has not so far been the object of criticism, namely the subject A. What we saw in the last section was that judging cannot consist in a relation between A and the proposition, because propositions cannot be related to anything. But in the putative analysis just considered, A is related not to a proposition but to its constituents. So although the previous argument is no doubt suggestive enough to give us reason to experiment with dropping A, it does not in itself prove that we must. It does not, that is to say, show that there is no more reason to think A is simple than that p is. But recall that ‘a’, §R, and ‘b’ combine to make the symbolizing fact §R(a,b). If we are to respect the bareness of the notion of a logical picture, we have to think of this as a mere notational variant on aRb. In that case it is hard to see how there can be any place for A. If ‘aRb’ is complete, there is no relating left to do. (This is another way of putting the point made in the last section that our analysis of ‘A believes that p’ cannot both express p and relate it to A.) We conclude, then, that when the judgment relation is fully analysed, the subject A will not occur as a term in it: so let us remove it. What this gives us is J§R (a,b). Of course, we would expect something similar to apply to other attitudes we might take to the proposition. So ‘A doubts that aRb’ might be D§R (a,b), and ‘A believes that aRb’ might be B§R (a,b). What strikes us immediately is that none of these is very different from §R(a,b). In other words, what it is for A to judge, doubt, or believe that a has R to b is very like what it is simply for the symbolizing fact §R(a,b) to occur. Once he had given up the idea that the subject A occurs as a term in judgment, the way was clear for Wittgenstein to reach the theory of judgment offered in the Tractatus. According to this theory, all that has to be added to the occurrence of the symbolizing fact in order for it to count as a case of someone’s believing something is that its constituents should occur ‘connected in his mind and accompanied by a feeling of belief’.8 Similarly, a case where someone doubts something will be analysed as the constituents of the proposition connected in the person’s mind accompanied by a feeling of doubt. And so on for other propositional attitudes. The occurrence of the constituents connected in the mind in this manner expresses the proposition, and so the analysis fulfils Wittgenstein’s requirement that it should express what it is that is judged. As he put it in the Tractatus, ‘ “A believes that p”, “A thinks p”, “A says p”, are of the form “ ‘ p’ says p”.’9 This account ‘shows that it is impossible to judge a nonsense’ because it involves us in expressing the proposition that is judged. If there were a judg8

Ramsey, FoM, 145.

9

5.542.



Wittgenstein’s theory of judgment

ment that the table penholders the book, for instance, it would have the same form as the expression, ‘The table penholders the book’ says that the table penholders the book. But this is nonsense. I can look at the string of words, ‘The table penholders the book’, but I cannot read it as saying anything. That is to say, I cannot discern in it a symbolizing fact. So I cannot configure any of the contents of my mind so as to represent such a symbolizing fact, since there is no such symbolizing fact to represent. The decisive step, then, is the one Wittgenstein took in Norway of realizing that judgment cannot be any sort of relation between the judging subject and either the proposition or its components, and hence that the subject has no place in the analysis of judgment at all. . Ramsey Whether or not it quite represents Wittgenstein’s own route to his Tractarian theory of judgment, the narrative just sketched has one significant virtue, which is to bring out the similarity between that theory and Russell’s multiple relation theory. This similarity is especially worth bearing in mind when we are assessing alternative accounts of Wittgenstein’s objection to Russell’s theory. It is true that Wittgenstein’s objection was general enough to apply to all the variants of that theory that Russell came up with: that is to say, it applies to any analysis of ‘A believes that p’ in which the judging subject A is multiply related to the components of p. But any account which represents Wittgenstein as offering an objection to any multiple relation theory whatever is very unlikely to be correct, for the straightforward reason that Wittgenstein’s is also a multiple relation theory. The difference is that in Wittgenstein’s theory what are multiply related are just tokens of the components of p: the judger A disappears on analysis. This is important because it permits these tokens to occur in the analysis of the judgment with their ordinary grammatical values. What happens when Othello believes that Cassio loves Desdemona is that there occurs as a belief in Othello’s mind a token of the thought that Cassio loves Desdemona. Or, more precisely, there occur in his mind tokens of the components of this thought, combined just as these components are combined in the thought. Among them is a token of the verb ‘loves’; and it occurs in the belief just as the verb occurs in the thought, i.e. as a verb, not as a noun. Thus the flaw Wittgenstein had identified in all of Russell’s various multiple relation theories is not present in his own. What the Tractatus offers may thus with some justice be termed a multiple relation theory of judgment. What connects the components in Wittgenstein’s analysis is not what he called a ‘relation in the ordinary sense’—such as a spatial relation, for instance—because one of the components is a form; but it

Ramsey  is at any rate more similar to Russell’s multiple relation theory than it is to any of the other theories of judgment that had been proposed. Surprisingly few commentators have remarked on this similarity. One who did was Ramsey, whose account of Wittgenstein’s theory in ‘Facts and propositions’ clearly presents it as a multiple relation theory. Ramsey’s treatment of Wittgenstein’s theory of judgment has other virtues too. One is that he recognized, even before he had discussed it with Wittgenstein himself, how little Wittgenstein cared about describing the particular features of the mental tokens involved in belief or doubt or judging. Another is that Ramsey recognized quite explicitly (again before he had discussed the matter with Wittgenstein) the close affinity Wittgenstein saw between his theory of judgment and his theory of meaning. In his critical notice on the Tractatus Ramsey remarks that Wittgenstein ‘explicitly reduces the question as to the analysis of judgment, to which Russell has at various times given different answers, to the question “What is it for a proposition token to have a certain sense?” ’10 If we ignore the feeling of belief or doubt or judging, then for someone to adopt a propositional attitude towards a proposition is for the constituents of the proposition to be connected in the person’s mind so as to express the proposition concerned. Only thus can the verb of the proposition judged occur as a verb rather than as a term. Because the judging subject A has disappeared, the kernel of Wittgenstein’s theory of judgment is now indistinguishable from his theory of meaning. ‘Here we have no co-ordination of a fact and an object, but a co-ordination of facts by means of a co-ordination of their objects.’11 The semantic task I face if I am to judge something is to correlate the components of what I judge with symbols that go proxy for them. Having done this, all that remains is for me to assemble those proxies so as to express what I am judging to be the case. What is operative in judgment is the relationship thus created between these proxies; this is just the same as the relationship that holds between the corresponding objects if what is judged is the case. 10

FoM, 274–5.

11

5.542.

Chapter 

The picture theory Everyone who writes about the Tractatus agrees that the picture theory is one of its central themes. There is less agreement, though, about just what is meant by the ‘picture theory’. Some use the phrase very broadly as a sort of catch-all for the logical doctrines of the book, others much more narrowly for the specific proposal that propositions are pictures of reality. Exponents of the resolute reading of the Tractatus sometimes take ‘the picture theory’ to be just what we recognize to be nonsense when we throw away the ladder. And a similar vagueness of reference is widespread in the literature on Wittgenstein’s later writings, where it is commonly (and dubiously) asserted without further specification that he ‘abandoned the picture theory’. The question whether the picture theory is already present in the Notes is correspondingly ambiguous. Here we shall piece together the various elements that make up Wittgenstein’s semantic theory in an attempt to disambiguate this question.

. Coincidence of structure It is of course trite to observe that the words we use might have meant something different. It is, for instance, an accident of etymology that ‘black’ does not mean white, and a blessing I owe only to my parents’ good taste that my name is not ‘Rocky’ or ‘Dwayne’. But this trite observation immediately prompts a less trite question. If it is contingent which words mean what, what ensures that our words mean anything at all? What constraints are there on our choice of words? One first attempt at a sketch of Wittgenstein’s answer would be to say that what constrains us is grammar. Grammar, according to him, is not contingent as word selection is. But to recognize grammar as an essential feature of any language that succeeds in talking about the world is not yet to offer an explanation for that success. We get closer to the essence of the picture theory when we note Wittgenstein’s belief that the grammar of language cannot help but be in a certain sense identical to the grammar of the world. We noted earlier that Wittgenstein derived his distinction between names and forms from Frege’s distinction between saturated and unsaturated expressions. But, although the distinction between saturated and unsaturated

Coincidence of structure  was prompted by linguistic considerations, and is at its most natural when applied to language, Frege did not restrict its application to language. He held that the things in the world to which saturated and unsaturated expressions refer are correspondingly saturated or unsaturated. Names, which are saturated expressions, refer to objects, which are saturated; concept words and relation words, which are unsaturated expressions, refer to concepts and relations, which are unsaturated entities. In the same way, Wittgenstein intended the distinction between two kinds of components of symbolizing facts to track a distinction between the components of facts in the world. When he said, ‘Components are forms and constituents’,1 he intended the remark to apply to all facts, not just those that symbolize. In the case of a non-symbolic fact, the constituents would be objects and the form would be the relation between them (or, in the case of one object, the property of it) whose holding constituted the fact. The idea that there is a harmony between the structure of a fact and the structure of the symbol that expresses it is quite explicit in the lectures Russell gave at Harvard in 1914, where he said that ‘the structure of the symbol must be identical with the structure of the symbolised’,2 and that ‘there is always a sort of fundamental identity between symbol and symbolised’.3 In 1918, moreover, we find Russell in ‘The philosophy of logical atomism’ hoping to persuade his audience that in a logically correct symbolism there will always be a certain fundamental identity of structure between a fact and the symbol for it; and that the complexity of the symbol corresponds very closely with the complexity of the facts symbolized by it.4

In the Notes, on the other hand, Wittgenstein does not express the point in this general manner. Instead he makes it laboriously case by case, at least in relation to elementary propositions: if an atomic fact consists of a form and one object, the elementary proposition which expresses it must be of subjectpredicate form; and similarly for other forms. Every proposition which says something indefinable about a thing is a subject–predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on.5

The point, though, is at root the same: it is because the structure of the symbol must be identical with the structure of what it symbolizes that a proposition which says something indefinable about a thing must be a subject-predicate proposition. The doctrine that ‘the structure of the symbol must be identical with the structure of the symbolized’ is certainly a key component of the picture theory; moreover, it is present in the Notes on Logic, and directly traceable to the symbolic turn that we noted in the letter Wittgenstein wrote to Russell shortly 1

C50.

2

Eliot’s notes, 7 Apr. 1914.

3

11 Apr. 1914.

4

CP, VIII, 175.

5

B77.



The picture theory

after his visit to Frege at Christmas 1912. On the other hand, although it is present in the Notes, I would hesitate to say that it is fully secure there, because there is no sign that Wittgenstein yet had the resources to establish the harmony between proposition and fact at any but the atomic level. Only when he had a developed account that could explain away the case in which an apparently complex expression such as ‘the object on the table’ refers to something simple would he be entitled to the claim in its full generality. . The picturing analogy It is, of course, natural to think of the essence of the picture theory as being the eye-catching claim6 that a proposition is a picture. But to decide whether that natural thought is right, we need to look at how far the eye-catching claim takes us beyond Russell’s remark in 1914 that the structure of the symbol must be identical with the structure of the symbolized. The picturing analogy itself is not present in the Notes; the analogy that guides Wittgenstein’s account of propositional expression there is more like that of a compass needle being held next to a magnetic field to detect its direction (see §16.3). In the Tractatus, that analogy is almost wholly absent. Although propositions still have sense, the punning suggestion that this is to be thought of as like the direction of an arrow has been all but dropped.7 What replaces it is a new analogy according to which a proposition is a picture. Straightaway, though, the analogy has to be interpreted with care. A proposition is still a fact, as it was in the Notes. And, it emerges,8 a picture is a fact too: it has to be, since propositions are a particular kind of picture (to wit, the logical kind). So what Wittgenstein means by a picture cannot be a pictorial complex, an arrangement of things on a canvas or wherever; it must rather be a certain fact that we see exemplified in the arrangement when we look at it in a certain way. As we have seen, the idea that the structure of the proposition is the same as the structure of what it represents is one that Wittgenstein acquired early. But so far the point is essentially negative. What the analogy of the picture aims to add to it is something positive—an explanation for the harmony. It is by standing in a certain configuration that the components of a proposition represent that that is how things are. We can be fairly definite about when Wittgenstein came upon the idea of using this analogy to express how a proposition represents a possible situation. It is not mentioned in the Notes on Logic or the Notes dictated to Moore, but it is at the centre of the discussions early in the wartime notebooks. ‘In the proposition a world is as it were put together experimentally. (As when in the law-court in Paris a motor-car accident is represented by means of dolls, 6

4.021.

7

Cf. 3.144.

8

2.141.

Truth  etc.)’9 There is little reason to doubt his later recollection that he was led to this thought by a description of a court case which he had recently read in a magazine. If we follow the popular practice of regarding this analogy as the crux of the picture theory, therefore, we may say with some confidence that it was born in August or September 1914. On the other hand, there is room to question how significant the analogy really is for the Tractatus. Wittgenstein later remarked that the Tractatus contains kitsch,10 and there is something to be said for the view that the picturing analogy is among the kitsch. For a spatial picture represents a way things might be configured in space, and it does so by configuring representatives of these things in space in just this manner. But a proposition is a logical picture. What this means is that the manner of combination in a proposition is not spatial or musical, or of any other particular kind. Rather is it whatever all these particular kinds of combination have in common. One way of putting this would be to say that the notion of a logical picture has a certain thinness to it: there is something insubstantial about the manner of combination, in contrast to the case of a spatial picture, for instance, in which the manner of combination itself plays a representational role. And this thinness is essential to Wittgenstein’s account. Only by insisting that there is nothing substantial to how the components of a proposition are combined could Wittgenstein ensure the generality of logic—keep faith, that is to say, with the motivating insight that ‘logic cannot treat a special set of things’.11 If, for instance, what symbolized in a proposition were the spatial relationship between its components, what the proposition represents could only be a spatial relationship between things—spatial things. And that would be to restrict the generality of logic. It is vital, therefore, that there should be nothing to the manner of combination of the parts in a proposition—nothing, that is, except the bare minimum that is present in any picture. But Wittgenstein does not explain why there should even be such a minimum. Why, that is, should we suppose that there is anything various particular kinds of representation have in common?

. Truth I said earlier that in the Notes Wittgenstein did not aim to explain the truth or falsity of a proposition directly but went via a comparison between the proposition and the world. That is why he offered in the first instance an account of what it is for a proposition to be of like or contrary sense to the facts rather than of what it is for a proposition to be true. So, to complete the explanation, we need to add the stipulation that (T) A proposition is true if and only if it is of like sense to the facts. 9

NB, 29 Sep. 1914.

10

Diary, 16 May 1930.

11

C9.



The picture theory

But at this point a worry emerges. I have just described (T) as a stipulation. Could we then have made the opposite stipulation? (T! ) A proposition is true if and only if it is of contrary sense to the facts. If we adopted this amended truth scheme but left everything else in our language unchanged, the effect would be to reverse the sense of every proposition. Although it is very similar, this is not in fact quite the same as the proposal we considered in §10.2 of expressing ourselves by means of false propositions rather than true ones. Both proposals result in its being appropriate to say ‘not-q’ just in case in the standard interpretation it is appropriate to say ‘q’, but the reversals occur at different points: the earlier reversal attempted, in effect, to swap the roles of truth and falsity; the one now under consideration leaves these unchanged but reverses the sense of every proposition. Wittgenstein correctly blocked the earlier reversal by insisting on the close link between truth and assertion: a proposition is true just in case things are as it asserts them to be. This observation is of no help in blocking the stipulation (T! ), however. If the link between truth and assertion is too tight for there to be any room for choice in the matter, what has been said so far does not make it clear that the same can be said of how the truth conditions for a proposition depend on the meanings of its components. It seems, that is to say, as though we have a genuine choice about how we fill in the dots in the truth-scheme ‘ p’ is true if and only if . . . There is plainly a difficulty here. At the very least, what we must not do is to think of the way in which the components of the proposition are put together as a further substantial element in it. I mentioned in chapter 10 that Wittgenstein distinguished psychology, which attaches meaning to names and senses to forms, from logic, which deals with how these fit together to form propositions. In terms of this distinction the danger is that of seeing anything substantial in the logical part of the process. This, of course, is another application of Wittgenstein’s fundamental thought, that logic has no subject matter. But it is in any case urged on us by the need to avoid a version of Bradley’s regress. If we call the way in which the pieces of a proposition are assembled its structure, that should not be taken to reify the structure but only to introduce a variant way of talking about how the pieces are assembled. The difficulty, then, is that Wittgenstein’s account of sense has two parts. First, a proposition is a standard to which facts behave. Wittgenstein spoke of facts as being either of like or of contrary sense to the proposition, but he might just as well have used the language of correspondence: he might, that is to say, have talked of a proposition as corresponding or failing to correspond to the facts. Second, the proposition is true if the facts are of like sense to it,

The identity theory  i.e. if they correspond to it. The proposition offers a standard to which we may compare the world; and the proposition is true if the comparison comes out one way rather than the other. But this is wrong: where it has two parts, Wittgenstein’s account should only have one. Although the variant truth-scheme (T! ) is not quite the same as the variant language of the inveterate liar, the mistake it exposes is the same. What a proposition expresses is already true or false, and if I assert the proposition, what I assert is just that—what it expresses. There cannot be an intermediate stage in the explanation where a proposition represents a possible situation while yet remaining neutral as to whether it represents the situation as obtaining or not: to represent a situation is the very same thing as to represent it as obtaining. And to assert the proposition is just to say that the situation does obtain. Wittgenstein’s talk of a proposition as being true to the facts is therefore faulty. What else could a proposition be true to?

. The identity theory The compass-needle analogy encouraged us to think of a proposition as a measure that has to be held up against reality to see whether reality corresponds to it. That, we have now seen, was a weakness, because it suggested that there was a gap to be bridged between what the proposition expresses and how reality has to be if the proposition is to be true. But if this is an error that the compass-needle analogy encouraged, the picturing analogy with which Wittgenstein replaced it does not on its own prevent the error. If we think of the picture as a measure that we hold up against reality, we fall into the very same mistake. We could just as well adopt the convention that pictures are to be read as saying ‘This is how things are’, or equally ‘This is how things are not.’ The difficulty is one that Frege pointed out eloquently. It might be supposed . . . that truth consists in the correspondence of a picture with what it depicts. Correspondence is a relation. This is contradicted, however, by the use of the word ‘true’, which is not a relation-word and contains no reference to anything else to which something must correspond. If I do not know that a picture is meant to represent Cologne Cathedral then I do not know with what to compare the picture to decide on its truth. A correspondence, moreover, can only be perfect if the corresponding things coincide and are, therefore, not distinct things at all. . . But this is not at all what is wanted when truth is defined as the correspondence of an idea with something real. For it is absolutely essential that the reality be distinct from the idea. But then there can be no complete correspondence, no complete truth. So nothing at all would be true: for what is only half true is untrue. Truth cannot tolerate a more or less.12 12

‘Der Gedanke’, 59–60.



The picture theory

Frege’s warning is directed so pointedly against the picturing analogy that it would be easy to imagine he wrote it in response to something in one of his conversations with Wittgenstein. But in fact the distinction Frege draws here between propositions and pictures is already present in the 1897 draft of ‘Der Gedanke’. The most that Frege could have done in response to Wittgenstein is to expand the discussion a little. Eventually, Wittgenstein himself seems to have realized the point that Frege was making, but it was not his adoption of the picturing analogy that led him to it. Even when that analogy was in place, Wittgenstein repeatedly worried about the apparent gap between the picture and what it represents. If the proposition is given, and congruence, then the proposition is true if the situation is congruent with it. Or: the proposition is given and non-congruence; then the proposition is true if the situation is not congruent with it.13

It took him some time to realize that the solution to the problem was to remove the gap that the notion of congruence (or non-congruence) was designed to fill. We might indeed ‘show how not to fence by means of fencing puppets’,14 but fencing puppets constitute only a pictorial complex, not a picture. The sense in which a proposition is a picture is a sense in which it already expresses how things are if it is true: it is true, namely, just in case things are as it represents them to be. It might well be argued, then, that what is essential to the picture theory of the Tractatus is not its appeal to the picturing analogy on its own, but rather the manner in which the theory takes to heart Frege’s admonition that correspondence comes in degrees but truth does not. What is essential, that is to say, is that it aims to be an identity theory, not a correspondence theory. That the elements of the picture are combined with one another in a definite way, represents that the things are so combined with one another. In order to be a picture a fact must have something in common with what it pictures. In the picture and the pictured there must be something identical in order that the one can be a picture of the other at all.15

There is little sign in the surviving texts that Wittgenstein advanced this far until perhaps the very end of 1914. Only then does he seem to have become quite clear that the representational form of the propositional sign does not merely correspond, but is identical, to the situation it represents.16 ‘Non-truth,’ he noted some months later, ‘is like non-identity.’17 And, he could presumably have added, truth is like identity. Or, as Ramsey crisply put it, ‘ “The fact that a has R to b exists” is no different from “a has R to b”.’18 13

NB, 3 Nov. 1914. 1915. 18 FoM, 143.

14

NB, 5 Nov. 1914.

15

2.15–2.161.

16

NB, 18 Dec. 1914.

17

NB, 27 Apr.

Possibility  . Possibility The picture theory claims, then, that language succeeds in representing the world because the structure of the proposition is identical to the structure of the state of affairs it represents. But there is one last gap in the account that needs to be closed. The structure of a proposition, in the sense just referred to, is grammatical structure: each component of the proposition belongs to a grammatical category, and they are combined together in accordance with grammatical rules in whose expression these categories figure. It follows, therefore, that the state of affairs represented also has a grammatical structure: its components belong to grammatical categories just as the words that refer to them do. But one might now be perplexed as to what this grammatical structure amounts to. What does it mean for the world to have a grammar? In the Tractatus Wittgenstein’s answer is that the grammar of the world consists in the possibilities of combination of its parts. The grammatical type of an object consists in ‘the possibility of its occurrence in atomic facts’,19 i.e. in its possibility of combination with other objects. The contrast between object and fact is thus represented as being that between the necessary and contingent features of the world: objects are what is constant; what can vary is which of these constant elements are configured to form atomic facts and which are not.20 But there is no trace of this answer in the Notes, where the notion of possibility is not discussed at all. We saw in earlier chapters how convoluted were the explanations Wittgenstein gave in an attempt to explain what different propositions of the same form have in common: repeatedly he resorted to actualworld quantification to obtain the generality he needed. However naturally the conception presented in the Notes of a world made up of facts, positive and negative, invites a conception of possible worlds as varying only according to which of the atomic facts obtain, there is no sign that Wittgenstein came upon such a conception until he was in Norway. If so, perhaps the timing was not coincidental. For both the suspicion of possible worlds and the attempt to use actual-world quantification in their stead, which made Wittgenstein’s account in the Notes so convoluted, are thoroughly Russellian. It was Russell, after all, who claimed that talk of possibility and necessity is simply a confused way of expressing the quantifiers: to say that f is possible, he maintained, is just to say ( x) f x; and to say that f is necessary is just to say ( x) f x. Perhaps, then, it was only the move to Norway that gave Wittgenstein the distance from Russell that he needed in order to see that the possibilities of configuration of the world are encoded in the grammar of a proposition rather than expressed by it—shown by language, not said in it. E

19

2.0141.

20

2.0271.

Chapter 

Tractarian objects As I stressed at the outset, the aim of this book has been to treat the Notes on Logic as a philosophical work in its own right, not merely as a source of some of the leading ideas of the Tractatus. Pursuant to that objective, I have not devoted much space to discussing how the ideas in the Notes developed in the transition to the Tractatus. Nonetheless, there is one change in Wittgenstein’s views on which I think misunderstanding is sufficiently widespread to be worth clearing up here. It concerns the analysis of elementary propositions. Recall first that in the Notes Wittgenstein (or Russell on his behalf) drew a distinction between components and constituents. ‘Components are forms and constituents.’1 ‘Every proposition which says something indefinable about one thing is a subject-predicate proposition.’2 Such a proposition contains, he says, ‘only one name and one form’,3 and therefore has one constituent but two components. In 1913, then, Wittgenstein analysed ‘Socrates is mortal’ into two parts of fundamentally different kinds. The first is the name ‘Socrates’: the second is what he called the form of the proposition. In the same way ‘every proposition which says something indefinable about two things expresses a dual relation between these things’:4 Wittgenstein analysed the proposition ‘aRb’ into three components, two names ‘a’ and ‘b’ and a form. What I want to consider now is what happened later: did Wittgenstein continue to argue for this analysis in the Tractatus? . Relations as objects We have already seen that it is quite hard to think of good reasons for Wittgenstein’s claim that there cannot be different kinds of things. It is in a sense no great surprise, therefore, to find that he eventually gave up that part of the analysis. Although he continued to hold that there are three components in ‘aRb’, his view in the Tractatus was that all three, not just ‘a’ and ‘b’, are names: the third element is a name too (and, correspondingly, what it refers to is an object). What I have just asserted is regarded in some quarters as controversial. Various authors5 have claimed over the years that in aRb there are only two 1

C50. 2 B77. 3 C48. 4 B77. 5 E.g. Copi, ‘Objects, properties, and relations in the Tractatus’, Carruthers, Tractarian Semantics.

Widening the scope  objects, and that what differentiates it from aLb is only the manner in which these two objects are combined. Yet it is rather puzzling that there should be this controversy, since the historical evidence that by the time of the Tractatus he took there to be three objects in aRb, not two, is overwhelming. Briefly, this evidence is as follows. First, in the early 1930s Desmond Lee asked Wittgenstein to explain proposition 2.01 of the Tractatus, ‘An atomic fact is a combination of objects (entities, things).’ Wittgenstein’s response was, ‘ “Objects” also include relations; a proposition is not two things connected by a relation. “Thing” and “relation” are on the same level. The objects hang as it were in a chain.’6 That is of course not in itself absolute proof of what he thought when wrote the Tractatus: after all, it is not from Wittgenstein’s own pen, and it is no doubt prudent to leave room for the possibility that on some aspects of the Tractatus his later memory about his intentions was not wholly accurate. On the other hand, Wittgenstein was certainly very ready in the 1930s to admit to changes of mind about some other things he said in the book—earlier in the same conversation, for instance, he told Lee that 1.12 was based on ‘an erroneous idea’—whereas in this case he seems to have been quite definite about what he had meant. The second piece of evidence provides some reassurance on this point because it dates from significantly earlier. Surviving notes of Ramsey’s lectures in 1925 show him unequivocally expounding, as if it is Wittgenstein’s, the view that the relation is an object. Ramsey had discussed the book in detail with Wittgenstein in September 1923, and it is surely very likely that this would have been one of the issues Ramsey raised then. His exegesis of the text therefore has some claim to authority. The third piece of evidence is earlier still: Russell’s Introduction to the Tractatus confidently asserts that ‘wise’ is a constituent of ‘Socrates is wise’.7 He too had discussed the book with Wittgenstein himself—they spent a week in The Hague in December 1919 going through the whole book ‘point by point’8 —and if Russell was by then in any doubt about the answer to the question, it is hard to believe he would not have asked it. The view that there are two objects in ‘Socrates is wise’, and three in ‘aRb’, is also explicit in Wittgenstein’s own pre-Tractatus writing. In 1915, he wrote in his notebook, ‘Relations and properties, etc. are objects too.’9 . Widening the scope What we can be certain of, therefore, is that something in Wittgenstein’s conception changed after 1913. In the Notes the relation is not a third object in the fact that aRb; from 1915 onwards, it is. In the Notes Socrates is the only object in the fact that Socrates is mortal; by 1915 he is not. Something had 6

Lectures, 1930–1932, 120. June 1915.

7

CP, IX, 104.

8

BR to Constance Malleson, 16 Dec. 1919.

9

NB, 16



Tractarian objects

changed, then, but what? If we read the letter of January 1913 as addressing only the question of whether mortality is an object, it may seem as if the Tractatus marks a return to the view Wittgenstein was rejecting in that letter. But this is wrong. The fundamental point of the 1913 letter was its rejection not of mortality but of the copula, and that rejection continued to hold force in the Tractatus. The doctrine in the Tractatus continued to be, that is to say, that the proposition ‘Socrates is mortal’ contains only two components: one of them is the name ‘Socrates’; the change was that the other was now to be called a name rather than a form. What Wittgenstein had given up, therefore, was the claim that there cannot be different kinds of things (and, correspondingly, that there cannot be different kinds of names). The second element in ‘Socrates is mortal’ continued to be of a different kind from ‘Socrates’, and it therefore continued to be ‘impossible to substitute the wrong way round’, because the two symbols are still ‘of a different kind themselves’. Indeed, there is every reason to think that the rejection of the copula was now even more important to Wittgenstein than it had been in 1913. For he had by now seen clearly the force of his conception of logic as contentless. I suggested in §5.7 that in the Tractatus he used this conception as a way of circumventing the problem which had led Kant to the transcendental deduction of the categories, and hence to transcendental idealism. But the rejection of the copula is simply a particular case of this. If there were such a thing as the copula, a substantial element whose presence in a proposition is essential to its unity, then it—the copula—could fairly be said to constitute a subject matter for logic. This would have two consequences. We would then require a transcendental deduction to explain the application of logic to propositions. And, more importantly, logic would no longer be maximally general. The transcendental deduction would, simultaneous with demonstrating the validity of logic in application to propositions, implicitly restrict logic to apply only to those propositions—only, that is to say, to those assemblies of names whose unity is attributable to the presence in them of the copula. And that would be to undermine fatally logic’s status as being of ‘a totally different kind than any other science’. The change after 1913 is thus in the first instance primarily a terminological one: Wittgenstein’s 1915 statement that ‘relations and properties, etc. are objects too’ was a declaration as to a widened use of the word ‘object’, to encompass properties and relations as well as what he had earlier called objects. On the other hand, calling Wittgenstein’s change terminological runs the converse danger of making it seem more trivial than it was. He did not widen the scope of the term ‘object’ randomly. His purpose was to make his theory more general. In the sort of language countenanced in the Notes, elementary propositions can have only two kinds of components, which he calls

Widening the scope  names and forms. Such a language is not ruled out by the Tractatus: a language of this sort still counts as a Tractarian language in which both names and forms are kinds of Tractarian names. What the Tractatus leaves open, however, is the possibility that there could be some further kinds of Tractarian names, or indeed that the unsaturatedness of the forms might be more widely distributed, so that all the kinds of names in the language bear some measure of unsaturatedness. There are thus several stages in Wittgenstein’s move away from the Fregean categorization he used in the Notes. First he came to think that it is to some extent arbitrary to analyse fa into a saturated name a and an unsaturated form f x, since we could equally analyse it into a name f and a form φ a. But in that case, we could also analyse it in such a way that both elements are unsaturated. And if it is in principle possible for all the components of elementary propositions to be unsaturated to various degrees, this finally removes the reason we offered earlier for supposing that there are only two kinds of component, since the bipartite classification was based on a distinction between the saturated and the unsaturated. Of course, this so far leaves open the possibility that there may be some other reason for a bipartite classification: it might, for instance, be merely a proxy for an independently motivated distinction between particulars and universals. So what motivated Wittgenstein’s decision to give up Frege’s bipartite classification? One sort of motivation is of course the one to be found in ‘Universals’,10 where Ramsey adopted the essentially negative strategy of showing that none of the arguments in the literature in favour of a bipartite classification is persuasive. Ramsey made it quite clear in this paper that he regarded his general approach as Wittgenstein’s; and he was surely right11 to do so. It was indeed, as Ramsey said, Wittgenstein’s position in the Tractatus that ‘about the forms of atomic propositions we can know nothing whatever’.12 On the other hand, some of the details of Ramsey’s argument are perhaps not Wittgensteinian in character. Another sort of reason to abandon the bipartite classification might be provided by Wittgenstein’s picture theory, according to which a proposition is a picture of an especially abstract kind. In pictures of other, less abstract kinds, such as photographs or gramophone records, we feel no temptation to impose a bipartite classification of the components. In order that the picturing analogy should help to explain the representational capacity of the proposition, the form of a proposition had to be the most general pictorial form—a form that all pictures of whatever kind share. Wittgenstein therefore had a reason to remove from his conception of the proposition those features of its structure that it does not share with photographs and gramophone records. 10

FoM, 112–34.

11

Contra Anscombe, IWT, 110.

12

FoM, 134.



Tractarian objects

. Facts in the Tractatus So much for objects in the Tractatus. What about facts? I suggested earlier that in the Notes Wittgenstein conceived of facts as sums of positive and negative facts. It is not as clear as it might be, however, whether this was still what he meant by ‘fact’ in the Tractatus. There is an alternative interpretation according to which he now restricted the word to apply only to sums of positive facts. This alternative interpretation undoubtedly has textual support. For he says in the Tractatus that what is the case, the fact, is the obtaining of Sachverhalte.13 But he also says that the obtaining of a Sachverhalt is a positive fact.14 So if we take this strictly, a fact is a sum of positive facts. Moreover, a letter he wrote to Russell from the prisoner of war camp at the end of the war supports that interpretation. Sachverhalt is, what corresponds to an Elementarsatz if it is true. Tatsache is, what corresponds to the logical product of elementary prop[osition]s when this product is true.15

However, taking Wittgenstein at his word is distinctly problematic. For one thing, it leads to the considerable linguistic awkwardness that negative facts are not now a kind of facts. For another, he says that the world is the total of reality,16 and that reality is the existence and non-existence of Sachverhalte, i.e. positive and negative facts.17 So the world is the sum total of the positive and negative facts. But it is also, famously, the totality of facts.18 This makes sense only if a fact is taken to be any sum of positive or negative facts. If we take this to be what he really meant, it at least has the virtue that it avoids the rebarbative consequence that negative facts are not a kind of facts. Whatever its difficulties, though, we must grant that Wittgenstein did at least flirt with the alternative interpretation according to which only sums of positive facts are facts. What might have led him to do this? It cannot simply have been an antagonism towards negative facts, since he explicitly mentioned them in the book and allowed that they are part of reality. Perhaps it was a desire to capture the idea that when what there is has been given, what there is not is implicitly determined by subtraction. Or perhaps it was an inept attempt to avoid the problem of the structure of negative facts discussed in §15.2. The account of the structure of the negative fact canvassed there, whether or not it was ever Wittgenstein’s, clearly could not survive when he no longer privileged one component of the positive fact as being its form and hence susceptible to pairing with a correspondingly privileged component of the negative fact. By focusing only on what there is, Wittgenstein may now have hoped somehow to dissolve this problem: what there is has a structure; what there is not has no structure, just because it is not. 13

2.

14

2.06.

15

19 Aug. 1919.

16

2.063.

17

2.06.

18

1.1.

Confusion?  Nonetheless, it has to be granted that the mere possibility of confusion over this rather fundamental point of Tractarian interpretation is a little surprising. It is hard to avoid the feeling that Wittgenstein’s lack of clarity on the matter betrays a lack of interest in a certain kind of constitutive metaphysics. If he had thought the distinction mattered very much, he would surely have made sure that the Tractatus was clearer about it.

. Confusion? It is a common view among commentators that Wittgenstein came in the 1930s to think that he had been confused about the distinction between complex and fact at the time when he wrote the Tractatus. This view is based on a reminiscence of Geach’s. Wittgenstein told me once that after the Tractatus was published Frege asked him whether a fact is bigger than each thing it is about: this criticism made no impression at the time, but eventually led him to abandon the notion of facts as complexes.19

On the basis of this reminiscence Geach bluntly claims that in the Tractatus ‘Wittgenstein confused a fact about A and B with a complex containing A and B’.20 This view evidently challenges what I have said here, both historically and philosophically: historically, because if the remark of Frege’s that led Wittgenstein to the realization was made after the Tractatus was published, my account of how Frege might have influenced Wittgenstein’s treatment in the Notes is certainly wrong; and philosophically, because I have placed considerable weight throughout my presentation of Wittgenstein’s ideas on a distinction about which he was, according to Geach, still wholly confused. So it is important for us to examine Geach’s reminiscence in some detail. What is clear straightaway is that Geach’s report cannot be right just as it stands. Frege never met Wittgenstein after the Tractatus was published (or even after it was written). The nearest in Frege’s surviving letters to the remark Geach reports is the following, in a letter written after Frege had read the typescript of the Tractatus (although not after it was published). You write, ‘It is essential to a thing to be able to be a part of a Sachverhalt.’ Now can a thing also be part of a fact? The part of the part is part of the whole. If a thing is part of a fact and every fact is part of the world, then the thing is also part of the world. . . I would like to have an example where Vesuvius is part of a Sachverhalt. Then also, it seems, parts of Vesuvius must be parts of this fact; the fact will also be made up of solidified lava. That just does not seem right to me.21 19

‘Saying and showing in Frege and Wittgenstein’, 67–8.

20

p. 67.

21

28 June 1919.



Tractarian objects

This does not quite amount to the question Geach reports concerning whether a fact is bigger than each thing it is about, but it is at least recognizably in the same territory. What is significant for our purposes, however, is that neither Frege’s letter nor Geach’s recollection amounts to the same point as the one that I quoted in §11.1 from Wittgenstein’s 1931 typescript on ‘Complex and fact’. That remark was about the mistake of regarding a red circle as a complex made up of redness and circularity, and pointed to a confusion about complexes, not about facts. No remark like this is to be found in any of Frege’s surviving letters to Wittgenstein, and it is therefore much more likely that he made it in a conversation held not after the war but before it, during one of Wittgenstein’s visits to him. It is indeed very hard to see how Wittgenstein’s remarks about red circles could amount to a criticism of the Tractatus. As Kenny gently observes, ‘If they had been written by anyone else one would say they betrayed a misunderstanding of that work.’22 Most of the typescript on ‘Complex and fact’ surely consists of exegesis, not criticism, of the Tractatus. Wittgenstein makes the point, for example, that facts are not spatial: but it would be needlessly obtuse to read the Tractatus as suggesting that they are; and it would be curious if Wittgenstein had forgotten so completely his accusation in the Notes that it was Russell who imagined every fact as a spatial complex.23 But if Wittgenstein was not in the Tractatus crudely imagining facts as spatial complexes, it does not follow that he did not somehow allow intuitions drawn from the spatial case to infect what he said about them. And there is one point in the note where he does seem to intend a correction of the Tractatus. A chain, too, is composed of its links, not of these and their spatial relations. The fact that these links are so concatenated isn’t ‘composed’ of anything at all. The root of this muddle is the confusing use of the word ‘object’. The part is smaller than the whole: applied to fact and component part (constituent) that would yield an absurdity.

In the Tractatus (as in the Notes) Wittgenstein does talk of objects as constituents of facts.24 So he does here intend what he says as a corrective to something he said in the Tractatus. And his talk of the part–whole relation is certainly redolent of Frege’s letter: the ‘absurdity’ Wittgenstein is referring to is therefore presumably the one Frege mentions there, namely that a fact about Vesuvius should have pieces of solidified lava as constituents. This makes it plausible that Frege’s letter was indeed what led him to the point he was making here. But what point was that? It cannot have been straightforwardly that he was here rejecting ‘the notion of facts as complexes’, as Geach maintains. The distinction between a complex and the various facts that are exemplified 22

Wittgenstein, 220.

23

C26.

24

2.011.

Confusion?  in it is something Wittgenstein explicitly comments on in the Tractatus. ‘To perceive a complex,’ he says there, ‘means to perceive that its constituents are combined in such and such a way.’ This, he suggests, may explain why we see the Neckar cube in two ways as a cube. ‘For we really see two different facts.’25 Here, quite plainly, is an attempt to make the distinction between the complex and the various facts which we can see as exemplified in it. If there is room for dispute about quite what point Wittgenstein later extracted from Frege’s letter to him, there can be little doubt what Frege himself intended: his target was not the distinction between complex and fact but the conception of simplicity. Indeed he had used an almost identical example in his letter to Jourdain in 1914. That part of the thought which corresponds to the name ‘Etna’ cannot be Mount Etna itself; it cannot be the meaning of this name. For each individual piece of frozen, solidified lava which is part of Mount Etna would then also be part of the thought that Etna is higher than Vesuvius. But it seems to me absurd that pieces of lava, even pieces of which I have no knowledge, should be parts of my thought.26

Frege was here objecting to Russell’s conception of propositions as composed of pieces of the world, ‘even pieces of which I have no knowledge’, rather than of the senses which according to Frege contribute to the structure of thoughts. Frege had made just the same point to Russell in 1904, although then his example was Mont Blanc and its snowfields rather than Vesuvius and its lava flows. (When he reflected on simplicity, it seems, Frege’s thoughts always turned to mountains.) At that time, a year before ‘On denoting’, Russell’s response (quoted in §7.1) was little more than to confirm that this was indeed his view. By the time Frege made the point to Jourdain, however, there was more he could have said in response: once the proposition was correctly analysed by means of the theory of descriptions, the simple entities of which it was composed would indeed turn out to be pieces of the world with which I am directly acquainted. And in 1919 Wittgenstein could have responded to Frege’s letter in an analogous manner: the components of the fact are not Fregean senses but Tractarian objects, whatever these turn out to be. If by 1931 Wittgenstein had come to think Frege’s criticism valid, that could well be because he no longer believed in the conception of simplicity on which this response relies. As he justly observed, ‘The root of this muddle is the confusing use of the word “object”.’ But, if this was Frege’s point, it need not have been Wittgenstein’s (not directly, at any rate). Something else that may by 1931 have begun to trouble him was his extension of the meaning of the word ‘object’ to cover both saturated and unsaturated constituents of an atomic fact—or, more precisely, the blurring of the distinction between saturated and unsaturated which this 25

5.5423.

26

PMC, 79.



Tractarian objects

change of terminology was designed to encourage. For, as we noted in the last section, that use is in danger of making the existence of negative facts seem especially problematic. If we are not to reintroduce the copula, we must not regard the manner of combination as a substantial element in the fact that some objects stand in combination. But what should we then say about the negative case? The more Wittgenstein stressed his rejection of the copula by means of the immediacy of the combination of the objects in an atomic fact, when it obtains,27 the less he could say in the case when it does not obtain. When the objects stand in combination, that they so stand is not a further component over and above them; and when they do not stand in combination, that they do not so stand is equally not a further component. What, then, is the difference between the fact that they do not so stand and the fact that some other objects do not stand in combination? The problem of negative facts rears its head yet again. On this second account, then, what Wittgenstein came to regret was that the analogy with spatial pictures had led him to abandon Frege’s bipartite classification of the parts of a fact: Wittgenstein’s use of the word ‘object’ was confusing because it obscured this distinction and hence rendered puzzling the conception we must have of the structure of a fact if we are to think of it as something which may or may not obtain. Each of the two diagnoses no doubt has something to be said for it. The first makes Wittgenstein’s 1931 objection to his earlier conception rely on a view— the rejection of the Tractatus’s conception of simplicity—which we know he by then held. The second is philosophically deeper and presents him as engaging more fundamentally with the ideas that motivated Frege’s views, but it is perhaps harder to render consistent with the historical evidence. It would take us too far out of our way to arbitrate between them here. Nor do we need to. All that is necessary for our current purpose is to note that the mistake Wittgenstein later believed he had made in the Tractatus does not on either account lie on the surface. We are in no way compelled to represent it as merely a crude confusion over the distinction between complex and fact—a distinction which, I have suggested, he first came to appreciate in 1913. 27

4.221.

Chapter 

Philosophy Outside the guilds of philosophical logicians and metaphysicians, the Tractatus is known almost exclusively for the allusive but elusive remarks that constitute its metaphilosophical frame—the Preface and the concluding couple of pages. The insights to which these remarks try to give expression afford a distinctively problematic status to the book that they surround, and hence render paradoxical the route by which these insights were acquired. There is no trace in the Notes of this attempt to render philosophy nonsensical, or of the relish for deliberate paradox that it betrays. Nonetheless, what is visible already is Wittgenstein’s concern for and distinctive conception of the role of philosophy, its limits, and its relationship to the natural sciences.

. Metaphysics Pinsent once noted in his diary, in what one suspects was actually a report of Wittgenstein’s views, that ‘really logic is all Philosophy. All else that is loosely so termed is either metaphysics—which is hopeless, there being no data—or Natural Science, e.g. Psychology.’1 In the Notes, on the other hand, Wittgenstein was somewhat less pessimistic, observing that ‘philosophy consists of logic and metaphysics: logic is its base’, but not going as far as to dismiss metaphysics as hopeless. It is true that there is rather little in the Notes that can be regarded as straightforwardly metaphysical, but what that indicates is not so much that Wittgenstein thought metaphysics was hopeless as that he had a specific view about how metaphysical conclusions might be reached. What we see in operation in the Notes—perhaps, indeed, because of the lack of relevant data—is a developing method of approaching metaphysics via the symbolism. This is an application of the symbolic turn mentioned in chapter 6. The features of the world are deduced from the features of the symbols used to represent it. The philosopher’s project of analysis, which we sketched in chapter 4, exhibits various complex signs that occur in language as disguised definite descriptions (or other incomplete symbols). The sentences in which these signs occur are to be rewritten so that their occurrences are eliminated. The answer to the 1

25 Aug. 1913.



Philosophy

metaphysician’s question, what there is, is revealed by this process: what there is is whatever corresponds in the world to the simple signs that remain once the complex signs have been eliminated. Something like this conception, according to which metaphysical questions are correctly pursued via, or can be seen as equivalent to, an analysis of language, can perhaps be discerned, at least dimly, in Boltzmann; but it is more likely, I suspect, that Wittgenstein derived it from Frege. The idea certainly plays a role, at least tacitly, in the Grundlagen, for instance. Dummett, in his interpretations of Frege, has gone much further, maintaining that this conception is not just tacitly but fully and centrally present in the Grundlagen. On his account, the role of the context principle in the Grundlagen is to rule out as illegitimate any attempt to ask what numbers are that goes beyond explaining the contribution number-words make to the senses of sentences in which they occur. It is ‘part of Frege’s thesis’, he says, that the nominalist is the victim of a superstition about what has to be done in order to confer a reference on a name. If an expression satisfies the ‘formal’ criteria for being a name, then there is no further condition which needs to be satisfied for the expression to be a name having a reference.2

The effect, on this view, is at a stroke to transform a metaphysical question— what are numbers?—into a linguistic one—how are number words used? Other readers, however, have hesitated to see in the Grundlagen such a clear endorsement of the language-centred view. One might indeed wonder whether Dummett has been influenced by Wittgenstein’s more thoroughgoing application of this view into attributing to Frege a clearer grasp of it than he really had. Even such an enthusiast as Dummett is willing to grant, indeed, that Frege ‘was not fully conscious of the thrust’ in the direction of ‘the investigation of thoughts through the analysis of language’.3 And Dummett acknowledges that Frege’s later writings hesitate over this question: Frege admits, for instance (perhaps under the influence of the set-theoretic paradoxes), that ‘the main task of the logician consists in liberating himself from language’.4 Perhaps it is more accurate to regard this as another case of Wittgenstein’s tendency (already visible, for instance, in his conception of logic as contentless) to see the implications of Frege’s views more clearly than Frege did himself. The insight which Frege drew from the context principle—that if a name has a significant use, there can be no further condition it has to fulfil in order to refer to an object—is one that in the Tractatus Wittgenstein made his own. The symbolic turn, which in the Notes takes the simplicity of a name as a criterion for the simplicity of the object it refers to, is there extended to encompass a rejection of a metalinguistic perspective from which it would even make sense to ask about the latter independently of the former. The picture theory is not 2

FPL, 497.

3

OAP, 6.

4

To Husserl, 30 Oct. to 1 Nov. 1906, my emphasis.

Psychology  presented there as imposing a constraint which the world must satisfy if it is to be representable in language. (For what sense could we make of the notion of an unrepresentable world?) . Psychology I have already remarked on Wittgenstein’s insouciance about how logical analysis is actually to be carried out. He told Russell, for instance, that I don’t know what the constituents of a thought are but I know that it must have such constituents which correspond to the words of language. Again the kind of relation of the constituents of a thought and of the pictured fact is irrelevant. It would be a matter of psychology to find out. . . Does a Gedanke consist of words? No! But of psychical constituents that have the same sort of relation to reality as words. What those constituents are I don’t know.5

On Wittgenstein’s view, therefore, the analysis of language into its simple constituents corresponds to the analysis of thought into its simple constituents. It is, it seems, for psychology to tell us what these constituents are. But what did Wittgenstein understand by psychology? In Cambridge there were two rather different models available to him. Moore, whose lectures on psychology tried with agonizing care to get clear about the correct use of terms such as ‘psychology’, hoped to arrive at the nature of sense-data by enquiry, not by experiment. But Wittgenstein also spent a great deal of time at the newly opened Psychological Laboratory experimenting on the perception of rhythm. I am inclined to think that at this stage he had more sympathy with the second model of psychological enquiry than with the first. If he regarded psychology as part of natural science, as Pinsent’s diary entry suggests, this might explain why he apparently thought philosophers were safe to ignore it. It is less clear, though, whether Wittgenstein really did ignore empirical psychology as completely as he sometimes liked to claim. If it were, as a matter of fact, possible for me to see the Neckar cube in both its configurations simultaneously, would he still have felt able to maintain6 that my world is the totality of facts, not of things? And Wittgenstein’s conception of what it is to see the symbol in a sign surely depends for its plausibility on the phenomenology of reading. What is notably absent in Wittgenstein’s letter to Russell just quoted, of course, is any attempt to make a distinction between psychology and linguistics. The constituents of my mind, which are private to me, ‘have the same sort of relation to reality as words’, and yet words are communally available in a manner in which the constituents of my mind are not. The reason for Wittgenstein’s failure to distinguish between what is private and what is public is 5

19 Aug. 1919.

6

1.1.



Philosophy

one I discussed in chapter 7. One way of putting the point (a way that emphasizes the private aspect) would be to say, as I did there, that Wittgenstein adhered to, or was at least indifferent to the consequences of, a certain kind of solipsism. It would equally be possible, though, to put the point so as to emphasize the public aspect: once a communally available realm of thought is acknowledged, it is dubious what point is served by attempting to contrast it with a private sphere of ideas available only to me. We do not find Wittgenstein himself putting the point in this second way until the 1930s, of course, but the private language argument of the Investigations has its origins in such a shift. The transition between Wittgenstein’s early solipsism and this theme in his later philosophy is most clearly visible in the Blue Book, where he explored at length the idea, perhaps not fully recognizable in the Tractatus, that from this second perspective solipsism is not the outlandish claim that Russell always regarded it as, but simply what common sense demands. Of course the world is my world: who else’s could it possibly be? . Epistemology Even if philosophers can ignore the details of what the simple signs in language actually are, and hence what objects there are in the world, there remains the question of what the relationship is between the former and the latter (what Russell called ‘acquaintance’). Moreover, there is a further question of how we acquire knowledge: Russell called the relationship by which we know that an atomic fact obtains ‘perception’. When Wittgenstein labelled the philosophy of psychology epistemology,7 he presumably had in mind the task of explaining these two relationships, acquaintance of objects and perception of facts. We have seen how little the Notes contribute to either of these tasks. But if this was the role of epistemology at the time of the Notes, it had undergone a significant change by the time of the Tractatus. When the judging subject dropped out of the judgment relation, acquaintance became an unsayable relation between a name and its bearer. In the Tractatus nothing that is not merely programmatic is said about how this relationship is established. For this reason, perhaps, many commentators have offered a reading which renders epistemology significant in the Tractatus only by its absence. According to Dummett, for example, the Tractatus is a pure essay in the theory of meaning, from which every trace of epistemological or psychological consideration has been purged as thoroughly as the house is purged of leaven before the Passover.8

But this is too strong. It may indeed be the case that epistemology in the sense in which Russell used the term, is absent from the Tractatus, but what there 7

B62.

8

FPL, 679.

Value



is instead is a recognition of the actual world as the world we perceive, and hence of perception as what distinguishes the actual world from other possible worlds. Epistemology therefore makes its appearance when I note those features which the actual world has to possess by virtue of being my world. The most striking of these is the one Wittgenstein used as an objection to Russell’s theory of identity. It is possible for two objects to have all the same properties. But a world in which this occurred could not be the actual world: if it were, the two objects could not have been recognized as being distinct, could not have been given different names, hence would not be for me distinct objects. The Tractatus, unlike the Notes, is built on a conception of our capacity to represent the world as based on the recognition of the possibility that it might have been different. But Wittgenstein does not commit the modal fallacy of supposing, once these other ways the world could have been are recognized, that each of them could have been the actual world. I can recognize the possibility of there being brains in a vat without being forced to admit that I might be one myself. Epistemology, in the Tractatus, is what we glean when we take that observation at full value. It is clear, though, that this is very far from what Russell understood by the term. In Wittgenstein’s hands epistemology is a way of privileging the actual world over its non-actual alternatives—a way of explaining, in the absence of anything in it amounting to a judging subject, what makes the world nonetheless mine. What the Tractatus does not do is to contribute to the epistemological project which Russell had been attempting until Wittgenstein persuaded him to abandon it in 1913. According to the picture theory, our task in representing the world through pictures presupposes that we have formed names which can go proxy for the objects in these pictures. As already noted, the Tractatus says very little about what these objects are. But it says even less about how the task of naming is to be effected. If, as has been suggested, Wittgenstein sometimes uses ‘kennen’ to translate Russell’s notion of acquaintance, he tells us nothing about what acquaintance with an object amounts to. For him, the relation of acquaintance is not (most of the time, at least) amenable to the kind of armchair experimental psychology so often favoured by Russell; rather is it simply presupposed by meaningful discourse and mysteriously inaccessible to it.

. Value What is common to Wittgenstein’s comment in the Notes and the entry from Pinsent’s diary entry quoted earlier is that they allow no place for a philosophy of value or of religion. And indeed whenever Russell strayed into this territory (which, encouraged by Ottoline Morrell, he did with some frequency),



Philosophy

Wittgenstein expressed his disapproval forcefully. Now there is of course a limited sense in which a rejection of the philosophy of value remains in place in the Tractatus. Statements of value are according to its doctrines nonsensical, hence not for philosophy to adjudicate upon. But that is equally true of metaphysics (and indeed logic, insofar as it goes beyond the mere enunciation of tautologies). Statements of value, although nonsensical and therefore in a sense insignificant, are nonetheless recognized in the Tractatus as playing an important role in guiding our ethical lives. Moreover, although it is not for philosophy to adjudicate on them one by one, the Tractatus does discuss their nature. It thus implicitly concedes to a kind of meta-ethics a place as part of philosophy—or, at any rate, as part of the prolegomenon to philosophy that the Tractatus represents itself as being. In what he wrote in the Notes or said to Pinsent, Wittgenstein does not seem to have been willing to grant even this much. Something changed, therefore, in Wittgenstein’s attitude to the philosophy of value. And it is clear, too, that Wittgenstein’s own religious views changed somehow in the course of the war: when Russell met Wittgenstein in The Hague in 1919, his observation that Wittgenstein had turned into a mystic was plainly a personal comment, and not merely an objection to his book’s attempt to connect mysticism with logic by means of the doctrine of saying and showing. So it would be easy, no doubt, to succumb to the temptation of trying to correlate this change in his view of the philosophy of value with a change in Wittgenstein’s own religious views. Just what these views were when he was at Cambridge has not been transmitted to us with any clarity, however. Although instructed as a child in the Catholic faith, he quite soon rejected it; but just what residue this rejection left is not clear.9 His horror when he discovered that one of his undergraduate acquaintances in Cambridge was a monk10 suggests a certain hostility to organized religion; and other remarks in Russell’s letters (e.g. that he was ‘terrible with Christians’)11 reinforce this impression. But it is difficult to discern what the rational basis for Wittgenstein’s hostility to clerics really was (if indeed there was one). His charge against them seems, oddly enough, to have been one of dishonesty.12 If this was not simply the childish assumption that anyone who disagreed with him was a hypocrite, perhaps it was that he thought of religious experience as essentially private, so that any attempt to manifest it publicly would inevitably fail. One of his objections to Russell’s 1912 article on the essence of religion was ‘that such things are too intimate for print’.13 (The idea that religion is inexpressible would of course represent a point of continuity with the conception offered in the Tractatus.) On this view, the dishonesty of the cleric consisted in a failure to recognize this distinctively 9

McGuinness, Young Ludwig, 43. 10 BR to OM, 17 Mar. 1912. Pinsent’s diary, 9 Nov. 1912. 13 BR to OM, 11 Oct. 1912.

12

11

To OM, 18 Mar. 1912.

Value



private character of his religion: the attempt to put his religious beliefs into words could not possibly do them justice. Even if he was hostile to clerics, while he was at Cambridge Wittgenstein was evidently far from denying the existence or the importance of a religious aspect to man’s relationship with the world. Otherwise he would hardly have admired, as we know he did, the text, ‘What shall it profit a man if he gain the whole world and lose his own soul?’14 He also recognized already the importance of private religious experience: in about 1910 he had had what he regarded as a sort of religious experience (a strong feeling of complete safety) while watching a play.15 And his diaries during the war are, from early on, suffused with religious language of a sort that suggests a long-standing familiarity with religious concepts as modes for expressing his emotional life. Perhaps his religious understanding was deepened and enriched by prayer during periods of personal danger, such as when he served on the Eastern Front during the Brusilov offensive of 1916; but there is nothing in his diary to suggest that he underwent anything like a religious conversion or transforming experience that summer. By contrast, the philosophical change that occurred in 1916, visible in his notebook entries, is clear and profound. This was the moment when his work ‘broadened out from the foundations of logic to the essence of the world’.16 The insights he came upon that summer led him to make significant changes to the structure of the Tractatus, already largely written. Before that summer it was a work in what we would now call philosophical logic. (Wittgenstein, of course, professed to think that there is no such thing.) It was recognizably a descendant of the Notes, and might still be seen as Wittgenstein’s attempt to answer the question ‘What is logic?’—the very question which Russell had abandoned in October 1912 when he realized that Wittgenstein was better able than he to answer it. Moreover, the book still had as its concluding paragraph a technical claim in logic, a statement of the general form of proposition. Only after the summer of 1916 did Wittgenstein transform the book, by adding the 6s (especially the 6.5s), into a work with much wider ambitions, dealing also with ethics, mysticism, and the soul; now too it concluded, as it had not done before, by throwing away the ladder and admonishing us to silence. But this change in the ambitions of his philosophical work does not seem to be straightforwardly correlated with any change in his own religious beliefs. The fatalistic humanism that Wittgenstein wrote into the Tractatus after 1916 is visible in the works of Tolstoy and Dostoevsky that he had read and been affected by before the war: he may not have come across The gospels in brief, Tolstoy’s curiously fatalistic account of Christian teachings, until the Autumn of 1914, but he read Hadji Murat (which one might view as a sort of 14

BR to OM, 29 May 1912.

15

Malcolm, Memoir, 70.

16

NB, 2 Aug. 1916.



Philosophy

extended hymn to fatalism) in 1912 and thought it wonderful.17 The influence of Tolstoy and Dostoevsky is detectable in his friends as well: almost the only literary reference in Pinsent’s diary entries about Wittgenstein is to Anna Karenina (‘very like Levin’,18 he perceptively observes); and it is probably not a complete coincidence that Moore was reading Dostoevsky during the period when his friendship with Wittgenstein was at its closest. 17

LW to BR, [Summer 1912] (CL, no. 6).

18

2 Sep. 1913.

Chapter 

Themes It is hard to regard the Notes on Logic as a finished work, even in the Costello rearrangement, and one can only guess at the struggles Johnson or Moore might have had if they had been asked to examine a BA dissertation based on them. The claim I made at the outset, however, was that treating the Notes as a work in its own right might pay both historical and philosophical dividends. The philosophical dividends are perhaps not of a kind to be swiftly summarized in a concluding chapter: the philosophical illumination one hopes to cast in a study of this kind is not a single transforming insight but a series of small clarifications of puzzles not hitherto sufficiently understood. The historical insights, on the other hand, are probably rather easier to identify. It is now time to reflect on what we have learned here about Wittgenstein’s way of working and of thinking, about the influences on and of his work, and about how the Notes influenced the Tractatus itself.

. Working methods Wittgenstein’s working methods seem to have remained remarkably constant throughout his life. Central to them was his habit, his discipline, or even perhaps his obsession, of writing up his thoughts (almost always in German rather than English, however long he spent in Britain) in a philosophical journal dayby-day. The two methods by which Wittgenstein liked to mine these journals for material are exemplified by the two parts of the Notes. The first method, by means of which I believe the Birmingham Notes were produced, was to mark in his notebooks which remarks he thought worth preserving and then read these out to a shorthand typist. Later in life he sometimes cut the resulting typescripts up into slips which he could sort and rearrange, and it is surely very likely that some of the runs of consecutive remarks on particular topics in the Prototractatus were first assembled in this way. The second method Wittgenstein had of advancing beyond his journal entries was to dictate in English, not in general simply translations of remarks from the journals but often rephrasings or explanations of these remarks. The



Themes

Cambridge Notes do not quite conform to the method he later found ideal, though. For one thing, he wrote part of them himself, and the remainder he dictated not to a student or friend, as he did later in his life, but to a shorthand secretary. For another, his interlocutor was Russell, who was capable on occasion of arguing back, whereas later his preference was for more quiescent dictation partners—people who (so he presumed) understood what he was saying well enough to look quizzical from time to time, but not the sort by whom he might ever feel philosophically threatened. The fact that it was Russell he was explaining his ideas to rather than someone more willing to listen dutifully (Moore, perhaps) is probably part of the explanation for the rather chaotic character of the Cambridge Notes, especially MS2. It is worth stressing, however, that Wittgenstein’s working method was normally far from chaotic. The Birmingham Notes, in contrast to the Cambridge ones, are a methodical report of the logical results he had obtained so far. Closely keyed as it was to his interest in achieving philosophical understanding without advancing arguments, Wittgenstein’s later working method placed a premium on the ability to rearrange his thoughts so as to create juxtapositions that achieve certain effects. The Birmingham Notes were already in a form that would make it possible for him to draw on them for this purpose (as he eventually did when he came to compile the Prototractatus). It is worth noting, too, that Wittgenstein had literary models for the form of expression that resulted from this method of working. The collection of aphorisms was a familiar literary form in Wittgenstein’s Germanic tradition; perhaps the fact that he conceived a philosophical work with a highly nonnarrative structure is partly a result of his exposure to, and hence his appreciation of the cumulative effect of, collections of aphorisms. Lichtenberg, for instance, was celebrated for such a collection (a copy of which Wittgenstein gave to Russell in the summer of 1913), and he will have been aware of numerous other examples (Schlegel, Nietzsche, Schopenhauer, Kraus).

. Characteristics Wittgenstein had, it must be said, a particular, if not peculiar, sort of mind. Again and again we see in his work ideas that have a forcibly striking combination of depth and simplicity. This combination, indeed, is so characteristic of his ideas, both early and late, that respecting it may be taken to be an interpretative constraint on the commentator. Wittgenstein’s ideas are often not easy to grasp, but once grasped they usually have a sort of directedness— obviousness, almost—that makes them, if not compelling, then at any rate compulsive. Accounts which represent the arguments for his claims as complicated are therefore rarely convincing.

Characteristics  A running theme in Wittgenstein’s work is his refusal to argue for his ideas in anything like the conventional manner of philosophers. This, as we can see from his period in Cambridge, started early. Russell tried to persuade him of the importance of argument to philosophy, but in vain. ‘He said arguments spoil its beauty, and he would feel as if he was dirtying a flower with muddy hands.’1 Nor was Russell the only one who tried: Frege recommended him to rework the Tractatus as a series of articles arguing for specific conclusions.2 In the interpretative literature there has been a tendency to confuse this with a different strand in his later philosophy, the strand which emphasizes that Wittgenstein was not concerned with advancing theses or making claims. Sometimes his aim is described as being that of encouraging us to give up false (or, more usually, nonsensical) claims that we have entertained in the past, without putting other (equally false or nonsensical) claims in their place. On other occasions the purpose is presented more positively as being that of leading us into a certain way of viewing a problem, where what we must understand is that the ‘way of viewing’ in question is by its nature not something that can be put into words. Whichever of these two ways of conceiving his project is stressed, however, the nature of the goal he was pursuing is taken by interpreters to explain why he did not proceed by the philosopher’s normal method of advancing arguments. Wittgenstein’s delphic mode of expression is represented as being his response to the peculiar difficulty faced by anyone who tries to construct an argument directed towards what cannot really be described as a conclusion at all. Just for this reason the later Wittgenstein has occupied a strikingly controversial place in the wider taxonomy of philosophy. If it is taken to be characteristic of analytic philosophy that it is concerned with making claims and assessing arguments, Wittgenstein is presented as a paradigm case of a nonanalytic philosopher. And yet it is equally often held that what is characteristic of analytic philosophy is the pursuit of clarification, and from this perspective the later Wittgenstein may, equally paradigmatically, be presented as belonging firmly in that tradition, even if the methods by which he pursued the goal of clarification were atypical. The tension between these two views of the later Wittgenstein has had its influence on assessments of his early work too. The early Wittgenstein was for a long time treated, of course, as an analytic philosopher––as co-founder along with his two great inspirations, Frege and Russell, of the tradition. In the years immediately after his death the connections between the Tractatus and the posthumously published Philosophical Investigations were not detectable 1

To OM, 27 May 1912.

2

To LW, 30 Sep. 1919.



Themes

by most readers. No great attention was given to his own request that his later work be viewed next to the early, with the result that the ‘two Wittgensteins’ view dominated the interpretative literature for more than thirty years. What made this view increasingly untenable was the gradual emergence into the public domain of Wittgenstein’s middle period works, the Blue and Brown Books, and especially the Philosophical Remarks and Philosophical Grammar, together with notes of his Cambridge lectures during the 1930s. Once the continuities of thought between early and late began to be understood, it became increasingly uncomfortable to treat early Wittgenstein as a founder of the analytic tradition and late Wittgenstein as its first postmodernist subverter. It became popular, therefore, to look for signs in Wittgenstein’s early espousal of a version of Russell’s analytic method of an intention to deconstruct it from within. To the extent that a study of the Notes can be expected to contribute to these larger debates, what it offers is a reminder of much more modest and less philosophically resonant interpretative points. The most obvious of these is that Wittgenstein’s tendency towards the oracular was not a response to the peculiar difficulty involved in arguing for conclusions that cannot, by their nature, be put into words. Philosophically unsatisfying as it may be, the truth is that he did it because he was constitutionally incapable of doing anything else. The inability to argue for their conclusions, the inability to explain what they mean, are not such rare phenomena in budding philosophers. When I come across cases in which this tendency is not corrected early, my inclination is to blame the supervisor. In Russell’s case, though, I suspect that this is unfair. If he had resolutely withheld fatherly approval until Wittgenstein had supplied an argument for every claim he made, the result would no doubt have been only that Wittgenstein left philosophy and went back to engineering, with undoubted loss to the former and (judging by his 1910 patent application) no great gain to the latter. To the wider debate about the role of the unsayable in the Tractatus, on the other hand, a study of the Notes can contribute comparatively little. The Notes do at least show, however, that Wittgenstein did not always believe that the claims made in the text of the Tractatus were nonsense. For many of them are already contained, sometimes in the very same words, in the Notes, and they are not there advanced ironically, ‘transitionally’, or for purely literary effect: in the Notes, if not in the Tractatus, Wittgenstein said what he did because he believed it was true. Pinsent reported in 1912 that Wittgenstein had solved a problem in the most fundamental logic that had puzzled Russell and Frege for years; and solving such problems was just what throughout his time at Cambridge Wittgenstein saw himself as doing.

What if  . What if In chapter 11, I mentioned Adolf Reinach, a student of Husserl and near contemporary of Wittgenstein who fought in the German army and was killed in 1917. Nowadays Reinach merits barely a footnote in the history of early analytic philosophy. It is perhaps one of the idler what-ifs of history to wonder how much attention Wittgenstein would now be accorded if he too had been killed during the war. More relevant to our current purpose, I suppose, is the corresponding question in the event that Wittgenstein had committed suicide in Norway, as Russell evidently feared.3 Whether this was really as serious a risk as Russell supposed is open to doubt. Talk of suicide seems to have been used much more frequently then than now as a sort of rhetorical trope. (Russell, for example, assures us in his autobiography,4 with patent implausibility, that he persuaded his wife not to cite Ottoline in their divorce by the simple device of assuring her he would commit suicide if she did.) On the other hand, three of Wittgenstein’s brothers did actually commit suicide, which would on its own be sufficient to explain the frequency with which he himself considered it. Nonetheless, it needs to be remembered that considering suicide—even first-personally—is not quite the same as being close to doing it. What we can say for certain is that if Wittgenstein had died in Norway that winter, Russell would have done what he could to publish Wittgenstein’s writings and bring them to wider attention. What effect that would have had is moot, however. Even as late as 1916 the Prototractatus was a radically different work from the Tractatus as it was eventually published. The transforming insight that his work had ‘extended from the foundations of logic to the nature of the world’5 had yet to turn the book into anything other than the narrow account of the nature of the proposition, and hence of the nature of logic, for which his work in Cambridge with Russell had been a preparation. But the Prototractatus, even in its 1916 form, at least contained several important theories which have had an important influence on twentieth-century philosophy. It presented, for example, a conception of the role of a proposition as being not merely to say how things stand but to contrast that with other ways they might have stood but do not. The book works out many consequences of this conception, the most prominent of which is that objects are to be thought of as what remain constant as the possibilities propositions allow for vary. But none of this is mentioned in the Notes. For that reason the Notes would surely not have attracted the attention of the Vienna Circle in anything like the way that the Tractatus did, since there is nothing in the Notes even to suggest (let alone confirm) that Wittgenstein might have had any sympathy with their positivist agenda. 3

To OM, 2 Oct. 1913.

4

Autobiography, 213.

5

NB, 2 Aug. 1916.



Themes

On the other hand, the Notes do reach several conclusions which are surely of central importance to that part of philosophical logic which Frege and Russell had been developing. They point out, for example, the incoherence of Russell’s attempt to treat propositions as incomplete symbols when they occur in the context of belief attributions. And with equal finality they expose Frege’s bewildering error of treating sentences as names of truth-values. At other points the Notes are more tentative. Wittgenstein had seen the need for his account of the sense of a proposition to make manifest the manner in which φ (a) says about a just what φ (b) says about b, but he had not really got clear about how to meet that need. And the difficulty there is in stating the theory of types without violating its very principles is so far only gestured at and not thought through. More generally, one has a sense that although Wittgenstein had already come to conclusions—that ‘no name is the name of a form’, for example, or that ‘facts cannot be named’—which point to instances in which language seems inevitably to stumble; and, although he regarded such cases as rather more than awkwardnesses to be treated with a pinch of salt, he did not yet appreciate the full significance of this observation. Moreover, some gaps in the Notes are obvious, and were so to Wittgenstein himself. Having dealt adequately, as he thought, with the process by which molecular propositions are formed from atomic ones, he needed to extend the treatment to include an account of quantified propositions. That account, when he produced it, might be expected quickly to expose the inadequacy of what he had so far said about Russell’s paradox and hence to point up the need for a more nuanced account of what it is for one proposition to depend on, or be contained in, another. Even if quantification could be dealt with, that would still leave identity to be explained (or, as it turned out, explained away). And, having demolished Russell’s theory of judgment, Wittgenstein did not yet have more than a plainly inadequate sketch6 of a theory of his own to put in its place. . Fundamental thoughts What is most striking about the Notes, however, is not the particular claims there made which survive into the Tractatus, but rather the general principles which inform that book and which we find already guiding Wittgenstein’s work in the Notes. The most evident of these principles, perhaps, is what Wittgenstein himself regarded as his Grundgedanke, namely that the logical constants do not represent. Already in the Notes this principle led him to reject Russell’s use of real variables and to conceive of the logical structure of a proposition as implicit in the structure of its components rather than as a further element set against 6

B55.

Influences on Wittgenstein  them. Soon the same principle would lead him to his account of tautologies as essentially contentless and hence trivial, and to a rejection of Frege’s conception of numbers as objects. Equally important, however, was what I have called Wittgenstein’s symbolic turn, his understanding of the criterion for an object to be simple as being no more than that it be the reference of a simple symbol. This understanding gave prominence in his work to the problem of detecting which features of language symbolize—which features, that is to say, have logical as opposed to psychological significance. It was this that gave rise to the central role played in the Tractatus by the distinction between sign and symbol, the transition between which is the contribution made by the self to the process of representation. The search for what is symbolically significant in the proposition—for the symbolizing fact—may be characterized as a search for a logically perfect language. At what stage Wittgenstein began to see this holy grail as unattainable is a central question for Tractarian exegesis. It is plain, though, that it is a quest which he had not yet abandoned when he wrote the Notes. For him this quest depended on a distinctive conception of analysis which he explicitly enunciated in the Notes and which in turn depended on the eliminative technique introduced by Russell in ‘On denoting’. The analysis itself Wittgenstein did not attempt, and this too was a consequence of one of his guiding thoughts. Which words have which meanings is a matter for psychology, and the philosophy of psychology is epistemology, which was Russell’s business. What is a matter for philosophy is therefore form, not content: logic is the study only of the structure of propositions and of how that structure enables them to mean what they do; or, to put it another way, of how, rather than what, they symbolize. Another central thought in the Notes is that what symbolizes in a proposition is not the sentential complex by means of which we express it, however that may be individuated, but a fact which that complex may be seen as exemplifying. It is this conception of the proposition as a fact which enabled Wittgenstein, so he thought, to combine what is attractive in both the function–argument and the part–whole models of propositional structure.

. Influences on Wittgenstein In 1931 Wittgenstein sought in his diary to explore the question—alien, perhaps, to many modern readers—of what his Jewishness amounted to. ‘It is typical for a Jewish mind,’ he suggested, ‘to understand someone else’s work better than he understands it himself.’



Themes

I think there is some truth in my idea that I really only think reproductively. I don’t believe I have ever invented a line of thinking: I have always taken one over from someone else. I have simply straightaway seized on it with enthusiasm for my work of clarification. That is how Boltzmann, Hertz, Schopenhauer, Frege, Russell, Kraus, Loos, Weininger, Spengler, Sraffa influenced me.7

We are not compelled, of course, to take such musings at face value, but in this case there does seem to be some truth behind the less than convincing facade of Wittgenstein’s thinking. None of the fundamental thoughts I have just mentioned is wholly original with Wittgenstein, but each of them is a way of bringing to prominence what is significant in an earlier idea. Overwhelmingly, though, the origin of these ideas lies with either Frege or Russell, and not with the other authors cited. Hertz is often quoted as a source for the picture theory and his claim that ‘there musts be a certain conformity between nature and our thought’8 may well have influenced the view implicit in the Notes that there is an essential harmony between the structure of a fact and the structure of any proposition that represents it. But if there is an influence at just this point, it seems to me to be rather superficial: for Hertz the conformity between nature and thought is something which we learn from experience and which could in principle fail to hold, whereas for Wittgenstein it is a condition on anything that may properly be called thought at all. The deeper influence of Hertz on the Notes surely lies not so much in the picture theory itself as in a methodological principle which one might think of as a sort of correlate of it, namely that philosophical problems are to be solved by finding an analysis in which the number of primitives corresponds precisely to the number of degrees of freedom of the system being described. Boltzmann may have had an influence on the conception the Notes offer of the place of philosophy above or below the natural sciences. Above all, though, what Wittgenstein drew from him (as from Hertz) was, I suspect, not so much any specific doctrine as a method—in Boltzmann’s case, the method by which philosophical problems are not to be solved but rather dissolved. One example is Boltzmann’s demonstration that what is said by a certain kind of solipsist need not differ from the ordinary way of expressing things.9 Another is his attempt to show that various apparent philosophical questions result from not knowing what we were really asking: even the theist and the atheist may, Boltzmann suggests, ‘think the same thought without suspecting it’.10 It has often been suggested that Wittgenstein was careful to list the influences on him in chronological order. If so, then Wittgenstein had not been influenced by Weininger by the time he was at Cambridge, since he did not meet Loos until 1914. Nor, I think, is it necessary to search the Notes for traces 7

CV, 18–19. Ibid. 75.

10

8

Prinzipien der Mechanik, 1.

9

Theoretical Physics and Philosophical Problems, 61–2.

Influences on Wittgenstein  of Kraus: Wittgenstein had already read Die Fackel by then, no doubt, but Kraus’s influence on him surely lies some time later. And although Wittgenstein had read Schopenhauer in his youth (at sixteen, according to Anscombe), and returned to some of his ideas in the summer of 1916, the subject matter of the Notes is rather distant from Schopenhauer’s concerns, and it is therefore no great surprise to detect no Schopenhauerian influence on them. Although the later development of Wittgenstein’s ideas is not my main concern here, I nonetheless want to caution against a faulty contrast between his period in Cambridge with Russell and what followed thereafter. I have warned already against overstating the degree to which the 1916 transformation of Wittgenstein’s work—the extension of its scope beyond the confines of philosophical logic—is narrowly a response to the Brusilov offensive, or correlates crudely with a transformation in his religious views. It seems likely that his move to Norway was a more or less deliberate attempt, unexpectedly lengthened by the war, to move away from Russell’s sphere of influence. The work Wittgenstein did during the war was not a response to a wholly new stream of ideas: the influences it integrates into his work—Dostoevsky, Tolstoy, Schopenhauer—are ones to which he had already been exposed long before. What the war did, however, was to give him a space in which he could integrate these influences into his philosophy away from a climate which he had begun to find restrictive. That both Frege and Russell influenced Wittgenstein’s thinking in the Tractatus is too obvious to need much elaboration. Not only is that what Wittgenstein himself said in the Preface, but it is manifest in the text: both authors are targets for substantial criticism there, but both are also responsible for some of the book’s central ideas. One school of interpretation has been keen to emphasize Frege’s influence on Wittgenstein rather than Russell’s. Dummett, for instance, regards the point as obvious. ‘Everyone knows,’ he says, that Wittgenstein was soaked in Frege’s writings and Frege’s thought. Doubtless many philosophers unnamed by Wittgenstein can be shown to have given him ideas. Others, to whom he does refer, provided him with material that he found interesting to reflect or comment on: but Frege is very nearly the only one whom he quotes with approval.11

It may be significant, at least if Wittgenstein’s own later view of the matter has any relevance, that those who knew him later in his life have generally shared Dummett’s opinion. Geach, for example, said that ‘the influence of Frege on Wittgenstein was pervasive and life-long’.12 (Part of the reason Geach knew this was that Wittgenstein had been instrumental in persuading him to translate Frege into English.) And only a few days before his death Wittgenstein wrote, ‘Frege’s style of writing is sometimes great.’13 11

FOP, 237.

12

‘Saying and showing in Frege and Wittgenstein’, 55.

13

6 Apr. 1951 (CV, 99).

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Themes

Another school, aiming perhaps to rehabilitate Russell’s place as the true father of analytic philosophy, has treated pronouncements such as Dummett’s as a sort of challenge, and endeavoured instead to detect Russell’s influence in Wittgenstein’s works at every possible turn. Recognizing, perhaps, that it would be wildly implausible to deny Frege’s influence completely, this school of interpretations has instead tried to place that influence as late as possible, suggesting, for instance, that Wittgenstein did not seriously study Frege’s works until he arrived at Olmütz in Autumn 1916. This school has also, by pointing out places where Wittgenstein allegedly misunderstood Frege’s ideas,14 encouraged the idea that whatever influence Frege had on him was not merely late but superficial. Perhaps the most dubious technique that has been used to support this view, however, is to find in Russell’s writings from 1914–19 prefigurations of the Tractatus, ignoring the fact that in the writings in question Russell often explicitly acknowledges Wittgenstein’s influence.15 Russell’s influence on the Tractatus was undoubtedly significant, but it is implausible that it could be established by citing chapter 2 of Our Knowledge of the External World, the lectures on logical atomism, or section 1 of ‘On propositions’. In all of these places the influence was plainly in the opposite direction: Russell was here very largely reporting the ‘vitally important discoveries’ of his erstwhile student, not his own ideas. Debates about whether one influence was greater than another are often sterile, especially when, as here, it is so evident that both were formative and important. ‘It is obvious,’ said Pinsent, ‘that Wittgenstein is one of Russell’s disciples and owes enormously to him.’16 Nonetheless, I hope that one thing this book has done is to make clear how misbegotten is the attempt to minimize the effect of Frege on Wittgenstein’s thought, not just in the Tractatus but from the summer of 1912 onwards. Indeed, Frege’s influence on the Notes is so pervasive and so manifest that it is almost superfluous to supply an argument for it. The main reason that I have nonetheless thought it worth providing such an argument here is because I wanted to bring to light one suggestive feature that this argument possesses. When we look for passages in Frege’s pre-1913 writings that illustrate the influence he had on Wittgenstein’s thinking as it is expressed in the Notes, it is notable how often it is the Posthumous Writings that we turn to rather than his published works. What this does is to hint at the enormous effect that his few visits to Frege must have had on Wittgenstein. He seems to have absorbed Frege’s conception of philosophical method at least as much through their conversations during these visits as through a careful study of his ‘great works’. 14 Goldfarb, ‘Wittgenstein’s understanding of Frege: The pre-Tractarian evidence’; Kenny, ‘The Ghost of the Tractatus’ (although Kenny’s aim is not to promote Russell’s influence over Frege’s). 15 E.g. Grayling, Russell, 96–102. 16 25 Aug. 1913.

Influence on Russell  The allegation that Wittgenstein mistook Frege’s ideas is in any case a dangerous one to wield as a way of showing that his engagement with those ideas was superficial. Wittgenstein, like many of the best philosophers, was not a careful reader of texts. By that I mean that his technique in reading a text was to think through its ideas for himself, as if from the inside. The practice of this technique may occasionally lead to exegetical errors, no doubt; but it is the opposite of superficial.

. Influence on Russell For a time, Wittgenstein’s influence on Russell was substantial. ‘Wittgenstein’s doctrines influenced me profoundly,’ he later recalled. ‘I have come to think that on many points I went too far in agreeing with him.’17 The war split this influence into ‘two waves’.18 By the time he wrote Analysis of Mind and the Introduction to the 2nd edition of Principia, Russell had read the Tractatus and discussed it in detail with its author. Yet the influence of the Tractatus itself on these works is curiously muted. What we can certainly detect is the influence of ideas Wittgenstein had in Norway in 1913–14: this influence is visible, for instance, in Russell’s discussion of propositional attitude statements, in his attempt to do without the axiom of reducibility, and in the disappearance of the self from his account of mind, which in turn was a prerequisite for his retreat into neutral monism. And the account in Analysis of Mind of propositions ‘pointing towards’ or ‘pointing away from’ facts is based19 on things in the Notes which Wittgenstein had all but abandoned by the time he finished the Tractatus. Much of what Wittgenstein added to the book during the war, on the other hand, always struck Russell as radically mistaken. He was never tempted by Wittgenstein’s doctrine of unsayability, for instance, preferring instead to avoid the paradoxes by the device, nowadays usually associated with Tarski rather than Russell, of an ever-expanding hierarchy of metalanguages;20 and the book’s mysticism always struck him as wrong-headed. If the specifically Tractarian influence on Russell is slight, this need not mean, however, that Russell did not study the Tractatus carefully, or that he did not understand it at all. It is at least as likely, it seems to me, that this is to be interpreted in the opposite sense. There is room, therefore, to question how profoundly the Tractatus itself influenced Russell. The effect on him of the prewar Wittgenstein, although certainly profound, has perhaps been misrepresented. While Wittgenstein was in Cambridge, Russell published (by his hectic standard) rather little in mainstream analytic philosophy: his two main projects, on matter and epistemology, were both stymied by his pupil’s opposition, and he in any case diverted 17

MPD, 112.

18

Ibid.

19

The Analysis of Mind, 272 n.

20

CP, IX, 111.

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Themes

much of his energy into other projects which he hoped would please his lover, Ottoline. Once Wittgenstein was in Norway, Russell was able to complete his project on matter free from that stultifying presence, but in the resulting paper ‘On the relation of sense-data to physics’ Wittgenstein would surely only have agreed with the rejection of the inference from sense-data to matter, not with the attempt to replace it with a logical construction. Where Russell’s writing does show his debt to Wittgenstein is in the three texts cited earlier: chapter 2 of Our Knowledge of the External World, the 1918 logical atomism lectures, and the first section of ‘On propositions’. But here what Russell says contains so much that is straightforwardly a report of Wittgenstein’s views that it does not seem quite right to describe it without qualification as ‘influence’. In the 1918 lectures, especially, Russell was reporting Wittgenstein’s logical results in order to bring them to public attention while Wittgenstein was unable to do it himself. We should represent these lectures as a phase in Russell’s own intellectual history, therefore, only in a heavily qualified sense. The influence of the prewar Wittgenstein on Russell was in fact largely negative. When Russell described Wittgenstein’s criticisms as ‘an event of firstrate importance in his life’, he was right, although (as I argued in chapter 13) what was important was not the one paralysing criticism which led him to abandon the book he was writing, but the cumulative effect of their collaboration. Russell’s move into the psychology of language use in 1918 is explicable partly as an attempt to find a research area to which he could still hope, after Wittgenstein’s destructive criticisms, to make a distinctive contribution.

. Influence on Frege The other person on whom we might hope to detect a direct Wittgensteinian influence is of course Frege, but in truth it is hard to find. Frege’s 1919 article ‘Der Gedanke’, for instance, has surely struck many readers as containing some of the most Wittgensteinian passages in his writing, and it would therefore be forgivable to speculate that it is influenced by the Tractatus. But it is not: Frege submitted it for publication shortly before he saw the Tractatus for the first time. Wittgenstein’s wartime letters to Frege have survived only in tantalizingly brief summaries, but they can hardly have been enough to explain the material in question. And even the prewar Wittgenstein cannot have influenced the article very much, since all its main contentions are already present in an earlier draft that was almost certainly written in 1897. So if parts of ‘Der Gedanke’ (roughly, those not devoted to the rather inept attack on a naive version of idealism) strike the reader as Wittgensteinian, that can only be further evidence of Frege’s influence on Wittgenstein rather than the converse.

Conclusion  Much the same is true of the other parts of the Logische Untersuchungen: they very largely expound views that Frege held long before he met Wittgenstein; the only significant stretch of new material in them, a lengthy and frankly rather tedious discussion of negation, is a response not to Wittgenstein but to Bruno Bauch (a neo-Kantian colleague of Frege at Jena and founder of the journal in which the Untersuchungen were published).21 Of course, that is not to say that Wittgenstein did not influence Frege at all. It is notable, for instance, that in his late writings Frege hesitated over the existence of logical objects. And he no longer laid any stress on the idea that the reference of a sentence is a truth-value. These could conceivably be signs that their prewar conversations had had some effect on him. But when Frege read the Tractatus, most of his criticisms of it were ploddingly pedantic, depressingly reminiscent of the catalogue of confusions that, he told Jourdain, prevented him from progressing further than the first few pages of Principia. What is clear at any rate is that if Wittgenstein did influence Frege, then Frege, unlike Russell, did not have the grace to admit it. The reference he was prepared to write to Bauch on Wittgenstein’s behalf was by any standards crushingly lukewarm. I could tell him that I have got to know you as a thoroughly serious thinker. On the book itself I cannot express a judgment, not because I don’t agree with its content, but because it is too unclear to me.22

Frege plainly felt affection for Wittgenstein and respected his abilities enough to want to discuss philosophy with him, but there is no sign in the surviving correspondence that he had as much regard for him philosophically as Russell did (or, indeed, that he even realized in how much esteem Wittgenstein held him). . Conclusion Wittgenstein did not commit suicide in Norway nor die on the Eastern Front, and Russell was not required to publish the Notes posthumously as Wittgenstein’s only philosophical legacy. They have therefore been viewed, ever since they were eventually published (at first only in Costello’s rearranged version) in 1957, through the lens of the book for which they were preliminary sketches. More recently, and less plausibly, that book in turn has increasingly been viewed as a sort of prolegomenon to his later writings rather than as a philosophical terminus of its own. My aim here, by contrast, has been to explain the Notes and assess them on their own merits—aware, of course, of what they later became, but not guided solely by that awareness. 21

See Schlotter, ‘Frege’s anonymous opponent in Die Verneinung’. 1919.

22

Frege to LW, 30 Sep.



Themes

Russell thought Wittgenstein’s ideas in the Notes were ‘as good as anything that has ever been done in logic’.23 He devoted considerable time and effort to coaxing those ideas onto paper in the form of the Cambridge Notes, to translating the Birmingham Notes into English, and then to bringing the views both sets of notes contain to wider attention at Harvard and in London. His attitude to the Notes was no doubt complicated by the deep personal (and fatherly) affection he felt for their author. But he was not a sentimental old fool. What he had recognized, and had the grace to admit, was that Wittgenstein was already closer to a correct account of the nature of propositions and their relationship to the world than he would ever be. And yet, however persuaded Russell was by the details of Wittgenstein’s conception, most of the underlying principles which guided it were so far from Russell’s way of thinking that he never quite understood what they were. The nearest he came, perhaps, was to be troubled by an ominous sense that Wittgenstein had seen further and more clearly than he had. These guiding principles Wittgenstein owed to Frege, not to Russell. Some we see presented in the Begriffsschrift and in the account of definitions in the Grundgesetze. Others emerge clearly only in the late flowering of Frege’s work in ‘Der Gedanke’. Others still lie just below the surface of his published work and are explicit only in his unpublished writings. What is notable about these principles, though, is not merely that they continued to guide Wittgenstein in the Tractatus as much as in the Notes, but that they constitute so much of what it seems right to regard as that book’s philosophical legacy. The Tractatus, it is worth recalling, is riddled with implausibilities: Wittgenstein later rejected several of the book’s central contentions, most notably its atomism, and hence its conception of analysis; the picture theory, its most renowned logical proposal, is highly problematic, the more so if the analogy with pictures is made to do any argumentative work; the book’s atomism is in danger of resurrecting Zeno’s paradox (which Russell hoped to have buried) and thereby rendering motion impossible; the account of the will leaves the connection between its transcendental and empirical manifestations certainly unexplained and probably inexplicable; the book’s insistence on the triviality of logic is in blatant tension with its manifest complexity; and the notion that the only part of mathematics whose application in real-world reasoning requires philosophical explanation is elementary arithmetic is patently absurd. What is of lasting worth in the Tractatus is therefore not these theories, nor even, perhaps, the book’s eventual (somewhat contrived) self-immolation, but rather the insights into the central themes of philosophical logic which it offers, insights which it derives preponderantly from Frege and which, I hope to have shown, Wittgenstein had already in very large part grasped by the time he left for Norway in October 1913. 23

To OM, 3 Oct. 1913.

Appendix A

History of the text Much of this book has been concerned with explaining the content of the Notes on Logic. In this appendix, on the other hand, my aim is to reconstruct the circumstances of composition of Wittgenstein’s first surviving philosophical work. Because none of the versions of the Notes that have survived is in Wittgenstein’s hand (barring corrections and emendations to one typescript), we shall have to engage in a considerable amount of detailed detective work. The purpose will be to settle which of these versions best represents his intentions, and indeed what those intentions were. What will emerge as a consequence, I hope, is that understanding how the Notes on Logic were composed is a useful aid to understanding their content.

A.1 Narrative In September 1913 Wittgenstein took a three-week holiday in Norway with his university friend, David Pinsent. During the holiday he made plans to return to Norway almost immediately and spend the following year there. He wrote to Russell without yet telling him about these plans but asking to see him as soon as possible. ‘Give me time enough to give you a survey of the whole field of what I have done up to now and if possible let me make notes for you in your presence.’1 Russell suggested a time (4 October at 1pm), but in the event Wittgenstein evidently could not wait that long and went to see him as soon as he got back to Cambridge on Thursday 2 October. Part of this meeting they spent reading together the manuscript of Whitehead’s latest work on space; and part on discussing Wittgenstein’s plan to spend the following year in Norway.2 Then, Russell said, Wittgenstein ‘stayed late . . . and read me bits of the work he has done. I think it is as good as anything that has ever been done in logic.’3 He was ‘explaining a number of very difficult logical ideas which I could only just understand by stretching my mind to the utmost’.4 It is worth noting, though, that, despite Russell’s pleadings for Wittgenstein to write some of these ideas down, there is no mention yet that he did: perhaps 1

20 Sep. 1913.

2

BR to OM, 2 Oct. 1913.

3

To OM, 3 Oct. 1913.

4

To OM, 6 Oct. 1913.



History of the text

this was the occasion when ‘his artistic conscience got in the way, and because he couldn’t do it perfectly he couldn’t do it at all’.5 Over the next four days Wittgenstein met Moore on three occasions (Thursday, Friday, and Sunday).6 Russell, on the other hand, seems to have been away in London from Friday afternoon until Sunday, and we have no evidence that Wittgenstein saw him again before he himself went off to Birmingham on Monday 6 October to stay with Pinsent’s family for a couple of nights. By then he had promised Russell that he would ‘leave a written statement of what he [had] already done before he start[ed] for Norway’.7 The following day (Tuesday 7th) he went to the Berlitz school of languages next to New Street railway station in the centre of Birmingham, where he succeeded in finding a German shorthand writer to whom he could dictate ‘extracts from his note-book’. Pinsent’s diary records that he left the house at 5.30pm and returned around 8.30, so we may guess that the dictation session itself lasted from 6pm until about 8. On Wednesday morning Wittgenstein returned to Cambridge: he will have arrived shortly before lunch. Russell had arranged to walk Wittgenstein out to Jourdain’s for tea, but apart from that, and a brief period when he had to deal with another visitor, they were together (presumably from lunchtime) until ‘near midnight’, he told Ottoline. (Moore’s diary records that he saw Wittgenstein again from 9.30 to 10.30 that evening: perhaps that is when Russell saw his other visitor.) There were, Russell said, newer things [than what Wittgenstein had dictated in Birmingham], and things not sufficiently explained. He said he would make a statement of them, and sat down to do it. After much groaning he said he couldn’t. I abused him roundly and we had a fine row. Then he said he would talk, and write down any of his remarks that I thought worth it, so we did that, and it answered fairly well. But we both got utterly exhausted, and it was slow.8

Before this marathon session Russell had thought it would be ‘a last dose’ of Wittgenstein, but as things turned out they arranged to meet one more time the following day. In the interim Russell managed to obtain the services of an (English) secretary who could take dictation in an attempt to speed up the process of getting Wittgenstein’s explanations written down. Jourdain’s secretary (the one who is prettier than Waterlow’s bride) is coming to take down our conversation in short-hand. Mercifully Jourdain sent her this morning to borrow a book of mine so I grabbed her. It is early closing day so no one can be got except as a favour. Tomorrow Wittgenstein goes to London, and Saturday to Norway. Today in the middle I have to have Lucy Donnelly’s young lady to tea—she will give a breathing-space.9

The secretary (the Miss Harwood Wittgenstein refers to in a later letter)10 5 9

BR to OM, 9 Oct. 1913. 6 Moore’s diary. 7 BR to OM, 6 Oct. 1913. To OM, 9 Oct. 1913. 10 To BR, [Nov. 1913] (CL, no. 29).

8

To OM, 9 Oct. 1913.

The manuscripts  will no doubt have delivered the result of this last dictation session to Russell a few days later. He also soon received the typescript dictated in Birmingham. Perhaps Wittgenstein had asked the German stenographer to send it direct to Russell, but this would mean that he trusted the stenographer to have copied the logical notation correctly. It is more probable, I suspect, that the typescript was posted to Wittgenstein, who quickly copied in the logical formulae and posted it on to Russell just before leaving for Norway. Wittgenstein wrote to Russell from the boat, ‘Hope you have got typewritten business all right.’11 However, he forgot to post the letter—someone else did so on his behalf— and he therefore wrote again a few days later. ‘Did you get the copy of my manuscript?’ he asked.12 The first of these enquiries might, I suppose, be about either of the two typescripts, but the second can surely only refer to the typescript dictated in Birmingham. Shortly thereafter, then, Russell must have had in his possession two typescripts, one produced in Birmingham by the German stenographer and the other in Cambridge by Jourdain’s secretary, together with whatever manuscript Wittgenstein had managed to produce by the end of Wednesday’s marathon session with Russell. On 25 October Russell sent the Cambridge typescript to Wittgenstein, who had by then arrived in Norway, for him to correct, together with a list of questions that had occurred to him since their meetings; we also know that he had the Birmingham typescript by then, since one of his questions quotes from it.

A.2 The manuscripts The typescript dictated in Cambridge on the Thursday has survived: it has been labelled by Russell as a ‘Summary’, and contains corrections in his hand as well as in Wittgenstein’s own. But the Birmingham typescript and Wittgenstein’s manuscript from the Wednesday are now lost. What we have instead is a manuscript in Russell’s hand, written in English, which evidently dates from February 1914. This was when he was, as he told Ottoline, ‘translating and copying and classifying the notes of Wittgenstein’s work, as I shall want them for lecturing on logic at Harvard—that takes a lot of time, but is now finished’.13 Russell’s manuscript is in four parts, which he has labelled ‘MS1’, ‘MS2’, ‘MS3’, and ‘MS4’. One question that we need to address, then, is which parts of the manuscript correspond to which of the sessions described earlier. The view I shall argue for is that MSS 1,3, and 4 are Russell’s translation into English of the typescript dictated in Birmingham, while MS2 is Russell’s 11

17 Oct. 1913.

12

29 Oct. 1913.

13

28 Feb. 1914.



History of the text

transcription of what Wittgenstein had written out (mostly, at least, in English) in his presence on Wednesday 8 October. One reason for believing this is that there are some notable differences between MS2 and the other manuscripts. Certainly the versions of MSS 1, 3, and 4 in Russell’s hand all contain clear signs that they are translations: words are crossed out and amended in ways that would be odd if all he was doing was copying out a text in English. No such emendations are present in MS2. Yet Russell’s own description of his activities in February 1914 refers to both translating and copying: the summary did not need to be copied as it was already typed, so MS2 is left as the only piece we have that Russell might have copied. Stylistically, too, MS2 is closer to the Summary than it is to the rest of the Notes: much of it consists of reformulations of things in MSS 1, 3, and 4 (the things, presumably, that Wittgenstein thought were ‘not sufficiently explained’); and it reads as halting and repetitive in comparison to them. A further feature which distinguishes MS2 and the Summary, on the one hand, from MSS 1, 3, and 4, on the other concerns how Wittgenstein later used the material they contain. There is a section of the Prototractatus14 which draws very closely, with only minor changes of wording, on remarks from MSS 1, 3, and 4 (broadly in that order). But the Prototractatus never draws on MS2 or the Summary word-for-word in this manner: there are occasionally sentences expressing the same ideas, but the formulations are always a little different. This fits well with the notion that MS2 and the Summary represent Wittgenstein’s attempts to explain his ideas to Russell in English, to be written down in one case by Wittgenstein himself and in the other by Miss Harwood, whereas MSS 1, 3, and 4 are Russell’s translations of remarks taken from Wittgenstein’s notebooks. Whether or not Wittgenstein was in possession of a copy of MS2 and the Summary when he compiled the Prototractatus, he did not make direct use of them in it because they had a wholly different origin. Did Wittgenstein and Russell usually speak to each other in German? Russell’s recollection forty years later (commenting on a German radio broadcast about their relationship) certainly suggests that they did. At a certain p[oin]t in our acquaintance we began to say ‘du’ to each other and in our correspondence we always did so. In the broadcast it is right to keep ‘sie’ down to and including the point at wh[ich] I told him not to become an aviator, but after that it w[oul]d be more true to history to say ‘du’ instead of ‘sie’.15

But the suggestion, implicit in Russell’s recollection, that he and Wittgenstein habitually spoke to each other in German is quite hard to accept. When Wittgenstein first arrived in Cambridge, we know that they conversed in English at Wittgenstein’s insistence, even though, according to Russell, Wittgenstein spoke ‘very little English’.16 (Since Wittgenstein had by then been living in 14

pp. 28(3)–34(3).

15

To O’Rourke, 22 Sep. 1955.

16

To OM, 18 Oct. 1911.

The manuscripts  England for three years, this was presumably just a sign of his nervousness on meeting Russell for the first time; at any rate we have no evidence from any other source that Wittgenstein’s spoken English was weak enough to be a problem.) It seems very likely that they continued to speak to each other in English for the remainder of Wittgenstein’s time in Cambridge. For his letters to Russell continued to be written in English until some time after he had arrived in Norway; and when he finally wanted to switch to German,17 he realized that he did not know whether to address Russell familiarly or not. This would be strange if they were by then used to calling each other ‘Du’ in speech. The presumption that they always spoke to each other in English is in any case independently likely: Russell had spent a year in Germany as a young man, but that was now almost twenty years in the past, so it is hard to imagine that his spoken German was by 1913 nearly as fluent as Wittgenstein’s English. (Of course, this does not exclude the possibility that Wittgenstein sometimes read out to Russell sentences in German from his notebooks.) Nonetheless, it is a little odd that on the account I am offering Russell chose to copy out MS2 by hand before sending it to be typed. Why did he do this? One possibility—conjectured, in effect, by McGuinness18 —is that he did not: that the document we possess was written by Russell in October 1913 to Wittgenstein’s dictation. But at which meeting? Not, presumably, on the Wednesday, when Russell was explicit that it was Wittgenstein who did the writing, nor on Thursday, when Miss Harwood was there to take dictation. And which, on McGuinness’s account, was the document Russell ‘copied’ the following February? If we therefore reject McGuinness’s conjecture and agree that the document we possess is Russell’s transcription of an original, now lost, written by Wittgenstein himself at the Wednesday meeting, we do admittedly have to explain why Russell bothered to copy it out. Certainly Wittgenstein’s handwriting was in normal circumstances quite legible. One possibility is that the manuscript he produced on this occasion, even if it was legible, was so disorganized that Russell did not feel Miss Harwood would be able to make enough of it to be able to type it out. The manuscript as we have it is, as noted earlier, halting and repetitive; but the account I am now offering suggests, even so, that it may already have undergone one stage of editing by Russell. Another possibility is that even if, for the reasons just explained, Wittgenstein’s manuscript was very largely in English, it may in a few places have lapsed into German (presumably where Wittgenstein was simply quoting from his notebooks) and hence required partial translation. If this deals adequately with MS2, let us turn now to the other three manuscripts. We can be confident at least that MS3 was dictated in Birmingham, 17

[Nov. 1913] (CL, no. 30).

18

Young Ludwig, 186.



History of the text

for we have the anecdote, recollected by Russell much later,19 that when Wittgenstein came to the sentence ‘ “A” is the same letter as “A” ’ (which occurs only in MS3), the stenographer remarked, ‘Well, that is true anyway.’ We know also that Russell had received the typescript of MS3 by the time he sent Wittgenstein his list of questions, since (as noted earlier) one of these questions quotes a sentence from the German of MS3. And although we cannot be equally certain that the typescript Russell received from Birmingham included MS1 and MS4 as well, my main reason for thinking that it did is that it is hard to think of any other hypothesis which explains the available information. At any rate the total length of MSS 1, 3, and 4 (about 4,000 words) could comfortably be dictated to a competent stenographer in two hours, even allowing for the occasional passages of logical notation which the stenographer would have had to copy out longhand. If this is right, the productivity of Wittgenstein’s session in Birmingham was in marked contrast to the following day, when he managed to write down only 600 words in something over six hours. Wittgenstein’s ‘artistic conscience’, which ‘prevents him from writing anything until he has got it perfect’,20 was evidently something that struck only selectively, and was especially debilitating in Russell’s presence. It is worth noting, though, that if my hypotheses are correct, the circumstances were very different on the two occasions: in Birmingham Wittgenstein was simply reading out to a stenographer marked passages from his notebooks; in Cambridge he was explaining his ideas laboriously to Russell, arguing about many of them, and writing down, often in ex tempore formulations, those Russell thought worth preserving.

A.3 Russell’s labelling Why, then, did Russell label the sections of the Notes as he did? The labelling of the Summary suggests that Wittgenstein had brought with him to that dictation session several of his notebooks. And a comparison with the other parts of the Notes does indeed provide us with passages in the Summary which are close enough in wording to show that he was translating (if perhaps freely) from the notebooks from which MSS 3 and 4 are drawn. Turning now to MSS1–4, we should note that Russell probably saw the text we are currently discussing only as an intermediate stage in an editing task that would eventually turn Wittgenstein’s apparently random arrangement of material into what has become known as the Costello version of the Notes (of which more shortly). With this in mind, the labelling of MSS 1–4 may indeed have been no more than that—a labelling, whose only purpose was to correlate parts of Russell’s version with the corresponding parts of Wittgenstein’s 19

McGuinness, Approaches, 257.

20

BR to Lucy Donnelly, 19 Oct. 1913.

Russell’s labelling  text; it may therefore simply have reflected the order in which Wittgenstein’s texts lay in the file when he drew it out in February 1914. On the other hand, it is worth noting that the order in which MSS 1, 3, and 4 are numbered corresponds to the order in which Wittgenstein drew on the propositions contained in them when he composed the Prototractatus. So Russell does seem at least to have kept the three parts of the Birmingham typescript in their original order when he labelled his translations of them. The labelling of MS2 suggests that perhaps on the Wednesday, unlike the following day, Wittgenstein only had one of his notebooks with him. Indeed, although MS2 contains reformulations of various remarks in MSS1, 3, and 4, none is close enough in wording to show that Wittgenstein must have had the relevant notebook with him. But that does not explain why Russell numbered it as he did, interposing MS2 into the sequence of the Birmingham typescripts. In this connection it is perhaps worth noting that MS1 is only about 600 words long and need not have taken a competent typist more than quarter of an hour to type. One possible explanation, therefore, is that the Birmingham stenographer typed MS1 straightaway on the Tuesday evening and gave it to Wittgenstein then and there (or, I suppose, did it after Wittgenstein had gone and left it somewhere for him to pick up on his way to New Street station the next morning). Wittgenstein would then have given this typescript to Russell at the beginning of their meeting the following afternoon. On this hypothesis the Birmingham stenographer finished the rest of the typing and posted it to Russell later, so that the ordering of MSS 1–4 simply reflects the order in which Russell received them. It is difficult, however, to supply a similar hypothesis to explain the division of MS3 and MS4 into two parts. At a pinch, I suppose, it might simply be that the Birmingham stenographer typed up his shorthand notes in two further batches and forwarded them separately to Russell. A more likely explanation, I think, is that Wittgenstein had instructed the typist to keep them distinct because they were drawn from different notebooks. We know that Wittgenstein had taken with him on his holiday in Norway a ‘portmanteau with all his manuscripts inside’.21 We know, too, that Wittgenstein often referred to his notebooks as ‘manuscripts’. We may conclude, therefore, that there were at this point several Cambridge notebooks that Wittgenstein still regarded as worth consulting (and indeed whose temporary loss threw him into a panic, as Pinsent relates in his diary). It would surely have been natural for Wittgenstein to extract remarks from more than one of these, and to instruct the stenographer to type them separately. However, there is a difficulty with this hypothesis. Wittgenstein certainly had the text of the Birmingham typescript available to him when he was com21

Pinsent’s diary, 30 Aug. 1913.



History of the text

piling the Prototractatus: as noted earlier, there is a section of that work in which almost every remark is drawn from it. And yet we know, for reasons I need not go into here,22 that the only prewar notebook Wittgenstein had in Austria during the war was the one he kept during his year in Norway. For this reason McGuinness has suggested that the Notes must be derived directly from this notebook. I am inclined to reject this suggestion, for a number of reasons. As just mentioned, if all of MSS 1, 3, and 4 were drawn from the same notebook, it is puzzling why Russell labelled them as distinct texts. Moreover, it is quite implausible that all these notes are the product of a short creative burst during Wittgenstein’s three-week holiday in Norway with Pinsent. They read much more like extracts from notebooks written over a considerable period— not, certainly, the whole of Wittgenstein’s period in Cambridge, but the content suggests that they extend back perhaps as far as the previous February. If so, McGuinness’s hypothesis requires us to suppose that one notebook lasted Wittgenstein from early in 1913 until the summer of 1914. This is not impossible—we know, after all, that it was a ‘large notebook’23 —but it does seem a little unlikely. Something else worth noting is the pattern detectable during the war according to which Wittgenstein started a new notebook when his life entered a new phase. Of course, this pattern is scarcely decisive on its own—one reason he had to start new notebooks during the war was so that the old ones could be deposited in Vienna, from where they were to be dispatched to Russell in the event of his death in action—but it is at least suggestive. Wittgenstein’s behaviour during his short period in England between the Norwegian holiday and the beginning of his longer Norwegian exile speaks very much of someone tidying up the loose ends of one stage of his life in order to start a new stage with as little baggage as possible, and it is quite plausible that he would have started a new notebook at this point. If this consideration carries any weight, though, it is open to question exactly when Wittgenstein saw the new phase of his life as starting. There is a hint in Pinsent’s diary that perhaps Wittgenstein had already formed his plan to spend a year in Norway earlier in the summer and treated the holiday in September as a means of investigating the practical implementation of the plan. An hypothesis I find quite attractive, therefore, is that Wittgenstein started his Norwegian notebook at the beginning of the holiday and continued it through the succeeding year. If that is what happened, the ‘newer things’ not in the Birmingham dictation which Wittgenstein explained to Russell in Cambridge will presumably be the product of the Norwegian holiday, written up in the new Norwegian notebook. 22

See McGuinness, Approaches, 261.

23

9 Aug. 1914.

The Costello version  This, though, is a point of relative detail. More important are the general conclusions we have reached, namely that MSS 1, 3, and 4 are drawn from Wittgenstein’s Cambridge notebooks, which he left behind when he departed for Norway and which after the war he instructed Russell to destroy; and that the dictation in Birmingham was an attempt to extract from these notebooks—for Russell’s benefit, no doubt, but also (perhaps primarily) for his own—everything he thought worth preserving in them. If this is broadly correct, and if MSS 1, 3, and 4 correspond to three distinct Cambridge notebooks, the route by which extracts from them reached the Prototractatus cannot be directly from the notebooks, to which he had no access for the duration of the war, but must rather be via the typescripts themselves. One possibility is that Russell passed them back to Wittgenstein after he had translated them: a natural opportunity for him to have done this was via Moore, who departed for his visit to Wittgenstein in Norway shortly after Russell had finished his translation. However, even if Russell did pass his copy of the Birmingham typescript back to Wittgenstein in this manner, it is unlikely to have been the only copy: it would be natural, especially if this typescript was the summation of his work in Cambridge, for Wittgenstein to have ensured that the Birmingham typist made two copies of the typescript, one for Russell, the other for Wittgenstein himself. By one or other of these routes (or perhaps both), therefore, I believe Wittgenstein obtained in Norway a copy of the Birmingham typescript, from which he later extracted remarks to include in the Prototractatus.

A.4 The Costello version We noted earlier that in the last week of February 1914 Russell was ‘translating and copying and classifying’ Wittgenstein’s notes. The manuscript in Russell’s hand discussed in the previous section represents, as we have seen, the translating and copying. A surviving typescript (which McGuinness calls the ‘second-stage typescript’ or T) provides us with evidence of how he then went about the classifying stage. This typescript was prepared for Russell by Miss Harwood, the secretary who had taken dictation in October: in the case of the Summary she was therefore redoing what she had already typed in October, but now incorporating Russell’s and Wittgenstein’s corrections to it; in the case of the remainder of the Notes she was working from Russell’s manuscripts. Russell now wrote a list of numbered headings on a separate sheet and on the new typescript put corresponding numbers against the paragraphs, classifying them according to their subject matter; two paragraphs24 he marked for deletion. 24

B49, B64.



History of the text

Russell told Ottoline that he was doing this editorial work on Wittgenstein’s notes because ‘I shall want them for lecturing on logic at Harvard’. And certainly he took either the marked up second-stage typescript just described or another prepared from it with him to Harvard. There Russell was allocated as his teaching assistant for his logic course a graduate student called Harry Costello; and as Costello later recalled, he ‘had with him some notes and excerpts, giving the opinions of a brilliant student of his, named Ludwig Wittgenstein. . . I copied this manuscript’.25 This copy, which was first published in 1957 (by Costello himself) has come to be known as the Costello version of the Notes. Producing it from the secondstage typescript Russell had marked up was not a purely mechanical process, though. The Costello version not only divides the Notes into seven sections according to Russell’s classification; it also rearranges the paragraphs within each section; in some (but by no means all) of the cases where the Birmingham and Cambridge versions duplicate one another it chooses which to include (more often the Birmingham version); and it makes various minor changes in wording. What is clear is that whoever first performed this task had not only the surviving copy of the second-stage typescript with Russell’s annotations but a carbon copy as well, since there are no signs on the surviving typescript of the sort of tick marks that would have been necessary in order to confirm that each paragraph had been copied across. The method used was probably to cut up the carbon copy into paragraphs, sort these into seven piles corresponding to Russell’s classificatory markings, reorder each pile so as to make the best sense, adapt the prose by hand in a handful of places, and then either copy the result out by hand or give the ordered piles of paper slips to a typist to retype. Who performed this editing process? The only two plausible candidates are evidently Russell himself and Costello. According to Costello’s own reminiscences Russell did hand round to those who attended his logic lectures pieces of Wittgenstein’s work, and there may well have been various opportunities for them to copy parts of them. If it was Costello who performed the editing, however, he can only have done it with Russell’s knowledge and approval. For, as just noted, it involved at the very least marking, and probably cutting up, Russell’s carbon copy of the notes. The process must have cost someone a great deal of work and study,26 and there is therefore a certain plausibility in the idea that Russell—a man who, as McGuinness drily observes, ‘had never tied up a parcel in his life’27 —might have asked his teaching assistant, Costello, to perform it: the purpose, presumably, was to make the resulting document available for further consultation by other students attending Russell’s logic course.28 In that case, though, it is somewhat curious that Russell 25

Costello, ‘Notes on Logic’, 230. 26 See Approaches, 252. 27 Ibid. 258. 28 I have been unable to trace what happened to Costello’s copy after he published it in 1957, and do not know whether it was typed or handwritten.

The Costello version  himself should have broken off the editing process when he did, with all the paragraphs classified but not yet rearranged: he told Ottoline on 28 February 1914, six days before he was due to sail for America, that the editing, although it had taken a lot of work, was ‘now finished’; yet on this account it was really only half done. The alternative hypothesis is that Russell himself, having classified the paragraphs of the second-stage typescript according to subject, completed the task by performing the rearrangement of the carbon copy just described. It will then presumably have been Miss Harwood who produced, just in time for Russell’s departure, the final typescript of this rearrangement. On this account Russell may well have taken only this typescript to Harvard, where he made it available for Costello to copy. This hypothesis has the advantage that it conforms with Costello’s later description of his version of the Notes as one he had ‘copied’ (not ‘compiled’) at Harvard. On this account, too, the remains of the process which have survived among Russell’s papers are explicable as being just those he left behind in Cambridge when he departed for America in March 1914. When he returned to England in June, he may simply have left behind at Harvard the copy of the Costello version which he had taken with him. The textual differences between the surviving typescript and the Costello version as it has been transmitted to us may also be thought to lend some plausibility to this second hypothesis. Some of these differences (‘causes’ for ‘cases’, ‘cannot’ for ‘can not’, a whole phrase omitted) are evidently just copying errors, for which either Costello or an intervening typist was no doubt responsible; the logical symbols have been especially badly mauled (‘q’ for ‘φ ’, existential quantifiers omitted). Other changes are the sort of unimportant minor variations (‘where’ for ‘when’) that many copyists make almost unconsciously. But some (‘may be symbolized’ for ‘symbolized’, putting ‘logical objects’ in scare quotes, or changing ‘know the meaning of names’ to the somewhat different ‘know that our names have meaning’) suggest a certain confidence, a degree of ownership over the text, that would be a little surprising in a graduate student. And one or two are explanatory rephrasings which surely betray a more expert hand than Costello’s. Would a graduate student producing a text for circulation have felt certain enough to gloss ‘ab functions’29 as ‘functions with sense’, ‘logical types’30 as ‘forms’, ‘constituent’ as ‘particular’, and ‘component’ as ‘particular or relation, etc.’?31 None of these considerations is decisive on its own, of course, but cumulatively they make it more likely than not, I believe, that the Costello version resulted from his copying of a typescript, now lost, which Miss Harwood compiled on the basis of Russell’s own instructions and which Russell then took with him to Harvard. 29

B73.

30

B16.

31

B37.

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History of the text

A.5 Wittgenstein’s dissertation At just the time when Russell was engaged in ‘translating, copying, and classifying’ the Notes, Moore was preparing to visit Wittgenstein in Norway. Russell met Moore to discuss Wittgenstein’s ideas and lent him a copy of the Notes to read.32 If Russell had by then completed the Costello version, it would be very natural for him to have given a carbon copy of it to Moore to take to Wittgenstein. One obvious reason for him to do this would have been that Russell intended to show the Notes to his students at Harvard, and one might think that courtesy required him to ensure that Wittgenstein saw the version to be circulated. The question of who compiled the Costello version is therefore not of purely antiquarian interest. For if it was a carbon copy of the Costello version that Moore took to Norway, it is possible that this is what Wittgenstein then wanted to submit as a BA dissertation at Cambridge. If so, then perhaps the rearrangement of Wittgenstein’s material that the Costello version represents had Wittgenstein’s blessing. This might explain, too, Wittgenstein’s touchiness when Moore told him that he would have to supply a signed declaration of the extent to which the dissertation was his own work. The degree of editing which the Costello version had undergone very plainly represents a greater input than one would normally expect from a supervisor, and so it would not have been straightforward for Wittgenstein to make the declaration in just the form stipulated in the Cambridge regulations. But none of this is certain; the available evidence does not exclude the alternative that it was the notes Wittgenstein dictated to Moore in Norway that he wanted to submit as a dissertation. And whoever rearranged the Notes to form what we now know as the Costello version, what is certain is that it was not Wittgenstein. For this reason I have chosen to reprint the earlier version of the Notes in this book and not to attempt to draw conclusions about Wittgenstein’s views from the arrangement of the Costello version.

A.6 Conclusion The central point I have been concerned to establish in this Appendix is that the Notes should be thought of as consisting of two distinct texts of rather different characters, one produced in Birmingham and the other in Cambridge. The first of these, the Birmingham Notes on Logic, consists of MSS 1, 3, and 4. They are extracts from Wittgenstein’s Cambridge notebooks which he dictated in German to a stenographer in Birmingham; the version we have of these Birmingham Notes is a translation made by Russell four months later. 32

Moore’s diary, 28 Feb. 1914.

Conclusion  The three parts most likely correspond to three distinct notebooks. We cannot now determine for certain whether the Birmingham Notes preserve the chronological order of the entries in these notebooks, but I see no strong reason to suppose that they do not. The second text, the Cambridge Notes on Logic, consists of MS2 and the Summary. MS2 was written (largely, if not entirely, in English) by Wittgenstein himself in Russell’s rooms in Cambridge; the Summary he dictated the following day entirely in English. This Cambridge text seems to contain a few extracts from Wittgenstein’s notebooks, and in the case of the Summary, at least, these will presumably be Wittgenstein’s own impromptu translations. Very largely, though, the document is a commentary on the Birmingham Notes. Only a very few of its remarks are not either translations of or explanatory glosses on remarks in the Birmingham version: these few additional remarks may be what Russell called ‘newer things’—ideas Wittgenstein had had only very recently, during his Norwegian holiday. Much of the route we have taken here in order to reach these conclusions is perhaps at best of biographical rather than philosophical interest. (If nothing else, it provides eloquent testimony of Russell’s extraordinary patience in dealing with Wittgenstein and of the esteem in which he held Wittgenstein’s work.) Nonetheless, the outline just given is worth bearing in mind when studying the Notes, since it suggests the approach that we should take when reading them: read the Birmingham Notes first and treat the Cambridge Notes, as Wittgenstein intended them, mainly as a source of alternative formulations that may in some cases be more illuminating or explanatory than the originals.

Appendix B

The Notes on Logic The Birmingham Notes

II V

I

V

First MS. B1. Indefinables are of two sorts: names, and forms. Propositions cannot consist of names alone; they cannot be classes of names. A name can not only occur in two different propositions, but can occur in the same way in both. B2. Propositions [which are symbols having reference to facts] are themselves facts: that this inkpot is on this table may express that I sit in this chair. B3. It can never express the common characteristic of two objects that we designate them by the same name but by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where then would be the common element in the designations? Nevertheless one is always tempted, in a difficulty, to take refuge in different ways of designation. B4. Frege said “propositions are names”; Russell said “propositions correspond to complexes”. Both are false; and especially false is the statement “propositions are names of complexes”. B5. It is easy to suppose that only such symbols are complex as contain names of objects, and that accordingly “( x,φ ) φ x” or “( x,R,y) xRy” must be simple. It is then natural to call the first of these the name of a form, the second the name of a relation. But in that case what is the meaning of (e.g.) “∼( x,y) xRy”? Can we put “not” before a name? B6. The reason why “∼Socrates” means nothing is that “∼ x” does not express a property of x. B7. There are positive and negative facts: if the proposition “this rose is not red” is true, then what it signifies is negative. But the occurrence of the word “not” does not indicate this unless we know that the signification of the proposition “this rose is red” (when it is true) is positive. It is only from both, the negation and the negated proposition, that we can conclude to a characteristic of the significance of the whole proposition. (We are not here speaking of negations of general propositions, i.e. of such as contain apparent variables. Negative facts only justify the negations of atomic propositions.) B8. Positive and negative facts there are, but not true and false facts. B9. If we overlook the fact that propositions have a sense which is independent of their truth or falsehood, it easily seems as if true and false were two equally E

III I

I I

E

II

The Birmingham Notes  justified relations between the sign and what is signified. (We might then say e.g. that “q” signifies in the true way what “not-q” signifies in the false way.) But are not true and false in fact equally justified? Could we not express ourselves by means of false propositions just as well as hitherto with true ones, so long as we know that they are meant falsely? No! For a proposition is then true when it is as we assert in this proposition; and accordingly if by “q” we mean “not-q”, and it is as we mean to assert, then in the new interpretation “q” is actually true and not false. But it is important that we can mean the same by “q” as by “not-q”, for it shows that neither to the symbol “not” nor to the manner of its combination with “q” does a characteristic of the denotation of “q” correspond. 3rd MS. B10. An analogy for the theory of truth: Consider a black patch on white paper; then we can describe the form of the patch by mentioning, for each point of the surface, whether it is white or black. To the fact that a point is black corresponds a positive fact, to the fact that a point is white (not black) corresponds a negative fact. If I designate a point of the surface (one of Frege’s “truth-values”), this is as if I set up an assumption to be decided upon. But in order to be able to say of a point that it is black or that it is white, I must first know when a point is to be called black and when it is to be called white. In order to be able to say that “ p” is true (or false), I must first have determined under what circumstances I call a proposition true, and thereby I determine the sense of a proposition. The point in which the analogy fails is this: I can indicate a point of the paper without knowing what white and black are, but to a proposition without sense nothing corresponds, for it does not designate a thing (truth-value), whose properties might be called “false” or “true”; the verb of a proposition is not “is true” or “is false”, as Frege believes, but what is true must already contain the verb. B11. The comparison of language and reality is like that of retinal image and visual image: to the blind spot nothing in the visual image seems to correspond, and thereby the boundaries of the blind spot determine the visual image—as true negations of atomic propositions determine reality. B12. Logical inferences can, it is true, be made in accordance with Frege’s or Russell’s laws of deduction, but this cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also be inferred from q, and the “manner of deduction” is indifferent. B13. Those symbols which are called propositions in which “variables occur” are in reality not propositions at all, but only schemes of propositions, which only become propositions when we replace the variables by constants. There is no proposition which is expressed by “ x = x”, for “ x” has no signification; but there is a proposition “( x) x= x” and propositions such as “Socrates = Socrates” etc.

I

I

III

IV

 IV

I

II

II

III

VI

The Notes on Logic

B14. In books on logic, no variables ought to occur, but only the general propositions which justify the use of variables. It follows that the so-called definitions of logic are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions; and similarly the so-called primitive ideas (Urzeichen) of logic are not primitive ideas, but the schemes of them. The mistaken idea that there are things called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which e.g. expresses that q follows from p and p ⊃ q is this: p p ⊃ q ⊃ p,q q. B15. At a pinch, one is tempted to interpret “not- p” as “everything else, only not p”. That from a single fact p an infinity of others, not-not- p etc., follow, is hardly credible. Man possesses an innate capacity for constructing symbols with which some sense can be expressed, without having the slightest idea what each word signifies. The best example of this is mathematics, for man has until lately used the symbols for numbers without knowing what they signify or that they signify nothing. B16. Russell’s “complexes” were to have the useful property of being compounded, and were to combine with this the agreeable property that they could be treated like “simples”. But this alone made them unserviceable as logical types, since there would have been significance in asserting, of a simple, that it was complex. But a property cannot be a logical type. B17. Every statement about apparent complexes can be resolved into the logical sum of a statement about the constituents and a statement about the proposition which describes the complex completely. How, in each case, the resolution is to be made, is an important question, but its answer is not unconditionally necessary for the construction of logic. B18. That “or” and “not” etc. are not relations in the same sense as “right” and “left” etc., is obvious to the plain man. The possibility of cross-definitions in the old logical indefinables shows, of itself, that these are not the right indefinables, and, even more conclusively, that they do not denote relations. B19. If we change a constituent a of a proposition φ (a) into a variable, then there is a class pˆ {( x) φ ( x) = p}. E

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This class in general still depends upon what, by an arbitrary convention, we mean by “φ ( x)”. But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is now not dependent upon any convention, but only upon the nature of the symbol “φ ( x)”. It corresponds to a logical type. B20. Types can never be distinguished from each other by saying (as is often

The Birmingham Notes  done) that one has these but the other has those properties, for this presupposes that there is a meaning in asserting all these properties of both types. But from this it follows that, at best, these properties may be types, but certainly not the objects of which they are asserted. B21. At a pinch, we are always inclined to explanations of logical functions of propositions which aim at introducing into the function either only the constituents of these propositions, or only their form, etc. etc; and we overlook that ordinary language would not contain the whole propositions if it did not need them: However, e.g., “not- p” may be explained, there must always be a meaning given to the question “what is denied?” B22. The very possibility of Frege’s explanations of “not- p” and “if p then q”, from which it follows that “not-not- p” denotes the same as p, makes it probable that there is some method of designation in which “not-not- p” corresponds to the same symbol as “ p”. But if this method of designation suffices for logic, it must be the right one. B23. Names are points, propositions arrows—they have sense. The sense of a proposition is determined by the two poles true and false. The form of a proposition is like a straight line, which divides all points of a plane into right and left. The line does this automatically, the form of proposition only by convention. B24. Just as little as we are concerned, in logic, with the relation of a name to its meaning, just so little are we concerned with the relation of a proposition to reality, but we want to know the meaning of names and the sense of propositions—as we introduce an indefinable concept “A” by saying: “ ‘A’ denotes something indefinable”, so we introduce e.g. the form of propositions aRb by saying: “For all meanings of “ x” and “y”, “ xRy” expresses something indefinable about x and y”. B25. In place of every proposition “ p”, let us write “ab p”. Let every correlation of propositions to each other or of names to propositions be effected by a correlation of their poles “a” and “b”. Let this correlation be transitive. Then a a accordingly “ba− −b p” is the same symbol as “b p”. Let n propositions be given. I then call a “class of poles” of these propositions every class of n members, of which each is a pole of one of the n propositions, so that one member corresponds to each proposition. I then correlate with each class of poles one of two poles (a and b). The sense of the symbolizing fact thus constructed I cannot define, but I know it. B26. If p = not-not- p etc., this shows that the traditional method of symbolism is wrong, since it allows a plurality of symbols with the same sense; and thence it follows that, in analyzing such propositions, we must not be guided by Russell’s method of symbolizing. B27. It is to be remembered that names are not things, but classes: “A” is the same letter as “A”. This has the most important consequences for every

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symbolic language. B28. Neither the sense nor the meaning of a proposition is a thing. These words are incomplete symbols. B29. It is impossible to dispense with propositions in which the same argument occurs in different positions. It is obviously useless to replace φ (a,a) by φ (a,b) a = b. B30. Since the ab-functions of p are again bi-polar propositions, we can form ab-functions of them, and so on. In this way a series of propositions will arise, in which in general the symbolizing facts will be the same in several members. If now we find an ab-function of such a kind that by repeated application of it every ab-function can be generated, then we can introduce the totality of ab-functions as the totality of those that are generated by application of this function. Such a function is ∼ p v ∼q. B31. It is easy to suppose a contradiction in the fact that on the one hand every possible complex proposition is a simple ab-function of simple propositions, and that on the other hand the repeated application of one ab-function suffices to generate all these propositions. If e.g. an affirmation can be generated by double negation, is negation in any sense contained in affirmation? Does “ p” deny “not- p” or assert “ p”, or both? And how do matters stand with the definition of “⊃” by “v” and “∼”, or of “v” by “.” and “⊃”? And how e.g. shall we introduce p | q (i.e. ∼ p v ∼q), if not by saying that this expression says something indefinable about all arguments p and q? But the ab-functions must be introduced as follows: The function p | q is merely a mechanical instrument for constructing all possible symbols of ab-functions. The symbols arising by repeated application of the symbol “|” do not contain the symbol “ p | q”. We need a rule according to which we can form all symbols of ab-functions, in order to be able to speak of the class of them; and we now speak of them e.g. as those symbols of functions which can be generated by repeated application of the operation “|”. And we say now: For all p’s and q’s, “ p | q” says something indefinable about the sense of those simple propositions which are contained in p and q. B32. The assertion-sign is logically quite without significance. It only shows, in Frege and Whitehead and Russell, that these authors hold the propositions so indicated to be true. “#” therefore belongs as little to the proposition as (say) the number of the proposition. A proposition cannot possibly assert of itself that it is true. B33. Every right theory of judgment must make it impossible for me to judge that this table penholders the book. Russell’s theory does not satisfy this requirement. B34. It is clear that we understand propositions without knowing whether they are true or false. But we can only know the meaning of a proposition when we know if it is true or false. What we understand is the sense of the proposition.

The Birmingham Notes  B35. The assumption of the existence of logical objects makes it appear remarkable that in the sciences propositions of the form “ p v q”, “ p ⊃ q”, etc. are only then not provisional when “v” and “⊃” stand within the scope of a generality-sign [apparent variable]. 4th MS. B36. If we formed all possible atomic propositions, the world would be completely described if we declared the truth or falsehood of each. [I doubt this.] B37. The chief characteristic of my theory is that, in it, p has the same meaning as not- p. B38. A false theory of relations makes it easily seem as if the relation of fact and constituent were the same as that of fact and fact which follows from it. But the similarity of the two may be expressed thus: φ a ⊃ φ,a a = a. B39. If a word creates a world so that in it the principles of logic are true, it thereby creates a world in which the whole of mathematics holds; and similarly it could not create a world in which a proposition was true, without creating its constituents. B40. Signs of the form “ p v ∼ p” are senseless, but not the proposition “( p) p v ∼ p”. If I know that this rose is either red or not red, I know nothing. The same holds of all ab-functions. B41. To understand a proposition means to know what is the case if it is true. Hence we can understand it without knowing if it is true. We understand it when we understand its constituents and forms. If we know the meaning of “a” and “b”, and if we know what “ xRy” means for all x’s and y’s, then we also understand “aRb”. B42. I understand the proposition “aRb” when I know that either the fact that aRb or the fact that not aRb corresponds to it; but this is not to be confused with the false opinion that I understand “aRb” when I know that “aRb or not-aRb” is the case. B43. But the form of a proposition symbolizes in the following way: Let us consider symbols of the form “ xRy”; to these correspond primarily pairs of objects, of which one has the name “ x”, the other the name “y”. The x’s and y’s stand in various relations to each other, among others the relation R holds between some, but not between others. I now determine the sense of “ xRy” by laying down: when the facts behave in regard to “ xRy” so that the meaning of “ x” stands in the relation R to the meaning of “y”, then I say that they [the facts] are “of like sense” [“gleichsinnig”] with the proposition “ xRy”; otherwise, “of opposite sense” [“entgegengesetzt”]; I correlate the facts to the symbol “ xRy” by thus dividing them into those of like sense and those of opposite sense. To this correlation corresponds the correlation of name and meaning. Both are psychological. Thus I understand the form “ xRy” when I know that it discriminates the behaviour of x and y according as these stand in the relation R or not. In this way I extract from all possible relations the

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relation R, as, by a name, I extract its meaning from among all possible things. B44. Strictly speaking, it is incorrect to say: we understand the proposition p when we know that ‘ “ p” is true’ ≡ p; for this would naturally always be the case if accidentally the propositions to right and left of the symbol “≡” were both true or both false. We require not only an equivalence, but a formal equivalence, which is bound up with the introduction of the form of p. B45. The sense of an ab-function of p is a function of the sense of p. B46. The ab-functions use the discrimination of facts, which their arguments bring forth, in order to generate new discriminations. B47. Only facts can express sense, a class of names cannot. This is easily shown. B48. There is no thing which is the form of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell’s theory of judgment. B49. It is very easy to forget that, though the propositions of a form can be either true or false, each one of these propositions can only be either true or false, not both. B50. Among the facts which make “ p or q” true, there are some which make “ p and q” true; but the class which makes “ p or q” true is different from the class which makes “ p and q” true; and only this is what matters. For we introduce this class, as it were, when we introduce ab-functions. B51. A very natural objection to the way in which I have introduced e.g. propositions of the form xRy is that by it propositions such as ( x,y) xRy and similar ones are not explained, which yet obviously have in common with aRb what cRd has in common with aRb. But when we introduced propositions of the form xRy we mentioned no one particular proposition of this form; and we only need to introduce ( x,y) φ ( x,y) for all φ ’s in any way which makes the sense of these propositions dependent on the sense of all propositions of the form φ (a,b), and thereby the justness of our procedure is proved. B52. The indefinables of logic must be independent of each other. If an indefinable is introduced, it must be introduced in all combinations in which it can occur. We cannot therefore introduce it first for one combination, then for another; e.g., if the form xRy has been introduced, it must henceforth be understood in propositions of the form aRb just in the same way as in propositions such as ( x,y) xRy and others. We must not introduce it first for one class of cases, then for the other; for it would remain doubtful if its meaning was the same in both cases, and there would be no ground for using the same manner of combining symbols in both cases. In short, for the introduction of indefinable symbols and combinations of symbols the same holds, mutatis mutandis, that Frege has said for the introduction of symbols by definitions. B53. It is a priori likely that the introduction of atomic propositions is funE

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B56. The ab-notation shows the dependence of or and not, and thereby that they are not to be employed as simultaneous indefinables. B57. Not: “The complex sign ‘aRb’ ” says that a stands in the relation R to b; but that ‘a’ stands in a certain relation to ‘b’ says that aRb. B58. In philosophy there are no deductions: it is purely descriptive. B59. Philosophy gives no pictures of reality. B60. Philosophy can neither confirm nor confute scientific investigation. B61. Philosophy consists of logic and metaphysics: logic is its basis. B62. Epistemology is the philosophy of psychology. B63. Distrust of grammar is the first requisite for philosophizing. B64. Propositions can never be indefinables, for they are always complex. That also words like “ambulo” are complex appears in the fact that their root with a different termination gives a different sense. B65. Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads to an impenetrable thicket. B66. Philosophy is the doctrine of the logical form of scientific propositions (not only of primitive propositions). B67. The word “philosophy” ought always to designate something over or under, but not beside, the natural sciences. B68. Judgment, command and question all stand on the same level; but all have in common the propositional form, which does interest us. B69. The structure of the proposition must be recognized, the rest comes of itself. But ordinary language conceals the structure of the proposition: in it, relations look like predicates, predicates like names, etc.

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B70. Facts cannot be named. B71. It is easy to suppose that “individual”, “particular”, “complex” etc. are primitive ideas of logic. Russell e.g. says “individual” and “matrix” are “primitive ideas”. This error presumably is to be explained by the fact that, by employment of variables instead of the generality-sign, it comes to seem as if logic dealt with things which have been deprived of all properties except thing-hood, and with propositions deprived of all properties except complexity. We forget that the indefinables of symbols [Urbilder von Zeichen] only occur under the generality-sign, never outside it. B72. Just as people used to struggle to bring all propositions into the subjectpredicate form, so now it is natural to conceive every proposition as expressing a relation, which is just as incorrect. What is justified in this desire is fully satisfied by Russell’s theory of manufactured relations. B73. One of the most natural attempts at solution consists in regarding “notp” as “the opposite of p”, where then “opposite” would be the indefinable relation. But it is easy to see that every such attempt to replace the ab-functions by descriptions must fail. B74. The false assumption that propositions are names leads us to believe that there must be logical objects: for the meanings of logical propositions will have to be such things. B75. A correct explanation of logical propositions must give them a unique position as against all other propositions. B76. No proposition can say anything about itself, because the symbol of the proposition cannot be contained in itself; this must be the basis of the theory of logical types. B77. Every proposition which says something indefinable about a thing is a subject-predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate proposition, and so on. An indefinable simple symbol can only be a name, and therefore we can know, by the symbol of an atomic proposition, whether it is a subject-predicate proposition.

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C3. Not only must logic not deal with [particular] things, but just as little with relations and predicates. C4. There are no propositions containing real variables. C5. What corresponds in reality to a proposition depends upon whether it is true or false. But we must be able to understand a proposition without knowing if it is true or false. (B41) C6. What we know when we understand a proposition is this: We know what is the case if the proposition is true, and what is the case if it is false. But we do not know [necessarily] whether it is true or false. (B41) C7. Propositions are not names. (B4) C8. We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other. (B20) C9. Symbols are not what they seem to be. In “aRb”, “R” looks like a substantive, but is not one. What symbolizes in “aRb” is that R occurs between a and b. (B57) Hence “R” is not the indefinable in “aRb”. Similarly in “φ x”, “φ ” looks like a substantive but is not one; in “∼ p”, “∼” looks like “φ ” but is not like it. This is the first thing that indicates that there may not be logical constants. A reason against them is the generality of logic: logic cannot treat a special set of things. C10. Molecular propositions contain nothing beyond what is contained in their atoms; they add no material information above that contained in their atoms. (B36) C11. All that is essential about molecular functions is their T-F schema [i.e. the statement of the cases when they are true and the cases when they are false]. C12. Alternative definability shows that the indefinables have not been reached. (B18) C13. Every proposition is essentially true–false: to understand it, we must know both what must be the case if it is true, and what must be the case if it is false. Thus a proposition has two poles, corresponding to the case of its truth and the case of its falsehood. We call this the sense of a proposition. C14. In regard to notation, it is important to note that not every feature of a symbol symbolizes. In two molecular functions which have the same T-F schema, what symbolizes must be the same. In “not-not- p”, “not- p” does not occur; for “not-not- p” is the same as “ p”, and therefore, if “not- p” occurred in “not-not- p”, it would occur in “ p”. (B22) C15. Logical indefinables cannot be predicates or relations, because propositions, owing to sense, cannot have predicates or relations. Nor are “not” and “or”, like judgment, analogous to predicates or relations, because they do not introduce anything new. C16. Propositions are always complex even if they contain no names. (B5) C17. A proposition must be understood when all its indefinables are understood. The indefinables in “aRb” are introduced as follows:

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C18. A complex symbol must never be introduced as a single indefinable. (Thus e.g. no proposition is indefinable.) For if one of its parts occurs also in another connection, it must there be re-introduced. And would it then mean the same? (B52) C19. The ways by which we introduce our indefinables must permit us to construct all propositions that have sense [? meaning] from these indefinables alone. It is easy to introduce “all” and “some” in a way that will make the construction of (say) “( x,y) xRy” possible from “all” and “ xRy” as introduced before. (B51) Summary C20. One reason for thinking the old notation wrong is that it is very unlikely that from every proposition p an infinite number of other propositions notnot- p, not-not-not-not- p, etc., should follow. (B15) C21. If only those signs which contain proper names were complex then propositions containing nothing but apparent variables would be simple. Then what about their denials? (B5) C22. The verb of a proposition cannot be “is true” or “is false”, but whatever is true or false must already contain the verb. (B10) C23. Deductions only proceed according to the laws of deduction, but these laws cannot justify the deduction. (B12) C24. One reason for supposing that not all propositions which have more than one argument are relational propositions is that if they were, the relations of judgement and inference would have to hold between an arbitrary number of things. C25. Every proposition which seems to be about a complex can be analysed into a proposition about its constituents and about the proposition which describes the complex perfectly; i.e., that proposition which is equivalent to saying the complex exists. (B17) C26. The idea that propositions are names of complexes suggests that whatever is not a proper name is a sign for a relation. Because spatial complexes1 consist of Things and Relations only and the idea of a complex is taken from space. C27. In a proposition convert all its indefinables into variables; there then remains a class of propositions which is not all propositions but a type. (B19) C28. There are thus two ways in which signs are similar. The names Socrates 1

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and Plato are similar: they are both names. But whatever they have in common must not be introduced before Socrates and Plato are introduced. The same applies to a subject-predicate form etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e., types are not indefinables. C29. When we say A judges that etc., then we have to mention a whole proposition which A judges. It will not do either to mention only its constituents, or its constituents and form, but not in the proper order. (B55) This shows that a proposition itself must occur in the statement that it is judged; however, for instance, “not- p” may be explained, the question, “What is negated” must have a meaning. (B21) C30. To understand a proposition p it is not enough to know that p implies ‘ “ p” is true’, but we must also know that ∼ p implies “ p is false”. This shows the bi-polarity of the proposition. (B55) C31. To every molecular function a WF2 scheme corresponds. Therefore we may use the WF scheme itself instead of the function. Now what the WF scheme does is, it correlates the letters W and F with each proposition. These two letters are the poles of atomic propositions. Then the scheme correlates another W and F to these poles. In this notation all that matters is the correlation of the outside poles to the pole of the atomic propositions. Therefore not-not- p is the same symbol as p. And therefore we shall never get two symbols for the same molecular function. C32. The meaning of a proposition is the fact which actually corresponds to it. C33. As the ab functions of atomic propositions are bi-polar propositions again we can perform ab operations on them. We shall, by doing so, correlate two new outside poles via the old outside poles to the poles of the atomic propositions. (B30) C34. The symbolising fact in a– p–b is that, say3 a is on the left of p and b on the right of p; then the correlation of new poles is to be transitive, so that for instance if a new pole a in whatever way i.e. via whatever poles is correlated to the inside a, the symbol is not changed thereby. It is therefore possible to construct all possible ab functions by performing one ab operation repeatedly, and we can therefore talk of all ab functions as of all those functions which can be obtained by performing this ab operation repeatedly.4 (B25) C35. Naming is like pointing. A function is like a line dividing points of a 2 W-F = Wahr-Falsch 3 This is quite arbitrary, but if we once have fixed on which sides the poles have to stand we must of course stick to our convention. If for instance “a p b” says p then b p a says nothing. (It does not say ∼ p) But a–a p b–b is the same symbol as a p b (here the ab function vanishes automatically) for here the new poles are related to the same side of p as the old ones. The question is always: how are the new poles correlated to p compared with the way the old poles are correlated to ∼ p. 4 [Note by B.R. ab means the same as WF, which means true-false.]

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plane into right and left ones; then “ p or not- p” has no meaning because it does not divide the plane. (B23) C36. But though a particular proposition “ p or not- p” has no meaning, a general proposition “for all p’s, p or not- p” has a meaning because this does not contain the nonsensical function “ p or not- p” but the function “ p or not-q” just as “for all x’s xR x” contains the function “ xRy”. (B40) C37. A proposition is a standard to which facts behave, with names it is otherwise; it is thus bi-polarity and sense comes in; just as one arrow behaves to another arrow by being in the same sense or the opposite, so a fact behaves to a proposition. C38. The form of a proposition has meaning in the following way. Consider a symbol “ xRy”. To symbols of this form correspond couples of things whose names are respectively “ x” and “y”. The things x/y stand to one another in all sorts of relations, amongst others some stand in the relation R, and some not; just as I single out a particular thing by a particular name I single out all behaviours of the points x and y with respect to the relation R. I say that if an x stands in the relation R to a y the sign “ xRy” is to be called true to the fact and otherwise false. This is a definition of sense. (B43) C39. In my theory p has the same meaning as not- p but opposite sense. The meaning is the fact. The proper theory of judgment must make it impossible to judge nonsense. (B33) C40. It is not strictly true to say that we understand a proposition p if we know that p is equivalent to “ p is true” for this would be the case if accidentally both were true or false. What is wanted is the formal equivalence with respect to the forms of the proposition, i.e., all the general indefinables involved. (B44) The sense of an ab function of a proposition is a function of its sense. (B45) There are only unasserted propositions. Assertion is merely psychological. (B32) In not- p, p is exactly the same as if it stands alone; this point is absolutely fundamental. Among the facts which make “ p or q” true there are also facts which make “ p and q” true; if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical, but in fact, their sense is different for we have introduced sense by talking of all p’s and all q’s. (B50) Consequently the molecular propositions will only be used in cases where their ab function stands under a generality sign or enters into another function such as “I believe that, etc.”, because then the sense enters. C41. In “a judges p” p cannot be replaced by a proper name. This appears if we substitute “a judges that p is true and not p is false”. The proposition “a judges p” consists of the proper name a, the proposition p with its 2 poles, and a being related to both of these poles in a certain way. This is obviously not a relation in the ordinary sense. (B55) C42. The ab notation makes it clear that not and or are dependent on one an-

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other and we can therefore not use them as simultaneous indefinables. Same objections in the case of apparent variables to the usual old indefinables, as in the case of molecular functions: The application of the ab notation to apparent-variable propositions becomes clear if we consider that, for instance, the proposition “for all x, φ x” is to be true when φ x is true for all x’s and false when φ x is false for some x’s. We see that some and all occur simultaneously in the proper apparent variable notation. C43. The Notation is:

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(2) it must show which components of a proposition of this type are constants. C50. [Components are forms and constituents.] C51. Take (φ ) φ ! x. Then if we describe the kind of symbols, for which φ ! stands and which, by the above, is enough to determine the type then automatically “(φ ) φ ! x” cannot be fitted by this description, because it contains “φ ! x” and the description is to describe all that symbolizes in symbols of the φ !-kind. If the description is thus complete vicious circles can just as little occur as for instance in (φ ) ( x)φ (where ( x)φ is a subject-predicate proposition).

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The Notes on Logic

Textual notes This text is based on three sources: R A 23-page manuscript of MSS1–4 in Russell’s hand (RA1.710.057823); S A 7-page typescript of the Summary with corrections by both Russell and Wittgenstein (RA1.710.057822); T A typescript prepared from both R and S as corrected, containing further corrections by Russell (RA1.710.057824). The text generally follows T as corrected by Russell. In the very few cases where T differs from the corrected versions of R and S from which it was compiled, I have chosen whichever made the best sense. In a few places (B10, B17, C12) clear errors which remained in T have been corrected. Readers seeking full diplomatic and normalized editions of the texts are recommended to consult those prepared by the Wittgenstein Archives at the University of Bergen, which are available online at http://wab.aksis.uib.no/. The text is presented here in two parts, which are labelled the Birmingham and Cambridge versions of the Notes: the justification for this presentation is contained in Appendix A. Paragraph numbers, as well as the parenthetical references in the Cambridge Notes to paragraphs of the Birmingham Notes covering similar material, have been added. Square brackets are present in the original texts and evidently represent Russell’s own insertions. Italicization of letters used as logical signs ( p, q, r, etc.), which in the source texts is carried out only fitfully, has here been silently made consistent. In the notes below, alterations made by Wittgenstein to the typsecript S are labelled LW; those made by Russell to S or to his manuscript R are labelled BR1; those made by Russell to the typescript T are labelled BR2. The sentences within a paragraph are referred to, where necessary, by letter. First MS. B3 designate] after deleted denote BR1 B7 what it] its B7 the proposition] inserted 3rd MS. B10 An analogy] after deleted A comparis BR1 B10g in] after deleted on BR1 B10 fails] after deleted depends BR1 B10 without knowing what white and black are] what is white and black R,T B14 (Urzeichen)] inserted BR1 B14 questioning] emphasized with query in margin BR2

B16 like] above deleted as BR1 B17 sum] product R,T B21 logical] inserted BR1 B21 aim at introducing into the function] inserted B21 only] before deleted contain BR1 B21 form] forms R B23 propositions] after deleted sentences BR1 B24 propositions—] propositions T B30 introduce] after deleted define BR1 B31 every] after deleted all BR1 B32 only] inserted BR1 B35 in the sciences] inserted BR1 B35 p v q] p or q

Textual notes  B35 are] inserted BR1 4th MS. B42 understand] R understood T B43 now] after deleted know B44 we] We R B49 very] inserted B50 only] inserted B51 justness] replaces justification BR2 B52 combinations] over deleted classes B54 quite] after deleted the BR2 B68 command and] inserted BR1 B68 all] after deleted and BR1 B69 structure] above deleted construction B69 proposition] above deleted sentence B71 sign] signs BR1 B77 symbol] above deleted sign 2nd MS. C12 definability] indefinability R,T Summary] added BR1 C20 One] after deleted The BR1 C21 those] written over these BR1 C23 Deductions] replaces The deductions BR1 C24 One] above deleted The BR1 C24 if] inserted BR1 C24 were,] comma inserted C24 would] above deleted that BR1 C25 its] after deleted those BR1 C25 the] above deleted a (twice) C25 the] after deleted a C26 suggests] replaces deleted between suggestions LW C26 spatial complexes] footnote added LW C26fn. Russell—for instance imagines] You—for instance imagine BR2 C26 Because . . . space.] added LW C27 is] above deleted has BR1 C28 a] omitted S C29 A judges] a judge is (twice) BR1 C29 constituents] constituent (twice) C29 meaning.] before deleted Always a question that is negated must have a meaning. LW adds in margin Rott! C29 the question . . . a meaning] added

LW C30 p implies ‘ “ p” is true’] replaces “ p implies p” is true BR1 C30 ∼ p implies “ p is false”] after deleted p also implies ‘ “not- p” is false’ C30 bi-polarity] polarity C31 W] w (four times) BR1 C31 F] f (four times) BR1 C31fn. W-F = Wahr-Falsch] added BR1 C31 the scheme correlates] corresponds C31 W] f C31 F] w C31 matters] after deleted it BR1 C31 the] its C31 And] inserted C31 function] functions BR1 C33 bi-polar] by polar C33 shall, by] will be LW C34 say] footnote inserted LW C34fn. arbitrary, but] arbitrary but S C34fn. (here] omitted S C34fn. ∼ p] p S C34 so] such C34 for instance] added LW C34 i.e. via whatever poles] added LW C34fn. Note by B.R.] inserted BR1 C34fn. ab] after NB. S C35 of a plane] added LW C35 the] a LW C36 “ p or not- p”] replaces “ p” or “not- p” LW C36 “for all p’s, p or not- p”] replaces for all p’s, “ p” or “not- p” C36 the] over a (twice) C36 “ p or not-q”] replaces p or “not-q” C37 facts] after deleted all BR2 C37 with names it is otherwise] that names it otherwise C37 thus bi-polarity] then by polarity C37 arrow] error (twice) C38 with respect to] inserted after deleted the one between C38 R] inserted before deleted of the other LW C38 R] inserted after deleted of LW C38 “ xRy”] x or y LW C40 In not- p, p is] If not- p is



The Notes on Logic

C40 true] inserted LW C40 have only meaning] over deleted do only mean BR1 C40 in such a case, to say] to know such a case, say LW C40 their] there S C40 then] inserted LW C41 a relation] the relation LW C42 makes] and apparent variables make LW C42 Same objections . . . indefinables] Some objections to old indefinables LW

C42 the usual] inserted BR2 C42 apparent-variable] apparently variable LW C42 becomes] become C44 more correctly] after deleted the BR1 C47 indefinable] above deleted important LW C49 apparent-variable] apparent C51 for which φ ! stands and] inserted LW C51 fitted] filled C51 in] omitted T C51 because . . . proposition).] inserted LW

The Costello version Below is an analysis of how the Costello version, as published in 1957, relates to the typescript T from which it was derived. As was explained in Appendix A, Russell allocated paragraphs to sections by markings (reproduced above) in the margins of T. Paragraphs B49 and B64, which Russell there marked for deletion (although they do not duplicate remarks elsewhere in the text), are omitted from the Costello version. Paragraphs C3, C7, and C23, which are not there clearly assigned to sections, are likewise omitted. Paragraphs B21c, B41ab, B42, B69, C8, C22, C38, C40ae, C42a, and C45 (all of them more or less duplicates of other paragraphs elsewhere in the Notes) are omitted from the Costello version even though they have been assigned to sections in T. Elsewhere the Costello version follows Russell’s classifications in T. In the analysis that follows, minor differences in punctuation have been ignored.

Preliminary B58 B67 B59 B60 B61 B62 B63 B66

it] it Philosophy] and logic] the former

only of primitive propositions] primitive propositions only B75 the] omitted I. Bi-polarity of propositions: sense and meaning, truth and falsehood.

B4 B70 named] named B74 logical objects] “logical objects” C5 C6 know[necessarily] ] necessarily know true or false] after inserted actually C13 to understand . . . false] omitted falsehood] falsity C32 B37 not- p] added (constituent = particular, component = particular or relation, etc.) B28 B34 meaning] meaning

The Costello version  sense] sense C30 B41cd B42 B44 C40b B7 to] about negations] the negations B8 B9 then] deleted B10 that it is white] it is white what] which B11 retinal] a retinal as] just as B15 At a pinch] deleted lately] recently B32 C40d B68 which does interest us] and that alone interests us C45b in them] omitted is. . . proposition]are. . . propositions C29 statement that] statement to the effect that However, for instance,] For instance, however C41 a judges p] A judges (that) p (twice) appears] is apparent a being] A’s being B33 C24 C26 suggests] has suggested Because . . . taken from space] Russell, for instance, imagines every fact as a spatial complex, and since spatial complexes consist of things and relations only, therefore he holds all do. B21ab At a pinch we are] We are very often overlook that] overlook the fact that B23 B73 One of the most natural attempts at solution] It is wrong to conceive every proposition as expressing a relation. A natural attempt at such a solution

the ab-functions] functions with sense (ab functions) B55 here we could] we could here ab notation] before added (later explained) say by making] by, say, making C37acb with] but with C39 my theory] this theory The proper theory] a proper theory C40c C40f–j for] and II. Analysis of atomic propositions: general indefinables, predicates, etc. B36 If . . . each [I doubt this] ] It may be doubted whether, if . . . each (Russell). B39 If a word creates a world so that in it] If there were a world created in which it thereby creates a world in which] in that world and similarly it could not create a world] No world can be created without creating its constituents] unless the constituents of the proposition are created also B1 can not] cannot B2 C1 B65 C17 B24 Just as little as we are] We are not a name] any specific name just so little are we concerned] and just as little proposition] given proposition but we want to know the meaning of names and the sense of propositions—as we introduce] We do want to know that our names have meanings and propositions sense, and we thus introduce so we introduce e.g. the form] or the form

The Notes on Logic

C2 C10 C11 when] where (twice) B53 B22 B26 C35 C36 B12 B6 B40 B35 B18 C15 B25 B45 B46 C31 WF] TF (three times) scheme] (or ab) scheme it] that it C33 ab] ab (TF) C34 Then] Then, given a p b, repeatedly] before inserted (cf., Sheffer’s work) B50 B30 B31 “∼”] “ ”

V. Principles of symbolism: what symbolizes in a symbol. Facts for facts. B5 suppose that] suppose ( x ,φ ) φ x ] ( x ,φ ) φ x ( x,R,y) xRy] ( x,y) xRy of (e.g.) ∼( x,y) xRy] e.g. of ∼( x,y) xRy C12 shows that] shows not been] not yet been B52 would be] could be B3 but by] but otherwise by B27 C14 note] observe schema] scheme C18 its parts] the parts of the complex symbol C19 B57 Not:] One must not say B47 C44 E

III. Analysis of molecular propositions: abfunctions.

IV. Analysis of general propositions B72 C21 were] are C16 C4 B13 only . . . when] do not . . . unless B14 the schemes] schemes cases] causes B54 C42b–d Same objections] There is the same objection old indefinables] indefinables C43 B51 justness] justification proved] established

E E

B43 But] deleted symbolizes] may be symbolized to these] to which among others] and among other relations they (the facts)] these facts B48 C9 is not] it is not symbolizes] is symbolized B16 logical types] before inserted (forms) would] would then B38 the similarity of the two may be expressed] there is a similarity of the two, expressible B17 apparent] deleted C25 Every proposition] To repeat: every proposition

E

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VI. Types B71 deprived of all properties except thinghood, and with propositions] omitted B77 and so on] etc. simple symbol] symbol C46 B19 C28 thus] omitted C46 and so on] etc.

The Costello version  says something] says something indefinable

C47 C49 components] components (forms and constituents)

C51 complete] completed for instance in] can for instance B20 Types can never be distinguished from each other] We can never distinguish one logical type from another

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Citations Notes on Logic B1 B3 B4 B5 B7 B9 B10 B11 B13 B16 B17 B18 B19 B21 B22 B23 B24 B26 B27 B29

111 82, 201 135 110 106 73, 97 89, 97, 172 143 55, 197 200, 273 44, 45 159, 175 178, 188 67, 175, 190 166, 213 154 99 213 64 208

B30 B31

B61 B62 B63 B64 B67 B69 B71 B73 B76 B77

B50 B51 B55 B57

159 159, 171, 189 95, 97 122 146, 148 132, 273 140 151, 152 151 99, 100, 153, 154 152, 153 111 115, 117, 125 138 153, 199 220, 254 113

2.225 3.12 3.14 3.142 3.143 3.1432 3.144 3.203 3.3 3.322 3.332 4.002 4.021 4.024

30 210 211 111 211 113 226 64 13, 178 82, 201 187, 189 68 226 151

4.0312 4.0621 4.063 4.1273 4.221 4.26 4.431 4.441 4.442 5.02 5.134 5.23 5.461 5.502

B32 B33 B36 B37 B40 B41 B42 B43 B44 B47 B48

C2 C5 C6 C9 C10 C11 C13 C14

57 99, 244 68 92 41, 57 68 56, 199 273 187 116, 189, 225, 232 148 73, 151 151 57, 227 148 170 174 170, 209, 210

C15 C25 C26 C28 C31

C41 C43 C44 C48 C49 C50 C51

175, 220 44 238 84 163, 170, 173 170, 173 139 153, 154 138 97, 99, 153, 175, 190 219 180 113 232 178 225, 232 190, 195

61 141 89, 172 193 240 146 166 174 160 113 13 165 176 191

5.512 5.514 5.522 5.5302 5.532 5.542 5.5422 5.5423 5.5521 5.5563 5.64 6.1203 6.375 6.3751

214 215 178 205 208 221, 223 123 239 57 212 76 164, 174 148 46

C33 C36 C38 C39 C40

Tractatus 1.1

150, 236, 243 1.12 233 2 236 2.01 233 2.011 238 2.0141 231 2.021 72 2.0271 231 2.06 141, 236 2.063 236 2.141 226 2.15 230 2.16 230

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Index

a priori, 27, 30, 40, 137, 149, 150, 193, 205, 206, 282 synthetic, 149, 150, 205 aboutness shifter, 14, 21 Alexander, S., 133 analogy black patch, 89, 277 compass needle, 155, 156, 226 Anscombe, E., 90, 98, 235, 257 arithmetic, see mathematics assertion, 55, 56, 86, 87, 89–91, 94–101, 166, 177, 196, 198, 199, 228 sign for, 280 atomism, logical, 36, 37, 76, 133, 134, 148–150, 205, 207, 225, 258, 260, 262

De Hevesy, G., 7 Dedekind, R., 101 deduction, see inference definite description, 241 Demos, W., 144 denotation, 12, 14, 20–23, 71, 159, 179, 213, 277–279, 290 Descartes, R., 32 description definite, 23, 69, 71, 135 disguised, 23, 25, 71, 135, 136, 241 Donnelly, L., 106, 264, 268 Dostoevsky, F., 247, 248, 257 Dummett, M., 67, 91, 98, 99, 112, 156, 242, 244, 257, 258

Baker, G., 112 Bauch, B., 261 Beethoven, L., 20 Berkeley, G., 11, 32 Blackwell, K., 33 Boltzmann, L., 242, 256 Bradley, F., 11–13, 114, 116, 228 Brentano, F., 105 Brusilov, A., 247, 257

Eccles, W., 9, 18 eliminativism, 102 Eliot, T. S., 123, 133, 225 Epimenides, 16 epistemology, 10, 14, 23, 33, 34, 37, 43, 61, 99, 120, 121, 126, 130, 133, 149, 206, 219, 244, 245, 255, 259, 283 Erdmann, J., 65 ethics, 246, 247 meta-ethics, 246 experience, 11, 24, 26, 30, 32, 35–37, 42, 57, 69, 73, 75, 103, 140, 149, 193, 215, 246, 247 external world, 12, 23, 24, 26, 32, 37

Carruthers, P., 232 Chadwick, J., 7 Coffey, P., 65 common sense, 25, 34, 37, 40, 74, 102, 244 concept, 11–14, 21, 81, 113, 117, 225 denoting, 20, 21, 42 conjunction, 44, 45, 51, 145, 161, 164, 166 constructional conjecture strong, 26, 28, 29 weak, 28, 39 context principle, 64, 65, 178, 242 contradiction, 15, 16, 67, 84, 99, 143, 185, 216, 280 Copi, I., 232 Costello, H., 133, 249, 261, 268, 271–274, 292 Davidson, D., 99

fact as a complex, 237, 238 atomic, 133, 137, 140, 143, 145–150, 157, 205, 206, 225, 231, 233, 239, 240 complex, 144–146 conjunctive, 142 disjunctive, 142–145, 214 general, 147–150 logical, 156, 157 negative, 127, 137, 141, 143–147, 154, 176, 206, 217, 236, 240, 276, 277 not complexes, 108, 114, 211 positive, 127, 137, 154, 206, 236, 277 signifying, 210–212, 214, 215

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Index

symbolizing, 93, 115, 170, 171, 175, 182, 189, 190, 209–212, 214–216, 220–222, 225, 255, 279, 280 totality of, 236 fatalism, 247, 248 Fine, K., 142 force assertoric, 87, 89, 94, 97, 100 indicator of, 94, 98 formalism, 43, 47, 91 Frege, G. Begriffsschrift, 15, 55, 61, 67, 86, 87, 91, 95, 112, 178, 213, 262 Grundgesetze, 15, 53, 59, 86, 88, 95, 156, 199, 262 Grundlagen, 64, 178 Posthumous Writings, 59, 60, 67, 91, 93, 97, 98, 100, 101, 258 ‘Der Gedanke’, 60, 72, 229 fundamental thought, 49, 54, 61–63, 79, 100, 143, 150, 228, 256 Gödel, K., 16, 88, 186 incompleteness theorems, 16 Geach, P., 95, 106, 112, 220, 237, 238, 257 Geiger, H., 7 geometry, see mathematics Goldfarb, W., 98, 101, 258 grammar, 23, 68, 78–81, 83–85, 87, 88, 94, 95, 99, 103, 117, 121, 137, 180, 199, 209, 211, 222, 224, 231, 283 Grattan-Guinness, I., 16 Gray’s Elegy argument, 21, 71 Grayling, A., 258 Grelling, K., 185, 194 Griffin, N., 12, 118, 129, 130 Hacker, P., 112 Hamilton, K., 7 Hardy, G., 18 Harwood, Miss, 264, 266, 267, 271, 273 Haworth, W., 7 Hegel, G., 11 Hertz, H., 33, 84, 85, 159, 256 Hesperus, 13, 71 Hicks, D., 30, 31 Hide, O., 11 Hitler, A., 6 Holmes, S., 23 humanism, 247 Hume, D., 31 Husserl, E., 66, 67, 157, 253 idealism, 10–12, 32, 61, 62, 66, 72, 74, 76, 77, 136, 234, 260 identity, 245 Russell’s theory of, 245

incomplete symbols, 22, 42, 51–53, 57, 69, 111, 119, 184, 241, 254, 280 indefinables, 84, 85, 92, 153, 159, 199, 278, 282–289, 292–294 inference, 198 deductive, 50, 61, 62, 234, 277, 283, 286, 291 inductive, 27 logical, 26, 40, 43, 95, 277 infinity, 14, 92, 101, 142, 150, 171, 200, 202, 214, 278, 286 internal relations, 12, 13 intersubjective, 99 intuition, 61, 190, 238 Johnson, W., 249 Jourdain, P., 16, 22, 50, 70, 71, 75, 101, 186, 239, 261, 264, 265 judgment stroke, 87, 88, 94, 96, 98, 99 Kant, I., 11, 61, 64, 205, 234, 261 Kaplan, D., 63 Kenny, A., 238, 258 Kienzler, W., 91 Klagge, J., 85 knowledge, 3, 7, 12, 23, 25, 31, 32, 37, 40, 59, 89, 129, 149, 195, 239, 242, 244, 272 Kraus, K., 250, 256, 257 Kremer, M., 98 Kripke, S., 25 Lamb, H., 7, 8, 18, 47 Landini, G., 118 language and the world, 68, 69, 72, 176 formal, 182, 199, 201 logically perfect, 67, 213, 215, 216, 255 ordinary, 23, 26, 43, 64, 66–68, 83, 87, 93, 119, 156, 175, 185, 190, 211–213, 279, 283 structure of, 65, 66 Lee, D., 233 Lewis, D., 146 Lichtenberg, G., 250 linguistic turn, 64–66 linguistics, 68, 99, 243 Linsky, B., 15 Littlewood, J., 8 Locke, J., 31 logic as a science, 58 as contentless, 58, 60, 61, 234, 242 foundations of, 247, 253 generality of, 57–59, 227, 285 laws of, 58, 59 nature of, 53, 56–58, 121, 253 polyadic, 61, 183

Index  quantified, 182, 214 universality of, 58 logical analysis, 23, 43, 67, 243 connective, 158, 175 constant, 51–54, 57, 61, 79, 143, 145, 254, 285 construction, 26, 27, 34, 39–41, 260 form, 22, 55, 218, 265, 283 positivism, 65 type, 184, 187–189, 197, 199, 200, 273, 278, 284, 285, 294, 295 logicism, 13–15, 157 Long, P., 100, 113 Loos, A., 256 Lotze, H., 65 Marbe, K., 105, 106 Marsden, E., 7 mathematics, 7–10, 14–19, 23, 40–42, 49, 50, 55, 61, 73, 114, 119, 134, 157, 160, 195, 206, 262, 278, 281 arithmetic, 13, 14, 88, 101, 262 geometry, 7, 58, 59, 67 McGuinness, B., 6–8, 53, 59, 202, 246, 267, 268, 270–272 McTaggart, J., 18, 106 meaning, theory of, 140, 141, 223, 244 Meinong, A., 22, 90 mereological sum, 104, 145, 146 meta-ethics, see ethics metalanguage, 168, 201, 203, 259 metaphysical subject, 77, 134 metaphysics, 29, 57, 65, 142–150, 206, 237, 241, 246, 283 Miah, S., 40 Milkov, N., 24 Monk, R., 6 Moore, G., 1, 11–14, 18, 19, 24, 31, 33, 42, 43, 46, 51, 63, 68, 71, 72, 83, 90, 96, 106, 130–133, 136, 162, 173, 194, 204, 206, 226, 249, 250, 264, 271, 274, 303 Morrell, O., 19, 20, 30, 47, 107, 116, 118, 120, 121, 126, 128, 129, 135, 245, 253, 260, 264, 265, 272, 273 Mozart, W., 107 multiple relation theory of judgment, 19, 118, 120, 129, 218, 222 mystical, 216 name, logically proper, 23, 25, 73, 135 natural science, 41, 46, 57, 61, 241, 243, 256, 283 Neckar cube, 239, 243 Nietzsche, F., 250 nonsense, 16, 80, 82, 123, 124, 126–128, 197, 221, 222, 224, 241, 246, 251, 252, 288

Nordmann, A., 85 Occam’s razor, 34, 40, 41, 71, 102, 143 Ogden, C., 47, 195 ontology, 39, 42, 102, 105 paradox, 57, 95, 184–196, 201, 203, 218, 242 Burali-Forti’s, 16 Grelling’s, 185, 186, 194 liar, 218 Russell’s, 15–17, 80, 81, 184, 186, 254 set-theoretic, 242 Zeno’s, 262 Parak, F., 65 Peano, G., 15 perception, 26, 27, 31, 37, 103, 243–245 Perkin, W., 7 Petavel, J., 8 phenomenology, 214, 243 philosophical logic, 17, 135, 241, 247, 254, 257, 262 philosophy of mind, 19 philosophy of psychology, 99, 244, 255, 283 philosophy of value, 245, 246 Phosphorus, see Hesperus physics, 7, 26–30, 33, 34, 36, 38, 39, 45, 46, 48, 57–59, 260 pictorial form, 235 picture theory, 1, 141, 216, 224–227, 230, 231, 235, 242, 245, 256, 262 Pinsent, D., 17, 50, 106, 118, 180, 186, 241, 243, 245, 246, 248, 252, 258, 263, 264, 269, 270 Platonism, 21 possible world, 138, 152, 207, 231, 245 postmodernism, 252 predicate calculus, 22, 196 present King of France, 20, 22, 23, 39, 144, 145 private language argument, 244 proposition as a complex, 51, 109, 110, 118 as a name, 92, 135, 136, 276, 284, 286 asserted, 95–98, 288, 289 atomic, 92, 103, 110, 116, 137, 143–146, 148, 162, 163, 171, 181, 188, 219, 276, 277, 281–284, 287, 293 complex, 171, 280 compound, 22, 146, 148, 174, 284 elementary, 13, 110, 116, 141, 148, 170, 171, 181, 205, 207, 225, 232, 234, 235 false, 96, 97, 119, 120, 132, 155, 228, 277 general, 147, 179, 192, 278, 283, 288, 294 generalized, 191 molecular, 141, 148, 171, 175, 181, 189, 191, 192, 219, 254, 288, 294 primitive, 55, 128, 198, 277, 283, 292

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Index

true, 21, 56, 96, 119, 132, 138, 140 unasserted, 95–97, 288, 289 unity of, 79, 80, 87, 109–111, 114, 115 propositional function, 15, 25, 44, 55, 56, 83, 102, 154, 177–179, 184–186, 188, 190–192, 194, 195, 202, 205 propositional meaning, 137, 141 propositional sign, 171, 210–213, 216, 230 psychologism, 59, 64–66, 100, 178 psychology, 12, 19, 31, 37, 47, 59, 72, 95, 96, 98–101, 118, 135, 181, 228, 243–245, 255, 260, 281, 283, 288 quality, 33, 34, 47, 59, 81, 82, 96, 108, 119 quantifier, 22, 23, 44, 52, 55, 81, 139, 163, 179, 181, 195, 198, 208, 231, 273 universal, 52, 55, 81, 198 Quine, W., 4 Ramsey, F., 18, 188, 213, 214, 221–223, 230, 233, 235 realism, 11, 28, 42, 66, 71–77 reducibility, axiom of, 40, 197, 198, 204, 259 reference and sense, 70, 74, 75 of a name, 73 of a simple symbol, 66, 255 theory of, 74 Reinach, A., 105, 106, 253 religion, 245–247, 257 Rhees, R., 10 Robinson, R., 7 Russell, B. Introduction to Mathematical Philosophy, 134 Our Knowledge of the External World, 133, 258, 260 Principia Mathematica, 26, 40, 47, 49, 50, 55, 56, 82, 97, 127–129, 134, 170, 174, 177, 179, 181, 190, 195–197, 201–204, 213, 218, 259, 261 Principles of Mathematics, 10, 11, 14, 15, 18, 20, 21, 50, 51, 57, 63, 90, 96, 97, 118, 119, 127, 153 Problems of Philosophy, 26, 30, 32 Theory of Knowledge, 110, 116, 120, 122, 219 Rutherford, E., 7, 8 saturated, 85, 110, 113–117, 224, 225, 235, 239 saying and showing, 194, 246 scepticism, 31, 33, 41, 42, 102 Schlegel, F., 250 Schlotter, S., 261 Schopenhauer, A., 10, 250, 256, 257 World as Will and Idea, 10 semantic externalism, 73 semantic theory, 12, 70–72, 87, 215, 224

one-step, 12, 72, 215 two-step, 12, 71, 215 semantic value, 152–154, 215 sensation, 24, 32, 34, 69 sense-data, 24–41, 45, 63, 69, 73–75, 205, 243, 260 sensibilia, 33–36, 40, 43, 63, 76 Sheffer, H., 159, 160, 165, 191, 294 simples, 25, 26, 37, 44, 46, 69, 103, 278 solipsism, 74–77, 244 Spengler, O., 256 Spielt, P., 7 Sraffa, P., 256 Stevens, G., 118 Stumpf, C., 105, 106 subjective, 99, 120 substance, argument for, 72, 136 Sullivan, P., 105, 112 symbolic turn, 63–70, 78, 80, 81, 93, 116, 187, 209, 225, 241, 242, 255 symbolism, 49, 80, 82, 83, 86, 109, 110, 113, 116, 178, 180, 182, 187, 213, 214, 217, 225, 241, 279, 294 tautology, 134, 156, 164, 174, 198, 213, 216, 217, 255 theory of descriptions, 22, 23, 42, 73, 207, 239 theory of types, 15, 19, 80, 82–84, 124, 129, 180, 184–187, 189–197, 199–201, 203, 211, 254, 289 Tolstoy, L., 247, 248, 257 transcendental, 77 deduction, 61, 62, 234 idealism, 61, 62, 234 truth and assertion, 97, 228 concept of, 61, 169, 173, 218 correspondence theory of, 141 identity theory of, 71, 96, 119 laws of, 59 logical, 30, 49, 56, 139, 140, 156, 157, 208, 216 truth-diagrams, 160, 162–164, 173, 174, 180, 181, 204, 208 truth-maker, 132, 144 truth-operation, 165–168, 170, 171, 173 truth-tables, 160, 164, 173 truth-value, 53, 54, 86–94, 139, 158, 160, 165, 166, 173, 254, 261, 277 turnstile, 86, 87, 94, 95, 97, 98 typical ambiguity, 81, 196, 197, 200–202, 211 universals, 25, 73, 235 unsaturated, 85, 110, 113–117, 224, 225, 235, 239 unsayability, 194, 200, 216, 259

Index  value-ranges, 53 verification, 28, 29, 41 vicious circle principle, 185–187, 189, 192 Vienna Circle, 46, 47, 253 von Wright, G., 10 Wahl, R., 118 Ward, J., 15 Weininger, O., 256 Weiss, B., 118 Whitehead, A., 18, 27, 33, 40, 47, 49, 54, 55, 82, 129, 177, 195–197, 200–203, 263, 280 Wittgenstein, H., 8

Wittgenstein, L. Blue Book, 244, 252 Brown Book, 252 Investigations, 244 Notebooks, 45, 53, 61, 110, 111, 193 Notes dictated to Moore, 1, 51, 174, 192, 207, 226 Philosophical Grammar, 252 Philosophical Investigations, 251 Philosophical Remarks, 252 Prototractatus, 164, 192, 194, 210, 249, 250, 253, 266, 269–271 Wrinch, D., 219

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Bibliography Primary sources

Letters Frege (excluding letters to Wittgenstein): G. Gabriel et al. (eds), Philosophical and Mathematical Correspondence (Oxford: Blackwell, 1980) Ramsey: The Modern Research Archive, King’s College, Cambridge Russell: The Bertrand Russell Archive, McMaster University Wittgenstein: Gesamtbriefwechsel, CD-ROM ed. by Brian McGuinness et al. (InteLex); some translated in Brian McGuinness and G. H. von Wright (eds), Cambridge Letters: Correspondence with Russell, Keynes, Moore, Ramsey, and Sraffa (Oxford: Blackwell, 1995) Letters are generally referred to by date, since many of them have also been published in various other places. In footnotes, authors and recipients are referred to by the following abbreviations: OM BR LW

Ottoline Morrell Bertrand Russell Ludwig Wittgenstein

Lecture notes Russell, Philosophy of Mathematics, Cambridge, 1911–12: notes taken by G. E. Moore, Add. 8875, 10/4/3, Cambridge University Library Moore, Psychology, Cambridge, 1911–12: Moore’s notes, Add. 8875, 13/2, Cambridge University Library Ramsey, Foundations of Mathematics, Cambridge, 1925: notes taken by L. H. Thomas, archive 210.4, Special Collections Research Center, North Carolina State University Library Russell, Epistemology, Harvard, 1914: notes taken by Victor Lenzen, Houghton Library, Harvard University Russell, Logic, Harvard, 1914: notes taken by T. S. Eliot, Am. 1691.14, Houghton Library, Harvard University

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