The Rise of Analytic Philosophy, 1879–1930: From Frege to Ramsey [1 ed.] 113801513X, 9781138015135

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THE RISE OF ANALYTIC PHILOSOPHY 1879–1930

In this book Michael Potter offers a fresh and compelling portrait of the birth of modern analytic philosophy, viewed through the lens of a detailed study of the work of the four philosophers who contributed most to shaping it: Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein, and Frank Ramsey. It covers the remarkable period of discovery that began with the publication of Frege’s Begriffsschrift in 1879 and ended with Ramsey’s death in 1930. Potter—one of the most influential scholars of this period in philosophy—presents a deep but accessible account of the break with absolute idealism and neo-Kantianism, and the emergence of new approaches that exploited the newly discovered methods in logic. Like his subjects, Potter focuses principally on philosophical logic, philosophy of mathematics, and metaphysics, but he also discusses epistemology, meta-ethics, and the philosophy of language. The book is an essential starting point for any student attempting to understand the work of Frege, Russell, Wittgenstein, and Ramsey, as well as their interactions and their larger intellectual milieux. It will also be of interest to anyone who wants to cast light on current philosophical problems through a better understanding of their origins. Michael Potter is Professor of Logic at Cambridge University, UK, and a Life Fellow of Fitzwilliam College. His studies in the history of analytic philosophy include Reason’s Nearest Kin (2000) and Wittgenstein’s Notes on Logic (2009). He is also noted for work in the foundations of mathematics, including Set Theory and its Philosophy (2004).

‘The book is an impressive achievement, and it will be an important contribution to the literature on Frege, Russell, Wittgenstein, Ramsey, and the history of early analytic philosophy. I thoroughly enjoyed reading it and learned a lot from it. It is not only a state-of-the-art contribution to scholarship but will also be a valuable textbook for courses on the history of early analytic philosophy, or on the work of one or more of the four philosophers discussed.’ –David G. Stern, University of Iowa ‘This book is a significant contribution to studies in the history of analytic philosophy and will benefit upper-level undergraduates studying this material for the first time, as well as active researchers in the area.’ –James Levine, Trinity College Dublin

THE RISE OF ANALYTIC PHILOSOPHY 1879–1930 From Frege to Ramsey

Michael Potter

First published 2020 by Routledge 52 Vanderbilt Avenue, New York, NY 10017 and by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business c 2020 Michael Potter

The right of Michael Potter to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this title has been requested ISBN: 978-1-138-01513-5 (hbk) ISBN: 978-1-138-01514-2 (pbk) ISBN: 978-1-315-77618-7 (ebk) Typeset in Bembo by codeMantra

CONTENTS

Acknowledgments Introduction

xvi 1

PART I

Frege

5

1

Biography

7

2

Logic before 1879 Stoic logic 10 Aristotelian logic 11 Supposition 12 Kinds of entity 13 Relations 14 Transcendental logic 15 Empiricism and idealism 17 Boole 19

10

3

Begriffsschrift I: Foundations of Logic The aim of the concept-script 21 The judgment stroke 22 The content stroke 24 Axioms, basic laws and rules 25

21

vi Contents

Tone and conceptual content 27 Function and argument 28 Greek and Latin letters 30 4

Begriffsschrift II: Propositional Logic Syntax 32 Semantics 33 Rules and basic laws 35

32

5

Begriffsschrift III: Quantification Syntax 37 Semantics 40 Rules and basic laws 42

37

6

Begriffsschrift IV: Identity Syntax 44 Semantics 44 Basic laws 47

44

7

Begriffsschrift V: The Ancestral Defining the ancestral 48 Logic as non-trivial 50 Impredicativity 50

48

8

Early Philosophy of Logic Psychologism and empiricism 52 Judgment and truth 53 Defining the scope of logic 55 Metalogic 56

52

9

The Hierarchy Levels 59 Applications 60 Concept and object 61 Objecthood and identity 63

59

10 Grundlagen I: The Context Principle Semantic incompleteness 65 The Lockean model 67 Concept and object again 68

65

11 Grundlagen II: Arithmetical Truth Game formalism 70

70

Contents vii

Leibniz 71 The a posteriori 72 Analytic or synthetic? 74 12 Grundlagen III: Numbers Numbers of what? 76 Numbers as non-linguistic 78 Numbers as objects 78 Numbers as self-subsistent 79 Numbers as non-actual 79 Numbers as objective 81 Numbers as abstract 82 Numbers as logical 83

76

13 Grundlagen IV: The Formal Project Numerically definite quantifiers 85 Hume’s Principle 85 Section 64 86 Deriving arithmetic 87 The Julius Caesar problem again 88 Other definitions 89

84

14 Sense and Reference I: Singular Terms Rejection of the Begriffsschrift theory 92 The sense of a singular term 93 Names and descriptions 95 Internalist and externalist conceptions of sense 96

92

15 Sense and Reference II: Sentences The reference of a sentence 100 Thoughts 102 Referring to thoughts 104

100

16 Sense and Reference III: Concept-Words Unsaturated senses 106 Concepts 107 The concept horse 109

106

17 Grundgesetze I: Types Syntax 111 Semantics 112 Wittgenstein’s objection again 114 Truth and taking as true 115

111

viii Contents

18 Grundgesetze II: Extensions Extensions and value-ranges 118 Semantics 119 The contradiction 121 Logical objects again 122

118

19 The Frege–Hilbert Correspondence The Foundations of Geometry 124 Deductivism 125 Uniqueness 127 Existence and consistency 128 Frege’s 1906 definition 130

124

20 Late Writings The structure of the realm of reference 134 Thought and language 135 Sameness of sense 136 Indexicals 137 Idealism 138

133

21 Frege’s Legacy Logic 140 Language 141 Mathematics 143

140

PART II

Russell

145

22 Biography

147

23 Bradley Absolute idealism 151 Russell’s conversion 153

151

24 Geometry Metric geometry 155 First deduction of metric geometry 157 Second deduction of metric geometry 158 Dimension 160 Geometry and physics 160 Deduction of descriptive geometry 161 Public and private space 163

155

Contents ix

25 McTaggart The Tiergarten programme 165 The paradoxes of relativity 166

164

26 German Mathematics The continuum 169 The infinitely large 171 Mathematical education 172

168

27 Whitehead Universal Algebra 174 A single paradox of relativity 176 A wrong turning? 177

174

28 Moore The existential theory of judgment 179 Thing and concept 181 The refutation of idealism 183 Against Kant? 184 Moore’s influence 185

179

29 Leibniz Analyticity and necessity 188 External relations 190 The paradox of relativity resolved 192 Matter and position again 193

188

30 Peano Classes 196 Relations 196 The 1900 draft of the Principles 197 Logic and abstraction 198

195

31 Early Logicism Russellian logicism 201 Characterizing logic 203 Knowledge of logic 205

201

32 Denoting Concepts The variable 207 Indefinite denoting concepts 209 Definite denoting concepts 210 Epistemology 211

207

x Contents

33 The Contradiction The paradoxes 213 Types 214 Reducibility 216

213

34 On Denoting The Gray’s Elegy argument 218 Against Frege? 220 The new theory of denoting phrases 220 Scope 221 Non-entities 223 Significance 224 The substitutional theory 226

218

35 Truth The coherence theory 228 The pragmatist theory 230 The primitivist theory 231 The correspondence theory 232

228

36 Types The hierarchy of propositional functions 235 The axiom of reducibility 238 Identity 240

235

37 Middle Logicism Continuities 243 The vicious circle principle 245 The universality of logic 246 Classes as fictions 247 The regressive method 248 Logical truth 250

243

38 Acquaintance Particulars 251 Complexes 253 Universals 254 The variable 255 The self 256

251

39 Matter On the notion of cause 258 Inference and construction 259

258

Contents xi

The constructional base 261 Public and private space 262 40 Pre-war Judgment The 1906 theory and the Frege point 265 The 1909 theory and the ontology of the Introduction 266 The 1911 theory and the direction problem 267 The 1913 theory and the verb 268 The existential proposal 269 Permutative complexes 270 Judging a nonsense 271

265

41 Facts Complex and fact 275 Negative facts 276 General facts 277 Names and particulars 278 Forms and universals 279

274

42 Late Logicism The subject matter of logic 282 The new theory of types 284 Propositions are not names 286 Bipolarity and truth 287 Logic and necessity 288

282

43 Post-war Judgment Language and vagueness 290 Rejecting the subject 291 Meaning 292 The feeling of belief 294

290

44 Neutral Monism Adopting neutral monism 297 A second wave? 299 The inference from percepts to events 300 Emergence 301

297

45 Russell’s Legacy The demise of absolute idealism 303 The external world programme 304 Logicism 305

303

xii Contents

Naturalism 306 Cambridge analysis 307 Ordinary language philosophy 308 Later reception 312

PART III

Wittgenstein

313

46 Biography

315

47 Facts Facts 319 Atomic facts 321 Molecular facts 322

319

48 Pictures Meaning 323 Truth 325 Relative inexpressibility 325 Possibility 326

323

49 Propositions Logical form 328 Sign and symbol 329 Absolute unsayability 331 Wittgenstein’s context principle 332

328

50 Sense Truth-possibilities 335 Tautology and contradiction 336 Independence 337 The general form of proposition 339

335

51 Wittgenstein’s Concept-Script The N-operation 341 Propositional variables 341 Direct enumeration 342 Propositional functions 343 Formal series 344 Wittgenstein’s vicious circle principle 345

340

Contents xiii

52 Objects Simplicity and elementary propositions 347 The argument for substance 349 Components 351 Forms of object 351 Thought and world 353

347

53 Identity Wittgenstein’s argument 355 Doing without identity 356 Identity and classes 358

355

54 Solipsism The argument for solipsism 360 The thinking subject 361

360

55 Ordinary Language Vagueness 364 Austerity 365 Philosophy 366

364

56 Minds Judgment 368 The empirical subject 369 Logic and psychology 371 Theory of knowledge 372

368

57 Logic The propositions of logic 374 Logic as transcendental 376

374

58 The Metaphysical Subject The eye and the visual field 379 The a priori order 380 Talking of the I 381

378

59 Arithmetic Numbers 384 Dependence on infinity 386

384

60 Science The mesh 388

388

xiv Contents

Principles 389 The independence of the will 390 61 Ethics Moral nonsense 393 The willing subject 394 The lecture on ethics 395 Religion 396

392

62 The Mystical The riddle 398 Quietism 400 The dialectical reading 401 The resolute reading 402 The frame 405 The preface 406

398

63 The Legacy of the Tractatus The independence of elementary propositions 409 The shape of space 410 Atomism 411 The picture theory 412 Philosophy 414

408

PART IV

Ramsey

417

64 Biography

419

65 Truth Propositions 423 Truth 424 Expressing and describing 426 Reverse semantics 428

423

66 Knowledge Reliabilism 431 Induction 433 Human logic 434

431

67 The Foundations of Mathematics I: Types Ramsey’s simple hierarchy 435

435

Contents xv

Saying the unsayable 437 A transcendental argument 438 68 The Foundations of Mathematics II: Logicism Propositional functions in extension 440 Wittgenstein’s objections 442 The demise of the transcendental argument 443

440

69 Universals Incompleteness 445 Narrow and wide ranges 446 Complex universals 447 Unigrade and multigrade terms 448

445

70 Degrees of Belief Logical probabilities 450 Betting 451 Synchronic Dutch books 452 Diachronic Dutch books 454 Degrees of belief and frequencies 455 Partial belief and desire 456

450

71 Facts and Propositions Chicken beliefs 458 Partial belief and chickens 460

458

72 Last Papers Laws 462 The Ramsey test 463 Causation 465 Theories 465

462

73 Ramsey’s Legacy Truth and meaning 468 Logicism 470 Universals 471

468

Bibliography Index

473 493

ACKNOWLEDGMENTS

I am grateful to Andy Beck at Routledge for proposing this project to me; for his gentle persistence in persuading me, despite my initial reluctance, to commit to it; and for his tolerance of its eventual scale. I am fortunate to have received comments on early drafts from Peter Smith, Jim Levine, Steven Methven, Peter Sullivan, Alex Oliver, David Stern, Chloé de Canson, Nicholas Williams and Fiona Doherty. I am grateful for prompt and generous assistance with queries from Jamie Tappenden, Keith Hannabuss, Matthew Wingate, Martin Pilch and Nick Denyer. For their forbearance as well as their feedback I thank audiences in Cambridge, Stirling, Dubrovnik, Vienna, Berlin and Kirchberg, on whom I have tested parts of the book. I am grateful to the Bertrand Russell Archives at McMaster University for permission to quote from previously unpublished letters held there. The notes on further reading at the chapter ends are aimed mainly at students, and therefore sometimes neglect authors from whose work I have benefitted over the years, my debt to whom I express collectively here.

INTRODUCTION

This is an introduction to the work of four philosophers, Frege, Russell, Wittgenstein and Ramsey, in the half-century from 1879. Why them, and why then? I start in 1879 because that was when Frege published Begriffsschrift and thereby inaugurated quantifier-variable logic. This transformed philosophy because it greatly expanded logic’s reach—what thought can achieve unaided—and hence compelled a consequent re-examination of everything philosophers had previously said about the resources that ground our thinking when logic gives out. To explain my choice of philosophers it would then be sufficient to point to their commonality of purpose. All four devoted much of their work to thinking through the philosophical problems which Frege’s discovery of quantifier-variable logic generated. How should judgment be analysed? What categorial distinctions among the components of a judgment does logic require? Can logic tell us anything about the structure of the world? How are logical paradoxes to be solved? How is logic related to mathematics? Because of this thematic unity, it makes sense to study these four philosophers together: one cannot understand Ramsey’s work if one is not aware of the Tractarian and Russellian background which it presupposes, and one cannot understand the Tractatus without first studying the ‘great works of Frege’. I shall try throughout to respect the order in which our four philosophers had their ideas. In Russell’s case, admittedly, his frequent changes of mind make such an approach all but inevitable; but in the others it has been less popular among previous commentators than one might expect. This genetic approach pays particular dividends in the case of the Tractatus, the final version of which was significantly different in character from the book Wittgenstein originally planned. Even in the case of Frege, who is generally supposed to have changed his mind very little, it will emerge that there are significant

2 Introduction

differences—largely ignored by Dummett, for instance—between his views at different stages of his career. Ramsey, of course, did not live long enough to have many changes of mind, but even in his case something of interest will emerge— namely how many of his views were already in place by his first year as a graduate student. Our philosophers’ effect on subsequent developments in philosophy was not wholly straightforward. Frege, for instance, was not widely read until after the Second World War. The effect of Wittgenstein’s Tractatus was limited not only by its opacity but by the rapidity with which its author moved away from some of its central claims. And Ramsey’s influence would no doubt have been much greater if he had not died so young. Of our four, only Russell had a large effect on philosophy straightaway; and even he would surely have been more influential if he had adhered longer to any single philosophical view. Nonetheless, the relevance of all four to current philosophy is unquestionable: Frege’s notion of sense, for instance, is central to modern discussions of the semantics of logic; Russell’s theory of descriptions is still regarded by many as a ‘paradigm of philosophy’; philosophers of language continue to be influenced by Wittgenstein’s picture theory even if many of them do not know what it is; and any modern discussion of the particular/universal distinction must take account of Ramsey’s argument against it. Not just in these cases, though, but also in numerous others we shall find as we proceed that popular accounts of their work often misstate their conclusions and misrepresent their philosophical insights. Our four philosophers played a major part in the birth of what became known as the ‘analytic’ method, in philosophy. This name—bestowed, it seems, by Russell in a 1922 book review—is not altogether happy, and it is now quite hard to identify exactly what the analytic method consists in. When Russell wrote, he had in mind an approach which aims by analysis to identify the fundamental concepts of some already existing sphere of discourse, and in the process to provide the vague meanings of some of the words we ordinarily use with more precise replacements. And when others (e.g. Wisdom 1931, Stebbing 1932) took up the phrase, this was what they meant. But much that is nowadays counted as ‘analytic philosophy’ does not employ the analytic method at all. So the phrase now expresses (unsurprisingly, perhaps, given how many philosophers there now are who self-describe as analytic) at best a family-resemblance concept. And it would be easy to become embroiled in turf wars over whether some philosopher or other—the later Wittgenstein, perhaps, or Rorty—should be counted as analytic or not. There have been some, indeed, who have met this difficulty by treating ‘analytic’ philosophy as if it were somehow merely the opposite of the ‘continental’ philosophy which proceeded in parallel with it for most of the 20th century. If it is indisputable that the ideas to be discussed in this book were central to the emergence of the analytic method, it would be absurd to suggest that they were entirely responsible for it. Any comprehensive history would of course have

Introduction 3

a much larger cast than I shall discuss here. It would hardly be possible in a single volume such as this to explain in any detail what each of them thought and why. It therefore seems preferable to choose only a few and follow through their ideas in detail. I shall in any case not be much concerned in this book with the extent to which our four philosophers were responsible for founding analytic philosophy. Why they were influential and others not is a large question, on which I have little to say. (And I have even less to say about why they had much less influence on philosophy in continental Europe than they did in English-speaking countries.) When it comes to influences in the other direction, however, the reader should be warned that none of our four was especially careful about citing their sources. Wittgenstein famously boasted in the Tractatus that he was not concerned with such trivia, and Frege, in particular, seems to have had difficulty giving others due credit for their ideas. (Other philosophers have displayed similar traits: Heidegger, for instance, was not above claiming specious novelty for his work.) Russell was the only one of our four who devoted any significant effort to intellectual autobiography, and his attempts in this direction were marred by a liking for neat binaries and for overstating for dramatic effect the nature of the revolution to which he contributed. Sometimes, therefore, commentators who relied too much on our philosophers’ own writings have made it seem as if analytic philosophy was born out of nothing in 1879. That is of course false: in a loose sense the analytic method can be traced back to the ancient Greeks; and it would certainly be possible to make a case for more direct precursors in the mid-19th century, such as Bolzano (in Frege’s case) or Sidgwick (in Russell’s). It would be an exaggeration, then, but an understandable one, to say that analytic philosophy was born in 1879. The problems that concerned our four philosophers flowed directly from the new quantified logic that Frege devised in Begriffsschrift. Without making any claim to completeness I hope nonetheless to be able here to convey a sense of the exhilarating progress they made, and of the extent to which modern analytic philosophy is in their debt.

Notation The reader is assumed to be familiar with the elements of formal logic. I use the following logical notations: ∼ negation ∧ conjunction ∨ inclusive disjunction ⊃ material conditional ≡ material biconditional ∀ universal quantifier ∃ existential quantifier

4 Introduction

I also use the following notations from the theory of classes: {x : φx} the class of all x such that φx ∈ class membership ∅ the empty class

Further reading On the development of Frege’s ideas see Bell (1981). On the birth of analytic philosophy see Potter (2008), Dummett (1993a) and Frost-Arnold (2017). For the history of analytic philosophy more widely see the collection assembled by Beaney (2013).

PART I

Frege

1 BIOGRAPHY

Gottlob Frege was born in 1848 in Wismar, a Baltic port in what was then Mecklenburg. His father, the headmaster of a girls’ secondary school, died when he was 18 and his mother ran the school for some years thereafter. He went to Jena for his undergraduate degree, seemingly at the encouragement of a young mathematics teacher called Leo Sachse who had studied there a few years earlier. Frege majored in mathematics, but also attended Kuno Fischer’s Kant course. He studied geometry as a doctoral student at Göttingen for two years (during which time he attended Lotze’s lectures on the philosophy of religion) before returning to Jena for the remainder of his professional career. Although many of his publications were in philosophy, his teaching appointment at Jena was in the mathematics department. Frege’s first major publication, in 1879, was Begriffsschrift (‘concept-script’), the book which began the modern era in logic. Although its reviews were by no means uniformly negative, he was disappointed by its reception. By then he was writing another book which continued the formal project initiated in Begriffsschrift of establishing a logical basis for arithmetic, but he was advised by a friend to postpone its publication and focus instead on explaining the motivation. The result was his 1884 book Die Grundlagen der Arithmetik (‘The Foundations of Arithmetic’), in which he sought to make it plausible that arithmetic is derivable from logic. The book largely avoids the pedantic and uncharitable criticisms of others that mar his later work, and is one of the most readable of the classics of philosophy. This was evidently a relatively happy time in Frege’s life: he was recently married and now had a secure (although not well paid) job at Jena. Although he published nothing between 1885 and 1891, he made significant changes to his

8 Frege

semantic theory, resulting in three papers in 1891–2: ‘On sense and reference’, in particular, remains one of the most widely read papers in analytic philosophy. In 1893 there appeared the first volume of Grundgesetze der Arithmetik (‘Basic Laws of Arithmetic’), intended as the culmination of the Begriffsschrift project. Frege said there (I, ix) that the delay in its publication was due partly to these changes, which had led him to put aside an almost complete manuscript, but he also mentioned once again his disappointment at the failure of his earlier work to have the effect it merited. In 1902, with the second volume of Grundgesetze already in press, Frege received a now famous letter from Russell informing him that the formal system on which his whole account of arithmetic was based contained a contradiction. He went ahead with publication, but added a short appendix proposing an amendment to Basic Law V (the axiom which gave rise to the contradiction). The second volume ends abruptly, and he evidently intended a third, which never appeared. After this, Frege took sick leave with what was probably depression. Some have speculated that the reason was the discovery that his life’s work was flawed— he realized quite soon (certainly by 1906) that his amended system is insufficient to ground arithmetic—but it is at least as likely that the reason was the death of his wife, after a long illness, in 1904. Wittgenstein said much later that he thought they had had children who died in infancy, but the records do not confirm this. Soon after his wife’s death, however, Frege took in an orphan called Alfred, a distant relative, whom he eventually adopted as his son. (He also gave financial support to Alfred’s sister, Toni.) After the collapse of his attempt to derive arithmetic from logic, Frege continued to offer a lecture course on his concept-script at Jena each year (generally to very small audiences), but since he now omitted any mention of value-ranges (the notion which, as we shall see, was responsible for the contradiction), the system he taught had no hope of grounding logicism. Carnap, who attended the course in 1910, recalled an extremely shy man who addressed his lectures almost entirely to the blackboard. Between 1911 and 1913 Frege received annual visits from Wittgenstein. These visits were formative for the young man, who held Frege in reverence for the rest of his life. They continued to correspond until Wittgenstein had him sent a typescript of the Tractatus in December 1918: Frege’s lack of comprehension of it was a bitter disappointment to him. After missing two semesters in 1913–14 and three in 1917–18 through further ill health, Frege retired from his university position in 1918 and, aided by a generous gift from Wittgenstein, moved to Bad Kleinen, close to his Mecklenburg birthplace. He had for many years been planning a book on the philosophy of logic—to be called Logische Untersuchungen (‘Logical Investigations’)— and in retirement he published several chapters of it. He also continued to look for an account of mathematical knowledge, but he now speculated that it has a geometrical rather than a logical source.

Biography 9

Frege had previously counted himself a liberal, but in the last year of his life he expressed in his diary both disenchantment with democracy and intolerance of various other groups such as Catholics, Jews and the French. Although he explicitly disapproved of Hitler’s recent failed coup, his desire for a leader with ‘youthful vigour to sweep away the people’ (Mendelsohn 1996, 324) makes for uncomfortable reading. These diary entries were omitted from the collected edition of his writings as not being philosophical in character. Dummett later expressed regret at their omission, saying that by reading them he had ‘learned something about human beings which I should be sorry not to know’ (1973, xii). When Frege died in 1925, he left his unpublished papers to his son, telling him that ‘even if all is not gold, there is gold in them’ (PW , ix). Alfred deposited the material at the University of Münster in 1935, but it was apparently destroyed in its entirety by an allied bombing raid in March 1945: although it matters little now, the raid was carried out not by the Americans, as Dummett (1973, 662) asserted, but by 175 aircraft of RAF Bomber Command. Fortunately for posterity, much of the content survives in copies which the editors of the Nachlass had made before the war.

References Frege’s main works will be referred to here by the following abbreviations: Bs Begriffsschrift (1879) Gl Die Grundlagen der Arithmetik (1884) Gg Grundgesetze der Arithmetik (2 vols, 1893–1903) His other surviving writings are translated in the following volumes: CN Conceptual Notation and Related Articles CP Collected Papers PMC Philosophical and Mathematical Correspondence PW Posthumous Writings L13 Lectures on Logic

Further reading Kreiser’s (2001) biography contains a great deal of circumstantial information but little insight. The fate of Frege’s Nachlass at Münster is described by Wehmeier & am Busch (2005). Of the single-volume introductions to his philosophy Kenny (1995) is the best for beginners; Kanterian (2012) focuses mainly on Begriffsschrift, Noonan (2001) on his later semantics. For an overview of Frege’s logic see Kneale & Kneale (1962, ch. 8).

2 LOGIC BEFORE 1879

‘Logic is an old subject, and since 1879 it has been a great one.’ (Quine 1952, vii) That the publication of Begriffsschrift was a key moment in the history of logic can hardly be denied, but if we are to appreciate Frege’s achievement, we need to survey previous developments. Modern logic is so much in his shadow, indeed, that it is something of a challenge to present these as more than a series of missed opportunities to grasp Fregean insights.

Stoic logic There were ancient traditions of Indian, Arabic and Chinese logic; but, for better or worse, these other traditions had rather little influence on European logic, which emerged from two distinct (and competing) schools in ancient Greece. The first of these schools, which was founded by Euclid of Megara (who studied under Socrates and was active in the early 4th century BC) before being continued by the Stoics, focused on propositional logic. Philo of Megara introduced the material conditional, nowadays written as p ⊃ q, around 300BC, and came tantalizingly close to the modern truth-table for it when he observed that there are three ways that it can be true and one that it can be false (Sextus Empiricus 2005, bk 2, §113). He argued against the attempt by his teacher, Diodorus Cronus, to treat the conditional as a tensed generalization—one that ‘neither could nor can begin with a truth and end with a falsehood’. Later Chrysippus recommended something closer to what we now call the strict conditional (p ⊃ q). Philo may have been an inch away from the modern conception of a truth-function, but his successors showed little recognition of its significance. Although Chrysippus distinguished between atomic and molecular propositions and included among the logical connectives the material conditional, conjunction

Logic before 1879 11

and (exclusive) disjunction (which are truth-functions), he also included causation (which is not). Nonetheless, the Stoics did at least systematize the rules of inference (Modus Ponens, Modus Tollens, etc.) now familiar in introductory courses on propositional logic. Later logicians called propositions formed by means of the connectives just mentioned ‘hypothetical propositions’, and the inferences which they license ‘hypothetical syllogisms’. The details of the Stoic theory are somewhat obscured, however, by the paucity of the available texts. Only fragments of Chrysippus’s logical writings survive, and he is nowadays famous for little more than having died laughing at one of his own jokes.

Aristotelian logic A second strand of logic began in the 4th century BC with Aristotle’s Prior Analytics, which studied the logical relationships between four kinds of proposition known as ‘categorical’. (A) (E) (I) (O)

Every Φ is a Ψ No Φ is a Ψ Some Φ is a Ψ Some Φ is not a Ψ

It was Aristotle who initiated the use of a letter of the (Greek) alphabet ‘schematically’, i.e. to stand for an unspecified piece of language of some appropriate grammatical type. We can treat the four Aristotelian forms as binary operators, written A(Φ, Ψ ), E(Φ, Ψ ), I(Φ, Ψ ) and O(Φ, Ψ ) respectively. So, for instance, the argument

∴

No Ψ is a Φ Some X is a Ψ Some X is not a Φ

would be formalized as

∴

E(Ψ, Φ) I(X, Ψ ) O(X, Φ)

(major premiss) (minor premiss) (conclusion)

Aristotle classified the various syllogisms and showed that all the valid ones follow from a small number of syllogisms and other rules. From the outset Aristotelian logic was seen more as a competitor than as a supplement to Stoic logic, and there emerged among the peripatetics an ambition—towards which, however, they made little progress—of assimilating the latter to the former. Because of this failure, logic continued right up to the modern era to be seen as the study of two distinct kinds of syllogism, hypothetical and categorical, neither of which could capture all the complexity of actual arguments in fields such as mathematics.

12 Frege

In the following centuries the pace of development of Aristotle’s logic was slow. When the study of logic resumed in Europe after the Dark Ages, progress was further hampered by the fact that the Prior Analytics had been lost: its rediscovery in the 12th century marked the transition from what came to be known as the ‘old’ to the ‘new’ logic. One purely notational device introduced in the Middle Ages was the labelling of the four forms A, E, I and O. The valid syllogisms were then given names which encoded (by means of the vowels) which forms they contained and (by means of the consonants) how they could be derived from other forms regarded as more basic. The syllogism mentioned earlier was named ferio, for instance, the vowels indicating that the major premiss, minor premiss and conclusion are of the forms E, I and O respectively. This encoding then permitted in turn a compact representation of the valid forms by means of a mnemonic. Here is one variant, thought to have been devised by William of Sherwood in the 13th century. Barbara celarent darii ferio baralipton Celantes dabitis fapesmo frisesomorum; Cesare campestres festino baroco; darapti Felapton disamis datisi bocardo ferison. The practical utility of this piece of doggerel lay in the requirement for mediaeval undergraduates to learn to distinguish valid from invalid syllogisms as part of the trivium (grammar, logic and rhetoric) that formed the core of their education.

Supposition Besides inventing a mnemonic, William of Sherwood initiated, more significantly, the study of ‘supposition’ (contextual reference). This arose from a tension in Aristotle’s logic. In his early work, De Interpretatione, Aristotle took as his basic sentence form the combination of a name (e.g. ‘Socrates’) and a verb (e.g. ‘is rational’), whereas in his later work what he took as basic were relations between two common nouns (‘man’, ‘rational animal’, etc.). In one respect this made logic simpler, since all the letters now stood for items of a single grammatical type, but it also had the less fortunate consequence that, because ‘Socrates’ is not a common noun, the argument

∴

All men are rational animals Socrates is a man Socrates is a rational animal

no longer counted straightforwardly as an Aristotelian syllogism, but had to be shoehorned into the framework by making names a special case of common nouns, in effect treating ‘Socrates’ as short for ‘everything that is Socrates’. This made it seem natural to treat the semantic contribution of a quantifier phrase as uniform with that of a name. Thus was born the mediaeval theory according to

Logic before 1879 13

which predicates plurally supposit (i.e. refer to) worldly entities, ‘every’ and ‘some’ specify the mode of supposition in question, and the ambiguity in ‘Every man sees some donkey’ is due to a difference in the mode of supposition of ‘some donkey’. Nowadays, of course, we would call this a scope ambiguity. William of Sherwood came within a whisker of this explanation when he suggested that the meaning depends on the order in which the phrases ‘arrive’ in the sentence, but later mediaeval writers did not follow his lead, preferring instead to devise implausibly elaborate theories of ‘confused supposition’. This was an instance of what Peirce (quoted in Geach 1962, 104) called the ‘damnable particularity’ of the mediaeval logicians. (Arabic logicians exemplified a similar pattern, studying what they called ‘unusual syllogisms’, but doing little more than enumerate valid forms.)

Kinds of entity One particularly troublesome aspect of the theory of supposition was its treatment of empty terms. Following the assimilation of quantifier phrases to names, empty names such as ‘Odysseus’ were treated on a par with vacuous phrases such as ‘every unicorn’. In order to explain how sentences involving empty terms can be true, logicians appealed to Aquinas’s distinction between a thin notion of being that applies to everything—memorably described by Austin (1962, 68n) as ‘like breathing, only quieter’—and a more metaphysically demanding feature (by the 19th century often called ‘existence’) which only actual (i.e. spatio-temporal, or at least temporal) entities possess. If such a treatment is to have any hope of coherence, however, we must recognize that existence is sui generis and not what Kant called a ‘real property’. Take any subject you please, for example, Julius Caesar. Draw up a list of all the predicates which may be thought to belong to him. . . . You will quickly see that he can either exist with all these determinations, or not exist at all. The being who gave existence to the world . . . could know every single one of these predicates without exception and yet still be able to regard him as a merely possible thing which, in the absence of that Being’s decision to create him, would not exist. (1979, 57–61) Kant then objected to the ontological argument on the ground that only real properties were candidates to be perfections. This was not the only point, though, at which logic was uneasily related to metaphysics. Aristotle had distinguished in the Categories between two worldly relations that might be represented in language by a predication: this raven is an instance of ravenhood, its blackness an instance of blackness; but the blackness inheres in, and is not an instance of, the raven. These two relations, instantiation and inherence, then generated corresponding binary divisions among entities: a division between particulars and the universals they instantiate; and another between substances and the attributes that inhere in them. The two divisions do not coincide:

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this raven’s blackness, for instance, is an attribute of it, but is nonetheless a particular, because there is nothing else that is an instance of it. Some disputed whether it made sense to posit ‘insubstantial’ entities—entities without qualities whose being is constituted not by their own nature but by their relations to others. Instantiation may be iterated—this raven instantiates the universal raven, which in turn instantiates the family corvid—but it was controversial among mediaeval and early modern logicians whether inherence iterates similarly. Locke (Essay bk 2, ch. 13, §19–20), on the other hand, preferred to ridicule the whole notion that the relation of inherence has a useful role in philosophy. Among attributes some philosophers drew a further distinction between ‘qualities’, which are internal features of a substance essential to its nature, and ‘accidents’, which are not.

Relations Into the early modern era most logicians continued to accept Aristotle’s claim in the Categories that relations are reducible to attributes of their relata: that John is heavier than Andrew, for instance, reduces to John’s weight of twelve stones and Andrew’s of eleven. Apparent counterexamples to this paradigm were dismissed as exceptional (e.g. because connected with the Christian doctrine of the Trinity); and no one could explain the non-exceptional cases, such as the reasoning about spatial relations to be found in Euclid, in syllogistic terms. Thus resulted the rationalist doctrine that reason can supply us with non-trivial (sc. non-syllogistic) knowledge. Of all the early modern logicians, Leibniz came closest to recognizing the logical significance of relations. He still regarded them as unreal, but he held that this was not a bar to analysing them logically. He was, for instance, aware (1981, 479f.) of such valid non-syllogistic arguments as Jesus Christ is God So the mother of Jesus Christ is the mother of God. He also noted with approval Jungius’s 17th-century attempts to analyse such arguments and made significant progress of his own in that direction. In the process he drew an important distinction between two goals, a precise universal language for expressing thoughts (characteristica universalis) and a mechanical method for reasoning with the thoughts thus expressed (calculus ratiocinator). Unfortunately, though, he published only the statement of intent, not his attempts at fulfilling it, and his regard for Jungius’s work was not widely shared. Reid, for instance, was content to regard the non-syllogistic argument Solomon is the parent of David So David is the child of Solomon as material, not formal, so that its validity was not a matter that logic should be expected to explain. It was not until de Morgan in the 1860s that the opposing view was taken seriously once more. He argued (1966, 230) that the relationship

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between a relation and its inverse is a matter of logic, and hence wholly formal; what is material, he suggested, is merely that the inverse of the parent relation happens to be spelt ‘child’ in English.

Transcendental logic The Renaissance was not kind to logic: after the Black Death it came to be seen as trivial—the word deriving from the trivium)—and the progress made by mediaeval logicians was largely forgotten. So it was perhaps understandable, although inaccurate, that Kant should have said in the Critique (Bviii), ‘Since Aristotle . . . logic has not been able to advance a single step, and is thus to all appearance a closed and completed body of doctrine.’ Kant agreed with his predecessors about the triviality of syllogistic logic—‘pure general logic’ as he called it—and used the word ‘analytic’ for the empty truths that it delivers. Kant made little attempt to innovate in his account of this part of logic, adhering to the traditional categorization of judgments as universal, particular or singular; affirmative, negative or infinite; categorical, hypothetical or disjunctive; problematic, assertoric or apodictic. He also explicitly rejected, for reasons that are obscure, the peripatetic proposal of systematically reducing hypothetical to categorical judgments (Logic, §24). He was already aware of the possibility of representing syllogisms by means of what are effectively Venn diagrams (some of which he wrote in one of the logic books in his library), and he took this as an indication that pure general logic is wholly formal and abstracts entirely away from any reference to objects. (He liked to put this point by saying that it is a canon, not an organon.) Pure general logic is therefore wholly universal, since it does not engage at all with the nature of objects but treats solely of relations between concepts. One consequence repeatedly emphasized by Kant is that pure general logic is trivial, ‘concise and dry’ (A54=B78). It cannot advance knowledge, but is, in his jargon, explicative, not ampliative (A7=B11). Kant also held that pure general logic ‘has no empirical principles, thus it draws nothing from psychology’ (A54=B78), but it is nonetheless constitutive of the understanding, not normative: in its exercise error is literally unthinkable; the understanding by itself cannot err, ‘since, if it acts only according to its own laws, the effect (the judgment) must necessarily be in conformity with these laws’ (A294=B350). This claim would no doubt be wildly implausible if it were not for the triviality just mentioned: his thought seems to have been that if we make a syllogistic inference, we cannot err as we have not really said anything new. We are only, as Frege nicely put it, ‘taking out of the box again what we have just put into it’ (Gl, §88). Kant’s chief logical innovation was that he responded to the triviality of formal logic by proposing to supplement it with a further subject called transcendental logic, which takes truth rather than mere consistency as its study. He claimed that ‘a sufficient and at the same time general criterion of truth cannot possibly be given’ (A59=B83), although his reasons for maintaining this are not easy to decode. Instead, he suggested, we should take as given our ability to make true judgments

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about the world and ask what is required in order that this should be possible. What results is transcendental logic, the ‘logic of truth’ (B87), which deals not merely with containment relations between concepts but with judgments about objects. This makes transcendental logic in one sense more restricted in scope than pure general logic, but it is nonetheless still in another sense wholly general: there are ‘special’ logics that deal with objects of particular kinds, but transcendental logic, although a special logic in this sense (since it deals with objects), is the most general such (since it deals solely with the features possessed by any object we can think about at all). It ‘concerns itself with the laws of understanding and of reason solely in so far as they relate a priori to objects’ (A57=B82), and is therefore applicable to any object which I am capable of cognizing. It has been controversial among commentators whether this should be regarded as a genuine limitation or not. What would it even mean for logic to apply to the unthinkable? Transcendental logic is characterized, then, on the one hand by its concern with truth, on the other by its employment of the general notion of an object. The link between the two is provided by the unifying concepts Kant called the categories—concepts which are essentially involved in any judgment we might make that aspires to be about an object. Without the categories, I may have an intuition, but cannot make from it a judgment capable of truth or falsity, because I cannot manufacture the unity that is required if it is to be an intuition of an object. What is required for this is that the intuition should be subject to a mode of combination called synthesis, which involves the application of the categories. ‘Thoughts without intuitions are empty,’ as the slogan goes. (B75). The centrepiece of the Transcendental Analytic of the Critique was a ‘transcendental deduction’ that aimed to demonstrate the applicability of the categories to our intuitions in the manner just described. Kant distinguished between the ‘transcendent’, which is wholly beyond experience, and the ‘transcendental’, which, although not strictly part of experience, is a structural feature immanent in it (A296=B353). He pioneered the use of a ‘transcendental argument’, which takes as its premiss not the fact of some particular experience but its mere possibility, and infers from this some structural condition on which this possibility depends. The transcendental deduction of the categories is only the most celebrated of a number of arguments in the Critique that take this form. ‘Some logicians’, Kant said in his Logic, presuppose psychological principles in logic. But to bring such principles into logic is as absurd as taking morality from life. If we took the principles from psychology, . . . we would merely see how thinking occurs . . . ; this would therefore lead to the cognition of merely contingent laws. In logic, however, the question is not one of contingent but of necessary rules, not how we think, but how we ought to think. . . . Logic shall teach us the right use of the understanding, i.e. the one that agrees with itself. (16)

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Was he here talking about transcendental logic or pure general logic? Presumably the former, since the contrast he was drawing between normative and descriptive would be difficult to fit with his conception of pure general logic as so trivial that error in its employment is impossible. He went to some lengths to try to show (in §§18–19 of the transcendental deduction of the categories) that transcendental logic is not subjective, because the categories are principles of synthesis imposed on any intelligent being by the very nature of discursive thought, rather than merely parochial features of the way we humans happen to think. The categories cannot, he claimed, be obtained by starting from our ideas, since that would only result in more that is subjective. He thus contrasted the subjective validity of a Humean association of ideas with the objective (or, at least, inter-subjective) validity such ideas acquire when transformed by means of the categories. Precisely because we lack a general criterion of truth, he claimed, we need to look to the categories to supply the required degree of objectivity to judgment. So when Kant said (B131f) that any genuine judgment (i.e. any judgment in which the categories have been correctly applied) depends on its being accompanied by the ‘I think’, he meant not an empirical, psychological unity but the abstract unity of reason as such. Instead of ‘I think’, therefore, he might have been better to say ‘it is thought’ (with an impersonal ‘it’).

Empiricism and idealism In the early 19th century Kant’s anti-psychologistic conception of logic gained followers, the most notable of whom was Herbart. It is ‘necessary in logic’, he said, ‘to ignore everything psychological’ (1813, §34). Thus arose the view that what is relevant to logic is not the act of judgment, which is psychological, but the content judged, which is abstracted entirely from the psychological question of how we grasp it. This was a reversal of Kant’s view, which had been that the content is psychological, and that the judgment attains objectivity only by virtue of the structuring role of the categories. There was also renewed pressure to extend the reach of formal logic beyond the classical syllogisms. One difficulty, though, was to find a new criterion of logicality that would delineate this wider study. In the Wissenschaftslehre (1837) Bolzano took the entities of which truth is predicated to be not propositions in the subjective sense but ‘propositions-in-themselves’—objective entities existing independent of our apprehension. Where the proposition as apprehended has mental entities called ideas as constituents, the proposition-in-itself has instead ‘ideas-in-themselves’. He supposed a proposition-in-itself to have a structure permitting the substitution within it of one idea-in-itself for another. He defined a proposition-in-itself to be logical if its truth value is invariant under substitutions among any of its non-logical referring constituents. (He held that any proposition containing a non-referring constituent is false, which is why he made the restriction to referring constituents in the permitted substitutions.)

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This was a significant advance on Kant, but it did not on its own settle the scope of logic, because different decisions as to which constituents are logical (and hence not to be substituted) make different propositions count as logical. Moreover, his book was intimidatingly long, dry, and little read. His definition therefore received little attention until it was rediscovered (with substitution conceived of at the level of sentences rather than propositions-in-themselves) by Tarski (1936). There had in any case been (at least in Germany) a reaction against postKantian idealism towards strict empiricism, and a consequent rejection of the anti-psychologistic conception of logic advocated by Herbart and Bolzano. This reaction originated in the late 18th century, when Cabanis claimed that the brain digests impressions and secretes thoughts just as the intestine does with food. The view was then popularized by Czolbe (1855), who argued that all thinking is explicable in terms of the physical properties of the brain, and Moleschott (1852), who, in allusion to the role of phosphorus in brain activity, adopted the memorable slogan, ‘No phosphorus, no thought.’ The advocate of psychologism most often mentioned nowadays is Benno Erdmann (1892), although his fame resides principally in the attacks from Frege (Gg, I, xvi–xxiii) and Husserl (1900– 01, I, §§40–41) that he attracted. Even among empiricists there were some, such as Mill, who granted that logic has a normative aspect that psychology lacks. So far as it is a science at all, [logic] is a part, or branch of Psychology; differing from it, on the one hand, as a part differs from the whole, and on the other, as an Art differs from Science. (1865, II, 145–6) The difficulty with this, though, is that it leaves the non-psychological part of logic—the part which Mill called an art, not a science—ungrounded. Inferences that depend only on the syllogism are in his terms merely verbal; but it is hard to see what resources he had to explain the normativity of non-syllogistic inferences, and his writings show little sign that he was even aware of the problem. By the 1870s Czolbe’s radical materialism in turn was in decline. Liebmann (a colleague of Frege’s at Jena) pointed out that it is absurd to think logic is beholden to physical properties of the brain. ‘What’, he asked, ‘have the protein, potash and phosphorus in brain material . . . to do with logic?’ (1876, 476) Increasingly popular now was a new and (in Germany, at least) more or less Kantian idealism: Liebmann (1865, 15) adopted the slogan, ‘Back to Kant!’ It was a common trope of late 19th-century philosophy that the psychological aspect of the study of thinking is descriptive; the study becomes normative only when we bring in truth. When Windelband coined the term ‘truth-value’ in 1882, his purpose was to emphasize the parallel, in point of normativity, between the goals of truth and goodness (see his 1924, 31). In Germany the most influential of the new idealists was probably Lotze, who popularized a broadly Kantian conception of transcendental logic as distinguished

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from psychology by its concern with truth. ‘Logic’, he said, ‘cannot derive any serious advantage from a discussion of the conditions under which thought as a psychical process comes about.’ (1884, 467) Lotze was Herbart’s successor as professor of philosophy in Göttingen and inherited his distinction between the subjective act of judgment and the objective content judged, but he did not entirely share Kant’s (realist) sense of the gulf between the subjectivity of our ideas and the objectivity that the demands of truth impose on us. Although he distinguished between mere ideas and what he called ‘auxiliary thoughts’ (Nebengedanken), allocating to the latter the role that in Kant’s account was played by the categories, he tended to regard them as essentially psychological in origin and did not explain how they could give our thoughts objective validity.

Boole 1879 is often regarded as the date when modern logic was born. The only other date that could with any plausibility be regarded as a competitor is 1847, when Boole published The Mathematical Analysis of Logic. Boole proposed a single algebraic calculus (nowadays known as Boolean algebra) with two interpretations, one applying to predicates (which he called ‘primary propositions’), the other to sentences (‘secondary propositions’). The formal aspects of this material were in fact largely a rediscovery of Leibniz’s unpublished work of two centuries earlier. Boole’s work was an early example of the modern axiomatic method, whereby intellectual economy is achieved by studying a set of axioms in which the primitive terms have multiple interpretations. Theorems proved from the axioms then have, correspondingly, multiple interpretations as truths concerning the various interpretations. Thus a single theorem of Boolean algebra can be interpreted as a truth of categorical or of hypothetical logic. From a philosophical point of view, however, his work was more problematic. One concern, noted by Jevons (1864, §§197–202), was that he started from formulae with a logical interpretation, manipulated these formulae as if they were about quantities, then interpreted the end-result as logical once more, but gave no justification for doing this except that it seemed to work. Note, too, that although he had devised an algebraic calculus of considerable sophistication for solving problems in categorical and hypothetical logic, it fell short of unifying the Aristotelian and Stoic traditions. One continuing barrier to this unification was resistance to Herbart’s actcontent distinction. Even in the late 19th century, logic textbooks typically had separate chapters on ‘categorical judgment’, ‘hypothetical judgment’, ‘negative judgment’, etc. If there was to be any prospect of unifying the two approaches, logicians had to allow for a single act of judgment that relates to a complex content—for instance, a Megarian material conditional with Aristotelian categorical propositions as components. Even then, however, there remained the question of the relationship between ‘primary’ and ‘secondary’ propositions. Boole made the latter a special case of

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the former by supposing there to be a hidden variable ranging over the times at which the proposition is true: that is to say, he interpreted ‘If p then q’ to mean ‘If p is true at time t, then q is true at time t’. Yet even if this were not problematic in itself, it would be of no help with the problem of integrating the two sorts of reasoning. As he himself acknowledged, we can infer from Either every inhabitant is European or every inhabitant is Asiatic to Every inhabitant is either European or Asiatic, but his account was powerless to explain why.

Further reading The account of pre-1879 logic in Kneale & Kneale (1962, chs 1–6) remains excellent, but on mediaeval developments in Aristotelian logic see also Bonevac (2012). On deduction in Greek mathematics see Netz (1999, 197). On whether 1879 really marks the beginning of modern logic, see Boolos (1993a). Skorupski (1998) discusses Mill’s account of logic. On the generality of Kant’s transcendental logic see Tolley (2012). The mediaeval notion of supposition is discussed by Geach (1962); Quine’s notion of ‘divided reference’ (1960, §19) is a modern descendant.

3 BEGRIFFSSCHRIFT I: FOUNDATIONS OF LOGIC

Begriffsschrift is one of the most remarkable books in the history of human thought. The fact that I shall devote the next five chapters to discussing it is a tribute to its originality. Its full title may be translated ‘Concept-Script: A formula language, modelled on that of arithmetic, for pure thought’, but I shall here call it by the shortened German title, reserving the English ‘concept-script’ for the particular formula language which it expounds.

The aim of the concept-script Although Frege intended his concept-script to be applicable to any scientific discourse, its origins lay in his interest in the foundations of arithmetic (Bs, Preface). When he attended Fischer’s Kant lectures, he no doubt learnt of the central role geometry and arithmetic played in the Critique. Kant argued that the axioms of Euclidean geometry are synthetic a priori truths about space as it appears to us, before offering a broadly parallel account of arithmetic; the principal difference between the two cases was that he took each arithmetical equation (2 × 2 = 4, 7 + 5 = 12, etc.) to be a primitive truth in its own right, so that the system of axioms for arithmetic, in contrast to that of geometry, is effectively infinite. The axiomatic method is sometimes called ‘Euclidean’, but Euclid’s own axiomatization of geometry in the Elements was insufficient to derive all the theorems of geometry: at various points in his proofs he appealed to properties that are obvious from the accompanying diagram but do not follow from the stated axioms. Perhaps Kant had an inkling of this insufficiency in Euclid, which had been pointed out by Wallis in 1693 (see Torretti 1978, 44), and Frege certainly knew about it (CN, 85). Does arithmetic parallel geometry in this respect? Frege

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formulated the project of testing ‘how far one could get in arithmetic by means of logical deductions alone, supported only by the laws of thought, which transcend all particulars’ (Bs, Preface). Concerned lest his derivations should contain surreptitious appeals to intuition, as Euclid’s had in the geometrical case, he designed his concept-script with the aim of making all the steps of an inference completely explicit and free from gaps, so that any appeals to intuition would be immediately visible. The word ‘Begriffsschrift’ is a coinage, but not Frege’s: although it had already been used by von Humboldt in the 1820s, Frege probably got it from Trendelenburg (1867, III, 4), who used it as a German rendition of Leibniz’s notion of a lingua characteristica, a formal language in which scientific thoughts could be precisely expressed. Taking his inspiration from this project, Frege now aimed to set up a precise ‘formula language for pure thought’ in which gap-free derivations of arithmetical truths could be presented. He did not suggest, though, that we might abandon English or German and use his concept-script as a universal language like Esperanto. Rather, he likened the difference between ordinary language and the concept-script to that between the eye and the microscope: for scientific purposes the eye is inferior, but we could hardly go about our ordinary lives viewing everything through a microscope (Bs, Preface).

The judgment stroke Frege thus conceived of his concept-script as a supplement to, not a replacement for, the languages of the various sciences. Modern logic books often take as their starting point a stock of ‘logic-free’ atomic sentences—sentences not containing any logical connectives or quantifiers—and some commentators imagine this feature to be somehow silently present in his account. Evans, for instance, claimed that Frege’s ‘entire semantic theory was built around an account of the functioning of atomic sentences’ (1982, 7). Yet what is striking is how little attention he paid to the atomic case. He did not specify any primitive names, functionsymbols, or relation-signs other than the identity sign, but contented himself with noting that ‘the logical relations occur everywhere, and the symbols for particular contents can be so chosen that they fit the framework of the conceptscript’ (CN, 89). His dominant concern was not with how sentences can be built up from simpler constituents, but with discerning logical structure in preexisting sentences. He therefore began—at the opposite end, so to speak—with ‘the distinction between the act of judging and the formation of a mere assertible content’ (CN, 94): he symbolized the occurrence of a judgment by writing a vertical line, which he called the ‘judgment stroke’, to the left of an expression for the content judged. What most previous authors had treated as different kinds of judgment—categorical, hypothetical, negative, disjunctive, etc.—he now treated as differences only of content. The one exception (Bs, §4) was modal judgments, which he took to result from a difference not of content but of the ground on

Begriffsschrift I: Foundations of Logic 23

which the judgment is based. As a result, the only sort of necessity he recognized was epistemic. Having thus corralled modality within epistemology (and outside the concept-script), he then displayed little further interest in it. He mentioned this exclusion of modality from logic casually, as if little hinged on it, but in fact it had a profound effect on his conception of logic, as will become increasingly clear as we proceed. In distinguishing between a judgment and its content Frege was of course agreeing with Herbart, but not, at this stage, for Herbart’s reason. Herbart had treated the act of judgment as psychological in contrast to the content, which is non-psychological and hence the sole concern of logic. In his later writings Frege did express himself in such terms, but in Begriffsschrift he was relaxed about whether the content is psychological, insisting only that the judgment aspires to objectivity: his stated reason (Bs, §2) for denying that the idea house is judgable was not that ideas are subjective, but that ‘house’ is not a sentence. The reason he distinguished between a judgment and its content was rather that in logic we may derive consequences from a thought without asserting it—in order to test whether it is correct, for instance. Geach (1965, 449) dubbed this ‘the Frege point’: a single sentence may occur at some times so as to carry a commitment to its truth, at other times not, but needs to be understood as conveying the same content in both cases. If I assert, if A then B, for instance, I do not express any commitment to the truth of A, whereas if I assert A, I do. We cannot explain this by saying that ‘A’ expresses something different in the two cases, since this would lead to a puzzle as to why B should follow from them. In order that the validity of Modus Ponens should not rest on a pun, therefore, we must treat the two occurrences of ‘A’ as expressing the same content, in one case judged, in the other not. And this ‘Frege point’ is quite independent of Herbart’s concern about whether either the content or the act of judgment is psychological. Once the distinction is granted (whether for Herbart’s reason or Frege’s), there remains a further question whether we need a sign to mark it. Ordinary language has none, and indeed it is hard to know quite how to read the judgment stroke in English. Frege rendered it ‘is a fact’ (Bs, §3), but this is no good, because it once more gives us a declarative sentence which can be considered without being judged, whereas what we want is a way of expressing a judgment that resists this linguistic manoeuvre. Kanterian (2012, 75) uses this difficulty of capturing the judgment stroke in ordinary language as a reason to question its coherence, but that is surely too hasty: the demands of precise expression may lead us to draw distinctions which ordinary language has not deemed necessary, and hence to introduce expressions that have no ordinary language equivalents. What is more to the point is that any sign, once introduced, can be used non-seriously (e.g. by an actor on stage): if we anyway have to rely on context to settle whether the stroke is being used to mark a genuine judgment, why use it at all? In Begriffsschrift Frege already affirmed the anti-psychologistic conception of logic for which he has become famous; but it is one thing to say that logic is

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not psychology, another to explain what makes the two distinct. He said nothing so far about the subject matter of logic, or indeed about whether it has a subject matter at all, but his practice certainly suggests that he took its subject matter to be judgment: the judgment stroke was the first sign of his conceptscript that he defined, and his definitions of the others alluded in various ways to the circumstances in which we may make judgments. If he wished his account to ground an anti-psychologistic view of logic, then, he had to conceive of judgment as non-psychological. This may explain his otherwise puzzling insistence that judging is factive. ‘Inwardly to recognize something as true is to make a judgment, and to give expression to this judgment is to make an assertion.’ (PW , 2) Judging, on Frege’s conception, does not merely aim at truth: it cannot miss its target. Perhaps he hoped that by insisting on such an externalist conception of judgment he could render it sufficiently non-psychological to permit its presence in a logic book. Similarly to Kant, he needed the judgment stroke to mean, in effect, not ‘I, Frege, judge that’ but ‘it is judged that’ (impersonal ‘it’). Logic, he would later claim, is an investigation ‘of the mind, not of minds’ (CP, 369), but he unfortunately did not pause to address whether such an impersonal conception of judgment makes sense.

The content stroke To represent a judgment, Frege did not simply juxtapose the judgment stroke and a sentence but linked the two with a horizontal line which he called the content stroke. He thus distinguished between judging A and merely considering A. Since in practice the judgment stroke never occurs without an immedi’) has come to ately following content stroke, the combination of the two (‘ be treated as a simple sign, and indeed Frege first introduced it (§2) as if it were. From a purely notational perspective the content stroke serves the useful purpose of spacing formulae of the concept-script, providing a hook on which to hang a complex expression; but it is rather less clear why Frege thought, as he evidently did, that it is required for coherence rather than just visual clarity. It means, he said, ‘that the content which follows it is unified’ (CN, 94), but he did not say why it needs unifying in the first place. No doubt part of the difficulty was once again the inadequacy of ordinary language, where the convention is that to write a sentence unadorned is to assert it. In order to cancel that convention, Frege recommended reading ‘ A’ as ‘the circumstance that A’ (ibid.). The trouble with this is that ‘ A’ then becomes ‘the circumstance that A is a fact’, leaving us with the impression, which his stumbling explanations do little to contradict, that the content stroke merely cancels the implicitly assertoric character of the sentence, only for the judgment stroke to reinstate it, so that the compound sign ‘ ’ is reduced to a notational idle wheel. We might try to resist this conclusion by ridding ourselves of the convention that unadorned sentences express assertions and viewing them instead as names of judgable contents. In

Begriffsschrift I: Foundations of Logic 25

that case, though, it would still remain to be explained what sort of unity Frege supposed these to lack that the content stroke might supply. Perhaps he was influenced by Kant’s view, central to his transcendental deduction, that only in judgment are the components of thought unified into something capable of truth or falsity. If so, he may have intended ‘ A’ and ‘ A’ to express what Kant had called the problematic and assertoric judgment of A, respectively, and thereby to implement Kant’s doctrine that whether a judgment is problematic or assertoric ‘contributes nothing to the content of the judgment . . . , but concerns only the value of the copula in relation to thought in general’ (A74=B100).

Axioms, basic laws and rules In any formal system of logic we may distinguish between rules of inference, which govern the transition from antecedents to conclusion in an argument step, and primitive truths, which require no antecedent. One way of presenting logic that is nowadays popular is the ‘natural deduction’ system, which has only the former, not the latter. How can a proof get started in such a system? It may use as its starting point a posteriori truths about the world, but that would not be a route to a priori knowledge such as that of logical truths. Accordingly, natural deduction systems generally depend for their utility on ‘conditional proof ’, the method of making an assumption and then applying the rules of the system conditionally, before discharging the assumption at the last stage thus: A .. . B A⊃B Frege’s system, on the other hand, does not permit conditional proofs: every line of a proof is asserted unconditionally. The turnstile ‘`’ of modern logic derives from his complex sign ‘ ’, but in modern usage it is permitted to have antecedents and hence to represent inference rather than judgment, whereas Frege, rejecting the notion of a conditional judgment, never permitted anything to precede his judgment stroke. A system that had only primitive truths and no rules of inference would of course be useless. Frege’s system therefore contains both. Systems of this sort are nowadays called ‘Hilbert-style’ (see Hilbert & Ackermann 1928). Depending on the system chosen, natural deduction proofs may be converted to Hilbertstyle and conversely, but the latter are much less convenient, because each line of a sub-proof has to be converted into a material conditional containing all the premisses of the sub-proof in the protasis. Frege conceded that in Hilbert-style systems proofs might ‘sometimes attain a monstrous length’ (CP, 311), but he insisted (not altogether accurately) that in his own concept-script this risk could be mitigated.

26 Frege

Why, then, did Frege outlaw conditional proofs, even while acknowledging their convenience? Part of the explanation no doubt lies in his strongly actualist attitude, which had the consequence that if a judgable content is true, it is incoherent to suppose the content false. This then hindered him from formulating the notion of logical consequence in full generality, and hence from exploiting one of the most striking features of logic, namely that whether an argument is valid is independent of whether its premisses are true. In one place, exceptionally, he did recognize this. ‘The task of logic is to set up laws according to which a judgment is justified by others, irrespective of whether these are themselves true.’ (PW , 175) Elsewhere, though, he insisted doggedly that ‘we cannot legitimately infer from sentences, but only from true thoughts’ (PW , 180). Inference, he held, is a process linking judgments, i.e. recognitions of truth, and so an inference from false premisses is not merely pointless but incoherent. ‘To make a judgment because we are cognisant of other truths as providing a justification for it is known as inferring.’ (PW , 3) Modern authors have universally regarded Frege’s opposition to conditional proof as at best a puzzling foible, but there remains the philosophical puzzle of accounting for reasoning from a false premiss. In a sub-proof we make what seem like, but are not, assertions, since they occur within the scope of an assumption. How can an act be conditional? Proof is a route to truth, but not the only one: we cannot prove everything. If we trace back the reasons for our knowledge, we arrive at some truths which do not have proofs. These may be particular facts given to us by sense perception, in which case they are a posteriori; or they may, Frege said, be general laws which ‘neither need nor admit of proof ’ (Gl, §3)—he probably borrowed the phrase from Leibniz (1981, bk 4, ch. 9, §2) or Lotze (1874, bk 2, ch. 4, §200)—in which case they are a priori. Frege further distinguished two kinds of a priori principle. Accepting Kant’s doctrine that geometry rests on intuition, he called its principles ‘axioms’ (see CP, 273), but he avoided using this word for the basic truths of logic: in Begriffsschrift (§13) he called these simply the ‘kernel’, but previously he used ‘basic proposition’ (CP, 57) and later ‘basic law’ (Grundgesetz). He refused to use ‘axiom’ for a hypothesis assumed for the sake of argument. In his later work he called this a ‘vicious confusion’ (PW , 278), insisting on the point with what Tappenden (2000, 6) has aptly called ‘the relentless outrage of a cranky great uncle who ruins every family gathering with his interminable denunciations of Mountbatten’s perfidy in launching the raid on Dieppe’. Our reasons for taking the axioms of a science to be indubitably true cannot, on pain of circularity, themselves belong to that science. Frege conceived of epistemology, like logic, non-psychologistically: it is concerned with the justification for the truth of a thought, whereas psychology is concerned with the causal explanation for our taking it to be true. Nonetheless, which truths we treat as basic is to some extent up to us: different axiomatizations are possible. ‘We cannot

Begriffsschrift I: Foundations of Logic 27

accept a thought as an axiom if we are in doubt about its truth’ (PW , 205)—if in doubt, we should try to prove it—but axioms in one formalization might be theorems in another. This, though, is in tension with Frege’s conception of justification as an objective relationship between truths, which might lead us to expect the axioms to lie at the bottom of an independently given tree of justification.

Tone and conceptual content To distinguish between a judgment and its content is not yet to say anything illuminating about the nature of the latter. An important step—arguably the most important in all his philosophy of logic—came when Frege sought to single out what he called conceptual content—the part of the content that is relevant to inference. Two sentences have the same conceptual content just in case the same inferences may be drawn from each. He called the conceptual content of a declarative sentence a ‘judgable content’. The part of the content that this notion of conceptual content excludes he variously called ‘colouring’ (Farbung), ‘illumination’ (Beleuchtung), or ‘scent’ (Duft); in English Dummett (1973) called it ‘tone’. In Begriffsschrift Frege instanced ‘and’ and ‘but’ as words differing only in tone, not conceptual content; elsewhere he cited various other examples, such as ‘horse’ and ‘steed’, or ‘dog’ and ‘cur’. The fact that he never settled on a single word for tone is symptomatic of his lack of interest in it: he only ever mentioned it in order to set it aside as irrelevant to logic. The importance of this distinction is that it enabled Frege to make prominent those features of the structure of sentences that are relevant to inference. It is part of the logician’s task, as of the grammarian’s, to study how sentences reveal the structure of the contents they express, but their differing interests lead to different conceptions of that structure. The two sentences ‘The Greeks defeated the Persians at Plataea’ and ‘The Persians were defeated by the Greeks at Plataea’ have the same conceptual content, because the same consequences may be derived from each, whereas grammar quite properly distinguishes between the active voice of the verb in the former and the passive voice in the latter. The notion of tone is thus applicable not only to the contents of individual words but to the manner in which they are assembled to form sentences: sentences with the same conceptual content may differ in emphasis, and hence in tone, by having different terms as their grammatical subjects. Frege’s criterion of sameness of conceptual content is not as precise as might at first appear, however. If sentences have the same conceptual content whenever they have the same logical consequences, then all logical truths have the same conceptual content and the notion is therefore too coarse to play the role he intended for it of exhibiting features of interest to logicians. If, on the other hand, sentences have the same conceptual content only when they have the same immediately derivable consequences, then the notion becomes too sensitive to which consequences we take to be immediate. You may find obvious an inferential step

28 Frege

that for me requires explanation, in which case this notion has no place in an account of logic that aspires, as Frege’s did, to be independent of psychology. Once a formal system is in place, there will of course be an objective criterion for the immediacy or otherwise of a deduction, but this will depend on the system chosen and hence be unsuitable to serve, as he intended his notion of conceptual content to serve, as part of the grounding for that very system. There is also a problem about what background knowledge we are entitled to bring to bear in drawing inferences. It is unclear, for instance, whether we should say that ‘Hesperus is a planet’ has the same conceptual content as ‘Phosphorus is a planet’: that someone ignorant of astronomy might believe the first but doubt the second is a psychological fact whose relevance to settling the question of conceptual content would require argument. Ordinary language is inadequate and misleads us—that much is a common complaint—but Frege did not think that it misleads us entirely, nor that it simply supplies us with labels for thoughts: its structure is at least a guide, albeit an unreliable one, to the structure of the thoughts expressed. What else should guide us, though, when ordinary language fails? The obvious answer to put in Frege’s mouth is that the concept-script should do so. For instance, active and passive sentences express a single judgable content, and so the concept-script has a single sentence to express both. It would be natural, therefore, to aspire to a conceptscript in which every sentence has the same structure as the judgable content which it expresses; but if this ambition was Frege’s, it was not until 1906 (PMC, 67) that he even got close to stating it.

Function and argument The example of sentences in the active and passive voice with the same conceptual content demonstrates the inapplicability in logic of the grammarian’s notion of the subject of a sentence: the logical subject, according to Frege, is relative to some particular decomposition of it. The short paragraph in which he articulated this thought is surely one of the most significant in the history of philosophy. If, in an expression (whose content need not be assertible), a simple or a complex symbol occurs in one or more places and we imagine it as replaceable by another (but the same one each time) at all or some of these places, then we call the part of the expression that shows itself invariant a function and the replaceable part its argument. (Bs, §9) Frege intended this account of function-argument structure to serve his treatment of quantification (to be discussed in Chapter 5), and its true power emerges only when the two are combined, but he stated it separately, and it would be possible to discern function-argument structure even in a language which did not have the notion of quantification at all. His example was ‘Cato killed Cato’: if we regard the first occurrence of ‘Cato’ as replaceable, the constant remainder is ‘. . . killed Cato’; if the second, ‘Cato killed . . . ’; if both at once, ‘. . . killed . . . ’.

Begriffsschrift I: Foundations of Logic 29

The positions marked here by ellipses cannot be thought of merely as gaps, though: we need to distinguish whether two occurrences are being treated as successively or simultaneously replaceable. In Begriffsschrift Frege used for this purpose the same upper-case Greek letters that he used in inference patterns, but here I shall follow his later practice of using lower-case Greek letters. Thus in the example just mentioned we might write the first remainder as ‘α killed Cato’, from which in turn we could obtain ‘α killed β’ by treating the second occurrence of ‘Cato’ as replaceable. Alternatively, we might regard both occurrences of ‘Cato’ in the original sentence as simultaneously replaceable, so as to obtain ‘α killed α’ and thus express the concept of suicide. One reason for thus distinguishing (in his later work, at least) between the function ‘α killed α’ and the schema ‘A killed A’ may be that Frege wanted the function to be literally part of the sentence ‘Cato killed Cato’, which the schema is not, and therefore held that the letter ‘α’ does not occur in the function ‘α killed α’. In that case, though, one might reasonably wonder how this function differs from ‘α killed β’. It seems suspiciously as if Frege’s doctrine was that the gaps are simultaneously external and internal to the function according to need. In response to this difficulty Russell (PoM, §482) made the useful suggestion—later taken up by Wittgenstein—that Frege’s functions might be replaced by classes: ‘α killed α’, for instance, might be thought of as a notation for the class consisting of the sentences ‘Cato killed Cato’, ‘Brutus killed Brutus’, ‘Caesar killed Caesar’, etc. Three features of Frege’s account are worth stressing. First, a single sentence may be decomposed in multiple equally legitimate ways, but he conceived of these multiple decompositions at the level of language, not of content: a function is obtained by removing part of an expression, not part of its content. Any Begriffsschrift function gives rise in an obvious manner to what Geach (1961, 151) has called a ‘linguistic function’, i.e. a function in the mathematical sense that takes a linguistic expression as argument and outputs another linguistic expression as value. Second, Frege phrased his account so as to permit cases in which it is the grammatical predicate, not the grammatical subject, that is replaceable: in ‘Socrates is mortal’, we may treat ‘Socrates’ as replaceable, and obtain the function ‘α is mortal’; or we may treat ‘is mortal’ as replaceable, and obtain ‘φ(Socrates)’. This freedom has its limits, however: by calling the replaceable part a ‘symbol’ he presumably intended to require that it play an identifiable role in determining the content of the whole sentence. (This is at least part of what mediaeval logicians meant by calling an expression ‘categorematic’.) We cannot, for instance, extract from ‘Cato killed Cato’ the pattern ‘Cato killed Caα’. Third, and most importantly, Frege’s account was top-down, not bottomup: he aimed to decompose, and hence discern function-argument structure in, already existing sentences, not to explain how these sentences acquired their meanings in the first place.

30 Frege

Greek and Latin letters I have been using upper-case Greek letters, as Frege did, schematically.1 ‘To each of these letters’, he said, ‘the reader may attribute an appropriate sense if I do not specifically define them.’ (Bs, §2) Schematic letters had been in use since Aristotle, but Frege was more careful than most earlier authors to distinguish them from letters used to express generality, for which he reserved lower-case italic Latin letters. For instance, Γ (a) asserts every instance of the function Γ (α), whereas Γ (A) represents schematically the assertion of some particular instance. Expressions involving Greek letters are ‘actually only empty schemata; and in their application, one must think of whole formulas in the places of [Greek letters]’ (CN, 97). Nowadays we would say that the Greek letters belong to the metalanguage, Latin to the object language. Frege did not have these words, which were coined by Tarski (1933), but he was plainly sensitive to the distinction, noting explicitly that the logical rules in whose statement Greek letters occur cannot be expressed in the concept-script because they ‘form its basis’ (Bs, §13). By using lower-case Latin letters to express generality Frege was following the widespread practice of mathematicians, but he declined to join them in calling these letters ‘variables’, because this suggested to him either a word that refers variably to different things at different times or, worse, to a single thing whose intrinsic character is variable. He had no time for this notion of an arbitrary object. One common account of general reasoning maintained that to reach a general conclusion about triangles, for instance, the mathematician must first grasp the idea of some particular triangle and then, by a kind of selective attention, throw away those features which distinguish it from other triangles, so as to arrive at the general idea of an arbitrary triangle. Frege repeatedly ridiculed the forgetfulness which this conception required. If, abstracting from the difference between my house and that of my neighbour, I were to regard them as the same, and consequently act in the other house as in my own, the erroneousness of my abstraction would soon be made clear to me. (Gg, II, §99)

Further reading For the history of the word ‘Begriffsschrift’ see Barnes (2002). On Frege’s judgment stroke see Dudman (1970) and Smith (2000). For an objection to the

1 NB Here ‘A’ and ‘B’ are Greek, not Latin letters.

Begriffsschrift I: Foundations of Logic 31

force-content distinction see Hanks (2007). Geach (1960) used the ’Frege point’ as an argument against moral theories (e.g. Hare 1952) which give content only to the expression of a moral view and not to the view itself. Hare’s response was the beginning of a large literature on what ethicists nowadays insist on calling the ’Frege-Geach argument’: for a summary consult Schroeder (2008), and for a defence of expressivism see Blackburn (1984, chs 5 and 6). Much of the modern discussion has focused not so much on Frege’s distinction between content and tone as on Grice’s (1961) somewhat similar distinction between what an utterance says and what it implicates. The best route into the considerable literature on Frege’s Begriffsschrift notion of a function is via Oliver (2010). On Frege’s impersonal conception of assertion see Pedriali (2017). Kremer (2000) argues, contrary to what I have claimed here, that Frege’s notion is not factive, i.e. that on his view one can intelligibly make a false assertion. On conditional proof see Welty (2011); on Frege’s antipathy towards it see Currie (1987). On the significance of Begriffsschrift §9 see Dummett (1973, ch. 2). Fine (1983) attempts to save arbitrary objects from Frege’s critique.

4 BEGRIFFSSCHRIFT II: PROPOSITIONAL LOGIC

Before Begriffsschrift, the most sophisticated available treatment of propositional logic was algebraic: Boole’s account appeared in 1847, and was improved subsequently by de Morgan and Venn before being expounded for German readers by Schröder (1877). Frege’s account is significantly different from Boole’s, however, and it is unclear whether he even knew about Boole until after Begriffsschrift was published.

Syntax The first of Frege’s innovations in propositional logic lay in his choice of primitive signs. Boole had aimed to make the algebra of propositions as formally similar as possible to the arithmetic of numbers, and had therefore chosen negation, conjunction and exclusive disjunction as his primitives, because their formal properties are similar to those of subtraction, multiplication and addition respectively. A second tradition emphasized instead the duality between conjunction and inclusive disjunction. Neither of these algebraic traditions stressed the use of logical laws in inference; Frege, by contrast, placed curious emphasis on formulating propositional logic so that it needed only ‘a single rule of inference’ (Bs, Preface), Modus Ponens. This led him to take the material conditional instead of conjunction or disjunction as primitive, so as to make Modus Ponens particularly easy to state. In the notation I am using in this book A ⊃ B stands for the material conditional and ∼A for the negation of A. In devising a notation to express logical structure it is of course essential to avoid ambiguity. Nowadays we use brackets, distinguishing A ⊃ (B ⊃ Γ ), for instance, from (A ⊃ B) ⊃ Γ . A second novelty of Frege’s presentation was his two-dimensional notation in which

Begriffsschrift II: Propositional Logic 33

A and

B A

took the place of ∼A and A ⊃ B, so that A ⊃ (B ⊃ Γ ) and (A ⊃ B) ⊃ Γ became, respectively, Γ and

Γ.

B

B

A

A

In 1924 Łukasiewicz devised yet another notation: he placed the binary connective before the letters for the sentences it connects rather than between them, writing ⊃AB (prefix) instead of A ⊃ B (infix). In this notation (nowadays known as ‘Polish’) the two sentences just mentioned become ⊃A⊃BΓ and ⊃⊃ABΓ . From a formal point of view there is little to choose between the three notations—it is a trivial matter to write a computer programme to convert from one to another— but what is easy for computers is not so easy for us. Although it was used in pocket calculators in the 1970s, Polish notation never achieved widespread acceptance, presumably for reasons of human psychology: we find it difficult to detect the structure of a sentence written with prefixes. (Whether this would be different if we had all been brought up reading Polish notation from an early age, I do not know.) Frege’s concept-script fared even worse: no one except him ever used it. In this case unfamiliarity no doubt played its part—many other logicians ignored his work because they could not read it—but book production methods compounded the difficulty: complex formulae not only took up a lot of space on the page but were enormously time-consuming to typeset. Frege himself remained implacably convinced that these disadvantages were outweighed by the visual clarity of his notation’s two-dimensional representation of a formula’s structure by means of what we would now call its ‘parsing tree’.

Semantics Nowadays we stipulate the intended interpretation of the primitive signs of the propositional calculus by means of truth tables.

A T F

∼A F T

A T F T F

B T T F F

A⊃B T T F T

Frege did not in fact employ this device (first used by Peirce, then rediscovered by Wittgenstein in discussion with Russell in 1913). Instead, he noted (Bs, §5) that there are four possibilities,

34 Frege

1. A affirmed, B affirmed, 2. A denied, B affirmed, 3. A affirmed, B denied, 4. A denied, B denied, and stipulated that A ⊃ B is to be true in all except the third case. Why did he not use ‘true’ and ‘false’ instead of ‘affirmed’ and ‘denied’? Perhaps he was influenced by Kant’s talk in the Critique (A74=B100) of the affirmation or denial of a content of judgment. It is at least clear that he cannot have meant affirmation to be the same as assertion: only a few lines later he wrote ‘is to be affirmed’ instead of ‘is affirmed’; moreover, the ‘Frege point’ was precisely that when a sentence occurs as an argument of a conditional, it is either true or false, even though it is not then being asserted. If there remained any doubt that by ‘affirmed’ and ‘denied’ he did really just mean ‘true’ and ‘false’, he dispelled it three years later by defining A⊃B as the negation of A and not-B (CN, 95). Frege did not mention the previous history of the material conditional: perhaps, indeed, he was unaware of it and came upon the idea unaided. Subsequently, a literature has developed on the ‘paradoxes of the material conditional’, which expose the error of supposing that it means the same as ‘if ’. He was from the outset commendably resistant to this error, noting explicitly that ‘a rendering by means of “if ” is not appropriate in all cases of linguistic usage’ (CN, 95). The key point here is that if we want to formalize ‘if ’ by means of a truth-function, the material conditional is the only plausible candidate. The fact that he chose it is thus an instance of his intention to devise a notation that expresses only conceptual content. Any content that ‘if ’ has but the material conditional lacks is irrelevant to logic. What the material conditional most signally fails to capture is counterfactual reasoning—in which, however, Frege throughout his career displayed a marked lack of interest. Plainly, then, Frege had at least an informal grasp of the modern notion of a truth-function. The two truth-functions he chose, negation and the material conditional, are adequate to express all the others, but it is unclear whether he knew this. He certainly knew that starting from a stock of basic sentences it is possible by using his two to build up expressions for other more complex sentences, and that in consequence there is no need for separate signs for conjunction and disjunction. He expressly noted, though, that other choices of primitive connectives would have been possible (Bs, §7). A semantics in which the content of an expression is specified by giving the conditions under which sentences involving it are true is said to be ‘truth-conditional’. Because Begriffsschrift does not mention the non-logical part of language at all, we cannot describe its semantics as determinately truthconditional, but Frege’s practice, at least, strongly suggests that he took the conceptual content of a sentence to be exhaustively characterized by the conditions under which it is to be affirmed or denied.

Begriffsschrift II: Propositional Logic 35

Frege explained the content of a compound sentence in terms of the content of the sentences from which it is formed: ∼A in terms of A, and A ⊃ B in terms of A and B. Many commentators consequently credit him with a ‘compositionality principle’ according to which the content of a complex expression must always be wholly explicable in terms of the contents of its components and their manner of combination. Later in his life Frege was much struck by our capacity to understand sentences we have never heard before—a capacity which depends on compositionality for its explanation—but notably he did not mention this until 1914 (PMC, 79), and in his early writings he did not comment on the principle at all.

Rules and basic laws The only rule of inference in Frege’s propositional logic is Modus Ponens: ‘from the two judgments A and B the new judgment A follows’ B (Bs, §6). Here, of course, he was talking about judgments, not making them. When he re-stated the rule in Grundgesetze, 14 years later, he put expressions containing conditional occurrences of the judgment stroke in quotation marks; but although in Begriffsschrift he used quotation marks to mention expressions of ordinary language, he did not yet do this with expressions of his concept-script, presumably because in practice the grammar of a sentence in which an expression of the concept-script occurs leaves little doubt whether it is being mentioned or used. Frege split the basic laws of propositional logic into two groups. Those in the first group involve only the material conditional: p ⊃ (q ⊃ p);

(1)

(r ⊃ (q ⊃ p)) ⊃ ((r ⊃ q) ⊃ (r ⊃ p));

(2)

(r ⊃ (q ⊃ p)) ⊃ (q ⊃ (r ⊃ p)).

(3)

Those in the second group also involve negation: (p ⊃ q) ⊃ (∼q ⊃ ∼p);

(4)

∼∼p ⊃ p;

(5)

p ⊃ ∼∼p.

(6)

It is natural from a modern perspective to ask whether this system of truthfunctional logic is complete—whether every logical truth expressible using only negation and the material conditional is provable in it. Some years later, it was shown that the answer is yes, and indeed that law (3) can be omitted without loss (Łukasiewicz 1929), but Frege himself did not discuss this question.

36 Frege

Further reading As an illustration of how proofs work in Frege’s system, consult the explicit proof in Kneale & Kneale (1962) of the redundancy of Frege’s axiom (3). On whether a soundness argument can be genuinely explanatory see ‘The justification of deduction’ in Dummett (1978).

5 BEGRIFFSSCHRIFT III: QUANTIFICATION

Although Frege’s treatment of propositional logic contained several innovations, what marked Begriffsschrift as revolutionary was its introduction of a notation of quantifier and variable to express generalizations.

Syntax It had long been a commonplace of mathematics to express generality with variables, e.g. writing ‘a2 − b2 = (a + b)(a − b)’ to mean that the equality holds for all a and b. Frege deployed a similar device in his concept-script, writing ‘ Γ (a)’ to mean that Γ (ξ ) is true for every argument. What, though, of the ‘Frege point’ that the content of a judgment may occur unasserted, for instance as a component of a more complex content? If we are to embed ‘Γ (a)’ in a larger context, we need a notational device—nowadays called a ‘quantifier’—to limit its scope. Frege’s key insight was that, since there may be more than one variable in its scope, the notation for the quantifier needs to indicate which of them it is limiting. This radically alters how we think of the expressive role played by the letter marking an argument-place. If we focused solely on the quantifier-free Γ (a)’, we might be tempted to think of ‘a’ as a kind of arbitrary notation ‘ name, conferring generality by referring indeterminately. In ‘ ∀xΓ (x)’, on the other hand, the second ‘x’ serves only to link the argument-place it fills to the quantifier marked by the first: they combine to form a single device for indicating scope. For this reason Frege insisted (Bs, §11) that a free variable makes sense only when used as part of a whole judgment. The issue arises already in natural language. There is an ambiguity in ‘All that glisters is not gold’ that is absent from ‘My watch is not gold’. One way to resolve the ambiguity would be to use the anaphoric pronoun ‘it’, distinguishing

38 Frege

‘All that glisters is such that it is not gold’ from ‘Not all that glisters is such that it is gold’. A pronoun is anaphoric if it inherits its reference (if any) from an earlier phrase—in this case, ‘all that glisters’. Where there is more than one to choose from, we can use ‘the former’ and ‘the latter’ anaphorically, distinguishing ‘Everything is such that something is such that the former is bigger than the latter’, for instance, from ‘Everything is such that something is such that the latter is bigger than the former’. Ordinary language does not often have to deal with more than two possible anaphoras: legalese (at least if we are to believe the Marx brothers) borrows a device from the Stoics, calling them ‘the party of the first part’, ‘the party of the second part’, etc. Frege’s notational device uses letters anaphorically to link argument-places in the expression of a concept to the relevant quantifiers, and hence to distinguish ∀x ∃y xTy from ∃y∀x xTy. When we discussed propositional logic, we noted that there were various notations that could achieve the same effect as brackets, and in the current case, too, there are alternatives. For instance (see Bourbaki 1954, Quine 1940, 70), we could link the gaps and the quantifiers with lines: ∀∃T I have described the notation of quantifier and variable as Frege’s, but in fact he used Fraktur letters instead of italics, and an indentation in the content stroke to express the universal quantifier, writing ‘ ..a........ Γ (a)’ instead of ‘∀xΓ (x)’. No one else ever adopted his notation, but this was because it was embedded in his two-dimensional way of representing propositional structure. Nonetheless, although modern usage preserves the essence of his idea, it neglects to differentiate notationally between what are nowadays called bound variables, for which Frege used Fraktur letters, and free variables (sometimes called ‘parameters’), for which he used italics; hence it underplays the difference between these two roles. In Begriffsschrift itself Frege explained the quantifier-free parametric notation in terms of the quantified one, treating ‘ Γ (a)’ as an abbreviation for ‘ ∀xΓ (x)’. In more informal expositions (e.g. PW , 11) he sometimes elided this part of the explanation, hence obscuring the important point that because the quantifier-free notation lacks any way of limiting the scope of the variable, it is strictly weaker in expressive power than the quantified notation. For example, ∃xΓ (x) can be defined using the universal quantifier as ∼∀x∼Γ (x), but is not expressible in the quantifier-free notation. In Chapter 3 I emphasized the importance of the account of functionargument complexity and the use of variables to mark position that Frege offered in §9 of Begriffsschrift. At that stage, the reader might well have felt short-changed by the relatively modest gain of being able to express the logical connection between murder and suicide. In combination with the quantifier, though, the true power of his account becomes apparent: what gives it its expressive richness is that if we apply a quantifier to a function extracted from a sentence, the process

Begriffsschrift III: Quantification 39

can be iterated to reveal further concepts, ‘as plants are contained in their seeds, not as beams are contained in a house’ (Gl, §88). (Consider, for example, how the concept of a prime number is obtained from that of one number’s being the product of two others.) The word ‘monadic’ is nowadays used for the part of logic in which no two quantifiers overlap in scope; Frege’s logic, by contrast, is ‘polyadic’. In monadic logic variables are inessential, because scopes do not overlap. Polyadic logic therefore involves objects in a way that monadic logic does not. This is of crucial importance in determining the complexity of the resulting system. As Frege presciently observed, ‘Boolean formula-language only represents a part of our thinking; our thinking as a whole can never be coped with by a machine or replaced by purely mechanical activity.’ (PW , 35) He did not himself prove this claim, but his instincts were correct: monadic logic is mechanically decidable, but polyadic logic is not (Church 1936, Turing 1936). This is what leads so many commentators to treat 1879 as the moment when modern logic was born: it offered for the first time a precise formal system which escaped the Aristotelian paradigm by codifying canons of reasoning that cannot reasonably be regarded as trivial. Its non-triviality is explained by its ability to reveal, as Aristotle’s cannot, multiple analyses of a single thought. That Frege himself sensed the power and importance of his invention must have made the incomprehension that greeted it all the harder to bear. The most important difference between Aristotle’s account of quantification and Frege’s is thus that Aristotle’s quantifiers operate on predicates, Frege’s on functional expressions containing argument-places: Frege in effect replaced Aristotle’s A(Γ, ∆), etc. with Ax(Γ (x), ∆(x)), etc. There is also the further difference, however, that Frege then replaced Ax(Γ (x), ∆(x)) with ∀x(Γ (x) ⊃ ∆(x)): Aristotle’s quantifiers were binary operators, Frege’s unary. Remarkably, Frege seems to have been the first logician to study a unary quantifier. Part of the explanation is that the reduction of the binary to the unary case depends on his decision to focus on the conceptual content of sentences. Only on this simplifying assumption is it plausible that ∀x(Γ (x) ⊃ ∆(x)) has the same content as Ax(Γ (x), ∆(x)): no one would think prior to reading a logic book that ‘Everything is either black or not a raven’ says the same as ‘All ravens are black’. (Even after reading several, Russell thought that it obviously did not.) Once we grant, however, that it is only conceptual content we are trying to capture, we can of course perform the reduction in the opposite direction too, provided that our concept-script contains a universal predicate. Although Frege’s choice of the unrestricted quantifier is one of his most notable innovations, he did not pause to justify it, allowing the reader to suppose that little hinged on which he used, and that the matter could be decided wholly on grounds of simplicity: treating the restricted quantifier as primitive requires extra axioms relating the properties of the quantifiers to those of the propositional connectives—axioms which are redundant if we take the unrestricted quantifier as primitive. Opting for the unrestricted quantifier is not harmless, however, since

40 Frege

it has the effect of treating objects as a single logical category: the equivalence between the two approaches depends, as just noted, on assuming that there is a universal predicate. We shall see in Chapter 18 that one possible response to Russell’s paradox is to reject just this assumption. Frege’s adoption of the unrestricted quantifier therefore conditioned his response to the paradox by blocking this way out.

Semantics Modern logicians distinguish two kinds of quantificational semantics, ‘referential’ and ‘substitutional’. Here, though, we shall be making finer distinctions of semantic level, and so I shall instead refer to the referential and substitutional accounts as ‘content-level’ and ‘language-level’ respectively. Which, then, was Frege’s? He explained the truth or falsity of a universal quantification in terms of the truth or falsity of its instances: if we apply a universal quantifier to a function, he said, the resulting sentence means that ‘the function is a fact whatever we may take as its argument’ (Bs, §11). In Begriffsschrift functions are conceived of linguistically, which makes this is a language-level account. It has to be admitted, though, that he did not express himself with total clarity: would he not have been better to say that the function expresses a fact, rather than that it is a fact? Among language-level accounts there is a further distinction between ‘narrow’ and ‘broad’. On the narrow account the condition for the truth of ‘∀xΓ (x)’ is that Γ (A) be true for every singular term ‘A’ already in the language; on the broad, the condition is that Γ (B) be true in every extension to a language to which a new (‘auxiliary’) name ‘B’ is added. In the case where every object in the domain already has a name (e.g. arithmetic in a formal language containing the numerals ‘0’, ‘1’, ‘2’, etc.), the two accounts are equivalent; but this is of course the exception. Neither in ordinary language nor in the usual formal languages for geometry and physics does every object have a name. Moreover, it is far from obvious why we might want to generalize only over objects that happen to have names. The emphasis placed on the narrow version of the account in the literature on substitutional semantics is therefore odd. In order to grasp a generalization, we do not need to grasp each of its instances. As Frege himself later noted, ‘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.’ (CP, 227) It is in any case unlikely that Frege would have countenanced a narrow language-level semantics in Begriffsschrift, given that the concept-script presented there contains no singular terms at all. This is presumably why he resorted, in explaining his semantics, to saying that we should think of the letter as replaced by ‘something definite’ (§11): no more precise form of words was available to him, because what definite expressions there are is not a matter that he envisaged being settled by his concept-script on its own; they ‘can be easily created as required’ (CN, 88).

Begriffsschrift III: Quantification 41

With the narrow account off the table, the difference between language-level and content-level treatments of the Begriffsschrift quantifier is relatively unimportant, since for each object the language has an extension in which that object has a name. (Which objects we have the resources to name in practice is of course a quite different matter.) The difference might matter in indirect contexts such as belief, but these will hardly have been at the front of Frege’s mind at this stage, since they do not occur in mathematics or science. Frege’s semantics for the universal quantifier is compositional in the sense that it specifies the content of a universal generalization in terms of the content of the concept which it generalizes. Some authors have sought to impose the stronger notion of ‘direct compositionality’, which requires that the content of a sentence be explained in terms of the contents of the component parts of that very sentence. Whether the semantics of the quantifier is directly compositional depends on its precise phrasing: if we say that ∀xΓ (x) is true just in case Γ (ξ ) is true for every argument, then it is (because ‘ξ ’ here simply marks a gap and is not a component of the concept-word); if we require instead that Γ (A) should be true for every name ‘A’, then it is not (because ‘Γ (A)’ is not literally part of ‘∀xΓ (x)’). The fact that Frege treated functions as containing gaps rather than schematic letters suggests that perhaps he wanted his semantics to be directly compositional, but it is hard to see why this should matter much. In order that the syntax of a language should be effective, what is required is that every well-formed string in the language should have a finite parsing tree. Compositionality is best thought of as the requirement that the semantic value of each such string be explained in terms of the semantic values of the strings lower down its parsing tree (the semantic values of the strings at the bottom nodes of the tree being assumed as given). It is unimportant whether the language has the further feature that a string is always parsed in terms of its own substrings. The domain of Frege’s quantifier thus includes not just entities that do have names, but those that could. He did not place any overt restriction on this domain, but it is unclear, nonetheless, quite how inclusive he took it to be. He conceived of his concept-script as ‘supplementing the signs of mathematics with a formal element’ (PW , 13), and perhaps he viewed this as contextually restricting the sort of objects that can be named. Aristotle’s treatment of universal and existential propositions invited us to treat them as restricted—e.g. ‘Every Γ is a ∆’ is A(Γ, ∆)—but left the connection with singular propositions unexplained. As we saw in Chapter 2, some mediaeval logicians tried to deal with this problem by treating the quantifiers as arguments, in effect writing ‘Every Γ is a ∆’ as ∆(every Γ ). Frege’s treatment was different. Consider the two propositions: 1. The number 20 can be represented as the sum of four squares; 2. Every positive integer can be represented as the sum of four squares.

42 Frege

‘The number 20’ and ‘every positive integer’, he suggested, are not ‘concepts of the same rank’. By analysing the two sentences as ∆(a) and ∀x(Γ (x) ⊃ ∆(x)), respectively, he explained the content of the latter without ascribing a separable content to ‘every positive integer’; this phrase ‘acquires a meaning only from the context of the sentence’, in contrast to ‘the number 20’ in the former.

Rules and basic laws In addition to Modus Ponens Frege now needed further rules of inference to handle the delicate relationship between the three kinds of letters he had introduced for expressing generality. For instance, he needed (and stated explicitly) the rule that we can change bound variables, provided that we avoid using letters that already occur in the same scope. Because he regarded the expression of a judgment using an italic letter as merely an abbreviation for the expression of a universally quantified judgment in which the quantifier has maximum scope, he did not need a rule to license the inference from Γ (a) to ∀xΓ (x). However, he did need a rule to allow for the introduction of a quantifier in a conditional: from A ⊃ ∆(a) infer A ⊃ ∀x∆(x), provided that the letter ‘x’ does not occur in A or in ∆(ξ ). He then stated as an axiom ∀xf (x) ⊃ f (a), where the Roman letter ‘a’ is, according to his conventions, a parameter, not a schematic letter, and so the formula is really an abbreviation for its universal closure. Frege also needed (but did not explicitly state) a substitution rule licensing the inference from Γ (a) to an instance Γ (A). It is at this point, though, that his failure to distinguish types of argument-place makes matters awkward: gibberish will result if we substitute a name for a function-symbol variable or a function-symbol for a name variable. His elliptical remark that ‘if the German letter appears as a function symbol, this circumstance must be taken into account’ (Bs, §11) needs to be charitably interpreted as meaning that in any application of the rule the expression ‘A’ to be substituted must be of the same type as the parameter ‘a’ which it replaces. This substitution rule then straightforwardly entails the ‘second-order comprehension principle’, ∃f ∀x(f (x) ≡ Γ (x)) (see Boolos 1985, 337). In Begriffsschrift Frege did at least adhere to the practice of always writing function-symbols with their argument-places marked, and he formulated this explicitly as a general rule a couple of years later. ‘A sign for a property never appears without a thing to which it might belong being at least indicated.’ (PW , 17), and perhaps he viewed this as contextually restricting the sort of objects He seems thus to have hoped that in a logically correct concept-script distinctions of logical type would go without saying. There remains a difficulty, though, about how to make use of propositional calculus when reasoning with a parameter. Suppose, for instance, that we have Γ (a) ⊃ ∆(a) and Γ (a). What licenses us to infer ∆(a) from this? Not Modus Ponens, because of the point about universal closures already

Begriffsschrift III: Quantification 43

mentioned: what we really need to do is to infer from ∀x(Γ (x) ⊃ ∆(x)) and ∀xΓ (x) to ∀x∆(x). What becomes apparent, then, is that Frege did want to be able to use parametric reasoning after all: he wanted a licence to ignore the fact that the letter ‘a’ indicates an indeterminacy, but to treat it as if it were a constant for the purposes of inference. He mentioned this explicitly in Grundgesetze, where he said (I, §17) that we extend the scope of the parameter beyond a single judgment to encompass a whole argument.

Further reading There is a large literature on unrestricted quantification: see Rayo & Uzquiano (2007) for a selection. On Frege’s treatment of quantification see Evans (1977, §2). Blanchette (2012, ch. 3) discusses whether Frege understood the range of his quantifier as contextually restricted.

6 BEGRIFFSSCHRIFT IV: IDENTITY

Frege usually called singular expressions that purport to refer to objects ‘proper names’, but in one place (PW , 118) he called them ‘singular terms’. I shall generally follow modern fashion by using the latter phrase, because Frege’s already has a much more restricted sense in ordinary usage.

Syntax In Begriffsschrift Frege assumed as given a stock of singular terms appropriate to whatever subject matter is under discussion; he specified no general method for forming such terms. The fact that he used the expression ‘conceptual content’ in relation both to whole sentences and to singular terms suggests that he presumed a uniformity of semantic machinery—that a singular term has content in just the way a sentence does. Yet he himself did not quite say this. (Not at the time of Begriffsschrift, anyway: as we shall see, he explicitly committed himself a decade later to such an assimilation.) Nonetheless, he did accentuate the impression of uniformity by using the single sign ‘≡’, explained as expressing identity of conceptual content, both between singular terms and between sentences. In the former case the modern practice is to use ‘=’, and I shall follow that usage here so as to avoid confusion with my use of ‘≡’ for the material biconditional. (Modern logic has no widely agreed sign for sameness of conceptual content for sentences, perhaps because a precise criterion for this has proved elusive.)

Semantics Usually, Frege held, the conceptual content of a singular term is the object which is its bearer: if A = B, then ‘Γ (A)’ and ‘Γ (B)’ express just the same content; and if ‘A’ has no bearer, then ‘Γ (A)’ does not express a judgable content at all (PW ,

Begriffsschrift IV: Identity 45

Bα

A

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

α

174). ‘The rules of logic’, he maintained, ‘always presuppose that the words we use are not empty, that our sentences express judgments, that one is not playing a mere game with words.’ (PW , 60) He did not pause to consider whether it would be possible to adopt different rules which make no such presupposition. We come up against a difficulty, however, if we apply this account of content to the identity sentence itself, since it entails that if A = B, then the sentences ‘A = B’ and ‘A = A’ express the same content. How can this be, given that some inferences seem to be licensed by the former but not by the latter? For instance, we can derive the conclusion Γ (B) from the premisses Γ (A) and A = B, but not from Γ (A) and A = A. The difficulty is ubiquitous in ordinary language, where instances in which two names refer to the same object abound. Frege, though, was in the business of creating a concept-script. Could he not avoid the difficulty by stipulating that here no object should be given more than one name? No, because this fails to take account of the fact that there may be different ways of determining the same object. In his later article, ‘On sense and reference’, he famously used the example of ‘the Morning Star’ and ‘the Evening Star’, two singular terms which both have as their content the planet Venus (CP, 158), but in Begriffsschrift he used a geometrical example instead. Let A be a point on the circumference of a circle and let the diameter of the circle through A be drawn. Now draw a line through A at an angle α to this diameter and let Bα be the point where this line intersects the circle. It is a fact of plane geometry that Bπ/2 = A. So the two singular terms ‘A’ and ‘Bπ/2 ’ are associated with different ways of determining a single point. This illustrates that when we give an object a name, we must have a way of determining which object it is, but need not yet be in a position to know whether the object so determined is the same as the object determined by some other singular term. So the proposed stipulation never to give one object two names in the concept-script is inoperable. Frege’s deployment of a mathematical example here was no doubt deliberate: he wanted to make the point that the issue arises even in a language designed narrowly for the purposes of science. It was also deliberate not to use a simple

46 Frege

arithmetical example, since if arithmetical equations are analytic, as he would soon claim, it is at least prima facie plausible that they might be reducible in a canonical notation to triviality. In the geometrical example, by contrast, he held that the identity is in Kant’s sense synthetic, and so we should not expect its truth to be revealed by considerations internal to our logical notation. Frege did not at this stage expand on what a ‘way of determining’ an object might be, and he shied away from the idea that it might be part of the conceptual content expressed by an identity sentence. Instead, he explained the equation ‘A = B’ as meaning that the signs ‘A’ and ‘B’ have the same conceptual content. So for an equation the judgable content involves the names in a way that for other sentences it does not. His account of the semantics of singular terms was therefore, like Mill’s (1843, I, 58), disjunctive. They stand ‘at times for their contents’, he said, ‘at times for themselves’. Although symbols are usually only representatives of their contents, . . . they at once appear in propria persona as soon as they are combined by the symbol for identity of content, for this signifies the circumstance that the two names have the same content. (Bs, §8) The disjunctive nature of Frege’s account was precisely its flaw, however. To see why, recall that it arose as a response to the difficulty that we can derive Γ (B) from Γ (A) and A = B, but not from Γ (A) and A = A. This presupposes, however, that the judgable content of Γ (B) is different from that of Γ (A), since if they were the same, we would already have Γ (B). So the difficulty was not in fact restricted to identity sentences (see Dummett 1993b, 24). Hence if Frege’s solution were correct, it would have to be applicable quite generally: the judgable content of Γ (A) would always have to be understood as saying something about the name ‘A’ as well as about the object A. Frege showed no sign of grasping this point. He was explicit that in contexts other than identity sentences singular terms ‘are mere proxies for their content, and thus any phrase they occur in just expresses a relation between their various contents’; it is only in identity sentences that they ‘appear in propria persona’ (Bs, §8). Yet on his account Γ (A) and ‘Γ (B) have the same judgable content whenever A = B, even though their inferential powers seem to differ. Consider, for instance, a famous example mentioned by Leibniz. It is a logical truth that if Jesus Christ is God, then Jesus Christ’s mother is the mother of God. Yet since Jesus Christ’s mother is Mary, this has, on Frege’s account, the same conceptual content as the judgment that if Jesus Christ is God, then Mary is the mother of God. When we express it in this form, however, we no longer have the resources to explain how it can be a logical truth. The semantics for identity that Frege proposed in Begriffsschrift is seriously awry, therefore: fixing the error eventually required him to revise his whole semantic theory (see Chapter 14). Whatever its failings, however, his early theory at least conceived of identity as a relation with two argument-places. He

Begriffsschrift IV: Identity 47

never seems to have shared the widespread puzzlement (from which Russell, for instance, took some time to disentangle himself) as to how an object may bear this relation to itself.

Basic laws The basic laws for identity in the Begriffsschrift are as follows: c = d ⊃ (f (c) ⊃ f (d))

(1)

c=c

(2)

If the remark that followed Frege’s statement of (1) was supposed to explain its validity, however, he omitted some crucial quotation marks. With these quotation marks re-instated, we need to find a route from the premisses that ‘c’ and ‘d’ have the same content, that f (ξ ) gives a true content with c as argument, to the conclusion that f (ξ ) gives a true content with d as argument. This does not make sense: it is not the variables ‘c’ and ‘d’ that have content, but the names that might instantiate them. This neatly encapsulates what is wrong with Frege’s semantics for ‘=’. He then compounded the difficulty by applying (1) in the case where f (ξ ) is the concept ξ = c, so as to obtain c = d ⊃ (c = c ⊃ d = c), whence c = d ⊃ d = c. Yet on his own view these two concepts are not of the same kind: the former takes an object as argument, the latter a name. Frege himself seems soon to have become uneasy about his Begriffsschrift account. Two years later (PW , 29), he proposed to replace (1) with a rule of inference that from A = B and Γ (A) may be inferred Γ (B). A few pages later in the same article he even suggested that he no longer regarded identity as a primitive sign, but ‘would define it by means of others’ (PW , 36). Presumably he had it in mind to define α = β to mean ∀f (f (α) ≡ f (β)), but he did not explain any further, and in his later work he reverted to treating it as primitive.

Further reading The difficulty in Frege’s Begriffsschrift account of identity is clearly laid out by Mendelsohn (1982). Dickie (2008) offers a different interpretation of Frege’s point.

7 BEGRIFFSSCHRIFT V: THE ANCESTRAL

Although Frege designed his concept-script primarily for expressing reasoning in arithmetic, it was not until Chapter 3 of Begriffsschrift that he addressed this case; and even then he restricted himself to the task of defining the ancestral of a relation.

Defining the ancestral Suppose that R is the relation of parenthood and R∗ that of ancestorhood. There is evidently a conceptual connection between these two relations, but can it be expressed in wholly logical terms? The obvious strategy for answering this questions is to proceed in stages. It is easy enough, with Frege’s account of quantification now available, to define the grandparent relation R2 in terms of parenthood: xR2 y =df ∃z(xRz ∧ zRy). Then define the great-grandparent relation R3 : xR3 y =df ∃z(xR2 z ∧ zRy). Now generalize this to define Rn for any natural number n: xRn+1 y =df ∃z(xRn z ∧ zRy). Finally define the ancestor relation R∗ : xR∗ y =df ∃n ≥ 1 xRn y. Notice, though, that this last definition contains a quantifier ranging over natural numbers. This could be made logically unobjectionable if we presupposed not

Begriffsschrift V: The Ancestral 49

only the natural numbers but the method of definition by recursion; but it is useless if the ultimate goal is wholly logical derivations of arithmetical truths, since in such a context it is obviously circular. It was Frege’s achievement that he saw an alternative strategy: instead of approaching the ancestral as in the previous proposal step by step (from the inside out, as it were), he defined it instead by intersection (from the outside in). More precisely, a property Γ (ξ ) is said to be R-hereditary if ∀x, y((Γ (x) ∧ xRy) ⊃ Γ (y)), and x is an R-ancestor of y if y has every R-hereditary property possessed by all x’s children: xR∗ y =df ∀f ((f is R-hereditary ∧ ∀z(xRz ⊃ f (z))) ⊃ f (y)). Unlike our previous attempt, this definition makes no use of natural numbers: so if we apply it in the case where R is the successor relation, we obtain a non-circular definition of the greater than relation on natural numbers. Moreover, the definition allows us to prove, using second-order comprehension, that the property of having x as an R-ancestor is itself R-hereditary. Since 1872 Dedekind had been working independently on a similar account of the ancestral a hundred miles away in Brunswick, but he did not publish it until 1888. (Peirce also made similar discoveries independently a few years later.) Dedekind’s account, when it appeared, went much further than Frege’s: he had mastered the considerable technical challenge of proving the principle of definition by recursion, by means of which addition and multiplication may be defined. Chapter 3 of Begriffsschrift, by contrast, is notably laconic. Frege offered only a few illustrative theorems, most notably (no. 98) that R∗ is transitive, and (no. 133) that if R is a function, i.e. ∀x, y, z((xRy ∧ xRz) ⊃ y = z), then R∗ satisfies the trichotomy law ∀x, y, z((xR∗ y ∧ xR∗ z) ⊃ (yR∗ z ∨ y = z ∨ zR∗ y)). If Frege’s goal was a logicist treatment of the natural numbers, he omitted to say so, and the only application of the theory of the ancestral that he mentioned was to the sorites paradox. If we apply theorems 98 and 133 to the successor function, we obtain the significant result that its ancestral, the greater than relation, is a strict total ordering of the natural numbers; but he did not mention this. Perhaps he did not want to draw attention to how far the sample theorems he had so far proved fell short of his ultimate goal: there is nothing in Begriffsschrift on addition and multiplication, for example. Moreover, the theorems the book does contain are conditional: if there is a successor relation, then its ancestral has such and such properties. Frege did not prove that there is a successor relation in the first place, and gave no inkling that he yet saw how to do this. He had claimed in his dissertation that arithmetic is independent of intuition (CP, 57),

50 Frege

but his logic singularly lacked any plausible route to a proof of an unconditional existence claim. He therefore offered only what would nowadays be called a ‘proof of concept’—a few sample theorems from the middle of a logicist account of arithmetic, omitting both ends.

Logic as non-trivial As we have noted, Frege was less than explicit in Begriffsschrift about how the machinery of logic should cope with the distinction between first- and secondorder reasoning. So he may not yet have had much grasp of their difference in strength. (Ironically, perhaps, this difference can be traced to the same source as the contradiction that would later prove his undoing.) Even so, it is second-, not firstorder quantification that is responsible for even the modest harvest of theorems 98 and 133 in Chapter 3 of Begriffsschrift. Frege positioned at the beginning of the chapter an eloquent passage in which he expressed the power he claimed for his logic. We see . . . how pure thought (regardless of any content given through the senses or even given a priori through an intuition) is able, all by itself, to produce from the content which arises from its own nature judgments which at first glance seem to be possible only on the grounds of some intuition. We can compare this to condensation by which we succeed in changing air, which appears to be nothing to the childlike mind, into a visible drop-forming fluid. The propositions about sequences developed in what follows far surpass in generality all similar propositions that can be derived from any intuition of sequences. (Bs, §23) Although Frege did not yet mention him by name in Begriffsschrift, Kant was plainly his intended target. In Grundlagen, five years later, he made an explicit contrast with Kant’s conception of general logic as trivial, emphasizing quantified logic’s ability to ‘extend our knowledge’ by not merely ‘taking out of the box again what we have just put into it’ (Gl, §88); but already in Begriffsschrift he was clear that his logic could deliver proofs of non-trivial results.

Impredicativity As Frege noted (Bs, §8), the property of being an ancestor of x is itself one of the R-hereditary properties in terms of which it is defined. This kind of circularity is now known as ‘impredicativity’, and any definition of the ancestral in pure second-order logic may be expected to suffer from it. (It is commonly the definition that is said to be impredicative, but strictly it would be more accurate to apply this term to the comprehension principle, since that is what permits the proof that the ancestral falls within the range of the quantifier in the definition.) Is the circularity vicious? Some impredicative definitions are unproblematic. ‘The tallest man in the room’ is harmless, but this serves only to pick someone out,

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not to conjure him into existence. In order to argue in the same manner for the harmlessness of Frege’s definition of the ancestral, we would have to claim an independent grasp of it. Is that not just what Frege was hoping to eliminate? Kerry (1887, 295) objected that it would be impossible ever to prove that Frege’s definition of the ancestral relation is satisfied: in order to show that y is an R-ancestor of x, we would have to check that it possesses each of a class of properties to which the property itself belongs. As Russell later observed, however, Kerry’s argument radically misconceives the nature of deduction. In deduction, a proposition is proved to hold concerning every member of a class, and may then be asserted of a particular member: but the proposition concerning every does not necessarily result from enumeration of the entries in a catalogue. Kerry’s position involves acceptance of Mill’s objection to Barbara, that the mortality of Socrates is a necessary premiss for the mortality of all men. The fact is, of course, that general propositions can often be established where no means exists of cataloguing the terms of the class for which they hold. (PoM, §496) Russell was thus making a version of Frege’s point about the Central African chief: grasp of a generalization does not, on a broad language-level understanding of the quantifier, require grasp of all its instances. Only on the narrow language-level understanding would a universal generalization mean the same as the conjunction of its instances.

Further reading For discussion of Frege’s account of the ancestral see Boolos (1985). On the impredicativity of definitions of the natural numbers in second-order logic see Parsons (1992) and Heck (2016).

8 EARLY PHILOSOPHY OF LOGIC

In Begriffsschrift itself Frege said little about what guided his choices in formulating his logical system. Two questions, in particular, remained unaddressed: how are logical truths distinguishable from others, and what justifies them? For clues to his answers to these questions we must consult what he wrote a little later: articles he wrote in 1881 and 1882, a draft entitled ‘Logic’, and the tantalizingly brief ‘17 key sentences’, seemingly written in the early 1880s in reaction to a logic treatise by Lotze, his former teacher at Göttingen.

Psychologism and empiricism Frege famously insisted throughout his career that logic is not psychology: in the ‘17 key sentences’, for instance, he noted that ‘no psychological investigation can justify the laws of logic’ (PW , 175). Anyone who read only him might get the impression that he stood alone in opposing to psychologism, but in fact the view originated with Kant and by the late 19th century was something of a commonplace. (At this point in the ‘17 key sentences’ he was agreeing with Lotze.) The argument against psychologism most often advanced by Frege’s contemporaries was that psychology, being descriptive, cannot explain the normativity of logic; it can only explain why we think as we do, not whether we are right to do so. Frege certainly shared this view, likening the role of truth in logic to that of goodness in ethics. Although our actions and endeavours are all causally conditioned and explicable in psychological terms, they do not all deserve to be called good. Here, too, we can talk of justification, and here, too, this is not simply a matter of relating what actually took place . . . . Certainly we say

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‘tout comprendre, c’est tout pardonner’, but we can only pardon what we consider not to be good. (PW , 4) Elsewhere, however, he made use of the Kantian distinction between reasons and causes. ‘Taking something to be true’, Kant said, ‘is an occurrence in our understanding that may rest on objective grounds, but that also requires subjective causes in the mind of him who judges.’ (A820=B848) This contrast between ‘the causes which merely give rise to acts of judgment’ and ‘the grounds which justify the recognition of a truth’ (PW , 2–3) remained a constant feature of Frege’s view thereafter (e.g. PW , 147). Since psychology is empirical and logic not, Frege inferred that empiricism is false. A farmer may mistake a frequent conjunction of events (kinds of weather and phases of the moon, for instance) for a genuine connection. Empiricists make the same kind of error more systematically, mistaking causal explanations of our acts of judgment for justifications of their correctness. Psychological laws ‘have no inherent relation to truth whatsoever’ (PW , 2). The way Frege attacked empiricism creates the impression that it was the dominant philosophical view, just like the psychologistic conception of logic on which he supposed it to be based, but by then it was in fact already in decline in Germany. Throughout his life Frege never tired of finding dead horses to flog. Although Kant distinguished between reasons and causes, he nonetheless conceived of the former as sufficiently subjective that my reasons for a judgment might be different from yours. Frege, by contrast, quoted with approval (Gl, §17) Leibniz’s remark in the New Essays (bk 4, ch. 7, §9) that the issue ‘is not about the history of our discoveries, which differs in different men, but about the connection and natural order of truths, which is always the same’. In the Preface to Begriffsschrift Frege drew just such a contrast. ‘How we have gradually arrived at a given proposition’ is a psychological matter, he said, and might well be different for different people, whereas ‘how we can finally provide it with the most secure foundation’ is the same for all. We must therefore distinguish two sorts of explanation, psychological and logical: the former may explain the fact that a proposition has ‘come to the consciousness of a human mind’; the latter concerns ‘the most perfect method of proof ’, which makes use only of ‘the laws on which all knowledge rests’. He thus gestured towards an impersonal conception of epistemology as ‘sharply distinguished’ (CP, 111) from psychology and concerned with an objective relation between facts by means of which one justifies another; but he never successfully articulated what this objective relation is.

Judgment and truth In Begriffsschrift Frege seems, somewhat obscurely, to have regarded judgment as objective, but by the time of his undated logic draft he had shifted to something more like Herbart’s view that it is a psychological act directed at an objective judgable content. In that case, though, the notion that the non-psychological

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science of logic might be about judgment could not be sustained. Instead of attempting to render judgment non-psychological by linking it as closely as possible to truth, he now short-circuited this strategy by saying that logic is already about truth. Psychology, he said, is only concerned with truth in the way every other science is, in that its goal is to extend the domain of truths; but in the field it investigates it does not study the property ‘true’ as, in its field, physics focuses on the properties ‘heavy’, ‘warm’, etc. This is what logic does. It would not perhaps be beside the mark to say that the laws of logic are nothing other than an unfolding of the content of the word ‘true’. (PW , 3) Frege thus characterized logic as concerned with the laws of truth. Since truth is objective—‘What is true is true independently of our recognizing it as such’ (PW , 2)—it follows that logic is objective too. There is thus a shift of focus: in Begriffsschrift he still sympathized with the Kantian view that only in judgment do ideas acquire the sort of unity that makes them candidates for truth or falsity; now he took the judgable content to be the main unit. In the ‘17 key sentences’ he was explicit that it is the mark of a thought—not a judgment, note—to be true or untrue. As Dummett (1981a) was the first to point out, Frege’s primary concern in these sentences was with opposing Lotze’s proposal that a combination of ideas could be made into a candidate for truth by the addition of ‘auxiliary thoughts’ (Nebengedanken). The combinations which constitute the essence of thinking are different in a peculiar way from the associations of ideas. The difference is not a matter of some auxiliary thought which adds the justifying ground for the connection. (PW , 175) Frege now made explicit, as he had not in Begriffsschrift, that this required the components of the thought to be objective too—‘things, properties, concepts, relations’ (PW , 174). Henceforth, he rejected the notion that there is a route to the objectivity of judgment which starts from a mentalistic base. As we have seen, part of Frege’s purpose was to reject the traditional assumption that every judgment has subject-predicate form: he was now suggesting that the content stroke plays the unificatory role traditionally assigned to the verb of a subject-predicate proposition. Perhaps he would have been better to express this by saying that the content stroke is the logical verb, even though ordinary grammar does not treat it as one. Even with this clarification, however, it is far from clear that he was right to follow Kant in this way. In the Tractatus Wittgenstein offered an argument intended to show that logic should not recognize such a separation between content and verb. An illustration to explain the concept of truth. A black spot on white paper; the form of the spot can be described by saying of each point of the plane

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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), a negative fact. If I indicate a point of the plane (a truth value in Frege’s terminology), this corresponds to the assumption proposed for judgment, etc. etc. But to be able to say that a point is black or white, I must first know under what conditions a point is called white or black; in order to be able to say ‘p’ is true (or false) I must have determined under what conditions I call ‘p’ true, and thereby I determine the sense of the proposition. The point at which the simile breaks down is this: we can indicate a point on the paper, without knowing what white and black are; but to a proposition without a sense corresponds nothing at all, for it signifies no thing (truth value) whose properties are called ‘false’ or ‘true’; the verb of the proposition is not ‘is true’ or ‘is false’—as Frege thought—but that which ‘is true’ must already contain the verb. (4.063) Wittgenstein’s complaint was that if we could separate the judgable content from the verb, as Frege held, it would be comprehensible that someone should be able to express the content without knowing what truth and falsity are, just as someone who did not know what white and black are could name a point on a sheet of paper. Wittgenstein held that this is absurd: whether a content is judgable is not a matter of happenstance but an intrinsic feature of it. What makes a content judgable just is that it is capable of truth or falsity. It does not need to be accompanied by the ‘I think’ in order to achieve this. This is perhaps a case (one of several we shall encounter) in which Wittgenstein grasped more securely than Frege an insight that was originally Frege’s; for his point was that Frege’s context stroke, like Lotze’s ‘auxiliary thought’, attempted, impossibly, to impose the structure of judgment on what otherwise lacked it.

Defining the scope of logic Even if we recognize logic’s distinctive concern with truth, we still have the difficulty of converting this into a criterion of logicality. Perhaps it is a property of the concept of truth that the content of the sentence ‘My book is on the table’ falls under it, but that is not sufficient to make this content logical. Such properties as this are to be excluded, because they are in some sense external to truth, not part of its nature, but Frege’s early writings give us no clue to the sense of externality in question. This is where his rejection of a non-epistemic notion of modality hampered him, by barring him from using the familiar modern criterion that a truth is logical if it is true in every possible world. Although Frege regarded generality as a mark of the a priori, he took logic’s generality to be of a different order from geometry’s: the axioms of geometry apply to everything spatial, whereas the basic laws of logic apply to anything

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whatever. Nonetheless, although logic is in this sense maximally general, he stopped short of claiming that this characterizes it, i.e. that all maximally general truths are logical. He seems instead to have hoped to characterize logic on the basis of its grounds: the axioms of geometry are grounded in intuition; the basic laws of logic, not. But if not in intuition, then what? He found it easier to say what logic does not depend on than what it does. He alluded more than once to the impossibility of denying a logical law—if we try, ‘complete confusion ensues’ (Gl, §14)—but confusion is a psychological state and can therefore be at best a symptom, not an explanation, of logical error. One might wish Frege had been more explicit. He gestured towards a Kantian explanation according to which what grounds the basic laws is that they are constitutive of thought itself. When a judgment is called a logical truth, this is not a judgment about the conditions. . . which have made it possible to form the content of the proposition in our consciousness; nor is it a judgment about the way in which some other man has come, perhaps erroneously, to believe it true; rather it is a judgment about the ultimate ground upon which rests the justification for holding it to be true. (Gl, §3) He thus seems to have held that one judgable content may be grounded in another, independently of us, and that our recognition that this relation obtains is what justifies us in judging it to be a logical truth.

Metalogic We have already noted Begriffsschrift’s occasional unclarity about the distinction between a sentence and the content it expresses. In the ‘17 key sentences’ a few years later, however, Frege drew this distinction explicitly and was willing to speak of a sentence’s being true only derivatively, i.e. when it expresses a true thought. Implicit there, too, is a somewhat subtler distinction between a mere string of words and a sentence that expresses a judgable content—or, at the sub-sentential level, between a mere word and what he later called a sign, i.e. a unit that expresses a content. He explained the intended interpretation of the components of his concept-script at the outset, and hence presented them not just as words but as signs. The laws of logic are those of truth, and hence apply to anything that aims at truth, i.e. anything thinkable. Some commentators have inferred from this that ‘no general meta-logic was, strictly speaking, possible for Frege’ (Hintikka 1988, 1). We can give different formulations of logic, formulations that differ with respect to what logical constants are taken as primitive or what formulas are taken as formal axioms, but we have no vantage point from which we can survey a given formalism as a whole, let alone look at logic whole. (Dreben & van Heijenoort 1986, 44)

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Why not? Their idea seems to be that to ‘survey a given formalism as a whole’ would be to survey thought, and hence to stand, impossibly, outside of it; but it is not clear either that this is so, or that Frege ever supposed it was. Nonetheless, it must be conceded that although he made various remarks about his choice of signs and stated explicitly that other choices would be possible, he did not in Begriffsschrift offer formal proofs of metalogical claims. He did not prove, for instance, that ‘∼’ and ‘⊃’ are adequate to express all truth-functions, or that his basic laws suffice to prove all propositional tautologies. Did it simply not occur to him to try, or did he, as some commentators have suggested, have a principled objection to the very idea? There is a possible explanation for Frege’s failure to engage in metalogical reasoning which does not require him to have had a principled objection to it. Some reviewers had compared his concept-script unfavourably to Boole’s on the ground that the latter had multiple interpretations and therefore seemed more widely applicable. In his attempt to explain why this extra generality was illusory, Frege stressed that his concept-script consisted of signs, not mere words. This defence of his system against Boole’s set him on a course that in one respect was unfortunate: he acquired a hostility to empty formalism which blinded him to the fruitfulness of studying a formal system in its own right, independent of its interpretation. This blindness is most apparent in his reaction to Hilbert’s work on metageometry. One who stands outside a particular formal system for geometry plainly cannot be accused of the kind of conceptual incoherence which, on the ‘no metalogic’ view, is involved in standing outside a formal system for logic. Frege himself explicitly acknowledged that ‘for purposes of conceptual thought’ (Gl, §14) the study of non-Euclidean geometry is possible. On this point his letters to Hilbert display not principled opposition but incomprehension: he was slow to understand how an uninterpreted formal system could fruitfully be studied as Hilbert proposed. Whatever the explanation for Frege’s failure to engage systematically in metalogical reasoning, there is in any case a considerable difference between surveying a particular formalism and surveying logic. When it comes to the latter, the ‘no metalogic’ view is on stronger ground. If logic is the science of the most general laws of truth, there is a ‘logocentric predicament’ (Sheffer 1926, 228) implicit in any attempt to critique it without presupposing the very thing under critique. One salient instance is the inscrutability of the notion of truth. ‘What true is’, Frege said, ‘I hold not to be explicable.’ (PW , 174, amended) The standard translation unhappily renders ‘explicable’ (erklärbar) here as ‘definable’, but whenever someone claims that a concept is indefinable, the obvious question is, in terms of what? Tarski, for instance, proved precise formal results to the effect that truth is definable in some circumstances, indefinable in others, but these are plainly not what Frege had in mind. After all, truth is trivially definable, e.g. by saying that a thought is true just in case its negation is false. The objection to this definition is that it could not be used to explain the concept to someone ignorant of it, since no

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one understands falsity who does not also understand truth. (This is a point about concepts, not words: an English learner might know the word ‘false’ without yet having come across ‘true’.) Logic is the study of the properties thoughts have by virtue of their relationship to truth, so any explanation of truth is as circular as a justification of logic’s laws. In neither case, though, does it follow that there is nothing useful to be said on the matter (see Dummett 1974). It is a consequence of the logocentric predicament that one kind of justification of a logical law—the kind that would persuade someone to whom logical reasoning was alien—is not to be had; but this leaves open the possibility of another kind of justification— one that by the manner in which it proceeds exhibits the law not merely as true but as logically true. As instances one might cite the explanations of the basic laws given in Begriffsschrift: the reader may confirm by inspection that these explanations appeal not to intuition but only to the meanings Frege has stipulated for the signs they contain.

Further reading On the dating of the ‘17 key sentences’ see Hovens (1997). Rein (1985) discusses Frege’s attitude to natural language. On Frege’s rationalism see Burge (1992). On the logocentric predicament see Nagel (1997). The ‘no meta-logic’ view is due to van Heijenoort (1967). The most influential modern discussion of it is by Ricketts (1985), and it deserves a more detailed critique than I have had space for here: see Stanley (1996), Shieh (2002) and Sullivan (2004a).

9 THE HIERARCHY

Frege based his account of quantification on the idea of an argument-place in which different expressions may be substituted. Although he recognized in Begriffsschrift that these argument-places may be of different kinds, he did not yet explain this, beyond a single, somewhat obscure reference to a distinction in ‘rank’ between ‘the number 20’ and ‘every even number’. Soon thereafter, however, he began to articulate a classification not only of expressions but also of their contents into ‘levels’.

Levels From 1882 onwards, Frege used the term ‘unsaturated’ (PMC, 101) (which he borrowed from chemistry) for the remainder that results from removing one expression from another in accordance with Begriffsschrift, §9: he called the remainder obtained from a sentence by removing a single expression a ‘concept-word’; by removing two, a ‘relation-word’. He then called the conceptual contents of such unsaturated expressions ‘concepts’ and ‘relations’ respectively. From 1882, too, he employed a distinctive locution according to which an entity ‘falls under’ a concept. In Begriffsschrift Frege already held that argument-places are not mere gaps: Greek letters are needed to indicate whether they are instances of the same or of different substitutions. The key step, though, was that in order to ‘take into account’ which expressions may be legitimately substituted he recognized, and classified hierarchically, different kinds of argument-place, nowadays called ‘types’: singular terms, being saturated, are at level zero; concept-words with gaps suitable for singular terms are of level one; those with gaps suitable for first-level conceptwords are of level two; and so on. (For simplicity, I here ignore complications

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arising from multiple argument-places.) Just when he devised this classification is uncertain, but it is at least fairly explicit in the Grundlagen (§53). In principle, there is no finite limit to the levels we might form, but Frege did not explicitly discuss levels higher than three. One reason for this is that in his later work he thought he had a technical device (to be discussed in Chapter 18) for replacing concept-words of higher levels with singular terms. Another reason, perhaps, is that to express the general notion of a level would require a prior grasp of the natural numbers. On Frege’s conception each entity occurs at only one level in the hierarchy. If, for instance, ‘Socrates’ is at level zero, then in ‘Socrates is wise’ Socrates falls under the first-level concept expressed by ‘ξ is wise’. In ‘Wisdom is a virtue’ this first-level concept falls in turn under a second-level concept. Notice, though, that the concept of being Socratic—under which fall just those first-level concepts satisfied by Socrates—is also of level two. ‘Wisdom is Socratic’ therefore places a first-level concept under a second-level one, but is in the end just a slightly more elaborate way of saying ‘Socrates is wise’. More generally, for any expression A of level n in the hierarchy there is a mirroring concept-word PA (γ ) of level n + 2 defined by PA (Γ ) =Df Γ (A). What in Begriffsschrift had been a difference of ‘rank’ between ‘Socrates’ and ‘every man’ now becomes a difference of level: the quantifier phrase is two levels higher than the name which instantiates it.

Applications On the basis of this classification into levels Frege offered explanations (which, like the classification, have since passed into logical orthodoxy) of confusions he detected in other writers. One is the confusion between ‘Socrates is mortal’ and ‘Man is mortal’: in the former case, he said, an object falls under a concept; in the latter, one concept is subordinate to another (see Gl, §47). The verb is plays quite different logical roles in the two sentences, as is apparent when we formalize them in the concept-script. One way of defining a concept is as a conjunction of more primitive concepts which Frege called its ‘characteristic marks’: we might, for example, define human by the characteristic marks rational and animal. It would then be an instance of the confusion just mentioned to regard the characteristic marks as properties of the concept human, rather than of the individuals who fall under it. On this analysis rational applies to objects, not to concepts, but there are some concepts which do apply to concepts, not to objects. For instance, human is a species, but each human is not. Species is thus a second-level concept, man a first-level one. This allowed Frege to provide a new account of the error in the ontological argument for the existence of God. Kant’s objection had been that existence is not a genuine concept (‘predicate’, in his terminology), whereas for Frege it is a genuine concept, but of the second level, not the first. Since the perfections that are characteristic marks of the concept God are of the first level, existence is not a candidate to be one of them (Gl, §53).

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Frege’s account also made a concept radically different in nature from an aggregate. An aggregate (nowadays often called a ‘fusion’ or ‘mereological sum’) is merely a ‘many’ conceived of as ‘one’, whereas a concept is a distinct entity at a different level from whatever falls under it. A concept may be empty, i.e. have nothing falling under it, whereas an aggregate ‘consists of objects’ and hence ‘must vanish when these objects vanish. If we burn down all the trees of a wood, we thereby burn down the wood.’ (CP, 212) It follows that there is a radical logical difference between concept-words and proper names. An empty concept-word (e.g. ‘round square’) refers to a concept under which nothing falls, whereas an empty proper name (e.g. ‘Sherlock Holmes’) is a logical error, he then thought, since it is a presupposition of discourse that a name designates something. Anyone who uses the name ‘Leo Sachse’ does not assert, but simply presupposes, that he exists (PW , 60). Failure to observe this distinction had led earlier writers to slide from a concept-word (e.g. ‘a positive square root of two’) to a putative singular term (e.g. ‘the positive square root of two’) without providing the required demonstration that the term designates something. Frege further detected (Gl, §57) a confusion between what are now often called the is of predication (‘Cicero is an orator’) and the is of identity (‘Cicero is Tully’), as a result of which it was widely, and implausibly, claimed that for a subject-predicate proposition to be true the subject had somehow to be equal to the predicate. The confusion had in fact been pointed out by Aquinas (see Weidemann 1986, 183), but few logicians had taken any notice. Frege distinguished analogously between ‘there are teachers’ and ‘Leo Sachse is’: the former ascribes the second-level property of being instantiated to the first-level property of teacherhood; the latter attempts to ascribe a property to an object, namely that of having being. A concept under which no object falls is in an obvious sense empty, but the concept under which every object falls is empty in a converse sense: it ‘can no longer have any content at all’ (PW , 63); it is ‘an apotheosis of the copula’ (ibid., 64), hence only a ‘quasi-concept’, not a concept proper. It is only in virtue of the possibility of something not being wise that it makes sense to say ‘Solon is wise’. The content of a concept diminishes as its extension increases; if its extension becomes all-embracing, its content must vanish altogether. (Gl, §29)

Concept and object Function-argument decomposition reveals structure in an already existing judgment. The concept-script, Frege claimed, commands a somewhat wider domain than Boole’s formula-language. This is a result of my having departed further from Aristotelian logic. For in Aristotle, as in Boole, the logically primitive activity is the formation of

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concepts by abstraction, and judgment and inferences enter in through an immediate or indirect comparison of concepts via their extensions. . . . As opposed to this, I start out with judgments and their contents, and not from concepts. . . . I only allow the formation of concepts to proceed from judgments. (PW , 15) How, though, do we ‘start out with judgments’? What are the resources from which they are formed in the first place? The account we have been considering explains how higher-level concepts are formed from lower-level ones, but not how concepts at the atomic level are to be distinguished from objects. Whether or not Frege had read Boole before he wrote Begriffsschrift, he certainly did so soon afterwards when reviewers suggested that his concept-script had no expressive advantage over Boole’s. Responding to this criticism led him for the first time to say something about atomic sentences, because his system’s advantage over Boole’s lay precisely in its ability to deal with an atomic sentence which is already articulated. We may infer from this that at least the properties and relations which are not further analysable must have their own simple designations. But it doesn’t follow from this that the ideas of these properties and relations are formed apart from objects: on the contrary they arise simultaneously with the first judgment in which they are ascribed to things. Hence in the concept-script their designations never occur on their own, but always in combinations which express contents of possible judgment. (PW , 17) The significant point here is that Frege now used the distinction between saturated and unsaturated, previously explained in connection with the decomposition of already given judgments, as a way of categorizing the contents from which those judgments are first formed. ‘I could compare this’, he said, ‘with the behaviour of the atom: we suppose an atom never to be found on its own, but only combined with others, moving out of one combination only in order to enter immediately into another.’ (PW , 17) Just as an oxygen atom does not occur naturally on its own but only in combination (in an O2 molecule, for instance), so a concept ‘is unsaturated in that it requires something to fall under it; hence it cannot exist on its own’ (PMC, 101). Dummett (1973, 27–33) has objected that when applied to simple sentences at the bottom of the hierarchy Frege’s account is circular. If the unsaturated expression ‘ξ is wise’ is obtained from ‘Socrates is wise’ by concept extraction, how are we supposed to understand the latter in the first place? Perhaps this objection is answerable (see the next chapter). There is also a converse objection, however. What we have just considered was a query as to how on Frege’s conception there could be any route to an understanding of an atomic sentence. The converse objection is that this conception allows there to be more than one such route. We have noted already that ‘Socrates is wise’, which is the result of instantiating the

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first-level concept-word ‘ξ is wise’ with the name ‘Socrates’, can equally be reexpressed as the instantiation of the second-level concept-word ‘φ is Socratic’ by the first-level concept-word ‘ξ is wise’. This is not yet troubling (because when written in the primitive notation the two analyses collapse into one), but why proceed in this way at all? Why not think of it as ‘wisdom is Socratic’, i.e. the instantiation of the first-level concept-word, ‘ξ Socratic’ by the name ‘wisdom’. Now we do, puzzlingly, seem to have two genuinely distinct analyses of a single judgable content. To resolve this difficulty, we would need an independent reason to categorize Socrates as an object and wisdom as a concept, not the other way round. Frege himself was uncharacteristically casual about this issue. Despite stressing the importance of the distinction between concept and object at the beginning of Grundlagen, he relegated his explanation of it to a footnote. ‘A concept is for me that which can be predicate of a singular judgment-content, an object that which can be subject of the same.’ (Gl, §66n) This scarcely helps, since it merely reduces the object-concept distinction to the subject-predicate one. (Readers of Begriffsschrift might also be surprised to see him here giving subject-predicate judgments a privileged role in his account.) Elsewhere in Grundlagen he approached the matter more linguistically: an object is what a singular term refers to, a concept what a concept-word refers to. Only when conjoined with the definite article or a demonstrative pronoun can [a general concept word] be counted as the proper name of a thing. . . . The name of a thing is a proper name. . . . As soon as a word is used with the indefinite article or in the plural without any article, it is a concept word. (Gl, §51) So ‘a man’ is a concept-word, and ‘the man’ a name. Frege thus used a linguistic distinction (between names and concept-words) as a guide to an ontological one (between objects and concepts). Yet he had already noted in Begriffsschrift that ordinary language obscures the difference in rank between ‘the number 20’ and ‘every even number’. Why trust it, then, to get the distinction between concept and object right? Might it not merely split up the sentence ‘according to our way of looking at it’?

Objecthood and identity To hold that logic distinguishes between concepts and objects is not yet to address whether it should also distinguish among objects. In his later work Frege explicitly denied that it should, claiming that objects form a single logical type and that the only distinctions logic need recognize are those that flow from the notion of unsaturatedness. Quite when he took this further step is somewhat obscured by the informal character of his writings during the 1880s: in ordinary language, after all, there are many contexts where singular terms cannot be inter-substituted, but it is a further question whether this is a logical distinction.

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Frege’s typing of argument-places automatically limits the applicability of any predicate to a single type. One consequence is that type-ambiguous generalization is impossible. Another is that identity is distinctively a relation between objects. For any two objects it is meaningful, even if false, to ask whether they are identical, whereas to ask this concerning concepts is strictly speaking meaningless: it is what some philosophers like to call a ‘category error’. In his early work, indeed, Frege did not even discuss how to individuate concepts at all, and therefore left it open whether there is even a relation analogous to identity for them. For Frege, then, there is an intimate connection between objecthood and identity: objects form the domain of the identity relation, and so an object is that for which there is a ‘criterion of identity’—a criterion for determining whether or not it is identical to any other object. ‘For every object there is one type of proposition that must have a sense, namely the recognition-statement.’ (Gl, §106) It might seem, then, that we could exploit this connection between identity and objecthood to generate a characterization of the latter in terms of the former: an object is whatever is in the domain of the identity relation. Such a characterization might save us from some implausible ontological extravagances, by giving us a reason to say that certain phrases which satisfy a purely linguistic criterion for being singular terms, such as ‘the sake of decency’ or ‘how I roll’, nonetheless fail to designate anything, because identity statements involving them have been given no content. It is difficult to see, though, how this strategy would help in deciding whether it is Socrates or wisdom that is logically the subject of ‘Socrates is wise’, since the problem of determining the domain of the identity relation is no easier to solve.

Further reading Sullivan (2010) enlarges on Dummett’s criticism of Frege’s application of the notion of function-argument complexity to atomic sentences. Ackrill (1957) argues that Plato distinguished the ‘is’ of identity from the ‘is’ of predication in the Sophist; Lockwood (1975) not only disagrees, but argues that we can explain the confusion without appealing to two uses of ‘is’; Sommers (1982, ch. 6) also questions the need for the distinction as Frege drew it. On Frege’s notion of selfsubsistence see McLeod (2017). Quine (1957, 20) coined the slogan ‘no entity without identity’ to highlight the relevance of identity conditions for objecthood.

10 GRUNDLAGEN I: THE CONTEXT PRINCIPLE

How are numbers given to us? The Grundlagen of 1884 was Frege’s attempt to answer this question. In the book’s Introduction he identified three principles which, he said, guided this attempt: always separate the psychological from the logical, the subjective from the objective; never ask for the meaning of a word in isolation, but only in the context of a proposition; never lose sight of the distinction between concept and object. The first and third of these we have encountered already, but the second is new. Dummett later dubbed it the ‘context principle’ and called §62 of the Grundlagen, in which Frege explained it, ‘arguably the most pregnant philosophical paragraph ever written’ (1991b, 111). There can be no doubting the importance Frege attached to the principle: having listed it in the Introduction as one of three principles that would guide his inquiry, he repeated it twice in the main text (§§60, 62) before mentioning it again in the conclusion (§106). We therefore need to examine what it amounts to.

Semantic incompleteness Let us call an expression ‘semantically incomplete’ if it has meaning only in the context of a whole sentence. The context principle is then the claim that all words are semantically incomplete. The manner in which Frege stated this suggests that he thought it a platitude. Why? He can hardly have been alluding to context sensitivity, since he proposed the principle in a book about arithmetic, where context sensitivity is little in evidence; and anyway what the meaning of a

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context-sensitive word typically depends on is not just the sentence in which it occurs but the circumstance of utterance. More plausibly, his concern was with conceptual content. The sentence is the smallest unit which is a candidate for truth or falsity and hence, given logic’s distinctive concern with truth, for logical assessment. The logical role of a sub-sentential expression, therefore, is exhausted by its contribution to determining the judgable content of the whole. So the context principle places a constraint on the compositionality principle: according to the latter, the content of a sentence is composed of the contents of its parts; according to the former, there is no more to the content of a part than the contribution it makes to determining the content of the whole. ‘It is enough if the proposition as a whole has a sense; it is this that confers on its parts also their content.’ (Gl, §60) We must not, of course, interpret this as saying that in order to understand a word I must understand every sentence in which the word occurs: I may understand this word’s contribution perfectly, but not understand the sentence as a whole because I do not understand some other word in it. My understanding of a word thus consists in a schematic understanding of a range (depending on its grammatical type) of contexts. Frege recommended using the context principle to explain ‘contextual’ definitions—definitions which allow us to eliminate an expression from whole sentences. These are to be contrasted with ‘explicit’ definitions, which give a sign meaning independent of context. Both kinds ‘bring about an extrinsic simplification by the establishment of an abbreviation’ (Bs, §24), but only a contextual definition can alter the grammatical structure of whole sentences. He instanced the infinitesimal calculus, where mathematicians were in the habit of giving a meaning to various sentences involving terms such as ‘dx’ or ‘dy’, e.g. df (x) = g(x)dx, without giving any meaning to the expressions on their own (Gl, §60). After 1890 Frege drew a further distinction (to be discussed in Chapter 14) between significance (‘sense’) and thing signified (‘reference’). Modern commentators therefore split the context principle correspondingly into two, but this is anachronistic in relation to Grundlagen, where objects and concepts are alike described as ‘objective ideas’ (§27n) and there is no trace of the later distinction. It is in his discussion of contextual definitions that this lack is most glaring. If he had already possessed the sense/reference distinction, he might have granted that the mathematicians’ rules had given ‘dx’ a sense, but not that they had given it a reference. At the time of Grundlagen, though, he conceived of the content of a singular term as its reference, that of a concept-word as more like its sense. He was therefore tempted to treat the contextual definition as introducing us to the content of an expression, but leaving it undetermined what kind this content is, when he might have been better to say, using his later distinction, that it introduces us to a sense but leaves undetermined whether this sense refers to anything.

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The Lockean model The context principle’s main role for Frege was to facilitate the rejection of a psychologistic conception of content. Although he insisted in Begriffsschrift on an anti-psychologistic conception of logic, and took this to be intrinsically related to the objectivity of judgment, he did not there insist that the content of judgment be correspondingly objective. In 1882 he was still content to speak of a symbol as a ‘focus about which ideas gather’; these ideas belong to an ‘inner world’ in which we can ‘move about . . . at will’ (CN, 83–4). In Grundlagen, however, he rejected this conception—often called ‘Lockean’, although Locke did not originate it—of words as standing ‘for nothing but the ideas in the mind of him that uses them’ (Essay, bk III, ch. ii, §2). Instead Frege now distinguished two kinds of idea, subjective and objective. An idea in the subjective sense is what is governed by the psychological laws of association; it is of a sensible, pictorial character. An idea in the objective sense belongs to logic and is in principle non-sensible, although the word which means an objective idea is often accompanied by a subjective idea, which nevertheless is not its meaning. Subjective ideas are often demonstrably different in different men, objective ideas are the same for all. (§27) Thereafter, Frege used ‘idea’ only for the subjective kind and insisted that they are no concern of logic. In Grundlagen Frege’s particular interest was in number-words, but he rejected the Lockean model not only for them but for all words, offering two sorts of argument. The first argument trades on a conception (often called ‘Cartesian’) of the mental as intrinsically private: ideas, being private, cannot suffice to explain the judgable content expressed by a sentence, since this is communicable. Locke himself used this as an explanation of our need for words as well as ideas. Frege held that words are still not enough, and that the conceptual contents they express are (often) public. We can distinguish, he said, between a private meaning of the word ‘blue’, which picks out my subjective sensation, and a public meaning, with respect to which it is an objective matter what colour an object is. ‘Often’, he said, ‘a colour word does not signify our subjective sensation’ (§26). Why did he not say ‘always’ rather than ‘often’? If he had, he might have been credited with formulating the private language argument that Wittgenstein famously advanced in the Investigations. Even in this partial form, however, Frege was making the point that the communicable content of a sentence cannot be explained wholly in terms of private ideas. What led him to limit himself to the partial statement was that he was dodging away from thinking through how, on his radically private conception of the mental, I can talk publicly about my sensations at all. His argument may also be contested, though, for reasons not connected to the private language argument. It is not as clear as he supposed that for information to be transmitted from you to me, some item must be available to both of us. Heat is transferred by

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conduction from one end of a metal bar to the other without any molecule travelling the whole length. We do of course talk of ‘sharing thoughts’ and ‘pooling information’, but Frege needed to show that this is not just a metaphor (as it is if I claim to feel your pain). Frege’s second argument against the Lockean model (Gl, §60) does not depend on the problematic privacy of the mental. Even in relation to some concrete objects such as the Earth, he suggested, our ideas are plainly inadequate. He noted that ‘the idea may be quite different in different men’ (Gl, §59), and hence gave the impression that he was merely making once again the earlier point about the privacy of the mental. In fact, though, he was now making the different point that whether my thought is genuinely about the Earth depends not at all on how accurate a mental image of it I am capable of bringing to mind. The image therefore cannot be the vehicle of meaning in the manner that Lockeans supposed. Although Frege conceded that we associate ideas with words, they are not a prerequisite for a word to have meaning. ‘That we can form no idea of its content is’, he suggested, ‘no reason for denying all meaning to a word, or for excluding it from our vocabulary.’ (Gl, §60) Wittgenstein made a similar point when he asked, ‘What makes my image of him into an image of him? Not its looking like him.’ (PPf , iii, §17) It is worth noting, though, that this second argument seems to apply only to treating images as vehicles of meaning, not ideas more generally. Locke himself could hardly complain at this treatment, given his talk of ‘pictures drawn in our minds . . . in fading colours’ (ibid., bk II, ch. x, §5), but a less pictorially minded ideationist (e.g. Lowe 2005, ch. 4) might resist.

Concept and object again In the last chapter I mentioned Dummett’s complaint that Frege had not explained how concepts can ‘arise simultaneously with the first judgments in which they are ascribed to things’. The context principle suggests a response: Frege could have said that our understanding of the concept-word ‘ξ is wise’ simply consists in a schematic understanding of the sentences of the form ‘A is wise’ from which it is extracted, and hence that understanding the concept-word is not prior to but simultaneous with understanding the sentence. On the other hand, it might be thought that in some cases we could explain our understanding of an expression ‘B’ simply by saying ‘This is B’ while pointing. In the literature this is sometimes called the ‘baptismal’ account or, following Wittgenstein (BB, 1), ‘ostensive definition’. The context principle then seems most plausible in the case of concept-words, least plausible in the case of baptism. Even so, it bears repeating that Frege asserted the context principle as applying unconditionally to all words, even baptismal names. Some commentators have resolved the perceived tension by reading the principle as making whether a term has content a question internal to language, to be settled by an inquiry into meaning. In its most extreme variant this interpretation renders the object named little

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more than a theoretical posit. It is surely not compulsory, though, to read the context principle in quite so internalist a spirit: alternatively one might explain baptism in a manner consonant with the context principle. In this connection it should surely give us pause that empirical concepts such as yellow seems to fall under both the preceding cases: we give meaning to sentences of the form ‘A is yellow’ by pointing at samples and saying ‘This is yellow’. Perhaps, then, baptism is not as antithetical to the context principle as first appeared. Even in the case of a concrete object the practice makes sense only against the background of previously accepted identity conditions for the kind of object being baptized. When I baptize an object, I should be thought of not as giving meaning to a name on its own, but rather as implicitly giving a content to a certain class of sentences in which the name may occur, namely ‘those which express our recognition of [it] as the same again’ (§62). In Grundlagen, of course, Frege was primarily interested in the case of numerals, which, he thought, belong to neither of the two kinds just mentioned: they are not concept-words; yet they cannot be given meaning by baptism, because their content is non-sensible. The context principle, he suggested, provides us with the only non-Lockean explanation for our ability to think about them. Dummett, in particular, has repeatedly (e.g. 1956) cited the principle in excoriation of nominalists. Even so, the real purpose of the context principle for Frege seems to have been to supply a uniform criterion, applicable to proper names and conceptwords alike, for whether a term has been given content, and hence to separate this from the question whether the content so given is an object or a concept.

Further reading On meaning and privacy see Craig (1982). On subjective senses of colour words see Dummett (1993a, ch. 9). Tappenden (forthcoming) discusses contextual definition in 19th-century mathematics. The locus classicus of the critique of ostensive definition is Philosophical Investigations, §§27–40. On the role of the context principle in Frege’s philosophy see Dummett (1991b, ch. 16).

11 GRUNDLAGEN II: ARITHMETICAL TRUTH

What grounds the truth of an arithmetical equation such as 7 + 5 = 12? Here are five possible answers: it is (a) not really true at all; (b) analytic in Kant’s sense; (c) a posteriori; (d) a priori but non-logical; or (e) logical. In Grundlagen Frege aimed to render plausible the last of these answers—arithmetic rests solely on logical grounds, so that no appeal to intuition, whether empirical or pure, is required to explain it—by the indirect strategy of refuting the others. Rather than treat the various positions in abstracto, however, he used previous advocates of them as his targets. His knowledge of the existing literature on the philosophy of arithmetic was probably not extensive—many of the passages he referred to were from a single collection of readings (Baumann 1868)—and therefore his choice of opponents was sometimes a little idiosyncratic. Moreover, the book betrays in places his philosophical inexperience as well as a distinct sense of haste. Nonetheless, many of his criticisms are devastating, and his writing justifies Dummett’s description of it as ‘the most brilliant piece of philosophical writing of its length ever penned’ (2007, 9).

Game formalism Frege should probably have started by refuting ‘game formalism’—the claim that arithmetic is a formal game not answerable to anything beyond the rules by which it is played, and an arithmetical sentence expresses not a thought but only a position in this game. Curiously, though, in Grundlagen he was content simply to assume that we do know some arithmetical truths. Perhaps it had not occurred to him that anyone might be daft enough to deny this. He contented himself with quoting Mill—‘The doctrine that we can discover facts . . . by an artful manipulation of language, is so contrary to common sense, that a person must have

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made some advances in logic to believe it’ (1843, bk 2, ch. 6, §2)—before noting dismissively that Mill is here criticizing a kind of formalism that scarcely anyone would wish to defend. Everyone who uses words or mathematical symbols makes the claim that they mean something, and no one will expect any sense to emerge from empty symbols. (§16) Some years later, however, Thomae explicitly canvassed a version of game formalism. Arithmetic is for the formal conception a game with signs which one may well call empty, meaning thereby that they do not have any content (in the calculating game) except what is attributed to them with respect to their behaviour under certain combinatorial rules (rules of the game). A chess player makes use of his pieces similarly: he attributes certain properties to them that constrain their behaviour in the game, and the pieces are only external signs for this behaviour. (1898, §1) Thomae went on to concede, though, that despite the similarity between arithmetic and chess, there is one point of difference. The rules of chess are arbitrary; the system of the rules of arithmetic is such that, by means of simple axioms, the numbers can be related to intuitive aggregates and consequently perform us an essential service in the knowledge of nature. (Ibid.) The difficulty with this, as Frege observed (Gg, II, §91), is that it leaves the ‘essential service’—the application of arithmetic in counting—wholly unexplained. Why should this one game be applicable when others are not? Moreover, game formalism could not be an adequate account of the whole of mathematics: even if chess itself is just a game, meta-chess is not: we can prove results about chess (e.g. that one cannot force checkmate with only a King and a Bishop), and these are not purely formal in character. In his later criticisms of the position Frege treated Thomae (a colleague of his at Jena and even a personal friend) with brutal and extended contempt—the journal editors had to delete from the published version of one of his polemics, a description of Thomae’s work as ‘bullshit’—but the underlying point survives Frege’s regrettable phrasing of it.

Leibniz Frege also gave short shrift to the view that arithmetical equations follow trivially from explicit definitions of the terms occurring in them, and hence are analytic on Kant’s narrow understanding of that word. As his exemplar of this view he used Leibniz, who had argued in the New Essays (ch. 4, §10) that if we define

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2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1, then 4 = 3 + 1 (by the definition of 4) = 2 + 1 + 1 (by the definition of 3) = 2 + 2 (by the definition of 2). The equation 4 = 2+2 therefore seems to follow from the definitions by nothing more elaborate than the transitivity of identity. Unfortunately, however, a step has been omitted: the middle line makes surreptitious use of the associativity of addition, obscured in the presentation just given by the omission of brackets: ‘2 + 1 + 1’ is ambiguous between ‘(2 + 1) + 1’, which is what we obtain from the previous line, and ‘2 + (1 + 1)’, which is what we need in order to derive the next. Leibniz’s proof does not demonstrate the analyticity of the equation, since we have been given no reason yet to think that the associativity of addition is analytic. Of course, merely to point out this fallacy, as Frege did in Grundlagen (§6), is not enough to show that no proof of the equation from explicit definitions is possible, but he did not have at his disposal the sort of metamathematical resources that would be needed to demonstrate such an impossibility. So he was in fact no more able than Kant had been to prove conclusively that arithmetic is not analytic in this restricted sense.

The a posteriori Consider now Mill’s view that arithmetical equations are generalizations from experience. This, Frege said, confuses an equation with its application. That 3+2=5 does not entail that three litres of fluid added to two make five, for instance, since if the first fluid is water and the second alcohol, we obtain less than five litres of mixed fluid (because the molecules intermingle). Frege’s argument was too swift, however. He did not consider how to respond to someone who concedes that the arithmetical equation has counterexamples, but insists that it is nonetheless confirmed in the overwhelming majority of cases. The point to stress if we want to fill out his sketch is that in practice there is no experiment we are willing to treat as a refutation of the equation. Quine of course suggested that this is merely a matter of degree: we choose whatever emendation mutilates our theory least, and in practice it is never basic arithmetic that we change. The crucial question is whether this is a matter of convenience or of principle. On Frege’s view there is a reason why an experiment is never taken to refute an arithmetical equation.

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We noted earlier that Frege conceived of thoughts as having ideal justifications which may well not be the same as the routes by which we come to know them. So although I may grasp numerical concepts initially through counting, or come to know an arithmetical equation by means of the empirical properties of my pocket calculator, this does not refute his claim, which is only that there is a route independent of such facts, not that it is the only possible route. Frege also objected that treating arithmetical equations as empirical would have the consequence that many of them are outside our ken, because we could not carry out the requisite experiments. The conclusion he drew was that the justification of arithmetic has to go via general laws, which allow us to establish the particular cases. The justification for these general laws cannot be enumerative induction, however. An inductive justification of a law requires that there be instances of the law having some other justification, but in the arithmetical case it is hard to see what these instances could be except numerical equations. So the question is pushed back to them; and, as we have seen, he objected to the idea that they can be justified empirically. It is worth noting, though, that although enumerative induction is most often invoked in connection with empirical generalizations, it could in principle be used within the realm of the a priori. Even if numerical equations are not empirical, there remains the possibility that we might justify an arithmetical generalization inductively via its instances. Suppose, for instance, that we had come to know a large number of instances of the commutative law of addition (2 + 3 = 3 + 2, 7+5 = 5+7, etc.). Would our knowledge of these equations give us any inductive ground to advance to the truth of the general law? Frege argued that it would not, on the ground that each natural number (unlike a point in space, for instance) has properties that are specific to it: from the truth of k instances Φ(n1 ), . . . , Φ(nk ) it is unsafe to advance to the truth of the generalization ∀nΦ(n), since we have no guarantee that the truth of the instances is not due to the particular properties of the numbers n1 , . . . , nk in question. In a puzzling passage (Gl, §10), Frege attempted an analogy with the temperature at various depths in a borehole. Even if the temperature in the borehole has been found to increase uniformly with depth, it would be unsafe, he said, to infer anything about the temperature at lower depths to which we have not yet bored. It is unclear, though, just what his objection was, unless he meant that the very existence of the bore may have altered the temperature; but in that case it is dubious how relevant the example is to the point at hand, since no one, presumably, is suggesting that in arithmetic the very act of calculation might alter the result. Induction from a finite number of instances can at best be a fallible guide to the truth of a generalization, since there are false generalizations the least counterexample to which is very large. For instance, Mertens’ conjecture of 1897 (that P √ | nk=1 µ(k)| < n for all n) is false, but the least counterexample is known to be larger than 1014 . Whether induction is any guide to truth in arithmetic

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is controversial. Certainly number theorists may use induction in formulating conjectures, but whether there is any sense in which they are then entitled to regard these conjectures as probable is a difficult question, which Frege’s remarks do not address.

Analytic or synthetic? To say that arithmetic is a priori is not yet, on Frege’s view, to differentiate it from geometry. In search of a point of difference, he focused on Kant’s claim that each numerical equation is grasped individually via intuitions of the numbers in question. Frege objected that whatever plausibility this might have for small numbers is wholly absent in the case of large ones. Notice, though, that the objection depends on limitations of our own human capacities. Whether it is effective ad hominem depends on whether Kant’s account was as psychologistic as Frege evidently supposed it to be. Whatever one thinks about that, there remains the question whether there might be an idealized sense in which each numerical equation is grounded in intuition. In a famous passage Frege went on to point out a further, and much more fundamental, difference between the ranges of application of the principles of geometry and arithmetic. The wildest visions of delirium, the boldest inventions of legend and poetry, where animals speak and stars stand still, where men are turned to stone and trees turn into men, where the drowning haul themselves up out of swamps by their own topknots—all these remain, so long as they remain intuitable, still subject to the axioms of geometry. (Gl, §14) In imagining anything spatial, we are thus constrained by the axioms of Euclidean geometry. Yet it is still possible ‘for the purposes of conceptual thought’ to study the consequences of negating one of these axioms. In arithmetic, on the other hand, we cannot, even for these narrow purposes, assume the contrary of one of the fundamental propositions. ‘Here, we have only to try denying any one of them, and complete confusion ensues. Even to think at all seems no longer possible.’ (Gl, §14) Frege inferred from this that the range of applicability of arithmetic is ‘the widest range of all; for to it belongs not only the actual, not only the intuitable, but everything thinkable’. It is, in fact, just the same range as that of logic. The obvious explanation for this coincidence, he suggested, is that arithmetic is part of logic. ‘Should not the laws of number’, he asked, ‘be connected very intimately with the laws of thought?’ (Ibid.) The word ‘Logizismus’ was not coined, it seems, until the 1920s (Fraenkel 1928, 263, Carnap 1929, 2–3), but logicism concerning arithmetic—the twofold claim that the concepts of arithmetic are expressible in purely logical terms and that the truths of logic are thereby rendered as logical truths—is due to Frege. It is worth noting, though, that he took the universality of arithmetic to be merely

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suggestive of its logical character. If he had taken it to characterize logic, as some commentators suppose, the universal applicability of arithmetic would have been sufficient to prove that it is part of logic. He recognized, however, that until he had supplied gapless proofs of all the basic laws of arithmetic, a legitimate doubt as to their status would remain. In Grundlagen, therefore, he claimed only that logicism concerning arithmetic is a plausible hypothesis, not a proven fact. In order to express his logicist claim, Frege appropriated Kant’s word ‘analytic’, using it for anything provable from definitions by logical means. He claimed thereby to be merely expressing what Kant had meant by the word, but it is very dubious whether that is so. Kant was, of course, working with a much more restricted conception of what logic might achieve, but his use of the word ‘analytic’ did not even encompass everything that he understood as logic: transcendental logic is not analytic in Kant’s sense. A truth is analytic according to Kant if it follows trivially from definitions by means of pure general logic. For Frege, on the other hand, the whole of polyadic logic is included. He found it hard, as we have seen, to delineate its scope precisely, but he stressed nonetheless its capacity to deliver non-trivial results. So for him an analytic truth may be ampliative, in direct contrast to Kant’s pronouncements on pure general logic. The key disagreement between them was therefore not directly about arithmetic, but about whether a truth which is synthetic in Kant’s sense but analytic in Frege’s may nonetheless be known independent of intuition. In his later writings Frege avoided using the word ‘analytic’ altogether, but in the 20th century the usage he introduced in Grundlagen became widespread, with the result that Kant’s narrower conception of analyticity was too often dismissed as a symptom of his ignorance of quantificational logic, rather than a coherent notion with a philosophical significance of its own.

Further reading Mainstream philosophers of mathematics have generally neglected the question whether inductive arguments have any validity in arithmetic. One exception is Baker (2007).

12 GRUNDLAGEN III: NUMBERS

Having conjectured that arithmetical truths may be proved by logical inferences from wholly general logical laws, Frege was not yet ready to make a positive proposal as to how this might be done, but instead embarked on a general discussion of the application of numbers in counting. He wished to claim that although arithmetic is a distinct science which does not merely consist in its applications, it is nonetheless conceptually grounded in them. Since its central application is in counting, he first investigated the relation, the number of A is B, with the aim of settling what kinds of entities the relata A and B are.

Numbers of what? What is a number a number of ? One crude answer, which Frege was concerned to discredit, is that a number is a property of an aggregate. Unadorned, the proposal is indeed hopeless, since an aggregate does not in general have a unique decomposition into parts, and so there is no single number associated with it. Mill had tried to get round this by saying that we count according to the characteristic manner in which the aggregate is assembled. This works for many of the aggregates that we count in our daily lives, where one particular decomposition into parts is salient, but not for all. If you silently hand me a deck of playing cards, I cannot begin to count because I do not know what you want me to count—cards, suits, packs (or even, Frege suggested, honour cards at skat). A slightly better answer would be that we count the aggregate relative to a unit of counting (cards, suits, packs). Frege objected to this answer too, though, because an aggregate cannot be empty, and so the proposal gives no explanation of cases where the answer is zero, since in these there is no aggregate. Instead Frege advocated an alternative proposal, traceable to Herbart: he may well have learnt

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it from his former schoolteacher, Leo Sachse, who studied Herbart extensively in the early 1880s (see Gabriel 1997). Herbart had proposed that numbers attach not to aggregates but to concepts. In the case considered earlier, we might decide to count relative to the concept card in the deck, or suit in the deck, or pack in the deck: each provides us with a means of classification that tells us what to count (cards, suits, or packs). This proposal deals smoothly with the number zero, which is simply the result of counting relative to a concept under which nothing falls. It also (a virtue in Frege’s eyes) distances arithmetic from empiricism: numbers now attach not to pieces of the world but to concepts. A competing threat, however, is that the account might be thought psychological: counting depends on how we think about the world, rather than on the world itself. The desire to fend off this threat no doubt reinforced Frege’s tendency (already visible in the ‘17 key sentences’) to conceive of concepts non-psychologistically. Herbart’s is not the only workable solution, however: we could also attach numbers to classes, since a class is distinguished from an aggregate by its being determinate what its members are (and an empty class makes sense, whereas an empty aggregate does not). The class which we would nowadays denote by ‘{x : Φ(x)}’ Frege called ‘the extension of the concept Φ’. He conceived of it as logically posterior to the concept, ‘constituted in being, not by the individuals, but by the concept itself ’ (CP, 224–5). He therefore dismissed as ‘futile’ the attempt (on which the iterative conception of class depends) to make the extension ‘rest, not on the concept, but on single things’ (CP, 228). Since the extension is thus in his view a derived notion, he saw no advantage in treating the number as attached to it rather than to the concept from which it is derived. Notice, though, that not all concepts are suitable for counting (Gl, §54). I can intelligibly ask how many books there are in the room, but not how many water. The word ‘sortal’ by which concepts that can be counted are nowadays known originates in Locke’s Essay (bk 3, ch. 3, §15), but was not widely used until it was popularized by Strawson (1959). Grammarians differentiate similarly between count nouns and mass nouns: ‘how many’ questions make sense for count nouns, ‘how much’ questions for mass nouns. (The English language is, or was, unusual in marking this distinction by attributing ‘fewer’ to count nouns, ‘less’ to mass nouns.) However, the correspondence between the grammatical distinction and the sortal/non-sortal one is inexact: for a sortal concept only a natural number can be appropriate, whereas we can also ask ‘how many’ questions in cases where the answer may be a fraction (e.g. how many miles is it to the station?). Here Frege’s decision not to use the concept-script in his presentation makes itself felt. The notion of a sortal concept, important though it is in the grammar of ordinary language, is not required in quantificational logic, which assumes that objects are individuated, so that all the concepts it considers are automatically sortal. One notable absence from Frege’s discussion is any consideration of the proposal that numbers attach to pluralities—what plural terms refer to. Because his logic was resolutely singular, he treated a plural term not as referring plurally

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to its instances but as a concept-word expressing the concept under which these instances fall.

Numbers as non-linguistic Suppose we agree with Frege that the first relatum of ‘the number of A is B’ is a (sortal) concept. What of the second? Some of Frege’s contemporaries held a position, now known as ‘term formalism’, according to which numbers are ‘certain tangible signs’ (Heine 1872, 173). In his criticisms of this position Frege focused on the case of the integers and the rational numbers, but it would of course be possible to make the proposal for the natural numbers too. (It would be less plausible for the real numbers, because of their uncountability.) Frege did not always distinguish clearly between term formalism and the game formalism that we considered earlier, but the positions are different: according to the term formalists, the sentences of arithmetic express judgable contents capable of being true or false. Frege accused them nonetheless of confusing sign and thing signified, but it is not clear that they were guilty of quite such an elementary mistake. They could surely grant the general distinction, but insist that some linguistic expressions are names of themselves: onomatopoeia is plausibly an example. It is also too glib to object against them that ‘6 + 6’ and ‘5 + 7’ are different signs but refer to the same number: they can respond that what is usually known as identity between numbers is really only an equivalence relation among token expressions. More problematic, though, is that if we want to retain ordinary arithmetic as standardly presented, we must include types of which no physical token has yet been written or uttered. So it is questionable whether the subject matter of arithmetic is really tangible signs at all. Why should a nominalist who objects to Frege’s abstract objects be any more comfortable with such abstracta as uninstantiated types? The principal objection to term formalism, therefore, is that it does not solve the problems it was designed to address.

Numbers as objects Frege claimed that numbers are objects, but despite its importance devoted remarkably little space to arguing for this claim. Indeed the thinness of his argument goes hand-in-hand with the previously noted inadequacy of his criterion of objecthood, namely whether we use a singular term to refer to it. The particular difficulty in the present case is that numerals are used in ordinary language both as singular terms (e.g. ‘The number of Galilean moons of Jupiter is four’) and as adjectives (‘Jupiter has four Galilean moons’).1 Frege thought it was sufficient

1 Frege used as his example ‘the moons of Jupiter’, meaning thereby the four discovered by Galileo in 1610. The first of Jupiter’s other, much smaller satellites was not discovered until 1892.

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to observe that the latter construction can be re-expressed as the former, and to conclude from this that the former is to be preferred. He did not address the question of whether, in the reverse direction, the substantival construction can be converted into the adjectival. Dummett(1991b, 105) has suggested that §56, in which Frege offered the argument just sketched, ‘may be stigmatized as the weakest in the whole Grundlagen’. Part of the difficulty is no doubt that Frege did not yet have a criterion of identity for concepts; absent such a criterion, he found it hard to take seriously the proposal that numbers are concepts. A decade later, having by then formulated identity conditions (or rather, strictly speaking, conditions analogous to identity conditions) for concepts, he did take the proposal sufficiently seriously to offer an argument against it, namely that numbers have to be objects in order to fulfil the requirement (already mentioned in Grundlagen §14) that we can count them in the same way that we count ordinary objects. We want to be able to say that the number of prime numbers less than 10 is equal to the number of Galilean moons of Jupiter, i.e. four. This argument does not settle the matter, however. Can we not count concepts also in just the same way? Is it not true, for instance, that the number of concepts under which no objects other than 0 or 1 fall is four, just as the number of Galilean moons of Jupiter is four? What Frege’s argument really shows is the difficulty of reconciling counting with the hierarchy of levels generated by his quantificational logic, which seems to require a distinct number-of operator for each level.

Numbers as self-subsistent In fact, Frege held not merely that numbers are objects but that they are ‘selfsubsistent’ (Gl, §57). In Grundlagen itself his usage permits the interpretation that he intended this word merely to intensify ‘object’, thereby emphasizing the contrast between objects and concepts. The following year, however, he claimed that the infinitesimal dx has ‘a certain self-subsistence’ (CP, 111, amended), which suggests that he thought some objects might fail to be self-subsistent. Perhaps, then, what he meant by ‘self-subsistent’ was substances—entities independent not merely from us but from other objects. So although he was explicit that the Earth’s axis is an object, the fact that there would be no such thing if the Earth did not rotate would on this view be enough to disqualify it from self-subsistence. What is most striking about Frege’s use of the word ‘self-subsistent’, indeed, is that he did not bother to explain it, perhaps because he did not think the issue relevant to the task at hand.

Numbers as non-actual Frege classified objects according to whether or not they are wirklich: in the first English edition of Grundlagen Austin translated this as ‘existent’, following

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a now-extinct English philosophical usage (employed, as we shall see later, by Russell); in later editions this was changed to ‘actual’. The German word is cognate with the verb wirken (to act or have an effect): what is actual is ‘what acts on our senses, or at least produces effects which may cause sense-perceptions as near or remote consequences’ (§85). It is therefore a somewhat broader category than what early modern philosophers such as Hobbes and Descartes called the corporeal, since this was typically glossed as ‘having mass’, whereas photons are perceptible but have no mass. It is also a broader category than what is directly sensible, since electrons, for instance, have perceptible effects although we do not sense them directly. Whether the actual coincides with the physical is more debatable: economic recessions have observable consequences, but it is open to question whether they are reducible to the physical; and the Judaeo-Christian God is supposed to be capable of acting in the world, but yet not physical. Modern philosophers often use ‘actual’ in a different sense that contrasts it with what is merely possible. This use is not Frege’s. As we noted in Chapter 3, he did not recognize the modal as a distinctive kind of proposition, but only as a ground of judgment, with the consequence that for him modal notions were epistemic. Nor did he oppose the actual to the fictitious. Just what his attitude to fiction was is hard to tell, since he usually mentioned it only in order to discount it: in his later work he at least allowed that sentences containing fictional names make sense, but at the time of Grundlagen he had not developed the notion of sense on which that view depends. Applying Frege’s criterion of actuality is not always easy. He himself claimed, for instance, that the Equator is a non-actual object (§26), but that is disputable. Why not also the surface of the Earth? I can surely see that. Examples such as this suggest that the boundary between the actual and the non-actual is vaguer (or at any rate more contested) than perhaps Frege recognized. Nonetheless, even if there is room to dispute where exactly the boundary lies, some objects lie incontrovertibly on the actual side of it. Frege maintained that numbers lie just as incontrovertibly on the other side, but his reason for holding this is somewhat compressed. He devoted considerable space to arguing against a specific instance of the contrary view, namely Mill’s proposal that the number three is the sum of all the three-membered collections of things, but gave rather short shrift to the general claim that numbers are actual. To refute this general claim, it was not enough for him to appeal to his previous conclusion that arithmetical truths are a priori, since there are a priori truths about actual objects (e.g., at one time, that the standard metre stick was one metre long). Instead, he appealed to the universal applicability of arithmetic, which he had earlier used in support of the suggestion that arithmetical truths are logical. It would be not merely inconceivable, he said, but even a category mistake for the sensible to be applicable to the non-sensible. It would indeed be remarkable if a property abstracted from external things could be transferred without any change of sense to events, to ideas, and to

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concepts. The effect would be just like speaking of fusible events, or blue ideas, or salty concepts, or tough judgments. (Gl, §24) Although this is at first sight only an argument against treating numbers as properties of physical objects, Frege at once extended it so as to become an argument against taking numbers to be actual. ‘It does not make sense’, he said, ‘that what is by nature sensible should occur in what is non-sensible.’ (Ibid.) He did not explain why this does not make sense, however. He seems to have supposed that the non-sensible is in some way conceptually prior to the sensible, but it is not clear what his reason was. Notice, moreover, that this whole argument depends crucially on Frege’s prior rejection of empiricism and would have no force against someone who believed that the empirical is all there is.

Numbers as objective Frege further distinguished between the subjective and the objective. The objective is what is ‘independent of our sensation, intuition and imagination, and of all construction of mental pictures out of memories of earlier sensations, but not what is independent of the reason’ (Gl, §26). It is a mark of the subjective, on the other hand, that it is intrinsically private—his emphasis on this became a recurring trope in his writing—but he was not the first to adopt this view: Locke said in the Essay (bk 3, ch. 2, §1) that ideas were ‘within his own breast, invisible and hidden from others’, and the notion is generally thought to originate with Descartes. For Frege, unlike many of his contemporaries, what is not actual may nonetheless be objective—the equator, for example (§26). Objectivity is also distinct from objecthood: ideas are objects, but not objective; concepts are objective, but not objects (§27n). Frege repeatedly compared mathematicians in highly realist terms to scientists, who discover what is true independently of them. ‘If number were an idea, then arithmetic would be psychology. But arithmetic is no more psychology than, say, astronomy is.’ (§27) Number is no whit more an object of psychology than, let us say, the North Sea is. The objectivity of the North Sea is not affected by the fact that it is a matter of our arbitrary choice which part of all the water on the Earth’s surface we mark off and elect to call the ‘North Sea’. This is no reason for deciding to investigate the North Sea by psychological methods. In the same way number, too, is something objective. If we say, ‘the North Sea is 10,000 square miles in extent’ then neither by ‘North Sea’ nor by ‘10,000’ do we refer to any state of or process in our minds: on the contrary, we assert something quite objective, which is independent of our ideas and everything of the sort. (§26) Like the geographer, the mathematician ‘can only discover what is there and give it a name’ (§96). Numbers must be objective, Frege held, since otherwise a shared

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pool of arithmetical knowledge would be impossible: your number two would be different from mine; arithmetic would be essentially solipsistic. One of the motivations sometimes offered for the view that arithmetic is analytic is that it increases our certainty that arithmetical truths are true. In the argument just mentioned, by contrast, Frege used the certainty of arithmetical knowledge as a premiss. He seems therefore to have aspired to identify the source of this knowledge, not to establish that we have it. There is a tension here notwithstanding. He wanted a definition of number-words that would reveal for the first time the true content of arithmetical sentences. So his appeal to a shared pool of arithmetical knowledge en route to that very definition looks suspiciously circular. This is an instance of what is nowadays known (following Langford 1942) as the ‘paradox of analysis’, but there is little sign that Frege had yet thought much about it.

Numbers as abstract None of the three distinctions just discussed—between objects and concepts, between actual and non-actual, and between objective and subjective—originated with Frege, but he seems to have been the first philosopher to argue for a distinctive class of objects that are neither actual nor subjective. They are nowadays called ‘abstract’, but philosophers rarely used the word in that sense before it was popularized by Goodman and Quine (1947). (The word ‘concrete’ is often used as the negative of ‘abstract’, with the slightly odd consequence that desires and hallucinations are thereby classified as concrete.) Proponents and opponents of abstract objects are often called ‘platonists’ and ‘nominalists’ respectively, regrettably hijacking two words that already have useful senses: modern platonism is not a view about Plato’s forms (which would count as actual according to Frege because they have sensible effects); modern nominalism has nothing to say about the existence of universals per se. That modern usage has largely supplanted the original senses of these words is an indication of the importance in 20th-century philosophy of a debate that originated with Frege. His view that numbers are abstract, if they exist at all, is widely accepted. Modern nominalists (e.g. Field 1980) therefore accept that numbers would be abstract objects, but deny that there are any, because they hold a causal theory of knowledge, from which it follows that no knowledge of abstract objects is possible. Frege would have been contemptuous of such a theory, because it confuses the explanation for my holding the beliefs I do, which may well be causal, with the objective justification for their truth. Even so, he did grant that there is prima facie a puzzle about how we can think about abstract objects, given that we cannot form mental images of them as we can of the corporeal objects of mundane experience. His response to this problem relied, as we shall see in the next chapter, on using the context principle to convert it into that of giving content to whole sentences in which names of abstract objects occur.

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Numbers as logical The claim on which Frege laid most stress was not that numbers are abstract but that they are logical; i.e. their existence and nature have a purely logical source. Not all abstract objects are logical: the Earth’s axis, for instance, is not. Most modern logicians reject Frege’s notion of a logical object, but for a disappointingly trivial reason. They conceive of logic as consisting of laws that are valid in reasoning about any domain whatever, however many or few objects the domain contains. Although there is some debate over whether it is a logical truth that there is at least one object—standard systems of first-order logic allow this to be proved, but some think this is a ‘defect in logical purity’ (Russell, IMP, 203n)—there is no debate about whether it is a logical truth that there is more than one: Boolos was speaking for the modern mainstream when he said (1998, 302) that ‘on any understanding of logic now available to us’ it is not. Frege’s conception of logic was different, in two respects. First, he generally conceived of the domain of interpretation of logic as fixed, not variable. Second, he held that reason has available to it resources that flow from its own nature; hence he did not think it absurd that there might be, as a matter of pure logic, infinitely many objects. So when he explained the fallacy in the ontological proof, he was careful to stress (Gl, §53) that he did not intend thereby to deny the very possibility of a purely logical existence proof. One especially interesting case is Dedekind’s (1888, no. 66) proof of the axiom of infinity. He held that I have an a priori grasp of the idea of my self, and that for every idea I can form the idea of that idea. Hence, he concluded, there are infinitely many objects available to me a priori. Frege saw no reason to object to Dedekind’s proof, provided only that the ‘ideas’ he referred to in it were understood as objective rather than subjective (PW , 136).

Further reading Sullivan (1990) discusses in detail Herbart’s account of numbers and its possible influence on Frege. The role played by Leo Sachse in the matter is explained by Gabriel (1997). On the difficulties of locating the boundary between actual and non-actual, see Lewis (1986, §1.7). Moltmann (2013) discusses linguistic evidence that, contrary to Frege’s claim, number-words should be treated as predicates. On Frege’s conception of numbers as objects see Hale (1984).

13 GRUNDLAGEN IV: THE FORMAL PROJECT

Frege had been working since the publication of Begriffsschrift to establish his logicist conjecture, and by 1882 was in a position to write to Carl Stumpf (who had been his contemporary at Göttingen),1 I have now nearly completed a book in which I treat the concept of number and demonstrate that the first principles of computation, which up to now have generally been regarded as unprovable axioms, can be proved from definitions by means of logical laws alone, so that they may have to be regarded as analytic judgments in Kant’s sense. (PMC, 99–100) He complained, however, of the difficulty he was finding in getting his ideas noticed. I find myself in a vicious circle: before people pay attention to my conceptscript, they want to see what it can do, and I in turn cannot show this without presupposing familiarity with it. So it seems that I can hardly count on any readers for the book. (PMC, 102) In response, Stumpf suggested that he should explain his line of thought ‘first in ordinary language and then—perhaps separately on another occasion or in the very same book—in concept-script’. Frege heeded the advice: although it was his central contention in Grundlagen that arithmetic is derivable from logic, he deferred his attempt at a detailed proof of this claim; even in the part of the book

1 There is some doubt over whether Frege addressed the letter to Stumpf or to Anton Marty. At any rate it was Stumpf who replied either to this or to a very similar letter.

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devoted to his positive proposal he merely outlined his approach and mentioned the concept-script only in passing.

Numerically definite quantifiers Before coming to the account of the natural numbers that he eventually favoured, Frege considered and rejected two others. The first of these formalizes the notion of a numerically definite quantifier by defining ‘There are n Fs’, or in symbols ∃n xFx, as follows: ∃0 xFx =df ∼∃xFx; ∃n+1 xFx =df ∃y(Fy ∧ ∃n x(Fx ∧ x 6= y)). This is an implementation of the ‘adjectival’ strategy mentioned in Chapter 12. We can obtain from this successively contextual definitions of ‘There is one F’, ‘There are two Fs’, etc. There are two sorts of difficulty with this sequence of definitions, however. The first concerns how we make the transition from particular numbers to the general case, ‘There are n Fs’. If we could presuppose the natural numbers as given, we could understand this as a recursive definition, but that would plainly be circular in the current context. The circularity could in principle be avoided, by adapting the method of Frege’s treatment of the ancestral, but he himself did not attempt this. Instead, he rejected the definition as soon as he had proposed it because of the second difficulty, namely that it fails to introduce numbers as selfsubsistent objects. Even if it furnishes an explanation of the expression ‘∃n xFx’ as a whole, it does not do so in such a way that we can treat the ‘n’ as the name of an object, because it does not settle which object it is. The way Frege made this point was to ask—somewhat eccentrically, one might think—whether Julius Caesar is a natural number. This question depends on his earlier claim that numbers are objects, since only on that assumption does it even make sense. This is nowadays known, because of Frege’s ‘crude example’, as the ‘Julius Caesar problem’.

Hume’s Principle Frege next considered introducing numbers by means of the principle that the number of Fs equals the number of Gs if and only if the Fs and the Gs are equinumerous. (Austin infelicitously translated Frege’s word gleichzahlig as ‘equal’, but ‘equinumerous’ has become the standard translation in the secondary literature, and I shall use it throughout.) If equinumerosity were defined to mean ‘having the same number’, this would be hopelessly circular, of course, but we can define two concepts to be equinumerous if there is a one-to-one correspondence between the objects falling under them, and this can be expressed in pure second-order logic without mentioning numbers. First define ∃!xFx =df ∃x∀y(Fy ≡ x = y),

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and then define the equinumerosity relation F ∼ G =df ∃R(∀x(Fx ⊃ ∃!y(Gy ∧ xRy)) ∧ ∀y(Gy ⊃ ∃!x(Fx ∧ xRy))). Here ‘∃R’ quantifies over relations, and so the definiens is a sentence of pure second-order logic. Now introduce a term-forming operator ‘N’, so that ‘NxFx’ is a singular term meaning the number of Fs for any sortal concept F. (Because all the concepts dealt with in Fregean quantificational logic are sortal, I shall ignore this restriction from now on.) The principle under consideration is then NxFx = NxGx ≡ F ∼ G. Frege credited Hume with the insight that equality of number may be derived from equinumerosity of concepts, and for that reason this equivalence is now widely known as ‘Hume’s Principle’. Frege acknowledged at once that Hume’s Principle is ‘a very odd kind of definition’ (Gl, §63), presumably because it is implicit, not explicit. In an attempt to mitigate its oddity, he compared it with other principles of a similar shape. In geometry, for instance, the notion of direction can be introduced by means of the ‘direction principle’, that two lines have the same direction if and only if they are parallel. In symbols, dir(a) = dir(b) ≡ a||b. (Here we need, of course, to adopt the convention that every line is parallel to itself. We also need to realize that what Frege here called the ‘direction’ of a line might be better termed its ‘orientation’.) More generally, if ‘∼’ denotes any equivalence relation, the formula 6(a) = 6(b) ≡ a ∼ b, is called an ‘abstraction principle’. We make no general restriction on what kinds of entities ‘a’ and ‘6(a)’ refer to. In order that the abstraction principle should be a candidate to provide a non-circular route to an understanding of its left hand side, ‘6’ must not occur in the definition of ‘∼’: Frege took some trouble in Grundlagen to criticize accounts of geometry which make use of the notion of direction in defining the relation of parallelism and hence violate this constraint.

Section 64 In §64 of Grundlagen Frege offered an argument that abstraction principles are in some sense analytic. The case he focused on was the direction principle, but he evidently intended his argument to apply to any abstraction principle. When we move from the right to the left hand side of the principle, he said,

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we replace the symbol || by the more generic symbol =, through removing what is specific in the content of the former and dividing it between a and b. We carve up the content in a way different from the original way, and this yields us a new concept. (Gl, §64) This talk of recarving is suggestive of Begriffsschrift §9, where Frege said that the content of a sentence may be divided up in different ways, but the cases are quite different for two reasons. First, Begriffsschrift §9 envisages that we may multiply decompose, and hence recognize various patterns in, a judgable content without calling into question that it has a unique analysis into atomic constituents. Grundlagen §64 threatens this conception, since the constituents of the two sides of the equivalence seem irresolubly different: one side contains identity, the other not; one side contains numbers, the other not. Second, Frege’s talk of recarving leaves out of account the requirement that the abstraction principle should deliver objects. In this respect he was ill-served by his choice of the direction principle rather than Hume’s Principle as his example, since the latter, unlike the former, is impredicative: the definition of the equivalence relation on the right hand side involves quantification over a domain to which the entities referred to by the terms on the left already belong. Frege’s metaphor leaves unaddressed why the recarving should result in entities that belong to this domain. This impredicativity is not easily fixable without destroying the deductive power of Hume’s Principle: a predicative abstraction principle will deliver a conservative extension of a system, and hence cannot hope to generate arithmetic. In the short term, the inadequacy of §64 did not matter much, since, as we shall see shortly, the account Frege ended up with did not require him to show that Hume’s Principle is analytic. Moreover, in the case of Hume’s Principle, the impredicative demand imposed by the abstraction principle can in fact be met. In the longer term, though, Frege’s failure to notice what is problematic about impredicative abstraction principles was to prove a costly mistake, as it contributed to his error of adopting the inconsistent Basic Law V in Grundgesetze.

Deriving arithmetic In §§74–83 Frege sketched how to define the natural numbers from Hume’s Principle. First he defined zero as 0 =df Nx(x 6= x). Then he defined the successor relation by xSy =df ∃F(y = NwFw ∧ ∃z(Fz ∧ x = Nw(Fw ∧ w 6= z))). Now he could make use of Chapter 3 of Begriffsschrift, where he had shown how to define the ancestral of a relation. Applied to the successor relation, this generates

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its ancestral S∗ (more familiarly written as 2’ (PW , 233); ‘Sea water is salty’ (PW , 252); ‘I smell the scent of violets’ (CP, 354). I shall refer to the claim that prefixing a declarative sentence with ‘it is true that’ makes no difference to the thought expressed as the ‘transparency schema’. (I shall avoid the term ‘redundancy schema’, by which it is often known, for reasons that will emerge shortly.) This is, of course, to be distinguished sharply from the disquotational schema, which concerns the truth of a sentence, not of a thought. In modern philosophy, the transparency schema is commonly associated with a deflationary view according to which it reveals all there is to know about truth. Frege, however, read it instead in the opposite, inflationary direction as showing that the word ‘true’ has a sense which, since it ‘contributes nothing to the sense of the whole sentence in which it occurs as predicate’ (PW , 252), is, mysteriously, already contained in every thought. This transparency, unique to truth, is what makes it, on Frege’s view, sui generis. Since logic’s concern is with truth, the thought expressed by a sentence consists in the conditions under which it is true (Gg, I, §32). Hence there can be no possibility of explaining what it is to understand a sentence without already implicating truth—since otherwise that understanding would reduce to nothing. If the whole explanation of the sense of the word ‘win’ consisted in a stipulation, for each game, of the conditions under which one player or another was said to have won, then a knowledge of what a particular game is could not involve knowing what it is to win that game: for the knowledge that, e.g., one wins a game of chess when one either check-mates one’s opponent or he resigns would amount to no more than the ‘knowledge’ that either one checkmates one’s opponent or he resigns when one checkmates one’s opponent or he resigns, which is no knowledge at all. (Dummett 1973, 459) In Begriffsschrift truth is a property of circumstances. On Frege’s later account it is something more puzzling: it pervades every thought and yet is not a separable component of any (since if we try to remove it, what remains still stubbornly contains it). It is not, he now held, a property of thoughts, but rather represents a transition from the level of sense to that of reference. Fiction and myth make frequent use of empty names—names without reference—but Frege took it as evident that sentences containing such names may be comprehensible, and hence have a sense. The reference principle then required him to hold that such sentences have no truth value, since a function can have a value only if it has an argument. In ‘On sense and reference’ he categorized the senses of such sentences as thoughts (CP, 163), but five years later he changed to calling them ‘mock thoughts’ (PW , 130). In ordinary language (although not, as we shall see shortly, in the concept-script) Frege thus tolerated truth-value gaps. Some have questioned whether this notion of a mock thought is coherent.

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‘What can it mean’, asked Evans (1982, 24), ‘on Frege’s, or on anyone’s, principles for there to be a perfectly determinate thought that simply lacks a truth value?’ The most plausible argument against mock thoughts comes from the transparency schema. (Frege never explicitly stated this schema for mock thoughts, but it is difficult to see why he should have resisted doing so.) From this it follows that if p lacks a truth value, then so does ‘It is true that p’. Many commentators (e.g. Dummett 1958, 145–6) find this implausible. It is clear, they claim, that the sentence ‘It is true that Vulcan is a planet’ is false. Fregeans, of course, will simply disagree.

Referring to thoughts The reference of a declarative sentence is its truth value, and yet the truth value of ‘John believes that p’ does not depend functionally on that of p (assuming that John is not omniscient). Frege responded to this apparent violation of the reference principle by amending his theory so as to make the reference of a sentence depend on context: in normal circumstance a declarative sentence refers to a truth value, but in an indirect context it refers instead to the thought which it would normally be taken to express. To those who might find this a rather ad hoc way of solving the problem, he offered two responses. First, indirect speech is not the only sort of context that changes the reference of a sentence: when it occurs in quotation marks, its reference is not the truth value but the sentence itself. Second, his proposal conforms with intuition, since in indirect speech we are indeed talking about a thought: in ‘George IV wished to know whether Scott was the author of Waverley’, for instance, we say something about the thought that Scott was the author of Waverley, namely that George IV wished to know whether it was true (which, as it happens, it was). Frege’s concern with indirect speech in ‘On sense and reference’ is an instance of the greater interest he now took in the workings of ordinary language. If his sole concern had been with grounding a concept-script adequate for mathematics, he could simply have ignored indirect contexts, because they do not arise there. It is curious, therefore, that in the Foreword to Grundgesetze, a book devoted to expounding just such a concept-script, he cited his semantic theory’s capacity to deal with indirect contexts as one of its chief virtues. Frege seems to have held, albeit tentatively, that the realms of sense and reference are disjoint. ‘A truth value cannot be part of a thought, any more than, say, the Sun can, for it is not a sense but an object.’ (CP, 158) Senses, then, seem not to be objects: that the Sun and the truth values are objects is apparently sufficient to show that they cannot be parts of a thought. Since in his metaphysics objects are the only saturated entities that we can refer to at all—anything in the realm of reference that is not an object is a function and hence unsaturated—it follows that we cannot refer directly to senses.

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One might be worried that Frege’s account of indirect speech conflicts with holding that the realms of sense and reference are disjoint, since the account requires that we should be able to refer to thoughts. It is preferable, I think, to hold that we can only refer to thoughts indirectly, and that to do so we are not required to grasp them by means of ‘indirect senses’. An independent reason for holding this is that it neatly avoids a regress which would otherwise threaten: to grasp a sense, we should have to grasp its indirect sense, hence its doubly indirect sense, and so on. Even independent of this concern, however, there seems little reason to posit indirect senses. The sense of a singular term consists in a way in which its referent is given to us, and we can think about or be aware of an object only by grasping such a sense. When we grasp a sense, however, it cannot be given to us in any one way rather than another; our mode of awareness of it is not constituted by our grasp of any other sense. That is because everything that contributes to determining the reference of an expression is part of its sense: we do not go to the reference via the sense, as one goes from Oxford to Leeds via Birmingham; the sense is the route, and not an intermediate station. (Dummett 1989, 13) Frege’s best view, then, was that the realms of sense and reference are disjoint. It should be conceded, however, that he did not hold this view consistently, suggesting elsewhere that it is possible to refer to a sense by means of the phrase ‘the sense of the expression “A” ’ (CP, 159). To keep the realms of sense and reference disjoint, he would have had to hold that this phrase misses its apparent target. In the next chapter we shall see another example in which he held that something similar does indeed happen.

Further reading Burge (1986) discusses the sense and reference of sentences further. Searle (1995) advocates blocking the slingshot by distinguishing among the references of logically equivalent sentences. On indirect senses see Kripke (2008), Burge (1979) and Skiba (2015). On Frege’s argument that truth is indefinable see Stepanians (2003). On his objection to the correspondence theory of truth see Dummett (1973, 442–4) and Baldwin (1995).

16 SENSE AND REFERENCE III: CONCEPT-WORDS

The distinction between concept and object, whose importance Frege was so concerned to stress in the Grundlagen, continued to hold a central position in his mature semantic theory, but now re-worked to accommodate a distinction between sense and reference for unsaturated expressions parallel to that he had drawn for saturated ones.

Unsaturated senses In Frege’s early writings the content of a proper name is the object it refers to, but that of a concept-word is more like what he later called its sense. Once he had distinguished sense from reference, however, he held that the sense of a sentence, a thought, is composed of the senses of the sentence’s meaningful components: the whole thought can thus be conceived of as having a structure which reflects that of the sentence. In particular, the distinction between saturated and unsaturated components of a sentence—between singular terms and concept-words or relation-words—corresponds to a distinction at the level of sense: the process by which a name is removed from a sentence mirrors that by which the name’s sense is removed from the thought the sentence expresses; the residual component, the sense of a concept-word, is unsaturated. Frege held that this notion of unsaturatedness solves the problem of explaining what is required, in addition to the senses of ‘Socrates’ and ‘wise’, to form the thought that Socrates is wise. The standard answer is that what is required is the copula is, but some have objected that this just invites the further question of what it is that binds this to the previous two senses. When expressed in this form, the problem is nowadays known as ‘Bradley’s regress’, but there is no evidence that Frege read Bradley, and in any case the problem dates back at least to Abelard.

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Frege diagnosed the difficulty as arising from a failure to distinguish between the saturated and the unsaturated. ‘Not all the parts of a thought can be complete,’ he claimed. At least one must be ‘unsaturated’, or predicative; otherwise they would not hold together. For example, the sense of the phrase ‘the number 2’ does not hold together with that of the expression ‘the concept prime number’ without a link. (CP, 193) It is dubious, though, whether unsaturatedness really can explain the unity of the proposition: not only is it, as he conceded (CP, 194), only a figure of speech, but it might well be suspected of depending on the very notion of unity he now sought to explain. After all, it originated in §9 of Begriffsschrift, where he used it to discern function-argument complexity in a pre-existing, hence already unified, thought. How, then, could this same notion explain that very unity? ‘It is like saying that we need to divide up the cake into two parts in such a way that they make up the whole cake.’ (Kanterian 2012, 188)

Concepts Frege now called the references of concept-words concepts. (In his earlier usage a ‘concept’ had been closer to being the sense of a concept-word.) How should concepts be individuated? Once we have determined the reference of ‘Socrates’ (namely a certain Greek philosopher), what more is required in order to determine the reference (i.e. the truth value) of the sentence ‘Socrates is mortal’ is only whether or not that object is mortal. The reference of the concept-word ‘ξ is mortal’ must be such as to supply this information, not only for ‘Socrates’ but for any term that might be inserted in its argument-place. The reference must, that is to say, consist in the information which things fall under the concept and which do not. Concepts are therefore to be individuated extensionally; they differ ‘only in so far as their extensions are different’ (PW , 118). So ‘ξ is a featherless biped’ and ‘ξ is a rational animal’ refer (supposedly) to the same concept. Frege’s choice of the word ‘concept’ (in German, Begriff ) for the extensional notion is unfortunate. Not only had other philosophers used it to mean something intensional, but talk of ‘grasping a concept’ comes easily, whereas on his terminology what we grasp is not the concept but the sense. In his published writings, moreover, he was curiously inexplicit about this, with the consequence that in early Frege scholarship (e.g. Grossmann 1961) it was controversial what his view was. Frege conceived of concepts as a kind of function: just as the sine function outputs the value 1 for π/2 as input, the concept German outputs the value True for Frege as input, False for Russell. He called the relation in which concepts stand if they have the same objects falling under them ‘mutual subordination’, not ‘identity’: it cannot be identity, because the level of a relation is determined by the types of its argument-places, and the argument-places of the identity predicate,

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being fit for objects, are not also fit for concepts. He was therefore forced into long-winded periphrasis. Coincidence in extension is a necessary and sufficient condition for the occurrence between concepts of the relation corresponding to identity between objects. (Note: For identity in the proper sense of the word does not occur between concepts.) (CP, 200) In a letter to Husserl (PMC, 63, amended) Frege used a diagram to clarify how he conceived of sense as a route from language to reference. Sentence

Singular term

↓ Sense of the sentence (thought) ↓ Reference of the sentence (truth value)

↓ Sense of the singular term ↓ Reference of the singular term (object)

Conceptword ↓ Sense of the concept-word ↓ Reference of the concept-word (concept)

→

Object falling under the concept

Given the historical importance of mediaeval nominalism, one version of which held that there are no universals but only particulars, we might expect Frege to have recognized that his commitment to the objective existence of concepts was controversial, but curiously he made no attempt in his published writings to defend it at all. What I have said about concepts thus far does little more than extrapolate the account of his semantic theory given in the previous two chapters, and this does not in itself involve any substantial ontological commitment: for all I have said so far, concepts might be merely part of the semantic machinery required to deliver an account of how the truth values of sentences are determined. Frege did in fact hold, however, that concepts are more than mere posits to improve the symmetry of the diagram. ‘Concepts and objects have the same objectivity,’ he said (PMC, 63). That is realism of a sort, but it does not distinguish them from the senses of concept-words. He also held, however, that concepts, like objects, may be the subjects of predication: we can talk directly about them, whereas, at least on his best view, we talk about senses only indirectly. Moreover, since in such a predication the concept-word is replaceable, it follows that we may quantify over concepts. So if Frege’s criterion of reality were Quine’s—our ontological commitments are revealed by the domains of our quantifiers—then concepts would count for him as real. (In his formal theory, Frege did not in fact make use of quantifiers of higher than second order, but there is no reason to think he would have opposed doing so if the need arose.)

Sense and Reference III: Concept-Words 109

Nonetheless, that concepts may be quantified over is still a rather thin sort of reality. Many philosophers have wanted another, thicker sort, but have struggled to articulate what it might amount to. Frege himself never even tried. This is not to deny, though, that he sometimes felt the want. In one place he made a brief attempt to express a sense in which concepts and relations are real. To say that Jupiter is larger than Mars is to say that the heavenly bodies themselves stand in a certain relation to one another, and this, he suggested, is comprehensible only if the relation belongs to the realm of reference too (PW , 193). This has some plausibility if we conceive of that realm as supplying truth-makers for our thoughts. Such a notion clearly tempted Frege, but it is far from clear whether he could consistently endorse it.

The concept horse However plausible it may be that the structure of thoughts mirrors that of language, it is plainly more contentious to hold, as Frege did in his middle period, that the distinction between saturated and unsaturated carries across from the realm of sense to that of reference, so that objects are saturated and concepts unsaturated, just as singular terms and concept-words are. I shall call this the ‘mirroring principle’. In most of the places where Frege asserted it he did so in language—‘accordingly’, ‘of course’—that suggested no argument was needed, and not until 1903 did he provide one. An object, e.g. the number 2, cannot logically adhere to another object, e.g. Julius Caesar, without some means of connection. This in turn cannot be an object but rather must be unsaturated. A logical connection into a whole can come about only through this, that an unsaturated part is saturated or completed by one or more parts. . . . Now it follows from the fundamental difference of objects from concepts that an object can never occur predicatively or unsaturatedly; and that logically, a concept can never stand in for an object. One could express it metaphorically like this: There are different logical places; in some only objects can stand and not concepts, in others only concepts and not objects. (CP, 281–2) Frege’s claim that the unsaturatedness of concepts is required in order to explain how they can cohere with objects so as to form a whole is of course strikingly similar to what he had already said about thoughts. In that context, though, there was at least a difficulty to which he was responding, namely that of explaining the unity of the thought. In order to see a corresponding difficulty at the level of reference, we would have once again to see it as containing states of affairs. And even then one might be surprised by his use of the words ‘logical’ and ‘logically’ (four times in the paragraph). What does logic have to do with the fact that Julius Caesar was a Roman emperor? Moreover, the mirroring principle has an uncomfortable consequence: since the phrase ‘the concept horse’ is saturated, it cannot refer to something unsaturated; in particular, it cannot refer to a concept.

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There are then two alternatives: either the phrase refers to something other than a concept, or it does not refer at all. Frege chose the former option, but what is this mysterious entity? He was at first tempted by the idea that the phrase ‘the concept F’ could be used interchangeably with ‘the extension of the concept F’. In his draft of ‘On concept and object’ there is a footnote (dropped from the published version) saying that whether to adopt this view is merely a question of ‘expediency’ (PW , 106).1 What if he had held instead that ‘the concept F’ does not refer to anything? He would then have been saved from the explicit paradox of having to admit that the concept horse is not a concept, but he would not have avoided the underlying difficulty, which is that if the mirroring principle is true, logic has features we are puzzlingly unable to express. Frege maintained, nonetheless, that this difficulty indicates only an inadequacy of language, not a deep incoherence. We are tempted to treat a concept like an object, contrary to its unsaturated nature. This is sometimes forced upon us by the nature of our language. Nevertheless, it is merely a linguistic necessity. (CP, 282n.) Rather than enquire into this ‘linguistic necessity’, he was content to downplay it. ‘My expressions, taken literally, sometimes miss my thought. . . . I was relying upon a reader who would be ready to meet me half-way—who does not begrudge a pinch of salt.’ (CP, 193) In Chapter 49 we shall discuss Wittgenstein’s refusal to grant him his pinch of salt.

Further reading There is an extensive literature on Frege’s paradox of the concept horse, much of it (e.g. Dummett 1973, 212–8) concerned with the peripheral issue of whether ordinary language can refer to it. For more relevant discussion see Dudman (1972), Noonan (2006), MacBride (2011), Proops (2013) and Trueman (2015). On the metaphysics of Fregean concepts see Currie (1984).

1 That he should have been toying with this identification presumably explains why he claimed around the same time (CP, 143) that the two sides of Basic Law V have the same sense.

17 GRUNDGESETZE I: TYPES

In the Grundlagen Frege offered only a sketch of the derivation of arithmetic from logic, and conceded that he had made its completion no more than ‘probable’ (Gl, §90). What was needed was a gapless derivation in his concept-script, so as to rule out any surreptitious appeal to intuition. The early part of Grundgesetze covered once again the same ground as Begriffsschrift, laying out a formal machinery for the gapless expression of logical arguments, so that no doubt as to their correctness might remain. In outward appearance the concept-script had changed little in the intervening 14 years, but in the detail there were significant changes.

Syntax One difference between Begriffsschrift and Grundgesetze is that whereas the former simply assumed a stock of singular terms and predicates given in advance, the latter included a definite description operator ‘ ’ for forming singular terms. Even so, Frege did not explicitly allow for non-logical expressions—his concept-script was in this sense ‘pure’—which is odd if his ultimate aim was to ground arithmetic as a tool for counting non-logical entities such as planets or Roman emperors.1 Quine (1940, §4) gave the names ‘use’ and ‘mention’ to two ways signs may occur, respectively outside and inside quotation marks, but Frege in Grundgesetze (I, 4) already insisted on the distinction without using these words. He continued, as he had in Begriffsschrift, to use Fraktur letters as bound variables and upper-case Greek letters to stand schematically for expressions of the concept-script, but he also now used lower-case Greek consonants to indicate the argument-place in a ι

1 In fact, Frege designed his definite description operator to work in concert with the value-range operator, to be discussed in the next chapter, but I shall ignore that complication here.

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concept-word. As he pointed out (Gg, I, §1), these Greek letters are not signs of the concept-script itself, but are used only to talk about it. If quotation marks surround an expression containing an upper-case Greek letter, Frege’s convention was that this names not the expression itself but what we get if we replace the letter with the expression it stands for: the letter has wider scope than the quotation marks. Quine (1940, §6), by contrast, used ordinary quotation marks only to name the signs they surround, and therefore devised a new kind—so-called ‘quasiquotes’—to capture Frege’s usage. A further point on which Grundgesetze improved on Begriffsschrift was the distinction between kinds of argument-place: previously Frege had said obscurely that this ‘must be taken into account’; now he introduced distinct notations for first- and second-level argument-places, making it possible to specify how. In Grundgesetze, then, Frege drew certain syntactic distinctions which he had not drawn in Begriffsschrift. In one respect, though, he removed a distinction. We have already noted that his mature semantic theory treated sentences as saturated expressions akin to singular terms in having both sense and reference; but that is not yet to hold that they are singular terms. Yet he did indeed take this further step, explicitly assimilating sentences to singular terms and maintaining that the result of substituting a sentence into a first-order argument-place of a function is syntactically well-formed. Why did he do this? One reason was that he was attracted by the formal simplicity which this permitted (Gg, I, x): he exploited to the full the scope that it provided to simplify the rules of his formal system. A more principled reason, however, was that he took the distinction between saturated and unsaturated that originated in Begriffsschrift §9 to be the only distinction of any logical significance (PW , 194). What he did not do, though, was to give any argument for this. It is surely possible to acknowledge the central importance of function-argument decomposition without holding that the whole of logic is derived from it. We shall therefore have to return shortly to the question whether the assimilation is philosophically defensible.

Semantics Now that Frege was clearer about the distinction between use and mention, we might hope that he would also have been clearer than in Begriffsschrift about the distinction between reference-level and language-level accounts of quantification. In Grundgesetze he did indeed offer a reference-level account of the first-order quantifier. Yet in 1897 he explicitly stated that this is just ‘another way of putting’ the language-level account. A sentence completed with a judgment-stroke that contains roman objectletters affirms that its content is true whatever meaningful proper names you may substitute for those letters, provided the same proper name is substituted for one and the same letter throughout the sentence. Since proper names are signs which mean one individual determinate object, another

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way of putting this is: such a sentence affirms that its content is true, whatever objects be understood by the roman object-letters occurring in it. (PW , 154) The fact that Frege regarded the two interpretations as equivalent suggests that he regarded the Kantian principle that we cannot grasp an object except in some particular way as applying to quantification too. Since Frege’s syntax distinguished explicitly between first- and second-level functions, he offered a correspondingly distinct statement of the semantics of the second-level quantifier. At first sight, it is a reference-level semantics: the expression ‘refers to the True provided the value of that second-level function is the True for every fitting argument’ (Gg, §24). But what it is for an argument to be ‘fitting’ is for its name to fit the argument-place in question (Gg, §23), and so the second-order quantifier ranges only over functions with names. As in Begriffsschrift, however, there seems to be no strong reason to read this narrowly, i.e. as limiting the range to those functions that already have names in the conceptscript. Frege now had the resources, which his early semantics lacked, to account for empty singular terms as having sense but no reference. He continued to hold, nonetheless, that in scientific language they must be avoided: the concept-script should be so constructed that every singular term has a reference. A partial explanation for his animus against empty terms is provided by the reference principle— that the reference of a compound expression is a function of the references of its component expressions—from which it follows that if one of the component expressions has no reference, then the compound expression has no reference. In order that each sentence of the concept-script should have one or other of the two truth values as its reference, Frege had to insist that a sentence cannot contain an empty term. This is not a decisive objection to free logics, however, but merely mandates a further slight complication in the rules of the formal system to cope with such cases. Frege was aided in his ban on empty terms in the concept-script by the fact that the examples he considered were mostly drawn from fiction. Even he had to admit, though, that mathematicians occasionally use empty terms (e.g. ‘the least rapidly convergent series’). In order to deal with such cases, Frege picked a ‘null object’ to act as the default reference of the phrase ‘the F’ in the case where there is not exactly one F. The object he chose was the extension of the concept F, so that in the case where there are no Fs ‘the F’ refers to the empty class. The same objection could be raised to the element of arbitrary choice that is involved here as in the earlier case of Frege’s explicit definition of numbers. (In ‘On sense and reference’, indeed, he had made a slightly different choice of null object.) In order to guarantee that there are no empty singular terms in the conceptscript, Frege had to insist that if a function-term is introduced, every filling of its argument-places with terms of appropriate types results in a singular term and

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hence has a reference. This is, of course, highly artificial: when defining addition of natural numbers, for instance, we do not ordinarily feel the need to stipulate the result of adding the Sun to the Moon (see CP, 148). Frege made various attempts, none of them very convincing, to argue that logical hygiene compelled this. He suggested, for instance, that otherwise we might be tempted to define the function again for the other cases and in doing so define it twice by mistake for some values. Even if some of his mathematical contemporaries were susceptible to this temptation, though, that is plainly at most a pragmatic objection to partial definitions: it would be just as feasible to meet it by making a rule that the domains of applicability of definitions should never overlap. A second argument Frege gave against partial definitions was that the conceptscript ought not to be infected by vagueness, but he was wrong to suggest that a definition is vague simply because it is partial. We may grant that in a conceptscript that aspires to precision it should be determinate which strings of signs are well-formed. We might even insist that it should be not merely determinate but mechanically decidable (in which case we would have to ban the phrase ‘the largest Mersenne prime’). But this is no reason to bar partial definitions tout court. He was wrong to elevate the avoidance of partial definitions from a desirable feature of his concept-script to a necessary one. Although Frege insisted, at the level of language, that every filling of the argument-places of a function should result in a singular term that refers, it is less clear whether he also insisted, at the level of reference, that a function should be everywhere defined. I have noted already how little he thought it mattered whether we give an account of quantification at the level of language or of reference; it is therefore no great surprise to find him similarly indifferent here. In Begriffsschrift Frege had held that the content stroke is needed to convert a name of a judgable content into something capable of being preceded by the judgment stroke, and had attempted to express it in ordinary language as ‘is a fact’. On his new semantic theory, however, a sentence expresses a thought capable of being judged true or false, and no operator is required to effect the required unification. In the case where it precedes a sentence the content stroke becomes a mere identity operator, and for this reason Frege renamed it the ‘horizontal’. What, though, of the case where the horizontal precedes a name that is not a sentence? In order for the function it refers to to be everywhere defined, he now had, artificially, to give it a value in this case too: ( T if A = T A =df F otherwise

Wittgenstein’s objection again Commentators have united in condemning Frege’s assimilation of sentences to singular terms, but the reason they usually cite, namely the violence the

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assimilation does to ordinary grammar, misses the point. He was of course aware that in ordinary language a sentence cannot substitute for a singular term without being ‘complementized’ (turned into a substantive): the usual method in English is to prefix it with ‘that’. The issue for us, however, is whether logic should distinguish between sentences and singular terms, not whether grammar does. A more persuasive objection to Frege’s assimilation is this. We sometimes grasp the sense of a singular term without having any independent hold on which object it refers to. So if sentences really were just a sub-class of singular terms, it would be comprehensible that we might grasp the sense of a sentence without knowing whether what it refers to is a truth value or a mountain. That is absurd: we cannot grasp the sense of a sentence—a thought—without realizing that it at least aims at truth or falsity. We noted in Chapter 15 that on Frege’s view the prefix ‘it is true that’ makes no difference to the sense of a sentence. What we have seen now is that the horizontal, by contrast, does make a difference, namely that of converting whatever follows it into a name of a truth value. The objection just considered shows that Frege’s definition of the horizontal is awry, because it attempts quite generally to turn whatever follows it into a judgable content and hence to manufacture thoughts where there were none. An alert reader may feel a sense of déja vu at this point, for the objection to the horizontal of Grundgesetze just given is essentially the same as Wittgenstein’s objection to the content stroke of Begriffsschrift considered in Chapter 8. Frege had indeed revised his semantics in the interim, but not so as to meet the complaint. Wittgenstein’s objection was that the expression of a thought cannot be secured in the opaque fashion that Frege’s conception, both early and late, apparently licenses. A correct theory of judgment, one might say, must show—and not merely stipulate—‘that it is impossible to judge a nonsense’ (TLP 5.5422). What is particularly striking about Wittgenstein’s argument, though, is its essentially Fregean character, for it was Frege’s recognition of the transparency of truth that showed it to be internal to a thought. We should certainly allow that it is external to the sense of a sentence whether it is true or false; we can even side with Frege against Evans by allowing that it is external whether it has a reference at all (i.e. whether it is a thought or only a mock thought); but we cannot allow— what Frege’s assimilation of sentences to singular terms requires—that it could be external to the sense whether it purports to refer to a truth value. Yet was that not essentially Frege’s own objection (PW , 175) to Lotze’s proposal that an ‘auxiliary thought’ could make an association of ideas truth-evaluable?

Truth and taking as true Logical laws, on Frege’s account, are those a priori laws that are knowable independent of intuition. It is because this characterization is essentially negative that he did not aspire to explain how our knowledge of them arises.

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As to the question, why and with what right we acknowledge a logical law to be true, logic can respond only by reducing it to other logical laws. Where this is not possible, it can give no answer. (Gg, I, xvii) Although Frege never wavered in his opposition to psychologism, he did amend his argument against it: previously he had conceived of it as resulting from a failure, derived from empiricism, to distinguish between reasons and causes, whereas now he diagnosed it as resulting from a failure to distinguish between being true and being taken to be true. The laws of logic, he said, are not psychological laws of taking to be true, but laws of being true. If it is true that I write this in my room on 13th July, 1893, while the wind is blowing outside, then it remains true even if all humans should later hold it to be false. If being true is thus independent of anyone’s acknowledgment, then the laws of being true are not psychological laws either but boundary stones which are in an eternal ground, which our thinking may wash over but yet cannot displace. (Gg, I, xvi) When Wittgenstein later spoke of the ‘glories’ of Frege’s Foreword, this was presumably one of the passages he meant. Frege’s aim here was to align logic with the natural sciences. In both, he thought, we can make sense of the possibility that we might all be wrong about something. What disproves the psychologistic conception of logic is not that truth is normative, but that it is objective. The word ‘law’ is ambiguous (Gg, I, xv). A scientific law is descriptive, and it is in this sense that there are laws of truth. Yet descriptive laws constrain how we ought to think if we aim at truth, and so normative laws are derivable from them. ‘The former are given along with the latter.’ (PW , 128) In logic’s case, indeed, it would be more accurate to call the normative variety laws of judgment, not of truth: they are ‘prescriptions for making judgments’ which ‘provide the norm for holding something to be true’ (PW , 145, my emphasis). The question we raised earlier, whether logic’s concern is with truth or with judgment, is therefore misleading: since judgment aims at truth, the correct deployment of the former is constrained by the properties of the latter. In this respect, though, logic is no different from other sciences. Every law stating what is the case can be conceived as prescriptive, one should think in accordance with it, and in that sense it is accordingly a law of thought. This holds for geometrical and physical laws no less than for the logical. (Gg, I, xv) A reasoner who used incorrect logical laws would thus be making a mistake of the same order as a physicist who used incorrect physical laws. The only difference would concern how one might go about convincing them of their error. This is where the logocentric predicament lames us (Gg, I, xvii): we might be unable to persuade a logical alien of his error, because the argumentative tools at our

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disposal were ones he did not accept; but whether we could convince him that he was wrong would be quite a different matter from whether he was wrong.

Further reading Heck (2012) discusses in detail the formal features of the Grundgesetze system. The claim that Frege intended a non-standard interpretation of the second-level quantifier is made by Hintikka & Sandu (1992) and refuted by Heck & Stanley (1993). Blanchette (2012) places considerable interpretative weight on her claim that Frege did not require functions to be everywhere defined. Sullivan (1994) gives another argument that Frege was wrong to treat sentences as names of truth values. On justifying the laws of logic see Taschek (2008) and Burge (1998). On the reasons for the delay that resulted from abandoning an ‘almost completed manuscript’ see Heck (forthcoming).

18 GRUNDGESETZE II: EXTENSIONS

In Grundlagen Frege had proposed to define the number of objects falling under a concept as the extension of a certain concept (that of being a concept equinumerous with the given concept). Yet, despite the crucial role it played in his account, he provided no explanation there of the notion of the extension of a concept, merely observing in a laconic footnote that he assumed it was known (Gl, §68n). The principal innovation in Grundgesetze, as compared to Begriffsschrift, is that Frege now aimed to fill this explanatory gap by treating this notion (or rather a generalization of it that he called a ‘value-range’) as logical.

Extensions and value-ranges In Grundlagen Frege assumed implicitly that extensions satisfy the abstraction principle {x : Gx} = {x : Hx} ≡ ∀x(Gx ≡ Hx) for arbitrary concepts Gξ and Hξ . Now that he regarded sentences as names of truth values, and hence concept-words as a particular case of function-words, he correspondingly treated the notion of the extension of a concept as a particular case of the more general notion of the ‘value-range’ of a function. (The German word Wertverlauf, which he coined for this, is also sometimes translated ‘course of values’.) In formal terms, he supposed that there is a term-forming operator ‘W’ such that ‘Wxgx’ is a singular term for every function-word ‘gξ ’. He then assumed the basic law that Wxgx = Wxhx ≡ ∀x(gx = hx) (V) for any functions gξ and hξ . (Consistent with this, his procedure when he came to the treatment of numbers was to define NxGx to be the extension of the concept

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of being a value-range equinumerous with the given concept.) In this notation the abstraction principle for extensions becomes WxGx = WxHx ≡ ∀x(Gx ≡ Hx),

(V∗ )

i.e. the restriction of (V) to concepts. In the formal development Frege used Basic Law V in two ways: first, it served his explicit definition of number; second, whenever there occurs a term referring to a concept, he could replace it with a singular term by applying the W-operator, thus obviating the need for quantification of order greater than two. Concepts are on Frege’s view objective, extensional entities. The purpose of the value-range operator was therefore not to convert the non-objective into the objective or the non-extensional into the extensional, but only to collapse the hierarchy of levels by projecting every concept down to an object. One might try to make this last point vivid by simplifying (V∗ ) to W(G) = W(H) ≡ G = H, but Frege would have objected to this formulation on two grounds: first, he regarded it as a ‘monstrosity’ to denote a concept with a sign that does not have its argument-places explicitly marked; second, he would have rejected the ambiguous use of ‘=’, referring on the left to a relation between objects, on the right to a quite different (although analogous) relation between concepts.

Semantics Basic Law V, like Hume’s Principle, is an impredicative abstraction principle. In §64 of Grundlagen Frege had argued that in any such principle the left-hand side is merely a recarving of the content of the right. In Grundgesetze he said, in similar vein, that in the transition from right to left we ‘are not really doing anything new’ (Gg, II, §147). His claim, though, was that (V) is a basic law of logic, not an implicit definition. He therefore had the task, which he had not had in Begriffsschrift, of laying down the semantics of value-range terms in such a way that Basic Law V could be recognized as a logical truth. In doing this he could not appeal to some prior grasp we might have of what value-ranges are, since that might depend on intuition. What he did instead was simply to stipulate that the reference of the equation on the left-hand side of an instance of Basic Law V should be the same as that of the equivalence on the right, thus making Basic Law V trivially true. Call this the ‘basic stipulation’. The difficult part of the task (to which he devoted §10 and §§29–32 of Grundgesetze) was to explain how, using this stipulation as a starting point, every singular term in the system could be given a reference. There are two issues to consider here, uniqueness and existence. Take uniqueness first. The basic stipulation certainly does not on its own give every singular term a reference, because it does not solve the Julius Caesar

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problem: it does not settle the truth-conditions of mixed identity sentences of the form Wxfx = A, where ‘A’ is a singular term but not a value-range term. Frege canvassed the idea of solving the problem by stipulation: if ‘A’ is not a value-range term, stipulate that Wx(x = A) = A. He noted, though, that this does not work in general, since if an object is not presented as a value-range, it may still be one. Even so, he offered an argument designed to show that in the case of the two truth values, this concern did not apply, and hence that he could indeed stipulate that Wx(x = T) = T and Wx(x = F) = F. Notice, though, that the Julius Caesar problem has two versions, narrow and broad. The narrow version requires us only to deal with cases in which ‘A’ is a term in the current language; the broad version requires us to settle the identity of the reference of a term uniquely. Frege had solved the narrow version of the Julius Caesar problem (since truth values are the only objects other than valueranges to which his concept-script permitted singular reference), but he had not solved the broad version. His response (Gg, §10) was to suggest, lamely, that other cases could be dealt with later as they arose. How, in that case, could he purport to prove in §§29–32 that his semantics had settled the issue? The answer is that in those sections of the book he assumed that every object has a name in the concept-script, and hence that the broad version of the problem collapses into the narrow. The difficulty with this, of course, is that the assumption could only be temporary: as soon as we want to apply the theory (e.g. in counting Roman Emperors), the problem will arise once again. What, now, of existence? The basic stipulation had given a truth value to every equation between value-range terms. Frege now tried to show, by what we should nowadays call a proof by recursion on the complexity of terms, that a reference can be given to every singular term in which a value-range term may legitimately occur. In the process he implicitly appealed to a version of the context principle. In Grundlagen he had stated the context principle in terms of an undifferentiated notion of content. Now that he had split his notion of content into two parts, sense and reference, the context principle split correspondingly in two. He now explicitly endorsed a context principle for sense. The sense of an individual name, he said, consists in the contribution it makes to the thought expressed by a sentence of which it is a part (Gg, I, §32). ‘One can ask after reference only where signs are components of propositions expressing thoughts.’ (Gg, II, §97) A corresponding context principle for reference would state that for a singular term to have a reference it suffices that every sentence in which it occurs has a truth value. In this form, though, it would privilege truth values over other objects, contrary to Frege’s assimilation of sentences to singular terms. So what he actually used in his recursive argument was a generalized context principle for reference, according to which for a singular term to have reference it suffices that every singular term in which it occurs has reference. The difficulty with this, though, is that a recursive proof needs a starting point. In order for the recursion to get going, he in fact assumed (§31) that there are two singular terms, ‘T’ and ‘F’,

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which do have a reference. So it turns out that the truth values are special after all, despite his apparent claim to the contrary. We have already seen, though, that Frege’s assimilation of sentences to singular terms was mistaken. This suggests that we might tidy up his attempted proof of referentiality for the singular terms of the concept-script so that it aimed to show only that every sentence containing a value-range term has a reference. This, though, would not save the ‘proof ’, as we are about to see.

The contradiction So far from being a logical law, as Frege claimed, Basic Law V is in fact inconsistent. To see why, first define the notion of membership: x ∈W y ≡ ∃G(y = WzGz ∧ Gx). Then prove, using (V∗ ), that x ∈W WzGz ≡ Gx.

(1)

Now let aW = Wx(∼x ∈W x), and deduce from (1) that ∀x(x ∈W aW ≡ ∼x ∈W x), whence aW ∈W aW ≡ ∼aW ∈W aW . Contradiction. The argument just stated is of course Russell’s. Frege learnt of it in a letter from him in June 1902 and immediately expressed his consternation. He had already conceded (Gg, I, vii) that if there was a basic law over which a dispute might arise, it was Basic Law V. This, though, was presumably not because he already suspected it of inconsistency, but rather because the semantics that underpinned it was so much more elaborate than for the other basic laws. As soon as he received Russell’s letter, indeed, Frege recognized that there must be a flaw in his proof of referentiality. Notice, though, that Russell’s argument uses only (V∗ ), not (V). So responsibility for the contradiction cannot lie with Frege’s assimilation of sentences to names. Even the tidied-up version of his proof of referentiality must therefore contain a fallacy. The problem with it is that even the restricted recursion in the tidied-up proof would still be circular: the application of the value-range operator gives us an object which falls in the range of the first-order quantifier and hence has the potential to disrupt the truth values of simpler sentences already dealt with in earlier steps of the recursion. Notice, too, that Russell’s argument does not identify some one particularly troublesome object which is mysteriously incapable of being the extension of a

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concept, nor does it show that some particular concept, specifiable independently of the value-range operator W, is incapable of having an extension. What it shows is merely a point about cardinalities: if there are κ objects, then there are 2κ first-level concepts, but by Cantor’s theorem κ < 2κ , and so there cannot be a one-to-one function from first-level concepts to objects. This shows what is wrong with trying to save the notion of extension by maintaining that only some concepts—call them ‘good’—have an extension, and restricting (V∗ ) to these. Russell’s argument merely exposes the general point that an impredicative abstraction principle cannot be consistent if there are more equivalence classes than there are objects. Frege was slow to recognize this: in the appendix he added to Volume II of Grundgesetze in response to the contradiction he proposed an amended version of Basic Law V which does not obey this constraint, and hence is still contradictory (see Quine 1955). Russell’s contradiction therefore shows that there is something fundamentally amiss with Frege’s attempt at giving a sense to value-range terms. If I stipulate that ‘Tully’ is to refer to Cicero, I thereby give the former the same sense as the latter. The present case is different, because the sign ‘Wxfx = Wxgx’ to which Frege’s basic stipulation attempted to give a sense is complex: ‘=’ is the already understood sign for identity between objects. The underlying difficulty, though, is that Frege was working with two distinct notions of reference. In ‘On sense and reference’ he started from the notion that the reference of an ordinary singular term such as ‘Hesperus’ is its bearer, and proceeded from there to an argument that the reference of a whole sentence should be taken to be its truth value. In Grundgesetze he proceeded in the opposite direction, taking the two truth values as known and then attempting in effect to give every singular term a reference simply by giving a truth value to each sentence in which it occurs. It is difficult to see how the two accounts can be made to cohere: the first seems fitted only to explaining reference to actual objects by means of the name-bearer prototype, the second only to abstract ones. What is lacking is a single account that explains both usages equally.

Logical objects again Frege soon realized that his amendment to Basic Law V had not saved the theory of extensions of concepts. ‘Set theory in ruins’, he noted ruefully in 1906. He consoled himself, though, that his concept-script was ‘in the main not dependent on it’ (PW , 176), and in later years he continued to lecture on this at Jena, simply omitting any mention of value-ranges. This might save logic, but it dealt a fatal blow to the logicist project to which he had devoted his effort for the previous 30 years. The collapse of Basic Law V barred his only available route to logical objects. It therefore seemed ‘to undermine not only the foundations of my arithmetic but the only possible foundations of arithmetic as such’ (PMC, 132).

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So although Frege still maintained that ‘mathematics has closer ties with logic than does any other discipline’ (PW , 203), because ‘in no other discipline does inference play so large a part’, he now conceded that the primitive truths on which those inferences rest might be specifically mathematical—e.g. the principle of mathematical induction—rather than basic laws of logic. He also began to doubt his earlier conviction that numbers are objects. Wittgenstein later recollected an incident that must have taken place in 1913. As we were waiting at the station for my train, I said to him ‘Don’t you ever find any difficulty in your theory that numbers are objects?’ He replied ‘Sometimes I seem to see a difficulty—but then again I don’t see it.’ Frege therefore returned to the treatment of numbers as second-level quantifiers that he had considered and rejected in Grundlagen. These second level concepts form a series and there is a rule in accordance with which, if one of these concepts is given, we can specify the next. But still we do not have in them the numbers of arithmetic; we do not have objects, but concepts. How can we get from these concepts to the numbers of arithmetic in a way that cannot be faulted? Or are there simply no numbers in arithmetic? Could the numerals help to form signs for these second level concepts, and yet not be signs in their own right? (PW , 256–7) He had, he now thought, been misled by ordinary language into accepting too readily that phrases such as ‘the number of Fs’ or ‘the extension of F’ refer to objects. He held instead that the logical source of knowledge ‘on its own cannot yield us any objects’ (PW , 279). The difficulty with this, however, is that treating numbers as second-level quantifiers will not give us arithmetic without a further assumption, e.g. that there are infinitely many objects.

Further reading Dummett (1991b) and Boolos (1993b) discuss Frege’s error. For the project of restricting (V) to ‘good’ concepts see Shapiro & Weir (1999). On the tension between Frege’s two conceptions of reference see Dummett (1973, 196–203, 1981b, ch. 7). On Frege’s later views about numbers see Klement (2012). On the basic stipulation see Heck (2012).

19 THE FREGE–HILBERT CORRESPONDENCE

In Grundlagen Frege considered but in the end rejected the proposal that Hume’s Principle be regarded as an implicit definition of the number operator. It is worth pausing now to consider implicit definition in greater generality. Frege called this ‘formalism’, but I shall here call it axiomatic formalism, to distinguish it both from the radical formalism that we considered and rejected in Chapter 11 and from another, more elaborate view, also often known as formalism, proposed by Hilbert in the 1920s.

The Foundations of Geometry Already in Grundlagen Frege discussed a version of axiomatic formalism that had been proposed by several 19th-century authors in relation not to the natural numbers, but to extensions thereof: integral, rational, real and complex numbers. At its crudest, the proposal was that we can postulate a system with any properties that we like. Some proponents of the view were slow to recognize that the properties in question need at least to be consistent. A similar criticism could be levelled at early 19th-century work on non-Euclidean geometry, which investigated the consequences of negating the axiom of parallels but provided no guarantee that this is even consistent. It was a key moment when Beltrami (1868) showed how to re-interpret non-Euclidean geometry in Euclidean space, since, as Hoüel (1870) observed, this shows that non-Euclidean geometry is at least consistent. The standard modern way of describing Beltrami’s work is to say that he reinterpreted the words ‘point’, ‘line’ and ‘between’ as they occur in the axioms of non-Euclidean geometry in such a way that the axioms become true, but in fact neither Beltrami nor Houel quite did this, and it was left to Poincaré (1882, 8) to make the idea explicit.

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Frege’s best known engagement with axiomatic formalism came in his 1899– 1900 correspondence with Hilbert (PMC, 34–51), which followed his reading of the latter’s monograph, The Foundations of Geometry. Hilbert there used Poincaré’s method to show that various of the axioms of geometry are independent of one another. What caught Frege’s attention was Hilbert’s claim that the words ‘point’, ‘line’ and ‘between’ are implicitly defined by the axioms in which they occur. As Hilbert told Frege, he did not invent his view because he ‘had nothing better to do’ (PMC, 51), but because he wanted to eliminate intuition from geometry. Euclid had attempted to define the notion of a point as that which has no extension, and although he did not use this definition explicitly in proofs, he did so tacitly, by appealing to geometrical facts which we take to be obvious because of our grasp of their meaning; Hilbert wanted to eliminate all such illicit appeals to geometrical intuition. One is looking for something one can never find because there is nothing there; and everything gets lost and becomes vague and tangled and degenerates into a game of hide and seek. (Hilbert to Frege, 29 Dec. 1899, in PMC, 39) A further advantage Hilbert claimed for his conception was that it gave Euclidean and non-Euclidean geometry equal mathematical status. Because the words ‘point’, ‘line’ and ‘between’ have different meanings according as they occur in the axioms of Euclidean or of non-Euclidean geometry, the sentence ‘The angles of a triangle sum to two right angles’ expresses different contents in the two cases—true in the former case, false in the latter.

Deductivism Hilbert might well have expected Frege to support his project, since it paralleled Frege’s attempt to eliminate intuition from arithmetic. Just this raised Frege’s suspicions, however, since he held that Euclidean geometry is distinguished precisely by its dependence on intuition. Some of his objections to Hilbert’s account are no more than terminological. So we need to clear these away in order to come to what was really at issue. The first was that for Frege an ‘axiom’ is a true assumption. Since he believed that space is Euclidean, he concluded that the ‘axioms’ of non-Euclidean geometry are wrongly so called. No man can serve two masters. One cannot serve both truth and untruth. If Euclidean geometry is true, then non-Euclidean geometry is false, and if non-Euclidean geometry is true, then Euclidean geometry is false. . . . Do we dare to treat Euclid’s elements, which have exercised unquestioned sway for 2000 years, as we have treated astrology? It is only if we do not dare to do this that we can put Euclid’s axioms forward as propositions that are neither false nor doubtful. In that case non-Euclidean geometry will have to be counted amongst the pseudo-sciences, to the study of which

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we still attach some slight importance, but only as historical curiosities. (PW , 169) For us to be able to recognize an axiom as true, it ‘must not contain any term with which we are unfamiliar’ (PW , 244).1 For Frege, therefore, our grasp of the axioms of Euclidean geometry depends on a prior understanding of the words ‘point’, ‘line’, ‘between’, etc. that occur in them; what Hilbert counted a virtue of his account—that it was neutral between Euclidean and non-Euclidean geometry—Frege regarded as a demonstration of its error. Frege’s second objection was to the notion of a conditional proof. For simplicity’s sake I shall suppose that the axioms are finite in number, and hence can be conjoined to form a single sentence Γ (point, line, between). (What to say about the case of an infinite list is an important question nowadays, because many of the first-order theories that interest modern logicians are of this form, but its importance did not emerge until later.) Frege held that the non-Euclidean geometer’s ‘theorem’ should be re-described as a conditional whose antecedent contains all its assumptions. Instead of saying that the geometer proved Φ(point, line, between) from Γ (point, line, between), we should say that he proved outright the conditional Γ (point, line, between) ⊃ Φ(point, line, between). Frege’s third objection was that Hilbert misused the word ‘definition’ when he said that the ‘axioms’ implicitly define the non-logical words (in the current case, ‘point’, ‘line’ and ‘between’). Frege held that these primitive terms are variables, so that the theorem just mentioned is really ∀P∀L∀B(Γ (P, L, B) ⊃ Φ(P, L, B)), where the bound variables P, L and B range over two first-order concepts and a first-order ternary relation respectively. Hilbert had given not an implicit definition of the first-level concepts point and line, but an explicit definition of a ‘nonEuclidean geometry’—a structure consisting of concepts related in a certain way. The acceptable core of Hilbert’s account, on Frege’s view, was a set of merely conditional claims about the properties of such structures. In a grudging concession to Hilbert’s way of speaking, Frege eventually (CP, 330ff) agreed to call Γ (P, L, B) a ‘pseudo-axiom’ in this context. I shall call this view ‘deductivism’. (It is also sometimes called ‘implicationism’ or ‘if-thenism’.) It maintains that the geometer should be understood as making only the conditional claim that if something is a non-Euclidean space, then it has some property, e.g. that the angles of any triangle in the space add up to

1 Strictly, Frege should have said that the expression of an axiom must not contain unfamiliar terms: the axiom, on his terminology, is the content expressed, not the sentence itself.

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less than 180 degrees. This is presumably what Frege had in mind in Grundlagen when he talked of studying non-Euclidean geometry ‘for the purposes of conceptual thought’ (Gl, §14). Deductivism is in fact quite a good account of large parts of modern mathematics. Where it stumbles is in relation to those parts of mathematics that are directly applicable to the world, such as arithmetic, since it fails to explain how we detach the antecedent so as to arrive at unconditional conclusions. We can make two further simplifications. First, it is largely irrelevant whether the entities being posited are objects or concepts. In the case under contention Hilbert was attempting to introduce concepts (point, line, etc.), but a similar issue would also arise if he had given an axiomatic treatment of something Frege would have regarded as an object (e.g. the line at infinity in projective geometry). Second, we can limit ourselves to the case in which only one term is being defined. Hilbert’s axiom system defined several terms (‘point’, ‘line’ and ‘between’) simultaneously. Frege called this a ‘monstrosity’ (PMC, 91), but on this point he was simply wrong: there is no more reason to object to Hilbert’s procedure with several primitives than with one. After all, we could just as well treat the pseudo-axiom as characterizing a single ordered triple (point, line, between) rather than its three separate components. Let us therefore simplify matters by focusing on a formula Γ (X), in which the variable X ranges over some logical type. The deductivist uncontroversially asserts the conditional ∀X(Γ (X) ⊃ Φ(X)). The Hilbertian axiomatic formalist additionally claims an entitlement to introduce a new simple expression ‘A0 ’ implicitly defined by the pseudo-axiom Γ (A0 ), and then to deduce the theorem Φ(A0 ). The Fregean, on the other hand, insists that we must first give the term ‘A’ a meaning, and only then recognize Γ (A) as an axiom (if grounded in intuition) or a basic law (if on purely logical grounds), before deducing the theorem Φ(A). Deductivism is thus a sort of neutral core, representing what the axiomatic formalist and the Fregean agree on.

Uniqueness Grundlagen §64 contains what is in effect an argument for the analyticity of Hume’s Principle, on the ground that the equivalence relation on the right may be recarved to form the equality on the left; but Frege nonetheless rejected Hume’s Principle as an implicit definition of number terms, because of the Julius Caesar problem. In his correspondence with Hilbert he turned this into a general objection to axiomatic formalism, the only difference being that the example he now used was his pocket-watch rather than Julius Caesar (PMC, 45). The Julius Caesar requirement has two versions, narrow and broad: the narrow requirement is that when we introduce a term ‘A0 ’ we should determine the truth-conditions of the equation ‘A0 = A’ for every singular term ‘A’

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already in the language; the broad requirement is that we should determine the truth-conditions of ‘A0 = B’ for every term ‘B’ that we might add to the language. If every object in the domain already has a name, then the two versions are equivalent. Notice, though, that the broad version is tantamount to requiring that our definition should fix the identity of A0 uniquely, i.e. that we should be able to prove the uniqueness property ∃!XΓ (X). If we insist on this, however, the method of implicit definition loses any advantage it may have had over explicit definition, since we can simply define A0 = XΓ (X). In Grundlagen (unsurprisingly, given the book’s informal character) Frege did not make it clear whether he intended the narrow version of the requirement or the broad. In his later work, however, he was explicit that a definition must fix the identity uniquely. ‘One must not’, he said, ‘explain a sign or word by explaining an expression in which it occurs and whose remaining parts are known’, because ‘the reference of an expression together with that of one of its parts does not always determine the reference of the remaining part’ (Gg, II, §66). In Grundlagen Frege responded to the Julius Caesar problem by adopting an explicit definition of the number-of function instead. Why did Hilbert not do something similar? Why did he not, for instance, just define non-Euclidean space to be the Beltrami model? Similarly, why should the advocate of Peano’s axioms not just define the natural number structure to be the particular simply infinite system which Dedekind defined in the course of his existence proof? Dedekind advised against artificially resolving the uniqueness problem by means of explicit definitions because of the extra properties this method gives to the objects so defined. On the structuralist conception mathematical objects, unlike empirical ones, are insubstantial: they have no properties not logically entailed by the pseudo-axioms that implicitly define them. So the problem of extra content cannot arise for them. For Dedekind it counted as an advantage of the structuralist method that it gave to A0 only those properties logically entailed by Γ (A0 ). For Frege, by contrast, A0 would always be intrinsically under-specified, which is why he sought instead an explicit definition of a suitable entity A for which Γ (A) could be recognized to be true. ι

Existence and consistency On the question of how to respond to the paradoxes Hilbert had a head-start on Frege. Russell discovered his paradox in 1901 and communicated it to Frege in June 1902, whereas Hilbert had already learnt in 1898 from Cantor of a contradiction involving the class of all alephs, and Zermelo, Hilbert’s colleague at Göttingen, discovered Russell’s paradox independently in 1899 or 1900. By the time he corresponded with Frege, therefore, Hilbert already had reason to doubt the reliability of the direct method he had previously used to prove the consistency of a mathematical theory by exhibiting a model, because that method used notions (such as that of a class) over which the paradoxes cast doubt.

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Hilbert’s response was quite different from Frege’s, however. While Frege abandoned hope of a route to arithmetical truths independent of intuition. Hilbert proposed instead to circumvent the paradoxes by means of the doctrine (already proposed by Poincaré) that in mathematics consistency entails existence. When Hilbert first recommended this strategy, in his 1899 Munich lecture (1900, 183), he mentioned as motivation not Russell’s paradox but the paradox of the class of all alephs, and then only in passing. The letter Frege wrote to Hilbert after reading the text of the lecture betrays no indication that he realized the significance of the point at this stage. Eventually, though, Hilbert’s proposal gave him cause to examine more closely what it means to call pseudo-axioms ‘independent’ or ‘consistent’. In his terminology one pseudo-axiom is independent of another just in case the conjunction of the latter with the negation of the former is consistent (letter to Liebmann, 29 July 1900, in PMC, 91). A single pseudo-axiom is consistent just in case its negation is logically vacuous, i.e. it is possible that the pseudo-axiom should be instantiated. In the terminology that is nowadays usual, Frege’s notion of a pseudo-axiom Γ (ξ ) being consistent would be described by saying that ∃XΓ (X) is logically consistent. Frege, however, was reluctant to apply the vocabulary in relation to complete thoughts in this way. If Γ (A0 ) is consistent, it does not follow that Γ (A) is. For instance, the property of being alive is consistent, but ‘The late King George VI is alive’ is not consistent. So if Hilbert showed that the pseudo-axioms of non-Euclidean geometry are consistent, this would not show that the axioms of Euclidean geometry remain consistent when the axiom of parallels is replaced with its negation, since these axioms are about the points and lines of actual space and hence may have a content that goes beyond what is explicitly stated in them. Logical consistency plainly does not always entail existence: it is logically consistent, but false, that there should be unicorns. That, though, is an empirical matter. A more interesting case, perhaps, is Frege’s example of the concept of an omnipotent, benevolent being: this concept may well be consistent, but that does not suffice to prove God’s existence. It should be stressed, though, that Hilbert’s claim relates only to mathematics. If mathematics is part of logic, we might expect to be able to infer from the truth of a mathematical claim to its logical truth, and hence from consistency to existence. Even so, there remains the question, raised by Frege in his correspondence with Hilbert, of how to prove the consistency of a concept other than by exhibiting an entity that falls under it. As he had already pointed out in Grundlagen (§94), it is not sufficient merely to try to find an inconsistency and fail. ‘It does not follow that because we see no contradiction there is none there.’ What we need is a proof. Frege insisted that the method Hilbert had used prior to the paradoxes, of exhibiting a model, is the only way of proving a pseudo-axiom consistent. ‘We can only establish that a concept is free from contradiction by first producing something that falls under it.’ (Gl, §95) To show that the concept

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unicorn is consistent, for instance, we would have to establish that no contradiction lurks beneath its surface; and how could we do that? At that stage Hilbert’s metamathematical ideas were still in their infancy, but in his 1904 Heidelberg lecture he sketched a project—worked out more fully in the 1920s, but then brought to an abrupt end, at least for the cases that most concerned him, by Gödel’s incompleteness theorem (1931)—of proving the consistency of a theory proof-theoretically, by demonstrating that no finite string of signs constitutes a proof of an inconsistency. Notice, though, that this method is heavily dependent on the rules and laws of the formal system being complete, i.e. sufficient to prove all the logical truths expressible in the system. Only in that case does the failure of the system to prove an inconsistency entail that the theory is indeed logically consistent. We can see now how radical Hilbert’s new notion of mathematical existence was. Hilbert (1905, 176) called the mathematical objects he was now positing ‘objects of thought’ (Gedankendinge); it might be better to call them ‘metalogical objects’, since they owe their being not to logic directly, but indirectly to the consistency of their defining concepts. This notion has certainly had its adherents— it underlies the attitude of some modern set theorists towards large cardinal axioms, for instance—but it is highly problematic, given that the defining concepts of different such objects may be individually consistent but jointly inconsistent.

Frege’s 1906 definition Even without Hilbert’s programme, the failure of Frege’s logicism would have rendered urgent the need for a criterion to determine whether a sentence expresses a logical truth or not. In fact, though, the lacuna had been visible for some time. Kerry had already noted (1887, 262) that it is ‘very regrettable that in none of his writings has Frege defined the concept of the logical’. In 1906 Frege took a hesitant step towards plugging this gap. He based this on the notion of what I shall here call a logic-fixing permutation, i.e. a permutation of the primitive signs of a language which takes each logical sign to itself and each non-logical sign to another sign of the same logical type (singular terms to singular terms, function-words to function-words of the same type, etc.). Any logic-fixing permutation of the signs can then be used to define a permutation of the sentences of the language: he talked rather casually of using the permutation as a translation manual, but he was implicitly making use of what is now called a definition by recursion on the complexity of a string of words. Let us say that two thoughts ‘have the same logical form’ if there is a logicfixing permutation of the language which takes a sentence expressing the first thought to one expressing the second. For instance, the thoughts expressed by ‘Socrates is mortal or wicked’ and ‘Plato is wise or old’ have the same logical form, because there is a logic-fixing function that takes ‘Socrates’ to ‘Plato’, ‘mortal’ to ‘wise’, and ‘wicked’ to ‘old’. Frege’s proposal amounted in effect to defining a

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thought to be a logical truth just in case every thought of the same logical form as it is true. The underlying idea is that if two thoughts are of the same logical form, so that there is a logic-fixing permutation of one into the other, then any proof of a contradiction from the first could be converted by means of this translation into a proof of a contradiction from the second. If the second thought is true, then there is no proof of a contradiction from it; hence there is no such proof from the first either. There are two errors in Frege’s definition, however. First, its plausibility depends on having a language rich enough to express all thoughts. If Socrates, who happened to be wise, were the only man with a name, and wisdom were the only concept for which the language had a concept-word, then the thought expressed by ‘Socrates is wise’ would come out as a logical truth on his definition, because the only logic-fixing permutation of such an impoverished language would be the identity function. This example may seem too extreme to be of significance; but he made no presumption, let us recall, even that the conceptscript contains a name for every object. When we were discussing his account of quantification, we met this point by allowing auxiliary names, but we can hardly do so here, unless we allow, awkwardly, that the domain of our logic-fixing function contains all possible extensions of our current language. The modern definition of logical truth overcomes this difficulty by generalizing over interpretations: a sentence is a logical truth just in case it is true under every interpretation of the non-logical words it contains. Frege’s difficulty here was a consequence of his antipathy to detaching words from their meanings in this fashion. ‘We must retain at all costs’, he said, ‘the univocity of signs.’ (CP, 318) The second error in Frege’s criterion concerns hidden complexity. Is the thought expressed by ‘Bystrouška is a vixen but not a female fox’ logically consistent? If ‘vixen’ has the same sense as ‘female fox’, then it ought not to be; but if we apply Frege’s criterion to the sentence as stated, it is, because it has the same logical form as the true sentence ‘Russell is a man but not a German engineer’. Frege’s error was to omit the requirement that a logic-fixing permutation act only on words that are logically simple. He should have assumed, that is to say, that the language had already been fully analysed to reveal explicitly any analytic entailments such as that from ‘vixen’ to ‘female fox’. At that point, though, he would also have had to deal with the difficulty of logically simple but synonymous expressions. Even if the flaws in Frege’s criterion noted so far were corrected, there would still be room to doubt its usefulness, for it depends on a prior categorization of language into logical and non-logical parts. After all, if we knew what logical truth was, we could define the logical part of language in terms of it. So we have not really exited from the circle of interdefinable notions. This is by no means trivial. After all, Frege had previously thought that the number-of function is logical. Now, because of Russell’s paradox, he doubted this. Logic, on Frege’s view, has its own subject matter, but how are we to recognize it?

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That Frege proposed his criterion of logical truth with notable hesitancy may be due partly to this difficulty. A further consideration, however, is that this notion is of a fundamentally different character from those we are used to in logic: to show that a truth is logical, we prove it from the basic laws of logic; to show that it is not logical, by contrast, we must enter the ‘unexplored territory’ (CP, 339) of a science in which thoughts are no longer just the vehicles but the subject matter. I mentioned in Chapter 15 Frege’s tendency to regard the realms of sense and reference as disjoint, which would preclude the systematic study of this subject. Having rejected the alternative of treating the signs as his subject matter, he therefore ended his short foray into metalogic with the bare observation that these questions ‘cannot be settled briefly’.

Further reading The Frege–Hilbert correspondence is discussed by Resnik (1974) and Blanchette (1996). On Frege’s 1906 treatment of consistency see Ricketts (1998) and Blanchette (2014). On the project of improving Frege’s definition see Antonelli & May (2000).

20 LATE WRITINGS

Frege published nothing between 1906 and his retirement in 1918, but thereafter he completed three articles, and began a fourth, intended to form a connected presentation of the philosophical foundations of logic. These Logical Investigations, as he called them, had been long in the making: the outline dated from 1906, and the first two were based on a draft of 1897. Moreover, much of the time they merely re-iterate, or gives further reasons for, views that he had held throughout his working life. To take one example, he explained his reason for not treating rejection as a primitive force on a par with assertion, namely that rejection can be explained in terms of assertion and negation, whereas the reverse is not true: we cannot explain negation in terms of assertion and rejection, because we need to be able to understand what it is to negate a thought independent of whether it is being asserted or rejected. Another modest extension of his earlier work concerns the ‘Frege point’, ‘the dissociation of assertoric force from the predicate’ (PW , 184), which Frege now generalized to allow for forces other than assertion. Consider, for instance, these three utterances: (1) ‘The door is closed.’ (2) ‘Is the door closed?’ (3) ‘Close the door!’ The second, Frege now said, attaches interrogative force to just the same thought that is asserted in the first. We might similarly claim that the third attaches imperative force, since it constitutes the command to bring it about that this thought is true, but Frege rather oddly resisted this further step without explaining why. In 1911 Frege met Wittgenstein, who visited him, either in Jena or Brunshaupten (a resort on the Baltic coast where he sometimes holidayed), on three occasions between then and the outbreak of the war. These visits certainly had a great effect on Wittgenstein, who revered Frege for the rest of his life. The extent to which there was influence in the other direction is less clear. Certainly

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his article ‘Thoughts’ is not a direct response to the Tractatus—he submitted it for publication shortly before he saw a typescript of the Tractatus for the first time and the letters he wrote to Wittgenstein thereafter suggest no deep engagement with more than the first couple of pages of the book—but it is quite conceivable that both there and in his other later writings he may have been responding to the discussions he had had with Wittgenstein before the war.

The structure of the realm of reference One point at which Wittgenstein’s influence may be visible concerns Frege’s conception of the nature of the two truth values. In ‘On sense and reference’ Frege had suggested that the True has a structure in which every true thought is somehow detectable as a part. Since Venus falls under the concept planet, both planet and concept are part of the True, in a ‘special sense’ at least. In his later work, however, he rejected this view. The complexity of a way of conceiving of an object does not, he now thought, entail any complexity in the object conceived of. That Stockholm is the capital of Sweden does not entail that Sweden is part of Stockholm (PW , 255). When we evaluate the complex thought that the capital of Sweden lies on the Baltic, we first identify the capital of Sweden, and then determine whether that city falls under the relevant concept: it is irrelevant to the second stage of the process by what means we picked out Stockholm in the first. The model of functional evaluation operative in Frege’s semantics does not preserve the complexity of the argument of a function in its value. In the form just stated, this is a point about functions, but since Frege held that a concept is simply a function whose range consists of truth values, he had in consistency to apply it to them too: although the True is the value for the argument Charles Dickens of the function ‘ξ wrote Hard Times’, Dickens is not, even in a ‘special sense’, part of the reference of the sentence ‘Charles Dickens wrote Hard Times’. The references of the parts of the sentence are not parts of the reference of the sentence. However: The sense of a part of the sentence is part of the sense of the sentence. (L13, 87, amended) Here Frege neither acknowledged that he was deviating from his earlier view nor paused to consider whether the emendation threatened any of his other doctrines. Even so, there are several reasons to think that it did. The first concerns his view that concepts are objective entities and not mere artefacts of a semantic theory. His argument for this (PW , 192–3) had been, in effect, that concepts must be part of reality in order that they may participate in making thoughts true. This had some force when facts were conceived of as parts of reality, since it is perhaps plausible that the parts of parts of reality should themselves belong to reality; but now that Frege had rejected this conception, it is hard to see what is left of it. A second effect of Frege’s emendation was to cast doubt on his conception of concepts as unsaturated. That conception, we noted earlier, was based on the

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idea that the structure of reality in some way mirrors that of language. With that abandoned, the claim that concepts are unsaturated (and, with it, the paradox of the concept horse in its original form) is left wholly unmotivated. Now it was the realm of sense whose structure mirrored that of language: the unsaturatedness of ‘ξ is a horse’ was mirrored by the unsaturatedness of the sense it expressed (see PW , 255). A third issue concerns Frege’s conception of the realm of reference. On the earlier view, there had been an aspect of the True that made a thought true. Now, though, all there is at the level of reference is objects and functions. Facts, we are now told, lie at the level of sense. ‘A fact’, he said briskly, ‘is a thought that is true.’ (CP, 368) Facts (and thoughts more generally) are not real in the way that physical objects are, but are capable nonetheless of being grasped by us and taken as true. This is plainly insufficient, however. Even if we agree with Frege that truth cannot be explained in terms of correspondence, that is not to say that it should not entail some sort of correspondence. His difficulty now was that his conception left the relation between facts (i.e. true thoughts) and the world quite opaque. One might well think that this ought to have led him to question the assumptions on which his conception of the realm of reference rested—most egregiously, the assumption that sentences are names, and concepts functions. He did not, indeed, repeat this view in the Logical Investigations, but the notion that he might have given it up is unfortunately refuted by its repetition in notes he wrote in 1919 (PW , 255–6).

Thought and language One task for a theory of meaning is to explain the striking fact that we finite beings are capable of understanding a potential infinity of sentences we have never heard before. This fact struck Frege himself, but not, it seems, until remarkably late. Not until 1914 do we find him observing that the possibility of our understanding sentences which we have never heard before rests evidently on this, that we can construct the sense of a sentence out of parts that correspond to words. (PMC, 79) We have noted already his persistent concern with the detection of complexity in a thought we already grasp, but only now was he taking an explicit interest in the converse process of grasping a thought by means of our grasp of its parts. In both Begriffsschrift and Grundgesetze Frege’s semantics for predicate logic was guided at the molecular level by the compositionality principle—that the content of a logically complex sentence is to be explained in terms of the contents of its parts—but he was notably silent about the structure of non-logical atomic sentences (those containing no logical signs). Only in his late work did he suggest compositionality for these too. ‘The world of thoughts has a model in the world of sentences, expressions, words, signs. To the structure of the thought corresponds the compounding of words into a sentence.’ (CP, 378)

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It is astonishing what language can do. With a few syllables it can express an incalculable number of thought, so that even a thought grasped by a terrestrial being for the very first time can be put into a form of words which will be understood by someone to whom the thought is entirely new. This would be impossible, were we not able to distinguish parts in the thoughts corresponding to the parts of a sentence. (CP, 390) Once again, one might wonder here whether Wittgenstein’s voice is audible in the background. There is a tension, though, between Frege’s claim that ‘the structure of the sentence serves as the image of the structure of the thought’ (CP, 390) and his earlier view that ‘different sentences may express the same thought’ (CP, 188). The words ‘made up of ’, ‘consist of ’, ‘component’, ‘part’ may lead to our looking at it the wrong way. If we choose to speak of parts in this connection, all the same these parts are not mutually independent in the way that we are elsewhere used to find when we have parts of a whole. (CP, 386) Frege’s problem was that he was deploying two competing notions of complexity: part-whole, to explain how the sense of a sentence is constructed from the senses of its primitive constituents; and function-argument, to explain how we discern other patterns in the senses so constructed.

Sameness of sense We have already noted the puzzle concerning how fine-grained Frege’s notion of sense was. It is not as obvious as one might at first suppose that ‘Hesperus is a planet’ and ‘Phosphorus is a planet’ have different inferential powers and hence express different thoughts. There was an instability in his criterion of identity for thoughts: in 1906 he coarsened the notion of sense so as to treat logically equivalent thoughts as identical. Yet in the same year he gave the apparently stricter criterion that anyone who understands them must recognize that they have the same truth value. The difficulty is that it is not obvious how much stricter this criterion really is: on the face of it, I might understand a complicated expressions without realizing that what it says is a logical truth; but it would be open to the objector to say that I do not really understand what it says until I do realize this. In ‘On sense and reference’ Frege granted that different people may associate slightly different senses with the name ‘Aristotle’ without destroying communication completely, but he explicitly noted that this sort of variation is to be excluded from the concept-script (CP, 158n). In 1914, on the other hand, he hypothesized two explorers seeing the same mountain from different valleys and naming it ‘Afla’ and ‘Ateb’ respectively (PMC, 80). Only when they plot the positions on a map do they discover that Afla and Ateb are one and the same. In order to account for this, Frege claimed, we have to posit distinct senses for the two names. It would be easy, bolstered by this example, to suppose that senses should be individuated

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so narrowly that whenever you and I see the same object from different directions, the senses we grasp are different. That might indeed be the consequence if we required the notion of sense to capture every difference of opinion, however implausible, which anyone might rationally entertain, but not if we required it only to encode the information ordinarily communicated by an utterance. If the two explorers had been standing side-by-side when they named the mountain, for instance, the slight difference in angle between their viewpoints would probably not have been sufficient to open up a reasonable doubt that they were talking about the same one. Not only did Frege never settle which of the two criteria for sameness of sense, narrow or broad, he intended, but it is in any case unclear which is appropriate. Should everyday discourse be understood as sufficiently fine-grained to account for esoteric doubts that none of us ordinarily entertains? Frege evidently intended the example of Afla and Ateb to make a point about communication, but it is not clear that he was right to do so. When Dummett re-told the story in the first edition of Frege: Philosophy of Language (1973), he misremembered it as involving only one explorer, who only later realizes that he has seen the same mountain from two different valleys. It is an illuminating mistake: Dummett’s story is no less intelligible than Frege’s. So the role of sense cannot be, as Frege supposed, solely to ground communication; the notion is also required if we wish to conceive of the world as ultimately constituted of entities that, like mountains, have multiple aspects. The issue just considered concerns different spatial perspectives on an object, but it has analogues for other dimensions of variation such as time. Suppose, for instance, that yesterday I gave that day the name ‘A-day’ and then today I give yesterday the name ‘B-day’. If I had lost track of time, I might not realize that they are the same day; but if I do realize this, does that suffice to make the senses the same?

Indexicals Words that ‘only acquire their full sense from the circumstances in which they are used’ (PW , 135) are nowadays called ‘indexical’. Note that on Frege’s view it is words, not senses, that are indexical. In his 1897 ‘Logic’ draft he claimed that a thought expressed with an indexical, such as ‘I am cold’, could also be expressed without it, ‘by using a name to designate the one who feels cold’ (ibid.). Even if we agree with Frege that your utterance of ‘I’ and mine have different senses, there is another sense in which the word ‘means the same’ each time. Modern philosophers of language often follow Kaplan (1989) in calling this constant element ‘character’, but there is no agreement on just how to define it: some hold, for instance, that the character of ‘I’ is a function taking as its value some suitably canonical mode of presentation of the speaker. Frege, however, did not pursue this idea—another indication, perhaps, that ordinary language was not

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his real concern. Mathematics is, and science aspires to be, wholly free of indexical expressions, with the consequence that Frege’s concept-script, designed as it was to serve these disciplines, was free of them too. The problem of sameness of sense considered earlier applies in particular to indexicals. Frege claimed (CP, 358) that an utterance today of the word ‘yesterday’ has the same sense as an utterance yesterday of the word ‘today’, and did not consider the case in which I have lost track of how long has elapsed since my previous utterance. It is easy to see why he might have been uncomfortable with individuating senses so narrowly that the mere possibility of error suffices to render them distinct; for on that view the sense of yesterday’s utterance of ‘today’ would be one that could only be grasped yesterday, thus making the thought I had yesterday, when I wrote in my diary ‘Today is sunny’, now irretrievable. Yet in ‘Thoughts’ Frege did argue for a narrow criterion, at least in the case of the indexical ‘I’. If you hear me utter this word in normal circumstances, you of course have a perfectly good grasp of who I am referring to, but our perspectives are different: mine identifies me in a way that only I can—first-personally. Frege inferred that the word has for me a wholly special sense not shareable with others. If I say ‘I am in pain’, there is a thought which you can grasp, and which you may re-express by saying back to me, ‘You are in pain’; but there is another thought, which I can grasp but you cannot, in which I think about myself, not as Michael Potter, but in a ‘special and primitive way’ (CP, 359) that is in principle private, and hence incommunicable. How can a word have a sense that only I can grasp? In ‘Thoughts’, Frege’s consideration of the private sense of ‘I’ arose out of his discussion of indexicals, but the issue arises for perceptual vocabulary too. If the public thought expressed by ‘I am cold’ has a private shadow, so do ‘The sky is blue’ and ‘The cake is sweet’. This indicates once again that Frege’s rejection of the Lockean model of meaning was only partial: he still held that the sensation played a role as a sort of private surrogate of the public sense.

Idealism In Chapter 17 we noted a shift in Frege’s argument against psychologism. As a result of this shift, he now regarded Berkeley’s subjective idealism, no less than Locke’s empiricism, as responsible for this ‘widespread sickness’ (PW , 105), and hence as deserving of refutation. His argument has two stages, one directed against solipsism, the other against idealism proper. For the first stage in his argument Frege introduced the premiss that it is in the nature of an idea to have exactly one owner: even if I do not assume the existence of other people, I can be sure that my ideas are mine. He then attempted to refute the solipsist, who insists that there is nothing except ideas, by noting that I—the self who owns these ideas—am not myself an idea.

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Then, with the existence of his own self established, Frege felt entitled to regard the existence of other selves as probable too. This was enough to generate the distinction between private and public on which his doctrine of the third realm of sense depended. It is in this second step, he thought, that we allow for mistakes. ‘Contrary to widespread views’, he said, ‘we find certainty in the inner world, while doubt never altogether leaves us in our excursions into the external world.’ (CP, 367) When truth is objective, error is possible. Earlier I suggested that Wittgenstein’s influence might be visible in Frege’s later work, but not here. ‘Thoughts’ is the one piece of Frege’s writing which Wittgenstein is known to have deprecated. ‘It attacked idealism on its weak side’, he told Geach (1977, vii). The reason for his low opinion is not hard to find. Both steps of Frege’s argument against idealism may be criticized: a solipsist who denies the distinction between me and other selves could simply deny that there is any sense to be made of his claim that every idea has a unique owner; and a resourceful subjective idealist can allow for the possibility of localized error (although not, perhaps, of our being wrong about everything). Wittgenstein’s own later work contained an extended critique of Frege’s underlying Cartesian supposition that the inner world is more certain than the external.

Further reading On Frege’s views about ‘I’ see Dummett (1981b, 119f.). For a more radical argument concerning the sense of ‘I’ see Anscombe (1975). On Frege’s partial rejection of the Lockean model of meaning see Kenny (1966). Perry (1977) discusses Frege’s account of indexicals.

21 FREGE’S LEGACY

Although he did not live in quite the total professional obscurity of popular myth, Frege did not in his lifetime receive the recognition which he (rightly) believed was his due. That he is nowadays treated as a major influence on Anglo-American philosophy is due to several factors. One is Wittgenstein’s high opinion of him: it was due largely to his encouragement that in 1952 Peter Geach published a volume of translations of Frege’s writings. Another is Austin’s English translation of Grundlagen, published two years earlier. The most important, though, is that one of the students at the Oxford course for which Austin originally made the translation was Michael Dummett, who was gripped by the experience of reading Frege and went on to devote a large part of his professional life to expounding and analysing his thought. After Dummett’s work, no one could reasonably deny the importance of a detailed knowledge of Frege’s writings for an understanding of logic.

Logic That much of Frege’s logic looks familiar to the modern eye is a measure of his influence: the apparatus of truth-functions and unrestricted singular quantifiers derives from Begriffsschrift; he articulated the hierarchy of levels very shortly thereafter; the importance of criteria of identity derives from Grundlagen; and in Grundgesetze he explicitly restricted variables in the modern way to levels in the hierarchy. Logicians since then have given us no serious reason to doubt these orthodoxies. Yet although the ideas are Fregean, modern notation is not: he was not widely read in the first half of the 20th century, and his ideas were transmitted largely via Peano and Russell, who chose a quite different symbolism. It was left to later logicians such as Quine to restore to logical syntax the Fregean precision that

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Russell eschewed. Even so, there is a danger that the success of Frege’s way with logical notions may have suggested that it is the only way, and hence obscured the extent to which his decisions were not obligatory. For instance, he treated plural terms as incomplete symbols, to be eliminated by various devices, rather than as legitimate expressions in their own right, and therefore hindered the development of plural logic. Frege’s persistence in insisting that logic is not psychology certainly paid off: not only has it become a commonplace with which few analytic philosophers dare to disagree, but he is widely, albeit wrongly, credited as the view’s originator, and by persuading Husserl to join the anti-psychologistic cause he gave it another sphere of influence. If his anti-psychologism is now orthodox, though, there is little else in his philosophy of logic of which this could be said. In particular, his use of the turnstile sign to mark the act of assertion has not found modern favour, and many logicians have ignored his characterization of the truths of logic as those that rest on no other ground, saddling him instead with the un-Fregean idea that they are characterized merely by their generality. The Scholastics followed Aristotle in privileging logic, but Descartes put epistemology in this role instead. Frege may reasonably be said to have returned logic to its privileged position, but his conception of it as resting on a kind of ideal epistemology has been largely ignored. More influential has been his view that ordinary languages are ‘not made to match logic’s ruler’ (PMC, 68), so that logic is needed to ‘break the power of the word over the human mind, uncovering illusions. . . , freeing thought from that which only the nature of the linguistic means of expression attaches to it’ (Bs, Preface). This view was enthusiastically endorsed in the 1920s and 30s by the more scientifically minded of the logical positivists. In the 1940s and 50s, they were displaced in turn by followers of the later Wittgenstein in Cambridge and by a (contemporaneous but distinct) Oxford school of ordinary language philosophers led by Austin, Ryle and Grice. Both groups took their philosophical inspiration more from what people actually say than from what logicians think they ought to say. More recently, however, the idea that logical analysis reveals a deep structure which surface syntax disguises has regained ground—fuelled, perhaps, by the prevalence of introductory logic courses in the compulsory core of degrees in analytic philosophy. Indeed respect for the sort of formalization of arguments that such a course usually includes has become, unfortunately, one of the distinguishing marks of modern analytic philosophy.

Language The logical positivists regarded Frege’s third realm with suspicion: in Language, Truth and Logic (1936), remarkably, Ayer contrived not to mention Frege at all. Quine regarded intensional entities as ‘creatures of darkness’ (1956, 180). ‘The theory of meaning’, he wrote, ‘strikes me as in a comparable state to theology.’

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(1951, 92) Thereafter, a whole generation of naturalistically inclined American philosophers apostasized meanings as the product of a faulty metaphysics. The influence of Oxford ordinary language philosophers declined sharply after Austin’s death in 1960—partly, perhaps, because of their failure to supply a theory of meaning—and was replaced by a narrower debate between Davidson and Dummett about the form such a theory should take—a debate in which both sides at least agreed that the theory should be compositional. Their debate led to a renewed interest in ‘On sense and reference’, which is nowadays regarded as a landmark in the philosophy of language. The fact that the notion of sense is more properly a contribution to the philosophy of logic than of language is conveniently ignored. Frege’s focus on sense to the exclusion of tone had made this aspect of his work unappealing to philosophers of Austin’s stripe, who emphasized that the role of an utterance is not exhausted by the conditions for its truth or falsity. There are two recognizably distinct traditions here: Frege’s success in devising polyadic quantified logic was due partly to his single-minded exclusion of tone, and this gave him a mistrust of ordinary language in fundamental tension with the Austinian approach. Many modern theories of meaning attempt self-consciously to conform to the compositionality principle—sometimes called ‘Frege’s principle’ (Hintikka 1984, 31). The principle is more popular among philosophers of logic than of language, though, because the subtle context-sensitivity of ordinary language makes providing a compositional semantics for it a daunting challenge. What motivates Frege’s principle is the need for a theory of meaning to explain our ability to understand new sentences. So far, though, that is only to recognize meaning as one philosophical problem among many. Frege has also been credited, however, with attributing to the theory of meaning a quite distinctive foundational role. Dummett repeatedly urged (e.g. 1993a) that the context principle marked the decisive moment in the birth of analytic philosophy, because it launched a ‘linguistic turn’—the phrase is Bergmann’s (1953)—to rival in importance Kant’s Copernican turn a century earlier: the context principle converts a metaphysical question—what are numbers?—into a linguistic one—how do numerical terms contribute to the meanings of sentences in which they occur? On this view, Frege’s Grundlagen prefigures one of the dominant methods of 20th-century analytic philosophy, the substitution of the formal for the material mode: questions about what there is are converted into questions about which words make sense in which contexts, questions about thoughts into questions about the language in which these thoughts are expressed. The most striking application of this method was Dummett’s attempt (e.g. 1982) to characterize realism in terms not of the mind-independence of the entities but of the validity of bivalence for sentences referring to them. Dummett (1993a) went so far, indeed, as to claim that the linguistic turn was not merely an important strand in analytic philosophy but a defining one. The most immediate casualty of this characterization is Evans’s argument for a

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‘language of thought’, a putative way of understanding the structure of thoughts independent of the language we use to express them; but numerous other philosophers in the analytic tradition—Russell, Moore, Chisholm, Popper, Sellars, Dennett, Lewis—do not easily fit the characterization either. There is reason to suspect, therefore, that Dummett’s proposal might be just as historically misleading as the practice, which he rightly criticized, of referring to analytic philosophy as ‘Anglo-American’. The analytic method spread so widely in the 20th century that we should not be surprised if no single criterion defines it. Moreover, Dummett’s putative characterization risked making his own research programme, which sought to convert metaphysical problems systematically into problems in the theory of meaning, seem like the inevitable culmination of analytic philosophy rather than merely one instance of it—an instance whose popularity has hardly outlived his own attempts to prosecute it.

Mathematics Geach (acting, it seems, on Wittgenstein’s instructions) included Frege’s criticisms of formalism (under the title ‘Frege against the formalists’) in his selection (1960) of Frege’s writings. Perhaps because of this, both game and term formalism are so widely seen by modern philosophers of mathematics as hopeless. Mathematicians, though, have been more resistant, continuing to espouse these doctrines long after Frege refuted them. Frege’s most influential contribution to the philosophy of mathematics has been the conception he proposed in the Grundlagen of numbers as abstract objects, neither mental nor physical. The principal opposition to this view in the 20th century came from Brouwer, who held that numbers are the reasoner’s own construction and hence mental. He himself probably never read Frege, but in any case he would probably have been untroubled by the objection that conceiving of numbers as mental makes arithmetical knowledge solipsistic. His intuitionistic alternative to classical mathematics enjoyed brief popularity in the 1920s, during the so-called crisis of foundations that followed the discovery of the paradoxes, but this was largely due to the misconception that intuitionistic arithmetic is safer (i.e. less likely to be contradictory) than classical arithmetic—a misconception due principally to the failure to distinguish intuitionism from finitism. Once Gödel (1933) had exposed the misconception by means of his ‘negative translation’ of arithmetic, Brouwerian intuitionism largely faded from view (although this was probably due also to the perceived awkwardness and inelegance of intuitionistic analysis). More recently, it has been widely accepted that if there are numbers, they are abstract. Opponents of abstract objects have focused instead on trying to account for arithmetic without them, typically by holding that arithmetic is a story whose sentences do not have truth values. (A more radical variant holds that the sentences are actually false.) Fictionalists struggle, however, to explain why

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arithmetic is applicable to the real world in a way that other stories are not. Most non-fictionalists, on the other hand, accept that numbers are abstract objects, and contest only whether they are logical. The majority view has been that Russell’s paradox shows that they are not, and that we must look elsewhere than logic for an account of the grounds of mathematics. An exception has been Wright’s (1983) proposal that numbers are given to us via Hume’s Principle, in the way that Frege considered but rejected in Grundlagen. Much of the debate about Wright’s ‘neo-Fregean logicism’ has focused on the Julius Caesar problem, to which Wright’s solution is that objects given to us via an abstraction principle cannot be of an already familiar kind such as people (or a fortiori Roman emperors), because the criterion of identity the principle furnishes for them differs from the criterion for people. What guarantee is there, though, that the terms on the left hand side of Hume’s Principle refer? If we relied on our grasp of the natural numbers, we could of course answer this easily, but we would not then have fulfilled Frege’s aim of demonstrating that numbers are logical objects. So Wright proposes instead to rely on a context principle for reference (see Hale & Wright 2001). Now, though, the challenge is to show that Hume’s Principle is consistent, and Gödel’s second incompleteness theorem demonstrates that this cannot be done without appealing to metamathematical resources that are at least as much in need of justification. There is no dispute that Hume’s Principle provides an alternative to Peano’s axioms as a basis for the formal development of second-order arithmetic. What is questionable is whether the former has a logical status not shared by the latter that gives us a privileged route to a grasp of numbers as logical objects. The underlying concern is that Hume’s Principle—like the inconsistent Basic Law V, but unlike the direction principle—is impredicative: the objects putatively introduced by its left hand side already lie in the range of the quantifiers on the right. Frege’s ‘recarving’ metaphor correctly captures the intuitive appeal of abstraction principles as a route to concepts—such as that of the direction of a line—that amount to new ways of thinking about some notion—in this case, parallelism— that we already grasp. Yet it leaves wholly unaddressed the challenge presented by impredicativity, of explaining how we can grasp the domain of the quantifiers on the right and yet still claim that the objects introduced on the left are ‘new’ in any sense that could be supported by the metaphor of recarving. It would be a shame if Frege were to be remembered by philosophers of mathematics chiefly for Grundlagen §64, one of his least happy paragraphs.

Further reading On Peano’s role in spreading Frege’s account of polyadic logic, see Nidditch (1963). Hanna (2006) is a rare example of an analytic philosopher willing to challenge the modern consensus that logic is not psychological.

PART II

Russell

22 BIOGRAPHY

Bertrand Russell was born into the liberal wing of the English aristocracy in 1872. His mother and sister died when he was two, his father when he was three, and his grandfather (who had been Prime Minister in two Whig administrations) when he was six, with the result that he was raised by his paternal grandmother in an atmosphere permeated with intimations of mortality. Up to the age of sixteen, he was educated at home by tutors. His elder brother introduced him to Euclid’s Elements when he was eleven, an experience he found ‘as dazzling as first love’ (ABR, I, 36). Until he went to Cambridge he was, according to his brother, ‘an unendurable little prig’ (1923, 38). There he spent three years studying for the Mathematical Tripos before switching to philosophy for one year; he was awarded a starred first in the Moral Sciences Tripos in 1894, studying ethics, history of philosophy and metaphysics. That autumn, against his grandmother’s wishes, he married Alys Pearsall Smith, an American Quaker. The following year he wrote a dissertation on the foundations of geometry, on the basis of which he was elected to a six-year research fellowship at Trinity. Had he failed, he had resolved to give up academic work for politics, not because he needed the stipend—he inherited £20,000 from his father’s estate on coming of age—but because he was determined to pursue a career in which he would be judged by external criteria as successful. The conditions of Russell’s fellowship did not require him to reside in Cambridge, and he and Alys generally spent only the Lent (i.e. spring) Term there, living the rest of the year in their marital home—first a rented cottage near Fernhurst in Sussex and then from 1905 a newly built house in Bagley Wood near Oxford. In Lent Term 1899 Russell gave a lecture course at Cambridge on Leibniz. The resulting book is his only major scholarly work in the history of philosophy: the originality of his much later History of Western Philosophy (1945)

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lies principally in its inaccuracies. Most of his time, however, was spent working on the philosophy of mathematics, and it was The Principles of Mathematics, the book in which he advanced the logicist proposal that pure mathematics is part of logic, that made his name as a philosopher. In the spring of 1902, shortly before this book was completed, Russell realized that he no longer loved his wife. His account of this—‘I went out bicycling one afternoon. . . ’ (ABR, I, 147)—has become notorious. They did not separate for several years, but he insisted that they slept in different rooms. He soon embarked on a long series of involvements with other women, some of them married. The Principles finally appeared in May 1903, by which time Russell thought it ‘foolish’. Perhaps it was emotional fallout from his marital break-up—‘the darkest despair that I have ever known’ (ABR, I, 145)—that led him to publish the book in a patently unsatisfactory state. Thereafter he was intermittently involved in politics—he spent several months in 1903–4 defending free trade, and in 1907 he campaigned extensively alongside his wife for women’s suffrage, standing for Parliament on this cause in an unwinnable seat—but his philosophical work was dominated by attempts to resolve the logical contradictions he had discovered while writing the Principles. His most celebrated article, ‘On denoting’ (1905), was a by-product, and Principia Mathematica, the three-volume work which he wrote with A. N. Whitehead in attempted vindication of logicism, was the culmination, mostly delivered to the publisher in October 1909, although revisions and proofs continued until 1912. In October 1910 Russell returned to Cambridge as a lecturer in the philosophy of mathematics at Trinity. In March of that year he had fallen in love with Ottoline Morrell, a prominent socialite and wife of a Liberal MP, and they began an affair which lasted in one form or other for five years. The Bagley Wood house was sold the following year, and Russell no longer shared even a residence with Alys, who went to live in Chelsea with her brother. She remained devoted to Russell all her life nonetheless: her later assessment—‘I think it was a want of vitality on my part to allow one person to spoil my life, as he undoubtedly did’—is heartbreaking. In the summer of 1911 Russell wrote his ‘shilling shocker’, The Problems of Philosophy. It remains one of the best introductions to the subject for the general reader, as well as a succinct account of the sense-datum theory which, although now out of favour, was widely popular for a time. This was the basis of his ‘external world programme’, which sought to construct an account of worldly knowledge in terms of entities that are immediately presented in sensation. That summer he rented a flat in Bloomsbury from which to conduct his relationship with Ottoline. Wittgenstein studied with him in Cambridge from October 1911 until October 1913, and by the end of that time was not just a close friend but a major philosophical influence. In 1914 Russell visited Harvard to give the Lowell Lectures, later published as Our Knowledge of the External World, as well as a lecture course on logic devoted largely to expounding Wittgenstein’s views. T. S. Eliot,

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a graduate student in the philosophy department at the time, memorably fictionalized him as Mr Apollinax, ‘Priapus in the shrubbery / Gaping at the lady in the swing’. During the war Russell gave up philosophy almost entirely. This was partly because he was now devoting much of his time to pacifist activism—his removal from his lectureship at Trinity in 1916 was a side effect—but perhaps he had been losing interest in philosophy anyway. Late in the war he gave two courses of popular lectures, on the philosophy of mathematics and on logical atomism, both subsequently published and widely read, but of interest principally for the evidence they offer of the influence on him of the pre-war Wittgenstein. (When he gave the lectures, he had not yet seen the Tractatus.) Shortly afterwards he was sentenced to six months in prison (for libelling His Majesty’s allies), but granted privileges which enabled him to spend much of the sentence working. Russell divorced Alys hastily in 1921 in order to re-marry and thereby legitimate the son he had by then conceived with Dora Black. (Because his elder brother had no children, he was heir presumptive to his grandfather’s earldom, and he wanted his son to inherit after him the seat in the legislature to which this would entitle him.) The most notable of Russell’s post-war philosophical books, The Analysis of Mind (1921) and The Analysis of Matter (1927), attempt a completion of his external world programme, but from a somewhat different, ‘neutral monist’, starting point. Thereafter, he did not follow a conventional academic path, but pursued other careers alongside his academic writing, most notably journalism, politics and running a school. In the 1950s and 60s he campaigned for nuclear disarmament and became one of the first public intellectuals of the radio and television age. His life was not only long but full of incident (one plane crash, two jail terms, three sackings, four marriages, and numerous court cases). He was the only one of the four philosophers discussed in this book to receive major public honours in his lifetime (the Nobel Prize for Literature and the Order of Merit).

References Russell’s books are referred to by the following abbreviations: EFG Essay on the Foundations of Geometry (1897) L The Philosophy of Leibniz (1900) PoM The Principles of Mathematics (1903) PM (With A. N.Whitehead) Principia Mathematica (three vols, 1910–13) PP Problems of Philosophy (1912) OKEW Our Knowledge of the External World (1914) AMi The Analysis of Mind (1921) AMa The Analysis of Matter (1927) OP An Outline of Philosophy (1927) IMT An Inquiry into Meaning and Truth (1940)

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MPD My Philosophical Development (1959) ABR The Autobiography of Bertrand Russell (three vols, 1967–9) The standard edition of his shorter writings is: CPBR The Collected Papers of Bertrand Russell Many of his letters are contained in: SLBR Selected Letters of Bertrand Russell (two vols) Others are in the Russell Archive at McMaster University.

Further reading Volume 1 of Monk’s biography (1996) can be recommended as an account of Russell’s life up to 1921; volume 2 (2000) is unfortunately too unsympathetic to its subject to be wholly successful. Russell’s own account, My Philosophical Development, is readable and engaging. Pears (1967) and Ayer (1971) are good single-volume introductions to his work.

23 BRADLEY

Russell’s first major influence was the British absolute idealists. Later, once he had rejected their doctrines, he often referred to them as Hegelians, but British interest in Hegel was due as much to the popularity of The Secret of Hegel (1865), a rather unreliable book by a Scottish doctor called Stirling, as to any widespread engagement with Hegel’s own writings. Russell himself did not read Hegel until a couple of years after graduating, and the absolute idealism he imbibed as an undergraduate stemmed largely from Bradley. Now that his influence has declined, it is perhaps difficult to appreciate the central place Bradley held in British philosophy at the end of the 19th century, when his work was widely read. Much of his philosophy of logic was negative—it is easier to say what he was against than what he was for—while his positive theories were highly complex, at least apparently contradictory, and often expressed in an extremely prolix (but at the time much admired) style that defies intelligible summary. Here, though, we shall focus on his influence on Russell—an influence which, it is worth saying, was not mediated by personal acquaintance. When they eventually met in 1902, Russell ‘loved the man warmly’ (CPBR, XII, 13), but by then his enthusiasm for Bradley’s philosophical views had waned.

Absolute idealism Subjective idealists such as Berkeley denied the existence of a mind-independent reality, whereas objective idealists such as Bradley accepted the substantial existence of reality (which they called the ‘Absolute’) but held that thought cannot fully describe it. Hegel said in the Lesser Logic that ‘everything that exists stands in correlation, and this correlation is the veritable nature of every existence’

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(1892, 235). This conception of the Absolute as an infinite, unconditioned whole about which discursive knowledge is impossible but for which our aesthetic sense nonetheless gives us an inexpressible yearning can be traced not just to Hegel but to earlier German Romantics such as Schlegel. It also influenced Beethoven, for instance, as well as 19th-century English Romantics such as Coleridge and Wordsworth. The British idealists inherited from this tradition its monistic conception of reality as an inter-related whole a description of any part of which must involve every other; but they also inherited the mystical associations which the Romantics had attributed to reality so conceived. The British idealists therefore opposed the empiricism of Locke and Hume; but they also followed Lotze in rejecting its psychologistic conception of logic and argued that judging involves more than a mere association of ideas. Bradley, in particular, distinguished between an idea and its ‘universal meaning’ (1883, bk 1, ch. 1, §9). The latter played a similar role in his logic to a Fregean sense, except that he did not insist, as Frege did, that it is independent of the idea. It consists, he said ‘of a part of the content . . . cut off, fixed by the mind, and considered apart from the existence of the sign’ (1883, bk 1, ch. 1, §4). He also agreed with Frege in holding an ‘identity theory’ of truth. A proposition is true, he maintained, just in case it coincides with what makes it true. In combination with the monistic conception of reality, however, this entailed that no proposition could describe reality accurately: for a thought to be absolutely true, it would have to coincide with reality, but since reality is not propositional in structure, this would require the thought’s ‘happy suicide’ (1893, 173) and our consequent absorption into the absolute (an absorption whose desirability was a common trope of 19th-century mysticism). ‘The content of the subject strives, we may say, unsuccessfully towards an all-inclusive whole.’ (Ibid., 177) According to Bradley, then, partial truth is not really truth at all. ‘A fraction of the truth . . . becomes entire falsehood, because it is used to qualify the whole.’ (1883, bk 1, ch. 2, §67) Nonetheless, because absolute truth is unattainable, what is relevant to logic is not the identity theory but a weaker correspondence theory (dubbed the ‘existential theory of judgment’ by Russell) which Bradley advanced in the Principles of Logic as applicable to the partial truth of the everyday. (He also said, rather confusingly, that ‘there are no degrees of truth and falsity’ (1883, bk 1, ch. 7, §2), but seems to have meant by this merely that there are no degrees of assertion.) A judgment is true, according to the existential theory, only to the extent that it corresponds to what ‘exists’—by which, it should be stressed, Bradley (1893, 317) meant what occurs in time (so that tables and chairs exist, but numbers, for instance, do not). ‘Wherever we predicate,’ he wrote, we predicate about something which exists beyond the judgment, and which (of whatever kind it may be) is real, either inside our heads or outside them. And in this way we must say that ‘is’ never can stand for anything but ‘exists’. (1883, bk 1, ch. 2, §2)

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Because of the inter-connectedness of reality mentioned earlier, it then follows that every judgment ends up being about the whole of it: whatever its grammatical subject, its logical subject is always just reality. The existential theory was Bradley’s account of what it would be for a categorical judgment to be (partially) true, but he also argued that whenever we try to make such a judgment, what we achieve is only hypothetical, because all our devices for referring to particulars—descriptions, demonstratives, and proper names—are infected by conditionality: reference by a description is conditional on there being a unique referent; reference by a demonstrative is conditional on the context of utterance; and reference by a proper name is conditional because no name can be (in Mill’s terminology) purely denotative (although it should be said that Bradley’s reasons for this last claim are obscure). Bradley took this conditionality to be a further obstacle in the way of holding our judgments about the world to be straightforwardly true or false.

Russell’s conversion Although absolute idealism was popular at Cambridge, as elsewhere in Britain, it was by no means inevitable that Russell himself would succumb to it: among his teachers, Stout may have been sympathetic, but Sidgwick was not, and Ward, his other undergraduate teacher, was a Kantian who wrote a fairly hostile review (1894) of Bradley’s Appearance and Reality. Russell later claimed that he had resisted the views of the Hegelians for much of his undergraduate year studying philosophy, but was converted just before his Tripos examinations in May 1894. Stout had persuaded him that ‘it all turned on the ontological argument’ (letter to Morrell, 28 Sep. 1911), by which he meant not the conventional ontological argument concerning God, but a Bradleian variant purporting to prove the existence of the Absolute. The argument is as follows. The existential theory entails that for anything to be true, something must exist. This does not yet demonstrate that any one thing exists, of course, since different judgments might have different truth-makers, but if we combine it with Bradley’s monistic conception of reality, according to which the truth-makers for different judgments are merely aspects of the Absolute, we reach the conclusion that for any judgment to be true, the Absolute must exist. Russell later claimed (ibid.) that his moment of revelation occurred as he walked down Trinity Lane. ‘Great God in Boots!’ he supposedly exclaimed. ‘The ontological argument is sound.’ In an essay he wrote for Stout, he explained his reason. Whatever we think we cannot get away from reality; if we judge at all, we must affirm some predicate of reality; even negative judgment is only possible owing to some positive incompatible ground, i.e. must be based on an affirmation; but if we try to deny reality as a whole, there is no positive ground left as basis of our denial. We must think the Absolute, and its

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essence involves existence; hence the ontological argument can, it would seem, only be met by complete scepticism, by abstaining from judgment altogether; which is a negligible alternative. (CPBR, I, 179) Whatever this argument’s merits, though, it plainly falls some way short of vindicating absolute idealism, since it at most shows that the existential theory of judgment and the monistic conception of reality jointly entail the existence of a reality for our judgments to be true about. It does nothing to validate either antecedent of the entailment. What Russell’s story omits, that is to say, is an explanation of how Stout convinced him in the first place that the correctness or otherwise of absolute idealism ‘all turned on the ontological argument’. It does, however, have the virtue of making clear the central role the existential theory of judgment played in Russell’s adoption of the view.

Further reading Wollheim (1959) is a good introduction to Bradley’s philosophy. The views of the British Hegelians more generally are described by Robbins (1982), and their logical views by Ferreira (2014), while Stern (2009) discusses the differences between their metaphysics and Hegel’s own. On the role played by the Absolute in German Romanticism, see Stone (2011). For more on Bradley’s influence on Russell, see Keen (1971) and Candlish (2007). Griffin (1991) and Hylton (1990b) discuss Russell’s idealist period in some detail. Spadoni (1976) describes his conversion to idealism via Bradley’s ontological argument. Schaffer (2010) is a rare modern defence of a monistic reality.

24 GEOMETRY

In Russell’s day Cambridge did not have the modern apparatus of research degrees. The Litt.D. and Sc.D. degrees had been created in 1882 to give recognition to more senior academics, but the Ph.D. was not created until 1919. Until then, the usual route into an academic career was via a college research fellowship, awarded on a combination of dissertation and written examination at some point in the three years following graduation. When Russell graduated in 1894, he resolved in consultation with Ward, the philosophy don at Trinity, to write his dissertation on the philosophy of geometry. This was an astute choice, because it was a direct extension of work he had done on the philosophy of geometry for Ward’s metaphysics course earlier in the year, and it enabled him to bring his mathematical training to bear on a philosophical problem. He was given a reading list on the mathematical aspects of the subject by Whitehead, one of his undergraduate mathematics teachers, and set to work. Although there were distractions along the way—a period in Paris as honorary attaché to the British ambassador, then his marriage to Alys in the autumn—he worked on geometry in the spring (while in Berlin attending lectures on economics), finished the dissertation back in England in August 1895, and was duly elected to the Fellowship in October. ‘The logic of geometry’, a chapter of the dissertation, appeared in Mind the following January.

Metric geometry In the article, Russell took as his starting point the account of geometry in the Critique. Taking the possibility of experience as given, Kant had attempted, in parallel with the previously mentioned transcendental deduction of the pure concepts of the understanding (categories), a further deduction of the pure

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intuitions of the sensibility (space and time). He then inferred that the axioms of geometry are knowable a priori. In the course of this deduction, however, he gave even less consideration to the individual axioms of geometry than he had to the individual categories, with the result that he fell well short of establishing that it is Euclidean geometry that is a priori. During the 19th century the development of non-Euclidean geometry made this lacuna in his argument increasingly egregious. Russell’s purpose, then, was to show that Kant had been partially right: some of the axioms of geometry are a priori, he argued, others not. Russell based his argument on ‘neutral’ geometry, which studies Euclidean and non-Euclidean geometry simultaneously by presenting them as particular cases of a more general kind of structure. There is more than one way of doing this, depending on which structural properties one generalizes. Russell focused on metric geometry (nowadays known as differential geometry), the study of which had been initiated by Riemann’s memoir of 1854 (published posthumously in 1867). Riemann defined what is now called a ‘differential manifold’, i.e. a structure M which in the neighbourhood of each point resembles n-dimensional Euclidean space, in the sense that it is the union of a patchwork of sets each of which has the structure of a subset of Rn . Formally, it is presented as the union of a family (Mα )α∈A of sets, together with a family of one-to-one functions iα : Mα → Rn such that each composite function iβ ◦ iα−1 : Rn → Rn is smooth on Mα ∩ Mβ . These functions transport the differential structure from Rn to M. In particular, they enable us to define the notion of a geodesic and hence to define the distance between two points in M as the shortest length of a geodesic between them. The easiest examples of differential manifolds to visualize are surfaces in Euclidean space. For these it is straightforward to define the notion of curvature: the radius of curvature at a point is the radius of the sphere through the point that best matches the nearby contours of the manifold. In practice, though, mathematicians generally use instead the inverse 1/r of the radius of curvature, which is called simply the curvature of the manifold at the point. Thus a surface is flat just in case its curvature is everywhere zero. The key step came when Riemann, generalizing work of Gauss, showed how to define curvature by appealing only to internal properties of the manifold, so that the definition now made sense even for a manifold that is not a surface. Thus was inaugurated ‘neutral metric geometry’, the study of finite-dimensional differential manifolds of constant curvature, from which various geometries may be retrieved as special cases: elliptic geometry corresponds to the case of positive curvature; hyperbolic geometry to negative curvature; and Euclidean geometry to zero curvature. Riemann’s treatment of neutral geometry was well known by the time Russell wrote his dissertation, but its philosophical interpretation remained controversial. Helmholtz (1876) had claimed that the notion of the congruence of figures in space rests on the ‘axiom of free mobility’, which asserts that rigid bodies can move about freely: this principle is empirical, he argued, because it is about the

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properties of physical objects. Since neutral metric geometry depends on the possibility of comparing figures in space, it is consequently empirical. Russell had already written for Ward an undergraduate essay on Helmholtz in which he defended the Kantian view that space is a priori, but offered no persuasive critique of Helmholtz’s empiricist alternative (CPBR, I, 126–7). Shortly after he began work on his dissertation, however, another of his undergraduate supervisors, Stout, serendipitously asked him to review for Mind a book in which a Dutch philosopher and psychologist called Heymans argued (1890-94, I, 211), contrary to Helmholtz, that if a body is unable to move without distortion, we are not compelled to ascribe this to a variation in the curvature of space: we could equally suppose that curvature is constant and the body’s inability to move has some other source such as the resistance of the ether. In his review, Russell noted Heymans’ point with approval. ‘Any apparent exception to [the axiom of free mobility] on the part of actual bodies is always to be regarded as due to physical rather than geometrical cause’. (CPBR, I, 253) In ‘The logic of geometry’, he repeated the point in almost the same words, but this time without crediting Heymans. What the axiom of free mobility asserts about real bodies is not that their shapes do not change, but that such changes of shape as they do undergo are due to physical, not to geometrical, causes. This makes the investigation of these physical causes possible, by the ordinary inductive methods. (CPBR, I, 271) The axiom of free mobility, which Helmholtz had taken to be empirical, is a claim about how matter behaves in space, namely that rigid bodies (i.e. assemblages of matter) can move without distortion, and therefore belongs to physics. It should be distinguished from the geometrical principle that congruent figures may occur in different parts of space. I shall call the latter the ‘axiom of congruence’. (Russell, confusingly, sometimes called it the axiom of free mobility.) We shall return later to the distinction between geometry and physics, but let us meanwhile grant it for the sake of argument. Russell’s Mind article attempted on that basis two distinct arguments for the a priority of the axiom of congruence, corresponding to two methods he found in Kant for discovering the a priori basis of a science, one taking as its premiss the science itself, the other its subject matter.

First deduction of metric geometry Russell’s first argument for the axiom of congruence had two parts: he argued first that measurement is essential to geometry; then that in order for measurement to be possible, figures in different parts of space must be comparable, i.e. the axiom of congruence must be true. Both steps of Russell’s inference may be questioned: the first because it is not clear that all of geometry does depend on measurement; the second because

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even if our method of measurement presupposes rigid motion (e.g. by using a rigid ruler), it does not follow that without this method measurement would be impossible. (If we always measured temperature with mercury thermometers, it would not follow that in a world without mercury there would be no such thing as temperature.) Moreover, even if Russell had correctly identified a necessary condition for the possibility of measurement, it would be a further question whether this was a condition for the possibility of experience. He attempted to make this link by claiming that if reasoning in a science such as geometry ‘is impossible without some postulate, this postulate must be essential to experience of the subject matter of the science’ (CPBR, I, 292). But this is plainly fallacious: if geometry as it is in fact practised depends on some postulate, we need a further argument to show that spatial experience would not be possible at all if the postulate were false. Even if Russell had correctly identified the presuppositions of the science of geometry, that would tell us no more on its own about their a priori character than the fact that I cannot make an omelette without breaking eggs tells me about whether I can break eggs. Even if he had succeeded in showing that the axiom of congruence is a condition of the possibility of measurement, this would not show that the axiom is a priori unless measurement were in turn a condition of the possibility of any experience whatever. The ‘regressive method’ for axiomatizing a science is best kept separate from the question whether the science may be transcendentally deduced as a presupposition of the possibility of experience. Perhaps the reason for Russell’s carelessness about this issue was that he agreed with Bradley that no truth is absolutely necessary, and therefore supposed that the most he could ever be expected to do was to show that one piece of knowledge depends a priori on another.

Second deduction of metric geometry In his correspondence with Leibniz (Alexander 1958), Clarke famously argued (on Newton’s behalf) that space is a substance independent of the matter that is located in it, and that spatial points are the relata of spatial relations; Leibniz, by contrast, took the relational view that the pieces of matter are themselves the relata of spatial relations, so that space itself drops out as an unnecessary posit. Heymans’ point depended on admitting a distinction between positions in space and the matter that may occupy them, and hence on disagreeing with Leibniz’s relational view, but it did not thereby commit him to the substantial view, since it is a further question whether positions are independent substances or are constituted merely by the relations they bear to other positions. Russell’s second argument for neutral metric geometry attempted to show that the possibility of experience requires geometrical positions, although distinguishable from the material points that occupy them, to be wholly constituted by their spatial relations and hence non-substantial. He then strengthened

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this non-substantial conception by arguing that the possibility of experience further requires any point in space to be indistinguishable from any other. He expressed this by saying that space is ‘relative’, by which he meant nonsubstantial and homogeneous. He then concluded that space is of constant curvature (since otherwise positions would be distinguishable, contradicting homogeneity). The key step here is the transcendental argument from the possibility of experience to space’s insubstantiality and homogeneity. In the article derived from his dissertation, Russell attempted to argue for both of these simultaneously. If space were not relative, he said, it would no longer be passive, but would exercise a definite effect upon things, and we should have to accommodate ourselves to the notion of marked points in empty space; these points being marked, not by the bodies which occupied them, but by their effects on any bodies which might from time to time occupy them. This want of homogeneity and passivity is, however, absurd; no philosopher has ever thrown doubt, so far as I know, on these two properties of empty space; indeed they seem to flow from the maxim that nothing can act on nothing, for empty space is rather a possibility of being filled than a real thing given in experience. We must, then, on purely philosophical grounds, admit that a geometrical figure which is possible anywhere is possible everywhere, which is the axiom of Congruence. (CPBR, I, 269) The obvious difficulty with this argument, though, is that it is so nearly circular: the substantialist will deny that empty space is merely ‘a possibility of being filled’. When Russell discussed the issue again at the Aristotelian Society in February 1896, he offered a different argument, which, he said, ‘seems the more convincing for exposition’ (CPBR, I, 292). Space must be relative if experience is to be possible at all, he now argued, because externality is an essentially relative conception—nothing can be external to itself. To be external to something, is to be an other with some relation to that thing. Hence, when we abstract a form of externality from all material content, and study it in isolation, position will appear, of necessity, as purely relative—it can have no intrinsic quality, for our form consists of pure externality, and externality contains no shadow or trace of an intrinsic quality. (CPBR, I, 302–3) This argument is scarcely better than the previous one, however. If space is independent of the observer, it is not external to itself, but Russell did not explain why it should follow that positions in space can have no intrinsic qualities.

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Dimension Having shown—to his own satisfaction, at least—that space is relative and homogeneous, Russell then attempted to deduce that it has a fixed finite number of dimensions. His argument was that since position is relative, it is determined by relations to other points, and hence, since space is homogeneous, the number of relations required to determine a point must be constant throughout: this constant number is the dimension. Where he struggled, though, was in attempting to argue that the dimension is finite. In the earlier article he had dismissed the notion of an infinite-dimensional homogeneous space as simply incomprehensible (CPBR, I, 277). Now he at least recognized that he needed to say more, but all he could think of was that in such a space we should have to suppose a system of relations perpetually reaching out into new worlds, and this, while it would throw doubt on the homogeneity of our form, would leave a scrappy world without systematic unity. (CPBR, I, 303) This is weak: an infinite-dimensional space would of course be very different from the three-dimensional space of our perceptual experience, but this does not show that it would therefore be ‘scrappy’ or lack ‘systematic unity’. The infinite divisibility of space is also crucial to geometry, but curiously Russell felt no need to argue for it and was content to observe in a footnote that it ‘has sometimes been supposed to involve difficulties, though I have never been able to feel their force’ (CPBR, I, 285). The explanation for his insouciance was that he thought relativity contradicts finiteness. ‘The very essence of space, as conceived by Geometry, is relativity and externality of parts, which makes the notion of an atomic unit of finite extension particularly preposterous.’ (Ibid.) He thought, that is to say, that if space were conceived of as made up of distinct points, each of these points would have its own distinct identity, contradicting relativity. It is unclear, though, why this should be so. It is easy to construct examples of finite homogeneous spaces. Russell’s failure to focus on this issue is especially disappointing, because anyone sympathetic with his project might well regard continuity as the most likely feature of space to be transcendentally deducible.

Geometry and physics According to Russell, then, it is a priori that space is a manifold of constant curvature. What the curvature is, on the other hand, he took to be an empirical matter that cannot be known with certainty: measurement shows that it is close to zero, but since (unlike dimension) it is a real number rather than an integer, no measurement can show that it is exactly zero. On this point, however, there is an asymmetry between the Euclidean and non-Euclidean cases: if space is nonEuclidean, and hence has non-zero curvature, a sufficiently precise measurement

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might reveal it to be so. The experimental issue is complicated, however, by the fact that a slight deviation of curvature would be apparent only at a large scale (much as the curvature of the earth is not obvious at ground level). If we wanted to test whether space is Euclidean by measuring the sum of the angles of a triangle—space is hyperbolic, Euclidean or elliptic according as their sum is less than, equal to, or greater than 180◦ —the triangle we used would have to be very large, since even in non-Euclidean space the sum tends to 180◦ as the size of the triangle becomes small. The chief weakness of Russell’s discussion, though, was his failure to note that the implied distinction between the inductive methods of physics and the allegedly a priori methods of geometry was really the point at issue, and hence that it is legitimate to wonder what principled reason there is for making it. Russell argued (EFG, 78–9) that geometry is prior to physics, because the laws of physics make use of geometrical concepts, but not conversely. This is plainly insufficient to establish the claim, however. Indeed, relativity theorists would soon deny precisely this priority, holding instead that the curvature of space varies according to the matter that occupies it. In 1889, several years before Russell’s dissertation, the Irish physicist FitzGerald had already posited what is now known as the Lorentz-FitzGerald contraction, a distortion which any body appears to undergo from the perspective of an observer relative to whom it is accelerating. (From the body’s own perspective there is no distortion.) The prediction Russell took from Heymans, that we would always choose to attribute such distortions to physical rather than to geometrical causes, has not proved correct: the orthodoxy since the 1920s has been that the apportionment of the principles between geometry and physics is a matter of convention and that it is arbitrary to single out some rather than others as preconditions of the possibility of experience. In later life, Russell accepted this orthodoxy and consequently regarded his argument for the a priori character of the geometry of physical space as a failure (MPD, 39–40).

Deduction of descriptive geometry Riemann’s was not the only proposal for unifying Euclidean and non-Euclidean geometry: whereas he took distance as the primary notion and only later retrieved the notion of a straight line as a geodesic arc, an alternative, ‘descriptive’ approach took the notion of a straight line as primary and distance as secondary. The central notion of the descriptive approach to geometry is that of projective space. In the Euclidean plane, lines may intersect or not: those that do not intersect are called ‘parallel’. The projective plane is obtained by adding a ‘line at infinity’, so that lines which in the Euclidean plane are parallel intersect in the projective plane at a point on this extra line. In the projective plane, therefore, any two lines without exception have a unique point of intersection. Three-dimensional projective space is obtained in an analogous manner by adding a plane at infinity to Euclidean space.

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One of the most important devices in projective geometry is the cross-ratio. For any four collinear points a, b, c and d, the cross-ratio (a, b; c, d) is a real number, so called because if the points are parametrized by real numbers, it may be defined as (a, b; c, d) =

(a − c)(a − d) . (b − c)(b − d)

Desargues proved in the 17th century that the cross-ratio is a projective invariant (i.e. independent of the choice of parametrization), and von Staudt in 1847 showed how to define it directly in projective terms without going via a parametrization at all. What enables projective space to function as a neutral geometry is the definition of distance. This depends on the choice of what is known as the ‘absolute conic’. Once this conic is chosen, we define the distance between any two points a and b as d(a, b) = log(a, b; c, d), where c and d are the points where the line through a and b intersects the conic. Euclidean and non-Euclidean geometries then result as specializations of projective geometry according to the type of conic chosen: if the conic is an ellipse, we obtain elliptic geometry; if a hyperbola, hyperbolic geometry; if a circle, Euclidean geometry. Descriptive geometry is therefore a neutral geometry, but generalizes different structural features of space from those treated in metric geometry. It has a much more algebraic and classical flavour than Riemann’s, and was consequently favoured in the late 19th century by English mathematicians such as Whitehead, who strongly preferred such methods. The dissertation Russell submitted in October 1895 has not survived, and by the time it was accepted for publication in September 1896, he had spent some time revising it. He delivered lectures based on the book in the U.S. that autumn; and the final version did not appear in print until 1897. In his earliest publications he had concentrated on metric geometry and barely mentioned the descriptive alternative at all, but by now Whitehead had persuaded him of ‘the philosophical importance of projective geometry’ (EFG, preface), and in its published form the Essay attempted a deduction of descriptive geometry from the form of externality parallel to the deduction previously offered for neutral metric geometry. Quite apart from Whitehead’s urgings, of course, Russell might have been encouraged to attempt a fresh deduction of a priori principles underlying geometry by the failure of his earlier deduction to disentangle neutral geometry from the possibility of physical motion. He thought that the basic notions of projective geometry, such as that of a straight line, are more straightforwardly separable from physics. Nonetheless, the new deduction still fell well short of explaining clearly what he understood by the ‘form of externality’ on which the possibility of experience was supposed to depend. Moreover, as Poincaré (1899) soon pointed out, he offered no argument at all for the key distinguishing axiom of projective geometry, that any two coplanar lines have a common point. Not

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only is this an egregious gap, but it is hard to see how it could be filled. Why might the non-existence of parallel lines be a condition imposed on space by the mere possibility of experience?

Public and private space Russell’s attempted transcendental deductions, whether of metric or of descriptive geometry, are no doubt flawed in detail, but it is a separate question whether a transcendental derivation of features of the structure of experience from its possibility is so much as possible. Poincaré, Russell’s most immediate critic, proposed instead a weakly Kantian position according to which the principles of geometry are conventions that are not wholly arbitrary but may be tested by their utility in explaining experience. Even this view presupposes, though, that it is possible to draw a principled distinction between the form of experience (the conventional part) and its content. We noted earlier that in practice relativity theory has not drawn such a distinction, but in fact the point is independent of this: if we measured a large triangle and found the sum of its angles to be different from 180 degrees, we would not be compelled to conclude that space is non-Euclidean, but could alternatively revise the laws of optics in such a way as to explain our measurements in a Euclidean setting. An alternative, proposed by Ewing (1938, 41f) and Strawson (1966, 281f), would be to fill out the distinction, sketched by Kant in his correspondence with Eberhard, between public (physical) and private (perceptual) space, and then to concede that the former is as relativity theory says while defending a loosely Kantian account of the latter. As we shall see in Chapter 39, Russell himself later attempted at least the first part of this proposal, but never really explained how to ground a theory, Kantian or otherwise, of private space.

Further reading For more detailed discussion of Russell’s transcendental arguments, see Griffin (1991, ch. 4). The neo-Kantian claim that spatial experience has a synthetic a priori component was also defended by Bauch and by his supervisee, Carnap, in his doctoral thesis Die Raum (1922). Torretti (1978) places Russell in the context of the philosophy of geometry in the late 19th century. On the division between geometry and physics, see Putnam (1975). In the modern literature what I have here called ‘relationalism’, i.e. the conception of mathematical structure as consisting in relations without relata, sometimes goes under the name ‘radical ontic structuralism’. On the geometry of perceptual space, see Craig (1969), O’Neill (1976) and French (1987).

25 McTAGGART

There were two streams in Russell’s early philosophical views, one flowing from the Bradleian absolute idealism popular in late 19th-century Britain, the other more directly from Hegel’s notion of dialectic. It is worth keeping these streams distinct, because although the absolute idealists were fond of Hegelian contradictions, few indulged in Hegelian dialectic explicitly. The former stream of ideas, we have seen, Russell adopted as an undergraduate; the latter he acquired principally from McTaggart, a philosopher at Trinity four years his senior (CPBR, X, 23) whose Fellowship dissertation (published as Studies in the Hegelian Dialectic in 1896) attempted to articulate Hegel’s view that our theories about the world are inevitably contradictory and must be revised in a continually evolving dialectical process. Under McTaggart’s influence, Russell read Hegel’s Logics (the lesser in March 1896, the greater a year later) ‘very carefully’, but he was ‘woefully disappointed’ by what he found there. It seemed to me that the dialectic process was at best the psychology of discovery, at worst a string of puns; and that Hegel frequently talked in a very superficial way about mathematical questions which he didn’t understand. This was the greatest disappointment of my philosophical life; for until then I thought philosophy would give me a religion. (Letter to Joachim, 22 May 1906, in CPBR, V, 434) Later he even called Hegel’s remarks on mathematics ‘many pages of sheer gibberish’ (CPBR, XI, 149). Hegel’s effect on him was therefore only to make him bitter; McTaggart’s, though, was longer lasting.

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The Tiergarten programme Although the inevitability of contradictions does not belong to absolute idealism’s core, Hegelians such as McTaggart attempted to deduce the former from the latter. Reality, they believed, is interconnected in such a way that a partial truth is inevitably a non-truth. ‘A thing which in itself is imperfect and irrational may be a part of a perfect and rational universe.’ (1896, §7) In making this Hegelian point, McTaggart alluded to Tennyson’s poem ‘Flower in the crannied wall’, according to which it is impossible to understand something as small as a single flower completely, ‘root and all’, without knowing ‘what God and man is’. Yet even if we accept that a partial truth is not the whole truth, we are still some way from the inevitability of contradiction. Part of the problem here is that some Hegelians did not distinguish clearly (or at all) between incompleteness and inconsistency, but supposed that an incomplete description of the world is ipso facto inconsistent. Hegel declared that ‘all things are in themselves contradictory’ (1812–16, II, 77) and, like many of the British absolute idealists, routinely asserted explicit contradictions. Bradley, for instance, claimed variously that every proposition is existential and non-existential, hypothetical and categorical. McTaggart, by contrast, strove to show that contradictions are genuine signs of error, demonstrating that there is something faulty in the concepts we are applying. Even so, the details of the process are murky. Those used to monotonic logic struggle to make sense of the process, because they hold that if a logical system contains a contradiction, the right strategy for resolving it must be to drop one of its putative axioms, not to add some more. In order to rescue the project from incoherence, therefore, we might contrast internal and external perspectives, and hold that a theory which is internally consistent may yet be externally inconsistent. Our aim would then be to embed our system in a larger one in such a way that the inconsistency was shown to have resulted from failing to take account of some aspect of the objects involved. Although McTaggart’s famous argument concerning the A- and B-series descriptions of time dates from rather later (1908), it is an instance of his version of Hegelian dialectic. He described it as an argument that time is unreal, but it would be more accurate to call it an argument that tense is unreal. It does not seek to derive a formal contradiction within our tensed discourse, but only to show that this discourse is contradictory when viewed externally. The point to stress, though, is that on McTaggart’s neo-Hegelian view it is not reality itself that is inconsistent but only the theories we abstract from it. The more concrete we make our theory, the fewer inconsistencies he thought it would contain. In the spring of 1895, while Russell was studying in Berlin, there took place an event of great significance for his philosophical career. During a walk in the Tiergarten (a municipal park), he formulated an ambitious programme, to be executed over the coming years, for a philosophical treatment of the sciences exemplifying McTaggart’s conception of the Hegelian dialectic.

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Every science may be regarded as an attempt to construct a universe out of none but its own ideas. What we have to do, therefore, in a logic of the sciences, is to construct, with the appropriate set of ideas, a world containing no contradictions but those which unavoidably result from the incompleteness of these ideas. . . . We have . . . first to arrange the postulates of the science so as to leave the minimum of contradictions; then to supply, to these postulates or ideas, such supplement as will abolish the special contradictions of the science in question, and thus pass outside to a new science, which may then be similarly treated. Thus e.g. Number, the fundamental notion of Arithmetic, involves something numerable. Hence Geometry, since space is the only directly measurable element in sensation. Geometry, again, involves something which can be located, and something which can move—for a position, by definition, cannot move. Hence matter and Physics. (CPBR, II, 5) Each science would turn out to be contradictory, Russell believed, to the extent that it abstracted away from reality. The contradiction, once identified, could be resolved only by advancing to the more inclusive perspective of a less abstract science. The sciences would thus be arranged in a hierarchy, starting with the most abstract (arithmetic), and advancing towards the ultimate (non-contradictory, but alas inexpressible) science of the Absolute.

The paradoxes of relativity By the time he completed his Fellowship dissertation on geometry in August 1895, Russell already conceived of it as merely one stage in his Tiergarten programme. He already expected that geometry, being an abstraction from reality, would contain contradictions which could only be resolved by proceeding to the next, more concrete science in the hierarchy. He was therefore not at all surprised to find a contradiction in his conception of it. ‘It is interesting to observe’, he said, that the relativity of space involves the most glaring contradictions. For . . . it is only possible to define points by their relations, i.e. by lines, while lines can only be defined by the points they relate. This involves either an infinite regress or a vicious circle, the penalty of our attempt to give thinghood to a mere complex of relations. (CPBR, I, 304) The homogeneity of space, which makes it possible for congruent figures to occur in different places, dictates that points in different parts of space should have exactly the same relational properties; yet the relative conception of space holds that a point is to be conceived of wholly in terms of these relational properties, with the consequence, Russell claimed, that no sense can be made of two points differing from one another. He diagnosed this ‘contradiction’ as the result of conceiving of space independently of the matter that occupies it, and supposed

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that it would be resolved once we proceed from pure geometry to kinematics, because different points in space would then be distinguishable by virtue of the different pieces of matter they contain. This relief would only be temporary, however: kinematics would in turn be susceptible to a similar ‘contradiction’, which could only be resolved by a transition to mechanics. Russell thus believed himself to have uncovered a hierarchy of contradictions of just the sort that his reading of McTaggart led him to expect. It is unclear, though, quite how Russell conceived of this hierarchy. He did at least avoid the earlier confusion between incompleteness, which merely invites us to extend our theory, and inconsistency, which requires us to resolve the contradiction by means of the extension (CPBR, II, 5), but he was still somewhat vague about what this ‘resolution’ would involve. If the rule of ex falso quodlibet is valid, then from an inconsistent theory every proposition is provable, making the theory useless. One way to retrieve the situation would be to restrict the logic so that this rule is not valid (see Priest 1995, ch. 7). A second, less radical approach would draw a distinction between the theory itself and its philosophical interpretation: if the theory is formally consistent, no restriction of classical logic is required in order that it should be usable. To make this point with any precision, though, would require a distinction between object language and metalanguage that still lay 25 years in the future. Russell’s remarks suggest, nonetheless, that he was closer to this second conception, and that what he was calling ‘contradictions’ were really no more than circularities of explanation. An alternative strategy, one might think, would be to re-examine the conception of space which gave rise to these circularities in the first place: if it could be repaired, the motivation for the dialectic would be removed before it even began. Russell did not attempt this, however, because he firmly believed that no wholesale elimination of contradictions from our conception of the world was possible. He even went so far as to suggest that elliptic space should be rejected precisely because it has a finite volume and hence illegitimately fails to engage with the paradoxes of the infinite. It accepts one side of the antinomy, and excludes the other by a device at once too economical and facile. If we can show, a priori, that space has to be substantialized, and that the consequences of this substantialization have to be contradictory, then any theory which renders these consequences non-contradictory is condemned. (CPBR, II, 330) The fault of elliptic geometry, that is to say, was that its very consistency would cut short the dialectic.

Further reading Griffin (1991) discusses the Tiergarten programme in some detail.

26 GERMAN MATHEMATICS

Once his book on geometry was ready for the press, the next stage of Russell’s projected encyclopaedia of the sciences was to have been a book on matter, an outline of which he sketched in 1897 (CPBR, II, 84). It remained unwritten, however, because, having spent much of that year struggling to make the possibility of motion consistent with the relativity of space, he no longer thought that physics was the right place to look for a resolution. Instead, he turned his attention to the part of the Tiergarten programme prior to geometry, namely the theories of natural and real numbers. On Quantity and Allied Conceptions, which he began to write early in 1898 was the first of a bewildering series of book drafts that culminated five years later in the publication of The Principles of Mathematics. During this period, however, his views underwent radical changes. Some were internal to his project—the result of his own attempts to formulate his views and work out their consequences. Any author learns the importance of judging when is the right time to stop thinking and start writing. Part of the reason for the plethora of drafts was that he had started too soon, before he understood what he was talking about. It is remarkable, though, how much his views were shaped not just by these internal pressures in his thinking but by his exposure to a succession of external influences. Some were personal: McTaggart was the first of a number of friends—Moore, Morrell and Wittgenstein, for example—whose views he sometimes seems to have adopted for no better reason than that they wanted him to. Other influences were intellectual: a series of chance occurrences put him in the path of ideas that transformed his conception of mathematics. His primary difficulty, though, was that he began writing his monograph on the foundations of mathematics while blithely ignorant of relevant work that had been going on in the German-speaking world for more than a quarter of a century.

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The continuum In his early writings Russell frequently discussed space and time in relation to the property of ‘continuity’, by which he meant that they are ordered in such a way that any two elements have a third element in between them. He followed the Hegelians in holding that the very notion of such an ordering is contradictory because of the ‘antinomy of infinite divisibility’. Differentiations involve parts, parts involve elements and a whole. But where are its parts? and where is the whole? Our continuum can be divided anywhere, and is wholly indifferent to its divisions. . . . Therefore the search for parts leads to the infinitesimal, in the hope that somewhere we may find an obstacle to further division. (CPBR, II, 57) Hence arises, he maintained, the self-contradictory notion of mathematical zero as ‘a quantum containing no quantity’, from which he concluded that ‘no continuum . . . can be regarded as a thing’ (ibid.). ‘The continuum as an object of thought is self-contradictory; whatever we treat as a continuum must really, if it is to be intelligible, be discrete.’ (CPBR, II, 53) The notion of a continuous ordering, ‘though convenient in dealing with many mathematical topics, is selfcontradictory and absurd’. Any other view ‘would be suicidal alike in mathematics and in philosophy’ (CPBR, II, 58). The idea that infinite divisibility is inherently contradictory was far from new—it had been much discussed by the ancient Greeks—and Russell’s attempts at expounding it were notably brief: each time, he stopped short of saying clearly just what the contradiction is. He seems to have simply assumed that infinite division is impossible in reality, so that an infinitely divisible continuum can only be ideal, and hence contradictory. What Russell then called a ‘continuous’ ordering is nowadays called ‘dense’: his way of using the word ‘continuous’ has died out, because there is a sense in which a dense ordering may nonetheless have gaps and hence not be√truly continuous: the rational numbers form a dense linear ordering, and yet 2 is irrational, for instance. What differentiates the continuum of real numbers from the merely dense ordering of the rational numbers is the additional property now known as completeness:1 an ordered set is complete if every non-empty bounded subset has a least upper bound and a greatest lower bound. The existence of irrational numbers was known to the ancient Greeks, of course, but the role of completeness in the foundations of the differential and integral calculus emerged only in the 1860s: by the time Russell was a student it was well understood by German mathematicians (but not, apparently, by him). Russell’s undergraduate course at Cambridge made extensive appeal to the ‘infinitesimal continuum’, which contains not only all the real numbers but

1 This is not the sense in which the word is used in logic.

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also, for each real number, infinitely many other numbers differing from it only infinitesimally. A function f was said to be continuous at a real number a if f (a + ) − f (a) is infinitesimal for every infinitesimal ; similarly, a real number that is infinitesimally close to (f (a + ) − f (a))/ for every infinitesimal  was said to be the derivative of f at a. This account of the foundations of the calculus in terms of infinitesimals had been ridiculed by Berkeley in The Analyst (1734), but his criticisms had little effect on mathematicians, who continued to use infinitesimals (with great success, it should be said) in their arguments. The move away from infinitesimals did not really begin until Cauchy, who in his Cours d’analyse (1821) spoke of variables decreasing indefinitely towards zero rather than of numbers infinitesimally close to zero. Cauchy continued to appeal to a geometrical conception of the continuum, however, and the elimination from the calculus of this appeal to geometrical intuition had to wait until Weierstrass, whose lectures at Berlin aimed to prove the theorems of the calculus without appealing to geometrical intuition. Weierstrass did not publish his lectures himself, but they were widely (if imperfectly) disseminated in treatises by disciples (e.g. Biermann 1887). His work was known at Cambridge (which awarded him an honorary doctorate in 1893) but had not yet percolated into the Mathematical Tripos. Although Russell studied in Berlin in 1895, the lectures he attended there were on economics, not mathematics, and he still knew nothing of Weierstrass. In the summer of 1896 he reviewed Couturat’s book, De l’infini mathématique, which mentioned Dedekind’s construction of the real numbers. Couturat had misstated the definition of completeness in a way that allowed Russell to dismiss it briskly as erroneous, but in December 1896 he read Dedekind’s book for himself, and presumably realized that the definition could not be dismissed so easily. Dedekind had shown how, assuming the natural numbers as a starting point, it is possible to construct a complete ordered field. He had not discussed how to derive the calculus from this starting point, but only because he assumed his readers would already know about Weierstrass. Even after reading Dedekind, therefore, Russell remained innocent of Weierstrassian methods (see CPBR, II, 50). It was not until his visit to the U.S. in the autumn of 1896 to deliver a lecture series on geometry at Bryn Mawr and Johns Hopkins that Russell was told by the mathematicians he met there about Weierstrass and the German tradition of rigour in calculus (CPBR, XI, 11). Even so, he was slow to master this material. In An Analysis of Mathematical Reasoning, the book draft he wrote between April and July 1898, he still struggled unsuccessfully to make sense of infinitesimals in terms of what he called the ‘fifth kind of zero’ (a kind of zero such that two quantities which differ by it may nonetheless be distinct). On his list of the books he was reading, the only one that might have given him a rigorous account of the foundations of the calculus was Dini (1892), which he did not read until May 1900. He later recalled having read ‘several French Cours d’Analyse’ (MPD, 30), but none of these is listed, so perhaps he did not read them until after

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March 1902 (when he gave up maintaining the list). One of the Cours in question was presumably Jordan’s celebrated textbook (1893-6), credited with introducing rigour to the French university curriculum, which he mentioned in a footnote in the Principles (§307). At some point he also read Stolz (1885), which treats similar material. Mastering this Weierstrassian material enabled him to avoid the problematic notion of the infinitesimally small, and hence exposed the antinomy of infinite divisibility as no more than a quantifier-shift fallacy. Moreover, it showed him how, if the natural numbers are assumed, the theory of real and complex numbers can be derived without any appeal to geometric intuition.

The infinitely large The most persistent of the contradictions Russell found in mathematics, however, were those connected with the infinitely large rather than the infinitely small. Cantor had published a theory of infinite cardinal and ordinal numbers in the 1880s, but, like Weierstrass’s treatment of the calculus, it was slow to make an impression on Russell. When he first came across it in March 1896, he thought it was ‘impossible and contradictory’, because the sequence of natural numbers is never-ending and so we could never reach ω, the first of the infinite ordinals that Cantor’s theory posited. ‘When a series has no upper limit, even the mathematician will hesitate to speak of anything larger than its upper limit.’ (CPBR, II, 52) That August he read in Couturat’s book a defence of the claim that there is nothing contradictory about a completed infinite totality or about an infinite number, but he was still unpersuaded. He read more of Cantor in July 1899, but even as late as the draft of the Principles which he wrote in 1899–1900 he denied that Cantor’s theory of ordinals solved the problem of the infinitely large. I cannot persuade myself that his theory solves any of the mathematical difficulties of infinity, or renders the antinomy of infinite numbers one whit less formidable. Like most mathematical ideas on the subject it consists of a skilful combination of the two sides of the antinomy in the proportions most useful for obtaining results. (CPBR, III, 119) He continued to claim that ‘the rejection of infinite numbers seems unavoidable’; the best course was therefore to adopt Leibniz’s view that ‘infinite aggregates have no number’ (L, 117n). As with the paradox of infinite divisibility, Russell never gave a very precise statement of what the supposed contradiction actually was, but it probably resulted from interpreting the principle that ‘the whole is greater than the part’ as entailing that a class has more members than any proper subclass. Not until he read Bolzano’s Paradoxes of the Infinite (1851) and re-read Cantor in August 1900 did he realize that the contradiction dissolves if we concede that this entailment does not hold for infinite classes.

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Russell may have been slow to resolve the contradiction, but he was quick, once he did so, to ridicule the view he had only just abandoned. Early in 1901 he confidently claimed that almost all current philosophy is upset by the fact (of which very few philosophers are as yet aware) that all the ancient and respectable contradictions in the notion of the infinite have been once for all disposed of. (CPBR, III, 372) And in the Principles he ascribed the difficulty of infinite numbers to ‘confusions due to the ambiguity in the meaning of the finite integers’ (§179). ‘Of all philosophers who have inveighed against the infinite,’ he announced loftily, ‘I doubt whether there is one who has known the difference between finite and infinite numbers.’ (§183) He omitted to mention how recently he had shared their ignorance.

Mathematical education It may seem scarcely credible that after three years studying mathematics at Cambridge Russell should have been wholly unaware of Weierstrass’s existence, but he was not alone. The addition of the rigorous treatment of calculus to the Cambridge Mathematical Tripos was still some years off. In this two Trinity mathematicians were particularly influential: Whittaker (a near contemporary of Russell’s) published Modern Analysis in 1902 to give advanced undergraduates a rigorous treatment of complex analysis; and Hardy aimed his Pure Mathematics of 1908 at first year undergraduates studying real analysis. Why, though, was Russell’s mathematical education at Cambridge ‘definitely bad’ (MPD, 38)? One diagnosis was that the practice of publishing the Tripos results as an ordered list encouraged undergraduate supervisors (known as ‘coaches’) to prioritize exam tricks over understanding. Hardy was one of those responsible for abolishing this practice in 1909, since when the title ‘Senior Wrangler’ has not been officially awarded by the examiners. (Bizarrely, though, despite this decision Chairmen of Examiners still indicate by doffing their caps who has come top when the results are read out in the Senate House.) It would be naive to suppose that this reform on its own could have effected a cure, though. Ignorance of German mathematics was not unique to Cambridge: if Russell had gone to Oxford (where the class list was never published in rank order), he would have fared even worse. The abolition of ordered class lists at Cambridge was only one of several Edwardian reforms, and the removal of ancient Greek as an entry requirement was probably more significant. Moreover, part of the explanation for the deficiencies was simply a difference of emphasis: British mathematicians in the 19th century preferred algebraic to analytic methods (as Whitehead’s preference for projective over metric geometry illustrates), and many of the achievements of group theory and linear algebra were British, not

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German. That said, Russell’s mathematical education was deficient in comparison with what he might have learnt at the best German universities. What is most remarkable, perhaps, is how quickly this was rectified: by the time Ramsey was at Cambridge in the 1920s, the undergraduate education there was much improved. The most immediate effect of Russell’s belated mathematical education on his philosophical views was a change in his attitude to Hegelian dialectic. As its ‘contradictions’ were exposed one by one as mere misunderstandings, his confidence in its soundness was inevitably shaken. His education in German mathematics also contributed to the impact of the Principles when it eventually appeared in 1903. A considerable part of it (almost all of Part V, for instance) is devoted to expounding material which would seem novel to an anglophone audience, but which had been well known to German mathematicians for a generation. Russell’s account of this material was probably more influential than it deserved. Many English philosophers remained ignorant of relevant parts of mathematics because, even after the abolition of the ‘Little Go’, they had come to philosophy via a classical education which did not equip them to understand mathematics. Bradley, for instance, told Russell that his ignorance of mathematics was due to ‘an irremovable incapacity for abstract reasoning’ (28 Jan. 1901, in CPBR, III, 288). Russell, by contrast, repeatedly stressed the clarity that modern mathematics could bring to philosophical problems. He planned to found a school of mathematical philosophy and for a time even hoped (implausibly) that Wittgenstein would be its leading member. His evangelical zeal owed much to his own earlier inadequacies: for the rest of his career he delighted in accusing other philosophers of an ignorance of technicalities that was in truth no worse than his own had been at first.

Further reading Dugac (1973) describes Weierstrass’s rigorization of the calculus. Since the reforms, this material has been a staple of the undergraduate mathematics curriculum, and there are now many textbooks that expound it, but Hardy (1908) remains a classic. The paradoxes of the infinite were much discussed by ancient writers, but have also been used more recently by idealists to argue for the incoherence of the realist conception of the world. Richards (1988) gives more detail on Russell’s mathematical education at Cambridge.

27 WHITEHEAD

Another major influence on Russell was A. N. Whitehead, a mathematics don at Trinity ten years his senior who had been one of his undergraduate teachers. I have already mentioned his role in persuading Russell in 1895 of the importance of the descriptive approach to neutral geometry. More significant, though, was Universal Algebra, the book he published in 1898.

Universal Algebra In his book Whitehead attempted to unify geometry with algebra by treating projective geometry and its hyperbolic, Euclidean and elliptic particularizations as algebraic systems of the kind nowadays called vector spaces, with an additional operation known as an exterior product. He presented these alongside Boole’s algebra of propositions as algebraic systems involving operations of addition and multiplication, and emphasized their similarities by providing a geometric interpretation of Boolean algebra. He intriguingly claimed to have a kind of representation theorem, to the effect that any problem of logic may be solved by geometrical means. Moreover, although he did not devote space to ordinary arithmetic, his presentation made it clear that this was another case of the general notion of an algebraic system. Whitehead’s attempt to present diverse mathematical structures as instances of a single kind may be traced Boole’s idea, in The Laws of Thought (1854), that one algebraic system (now known as a Boolean algebra) has two valid interpretations, in one of which its members are propositions, in the other propositional functions. Whitehead’s book did not in the end quite deliver the unified account of arithmetic and geometry which it promised—this was intended for a second volume which never appeared—but it contained presentiments of more recent attempts at

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unification such as Bourbaki’s. Whitehead’s chief failing was that he did not utilize the general notion of class to be found in the works of Cantor, but supposed that the elements of a class (or ‘manifold’, as he called it) are always obtained by abstraction from objects viewed under some mode. The relation between these elements that he wrote as ‘=’ was not identity but equivalence in some respect. At this point his treatment cries out for the notion of an equivalence class (as well, it should be said, as the use/mention distinction). Notably, though, Whitehead characterized mathematics in terms not of its subject matter but of its method. The reasoning is formal in the sense that the meaning of propositions forms no part of the investigation. The sole concern of mathematics is the inference of proposition from proposition. The justification of the rules of inference in any branch of mathematics is not properly part of mathematics; it is the business of experience or philosophy. The business of mathematics is simply to follow the rule. In this sense all mathematical reasoning is necessary, namely, it has followed the rule. (1898, vi) He distinguished applied mathematics, where the symbols are interpreted and each definition has existential import, from pure mathematics, where a definition merely ‘sets before the mind by an act of imagination a set of things with fully defined self-consistent types of relation’ (1898, vii). In pure mathematics he envisaged no constraint on what might legitimately be studied other than those of consistency and imagination. Mathematical reasoning is formal, he said, in the sense that ‘the meaning of propositions forms no part of the investigation’ (1898, vi). One difficulty for this formalist view was to explain the reliability of conventional mathematics in applications. Whitehead claimed that conventional mathematics is reliable because its rules have the same form as those of the interpreted systems which it generalizes. The laws regulating the manipulation of the algebraic symbols are identical with those of Arithmetic. It follows that no algebraic theorem can ever contradict any result which could be arrived at by Arithmetic; for the reasoning in both cases merely applies the same general laws to different classes of things. If an algebraic theorem is interpretable in Arithmetic, the corresponding arithmetical theorem is therefore true. (1898, 11) This would nowadays be called a conservativeness claim. (The specific example Whitehead mentioned was the extension of the arithmetic of real numbers by adding a symbol i such that i2 = −1.) The best that can be said on his behalf is that he was not alone in holding this view, which was widespread among 19th-century British mathematicians. What he said was a variant of Peacock’s (1830, §132) ‘principle of permanence of forms’, which held that the laws of arithmetical algebra may be abstracted to the setting of symbolical algebra, where

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they apply to ‘combinations of arbitrary signs and symbols’ (ibid., §78), but that the results of symbolic manipulations may then be interpreted arithmetically. The problem, of course, is that Peacock’s principle is ‘simply a mistake’ (PoM, §357): to ensure that an extension of arithmetical methods is conservative, more is required than that the algebraic laws governing it are syntactically the same.

A single paradox of relativity One immediate consequence for Russell when he read Whitehead’s book in proof was to cast doubt on the conception of the dialectic of the sciences that he had sketched in the Tiergarten. Once geometry is presented as sharing a common structure with Boolean algebra, it becomes natural to wonder whether whatever contradiction is to be found in geometry is already present in this common structure; if so, its resolution could hardly depend, as he had supposed, on geometry’s position in the hierarchy of the sciences. In On Quantity and Allied Conceptions, the book he was then attempting to write, Russell therefore gave up arguing for a series of contradictions specific to various disciplines and instead suggested that the whole of mathematics is infected by a single paradox of relativity, which is ‘the mark of a certain kind of relation, namely a relation which is asymmetrical and is not connected with intrinsic adjectives of its terms’ (CPBR, II, 132). ‘Relations of this type pervade almost the whole of Mathematics, since they are involved in number, in order, in quantity, and in space and time.’ (CPBR, II, 225-6) During 1898 Russell referred repeatedly to this paradox, which arises, he suggested, if we have two terms A and B, with a relation R which transforms them into Aβ and Bα. Neither can be expressed without this reference, and α and β differ in content. But A and B, considered without reference to the relation R, have no differences of conception corresponding to the differences α, β. Either α or β alone may, however, be considered as expressing a difference between A and B: β, in fact, gives to A the adjective of differing from B in a certain manner, and α expresses the same difference with B as startingpoint. We have thus a difference between A and B, namely that expressed by either α or β, but we have no corresponding point of difference. We cannot use the difference between α and β to supply the point of difference, for both α and β state a difference, and therefore presuppose a point of difference. We must, in fact, have a difference between A and B, without there being any corresponding point in A by which it differs from B, and vice versa. Thus we have a difference without a point of difference, or, in the old formula, a conception of difference without a difference of conception. (CPBR, II, 225–6) Russell had identified a circularity in his conception not just of geometry but of any abstract science involving asymmetric relations: the identity of its objects

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is constituted by their relations to one another, but these relations in turn are grounded in the properties of their relata. It would be another year before Russell worked out how to resolve this paradox.

A wrong turning? Whitehead was hampered in articulating his view by his ignorance of modern logic. On ‘pure’ (i.e. philosophical) logic he was guided by ‘Mill, Jevons, Lotze and Bradley’, on formal logic by ‘Boole, Schröder and Venn’ (1898, x). He briefly mentioned papers by two students of Peirce (Mitchell and Ladd-Franklin), but he had not read Frege and his account of symbolic logic was concerned principally with representing syllogistic logic by means of Boolean algebra. His account of generality (typically for a late 19th-century mathematician) treated variables as indeterminate names, so that a letter x denotes any one of an assigned class with certain unambiguously defined characteristics. In the same series of operations the sign must always denote the same member of the class; but as far as any explicit definitions are concerned any member will do. (1898, 4) To explain why Whitehead was not yet tempted by logicism it is sufficient to note his ignorance of polyadic logic: in relation to the restricted logic he knew about—that of ‘Boole, Schröder and Venn’—logicism is plainly false. Nonetheless, Whitehead’s book prompted Russell (for a time, at least) to consider the relevance of logic to mathematics. He had begun the previous draft of his book on the foundations of mathematics by discussing number. In April 1898, after re-reading the parts of Whitehead’s book that interested him, he started on a new book draft, An Analysis of Mathematical Reasoning, which began instead with a section on the logic of predication. Here he tried to develop a sort of duality between, on the one hand, multiple predications of a single subject (such as ‘Socrates is both mortal and wise’) and, on the other, a single predication of multiple subjects (such as ‘both Socrates and Plato are wise’). He intended by this means to treat Whitehead’s manifolds as, in effect, plural subjects. In the short term, at least, this foray into plural logic proved abortive, as in the next draft of his big book the following year (and in several further attempts in the following two years) Russell reverted to his original plan of beginning not with logic but with numbers. Another reason for his turn away from formal logic, though, was that An Analysis of Mathematical Reasoning contained the first signs of a major shift in his thinking for which Moore, not Whitehead, was responsible, namely his rejection of absolute idealism. Whitehead’s influence on him resumed in August 1900, however, when they attended a conference in Paris together and went on a joint holiday afterwards with their wives, thus inaugurating a decade of close collaboration. The stated reason for this collaboration was that the projected second volume of Whitehead’s Universal Algebra overlapped so substantially with

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what Russell intended for the second volume of his Principles that a joint work seemed appropriate. Perhaps it is not coincidental, though, that Russell’s infatuation with Mrs Whitehead probably began soon after their holiday together. (In his Autobiography Russell was candid about many of his romantic and sexual liaisons, but chose not to mention his feelings for her at all.) For some years thereafter the two families were extremely close. From January 1901 until the middle of 1902 the Russells even shared a house with the Whitehead family in Cambridge, and they went on another joint holiday that summer. Russell first formulated the central claim of the Principles, that mathematics is part of logic, at just this time.

Further reading Whitehead’s collaboration with Russell has been rather little studied.

28 MOORE

It is widely held that analytic philosophy had two birthplaces, Jena and Cambridge, and that if the first birth occurred in 1879, then the second should be dated to 1898, when Russell and Moore rejected absolute idealism and replaced it with a kind of platonic realism in which a new conception of propositions played a crucial role. If it is agreed what the outcome was, though, the respective contributions of Russell and Moore in bringing it about are much less clear. G. E. Moore (who disliked and never used his first name, George) went up to Trinity to read Classics, but changed to Moral Sciences in his final year, partly on the encouragement of Russell, two years his senior, with whom he was by then friendly. Moore soon supplanted McTaggart as the dominant philosophical influence on Russell, who for a few years regarded him with reverence. They grew apart thereafter, however, as Moore came increasingly to resent Russell, whether because of the former’s (considerable) intellectual insecurity or the latter’s (equally considerable) public confidence.

The existential theory of judgment The first visible sign of Russell’s turn away from absolute idealism came early in An Analysis of Mathematical Reasoning, when he rejected Bradley’s existential theory, that the criterion of the truth of a judgment is its ‘reference to reality’, i.e. its (partial) correspondence with the Absolute. Russell objected that various claims—e.g. that 2 is numerical, that 2 differs from 3, or that redness differs from blueness—‘would be inexpressible if we refused to regard such terms as subjects’ (CPBR, II, 168). Since the terms in question are non-temporal and hence do not in Russell’s sense ‘exist’, the existential theory was unable to explain how these claims could be not only meaningful but true. Rather, he thought, we have to

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recognize a category of entities—the number 2, redness, etc.—that do not exist but only ‘subsist’.1 Russell’s argument is hardly persuasive, though. It would be open to his opponent to argue that ‘2 is numerical’ is not, despite appearances, really meaningful, or that its apparent structure is misleading. Aware, perhaps, of these lacunae in his argument, he noted enigmatically that the full proof of his claim would ‘be found at large throughout the present work’ (CPBR, II, 168), before moving swiftly on to other matters. If his stated argument was so weak, then, what was Russell’s real reason for rejecting the existential theory? One possibility, to which his later recollection lends some support, is that the initial motivation came not from him but from Moore. ‘He took the lead in rebellion; and I followed, with a sense of emancipation.’ (MPD, 12) A crucial document here is Moore’s Fellowship dissertation, submitted unsuccessfully in 1897, then resubmitted, this time successfully, the following summer with two new chapters. Later he extracted the bulk of these two chapters to form an article, ‘The nature of judgment’ (1899a), with the result that they are largely missing from his manuscript (the only surviving copy) of the dissertation. The published article is thus our principal source for the views he adopted in 1898. There he offered a variant of Russell’s argument against the existential theory. Not all propositions, he said, have the relation to reality that the existential theory demands. For example, 2 + 2 = 4, whether there exist two things or not. Moreover it may be doubted here whether even the concepts of which the proposition consists, can ever be said to exist. We should have to stretch our notion of existence beyond intelligibility, to suppose that 2 ever has been, is, or will be an existent. (Ibid., 180) This argument, though, would be no more likely than Russell’s to persuade the absolute idealist. Moore also offered a second argument against the existential theory of judgment—more specifically, against defining truth as correspondence with reality. It is, he suggested, impossible that truth should depend on a relation to existents or to an existent, since the proposition by which it is so defined must itself be true, and the truth of this can certainly not be established, without a vicious circle, by exhibiting its dependence on an existent. (Ibid., 181) This bears a striking similarity to Frege’s treadmill argument of 1897, although Moore must have come upon it independently. (Frege did not publish his argument until 1919.) Perhaps Moore was right that to understand a definition of

1 I here follow Russell’s terminology, although confusingly Aquinas, who originated the distinction, used ‘subsist’ for the more substantial sort of being, not the less.

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truth I must already have an implicit grasp of the notion, but it is not clear why this should tell especially against the correspondence theory. Moore was perhaps on stronger ground in the specific case of Bradley’s existential theory, which held that there is only one substance, namely reality, to which any judgment ascribes attributes—‘adjectives’, as he called them. In Appearance and Reality Bradley attempted to argue that their dependency on the whole makes these adjectives unreal. To this Moore objected that a component of a judgment is not in any intelligible sense an ‘adjective,’ as if there were something substantive, more ultimate than it. For we must, if we are to be consistent, describe what appears to be most substantive as no more than a collection of such supposed adjectives: and thus, in the end, the concept turns out to be the only substantive or subject, and no one concept either more or less an adjective than any other. (Ibid., 192–3) The ‘substance’ in Bradley’s account, that is to say, is reducible to its adjectives, which, in the absence of the contrast with reality, we no longer have any reason to idealize. Soon afterwards, Russell used a variant of Moore’s argument to oppose Bradley’s conception of reality as a substance. Either a substance is wholly meaningless, and in that case cannot be distinguished from any other: or a substance is merely all or some of the qualities which are supposed to be its predicates. (L, §25) His objection was in effect that the ‘reference to reality’ was an idle wheel in Bradley’s account. It was, he suggested, ‘the invariable result of admitting, as elements of propositions, any terms which are destitute of meaning’. This then led him to ‘share Locke’s wonder’ (L, §21) that terms without meaning could have any logical use. We should therefore, he concluded, reject Bradley’s view that whatever its grammatical subject may be, the logical subject of a judgment is always reality. Of course, even if this is right, it does not show that the logical subject is the same as the grammatical subject. Yet this seems to be what Russell now increasingly supposed. ‘Grammar,’ he wrote, ‘though not our master, will yet be taken as our guide.’ (PoM, §46)

Thing and concept Once he had rejected Bradley’s existential theory, the widest word in Russell’s logical vocabulary was ‘entity’. An entity, he said, is anything whatever that ‘may be the subject in true judgments’, and ‘everything that can be thought of, or represented by a word, may be a logical subject’ (CPBR, II, 168). Russell often called entities ‘terms’, but I shall avoid this usage here, because to a modern ear it makes them sound linguistic, which they are not. This is an instance of a

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general difficulty of interpretation arising from his habit of courting use/mention confusions. A ‘predicate’ or a ‘verb’, for instance, was for him not a piece of language but what that piece of language means. Having rejected the substance/attribute distinction in ‘The nature of judgment’, Moore made no immediate attempt to replace it with any other categorial distinction among entities, leaving Russell to remind him, on reading his dissertation, of the need at least to distinguish between subject and predicate (1 Dec. 1898, in SLBR, 191). Russell thus wished to draw not only the metaphysical distinction already mentioned between existents and subsistents, but also a different, logical distinction between things and concepts (PoM, §48). A thing may only occur as the subject of a proposition; a concept may occur either as subject (‘denotation’, as he later put it) or as predicate (‘meaning’). The concept wisdom, for instance, occurs as meaning in ‘Socrates is wise’, as denotation in ‘Wisdom is a virtue’. Why, though, did Russell hold to this ‘curious twofold use’ (ibid.) and not regard wisdom and wise instead as distinct entities? His answer (ibid., §49) was that the latter view is self-refuting, since even to state it we have to make wise a logical subject. Russell maintained, moreover, that the two distinctions differ extensionally: there are things that do not exist but only subsist. ‘Numbers, the Homeric gods, relations, chimeras and four-dimensional spaces all have being, for if they were not entities of a kind, we could make no propositions about them.’ (PoM, §427) ‘My queer zoo’, he later recalled, ‘contained some very odd monsters, such as the golden mountain and the present King of France—monsters which, although they roamed my zoo at will, had the odd property of nonexistence.’ (MPD, 42) The example of chimeras was originally Moore’s. ‘When I say “The chimera has three heads”, ’ he claimed, ‘what I mean to assert is nothing about my mental states, but a specific connexion of concepts.’ (1899a, 179) But what is ‘The chimera has three heads’ about? Is it a generalization about particular chimeras, real or imagined, or is it about the concept chimera? Moore was notoriously unclear about the difference, saying in Principia Ethica that ‘you can give a definition of a horse, because a horse has many different properties and qualities, all of which you can enumerate’ (ch. 1, §7). There is room to doubt whether Russell needed to commit himself to the curious being of a non-existent chimera, rather than to the somewhat less curious being of the concept chimera; he was in any case less interested in the ontological status of chimeras than in abstracta such as numbers and relations, for the obvious reason that these were the entities that his developing views about mathematics required. Although Russell’s ontology now included non-existent entities, he did not argue for the stronger view that there are contradictory entities such as round squares. Since all entities are capable of being logical subjects, all must conform to the laws of logic. His difficulty, though, was to explain why he was not in fact committed to the stronger view, since it seems that on his account the existing present King of France both does and does not exist. To resist this awkward

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conclusion he would presumably have had to treat existence as somehow sui generis, and hence to contradict Moore’s claim (1899a, 180) that it is just another concept (albeit a ‘peculiarly important’ one), but it would be some years before he acknowledged this.

The refutation of idealism Moore proposed to avoid the now-problematic correspondence between proposition and reality by identifying the former with the latter: the world consists of the true propositions, and there is no difference between a true proposition and the fact that makes it true. Truth collapses into identity and no longer has to be defined. To say that is not yet to decide between realism and idealism, however. Some of the earlier philosophers who distinguished between existent and nonexistent beings had been inclined to treat the latter as in some way ideal, whereas for Russell and Moore they are just as real as existent ones. What made this ‘the most platonic system of modern times’ (letter to Desmond McCarthy, 14 Aug. 1898) was that Moore took propositions to be independent of the mind. Their components may come into relation with a thinker; and in order that they may do anything, they must already be something. It is indifferent to their nature whether anybody thinks them or not. They are incapable of change; and the relation into which they enter with the knowing subject implies no action or reaction. (1899a, 179) The notion that the object of a judgment can be treated as an entity in its own right is not mandated merely by conceiving of judging as an act: we do not feel a similar temptation, in the case of an act of leaping, to regard the leap as independent of the act. Moore’s motivation for this further step seems to have been Herbart’s, not Frege’s: we find no trace here of the ‘Frege point’ that only by distinguishing the act of judgment from its content can we explain what the judgment that p and the judgment that if p then q have in common. Rather, Moore hived off the psychological act of judgment, like Herbart, in order that the mind-independent proposition that remains should be capable of playing its part in his anti-psychologistically conceived logic. So far, though, this is all mere assertion. The argument Moore (1899a, 177–8) offered against idealism took as its target Bradley’s distinction between the ‘universal meanings’ that occur in judgment and the ideas that occur in the mind. Moore focused his attack on Bradley’s statement that the universal meaning is ‘cut off ’ (i.e. abstracted) from the idea, complaining that this was circular because in order to cut it off, we would need first to have grasped a proposition about the idea, in which case we would already have the universal meaning. The overall shape of Moore’s complaint is clear enough—nothing we derive from mental entities by a process of abstraction can attain the independence from mind that is

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required by logic—but the details are obscure. In ‘The nature of judgment’ he never succeeded in explaining why a process of abstraction that started from the mental could not lead to something non-mental. Four years later, in ‘The refutation of idealism’ (1903, 444–5), Moore attempted more ambitiously a general refutation of the idealist claim that esse est percipi. A sensation of blue and a sensation of green have something in common: call this common element consciousness. This consciousness that is part of my sensation of blue cannot be just the same thing as the blue (by which Moore meant the instance of blueness that is sensed), since the former is also present in my sensation of green, whereas the latter is not. Hence, he argued, the sensation of blue is not identical to blue, contradicting the idealist. The flaw in this argument, though, is that it simply assumes that sensations are to be analysed in terms of part-whole complexity: there are plainly other ways in which a resourceful idealist might conceive of them.

Against Kant? Moore’s commentators have tended to focus on his objections to Bradley’s absolute idealism, but really he was at least as concerned with attacking Kant. A notable instance is his highly critical review (1899b) of Russell’s geometry book. He objected to Russell’s argument that experience presupposes the form of externality. That argument was eminently disputable in its details, as we saw in Chapter 24, but Moore criticized instead its overall form, and in doing so displayed what seems to amount to a serious misunderstanding of the nature of transcendental arguments: he interpreted Russell as attempting an inference from the truth of the judgments of experience, when in fact Russell assumed only their possibility. Moore also attacked Russell’s attempt to purge Kant’s notion of the a priori of psychologism. Russell had tried to distinguish the a priori from the subjective by insisting that the former is a logical notion—what is justifiable without appeal to sensation—whereas the latter is psychological—what is causally independent of sensation. In his first philosophical publication, for instance, he had criticized Heymans for basing his account of geometry on the motor-sense and hence making it subjective, not a priori. In his review Moore claimed that Russell had nonetheless failed in his attempt to elevate geometry above the subjective. Russell also maintained that even if Kant’s account of geometry was infected with psychologism, the Kantian categories were genuinely a priori, and hence logical, concepts bearing no causal relation to perception. Here, too, Moore insisted that Russell was wrong, maintaining that the categories are deduced from experience, not from its possibility, and that transcendental logic is therefore part of psychology. In thus accusing Kant of subjectivism he was certainly not unusual: in the late 19th century there was a widespread view (originating, ironically, with Hegel) that Kant’s attempt at a distinctively transcendental psychology—a

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study of how any mind capable of making judgments about experience must be configured—collapses into an empirical commentary on how we as a matter of fact judge. There is much Russell could surely have said to defend himself against Moore’s barrage of criticisms, but, curiously, he did not say it, meekly conceding that ‘on all important points I agreed with it’ (letter to Moore, 18 July 1899). Thereafter, he maintained that Kant held an error theory according to which whatever we cannot help believing must be false. What we cannot help believing, in this case, is something as to the nature of space, not as to the nature of our minds. The explanation offered is, that there is no space outside our minds; whence it is to be inferred that our unavoidable beliefs about space are all mistaken. (PoM, §430) It need hardly be said that this is a crude simplification of Kant’s claim that spatial relations are imposed by the mind on experience. After Moore’s criticisms Russell simply ignored Kant’s attempt in the transcendental deduction to secure the intersubjective validity of a priori judgments. What he got from Moore, then, was not an anti-psychologistic conception of logic, which he had had several years earlier, but the idea that Kant’s view could not be used to underpin it.

Moore’s influence How did the revolt against absolute idealism take place? Russell attributed it unambiguously to Moore. On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not reducible to adjectives of their terms or of the whole which these compose. (PoM, Preface) During the Lent Term of 1898 they were both in Cambridge and attending the same lectures, so had ample opportunity for discussion. Russell read a paper (now lost) to the Moral Sciences Club in March, and this led to a lengthy discussion between the two of them on the subject of existence. Moore also went to stay with the Russells in Kent during the Easter Vacation. During the following Easter Term, however, Russell was not resident in Cambridge and the only known meetings between them were during two visits (on 10th May and 28th June). It would be natural to conjecture that they rejected the existential theory in the course of these conversations. In that case, though, it is puzzling that Moore wrote to him in September as if he expected Russell to have no prior knowledge of his dissertation’s main proposal.

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My chief discovery which shocked me a good deal when I made it, is expressed in the form that an existent is a proposition. I see now that I might have put this more mildly. Of course, by an existent must be understood an existent existent—not what exists, but that + its existence. (11 Sep. 1898) Perhaps, then, their respective roles in the rejection remain tantalizingly out of reach. Russell later gave Moore’s views part of the credit for his progress in the philosophy of mathematics. Before learning these views from him, I found myself completely unable to construct any philosophy of arithmetic, whereas their acceptance brought about an immediate liberation from a large number of difficulties which I believe to be otherwise insuperable. The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. But I must leave it to my readers to judge how far the reasoning assumes these doctrines, and how far it supports them. Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour. (PoM, Preface) Russell’s reason for accepting Moore’s views thus seems to have been ‘regressive’: they allowed him to explain what he found otherwise inexplicable, namely the truth of mathematics. Russell himself later said of Moore’s ‘The nature of judgment’ that it ‘gave conclusive proof of philosophical genius’ (CPBR, XI, 209). Even if we allow for the context of this remark—in an obituary—it is notably strong. ‘It is difficult’, he said, for the present generation to realize what academic philosophy was like when he and I were young. . . . Those who are too young to remember the academic reign of German idealism in English philosophy after T. H. Green can hardly appreciate what Moore achieved in the way of liberation from intellectual fetters. (CPBR, XI, 211–12) Moore, whose education had been classical rather than mathematical, was certainly no expert on formal logic. When Russell said that on ‘fundamental questions of logic’ he hoped that Moore ‘may find solutions where I see none’ (CPBR, III, 218), he presumably meant only philosophical logic: the only formal logic he knew then was ‘the Boolean stuff ’, which he found ‘useless’ (letter to Jourdain, 15 Apr. 1910, in Grattan-Guinness 1977, 134). Even so, the extent to which he deferred to Moore’s logical views is surely remarkable. The puzzle only deepens when we read ‘The nature of judgment’ itself, a paper notable for the absence of arguments for many of its claims and the poverty of those arguments it does contain. It is a decidedly curious paper to have sparked a revolution.

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Further reading Ryle (1970) provides an enjoyable and astute critique of Moore’s paper on ‘The nature of judgment’, while Ayer (1971, ch. 6) discusses ‘The refutation of idealism’. Candlish (2007, ch. 5) discusses Russell’s changing views on the transparency of surface grammar. Grayling (1996) discusses Moore’s attack on Russell’s transcendental argument. On Moore’s strained relationship with Russell see Preti (2009). Russell’s ‘queer zoo’ of non-existent entities is often called ‘Meinongian’, although Meinong’s own view was more nuanced. For modern discussions see Parsons (1980) and Crane (2015). Moore developed the anti-Kantian argument in more detail in (1904); on his attribution of subjectivism to Kant see Baldwin (1984); on whether the attribution is merited see Allison (2004).

29 LEIBNIZ

In the summer of 1898 Russell’s work took another chance turn: McTaggart, who had been due to lecture on Leibniz for the Moral Sciences Tripos, was granted a year’s leave from Cambridge, and Russell agreed to give the lectures in his stead. He threw himself into the task with characteristic energy, spending much of the summer and autumn reading Leibniz’s published writings. He delivered the lectures in the Lent Term 1899 and published them, with some revisions, the following year. The book was long regarded as a significant contribution to Leibniz scholarship, but here I shall discuss only two topics on which Russell’s confrontation with Leibniz’s views had lasting consequences for his own philosophy.

Analyticity and necessity Leibniz conceived of reality as consisting of substances, each of which is fully described by its ‘complete concept’. Since he supposed that every proposition is of subject-predicate form, he inferred that any true proposition ascribes to the subject a property entailed by its complete concept. In his exposition of this aspect of Leibniz’s views Russell distinguished between existential and non-existential propositions: in the latter case (i.e. where the proposition is not about particular spatio-temporal objects), he correctly described Leibniz’s view as being that the proposition is analytic in Kant’s sense, i.e. that the predicate is contained in the concept of the subject; but he also misascribed to Leibniz the view that existential propositions are not analytic. Shortly after his book appeared, Couturat (1901, ch. 6, §17) convincingly argued that Leibniz had in fact regarded all true

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propositions as analytic: the difference was only that in the case of an existential proposition the proof that it is analytic would be infinite and hence beyond our capacity (but not, of course, God’s) to comprehend. Russell’s error was, in any case, relatively minor: he correctly noted that Leibniz held contingent propositions to be a priori. It is part of Julius Caesar’s complete concept, for instance, that he crossed the Rubicon; his apparent freedom to choose whether to do so was an illusion born of ignorance of what his own complete concept really was. Some may find the semblance of free will thus allowed to Caesar inadequate, and at times Leibniz seemed to agree, granting him more by way of free will than the account really allowed. Nonetheless, the idea that free will is an illusion born of ignorance has proved popular subsequently. To a theist such as Leibniz, however, the account has more troubling implications for God’s free will than for Caesar’s. Caesar may not have known in advance that he would cross the Rubicon; but God did, and hence had no freedom to prevent it. Leibniz saved his account from such a theologically unpalatable conclusion by positing a plethora of possible worlds in addition to the actual one: God’s freedom consisted in His choosing to actualize the world containing Caesar, rather than one containing a counterpart who resembled him but did not cross the Rubicon (see Mason 1967, 16). Russell, however, saw this as an instance of Leibniz’s regrettable desire to harmonize his philosophy with Christian dogma. Unconstrained by such a desire, Russell recognized no genuinely metaphysical modality. Like Frege (although quite independently) he interpreted modal claims by means of quantifiers: ‘A cold can lead to pneumonia’, for instance, means only that some colds do lead to pneumonia. One might suspect that, having detected a theological source for Leibniz’s belief in possible worlds, he did not look as carefully as he should have done for any other reason to believe in them. Moore held a similar view to Russell’s, but oddly expressed it by saying that all propositions are necessary (1899a, 189), which led Ryle to suggest that he ‘here seems to have gone temporarily crazy’ (1970, 99). Moore’s excuse was that, like many of the ancients, he interpreted necessity as meaning that a proposition is always true. So this was his way of expressing his platonistic conception of propositions as timeless. (The idea that perpetual and timeless truth might be worth distinguishing does not seem to have occurred to him.) Of course, when Russell first asserted his actualist view, his ‘zoo’ still contained a sufficiently varied collection of non-existent monsters that doing without possibilia may not have seemed like much of a hardship; the more monsters he culled, the more restrictive his rejection of genuine modality became. Nonetheless, he asserted his actualist view repeatedly between 1903 (CPBR, III, 518) and 1940 (IMT, 37), and accompanied it, indeed, with a marked hostility to counterfactual reasoning. ‘It may be questioned’, he said, ‘whether . . . there is any sense in saying, of a true proposition, that it might have been false.’ (L, 25)

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There seems to be no true proposition of which it makes sense to say that it might have been false. One might as well say that redness might have been a taste and not a colour. What is true is true; what is false is false; and concerning fundamentals, there is nothing more to be said. (PoM, §430) This, though, is not so much an argument for the view as another way of stating it. What Russell did not do was to explain those cases of counterfactual reasoning that resist straightforward paraphrase. How, for instance, are we to understand the claim that if Nelson had not existed, the outcome of the Battle of Trafalgar would have been different?

External relations It was when working on Leibniz, Russell later said, that he ‘realized the importance of the question of relations’ (MPD, 61). The claim is at first glance surprising. Were not spatial relations at the centre of his concerns in his geometry book? At that time, though, he assumed that all relations are in some sense reducible to properties of their relata. What caused him to doubt this assumption was seeing in Leibniz’s work the implausibilities to which it led. Because Leibniz held that all propositions are reducible to subject-predicate form, he maintained that all the relations that a substance bears to other substances belong to its complete concept. A man in India changes if his wife dies in Europe, because he gains the property of being a widower. ‘Nothing happens anywhere in the universe which does not affect every existent thing in the universe.’ (In Mates 1986, 225) That reality consists of individual substances with their own properties and that relations are ideal were, as noted earlier, mediaeval scholastic orthodoxies, but Russell found in Leibniz an unusually clear statement of them. He noted, for instance, that according to Leibniz ‘paternity in David is one thing, and filiation in Solomon another, but the relation common to both is a merely mental thing, of which the modifications of singulars are the foundation’ (L, 276). Of ‘capital importance’ (L, 13), too, was a passage in the fifth letter to Clarke. The ratio or proportion between two lines L and M, may be conceived three several ways: as a ratio of the greater L, to the lesser M; as a ratio of the lesser M, to the greater L; and lastly, as something abstracted from both, that is, as the ratio between L and M without considering which is the antecedent, or which the consequent; which the subject, and which the object. . . . But, which of them will be the subject in the third way of considering them? It cannot be said that both of them, L and M together, are the subject of such an accident; for if so, we should have an accident in two subjects, with one leg in one, and the other in the other; which is contrary to the notion of accidents. Therefore we must say, that this relation, in this third way of considering it, is indeed out of the subjects: but being neither a substance, nor an accident, it must be a mere ideal thing, the consideration of which is nevertheless useful. (In Alexander 1958, 71)

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Here, Russell said, Leibniz ‘seemed, for a moment, to realize that relation is something distinct from and independent of subject and accident’, but ‘thrust aside the awkward discovery’. At the Cambridge Moral Sciences Club in January 1899 Russell took the step Leibniz had ‘thrust aside’: he rejected the ‘doctrine of internal relations’ and adopted instead the contrary doctrine that ‘all relations are external’ (CPBR, II, 143), by which he meant roughly that their holding makes no difference to the related terms. (We shall see shortly that how to define the notion precisely was to be an ongoing difficulty for him.) Thus was born his atomistic conception of distinct things externally related to one another. In the Principles he gave Moore the credit for this, but there is some reason to doubt the attribution, since the reference he gave (§27) was to Moore’s paper, ‘The nature of judgment’, which does not in fact address the internal/external distinction at all. In rejecting the doctrine of internal relations Russell took himself to be arguing against Bradley, who discussed the notion at length in Appearance and Reality (1893, bk 2, ch. 3). Bradley contrived to leave it unclear, though, whether he did hold that relations are internal—it depends on what he meant by saying that a relation ‘essentially penetrates the being of its terms’ (1893, 392)—and indeed Moore was reduced, in seeking to attribute the doctrine to him, to citing an entry in the index to Appearance and Reality. In fact, Bradley’s real purpose was to problematize both the internal and the external conceptions of relations, and hence to conclude that the mediaeval logicians had been right: relations are ideal and ‘our experience, where relational, is not true’ (1893, 29). There were in any case several difficulties with Russell’s new account, irrespective of whether he had misunderstood Bradley. One was that he had supposed without argument that there is a binary choice between the two doctrines: either all relations are internal or all external. Later he rowed back quietly to the more moderate position that relations are not in general internal (CPBR, VI, 355): although a man’s relation to his wife is external, that to his father, for instance, might be internal. Another difficulty was an underlying unclarity in the distinction between internal and external on which Russell’s account depended. The most obvious way to explain it is modal—an object has internal properties necessarily, external ones contingently—but Russell’s actualism led him to think that such an explanation ‘raises irrelevant difficulties’ (CPBR, VI, 128). Over the next few years he repeatedly struggled to give a non-circular criterion for internality. The most serious difficulty with external relations, however, was Bradley’s (1893, 27–8) regress argument: if the relation is external to its relata, he argued, there has to be a copula to act as logical glue, linking it to them; but this copula is in effect another relation—a ternary relation between the original relation and its two relata—and therefore needs a further element to link it to its relata; and so on. Frege had attempted to counter this regress by conceiving of relations as having gaps into which their relata fit without needing any glue. It would be a couple of years before Russell read this account, and even when he did, he rejected it on

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the ground that he could not see how the sort of gap to which Frege appealed made sense. In the meantime, though, he had nothing convincing to put in its place.

The paradox of relativity resolved Russell’s rejection of the doctrine of internal relations was a significant step on his journey away from Hegelian dialectic, but he had not yet reached his destination. In a new book draft, Fundamental Ideas and Axioms of Mathematics, written in the summer of 1899, he continued to hold that there are contradictions in the infinite—‘there is, and is not, a number of numbers’ (CPBR, II, 267)—and in causality—‘each element has an effect, but no effect can be asserted apart from the whole’ (CPBR, II, 271). Quite soon, though, Russell became dissatisfied with this draft and embarked on another (the first to carry the title Principles of Mathematics), which occupied him until the summer of 1900. In this 1899–1900 draft he explained the consequences of rejecting the doctrine of internal relations, most importantly that a partial description of the world may nonetheless be true. Qualities are internal, relations external. If an entity’s complete concept consists only of its qualities, that of the man in India will not mention the death of his wife in Europe, because marriage is external and marital status therefore not a quality. This permitted Russell to abandon the monism which had been the last remaining pillar of his absolute idealism, since he could now explain how predicating something of only part of reality is possible. (Curiously, though, he did not abandon the surrounding mysticism, which recurred at various points in his subsequent writings.) Russell’s new-found realism concerning relations led him at first to conceive of a relation-instance aRb as a complex made up of a, R and b, and this presupposes that a and b are numerically diverse. ‘There can be no relation without at least two terms.’ (CPBR, II, 141) Numerical diversity, he inferred, is logically prior to other relations; in particular, it is prior to, and hence must be distinguished from, qualitative difference (CPBR, II, 286). This showed the error in his previous supposition that every relation is grounded in qualities of its terms. In the course of writing his 1899–1900 draft, he realized that his derivation of the paradox of relativity in An Analysis of Mathematical Reasoning had been wholly due to his use of this supposition as a premiss, and did not survive its abandonment. With elegant economy, therefore, he extracted the relevant pages from his earlier draft and re-used them as a reductio ad absurdum of the doctrine of internal relations (CPBR, III, 93). To say that numerical diversity does not mean the same as qualitative difference leaves open the possibility that the former might still entail the latter. It was left to Moore (1901) to deny this. We can, he said, coherently imagine two things in distinct places with exactly the same qualities. Such things would be distinguishable only externally (by virtue of their differing relations), not internally. Moore

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might have saved himself some time, however, if he had noticed that just this point had already been made by Kant (A263–4=B319–20). Moore concluded from this that things cannot be mere bundles of qualities: this white thing and that are distinct entities which both have the property of whiteness, not two whitenesses. What, though, about the insubstantial entities—entities with no qualities, constituted wholly by their relations—that Russell had in his Essay taken spatial points to be? Now he rejected the whole notion of such entities. ‘If they are to be anything at all,’ he said, entities ‘must be intrinsically something’; they must have a ‘genuine nature of their own’ (PoM, §242). Why, though, did he maintain this? After all, if we accepted Moore’s view that entities may differ without differing in internal properties, why not also admit entities that differ without having any internal properties? What Russell needed at this point, but did not supply, was an argument to show that the notion of an insubstantial entity without internal properties is incoherent. The paradox of relativity was a circularity of explanation that resulted from combining two views: that relations are internal to their relata, and that these relata are, in the case of mathematical entities such as spatial positions, constituted wholly by their relations. To resolve the paradox he needed to give up one of these views, but in fact he gave up both.

Matter and position again The Leibniz-Clarke correspondence concerned a dispute between a substantial conception of space as consisting of substantial spatial positions, each with its own identity, and a relational conception of it as consisting in relations between material points, not spatial positions (so that empty space is inconceivable). In his earlier work Russell had dismissed the latter view, as a necessary condition for conceiving of geometry as prior to physics, and had therefore reconfigured the dispute so as to become one between substantial and insubstantial conceptions of spatial positions. Now that he had rejected the insubstantial conception, for reasons unconnected with geometry, the dispute was restored to its original form. In 1901 Russell therefore reconsidered it in these terms, and now rejected the Leibnizian view (CPBR, III, 270–2). Instead, he held that geometry requires lines that cross to intersect, but that on the relational view the intersection would have to be a material point. Whether there is always such a point in the right place— whether space is a ‘plenum’—was an empirical question, he now thought, which ought not to be answered in the affirmative just to get us out of a logical difficulty. He then considered the idea that in order to preserve the relational account we should posit possible material points as well as actual ones. He granted that this was feasible, but it no longer had any advantage over the non-relational alternative. In a choice between two different sorts of posit—possible material points on the relational view, positions on the non-relational—the latter, he thought, should be preferred on grounds of simplicity.

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Further reading On Leibniz’s views about relations see Mates (1986, ch. 12). Griffin (2013) goes into Leibniz’s influence on Russell’s rejection of the doctrine of internal relations (although I do not agree with his contention that the evidence compels us to place that rejection earlier than January 1899). Humberstone (1996) summarizes ways of understanding the internal/external distinction. The Leibniz-Clarke correspondence (Alexander 1958) remains one of the best introductions to the issue of substantial and relational conceptions of space; Teller (1991) gives a lucid account.

30 PEANO

The most significant of the external events that shaped The Principles of Mathematics occurred while Russell was in Paris in August 1900 to deliver a version of his argument for absolute space at the International Congress of Philosophy. This was, he later said, a turning point in my intellectual life, because I there met Peano. I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation. In discussions at the Congress I observed that he was more precise than anyone else, and that he invariably got the better of any argument upon which he embarked. As the days went by, I decided that this must be owing to his mathematical logic. I therefore got him to give me all his works, and as soon as the Congress was over I retired to Fernhurst to study quietly every word written by him and his disciples. It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years, and that by studying him I was acquiring a new powerful technique for the work that I had long wanted to do. (ABR, II, 147) Peano was an Italian mathematician noted for his work on the theory of differential equations and for demonstrating the existence of space-filling curves. He had been engaged for some years in editing an encyclopaedia called the Formulaire, which rendered large numbers of mathematical theorems in a precise symbolism and sketched proofs of some of them. Russell quickly adopted Peano’s symbolism, from which many of the notations in Principia—such as the use of dots instead of brackets, and the labels ‘Pp’ for primitive propositions and ‘Df ’ for definitions— were derived. It was also Peano, as it happens, who introduced Russell to the device, later taken up by Wittgenstein in the Tractatus, of numbering statements

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with decimals to ease later revision. In the next three chapters we shall examine other, rather less superficial effects on Russell of this, his first encounter with modern logic.

Classes The most immediate effect of Russell’s encounter with Peano (see MPD, 66–7 ) was that he learnt the distinction between membership (for which he adopted Peano’s ‘ε’, since amended to ‘∈’) and inclusion (for which he adapted Peano’s ‘C’, nowadays ‘⊂’). This in turn depends on the distinction between aggregate and class: an aggregate is just the things considered as a single entity, whereas the singleton class {s} is distinct from its unique member s. Even by the time he published the Principles, however, Russell’s account of the difference between a class (‘class-as-one’, as he then called it) and an aggregate (‘collection’) was hesitant. It is usually held that although there is no empty aggregate, there is an empty class (for which we now use ‘∅’ instead of Peano’s ‘3’), but according to Russell this gives rise to the ‘grave logical difficulty’ that ‘a class which has no terms fails to be anything at all: what is merely . . . a collection of terms cannot subsist while all the terms are removed’ (PoM, §73). He therefore concluded that ‘there is no actual null-class’ (ibid.), and that talk of the empty class should be re-interpreted by means of some technical device. What is less clear is what his objection to the empty class really was: one might suspect that he was still thinking in terms of aggregates. Russell further distinguished between the aggregate of the φs, which is singular, and the φs, which are plural: he called the latter a ‘class-as-many’. Being plural, it is not an entity, and so ‘entity’ ceased to be, strictly speaking, the widest word in his vocabulary. This, he observed enigmatically, ‘raises grave logical problems’ (PoM, §58n)—presumably, although he did not elaborate, because it threatens the universality of Peano’s singular logic—but in his later work he made only minimal attempts to pursue the plural logic to which he was here gesturing. The sign ‘∃’ that is used nowadays for the existential quantifier is also derived from Peano, although he did not have this quantifier in his system, and wrote ‘ a’ to mean that the class a is non-empty, thus perhaps helping for a time to obscure from Russell the distinction between quantificational logic and the theory of classes. E

Relations In October 1900 Russell wrote out an account of relations in Peano’s style. (The logic of relations had already been treated extensively by Peirce and Schröder, but they had treated relations as a kind of aggregate.) There were gaps in Peano’s account, no doubt, but did Russell need to take ‘relation’ as well as ‘class’ as a primitive idea? Peano (1897) had instead taken the ordered pair (x, y) as a primitive notion, governed by an axiom

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(x, y) = (z, t) ⊃ (x = z ∧ y = t),

(1)

and treated relations simply as classes of ordered pairs. Russell resisted this, because he thought that it failed to recognize that relational propositions are just as ‘ultimate’ as class-propositions (PoM, §27), and hence that relations-in-intension, not relations-in-extension, should be taken as primary. What he meant, presumably, was that if all relations were reducible to monadic concepts—what Russell sometimes called ‘class-concepts’— that would make them internal and hence contradict the doctrine of external relations to which he now adhered. Russell did much later abandon his suspicion of relations-in-extension, at which point it would have been open to him to reduce them to classes via the primitive notion of the ordered pair, but he did not do this, instead giving in Principia a laborious parallel treatment. The alternative approach of reducing relations to classes of ordered pairs was significantly simplified when Wiener, briefly Russell’s student at Cambridge, showed (1914) that we need not take the ordered pair as a separate primitive but can instead define (x, y) = {{{x}, {∅}, {{y}}}}, from which (1) follows as a theorem. Later still, Kuratowski (1921) showed that the same holds for the even simpler definition (x, y) = {{x}, {x, y}}. Oddly, though, Russell showed no ‘particular approval’ of this innovation (Wiener 1953, 191) and omitted to mention it in the second edition of Principia at all.

The 1900 draft of the Principles Russell now rewrote what would become parts III–VI of the Principles using what he had learnt from Peano. On what he pedantically described as ‘the last day of the century’ (because he maintained that 1900 was in the 19th century, not the 20th) he told Helen Thomas (his wife’s cousin) that the book was complete. He often boasted (usually to women) about his productivity, and on this occasion he proudly claimed to have written 200,000 words since October. This was an exaggeration—the version he wrote in the last three months of 1900 had, in conformity with his usual practice, made significant use of pages from previous drafts—but even so he had undoubtedly undertaken one of the prodigious bouts of productivity of which he was periodically capable. Usually, too, Russell was supremely confident at the end of these prolific periods of the quality of what he had done. On this occasion he told Thomas that he had ‘invented a new subject’ and as a result ‘for the first time treated [mathematics] in its essence’. Perhaps, though, he was overstating the novelty of what he had done. One of the technical achievements of the late 19th century had been to plug the gaps in the proofs of the theorems of geometry, so that they

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did not depend on geometrical intuition, but Russell had played no part in this. Nor had he invented the rigorous development of the differential and integral calculus from the axioms for a complete ordered field: this had been done by Weierstrass in the 1850s and published by others since then. There remained, both for geometry and for the calculus, the question of how to formalize the newly rigorous proofs. This could not be done using the monadic logic Russell had experimented with in 1898, but Peano’s Formulaire suggested in outline how to achieve it. There were no doubt many gaps in his account, but Russell had not yet plugged them. Nor had he invented what is nowadays called the arithmetization of the real numbers—the construction of a complete ordered field, using the natural numbers as building blocks—which had been done in different ways by Dedekind, Weierstrass and others. Nor, finally, had he invented the rigorous treatment of arithmetic by means of an axiomatic characterization of the natural numbers, which, as he knew, had been done by Dedekind in Was sind und was sollen die Zahlen? (1888).

Logic and abstraction The one step in the deductive chain on which Russell might plausibly have taken himself to be saying something new was the claim that the initial building blocks, the natural numbers, can in turn be constructed as logical objects. In the previous draft of the Principles he had said it was ‘reasonable to suppose that numbers are indefinable’ (CPBR, III, 18). To overturn this supposition, he needed to define a function Card such that Card(a) = Card(b) ≡ a is equinumerous with b. In more general terms, the problem was to derive from any equivalence relation R a function 6 satisfying the ‘principle of abstraction’, 6(a) = 6(b) ≡ aRb. Now that his treatment of the natural numbers proceeded via an instance of this principle, it was crucial to settle whether its status was any different from that of Peano’s axioms: if not, then no great advance would have been secured by the new account. In the October 1900 draft of his article on relations Russell called the principle of abstraction a ‘primitive proposition’, by which he presumably meant a logical axiom, but his only gloss on this was the regressive argument that it was presupposed in the procedure of definition by abstraction, which could not be valid without it. This proposition states that, in all cases where definition by abstraction is formally allowable, there exists the entity which is defined. (CPBR, III, 593)

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This plainly falls well short of an independent argument for the truth of the principle, since it presupposes that the procedure of definition by abstraction is valid. Russell also worried that even if there is a function 6 with the required property, its lack of uniqueness disqualified the principle of abstraction from furnishing a definition of cardinality, and he had therefore been right to think that number is indefinable. This was in essence Frege’s Julius Caesar problem, although he never formulated it in quite these terms. Soon, however, he realized that if we put 6(x) = {y : xRy}, the principle of abstraction follows straightforwardly. It is not known exactly when he saw how to do this, but the proof is in the revised version of the article which dates from January or February 1901. This is an early instance in Russell’s work of what would later become a fruitful method, whereby logical constructions (generally classes of some kind) are substituted for entities that would otherwise have to be inferred. Of course, this proof deals only with the existence problem and not with uniqueness. Perhaps the naturalness of the definition silenced Russell’s worries on the latter score, although it is far from clear that it should have done so. Russell’s definition of cardinality was now, allowing for differences in terminology, essentially the one Frege had proposed in the Grundlagen: the cardinal of a class is the class of all classes equinumerous with it. Russell later claimed that in 1901 Frege’s writings were still ‘unknown’ to him (CPBR, III, 368), but this is not quite true: in the autumn of 1900 he mentioned them several times (CPBR, III, 207, 353, 354), and even lamented (CPBR, III, 352) that they were not better known. Russell now held that we can derive the objects mentioned on the left-hand side of the principle of abstraction from the relation on the right. In contrast, his argument for absolute space treated the left-hand side (in this case positions in space) as primary and the relation on the right (distance) as derived; similarly his account of quantity treated the left-hand side (quantities) as primary. At the proof stage Russell attempted to resolve this tension in the Principles by claiming that the notion of quantity does not really belong to pure mathematics. His whole discussion of it (Part III of the book) was merely, he said, a concession to tradition; for quantity, as we shall find, is not definable in terms of logical constants, and is not properly a notion belonging to pure mathematics at all. (PoM, §150) Numbers, that is to say, are definable in purely logical terms, whereas quantities (and positions in space) are not, and hence do not belong to pure mathematics. That he should have set quantities to one side in this manner was in marked contrast to his attitude during his idealist period, when he thought that the relation of number to quantity is one of the ‘most fundamental questions of mathematical philosophy’ (CPBR, II, 70).

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Further reading Levine (1998b) details Peano’s influence on Russell and the tension in his philosophy of mathematics to which it led. For a defence of Russell’s hesitation concerning the empty class see Oliver & Smiley (2006).

31 EARLY LOGICISM

In the form in which the Principles existed at the end of 1900, the book left unaddressed the status of the foundations on which its account of mathematics depended. By the time it was published, however, it argued for logicism concerning mathematics.

Russellian logicism Russell first announced his logicism in a popular article written in January 1901. Though there are indefinables and indemonstrables in every branch of applied mathematics, there are none in pure mathematics except such as belong to general logic. . . . All pure mathematics—Arithmetic, Analysis, and Geometry—is built up by combinations of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic. (CPBR, III, 366–7) There are two distinct claims here: that the subject matter of pure mathematics is built up from logical notions; and that its theorems are provable using logic alone. The claim that pure mathematics can be expressed in a precise formal ideography had been made already by Peano, but he denied that mathematics is part of logic, because he held (1891) that its primitive notions (such as the successor function in arithmetic) are indefinable. The first component of Russell’s logicism, then, was the claim that the formalization of pure mathematics requires no nonlogical primitive signs. The second concerned not expressibility but proof: it was the claim that once the concepts of mathematics were defined in logical terms,

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the theorems would be converted thereby into theorems of logic. This second component was certainly not achieved in Peano’s Formulaire, which did not aspire to Frege’s ideal of gaplessness. Thus far, Russell’s logicism resembled Frege’s. He wished, however, to apply the position not only to arithmetic and analysis, as Frege had done, but also to Euclidean and non-Euclidean geometry (although not, at least when he was speaking carefully, to applied mathematics). He therefore weakened it to the merely conditional claim that pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. (CPBR, III, 366) Thus Russell’s famous summary, ‘Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.’ (Ibid.) It is this conditional view that he repeated, rather less glibly, as the opening of Part I of the Principles, the first draft of which he wrote a few months after the popular article. ‘Pure mathematics’, he now said, is the class of all propositions of the form ‘p implies q’, where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. (PoM, §1) To make sense of this, however, we need to make two charitable emendations. First, by the time the book was published the phrase ‘propositions containing one or more variables’ was out of date: what he had previously called a ‘proposition containing a variable’ he in May 1902 renamed a ‘propositional function’ (see Beaney 2009). (He later suggested, somewhat misleadingly, that his imprecise formulation ‘may be excused on the ground that propositional functions had not yet been defined’ (1937, vii).) Second, when he said ‘all propositions’, he meant all true propositions. By broadening Frege’s logicism to include geometry as well as arithmetic within its scope, Russell had also weakened it to become what in Chapter 19 I called deductivism. It was clear, he said that Euclidean and non-Euclidean systems alike must be included in pure mathematics, and must not be regarded as mutually inconsistent; we must, therefore, only assert that the axioms imply the propositions, not that the axioms are true and therefore the propositions are true. (1937, vii) Notice, though, that the Dedekind-Peano axiomatization of arithmetic already allows us to present arithmetic in conditional form. So the version of logicism which Russell adopted in 1901 did not in fact depend on the success of the

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logical construction of the natural numbers as equivalence classes that he had been working on the previous autumn. What that construction does do, of course, is to prove the consistency of the protasis of the relevant conditionals; but Russell did not explain why anything other than their non-triviality should hinge on this. Notice, too, that although Russell’s logicism was conditional, and hence did not in itself demand that there should be any purely logical entities, he nonetheless committed himself to the subsistence of a variety of such entities, most notably logical constants, variables, and classes. He conceived of these as supplying a subject matter for logic somehow parallel to the existent entities that constitute the subject matter of physics.

Characterizing logic Russell’s study of Peano gave him no hint that the structure of propositions might be interestingly different from that of the sentences of the formal language which express them. Peano was a mathematician, not a philosopher, and generally abstained from expressing philosophical views, but one of the few that he did express concerned the goal of his ideography. It is not, he said, just a conventional abbreviated writing, or tachygraphy, since our symbols do not represent words but ideas. For this reason one must write the same symbol wherever the same idea is found, whatever the expression used in ordinary language to represent it, and distinct symbols must be used where the same word is found when, because of its position, it represents a distinct idea. In this way we establish a one-to-one correspondence between ideas and symbols, a correspondence which is not found in our ordinary language. (1973, 191) Peano claimed in consequence that ‘there cannot be two substantially different ideographies’, since the structure of any ideography must mirror that of thought. Eliminating the ambiguities of natural language had of course been one of the aims of Frege’s concept-script in 1879, and it was he who coined the phrase ‘logically perfect language’ (CP, 169) to describe the result, but he notably refrained from claiming that this structure is straightforwardly identical to that of a sentence either of his concept-script or of any plausible refinement thereof. Russell, on the other hand, tended to assume that a sentence in his Peano-inspired ideography accurately represents the structure of the proposition it expresses. Although Russell now thought of logic as expressed symbolically, then, he regarded the symbolism as ‘merely a theoretically irrelevant convenience’ (PoM, §11). Logic is no more about the language in which it is expressed than chemistry is. Nor is it about the meanings of words: ‘Meaning, in the sense in which words have meaning, is irrelevant to logic.’ (PoM, §51) Rather, it is about the entities words such as ‘not’, ‘all’, etc. refer to. It therefore has as its subject matter certain abstract entities, with which we are somehow capable of coming into an

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epistemic relation. Consequently, it ‘is just as synthetic as all other kinds of truth’ (PoM, §433). Given his lack of interest in the symbolism, it is unsurprising that Russell showed little interest in metalogical reasoning. He was familiar with the idea of reinterpreting a formal system, and was not averse to using this as a route to results about the system, but he did not regard such results as impacting on logic. Indeed, he regarded the method of proving axioms independent by supposing them false to be ‘subject to special fallacies’ (PoM, §17). Russell was disinclined to characterize logic in terms of its grounds, because he took these, following Moore’s attack on his earlier Kantianism, to be subjective. When Russell first adopted logicism, he characterized it instead by its generality. It may be defined, he wrote in 1901, as ‘the study of what can be said of everything, i.e. of the propositions which hold of all entities’, together with the study of the constants which occur in such propositions (CPBR, III, 187). There are two obstacles to rendering this precise, however. First, it is not obvious how a proposition can be about everything. In Peano’s logic, what makes a proposition wholly general is the presence in it of a variable. So the conception of a proposition as containing the entities it is about, which Russell and Moore had adopted in 1898, required revision. ‘The notion of the variable is one of the most difficult with which Logic has to deal, and in the present work a satisfactory theory as to its nature . . . will hardly be found.’ (PoM, §6) The second obstacle is that the propositions of logic do not contain only variables: we need an explanation of why the occurrence in a proposition of an ordinary concept such as man or good deprives it of the generality characteristic of logic, and yet the same is not true if it contains a logical concept such as negation or implication. Russell did try, unsuccessfully, to formulate a criterion for identifying the logical constants; but in the final version of the Principles he was forced to concede that they ‘are to be defined only by enumeration’ (§10). Having said this, though, he then rather oddly omitted the enumeration, contenting himself with the offhand remark that there appear to be ‘eight or nine’ (PoM, §12). Even if he had enumerated them, though, his account would still have been inadequate, because a mere list is not an explanation of why just these concepts may occur in a proposition without its generality being thereby restricted. Although Russell would probably have agreed readily enough with the correction I made earlier to his definition of pure mathematics, namely that he ought to have described it as consisting of true conditionals, he would not have agreed that he needed to say ‘logically true’. This is because he thought that the word ‘logic’ picks out a special kind of proposition, not a special kind of truth: a logical truth is just a logical proposition that is true (CPBR, III, 187); logicality was for him a property of propositions, not, as for Frege, of their grounds. Now, though, his failure to characterize the logical constants except by enumeration was all the more egregious, since it was in terms of them that he characterized logical truth.

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Knowledge of logic According to Russell the nature of logic’s subject matter entails that its truths are a priori. ‘The fact that all mathematical constants are logical constants, and that all the premisses of mathematics are concerned with these, gives . . . the precise statement of what philosophers have meant in asserting that mathematics is a priori.’ (PoM, §10) This supplied his early logicism with one of its motivations, namely to explain how it is possible to know mathematical truths. We must obviously be careful how we state this, though, if we are not to suggest that previous generations knew no mathematics. Are we to take the logicist definition of the number two as the class of all two-membered classes literally, as telling us what we really meant by ‘two’ all along, or more weakly, as only an adequate proxy? In his 1901 article Russell did indeed say, provocatively, that ‘pure mathematics was discovered by Boole’ (CPBR, III, 366), but by the time he sent the Principles to the publisher in May 1902 he distinguished more cautiously between formal and philosophical definability: only formally, he suggested, should numbers be taken to be classes of equinumerous classes, whereas ‘what philosophy and common sense recognize as numbers’ are ‘indefinable entities’ (Byrd 1987, 69). At the proof stage, though, Russell deleted this passage (bolstered, perhaps, by having now read Frege) and replaced it with the straightforward claim that numbers are classes of equinumerous classes. He did not at this stage show much interest in the paradox of analysis—unlike Moore, who took him to task on this point in his (still unpublished) review of the Principles. Russell thus held that logicism could transfer the a priori status of logic to mathematics, but he did not have much to say about why logic is a priori. He presumably thought that acquaintance with a logical constant is not an experience, but he was conspicuously silent about what it might be instead. He expressed the hope that he might enable his readers to see the indefinables of logic clearly, ‘in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of pineapple’ (PoM, xv), but it is questionable whether he succeeded. Perhaps his failure to offer an account of the epistemology of logic was of a piece with his inability to characterize its scope: one might well think that the latter is a precondition for the former. Some commentators (e.g. Hylton 1990a) stress Russell’s claim that his logicist hypothesis dealt ‘a fatal blow to the Kantian philosophy’ (CPBR, III, 379), but in truth he had already left Kant behind before he adopted it, as his later description of it as a ‘parenthesis in the refutation of Kant’ (CPBR, XI, 12, my emphasis) suggests. Moreover, although his logicism certainly required him to disagree with Kant, the nature of the disagreement was not the same as Frege’s: Russell did not regard logicism as contradicting Kant’s view that mathematics is synthetic. For many of Kant’s opponents, the crucial point was whether mathematics depends on intuition, but in order to place Russell in this tradition we would need to hear more than he had yet said about what the ground of our knowledge of logic is. At least as important to Russell was logicism’s role in refuting the empiricists. They

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held that any application of mathematics would be an ‘approximation to some exact truth about which they had nothing to tell us’ (PoM, §3), whereas logicism, he hoped, would show that mathematics is exactly true. For a few happy months in 1901, indeed, logicism did seem to him to vindicate mathematics as a body of certain knowledge. In the whole philosophy of mathematics, which used to be at least as full of doubt as any other part of philosophy, order and certainty have replaced the confusion and hesitation which formerly reigned. (CPBR, III, 369) In May 1901, however, Russell discovered the contradiction (to be discussed in Chapter 33). One might expect that this would have muted his confidence in logic’s epistemic reliability. In ‘The study of mathematics’, written in October 1902 but not published until 1907, he still claimed optimistically that ‘the edifice of mathematical truths stands unshakeable and inexpugnable to all the weapons of doubting cynicism’; it refutes those who say that ‘there is no absolute truth, . . . only truth for me, for you, for every separate person’ (CPBR, XII, 92–3). His ‘retreat from Pythagoras’ (MPD, 208)—from the hope that logic would confer certainty on mathematics—was by no means immediate.

Further reading Russell’s ‘Recent work on the foundations of mathematics’ (later reprinted under the title ‘Mathematics and the metaphysicians’) is a good introduction to his early logicism, despite its irritatingly facile tone. I have argued here, in agreement with Proops (2006), for the view that the advantages Russell originally saw in logicism were primarily epistemological. We shall see in Chapter 42 that he later diverged from this view of logicism as conferring an epistemic gain. For a discussion of the view that he never really regarded it in this light, see Hylton (1990a). For more on the sense in which he regarded logic as universal see Proops (2006).

32 DENOTING CONCEPTS

In May 1901 Russell wrote, and a year later he re-wrote, a new Part I of the Principles largely devoted to a philosophical account of Peano’s logic.

The variable Central to Peano’s logic were what Russell (PoM, §15) called the material and formal conditionals. Like Peano, he used an inverted letter ‘C’ (which later turned into the symbol ‘⊃’ still widely used today) for the material conditional, and subscripted it with a variable to denote a formal conditional, so that ∀x(φx ⊃ ψx) was written as φx⊃x ψx. Should this ‘formal conditional’ be understood, though, as the relative quantifier which in Chapter 2 we wrote as A(φ, ψ), or as a compound expression obtained by first forming the material conditional φx ⊃ ψx and then applying unrestricted quantification to that? Russell correctly noted the practical disadvantage of the former reading, namely that it prevents us from appealing to the properties of the material conditional when deriving the properties of the new notation, and for this reason he settled on the latter. Why, though, does it matter either way? On Frege’s view the two readings differ only in tone, not conceptual content (or, in his later terminology, sense), since we can draw just the same logical inferences from either; Russell, by contrast, maintained (PoM, §41) that although materially equivalent—both true or both false—the two readings differ in content: the former only says something about φs, whereas the latter says something about each thing there is (namely that it is either a non-φ or a ψ). Significantly, this prevented him from subsuming the restricted quantifier phrases ‘every φ’ and ‘some φ’ under the unrestricted phrases ‘everything’ and ‘something’; instead he treated the latter as special cases of the former, making his account of ‘φx ⊃x ψx’ subsidiary to an account of the general notion of asserting a propositional function.

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This in turn obscured from Russell the role of scope in connection with generality, since in asserting a propositional function the scope of the variable is automatically the whole formula. Of course, if he had focused carefully on how Peano’s notation operates, he might still have noticed that when ‘φx ⊃x ψx’ occurs within a more complicated proposition, the scope of x coincides with that of the implication symbol which it subscripts, but even this would not have been sufficient to persuade him that scope is relevant to the analysis of ‘Every φ is ψ’, because he held that this expresses a different proposition. Russell also followed Peano (1897) in distinguishing between ‘real’ and ‘apparent’ occurrences of a variable (nowadays called ‘free’ and ‘bound’ respectively), and in at first calling φx a ‘proposition containing a variable’ (strictly, he should have said ‘a real variable’). As we noted in the last chapter, though, he switched in May 1902 to calling this a ‘propositional function’. Just as ψa was for Russell a complex containing the entity a, so by parity of reasoning ψx was a complex containing the variable x. For him, therefore, the variable was not a letter but what that letter indicates—a logical entity capable of occurring in a proposition. One consequence was that his understanding of quantification was referential, i.e. at the level of the world, not of symbols. Another was that he was not tempted to assimilate the concept, redhead, to the propositional function, x is a redhead: the former is also a component of the proposition, Socrates is a redhead, but the latter is not. What, though, if we simply remove Socrates from the proposition without replacement? Russell called the result an ‘assertion’ (an awkward choice of terminology, since he also used ‘assertion’ more familiarly for the act of assenting to the whole proposition). This is of course reminiscent of Frege’s notion of unsaturatedness but with the significant difference that if several entities are removed, Frege held that the various gaps retained distinct identities (marked by means of Greek letters in the metalanguage), whereas Russell (PoM, §482) found this unintelligible. On Frege’s account, he thought, the identity function could not be an entity, since it would be ‘all gap’, as it were. He soon lost interest in this notion of assertion, however, when he saw that it did nothing to respond to the need Frege’s notion was attempting to address, namely that of explaining the difference between ‘x kills x’ and ‘x kills y’. Russell’s whole discussion was marred, however, by his apparent lack of a criterion for settling which distinctions matter to logic and which do not. To the modern eye, he made four clear mistakes: he treated the variables occurring in propositional functions not as mere symbolic devices but as genuine entities; he individuated propositions more finely than was required to explain inference; he failed to appreciate the significance of the notion of scope; and he assumed too readily that a concept may occur in subject position. On all these points Frege had done better. Russell had been given a copy of Begriffsschrift by Ward (probably in 1896, although the exact date is uncertain), but evidently had not grasped its importance.

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Indefinite denoting concepts Russell still needed an account of ‘every φ’ and ‘some φ’ for an arbitrary concept φ. There might still have been a way for him to adopt at least some elements of the Fregean account if he had regarded these as complex concepts arrived at by applying the functions every and some to the concept φ, but he rejected this. Although ‘ “all men” and “all numbers” have in common the fact that they both have a certain relation to a class-concept, namely to man and number respectively’, they are not ‘validly analysable’ into all and men or all and numbers, he thought, because ‘it is very difficult to isolate any further element of all-ness which both share, unless it be the mere fact that both are concepts of classes’ (PoM, §72). His thinking here seems similar to Moore’s when arguing against idealism: the only sort of analysis he considered was one according to which ‘all φ’s’ is some kind of sum of ‘all’ and φ. If, as Russell had held since 1898, the proposition ‘I met John’ contains John himself, what about ‘I met some man’? It cannot simply contain John, even if he is the man I actually met, since that would make it the same proposition, which it plainly is not. Russell therefore posited an entity, meant by the phrase ‘some man’, which occurs in the latter proposition in the position that John occupies in the former. He called this entity a denoting concept; the relation between it and the man himself he called denoting. Similarly, there is a denoting concept ‘every man’. Denoting concepts do not exist, in the sense in which Russell used that word in the Principles: they are not temporal but logical entities whose sole role is to occur in propositions. ‘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.’ (PoM, §56) At this point, however, Russell complicated his theory by treating ‘every φ’ and ‘some φ’ as part of a larger group of concepts including also ‘any φ’ and ‘a φ’. He then tried to use the different manners in which these concepts denote to explain the difference in meaning that he detected between, for instance, ‘There is a prime number greater than any number’ and ‘There is a prime number greater than every number’, or between ‘Every woman loves a man’ and ‘Every woman loves some man’. He proposed in effect a revival of the mediaeval theory of supposition, but he seems to have been unaware of this, claiming incorrectly that he was treating ‘a subject upon which logicians, old and new, give us only the scantiest discussion’ (PoM, §58). As if the theory was not already elaborate enough, Russell then complicated it even further by considering also ‘all φs’, which corresponds in the finite case to collective rather than distributive predication. He proposed to deal with this by saying that f (all φs) predicates f of the class of φs. He then suggested that the other cases be dealt with similarly, i.e. by positing different kinds of combinations of the φs to serve as denotations. Confusingly, then, two competing theories remain visible in the final version of the book: according to the first, there are just the φs, but various denoting relations—various ways concepts may be related to the φs; according to the

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second (PoM, §59), there is just one denoting relation, but there are many ways of combining the φs to form entities capable of being denoted. Russell regarded ‘everything’ as the particular case of ‘every φ’ in which φ is the concept ‘entity’. The same ambiguity between two competing accounts is visible once again, however: at one point he said that the variable is a distinct entity denoted by the concept any term; at another he cautioned, in seeming contradiction to this view, against there being, in addition to all the numbers, a mysterious further entity denoted by any number.

Definite denoting concepts In amendment to Russell’s 1898 conception, there were now two ways in which a proposition could be about an entity: directly, by containing the entity itself; or indirectly, by containing a denoting concept which denotes it. There was then a corresponding distinction at the level of language between two kinds of singular term: a proper name refers directly to an object; a denoting phrase means a denoting concept and denotes (if at all) only indirectly via the concept. In his May 1901 draft Russell treated only indefinite denoting concepts: every φ, any φ, some φ, a φ, all φs. When he revised the material the following year, he added to the list the definite denoting concept, the φ. Denoting may relate an indefinite denoting concept to various objects or, he now held, it may relate a definite denoting concept to a single object. One advantage Russell saw in this account was that it explained the informativeness of identity statements. If we say ‘Edward VII is the King’, we assert an identity; the reason why this assertion is worth making is, that in the one case the actual term occurs, while in the other a denoting concept takes its place. (PoM, §64) The role of definite denoting concepts in his theory was in this respect like that of senses in Frege’s, except that Russell applied his theory only to descriptions and not, as Frege had done, to all singular referring expressions. When, a few months later, he got round to reading ‘On sense and reference’ in order to write an appendix to the Principles on Frege’s work, he noted this difference, but did not pause to consider whether it was a point in Frege’s favour. A further virtue of his theory of denoting, Russell thought, was that it explained our ability to talk about what is not there. ‘A concept may denote although it does not denote anything.’ This occurs ‘when and only when “x is an a” is false for all values of x’ (PoM, §73). ‘The present King of France’, for example, means a denoting concept which does not denote anything. In the Principles, however, the idea that denoting might avoid the need for an ontology of non-beings remained unexplored. The material on definite denoting concepts was one of the last things added to the manuscript before he delivered it to

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his publisher in June 1902, and he left unaltered from the earlier version his commitment to the ‘queer zoo’ of non-existent things.

Epistemology Once the Principles had appeared, Russell returned to the issue of singular terms that do not denote. He now suggested that in fiction what ordinary grammar treats as proper names are only ‘substitutes for descriptions’, and that this explains how we can talk about chimeras or Greek gods. In a manuscript of 1903 (two years, therefore, before ‘On denoting’), he said that ‘Apollo’ is not a proper name like Aeschylus. . . . Imaginary proper names are really substitutes for descriptions. . . . A phrase which has meaning, and unambiguously denotes an individual, is always, in its logical essence, though not necessarily in linguistic form, compounded of a phrase which only has meaning, together with the word the. Thus when we ask: ‘Was there such a person as Homer?’, the word Homer is a synonym for ‘the author of the Iliad and the Odyssey’. (CPBR, IV, 285) This proposal that grammar and logic might differ in respect of what they categorize as names turned out to be one of Russell’s most fertile (but also most contested) ideas. He did not himself call a singular term which is thus replaceable a ‘disguised description’, as is nowadays common: the nearest he came was to call it a ‘truncated or telescoped description’ (CPBR, VIII, 213). Nor did he initiate the practice of calling one that resists this treatment a ‘logically proper name’: this phrase was coined by Stebbing (1930, 25). Russell took it to be a truism that we can make a judgment only if we have some grasp of the entities that compose it. In 1900, for instance, he wrote that ‘whatever can form part of a judgment which we make must be the object of one of our ideas’ (CPBR, III, 229). Later he gave the name ‘acquaintance’ to the relation that is required in order to grasp the object, after which he called the truism the ‘principle of acquaintance’. Already in the Principles he gave denoting a distinctive role in enabling us to make a judgment concerning something that is not in this sense part of the judgment: the domain of the denoting relation contains the entity we apprehend (the denoting concept); its range contains the entity or entities we are thinking about (the denotation). The relation thus straddles an awkward epistemological gap and explains the ‘inmost secret of our power to deal with infinity’ (PoM, §72). We finite beings cannot grasp the infinite directly, but only via the mediacy of a denoting concept. ‘The logical purpose which is served by the theory of denoting is, to enable propositions of finite complexity to deal with infinite classes of terms.’ (PoM, §141) A proposition about the natural numbers, for instance, contains not the numbers themselves, but the denoting concept all numbers—a finite entity whose role is to point to all the infinitely many numbers that the proposition is about.

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Although denoting thus played a central role in Russell’s epistemology, he needed, lest it be infected by the psychologism of Bradley’s universal meanings, to insist (PoM, §56) that it is a logical relation. Meaning, in the sense in which words have meaning, is irrelevant to logic. But such concepts as a man have meaning in another sense: they are, so to speak, symbolic in their own logical nature, because they have the property which I call denoting . . . Thus concepts of this kind have meaning in a non-psychological sense. (PoM, §51) So the epistemological role of allowing us to think about what would otherwise be inaccessible to us was not, as Levine (1998a, 422) has claimed, the ‘primary reason’ for Russell to introduce denoting concepts, but rather a grace of fate: the epistemological gap which denoting concepts filled may be a by-product of our psychological limitations, but their logical role is not. Of course, there remains the question of whether a thinker not subject to our finite limitations would have any need of denoting concepts. Such a being could, for instance, directly grasp a proposition containing all the numbers without going via the denoting concept all numbers. This proposition would itself be infinite, it is true, but Russell explicitly declined (PoM, §141) to rule out such propositions in principle.

Further reading Cartwright (2005) raises interesting issues concerning Russell’s notion of propositional function. Bostock (2009) offers a clear analysis of the inconsistencies in his early theory of indefinite denoting concepts, while Geach (1962) usefully compares his account with mediaeval treatments.

33 THE CONTRADICTION

Although he did not write his famous letter to Frege until May 1902, Russell discovered the paradox that now bears his name a year earlier, while writing the first draft of Part I of the Principles.

The paradoxes The paradox stemmed from Cantor’s theorem of 1874 that the cardinality of the power-class (class of all sub-classes) of a class a has a greater cardinality than a. In particular, if a is the universal class, its power-class has a greater cardinality than it; but this is absurd, since the former is a subclass of the latter. Russell noticed this in December 1900, but at first (SLBR, I, 211–12) thought merely that it refuted Cantor’s theorem—even though he had identified no specific error in its proof. In May 1901, however, he re-examined the proof. Cantor had argued that if f is any one-to-one function from the power-class of a to a, we can let b = {x ∈ a : ∃w(x = f (w) ∧ x ∈ / w)}, and then deduce f (b) ∈ b ≡ ∃w(f (b) = f (w) ∧ f (b) ∈ / w) ≡ f (b) ∈ / b (since f is one-to-one), which is a contradiction. To obtain the paradox in its now-familiar form, Russell had merely to consider the particular case in which a is the universal class and f is the identity function. There is of course a certain irony in the position Russell now found himself in. Having begun the Tiergarten programme of identifying the inevitable contradictions in the various theories of mathematics, he had slowly and painfully realized

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that they were all really just misunderstandings. Yet no sooner had he replaced the dialectic of the sciences with a logicist programme for deriving mathematics from logic than he found a contradiction in the latter. It is notable, then, that he did not revert to Hegelianism as a consequence, but insisted robustly that the contradiction ‘demands an answer’ (PoM, §105); nor was he tempted to abandon logicism, but noted enigmatically that ‘it is the distinction of logical types that is the key to the whole mystery’ (§104). To make matters worse, Russell soon realized that the problem is not restricted to classes: the notions of class-concept, propositional function and assertion generate similar paradoxes. For instance, the class-concept ‘not predicable of itself ’ is predicable of itself if and only if it is not (CPBR, III, 195). In the revised version of Part I of the Principles, written in May 1902, he therefore suggested that such apparently paradoxical notions do not give rise to ‘distinguishable entities’. The φ in φx is not a separate and distinguishable entity: it lives in the propositions of the form φx, and cannot survive analysis. . . . If φ were a distinguishable entity, there would be a proposition asserting φ of itself, which we may denote by φ(φ): there would also be a proposition notφ(φ), denying φ(φ). In this proposition we may regard φ as variable; we thus obtain a propositional function. . . . Can the assertion in this propositional function be asserted of itself? . . . If it can be asserted of itself, it cannot, and if it cannot, it can. This contradiction is avoided by the recognition that the functional part of a propositional function is not an independent entity. As the contradiction in question is closely analogous to the other, concerning predicates not predicable of themselves, we may hope that a similar solution will apply there also. (PoM, §85) The danger of this sort of strategy, though, was that it would cripple logicism by in effect restricting logic to the first order. Elsewhere in the book, therefore, Russell canvassed (§103) a less draconian proposal that propositional functions may sometimes be the values of variables, but the account he offered was, to say the least, obscure. He was reluctant to abandon the notion of an unrestricted variable, without which, he said, mathematics would be ‘abolished at one stroke’ (§105), but he had little to offer by way of a positive proposal.

Types In the preface to the Principles (vi) Russell announced a second volume, to be written jointly with Whitehead, in which the derivation of the foundations of pure mathematics from logic would be presented. Before writing this second volume, however, they had to find a formalism which excluded the paradox. Whitehead thought it unlikely ‘except by a miracle’ that they would do this without ‘a philosophical analysis of the subject’: ‘the better the analysis, the better the P[rimitive] p[roposition]s.’ (30 Apr. 1904, in CPBR, IV, xxxiv) He thus aspired

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to a diagnosis of the paradoxes, not just a way of avoiding them. Each of the proposals they considered had disadvantages, however. The surviving manuscripts bear witness that when Russell claimed to ‘have tried about a hundred hypotheses’ (CPBR, IV, 77), he was not exaggerating greatly. Until 1907 his life was dominated by a series of attempts to ‘solve the contradiction’, in the course of which he oscillated between the ‘philosophical’ approach recommended by Whitehead and the more ‘technical’ or ‘formal’ one of finding a system which seems to deliver mathematics efficiently while somehow avoiding the contradictions. If he was not to restrict himself to first-order logic, Russell needed to distinguish the case when a propositional function ‘lives in’ a proposition (or occurs ‘as meaning’, to use his later terminology) and the case when it is the logical subject (or occurs ‘as denotation’). For the latter case he tried several different notations: the one he eventually settled on, φ xˆ , was in fact suggested to him by Whitehead in April 1904. Nowadays logicians use for this purpose Church’s lambda-notation, λxφx, which lends itself better to indicating scope. One of the first proposals Russell considered, in the autumn 1902, was to divide entities into ‘types’, which are the ranges of significance of propositional functions. If we still permitted propositional functions with unrestricted ranges, though, we could define W (φ xˆ ) =df ∼φ(φ xˆ ), and hence obtain the contradiction W (W xˆ ) ≡ ∼W (W xˆ ). Russell proposed to block this paradox by stratifying propositional functions into types according to the types of their variables, so that W xˆ is of the next type above, and hence not a legitimate instantiation of, its argument. Thus far, Russell was merely following Frege’s division of concepts into levels in accordance with the types of their argument-places. On this ‘simple’ theory of types propositions, being in effect degenerate propositional functions, all fall into a single logical type. Soon, however, Russell discovered a further ‘propositional’ paradox—nowadays known as the Russell-Myhill paradox following its rediscovery by Myhill (1958)—to which the simple theory of types is susceptible. For any class m of propositions let pm be the proposition (∀p ∈ m)p. Then pm = pm0 ⊃ m = m0 .

(1)

So if we let w be the class of those propositions pm such that pm ∈ / m, then pw ∈ w ≡ pw ∈ / w, which is a contradiction. Notice, though, that the implication (1), on which the contradiction depends, follows from the definition of pm only because of Russell’s finely-grained individuation of propositions. If we individuate propositions extensionally, as in Frege’s coarser-grained hierarchy, the simple theory of types is not threatened, and the paradox shows merely that there cannot be a one-to-one function from classes of propositions to propositions. Russell, however, felt unable to take this course. Instead, the moral he drew from the Russell-Myhill paradox was that the totality of all propositions involves ‘a fundamental logical difficulty’. To solve it we might hold that ‘propositions

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themselves are of various types’, but this, he said, ‘seems harsh and highly artificial’ (PoM, §500). What he found implausible was not so much that entities should divide into types but that propositions should. Although he sketched the theory of types in an appendix added to the Principles late in 1902, and was still considering some variant of it in late 1903 (CPBR, IV, 52), in 1904 he focused instead on the alternative strategy of restricting which propositional functions give rise to classes. One side effect was that he re-examined the logic of sentences containing denoting phrases which do not denote anything. The example he used—‘The present king of France’—has since become famous. In April 1904 he wrote to his wife, Alfred and I had a happy hour yesterday when we thought the present King of France had solved the Contradiction; but it turned out finally that the royal intellect was not quite up to that standard. (Griffin 1992, 269)

Reducibility Russell’s challenge was now to come up with a general criterion to decide which propositional functions are ‘reducible’, i.e. give rise to a class. In 1904 he considered the ‘zigzag’ theory, already tentatively mentioned in the Principles (§103), which held that a propositional function is reducible provided that its syntactic form is ‘simple’ (in some sense to be determined); non-self-membership then offends by being in the relevant sense too complex. One notable feature of this theory is that if a propositional function is reducible, its negation is reducible too, and so every class has a complement. The modern inheritor of this idea is Quine’s system NF, in which a propositional function is reducible just in case it is possible to decorate the variables occurring in it with type markers in such a way that the function becomes legitimate in the theory of types. The difficulty with such a one-step-back-from-disaster strategy, whether in Russell’s implementation or in Quine’s, is that our confidence in its consistency seems wholly inductive. The axioms it generates ‘have to be exceedingly complicated, and cannot be recommended by any intrinsic plausibility’. Russell had found no guiding principle except the avoidance of contradictions; and this, by itself, is a very insufficient principle, since it leaves us always exposed to the risk that further deductions will elicit contradictions. (CPBR, V, 75) Russell also considered a second variant of the reducibility strategy according to which a propositional function is reducible provided that the number of things satisfying it is not too large. This ‘limitation of size’ proposal was in effect made by Russell’s friend Jourdain (1904), although his exposition of it was confused: he attempted to retain the assumption that every propositional function gives rise to a class and deny only, in the case of those with too many satisfiers, that this class has a number. As Russell pointed out to him, that might deal with the Burali-Forti paradox, but would not help with the paradox of classes.

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Russell noted, however, that these two paradoxes can be given a common form. He called a propositional function φ xˆ self-reproductive (CPBR, V, 72) if there is a function f which for every class w of φs gives an object f (w) ∈ / w satisfying φ xˆ . (Nowadays it is more usual to use for this Dummett’s term ‘indefinitely extensible’.) Let us in these circumstances call f a ‘reproduction function’ for φ. Russell showed that no self-reproductive propositional function is reducible. For if u = {x : φx} and f is a reproduction function for φ, then φ(f (u)) and hence f (u) ∈ u; but also f (u) ∈ / u by hypothesis, which is a contradiction. Russell’s paradox is obtained by taking φx to be x ∈ / x and f (w) = w. Burali-Forti’s paradox is obtained by taking φx to be ‘x is an ordinal’ and f (w) to be the least ordinal greater than all the members of w. Russell had thus proved that all self-reproductive propositional functions are irreducible. Now he conjectured, conversely, that self-reproductivity is the sole characteristic mark of the paradoxes, i.e. that all irreducible propositional functions are self-reproductive. The connection with limitation of size is that if f is a reproduction function for φ xˆ , then we can recursively define a one-to-one function g from ordinals to φs by g(α) = f ({g(β) : β < α}), and so the number of φs is at least as great as the number of ordinals. If we accepted Russell’s conjecture that self-reproductivity is the sole source of the contradictions, it would follow by contraposition that if there are fewer φs than ordinals, then φ xˆ is reducible. Nonetheless, limitation of size would remain, until more were said, no more than the skeleton of a theory: to put flesh on it we would first have to develop the autonomous account of the ordinals which the treatment presupposes, and Russell never attempted this.

Further reading The idea of limitation of size had in fact been proposed by Cantor, in letters to Hilbert and Dedekind, some years before Russell considered it, but he did not publish this work and the idea was not developed in any detail until Von Neumann (1925). Klement (2004) attempts to explain Russell’s apparent slowness in recognizing the constraint the paradox of propositional functions placed on his account. Church (1984) derives the Russell-Myhill paradox in a formalized version of the theory of types Russell put forward in Appendix B of the Principles.

34 ON DENOTING

In 1903 and 1904, as we have seen, Russell re-examined how the theory of denoting in the Principles dealt with definite denoting phrases that do not denote anything, in the hope that this would illuminate the case of class-expressions that do not denote a class. By 1905, however, he had become disaffected with the reducibility strategy and reverted to the view that propositional functions must sometimes be capable of occurring as ‘denotation’. This suggested a parallel with the theory of denoting, which posited a sort of logical entity called a denoting concept whose role, when it occurred in a proposition, was to make the proposition be about some other term or terms not occurring in it. To point up the parallel, Russell sometimes called this denoting concept the ‘meaning’ of the denoting phrase and the other term or terms the ‘denotation’. If a propositional function occurred as meaning, it behaved like a denoting concept, in that the proposition was not about the function. This parallel gave Russell a further reason to scrutinize his theory of denoting.

The Gray’s Elegy argument In the summer of 1905 Russell came upon what seemed to him to be a fatal objection to his theory of denoting. This has come to be known as the Gray’s Elegy argument, because when he expounded it in ‘On denoting’ he chose (for reasons that will emerge shortly) to use as his central example the first line of Gray’s Elegy, ‘The curfew tolls the knell of passing day’. Russell’s objection concerns the relationship between the meaning and the denotation of a phrase (namely that the former denotes the latter). A concept denotes if, whenever it occurs in a proposition, the proposition is not about it but about something else. To form a proposition that is about a meaning, therefore,

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we cannot simply insert the meaning itself, since if we did, the proposition would be about the denotation, not the meaning. In fact, Russell soon came to the conclusion that there is no proposition saying that the meaning denotes the denotation, but in order to reach that conclusion he had to consider more generally the various strategies for forming such a proposition that one might employ. In his discussion of the general case he used ‘C’ to stand for an arbitrary denoting phrase. The challenge was then to find an expression which refers to its meaning. One might at first think that we could use ‘the meaning of C’ to refer to it, but this cannot be right. For the letter ‘C’ that occurs in this phrase has been introduced to stand for a denoting phrase, and the occurrence of this phrase here will denote the denotation of that phrase. So ‘the meaning of C’ will denote the meaning (if any) of the denotation, which is not what was wanted. Russell evidently expected us to find this last point difficult to grasp, since he took the trouble to explain it more fully. He diagnosed our difficulty in grasping it as arising because in most of the cases that we deal with in practice the denotation has no meaning, with the result that when we are confronted with the phrase ‘the meaning of C’ we unconsciously adjust so as to read it as referring to the meaning that we are trying to get at. In order to make this mistake vivid, therefore, Russell needed a phrase with the unusual feature that its denotation does have a meaning. The one he chose was ‘The first line of Gray’s Elegy’. In this case the phrase ‘the meaning of C’ does indeed denote something. The trouble is that what it denotes is not the meaning we wanted, namely the meaning of ‘The first line of Gray’s Elegy’, but rather the meaning of ‘The curfew tolls the knell of parting day’, which is something else entirely; so we have missed our intended target. If ‘the meaning of C’ does not hit the right target, though, might some other function of C do the trick? The trouble with this is that the relationship between denotation and meaning is one-many, and hence there is no prospect of expressing the latter as a function of the former. For instance, ‘the smallest prime number’ and ‘the positive square root of four’ have the same denotation but different meanings. There is, as Russell put it, no ‘backward road’ (CPBR, IV, 422) from denotation to meaning. No function of C can provide us with a general method of referring to the meaning we want. By now, though, the reader may suspect that Russell had missed something glaringly obvious. The challenge was to refer to the meaning of ‘The first line of Gray’s Elegy’. Have I not just done exactly that? Russell rejected this solution, however, because he held that the relationship between meaning and denotation that we are trying to get at is logical, not psychological, and so cannot be, as he put it, ‘linguistic through the phrase’ (CPBR, IV, 421). The trouble with ‘The meaning of “The first line of Gray’s Elegy” ’ is therefore that it mentions, rather than uses, the phrase ‘The first line of Gray’s Elegy’. There is an obvious similarity here to Frege’s objection to his Begriffsschrift theory of reference, namely that that theory, by mentioning the words involved, made it a linguistic rather than an astronomical fact that Hesperus=Phosphorus.

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One might well think, though, that logic is closer to linguistics than astronomy is, and hence that it is less obvious whether the relationship between meaning and denotation is linguistic. Russell’s argument at this point relies strongly on his platonistic conception of logical entities as non-linguistic. Russell concluded that there is no general method for expressing the relationship between a meaning and a denotation that is not ‘linguistic through the phrase’. ‘The meaning cannot be got at’, he said, ‘except through denoting phrases.’ (CPBR, IV, 421) The theory of denoting concepts had led to an ‘inextricable tangle’ (CPBR, IV, 422), and should therefore be rejected. Russell expressed his objection as if it were directed only against definite denoting concepts, but it carries over to the indefinite case as well, since what it depends on is the denoting concept’s alleged property of making propositions in which it occurs be about something else, and this is supposed to hold of indefinite concepts just as much as of definite ones.

Against Frege? I have presented the Gray’s Elegy argument as an objection to the theory of denoting in the Principles, and Russell’s own working notes suggest (CPBR, IV, 361) that this is how he originally conceived of it. In ‘On denoting’, however, he turned it into an objection to any theory, including Frege’s, which distinguishes between the meaning and the denotation of a singular term. Even if the argument succeeds against his own earlier theory, it is a further question whether it also succeeds against Frege’s, because for Russell the meaning and the denotation are on the same logical level, whereas for Frege they are not. In the Principles Russell held that there are two kinds of proposition about an object: those containing the object itself as a constituent; and those containing not the object itself but a denoting concept (meaning) which denotes the object (denotation). Meanings and denotations may both occur in propositions, whence Russell’s difficulty. For Frege, by contrast, there are only propositions of the second kind, not the first. This observation leaves open to Frege a defence that was not open to Russell: he could deny that the relationship between meaning and denotation (sense and reference, in his terminology) is expressible propositionally at all. I have said that Frege ‘could deny’ this, not that he did. He certainly held that some things are unsayable, such as what we intend when we say, ‘The concept horse is a concept’, but this is because of his views about the unsaturatedness of concepts. To apply the doctrine of unsayability in the current case goes beyond anything his texts license. Nonetheless, it is, as Dummett (1991a, 238f) has suggested, ‘not only consonant with Frege’s ideas, but almost required for a coherent statement of them’.

The new theory of denoting phrases So far, we have been considering the negative part of ‘On denoting’, in which Russell rejected his own earlier theory of denoting, but the article has become

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famous for the positive part, in which he proposed ‘reductions’ (i.e. analyses) of sentences containing denoting phrases such as ‘the f ’, ‘every f ’ or ‘some f ’ by means of ‘definitions in use’ (i.e. contextual definitions): g(the f ) =df ∃x(∀y(fy ≡ x = y) ∧ gx); 1 g(every f ) =df ∀x(fx ⊃ gx); g(some f ) =df ∃x(fx ∧ gx). In order that this sort of logical analysis should be so much as possible, though, Russell had to revise his views about the identity conditions for propositions. In the Principles (§41), let us recall, he had held that ∀x(fx ⊃ gx) cannot be a correct analysis of ‘All ravens are black’, because the former is about all the things there are, the latter only about ravens. Changing his mind about this was crucial to the project of logical analysis which ‘On denoting’ instigated. So it is bizarre how casually Russell announced the change, merely referring the reader in a footnote to Bradley’s Logic (1883, bk 1, ch. 2). The reference is curiously non-specific: the chapter of Bradley in question is almost 70 pages long, and very little of it is about the analysis of quantifier phrases. The one relevant argument distinguishes the hypothetical judgment ‘All ravens are black’ from the categorical judgment that conjoins the assertion of the blackness of each existent raven: only the former, Bradley says, commits us to the blackness not just of current ravens but of any we might encounter. This argument, though, falls well short of establishing the conclusion Russell drew from it. Bradley’s issue was not whether ‘All ravens are black’ says anything about non-ravens—Russell’s original concern—but rather whether it is, in Johnson’s terminology (1921-4, III, ch. 1), a universal of law or of fact: if the former, it will not be captured by Russell’s rendering of it as a generalized material conditional any better than by a conjunction. One might wonder whether perhaps the author who persuaded Russell to adopt a looser criterion of identity for propositions was not Bradley but Frege. It was he, after all, who emphasized in Begriffsschrift that logic should pay attention only to those differences in content that are relevant to inference. Relative to this criterion ‘All ravens are black’ has (when read as a universal of fact) the same content as ‘Everything is either black or not a raven’.

Scope ‘A logical theory’, Russell said, ‘may be tested by its capacity for dealing with puzzles.’ (CPBR, IV, 420) The puzzles against which he tested his theory of descriptions were ambiguities in sentences of the form Bg(the f ), where B is a sentential operator. Such a sentence, he suggested, is ambiguous between ∃x(∀y(fy ≡ x = y) ∧ Bgx), and B∃x(∀y(fy ≡ x = y) ∧ gx).

222 Russell

In the former case, the phrase ‘the g’ has what he called a primary occurrence; in the latter, secondary; more recent writers often call these wide and narrow scope occurrences respectively. The first case Russell considered was that in which B is negation. He used once again the example he and Whitehead had been thinking about the previous year, ‘The present King of France is not bald’. If the definite description has primary occurrence, this is analysed as ∃x(∀y(fy ≡ x = y) ∧ ∼gx) (in words: There is exactly one King of France and he is not bald), which is false. If the definite description has secondary occurrence, it is analysed as ∼∃x(∀y(fy ≡ x = y) ∧ gx) (in words: It is not the case that there is exactly one King of France and he is bald), which is true. The example is perhaps not ideal, though, because it is not really ambiguous: the definite description here clearly has a primary occurrence. Contrast ‘I have not met the present King of France’, which most of us would probably regard as true. The next case Russell considered was that in which B expresses a propositional attitude such as belief or doubt. His now-celebrated example was ‘George IV wished to know whether Scott was the author of Waverley’. Here the definite description ‘the author of Waverley’ might, Russell claimed, have a primary occurrence (if, for instance, George IV saw Scott in the distance and was unsure who it was), in which case it would be read as saying: Exactly one person wrote Waverley, and George IV wondered whether that person was Scott. Alternatively, the definite description may be read as having a secondary occurrence, in which case the sentence is analysed as: George IV wished to know whether exactly one person wrote Waverley and that person was Scott. In any but the most implausible circumstances, the latter reading is more charitable to the ‘first gentleman of Europe’ (who did indeed want to know whether Scott was the author of Waverley, because the book had been published anonymously and its authorship was a matter of lively interest not just to the King but to much of literary society). The example is once again not ideal, though, because we do not easily hear the ambiguity. A better example might be ‘Smith believes that the murderer is depraved’. If Smith’s belief is based on personal knowledge of Jones, the perpetrator, the definite description has a primary occurrence; if on the condition of the bloody corpse, secondary. If the description has primary occurrence, the propositional attitude is often said to be de re (‘about the thing’); if secondary, de dicto (‘about what is said’). This terminology is not altogether happy, though, because it suggests a binary distinction, whereas if there is more than one operator, there will be more than two possible scopes. In ‘Sarah predicts that John will want to marry the next Prime Minister’, for instance, ‘the next Prime Minister’ has three possible scopes.

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The de re/de dicto terminology was originally devised by mediaeval logicians for the case of modal operators, and Russell’s analysis may be applied in such cases as well. ‘The Prime Minister could have been bald’, for instance, can be read de re, to say that the person who is in fact the Prime Minister could have been bald; or de dicto, to say that we could have had a bald Prime Minister, e.g. by electing a different government. Russell himself did not discuss such examples in ‘On denoting’, however, presumably because of his general hostility to counterfactual reasoning. The idea of using scope to explain ambiguities works just as well for indefinite as for definite denoting phrases: Bg(every f ) is ambiguous between ∀x(fx ⊃ B(gx)) and B(∀x(fx ⊃ gx)); and Bg(some f ) is ambiguous between ∃x(fx ∧ B(gx)) and B(∃x(fx ∧ gx)). Thus ‘Anne wants to kill some pheasants’ has a de re reading on which she has some particular pheasants in mind, and a de dicto reading on which any ones will do. Why, then, did Russell focus in his essay on puzzles involving definite, not indefinite, phrases? The reason is presumably that he had little new to say about the latter, which had been adequately explained by Frege 25 years earlier. In that case, though, it might have helped his readers if he had mentioned Frege at this point in the essay.

Non-entities The de re/de dicto distinction also gives us the resources for an account of belief about non-entities. I cannot have a de re belief that the present King of France is bald, because there is no such person for me to have beliefs about; but I might (erroneously) have such a de dicto belief if I were ignorant of French history. In order to apply this account in the case of names, however, we need to invoke Russell’s idea from 1903 that what grammar treats as a name may for logical purposes be a disguised description. If ‘S wonders whether there is such a thing as A’ is to make sense, ‘A’ must be a disguised description, not a logically proper name. Although Russell had culled some of the animals in his ‘queer zoo’, it still contained, as well as things that exist, other entities that only subsist: what had changed was that the subsistent entities were now only universals such as mortality or redness, not particulars such as Zeus or the present King of France. The distinction between existent and subsistent (or concrete and abstract) now coincided with that between thing and concept (or, as he would soon call them, particular and universal). Just as one distinction—between existence and thinghood—was collapsing, though, another was opening up. Some of Russell’s experimental solutions to the paradoxes required him to abandon the unrestricted variable and to distinguish between individuals, which fall in the range of the first-order variable, and those other entities that fall in the ranges of higher-order variables.

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One curious omission from ‘On denoting’ is an explicit analysis of ‘The f is an individual’. It is easy, though, to derive this from the previous analysis: the f is an individual ≡ ∃z(z = the f ) ≡ ∃x(∀y(y = x ≡ fy) ∧ ∃z(z = x)) (by the previous analysis) ≡ ∃x∀y(y = x ≡ fy) (by predicate calculus) Russell certainly knew this, and mentioned it explicitly in a letter to Jourdain (13 Jan. 1906, in Grattan-Guinness 1977, 70). Why, then, did he not mention it in ‘On denoting’? Perhaps he simply did not want to open the can of worms involved in explaining the difference (if any) between an entity and an individual.

Significance ‘On denoting’ is often said to have been a pivotal moment in Russell’s development, but not always for the right reasons. It was not, for instance, the moment when he abandoned his Meinongian ontology of non-existent things: that happened two years earlier, when he had the idea that what grammar classifies as proper names are often from a logical perspective disguised descriptions. In relation to non-existent things, the significance of the new theory was rather that it enabled an explicit rendering of the disguised descriptions in question. Nor was the essay the origin of Russell’s notion of acquaintance. Although in the penultimate paragraph he described the doctrine of acquaintance as a consequence of his new theory of denoting, the notion of acquaintance, if not the term, had in fact been implicit in his work for some time. The point is rather that although he previously took there to be a relation between the judger and the components of the proposition judged, he had given little thought to what it involves. It was ‘On denoting’ that gave him a criterion for acquaintance, namely that it should confer indubitability on the entity. The contrasting case, in which a proper name is really a disguised description, must, on Russell’s account, be invoked whenever the judger is in any doubt as to the existence of the object concerned. As Ramsey observed, A theory of descriptions which contented itself with observing that ‘The King of France is wise’ could be regarded as asserting a possibly complex multiple relation between kingship, France, and wisdom, would be miserably inferior to Mr Russell’s theory, which explains exactly what relation it is. (FoM, 142) ‘On denoting’ thus instigated what would now be called a ‘research programme’, the goal of which was to determine which the objects of acquaintance are, and how they are related to those with which we are not acquainted. Russell’s analyses do not preserve the subject-predicate grammatical structure of the analysanda. Some commentators have suggested, therefore, that it was ‘On denoting’ that gave him the idea that a proposition may have a different form from

On Denoting 225

the sentence which expresses it. Was it not, as Wittgenstein said, ‘Russell’s merit to have shown that the apparent logical form of a proposition need not be its real form’ (TLP, 4.0031)? Yet this was by no means Russell’s first encounter with that idea: in his idealist phase, for instance, he had for a time accepted Bradley’s ‘existential theory’ that any categorical judgment should really be understood as predicating something of the whole of reality. In the early years of his revolt against absolute idealism, however, he had supposed that most sentences correctly represent the forms of the propositions they express. His argument against the existential theory of judgment, for instance, had been that the arithmetical truth 2 + 2 = 4 is plainly about something that does not exist, namely the number two. His adoption of the new theory of denoting marked at most a return to the idea that grammar misleads us as to the true form of our judgments. Philosophical clarification familiarly aspires to offer precise thoughts as replacements for the vague, more or less inchoate ones with which we ordinarily make do. Russell’s theory maintains, though, that even when we think we have expressed ourselves precisely, we may nonetheless be unaware, if our sentence involved a definite description, of the true structure of the thought expressed. He was thus making the radical claim that I may judge a proposition while being ignorant of its true structure or of the identity of the entities with which I must be acquainted if I am to judge it. One salient feature of Russell’s theory was its employment of ‘definitions in use’, i.e. contextual definitions. He took these definitions to show that denoting phrases are semantically incomplete: they ‘never have any meaning in isolation, but . . . every proposition in whose verbal expression they occur has a meaning’ (CPBR, IV, 416). It was not until 1909 that he coined the phrase ‘incomplete symbol’ (PM, I, 69) to describe an expression whose meaning is given by a contextual definition, but he had already articulated the method of elimination in 1906 without giving it a name (CPBR, V, 243–4). He did not, of course, invent this method—Bentham (1838, VIII, 246), for instance, described how ‘paraphrasis’ may be used to talk about ‘fictitious entities’—but he did elevate it into a systematic tool for dealing with philosophically problematic entities. A contextual definition permits the elimination of an incomplete symbol from the propositions in which it seems to occur. Russell seems to have been slow to realize, though, that it is a separate question whether the entity the symbol denotes is eliminated from the world. In 1906 he described both the present King of France and the present King of England as ‘false abstractions’ (CPBR, V, 243), obscuring the important point that whereas ‘The present King of France is bald’ was false without there being any man the proposition was about, what made ‘The present King of England has a beard’ true was a certain person (Edward VII) who fell in the range of the variable. The theory of descriptions is thus ontologically eliminative only in the former case, not the latter. Russell later called the apparent entities ‘logical fictions’ (CPBR, VIII, 234), but still did not distinguish clearly between those that are, and those that are not, eliminated from the world

226 Russell

(rather than just from the propositions apparently about them). Nor were matters helped by his habit—for which, at Moore’s prompting, he was later excoriated by Stebbing (1930, ch. 9, §4)—of saying that certain entities are incomplete symbols, when what he should have said was either that the entities are logical fictions or that the expressions purportedly denoting them are incomplete symbols. Russell also fumbled how to tell which symbols are incomplete. In ‘On denoting’ he stressed his theory’s role in differentiating between primary and secondary occurrences, and one might therefore expect him to have pressed this point in generalizing to incomplete symbols, but in fact he gave it curiously little emphasis. The argument in the Introduction to Principia to show that the definite description in ‘Scott is the author of Waverley’ is incomplete relied only on the contrast between what he took to be the obvious non-triviality of this sentence and the triviality of ‘Scott is Scott’: he neglected to mention that George IV had wondered only about the former, not the latter, and hence did not introduce a context for a scope ambiguity. The criterion he offered instead was semantic incompleteness: an incomplete symbol ‘is not supposed to have any meaning in isolation, but is only defined in certain contexts’, whereas a complete symbol corresponds to a constituent of the proposition; and it follows from the principle of acquaintance that in the latter case if I grasp the proposition in question, then I must be acquainted with the constituent. He thus conceived of acquaintance as a route to bestowing meaning on a symbol ‘in isolation’, in opposition to Frege’s context principle.

The substitutional theory Whatever its longer term significance, the immediate effect of ‘On denoting’ was to suggest to Russell a new approach to solving the paradoxes, namely to treat expressions that appear to refer to classes as incomplete symbols, to be eliminated by means of suitable contextual definitions. In fact, he went a step further, treating propositional functions as incomplete symbols too: every statement that appeared to be about a propositional function was really, he proposed, a shorthand for a claim about the result of substituting one entity for another in a proposition. This ‘substitutional’ theory had many of the technical advantages of a simple theory of types, with the additional benefit that, because it treated propositions as entities, it allowed a purely logical proof of the axiom of infinity. ‘All went swimmingly,’ he reported (letter to Jourdain, 15 Mar. 1906, in Grattan-Guinness 1977, 80), and in April 1906 he submitted an account of this theory for publication. He believed himself to be in the happy position of having found, in the method of incomplete symbols, the uniform explanation for the paradoxes that he had been searching for, namely that the paradoxical sentences all contained incomplete symbols in contexts to which the contextual definitions did not apply. ‘On denoting’ had therefore been the ‘key to all [his] subsequent progress’ (ibid., 79). We shall see shortly, however, that his optimism once more turned out to be misplaced.

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Further reading The eight paragraphs in which Russell laid out the Gray’s Elegy argument have spawned a remarkable number of competing interpretations. The discussion by Cartwright (1987, 95–133) is excellent. Makin (1995) offers a sympathetic exegesis of the argument as a whole, as does Cassin (1970). Noonan (1996) argues, contrary to the reading I have offered here, that Russell’s argument assumes that we have acquaintance with the concept. Urquhart (2005) attempts to rehabilitate the old view that the Gray’s Elegy argument is irredeemably confused. The question whether the argument refutes Frege’s theory of sense and reference has also been much discussed. Blackburn & Code (1978) argue that it does. Frege has been defended by Geach (1978) and Levine (2004). For the application of the saying/showing distinction to Frege’s notion of sense, see Geach (1976). Russell’s curious attribution to Bradley of his view that ‘All f s are gs’ should be analysed as ∀x(fx ⊃ gx) is remarkably little discussed in the literature: Scarrow (1962) is a rare exception. For a defence of Russell’s theory of descriptions against criticisms see Neale (1990). Landini (1987) analyses Russell’s substitutional theory of 1905–6. Klement (2004) discusses his use of the method of incomplete symbols to solve the paradoxes; on the method more generally see Sainsbury (1980).

35 TRUTH

During the decade that Whitehead and Russell collaborated, both showed a considerable capacity for multi-tasking. In term time Whitehead was probably too occupied with teaching to contribute much, but he also pursued his own writing projects—two books on geometry (1906a, 1907) and a paper (1906b) on the definition of matter. Russell meanwhile devoted considerable attention from 1904 onwards to the problem of truth. One reason for this was polemical—it was an interpretative crux in his defence of realism against his absolute idealist and pragmatist opponents—and another was that he had begun to see problems with the identity theory he had got from Moore; but he also hoped that analysing truth might solve the liar paradox.

The coherence theory Some British idealists held that truth resides only in the Absolute and cannot be defined in terms of anything external to it, but should instead be conceived of as internal coherence. Joachim, for instance, said1 that truth ‘is in its essence conceivability or systematic coherence’ (1906, 68), ‘the systematic coherence which characterize[s] a significant whole’ (78). One attraction of this account, at least for the more mystically inclined idealists, was that it contributed to their teleological conception of a world each part of which was to be explained by its contribution to a self-purposed whole. When Russell first attacked this view in 1905, he focused on Joachim’s doctrine of inter-connectedness—an instance of the monism he had by now rejected—and on Joachim’s belief that propositions

1 Bradley is sometimes said to have adopted a coherence theory of truth in his later writings (e.g. 1909), but he seems to have regarded coherence rather as a criterion of truth than a definition.

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are only partially, not wholly, true. The claim that the true propositions form the only self-coherent class is separable from these other Hegelian doctrines, and at this stage Russell accepted this claim as extensionally correct (CPBR, IV, 497), objecting only, on grounds of circularity, to using it as a definition. The following year, however, he changed his mind. There is ‘no evidence’, he now said, ‘that a system of false propositions might not, as in a good novel, be just as coherent as the system which is the whole of truth’ (CPBR, V, 443). To make the point vivid, he cited Bishop Stubbs, a cleric and historian of unimpeachable morals who had died peacefully in his bed in 1901, but about whom one might construct a coherent story in which he was hanged for murder. Once we concede that fiction may be just as coherent as fact, it becomes hard to prevent the coherence theory from collapsing into the kind of soft relativism that sees no conflict between something’s being true-for-me and false-for-you, provided that our respective belief sets are internally coherent. This shift to rejecting the coherence theory was not originally Russell’s, however: he was in fact repeating a point Moore had made five years earlier. It seems to be frequently implied that the truth of a proposition may consist in its relation to other propositions—in the fact that it ‘fits into a system.’ This view, however, simply neglects the admitted fact that any logical relations which hold between a set of true propositions will also hold between a set of false ones; i.e. that the only kind of system into which a true proposition will fit, and a false one will not, is a system of true propositions. The view derives its plausibility merely from the fact that the systems of propositions considered are ones to which we are so thoroughly accustomed that we are apt to regard their contradictories as not merely false but self-contradictory. (In Baldwin 1901–5, II, 717) (In the same article, incidentally, Moore went on to attribute the coherence theory to Kant, whom he blamed for failing to distinguish sufficiently between truth and objectivity.) One might be forgiven for supposing that the Bishop Stubbs objection marked Russell’s rejection of the modal actualism he had adopted in 1899, since he now had to grant a sense in which what is actually false—that Bishop Stubbs was hanged for murder—is nonetheless a coherent possibility. It is disconcerting, therefore, to find him in the same 1906 article repeating his earlier view that counterfactual reasoning is always vacuous, because an hypothesis cannot cancel a fact. If two terms are related in a certain way, it follows that, if they were not so related, every imaginable consequence would ensue. For, if they are so related, the hypothesis that they are not so related is false, and from a false hypothesis anything can be deduced. (CPBR, V, 448).

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He did not trouble to explain how we can coherently entertain the false hypothesis that Bishop Stubbs was hanged for murder if, as he insisted, anything whatever is deducible from it. The coherence theory remained popular for a time not only among the diminishing ranks of absolute idealists (e.g. Blanshard 1939) but also among logical positivists suspicious of the relation between language and world posited by the correspondence theory. Hempel, for instance, held that that system of protocol statements which we call true . . . may be characterized by the historical fact that it is the system which is actually adopted by mankind, and especially by the scientists of our cultural circle. (1935, 57) Eventually, though, the Bishop Stubbs objection largely prevailed against the coherence theory: Ramsey, for instance, suggested that ‘after Mr Russell’s amusing essay . . . it is difficult to see how anyone can still cling to it’ (OT, 25). Even so, the theory has been resurrected by modern fictionalists about mathematics (e.g. Leng 2010) not as an account of truth—which, like the absolute idealists, they take to be unattainable in the mathematical case—but as an explanation of why we choose one mathematical fiction over another. The challenge for them is then to give a non-circular explanation of what it is for a set of sentences to cohere. (They cannot, for instance, explain it as the possibility that they might all be true.) Versions of the coherence theory also continue to be popular outside the analytic tradition. For instance, the post-modernist view that what we call truth is merely one narrative among many and that every reading of a text is inevitably a misreading can be viewed as a modern inheritor of this strand of absolute idealism.

The pragmatist theory The other internalist account of truth Russell encountered was the pragmatist theory advanced by Schiller (1906) and James (1907), according to which a judgment is true just in case it is useful. In a talk delivered in Oxford early in 1907, Russell objected against this theory that it is sometimes useful to believe what is false. James attempted to circumvent this objection by recommending a shift (much like that from act- to rule-consequentialism) to assessing usefulness ‘in the long run and on the whole’ (1907, 222) rather than case by case. It may be useful for me to believe falsehoods occasionally, but it is surely implausible that my life would go better if I did so systematically. Plainly, though, this amendment still does not suffice to pick out the class of truths uniquely. James went some way towards acknowledging this difficulty by admitting that usefulness is not the only criterion of truth and allowing verifiability as an alternative, usefulness being applicable in cases (such as God’s existence) where verification is impossible. ‘On pragmatistic principles,’ he said, ‘if the hypothesis of God works satisfactorily in the widest sense of the word, it is true.’ (1907, 299) For Russell (as, later, for

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the militantly atheistic Ramsey), this restriction was scant comfort: we must, they thought, leave room for the possibility that religious beliefs, like others, might be useful but false. James also appealed to the pragmatist account of truth to explain theory selection in science. In this case, Russell was sympathetic (CPBR, V, 484) to a structurally similar view, but with the significant emendation that for a scientific hypothesis to ‘work’ should mean not that the effects of believing it are good but that it has many verifiable, and no refutable, consequences. The emendation is crucial because it relies on a prior understanding of truth and falsity for the consequences of the theory, and hence does not offer a route to a thoroughgoing pragmatism about truth. Russell also advanced a further criticism of the pragmatist theory, namely that even if believing a truth were always more useful than believing a falsehood, still truth would not mean usefulness. ‘It seems obviously putting the cart before the horse’, he argued, to say that when the train is announced as starting from Platform No. 3, that means that it is useful to think so. Indeed is it not obvious that the fact that the train starts from Platform No. 3 is a fact about the train, about the engine and the carriages, not a fact about the utility or disutility of beliefs? (CPBR, V, 463) The pragmatists’ mistake (CPBR, VI, 274) was to confuse two senses of ‘meaning’: in one sense—later dubbed ‘natural meaning’ by Grice (1957)—clouds mean rain; in another—Grice’s ‘non-natural meaning’—pluie means rain. Even if the natural meaning of truth was usefulness (e.g. because the two were causally linked), that would not entail that it meant this in the non-natural sense that is the logician’s concern. At base, Russell’s objection to the pragmatist theory, as to the coherence theory, was that it made truth too internal—too dependent on us. Whatever else in Russell’s attitude to truth may have changed, what remained constant was his conviction that it involves something external to the believer.

The primitivist theory In Chapter 28 we discussed Moore’s argument against the correspondence theory of truth and his consequent adoption of an identity theory, according to which ‘a truth differs in no respect from the reality to which it was supposed merely to correspond’ (in Baldwin 1901–5, II, 717). How, though, is such an identity theory to account for false beliefs? If I believe that Charles I died on the scaffold, the object of my belief is a fact, but if I believe that he died in bed, there is no such fact, and hence, apparently, nothing for me to believe. What do we believe when we believe a false proposition? We believe in a relation (say) between two terms which, as a matter of fact, are not so

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related. Thus we seem to believe in nothing: for if there were such a relation as we believe in, the belief would not be erroneous. (CPBR, IV, 444–5) The identity theory has two variants, depending on its account of falsehood. The first is a ‘disjunctive’ theory: true judgments relate the believer to what Meinong (1902, ch. 7) had called an ‘Objective’; false judgments are internal states of the believer with no Objective. Some of the time, at least, Russell recognized the attraction of this sort of asymmetry between true and false judgments, on the ground that truth is objective, error subjective. ‘We feel that there could be no falsehood if there were no minds to make mistakes.’ (CPBR, VI, 119) The obvious difficulty with this theory, though, is that it holds out the implausible hope that we might be able to tell merely by introspection whether or not an empirical proposition is true. So in 1904 Russell preferred instead a second, ‘primitivist’ variant of the theory, according to which the objective realm of propositions falls into two parts: true propositions (or facts), and false propositions (or fictions). ‘There is no problem at all in truth and falsehood,’ he declared. ‘Some propositions are true and some false, just as some roses are red and some white.’ (CPBR, IV, 473) This overcomes the implausible internal asymmetry of the disjunctive theory, but at the price of the opposite flaw of making truth and falsity perfectly symmetrical. It ‘seems to leave our preference for truth a mere unaccountable prejudice, and in no way to answer to the feeling of truth and falsehood.’ Russell was therefore left to conjecture, rather weakly, that our preference for truth must be based upon an ultimate ethical proposition: ‘It is good to believe true propositions, and bad to believe false ones.’ This proposition, it is to be hoped, is true; but if not, there is no reason to think that we do ill in believing it. (CPBR, IV, 474) The implausibility of this response lies in the fact that on the one hand, we have agreed that no dissection of the proposition will reveal a difference between truth and falsity; but on the other, we have cut off the route to any other explanation of it by our very insistence that the properties in question are primitive. We have encountered the idea that our preference for truth over falsity is ‘ethical’ already in Chapter 2, where we noted that some 19th-century idealists used it to motivate the normativity of logic in contrast to psychology. To call the preference ‘ultimate’, as Russell now did, was to concede that it is an unaccountable prejudice.

The correspondence theory Eventually Russell abandoned the primitivist theory of truth because of its inability to explain our preference for truth over falsity and the implausibility of positing a realm of objective falsehoods. In 1909 (CPBR, VI, 11–12) he adopted instead a correspondence theory of truth, according to which a judgment that Charles I died in bed would be true just in case the king, his bed and dying were related

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so as to constitute what he called the ‘corresponding’ complex; it is because there is no such complex that the judgment is false. In what we can now recognize as a familiar pattern, he found, once he had given it up, that his belief in objective falsehoods had been ‘the very reverse of plausible’, ‘in itself almost incredible’, and ‘unsatisfactory’ (CPBR, VI, 118–19); by 1918 he even thought it ‘monstrous’ (CPBR, VIII, 197). If the primitivist theory was so implausible, though, why did Russell not give it up publicly until five years after he first articulated its inadequacies? And why, even then, did he not trouble to explain why the objection he had previously made against correspondence theories was not fatal to the one Moore now advocated? Part of the explanation, perhaps, was that the sense in which his new theory was a correspondence theory was not quite that in which the earlier one had been an identity theory: what he now maintained was that there was a correspondence between a judgment and the complex that makes it true, whereas he had previously claimed an identity between the object of the judgment and the complex. Nonetheless, it would have been helpful if he had explained why defining the truth of a judgment as its correspondence with an existing complex does not trigger a regress. While he adhered to the primitivist theory of truth, Russell had conceived of the world as consisting of complexes, some of them facts, others fictions. Now he abandoned fictional complexes, leaving only factual ones, which he now referred to indifferently as ‘complexes’ or ‘facts’: here I shall use the former term. That truth consists in correspondence still falls some way short of constituting a theory of truth, however, until the exact nature of the correspondence is specified. Russell proposed a hierarchy of correspondences giving rise to different kinds of truth. The simplest kind, ‘first truth’, consists in a direct correspondence between a judgment and some particular complex: he called judgments capable of this kind of truth ‘elementary’. A generalization of an elementary judgment is then said to have ‘second truth’ just in case all its instances have first truth. And so on. (CPBR, VI, 12) One virtue of this account, in Russell’s eyes, was that it solved the liar paradox. There is no single notion of truth applicable to all propositions: Epimenides cannot assert that everything he himself says is not true, but only everything of level n; this is no contradiction, since it is a candidate only for truth of order n + 1 and hence says nothing about its own truth or falsity. Another virtue was that the account did not require Russell to posit ever more elaborate complexes to serve as truth-makers for higher-order propositions: the only ones required were those corresponding to true elementary judgments. Still another virtue was that he hoped the hierarchy of kinds of truth might match up with his emerging epistemological views: the first truths would be just those we could in principle come to know directly via acquaintance. Which, though, are these? The theory had two variants, both problematic. On the first variant the base class included not just atomic judgments but also molecular ones, such as disjunctive or negative judgments. The trouble with this was that it required there to be disjunctive complexes to act as the truth-makers

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for disjunctive judgments. If you judge that Charles I died either on the scaffold or in bed, this is made true by a disjunctive complex consisting of his death on the scaffold disjoined with his death in bed. In that case there must presumably be a sub-complex consisting of his death in bed, and we are back at the objective falsehoods which it was the whole point of the theory to avoid. In 1906 he noted that this difficulty ‘leaves only a doubt, not a certainty’ (CPBR, V, 453). Unfortunately, however, the alternative theory, according to which the base class consists only of atomic judgments, is just as bad, implausibly requiring us to apply the hierarchy of notions of truth even to propositional connectives—i.e. to say that the negation of an atomic judgment has second truth, the double negation has third truth, etc. The doctrine of orders of truth just sketched has a murky history in any case. It is contained in a manuscript entitled ‘Truth and ambiguous assertion and general assertion’ which is undated but most likely written in 1906. Russell did not publish the manuscript then, but oddly incorporated it later into an article that he submitted for publication in 1909 (CPBR, VI, 10–14) and re-used in the Introduction to Principia. What makes this odd is that by then he had adopted a quite different solution to the liar paradox (to be discussed in Chapter 36), and it is difficult to see how the two could be made to cohere (see Goldfarb 1989, 37). One might worry, moreover, that the whole idea of different orders of truth is problematic for reasons unconnected with the theory of types. How could truth fail to be a univocal notion? In the same manuscript in which he outlined the idea of orders of truth, Russell also proposed an analysis of judgment (to be discussed in Chapter 40), but the latter proposal does not depend on the former, and it is notable that although he persisted with variants of the latter for several years, he did not take the notion of orders of truth any further.

Further reading Butler (2002) describes the context in which the post-modernist conception of truth arose. Candlish & Damnjanovic (2013) claim that the coherence theory which Russell demolished was his own invention, not Joachim’s. Stern (2004) explains Bradley’s use of coherence as a test for truth. Rorty (2003) defends a coherence theory. Walker (1989) discusses an attempt to avoid the Bishop Stubbs objection by requiring truths to form a coherent set of beliefs rather than merely of propositions, but concludes that this theory is susceptible to an explanatory regress in the case of beliefs about beliefs. Russell’s friend Hawtrey (1908) also attacked the pragmatist theory of truth; James (1909, ch. 14) responded (unconvincingly, it seems to me). On the identity theory which Moore and Russell endorsed up to 1906, see Cartwright (1987, 71–93). The idea of defining truth recursively is independent of Russell’s stratification into orders, and later became a logical commonplace following its use by Tarski (1933).

36 TYPES

By February 1906 Russell felt ‘hardly any doubt’ that the substitutional theory afforded the ‘complete solution’ to the paradoxes (CPBR, V, 89). Soon, however, he realized that the Russell-Myhill paradox, which had stymied the simple theory of types in Appendix B of the Principles, could be reformulated in the substitutional theory. He therefore withdrew from publication the paper in which he described the theory, saying that it ‘wanted correction’. That summer he for the first time described the project he was working on with Whitehead as a separate book entitled Principia Mathematica, rather than merely a second volume of the Principles. In ‘Mathematical logic as based on the theory of types’, submitted for publication in June or July 1907, he sketched both a substitutional account and one in terms of a hierarchy of propositional functions, suggesting that only technical convenience led him to prefer the latter. At first he planned to ignore the substitutional theory in the text of Principia and relegate it to an appendix (CPBR, V, 516), but by the time the book was completed, all mention of the substitutional theory had gone; what remained was an amended version of his earlier theory of types.

The hierarchy of propositional functions In Appendix B of the Principles Russell had rejected the theory of types because the simple version, which typed propositional functions only according to the types of their real variables, fell foul of the Russell-Myhill paradox, whereas the finer typing required to avoid this paradox led to a ‘harsh and highly artificial’ division of propositions into types. As late as 1906 he still found this ‘intolerable’ (CPBR, V, 185). Once he realized that the substitutional theory was susceptible to a similar paradox, though, he overcame his disgust and ramified the hierarchy,

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typing propositional functions not only, as in the simple theory, according to the types of their real variables, but also according to the types of their apparent variables. I shall call the more finely grained logical types of this ‘ramified’ theory ‘orders’. (The hierarchy becomes still more complicated if we include functions of more than one variable, since these may be of different levels.) In the ramified theory propositions, too, are typed according to the types of their apparent variables, with the (previously intolerable) consequence that no generalization over all propositions is permitted. In modern treatments, the type of a variable is indicated by attaching a subscript to it, but Whitehead and Russell instead adopted the convention that an expression without type indicators is ‘typically ambiguous’ (PM, I, ∗65): if we assert it, we should be taken to be ambiguously asserting all the various statements obtainable by disambiguating the types of the variables appropriately. By this means they attempted to circumvent the awkwardness that the restriction on variables prevents us from stating a wholly general logical truth, and hence that what occur in the text of Principia are not, strictly speaking, sentences expressing logical truths but only schemas specifying patterns which the expressions of various logical truths share. In his 1907 article Russell called a propositional function predicative if its order is next above that of its argument. So, for instance, a function of an individual variable is predicative just in case it is of the first order. He expressed a predicative function by means of an exclamation mark, writing φ!x for a predicative function of an individual variable; f !(φ!ˆz) for a predicative function of a first-order predicative variable; etc. Russell’s ramified hierarchy is also sometimes described as ‘predicative’, because the only permitted propositional function variables are predicative. (In the text of Principia, although not in the Introduction, Russell confusingly adopted a slightly different form of ramification, in which the variables are restricted to quantifier-free functions.) Given how intolerable Russell had recently found the division of propositions into types, one might expect him to have given a careful explanation of why the paradoxes compelled it. Why, then, did he not now mention the RussellMyhill paradox, which had persuaded him in 1902 that a simple theory would not do? Perhaps he was worried that this paradox, at least in the form in which he expounded it in the appendix to the Principles, depended on forming arbitrary classes of propositions, whereas in the system of Principia propositions are not individuals and are therefore not candidates to be members of classes. Instead Russell’s justification for ramifying the theory relied principally on the liar paradox. When a man says ‘I am lying’, we must interpret him as meaning: ‘There is a proposition of order n, which I affirm, and which is false.’ This is a proposition of order n + 1; hence the man is not affirming any proposition of order n; hence this statement is false, and yet its falsehood does not imply,

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as that of ‘I am lying’ appeared to do, that he is making a true statement. This solves the liar. (CPBR, V, 604–5) In the context of his 1907 paper this justification of ramification by means of the liar may have been fair enough, but in Principia itself he weakened its dialectical force by including in the Introduction a quite different solution to this paradox (discussed in Chapter 35) that used a single type of propositions but a hierarchy of truth-predicates. The reader might feel entitled to wonder what the point was of solving the liar twice over. Russell did also try to motivate ramification by means of an assortment of other puzzles such as Berry’s paradox of the least integer not nameable in fewer than nineteen syllables and König’s (1905) paradox of the least indefinable ordinal. These, though, he seemed already to have solved by observing (CPBR, V, 591) that ‘all names’ and ‘all definitions’ are illegitimate notions. He did not explain the illegitimacy in any detail, but he did refer in the course of his discussion to the paper in which Peano suggested that these paradoxes involve notions belonging ‘not to mathematics but to linguistics’ (1906, 157) and hence require the use of ordinary language in their formulation. All told, then, the case Russell presented for ramification as a way of avoiding paradoxes was weak. Perhaps a more telling example, had he yet known of it, would have been the paradox of heterologicality, due to Grelling (1908). An adjective is said to be heterological if it does not apply to itself. ‘Short’, for instance, is a short word, hence is not heterological; ‘long’ is not a long word, hence is heterological. Is ‘heterological’ itself heterological? It is if and only if it is not. Contradiction. So far, of course, this is only a contradiction in ordinary language. What makes it troubling is that it can be formalized in the simple theory of types. If we define Het x =df (∃φ)(x means φ zˆ ∧ ∼φx), then ‘Het’ means Het zˆ , and so Het(‘Het’) ≡ (∃φ)(‘Het’ means φ zˆ ∧ ∼φ(‘Het’)) ≡ ∼Het(‘Het’), In the ramified theory, on the other hand, the apparent variable φ in the definition of Het xˆ is restricted to range over some order; but Het xˆ is of the next order up, and hence cannot be a legitimate substituend for it. This paradox is more troubling than Berry’s and König’s because, although it admittedly involves the semantic notion of ‘meaning’, it does not mention language as a whole and hence does not obviously appeal to any of the notions such as ‘all names’ that Russell thought were illegitimate. Even if Russell did not yet know of Grelling’s paradox in 1907, he did at least know that the simple theory of types is incompatible with a finely grained individuation of propositions. Why, then, did he not just keep the simple theory,

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adopt a coarser individuation, and restrict himself to an extensional hierarchy of functions? If his purposes had been purely mathematical, he would have done. ‘Being confined to propositions in this hierarchy’, he wrote, ‘is a characteristic of mathematics.’ (CPBR, V, 568) The problem was that this left still to be dealt with those paradoxes in which ‘something beyond the truth or falsehood of values of functions is relevant’ (CPBR, V, 608). What he had in mind here were intentional operators such as belief. From a technical perspective, therefore, the explanation for his adoption of a ramified rather than a simple theory of types was that he was committed to the inclusion within logic of such operators, which discriminate more narrowly among propositions than a simple theory can consistently permit. The other notable assumption Russell made in his theory of types was that types are disjoint. If our only aim is paradox avoidance, we can assume that types are cumulative, as his friend Hawtrey pointed out to him in 1908. ‘This would be very revolutionary, but there is nothing in the reflexive paradoxes to prevent it.’ (7 June 1908, in CPBR, V, 627–8) The advantage of cumulative types, Hawtrey noted, is that they can then be iterated into the transfinite. Russell responded to this proposal merely by appealing to his ‘symbolic instinct’. If only he had accepted Hawtrey’s suggestions, he would have had a cumulative type theory essentially equivalent to the iterative account of set theory that later became widely accepted.

The axiom of reducibility Even after giving up the substitutional theory, Russell retained one aspect of it, namely the idea of treating class-expressions as incomplete symbols. The details of how to do this depended on the nature of the hierarchy of propositional functions. In the simple theory of types it could be done, whenever f (φ zˆ ) is a higher-order function, by the contextual definition f ({z : ψz}) =df ∃φ(∀x(φx ≡ ψx) ∧ f (φ zˆ )).

(1)

It is then possible to prove ∀x(φx ≡ ψx) ⊃ {z : φz} = {z : ψz}. If we further define y ∈ φ zˆ =df φy, we can prove y ∈ {z : ψz} ≡ ∃φ(∀x(φx ≡ ψx) ∧ φy). In the ramified theory, however, this procedure is barred, because (1) contains an apparent variable φ ranging over all propositional functions of the required level. The device Russell hit upon was, for any function f (φ!ˆz) of a variable predicative function, to use the contextual definition f ({z : ψz}) =df ∃φ(∀x(φ!x ≡ ψx) ∧ f (φ!ˆz)).

(2)

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This contextual definition of class-expressions is ontologically eliminative, enabling Russell to be agnostic about whether there is a class {z : ψz} corresponding to a propositional function ψ zˆ . What we can ask instead, though, is whether the incomplete symbol ‘{z : ψz}’ occurs in the expression of any true propositions: if it does, let us call ψ zˆ reducible (echoing the terminology of Chapter 33). From (2), the condition for ψ zˆ to be reducible is that there should be a predicative function formally equivalent to it. So in the absence of a further axiom, only predicative functions would be reducible. To develop a satisfactory theory of classes, therefore, Russell added what he called an ‘axiom of reducibility’ (or, sometimes, ‘axiom of classes’), asserting that every propositional function is reducible. In symbols: ∃φ∀x(ψx ≡ φ!x), where ψ xˆ is an arbitrary (typically ambiguous) propositional function. This schema achieves, he thought, ‘what common-sense effects by the admission of classes’ (CPBR, V, 606), because if there is a class α = {z : φz}, then φ xˆ is formally equivalent to xˆ ∈ α, and the latter, he said, is predicative (PM, I, 173). This is a delicate argument, though, because it depends on how the sign ‘α’ is introduced. If it is conceived of merely as an abbreviation for ‘{z : φz}’, Russell’s claim that it ‘involves no allusion to a variable function’ will be false: it will inherit whatever quantifiers occur in φ zˆ . His argument depends, therefore, on conceiving of ‘α’ as a simple name. But we cannot, on pain of inconsistency, assume that every class already has a name. So the argument requires us to add ‘α’ to the language as a new name. In that case, though, it is an argument not for the axiom of reducibility as stated, but for the weaker claim that for each propositional function the language has an extension in which there is an expression for a formally equivalent predicative function. Chwistek (1922) made use of a similar idea to this when he attempted to show that Russell’s theory is susceptible to a slightly more elaborate version of the paradox of heterologicality. Define a second-order propositional function Het2 xˆ by Het2 x = ∃φ(x means φ!ˆz ∧ ∼φ!x), where the exclamation mark indicates that φ ranges only over predicative functions. Use the axiom of reducibility to obtain a predicative function Het!ˆx formally equivalent to Het2 xˆ . Then Het!ˆx is a legitimate substituend for the variable φ in the definition of Hetˆx, and so we can seemingly derive a contradiction. Chwistek’s argument is fallacious, however, because it tacitly assumes that the predicative function generated by the axiom of reducibility has a name (see FoM, 28). What it shows, therefore, is only that the axiom of reducibility is inconsistent with the assumption that every predicative propositional function has a name.

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It has often been alleged (e.g. Quine 1936, Copi 1950) that the axiom of reducibility re-instates the simple theory of types, reducing the ramification of the hierarchy to an unnecessary elaboration. The effect is largely to restore the possibility of impredicative definition which the distinction of levels was designed to eliminate. Indeed, as many have urged, the true choice would seem to be between the simple functional calculi and the ramified functional calculi without axioms of reducibility. It is hard to think of a point of view from which the intermediate position represented by the ramified functional calculi with axioms of reducibility would appear to be significant. (Church 1956, 355) As Russell noted (CPBR, V, 545ff), this would indeed be the case if we were restricted to the truth-functional hierarchy obtained by applying truth-functions to elementary propositions. If our logic permits us to express intensional notions such as belief, however, the axiom of reducibility does not collapse the ramified theory into the simple one. The axiom holds only that for every propositional function there is a formally equivalent predicative function, and in intensional contexts these may be distinguished.

Identity The axiom of reducibility is required if we are to develop higher mathematics in the ramified theory of types. In the system of Principia, however, the axiom is needed much earlier, because of the oddity that identity is not treated there as a primitive. In a simple theory of types we can define identity by x = y =Df ∀ψ(ψx ⊃ ψy), but in Russell’s ramified theory this is barred, because the variable ψ ranges illegitimately over all orders. So he defined it instead by x = y =Df ∀φ(φ!x ⊃ φ!y), the functional variable now ranging only over predicative functions. The axiom of reducibility is then required in order to deduce x = y ⊃ (ψx ⊃ ψy)

(3)

for arbitrary ψ. Without reducibility, two things might share all their predicative properties but differ as to their higher-order properties. Notice, though, that this use of the axiom of reducibility is an artefact of the decision to treat identity as a defined notion. An obvious alternative would be to add it to the system as an extra primitive idea and treat (3) as an axiom schema. Of course, this would only delay the need for the axiom of reducibility, not eliminate it, but we may still ask why Russell did not take this course. One

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reason may have been the desire to keep the number of primitive ideas of the system to the minimum. Indeed there is independent evidence that this concern weighed with him: after Sheffer (1913) showed how to replace negation and disjunction with a single binary connective, Russell described this, implausibly, as ‘the most definite improvement . . . during the past fourteen years’ (PM 2nd edn, xiii). A quite different reason, however, was that his atomism problematized the very notion of a primitive relation of identity between individuals, since his account left unexplained what it would be to be acquainted with the same sense datum on two different occasions. If, as he maintained, ‘a exists’ is meaningless when ‘a’ is a logically proper name, by the same token ‘a = a’ should be too. Perhaps he hoped that defining identity might somehow get him out of this difficulty.

Further axioms In the Principles Russell had claimed that the axiom of infinity is a law of logic. As late as May 1906 he still thought that Dedekind’s argument for it was ‘fairly conclusive’ (CPBR, V, 271); and in June 1906 he derived it formally on the assumption that propositions are entities (CPBR, V, 288). With the demise of the substitutional theory, however, this assumption was no longer available to him, with the result that by September 1906 he thought that the axiom of infinity (restricted now to individuals) might not even be true, let alone provable. ‘No reason appears why the number of individuals in the universe should not be finite; nor is there, I believe, any reason except an empirical one.’ (CPBR, V, 365) Since individuals are non-logical entities, whether there are infinitely many of them cannot be a matter for logic to settle. In Principia, therefore, infinity is not treated as a logical law, but added explicitly as a premiss in the statement of every theorem which depends on it (although, rather oddly, it is still called an ‘axiom’). Moreover, in order to postpone its use as long as possible, Whitehead and Russell adopted the convention that a typically ambiguous proposition is assertible provided that there is some type in the hierarchy such that it is assertible at that type and all higher types. This permitted them to claim that quantifier-free arithmetic does not really depend on the axiom of infinity, since every quantifier-free equation or inequation is determinately true or false at sufficiently high types. The other existence assumption for which Whitehead and Russell found a need was that for any family of non-empty classes there is a class having as members exactly one member of each class in the family. They called this the ‘multiplicative axiom’, but nowadays it is always called the ‘axiom of choice’. Among the results whose proofs make use of it are ab = max(a, b) for all infinite cardinals a and b. a < b or a = b or a > b for all cardinals a and b.

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Their paper (Whitehead 1902) was one of the first to identify explicitly the role the multiplicative axiom plays in the arithmetic of infinite cardinals. Later work showed, in fact, that the above two statements are not merely entailed by the multiplicative axiom but logically equivalent to it. The regressive argument for the multiplicative axiom is relatively weak, since the arithmetic of infinite cardinals cannot be said to require the axiom, but only to be simplified by it. In other parts of abstract pure mathematics, too, the role of the multiplicative axiom is generally to simplify rather than to transform. It is never used at all in the arithmetic of natural numbers, and in the Weierstrassian theory of real numbers it makes matters easier but can hardly be said to be crucial. Moreover, the parts of mathematics where it is essential are of no practical utility. In Principia the multiplicative axiom, like the axiom of infinity, is stated explicitly as a premiss when required. The difference between them is that the axiom of infinity is ‘purely empirical’, whereas ‘the considerations which bear on the truth or falsehood of the multiplicative axiom are considerations of logic, a priori considerations’ (CPBR, VI, 52). Russell did not explain why in that case the multiplicative axiom is not a logical law, if true.

Further reading Copi (1971) explains both simple and ramified theories. Goldfarb (1989) discusses Russell’s reasons for adopting a ramified theory.

37 MIDDLE LOGICISM

In its essentials, the theory of types to be used in Principia was settled by June 1907. What remained was the labour, begun in September 1907, of writing out the book. Most of the main text—4000 pages in two crates—was delivered to Cambridge University Press on 19th October 1909. By then, though, Russell had given up several of the claims about logic made in the Principles: that classes are logical entities; that it is possible for us to come to know logical truths with certainty; that the logical truths are the same as the true logical propositions; and that logical truths are about everything.

Continuities Principia was originally conceived as a proof of logicism, and yet logicism, oddly, is not mentioned at all in the published version. Why not? The explanation cannot be that Russell had abandoned the doctrine: he re-affirmed in 1911 that pure mathematics ‘can be expressed and proved entirely in terms of the ideas and axioms of logic’ (CPBR, VI, 43). What is uncertain, therefore, is not whether he still accepted the doctrine, but only whether he was wise to do so, given the extent to which the logicist reduction depended on the axioms of infinity and reducibility (and, to a lesser extent, the multiplicative axiom), for which the a priori justification was weak. If the omission from Principia of an explicit commitment to logicism was not a simple oversight, then, it can only have been due to Whitehead’s caution, not Russell’s. Nor can there be any doubt of Russell’s continued opposition to psychologism: the laws of logic, he said in 1911, ‘are just as objective, and depend as little on the mind, as the law of gravity’ (CPBR, VI, 136); and he still thought of these

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laws as grounded in a correspondingly mind-independent realm of logical entities, drawing a contrast between ‘being[s] in the actual world’ and ‘beings in the logical world’ (CPBR, VI,44). Moreover, he still distinguished between reasoning about a formal system and metalogic proper, regarding the latter with suspicion. In relation to principles of inference, for instance, he said that ‘you can’t tell what follows from supposing them false, since if they are true, they must be used in deducing consequences from the hypothesis that they are false’ (letter to Jourdain, Apr. 1909, in Grattan-Guinness 1977, 117). Russell also still distinguished, as he had in the Principles, between particulars (what he previously called ‘things’), which may only occur as logical subjects, and universals (previously ‘concepts’), which may occur either as subjects or as predicates or relations-in-intension. And he continued to rule out treating a particular as a bundle of universals on the ground that ‘it is logically possible for precisely similar things to co-exist in two different places’ (CPBR, VI, 173). By ‘precisely similar’ here he presumably meant only sharing all their internal properties. After all, in Principia he defined individuals to be identical just in case they share all their properties. Internally indistinguishable things may be externally distinguishable, then, by dint of being in different places. Since 1905 Russell had aligned the particular/universal distinction with that between (concrete) existents and (abstract) subsistents, hence reducing the logical question of whether there are universals to the metaphysical one of whether there are abstract entities. His argument that there are did not really engage with mediaeval discussions of nominalism, but only with Berkeley and Hume, who had attempted to reduce a universal such as redness to similarity with some sample of it. Russell objected (CPBR, VI, 173) that this still required there to be at least one universal, namely similarity, which could not, on pain of regress, be explained away in turn. He concluded that they had fallen short of the nominalism to which they aspired. It is a further question how, if at all, the distinction between particulars and universals aligns with the hierarchy of types. Russell called the members of the lowest type—the entities that can occur as subject in a quantifier-free proposition—‘individuals’. His long-held view, repeated as late as 1911 CPBR, VI, 150, was that universals may occur as logical subjects. He defined an individual in 1907 as ‘something destitute of complexity’, and amended this in Principia to ‘something which exists on its own account’ (PM, I, 169). Neither definition rules out universals. One might wish, though, that he had been more explicit about this. Perhaps he was deferring to Whitehead’s wish in composing the book ‘to be as non-committal as possible and to point out its adaptability to widely divergent lines of thought’ (letter to Russell, 16 June 1907). Of an early draft of the material on individuals Whitehead wrote approvingly, ‘It is exactly the sort of non-committal statement which we want.’ (6 Jan. 1908, in CPBR, V, lxxix) Another possibility, though, is that Russell was uncertain about which universals would be revealed by analysis as ultimate: it is implausible, for instance, that

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redness might belong to the ‘world of logic’, but if redness were analysable as similarity to some suitable colour sample, it would not then be quite so implausible to treat similarity as logical.

The vicious circle principle The only argument in favour of the theory of types that we have discussed so far concerns paradox avoidance. It is, on the taxonomy Russell sometimes used, a mathematical, not a philosophical, reason. In order that the various ways of avoiding the contradictions should not seem unacceptably ad hoc, Russell needed to present them as having a single source. For this, borrowing an idea of Poincaré’s (1906), he enunciated the ‘vicious circle principle’: whatever involves all of a collection must not be one of the collection. This principle, however, is schematic at best, since it leaves open what notion of ‘involvement’ is in play. In various places Russell stated variants of it with ‘is definable in terms of ’ or ‘depends on’ in place of ‘involves’. In particular, the principle does not on its own settle the issue between the simple and the ramified hierarchies: whether the hierarchy should be ramified depends on whether a propositional function should be said to ‘involve’ all the values of the apparent variables occurring in it. The argument from the vicious circle principle for the ramified theory of types may conveniently be split into two parts: P1 A variable involves everything in its range; P2 A propositional function involves the real and apparent variables occurring in it. Assuming that involvement is transitive and irreflexive, these give us the desired conclusion: C Therefore no propositional function belongs to the range of any real variable occurring in it. What is required in order that P2 should be plausible, i.e. that a propositional function should ‘involve’ the apparent variables occurring in it? It is commonly held that this is plausible only if we conceive of a propositional function as constituted by its definition, since only on this view does ‘involves’ collapse into ‘is definable only in terms of ’. Those who take this view take Russell to have ramified his hierarchy needlessly, since he did not at this stage have a constructivist conception of propositional functions. A realist, they say, can distinguish between the being of an entity and our way of grasping it. For Russell, though, the issue was one concerning individuation, not realism. He held that in intensional contexts we are required to distinguish propositional functions on the basis of their apparent as well as their real variables, and therefore to recognize these as contributing to their being.

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Notice, moreover, that premiss P1 is far from uncontroversial. In defending Frege’s impredicative definition of the ancestral in the Principles, Russell had argued, in effect, that there is a sense of involvement on which a variable does not involve all its instances.

The universality of logic Russell had searched strenuously for a system of logic that avoided the paradoxes without abandoning an unrestricted variable ranging over everything. In 1906, while he still hoped that the substitutional theory could be made to work, he criticized ‘the older symbolic logicians’ for their use of relative quantifiers, ‘setting, as it were, bounds of decency, outside which no well-conducted variable would wander’ (CPBR, V, 289). The following year, having admitted defeat in the substitutional project, he adopted the view that there is a hierarchy of variables, each restricted to a single logical type by just the bounds of decency he had earlier ridiculed. One difficulty with this was that it was now a somewhat more delicate matter to explain the sense in which logic is maximally general. Although a logical proposition may not be about absolutely everything, he suggested, it is nonetheless as general as possible. Every logical notion, in a very important sense, is or involves a summum genus, and results from a process of generalization which has been carried to its utmost limit. This is a peculiarity of logic, and a touchstone by which logical propositions may be distinguished from all others. A proposition which mentions any definite entity, whether universal or particular, is not logical: no one definite entity, of any sort or kind, is ever a constituent of any truly logical proposition. (CPBR, VII, 97–8) (The logical constants presumably do not count as ‘definite entities’ here.) Some commentators (e.g. Klement 2010, Landini 1998, Stevens 2005), it should be said, question the account just given. They maintain that by the time of Principia Russell already regarded propositional functions as mere symbols and interpreted higher-order quantification nominalistically, with the consequence that all the entities in his ontology were individuals. These commentators will presumably also have to insist, somewhat awkwardly, that Russell’s talk of a ‘logical world’ should not be taken literally. The principal difficulty with their account, though, is the frequency with which Russell described types as limitations on the ranges of variables. There is no doubt that he eventually adopted a linguistic conception of propositional functions (for reasons we shall discuss in Chapter 42), or that there are places in Principia where he speaks in this way, but there are none which could not be instances of his notoriously casual attitude to the use/mention distinction. In fact, the strongest evidence for this nominalistic reading of Principia is a much later (1959) recollection of Russell’s that ‘Whitehead and I thought of a propositional function as an expression containing an undetermined variable and

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becoming an ordinary sentence as soon as a value is assigned to the variable’ (MPD, 92, my emphasis). Since the two men did not collaborate after the first edition of Principia, Russell must have been referring here to their views during that work’s composition. So anyone who, like me, rejects the nominalistic reading of the first edition must hold that here his recollection was simply mistaken. (It is germane to note that in the very next sentence after the one quoted he says that propositional functions ‘are constituted by intensions except as regards the variable or variables’, which would surely be wrong if they were symbols.) There is, however, one further twist to this. Shortly before Principia was published Russell adopted a new account of propositions with radical consequences for his ontology. I shall postpone consideration of this account to Chapter 40.

Classes as fictions Whatever his attitude to propositional functions, in Principia Russell retained from the substitutional theory the idea of treating classes as logical fictions, the contextual definitions for which were ontologically eliminative, in the sense that they allowed a sentence containing a class-expression to be true without there being a class for the expression to denote. It might seem, then, as if we could straightforwardly conclude that he had no ontological commitment to classes. The difficulty with this, however, is the justification Russell attempted for the axiom of reducibility. Formally, this axiom plays the same role in Principia as the axiom of separation plays in Zermelo-Fraenkel set theory. The sense in which it can be thought of as a class-existence axiom is somewhat complicated, however, by the fact that Russell treated classes as incomplete symbols. On first adopting this view in 1906, he glossed it as meaning that ‘there are really no such things as classes’ (CPBR, V, 244), but the following year he weakened this to the claim that ‘there is no advantage in assuming that there really are such things as classes’ (CPBR, V, 606). What had changed was that he now hoped to appeal to classes in support of the truth of the axiom of reducibility: the axiom expresses what would be true if there were classes. Needless to say, it is hard to decode the modality here. Part of the difficulty, no doubt, was Russell’s previously mentioned lack of clarity about the ontological commitments of a contextual definition. Why, then, did Russell still treat classes as logical fictions? There was no compelling technical reason to do so. While he adhered to the substitutional theory, Russell had regarded the method of incomplete symbols as a unified explanation for the paradoxes. Now, though, his explanation for the meaninglessness of ‘a ∈ a’ simply piggy-backed on the meaninglessness of ‘φ(φ xˆ )’. This is the point at which Russell’s lack of a good criterion for the incompleteness of a symbol was egregious. The only argument Russell in Principia offered against there being classes was a brief allusion to ‘the ancient problem of the One and the Many’.

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If there is such an object as a class, it must be in some sense one object. Yet it is only of classes that many can be predicated. Hence, if we admit classes as objects, we must suppose that the same object can be both one and many, which seems impossible. (PM, I, 75n) The obvious riposte to this, though, is that it merely confuses the plurality of φs, which are many, with the class of φs, which is one.

The regressive method When Russell first proposed in January 1901 that mathematics is part of logic, he trumpeted the certainty this conferred on mathematics. A decade later he was unsurprisingly less sanguine, conceding that ‘infallibility is never attainable’ (CPBR, VI, 31). In Principia, the focus of his uncertainty lay with the axiom of reducibility, which he had to admit was ‘not self-evident’ (CPBR, VI, 30). He now proposed—sketchily in 1906 (CPBR, V, 280–1), then in more detail the following year (CPBR, V, 572–80)—a new regressive method of justifying logical axioms. He based this on the observation that many of the theorems proved in Principia were known to us already—to take an extreme example, 1 + 1 = 2 is proved in volume II after several hundred pages—and we are more certain of them than we are of the axioms. This, he argued, suggests a difference between the logical and the epistemological orders: truth flows logically from axioms to theorems; but justification flows in the opposite direction, from theorems to axioms. The regressive method has some similarity to the one-step-back-from-disaster strategy (see Maddy 1988) which Zermelo (1908) used to justify his axiom system for the theory of classes. Given that the naive conception of class is contradictory, Zermelo said, there is nothing left for us to do but to proceed in the opposite direction and, starting from set theory as it is historically given, to seek out the principles required for establishing the foundations of this mathematical discipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to exclude all contradictions and, on the other, take them sufficiently wide to retain all that is valuable in this theory. (van Heijenoort 1967, 200) We could, that is to say, treat it as a purely technical challenge to place restrictions on the formal theory sufficient to block derivations of the known paradoxes but not sufficient to block the logicist reduction of mathematics. The main difference, however, is that Zermelo focused on avoiding paradox, whereas Russell had a more demanding constraint on the data that the theory should be capable of deriving. The regressive method is, Russell claimed, ‘substantially the same as the method of discovering general laws in any other science’ (CPBR, V, 573). The difference between the cases is that whereas the axiomatization of an

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empirical science aims to generate observational data—typically the results of experiments—as consequences, Russell’s axiomatization of logic aims to generate mathematical propositions: he retained the logicism he had advanced in the Principles, but now gave up the hope that introspection would reveal the properties of the logical constants as successfully as perception does the taste of pineapple. The principal difficulty with the regressive method concerns the justification of the propositions that are to serve as data. (It would of course be circular to appeal at this point to logical proofs of them.) In his essay, Russell not only referred to the data as the ‘empirical premiss’ of his argument, but gestured towards an empiricist account of how we come to know them. Yet this could at most justify those arithmetical propositions that have actually been tested empirically— ‘inferred from the case of sheep and other concrete cases’, as he vaguely suggested (CPBR, V, 572). These form a finite class, and hence there is an upper bound N to the numbers occurring in them. N is, for instance, far less than the number of baryons in the observable universe (about 1080 ). This finite class of empirically verified arithmetical propositions radically underdetermines even the class of arithmetical truths, let alone the logical theory capable of proving them. Empirical evidence on its own gives us no reason to prefer standard arithmetic to the variant in which all sums involving numbers greater than N come to 0 (say). So Russell proposed that before we seek the logical axioms that will generate the data, we should first enlarge the set of data from the finite set of propositions that have been empirically tested to an infinite set that includes all of quantifier-free arithmetic. He was rather vague, however, about what justifies this enlargement. Moreover, if the basis of the data were empirical, the logical principles derived by means of the regressive method would also be empirical. It is unclear, however, how seriously he meant this empiricist account of mathematical data. Elsewhere in the essay, he contrasted the empirical obviousness of the data of science with the ‘a priori obviousness’ of the data of pure mathematics; and in later writings he still maintained that pure logic is ‘wholly a priori’ (OKEW , 53). Russell sought to apply the regressive method to all the laws of logic—he even suggested, implausibly, that the law of non-contradiction ‘must have been originally discovered by generalizing from instances’ (CPBR, V, 573)—but it is difficult to see how a regressive justification of the ramified hierarchy can co-exist with a justification by means of the vicious circle principle. Perhaps he should have divided the logic of Principia into two parts—one consisting of the ramified theory of types, the other of the theory of classes—and applied the regressive method only to the latter. Whether this would still count as logicism depends on whether the claim that the theory of classes is part of logic survives the separation. Despite its difficulties, Russell maintained this regressive conception of the justification of mathematics in his later writings. If it were accepted, what would remain of the epistemic benefit of logicism? He claimed that organizing our knowledge by means of an axiomatic system allows us to test the coherence of the whole, and hence may increase our confidence in it, on the doxastic principle

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that we should prefer a more to a less coherent theory. What is less clear, though, is why he thought we should do this.

Logical truth In the Principles, as we have seen, Russell was reduced to defining the logical constants by enumeration. In 1911 (CPBR, VI, 35–7) he tried instead to characterize them as those items in a deduction that cannot be replaced by variables while preserving the deduction. This is hardly an improvement, though, since it simply transfers the problem to characterizing the notion of deduction. In any case, even if he had found a non-circular characterization, he would have faced a further difficulty in characterizing logical truth. As we saw in the last chapter, he conceived of the individuals as empirical rather than logical and concluded from this, in the summer of 1906, that it is an empirical, not a logical, question whether the axiom of infinity (that there are infinitely many individuals) is true. Yet this axiom can be stated in pure second-order logic. So a logical truth cannot just be, as he had earlier supposed, a logical proposition that is true. It seems, however, that he did not at first pay much attention to this consequence of his views. As late as 1911 he still said that ‘pure mathematics—which is usually called “logic” in its elementary parts—is the sum of everything that we can know, whether directly or by demonstration, about certain universals’ (CPBR, VI, 39– 40). The universals he was referring to were presumably the logical constants and variables, in which case the axiom of infinity should count according to this statement as part of logic (unless he now thought that we are incapable of knowing whether it is true). Not until 1918 did Russell explicitly acknowledge that not every proposition consisting wholly and solely of variables is logical (CPBR, VIII, 208). There then arose the question of what more is required, to make a truth logical, than that it be expressible in pure logic. Russell’s failure to address this question was no doubt due in part to the fact that his other views gave him very few options for doing so: his hostility to possible worlds ruled out one otherwise promising route (later exploited by Wittgenstein); his conception of epistemology as having a psychological component ruled out another (favoured by Frege).

Further reading Mayo-Wilson (2011) discusses Russell’s views on the benefits of adopting coherent theories. On Russell’s regressive method, see Irvine (1989) and Lakatos (1978, ch. 1). There is now a large literature on whether genuinely universal (as opposed to maximally general) quantification is coherent. On classes as logical fictions, see Boer (1973) and Demopoulos (2007).

38 ACQUAINTANCE

I have referred several times already to a putative relation, called ‘acquaintance’ by Russell, between mental and non-mental entities.

Particulars In a famous passage in The Problems of Philosophy (ch. 1), Russell looked round his study and asked what he could be certain of: not the table, since he did not experience that directly, but only the brown patch in his visual field—he could not be mistaken about that. The relation of acquaintance thus exemplified is, he supposed, sufficiently immediate to make it inconceivable that its object should be illusory. Russell called the particulars with which we are acquainted ‘sense data’. The term was coined by Royce (1885, 321), but in the sense in which he now used it Russell was following Moore, who discussed sense data under this name in his lectures in the autumn of 1910 (see Moore 1953, 32). Russell read these lectures shortly before writing the Problems, as he acknowledged, and they evidently influenced the sense-datum theory he expounded there. In the Problems, Russell said little about what kind of thing a sense datum is, and what he did say hardly discouraged the notion that it is mental. Berkeley was right in treating the sense data which constitute our perception of the tree as more or less subjective, in the sense that they depend on us as much as upon the tree, and would not exist if the tree were not being perceived. (PP, 64) If he had conceded that these ‘more or less subjective’ sense data are mental, however, his view would have collapsed into subjective idealism, since he also held that it is only via them that we have any empirical knowledge at all. Later he

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claimed indignantly that he had never intended to suggest this: only the apprehension of the sense datum is mental, he now insisted, not the datum itself (CPBR, VI, 186–7). In order for this to be at all plausible, however, there were two troublesome counter-arguments to which he needed to respond. First, we might argue that the sense datum is mental because it exists only for the short period that it is sensed. Even if we accept, as Russell did, that not everything non-mental is permanent, realism requires that non-mental entities are at least capable of existing without being perceived. He responded that it is not intrinsic features of the sense data themselves that make them go out of existence when we stop observing them, but only ‘empirical reasons of detail’ (CPBR, VI, 186). Unfortunately, he did not explain what these reasons are. A second argument is that the sense datum is mental because it is private: no two of us have ever been acquainted with the same sense datum, since even if we look at the same object, we inevitably do so from slightly different angles and hence sense slightly different data. This inference from privacy to mentality seems to have worried Russell less than that from impermanence: in the Problems he observed without comment that ‘sense data are private to each separate person’ (PP, 32). Not until two years later did he think it worth mentioning that this privacy is not intrinsic but an accidental consequence of the shape of our skulls: sense data are publicly available, even though each is as a matter of fact presented to only one person (CPBR, VII, 30). Moore was rather more alert to the issue, claiming (at a meeting of the Aristotelian Society in 1909 which Russell attended) that privacy is not the mark of the mental. Moore rejected the view that the objects of perception are mental—a view which he diagnosed as resulting from a failure (of which he accused Hume) to distinguish between the datum itself and the act of which it is the object. In his argument for this distinction, however, Moore followed Hamilton (1861, I, 288) by relying heavily on first-person testimony. I cannot persuade myself that a blue colour which I see is related to me in exactly the same manner in which my seeing of it is related to me. It seems to me to be related to me in no way at all except by the fact that I am conscious of it. But my consciousness of it is related to me in quite a different manner from this. Of my consciousness I am not by any means always conscious. Its relation to me is simply that it is my consciousness, an act of consciousness of mine: and the blue which I see certainly does not seem to me to be ‘mine’ in this sense, whatever that sense may be. (1909, 59) One might suspect that Moore was equivocating here between the shade of blue, which indeed is not in the relevant sense mine, and the patch in my visual field, which is. Perhaps, too, he was aided by his tendency to focus on the cases of vision and sound, where the distinction has at least a prima facie plausibility, in contrast to the other sensory modalities, where it has none: we feel no temptation to

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distinguish between my feeling of pain and the pain itself, or between my feeling of tiredness and the tiredness itself. For any dualist, there is an awkward transition between mind and world: for Frege, it comes when the mind grasps a mind-independent sense; for Russell, it comes when the mind is acquainted with a sense datum. It is curious how little attention Russell gave to Frege’s theory. The explanation seems to be that he took Frege’s notion of a mediating element standing between subject and object to have been refuted, whether by Moore’s argument against ‘universal meanings’ or by his own Gray’s Elegy argument, and supposed as a consequence that any such theory would place a veil between us and the world. He concluded, therefore, that ‘the actual objects concerning which we judge . . . are constituents of the complex which is the judgment’ (CPBR, VI, 156).1 He thus continued to think that in spite of all its snowfields Mont Blanc itself is a component part of what is actually asserted in the proposition [Satz], ‘Mont Blanc is over 4000 metres high’. . . . If we do not admit this, then we get the conclusion that we know nothing at all about Mont Blanc.’ (Letter to Frege, 12 Dec. 1904, in PMC, 169) It need hardly be said, however, that Russell’s reasons for thus rejecting the Fregean account were inadequate, because they depended on his dubious assimilation of it to quite different views (whether Bradley’s or Russell’s own).

Complexes Some of our worldly beliefs justify others, Russell thought, but there must, on pain of infinite regress, be some that are grounded not in other beliefs but directly in acquaintance. If, for instance, A and B are adjacent discs in my visual field, and C is the complex A-above-B, it is my acquaintance with C that grounds my belief that A is above B. This conception of acquaintance as supplying our knowledge of the external world with an indubitable basis is often described as ‘foundationalist’ or ‘Cartesian’. Notice, though, that what it takes to be indubitable is only that I am acquainted with C, not what I take to be its structure. In passing from the perception to the judgment, it is necessary to analyse the given complex fact. . . . In this process it is possible to commit an error; hence . . . a judgment believed to correspond to the fact is not absolutely infallible, because it may not really correspond to the fact. But if it does correspond, . . . then it must be true. (PP, 214) Hence ‘all or any of our beliefs may be mistaken’ (PP, 39). Am I not certain at least in believing that C exists? No, because ‘C’ is a logically proper name, and so

1 Here Russell was in fact considering the proposal that the mediating element is mental, but this does not affect the underlying point.

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‘C exists’ is, strictly speaking, meaningless. One might wonder, though, whether introspection is a (limited) source of certain knowledge. Do I not know that I am acquainted with C (or, even more modestly, that this is being sensed)? Although Russell mentioned examples of this sort, he did not make it clear how he thought they could be in error. I may be mistaken not only about the spatial relations of sense data, but also about their temporal relations. Russell tried at first to maintain that I may by means of memory have acquaintance with a sense datum that was once, but is no longer, present to me. This must be a mistake, however, since it is surely intelligible that my memory might be faulty, in which case the thing that I now incorrigibly name cannot be the very same sense datum that I was acquainted with formerly but must rather be my current (possibly erroneous) memory. At first he tried to head off this difficulty by restricting acquaintance to the case where the memory is genuine (PP, 183), but it is hard to see how this coheres with the rest of his account, since if we are willing to grant this restriction, by the same token we must surely allow ‘genuine perception’ of tables and chairs. Later Russell conceded this point and held instead (CPBR, VII, 66–70) that I can incorrigibly name an object only during what he called—using a term popularized by James (1890)—the ‘specious present’, i.e. the short time interval of which, at each moment, I am immediately aware. The fundamental difficulty was that Russell had two notions of simplicity in play. In our earlier example, I gave the name ‘C’ to a sense datum, and thus ensured that logic would treat it as being in a certain sense simple; ‘C’ is not just an abbreviation for ‘A-above-B’, since if it were, I would not then be able to express my doubt as to whether this analysis is correct. Nonetheless, since C is Aabove-B, it is in another, equally valid, sense complex. Lecturing in Paris in 1911, Russell claimed (CPBR, VI, 134–5) that everything is composed of simples, but did not make clear which sense of simplicity he meant.

Universals Our judgments are about universals as well as particulars, Russell thought, and so the principle of acquaintance entails that we are acquainted with these too. He considered briefly the alternative proposal that we are acquainted only with the complexes in which a universal occurs, not with the universal itself; but, this view, he said, is difficult to reconcile with the fact that we often know propositions in which the relation is the subject, or in which the relata are not definite given objects, but ‘anything’. For example, we know that if one thing is before another, and the other before a third, then the first is before the third; and here the things concerned are not definite things, but ‘anything’. It is hard to see how we could know such a fact about ‘before’ unless we

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were acquainted with ‘before’, and not merely with actual particular cases of one given object being before another given object. (CPBR, VI, 150) Nonetheless, it is difficult, in the empirical case at least, to see what acquaintance with a universal, independent of its instances, might be like. If I had never seen a white object, I would presumably have no acquaintance with whiteness, but how, in that case, would I analyse the first white patch I see? Russell proposed an uneasy compromise according to which my acquaintance with the universal is acquired gradually, ‘by seeing many white patches’ (PP, 158). Once acquired, it may then outlast my acquaintance with the white particulars that prompted it. We might wonder, though, whether this gradual process is sufficiently direct to confer the indubitability distinctive of acquaintance. When it comes to a priori concepts, the problem seems even worse, since it is not even clear what the instances are from which I am supposed to abstract. Is the idea supposed to be that I become acquainted with negation, for instance, via acquaintance with instances of absence, such as seeing that there is no rhinoceros in the room? The alternative, of course, would be to claim that acquaintance with a priori concepts does not depend on instances. The logician’s task would then be to help us to ‘have that kind of acquaintance with them which it has with redness or the taste of pineapple’ (PoM, xv), but many have found this sort of a priori acquaintance wholly mysterious.

The variable ‘True proper names’, Russell claimed, ‘can only be conferred on objects with which we are acquainted.’ (CPBR, VII, 37) This is because the theory of descriptions analyses the sentence ‘The f has being’ but provides no corresponding analysis of ‘A has being’ where ‘A’ is a logically proper name. If I am not acquainted with A, the mere fact that I can doubt whether it has being is therefore enough to show that in my idiolect ‘A’ is a disguised description, not a logically proper name, since otherwise the theory would be unable to analyse my doubting. We should note the role that Russell’s rejection of the notion of modality played in his conception of sense data. Suppose, for instance, that A is the brown patch currently in my visual field as I look at the table before me. Those who grant the coherence of genuinely modal talk will probably wish to say that A might not have existed (since I might, for instance, have painted the table blue yesterday), and will therefore have to treat ‘A’ as a disguised description. In that case, though, the theory will be in danger of collapse, since few of the particulars presented to me are both indubitable and necessary. Russell could resist this conclusion because he denied that ‘A might not have existed’ is a straightforwardly meaningful modal claim. Russell had used his original theory of denoting concepts to explain how an object can be thought about other than by being a constituent of a proposition. This idea survived the theory’s demise, but now he spoke of two ways of knowing

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an object—by acquaintance and by description. He was not the first to make such a distinction—something similar occurs, for instance, in James (1890, 221)—but he was the first to link it to two ways of referring to an object: in the former case, directly; in the latter, indirectly via a variable in whose range it lies. So the particulars that occur in your visual field are known to you by acquaintance, but to me only by description. On Russell’s account, then, the variable played the problematic role of enabling me to think generally about objects which I cannot think about individually. On reading ‘On denoting’ Moore wrote to ask Russell what sort of entity the variable is, and he replied: I only profess to reduce the problem of denoting to the problem of the variable. This latter is horribly difficult, and there seem equally strong objections to all the views I have been able to think of. (25 Oct. 1905, in CPBR, IV, xxxv) As his working notes show (CPBR, IV, 387–8), Russell had indeed been aware already that his new theory did not in fact eliminate denoting concepts, but merely reduced them all to the single case of the variable. This was an indefinite, not a definite, denoting concept, but that scarcely helped, since, as we noted in Chapter 34, the Gray’s Elegy argument applied to these with just the same force. The theory of descriptions offered an explanation of how (indirect) reference to objects other than sense data is possible, but only by using the variable as an epistemological magic bullet, enabling us to quantify over entities we have not named.

The self Russell had long accepted a dualism between mind and matter, but had based his acceptance, it seems, on little more than Moore’s introspective report of a recognized distinction between a (psychical) sensation and its (physical) object. In 1912 Russell was asked to review for Mind a collection of essays in which James (1912) rejected this distinction; but in response Russell did little more than repeat Moore’s claim that ‘mere inspection of experience’ (CPBR, VI, 303) demonstrates the relational character of perception. The following year, however, Russell offered a further, ‘most conclusive’, argument in favour of dualism, namely that monism denied the existence of a simple Self and hence rendered indexical thought inexplicable. ‘To me it seems obvious’, he said, ‘that such “emphatic particulars” as “this” and “I” and “now” would be impossible without the selectiveness of mind.’ (CPBR, VII, 40–1) The monist, he thought, had no way of generating a self-centred language using only neutral resources. Even if a simple Self exists, it is a distinct question whether I am acquainted with it. In 1911 Russell thought it ‘probable, though not certain, that we have acquaintance with Self, as that which is aware of things or has desires towards things’ (PP, 81). The reason he hesitated was that introspection, so often his

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favoured method of determining the facts of psychology, did not seem to reveal such a thing. ‘When we try to look into ourselves we always seem to come upon some particular thought or feeling, and not upon the “I” which has the thought or feeling.’ (PP, 78) By 1913, though, hesitation had turned into rejection (CPBR, VII, 40). Reflection, he now held, reveals to me merely that something is acquainted with the object, not what that thing is. Russell thought it obvious that ‘ “I” itself is not a universal’ (CPBR, VII, 36), and hence did not seriously consider the alternative that, like whiteness, it is revealed by repeated acquaintance with individual acts of sensation (see Taylor 1994). The simple Self which his dualism required was therefore a particular known only by description on the basis of inference, not directly by acquaintance. Whether this inference was fallible was a question to which he would shortly return.

Further reading For detailed discussion of Russellian acquaintance, see Pears (1967, ch. 3), Sainsbury (1986), Savage (1989) and Bostock (2012). In the case of universals, Jackson (1986) used the distinction between knowledge by acquaintance and by description in an argument against physicalism.

39 MATTER

Wittgenstein turned up at Russell’s rooms at Trinity on 18 October 1911 announcing his desire to work on the philosophy of mathematics at Cambridge. After hesitant beginnings, he soon turned from an annoying student into not only a personal friend but at least Russell’s philosophical equal. Russell was then about to begin the project, only hinted at in the Problems, of investigating in more detail the relationship between sense data and ordinary physical objects: whereas a psychologist investigating perception might hope to explain the former in terms of the latter, he hoped to do the opposite. This was the first of several issues on which, as we shall see, Wittgenstein soon affected his views.

On the notion of cause ‘On the notion of cause’, written in September 1912, is known for the Humean scepticism towards causation which Russell there advanced. His primary target was the ‘same cause, same effect’ model. His idea was that if the events concerned are specified with sufficient exactitude, it is highly unlikely that the same event will ever recur. He granted that science contains ‘laws of association’—empirical generalizations about observed regularities in the behaviour of physical systems. In the ‘advanced sciences’ he had in mind, these generalizations—e.g. Newton’s laws of motion or Maxwell’s field equations—are typically expressed as differential equations, but this is incidental. What matters is that they are time-reversible and hence cannot give rise to the time-asymmetry of causation. Russell’s claim that laws of causation cannot be deduced from laws of association has been widely accepted. It requires a separate argument, though, to show that science does not make use of laws of causation in addition to laws of association, and on this point his argument was shakier.

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Russell’s scepticism about causation was in one respect more nuanced than Hume’s. Rather than denying the notion outright, he claimed more narrowly that it does not occur in gravitational astronomy, which he took to be an exemplar of advanced science, and that its occurrence in the formulation of other sciences should be regarded as a sign of their relatively primitive state of development: as they become more advanced, he predicted that their appeal to the notion might be expected to diminish. One objection to this, perhaps, is that Russell’s prediction has so far turned out false. Another (see Cartwright 1979) is that he ignored the role causes play in guiding our actions. Causal laws allow us to predict how a system will behave if we interact with it. The reason why gravitational astronomy does not use the notion of cause is not that it is more advanced than other sciences, but that we cannot intervene to manipulate the paths of stars. The test of causation’s usefulness is not how primitive a science is but how local: causation comes into its own when we study small-scale systems in which we can intervene; its explanatory role diminishes in proportion as the system becomes larger and our capacity to manipulate it less. An alternative proposal is that the notion of cause is not so much local as agent-dependent. The time-asymmetry of causation can then be explained by the time-asymmetry of agents’ memories. The difficulty, though, is to explain how this applies in cases (such as earthquakes) where we use the language of causation even though they are beyond the capacities of any worldly agent to bring about (see Menzies & Price 1993, §5). There is an obvious risk at this point of making physics dependent on a kind of theism, since we shall have to posit an agent in these cases too.

Inference and construction Russell’s work on the problem of matter stemmed from his dissatisfaction with the inference from sense data to physical objects. One reason for this was that he now rejected the idea that simplicity can be a criterion of true explanation: unless the world was ‘created for the purpose of delighting mathematicians’ (CPBR, VI, 86), we have no right to prefer simple explanations to complex ones. Another reason was that physical objects seemed to be of a different kind from sense data. One might be forgiven for supposing—as some readers of the Problems did—that physical objects, unlike sense data, are incapable of being presented to us; and one might be further encouraged in this by Russell’s willingness to write as if the distinction between sense data and physical objects were analogous to that between appearance and reality. When challenged on this point by Dawes Hicks (1912), he tried to resist the analogy by claiming that he did not ‘know of any reason why the mind should be “disqualified” from knowing the physical thing’ (CPBR, VI, 186). This is a difficult position to maintain, however, since if sense data are of the same intrinsic nature as physical objects, it is puzzling why physics never mentions them. Eventually, therefore, Russell acknowledged that his

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inference from sense data to the physical object should be eliminated if possible and replaced with a logical construction. This is an instance of a general method. In the typical case there will be a class of assumed entities, treated for present purposes as unproblematic, and a class of doubtful entities (those spoken of in ordinary discourse, perhaps, or in physics), of whose existence we are uncertain. Instead of inferring the doubtful entities from the assumed ones, as he had done previously, he now proposed to define various constructed entities (classes, relations, etc.), using the assumed entities as a base, and then show that these constructed entities can perform the role previously assigned to the doubtful ones. At first Russell considered the instance of this method which he called ‘naive realism’, where ordinary objects are replaced by sets of sense data. A cat, for instance, might be replaced by the set of its appearances. He had two objections to this proposal. First, the cat cannot have ever been in any place where I did not see it; thus we shall have to suppose that it did not exist at all while I was not looking, but suddenly sprang into being in a new place. (PP, 36) Notice, though, that this is an objection only to taking the basis of the construction to be sense data, not to the method of construction itself. His second objection was more general. The cat, he said, ‘cannot be hungry, since no hunger but my own can be a sense datum to me’. This is a little too swift, however. The naive realist no doubt has a substantial task, to show how the cat’s hunger is reducible to sense data, but Russell had given no conclusive reason to suppose that it cannot be completed. Eventually he himself realized this and withdrew his objections to the method of construction. His first announcement in print of this change of mind was in 1914, but he had already made it in a paper ‘On matter’ which he wrote, but did not publish, in 1912. He said that he had got the idea for the method from Whitehead, and perhaps he had: it was an extension of a method which they had already applied in Principia, where they proposed that numbers should be replaced by certain classes. The difference was that now the objects to be replaced were physical, not mathematical. The idea of constructing physical objects out of classes is prima facie rebarbative to common sense, which insists that I feel pain in my toe because I stub it on a chair leg, not because of some complicated relationship between various classes of sense data. Notice, though, that Russell did not deny the existence of the chair leg, but merely declined to assert it. So he was not required to show that our common sense belief in the chair leg is faulty, but only that it lacks adequate justification. The method of construction requires for its utility merely that the existence of the assumed entities should be more certain than of the dubious ones. As a matter of epistemic caution, therefore, Russell held that we ought to prefer the constructed entities to their proxies. In the case to which he applied his method, however, it is a little difficult to articulate the threat to which this epistemic

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caution was supposed to be a response. The construction demonstrates at most that physics is consistent, not that it is true.

The constructional base Suppose we grant the method of construction. Which are the assumed entities that form the constructional base? Russell considered three positions. The solipsist starts only from the data of his own acquaintance. What Russell called the ‘phenomenalist’ also includes among the assumed entities the data of other people. Finally, the indirect realist additionally includes entities of the same kind that are not in fact sensed, because they occur in places where no sense organ is present. At first (CPBR, VI, 185), he called these putative entities ‘qualities’, but in 1914 he coined the term sensibilia (singular: sensibile). The first of these three positions, that of the solipsist, Russell showed no inclination to take at all seriously until it was urged on him by Wittgenstein in 1912. I argued about Matter with him. He thinks it is a trivial problem. He admits that if there is no Matter then no one exists but himself, but he says that doesn’t hurt, since physics and astronomy, and all the other sciences could still be interpreted so as to be true. (23 Apr. 1912) The context is an article Russell was trying to write, ‘On matter’, in which he addressed whether physical matter could be constructed logically out of sense data. He discussed it with Wittgenstein more than once, and this may explain his uncharacteristic hesitancy in the article (which he never published) about whether the construction is possible or not. Wittgenstein’s sympathies evidently lay on the negative side of the issue. ‘I cannot imagine your way of working from sense data forward,’ he told Russell (CL, no. 10). Central to Russell’s conception was a distinction between the part of the world I am acquainted with and the part I am not. What enables me to make judgments about the latter is the variable, which served for Russell as a kind of epistemological magic bullet, enabling me to quantify over what I cannot name. For Wittgenstein, by contrast, a variable is merely a piece of symbolism, a way of collecting together propositions for notational purposes, and cannot play any such substantive epistemological role as Russell had allocated to it. He deduced that Russell’s notion of knowledge by description is illusory, and hence that the only things I can talk about are those I can name. If those were just the things I was acquainted with, as Russell supposed, this would lead to the conclusion that my world consists of what I am acquainted with. Each of us would be trapped, in practice if not in principle, in a private language of acquaintance. To Russell, this solipsism of acquaintance was too extreme to be plausible. He therefore felt confident of the phenomenalist inference to data sensed by other humans. To him the question was only whether this phenomenalism was a stable resting place on the way to full-blown realism. After all, the evidence for the

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existence of other humans is no stronger than for the existence of matter (the former being an instance of the latter). And if human testimony is ignored, the evidence for the existence of the sense data of other humans is no stronger than the evidence for unsensed sensibilia. That there are some unsensed sensibilia is perhaps plausible enough once we grant that sensibilia are non-mental: it would surely seem like magic if sensibilia were mind-independent and yet each of them was sensed by someone. For the proposed construction of the external world to have any chance of success, however, we must assume not just some but a continuous network of them. If I shut my eyes briefly while looking at a light, for instance, there will have to exist not only the two sense data with which I am acquainted beforehand and afterwards, but also the unsensed sensibile with which I would have been acquainted if I had kept my eyes open in the meantime. Russell now rejected the inference he had made in the Problems from sense data to physical objects, and yet his new position depended instead on the inference from sense data to unsensed sensibilia. Why was this an improvement? In the terminology popularized by Ayer (1971, 57), the former was a ‘vertical’ inference, the latter ‘horizontal’. The horizontal inference is preferable, Russell thought, because it ‘avoids an unknowable noumenon, since matter will consist entirely of things of the kind with which we are acquainted’ (CPBR, VI, 94). He could not pretend that the new ontology was more modest than the old, perhaps, but he could at least claim that it was simpler.

Public and private space Once Russell had overcome his objections of principle to the method of construction, it was still not clear to him whether a construction of the matter of physics was possible. His difficulty was that although the sense data we receive from nearby objects may plausibly be said to have distance encoded in them— stereoscopic vision enables us, for instance, to estimate how far away a nearby object is—this is not true for distant objects. Stars in the night sky, for instance, appear to us merely as points of light on the celestial sphere: their angle of presentation is perceptible, but the radius of the vector (the distance from us to the star) is not. So the relationship between matter and sense data is (in astronomy, at least) many-one, not one-one. In 1912 he concluded accordingly that the distance scalar has no direct basis in experience, and hence that no general reduction of the discourse of astronomers to talk about sense data is possible. In 1914, however, Russell enriched his conception of the information presented to us in acquaintance. In relation to nearby objects, at least, my sense data have a magnitude encoded in them, so that my view of the nearby world at a given moment is three-dimensional. It is, in practice if not in principle, private to me, since no one else has just this view, but there are also, he supposed, three-dimensional worlds adopting perspectives which neither I nor anyone else

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in fact occupies. In order to construct the physical world from these resources, he needed to be able to patch together different perspectives by positioning them in a three-dimensional space that he called ‘perspective space’. His method for doing this was rather like assembling a jigsaw: if the sensibilia of two perspectives are similar, he supposed that the points of view of the two perspectives are near together in perspective space (OKEW , 88). If I succeed in positioning the various private worlds in perspective space correctly, I can plot two positions for each sensibile, that from which and that at which it is located. It is then possible to construct matter by bundling together components of different worlds that are located at the same place in perspective space: a momentary piece of matter is a class of sensibilia drawn from different perspectives but directed towards the same position; and a ‘thing’ is a continuous temporal series of such momentary fragments. However, I can also construct perspectives as classes of sensibilia directed from the same position towards different positions. Russell called a continuous temporal sequence of such perspectives a ‘biography’. Objects and biographies are thus both constructed out of sensibilia, but according to different rules of classification. Russell likened this to the post office directory, which used to list the same people twice, first by name, then by address (CPBR, VIII, 195). In both cases, the construction over-generates: a ‘thing’ may, but need not, coincide with what physicists regard as a physical object; and a ‘biography’ may, but need not, constitute ‘the whole of the data of one percipient throughout his life’ (CPBR, VIII, 85). Notice, though, that there is considerable difficulty in the procedure. Perspective space is an inference (OKEW , 88), and hence fallible like other non-deductive inferences. Sometimes we shall want to put two perspectives close together even though they are very different, perhaps because there is a discontinuity in the way the world looks, or because we decide that one of the perspectives involves a hallucination. (You see a dagger before you, but I, standing next to you, do not.) Conversely, distant parts of the world might look the same, in which case we could be misled into treating the perspectives as close together when they are not. (Anyone who has done a jigsaw involving large expanses of clear blue sky will understand the problem.) And even supposing that I have located the private worlds correctly in perspective space, I must then, in order to determine where a sense datum is located, convert it from an element of a private world into a vector in public (i.e. perspective) space. It is far from clear how this is to be done, since my private world does not come with a handy ruler to mark the scale and I cannot move about to measure the distance without adopting a different perspective. The story Russell told later (ABR, I, 325) was that the idea for the construction of public space out of perspective spaces burst upon him fully formed on New Year’s Day, 1914, when his secretary arrived to take dictation. At least as significant as the inspiration of the moment, though, was that three months earlier he had read a manuscript by Whitehead proposing a rather similar theory. Russell acknowledged a debt to him in the preface to Our Knowledge (vi), but

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Whitehead nonetheless resented the manner in which his ideas had been appropriated (letter to Russell, 8 Jan. 1917, in ABR, II, 78). He therefore stopped showing Russell his unpublished work, bringing to an end a decade’s otherwise harmonious collaboration.

Further reading On the genesis of Russell’s construction of matter out of sensibilia, see Blackwell (1973). Bostock (2012) discusses the construction in some detail.

40 PRE-WAR JUDGMENT

In May 1913 Russell began writing a book on the relationship between acquaintance and knowledge. He never completed it, partly as a consequence of Wittgenstein’s criticisms of the theory of judgment he presented there, but the exact nature of these criticisms has been the subject of much subsequent controversy.

The 1906 theory and the Frege point The dispute’s origins lie seven years earlier in Russell’s dissatisfaction with analysing Othello’s judgment that Desdemona loves Cassio as a binary relation between judger and proposition. How could there be such a proposition, given that she did not in fact love him? In 1906 Russell therefore put forward an alternative theory for consideration. We may take . . . the view that only true propositions subsist objectively, while mistaken beliefs have no objects. The first difficulty in such a view is to explain what we mean by belief in what is false. In order to explain this we shall have to regard what is called belief in a proposition as not a thought related to the proposition, but rather a thought related to the constituents of the proposition. I.e. if I believe that A is B, I have a thought related to A and to B, but not to the proposition ‘A is B’ (except derivatively, where A is B, so that there is such a proposition). (CPBR, V, 321) On this analysis a belief is an instance of a multiple relation, ‘a certain kind of relation to several objects’ (CPBR, V, 542). Othello’s judgment that Desdemona loved Cassio, for instance, would be analysed as J(Othello, Desdemona, loving, Cassio),

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where J is the (multiple) relation of judging. What dissuaded Russell from adopting this multiple relation theory of judgment straightaway in 1906 was that it fell foul of a version of the ‘Frege point’. ‘The proposition “p implies q” will be all right when p and q are true,’ he said, ‘but will need a new interpretation when p is false.’ (Ibid.) He therefore had to concede that the theory’s difficulties leave ‘only a doubt, not a certainty’ (CPBR, V, 453). ‘It seems as though, for the sake of homogeneity, we must allow that a proposition . . . subsists equally when true and when false.’ (CPBR, V, 542) In the article he published in 1906, therefore, he only reported, and did not endorse, the multiple relation theory; and in 1908 he repeated his old view that a proposition is a complex which ‘can be the object of a belief . . . and must have one or other of two opposite properties’ (CPBR, V, 738). Eventually, though, Russell came to think that he could deal with the Frege point by treating the fact that someone understands whether a has R to b as a suitable substitute for the proposition that a has R to b, i.e. replacing the proposition aRb with ∃AU(A, a, R, b), where U is the relation of understanding. He was here using ‘understanding’ in the occurrent, not the dispositional, sense, so that the fact obtains only as long as someone is actively considering whether a has R to b. Since he was not minded to appeal to a Berkeleian God to do the understanding when we are not, he instead suggested (somewhat optimistically, one might think) that his account ‘provides propositions, both true and false, as fast as we can think of them’ (CPBR, VII, 115).

The 1909 theory and the ontology of the Introduction Russell first committed himself in print to the multiple relation theory in an article completed in the Autumn of 1909, around the time that most of Principia was delivered to the Press. The passage in which he adopted the theory (CPBR, VI, 10–11) was a late addition, however, which re-used a four-page manuscript apparently dating from his earlier consideration of the theory in 1906—the same manuscript, in fact, in which he floated the notion of orders of truth as a solution to the liar paradox. Although he alluded briefly to the multiple relation theory in the main text of Principia (I, 169), this was an isolated remark (most likely added in proof), and elsewhere in the text he not only continued to speak freely of propositions, but still said that an individual is ‘whatever there is that is neither a proposition, nor a function’ (PM, I, 97)—an unhappy formulation if propositions are incomplete symbols that are to be eliminated on analysis. Russell re-used the 1909 article with only minor alterations in the Introduction to Principia—unhappily, because in it he described a version of the theory of types that differed somewhat from the one that by then appeared in the main text. Even within the Introduction, though, the added passage reads awkwardly, not only because, as already noted, the solution it proposes to the liar paradox is different from the one implied by the ramified theory, but also because it is difficult to see how that theory can be made to cohere with the multiple relation

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theory of judgment. After all, according to the theory of types, propositional functions, no less than propositions, are capable of being judged, and hence ought to be treated as incomplete symbols, to be eliminated on analysis in accordance with the multiple relation theory. There is no sign, though, that Russell had seen this. Indeed according to a passage Whitehead added to Principia in 1911, ‘the assumption that propositions are incomplete symbols excludes’ their forming the range of a variable (II, viii). By parity of reasoning, therefore, treating propositional functions as incomplete symbols should presumably entail that they cannot form the range of a variable either, in which case the theory collapses (see Church 1976, 748).

The 1911 theory and the direction problem In 1911 Russell repeated the multiple relation theory in the Problems, but by then Stout had written to him with an objection nowadays known as the ‘direction problem’, namely that his theory could not explain the difference between Othello’s judgment that Desdemona loves Cassio and the converse judgment that Cassio loves Desdemona. On the face of it, the objection is hardly compelling. As Russell noted in response, The judging alone may arrange the terms in the order Mind, A, r, B, as opposed to Mind, B, r, A. This has the same effect as if r had a sense in the judgment. (In Stout 1911, 203) Yet Stout’s remarks in his 1911 paper suggest, albeit obscurely, a slightly different objection, namely that Russell was sliding between two positions: in order to explain the possibility of error, r must not be related to A and B, but in order to explain how it can occur with sense, it must be. Stout did not press the point with any clarity, but his thought may have been connected with the grammatical point that it is not the verbal noun loving that has a direction but the verb loves. So in order for loves to have a direction, it must be used as a verb. That, at any rate, seems to be how Russell now took the objection. We shall see shortly how this concern soon became salient. Russell called judging a ‘multiple’ relation, but the point is not merely that it has more than two terms, but that the number depends on what is being judged. There are two ways one might explain this: according to the first, the word ‘judgment’ is ambiguous between a sequence of relations of fixed adicity suitable for making judgments of some particular form (subject-predicate, relational, etc.); according to the second, there is a single judging relation, but its adicity is variable. Nowadays relations of variable adicity are called ‘multigrade’, but it remains controversial whether logic should admit them. Russell himself was curiously reticent in his published discussions about which of the two explanations he preferred. In 1911 he said that judging is a constituent ‘shared by all my judgments’ (CPBR, VI, 154), which strongly suggests that he held the second view. Yet in a letter to Broad the following year he was explicit that he held the first.

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I have been always inclined to suppose there were different rel[ation]s of belief, 3-term, 4-term, etc. But then, as you say, one wants to know what they all have in common, & to that I don’t know what to answer. Perhaps only an associated feeling. The question is serious & I should be glad to know the answer. (31 Jan. 1912, in RA3.17C) Given this difficulty for the first view, one might wonder why Russell did not simply adopt the second. If he had a general objection to multigrade relations, he did not in the letter say what it was (presumably because he assumed that on this point Broad agreed with him). One possible explanation is mere unfamiliarity: the logic Russell inherited from Frege and Peano did not allow for multigrade predicates. If he wanted examples, however, he had only to look to natural language, in which multigrade predicates are the rule rather than the exception.

The 1913 theory and the verb On 20 May 1913 Wittgenstein visited Russell ‘with a refutation of the theory of judgment which I used to hold’—i.e. presumably, a refutation of the multiple relation theory as it appeared in the Problems. Russell consoled himself that ‘the correction required is not very serious’; and when he reached the theory of judgment a few days later, he did indeed make a significant amendment to it. To believe that a complex exists requires, he now held, acquaintance with its form as well as its constituents. 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 complex is true. (CPBR, VII, 116) Thus Russell now analysed Othello’s judgment that Desdemona loves Cassio as ˆ y, Desdemona, loving, Cassio), J(Othello, xˆ Rˆ ˆ y is the form of all relational complexes. And Plato’s judgment that where xˆ Rˆ Socrates is mortal, for instance, would be analysed as J(Plato, φˆ xˆ , Socrates, mortality). We can see how this emendation might be regarded as a response to Stout’s direction problem, and hence something Russell was already contemplating independent of Wittgenstein’s complaint: acquaintance with the form of binary relations, Russell might have thought, somehow supplies us with the notion of direction which loving on its own lacks. Even in this narrow role, though, the proposal looks shaky, because it leaves unexplained how we are supposed to obtain the required grasp of the form.

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In any case, the passage in which Russell introduced this new proposal is uncharacteristically hesitant, as if it is an attempt to explain an objection of Wittgenstein’s that he had not wholly understood. He even ended the paragraph by lamely pleading, 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. (Ibid.)

The existential proposal ˆ y’, but what do these mean? Previously I have written the forms as ‘φˆ xˆ ’ and ‘ˆxRˆ when Russell discussed forms of complexes (‘modes of combination’, as he then called them), he could perhaps think of them as propositional functions; but this was no longer open to him after 1909, as propositional functions had been abolished, along with propositions, by the multiple relation theory. In October 1912 he maintained that the form of a proposition ‘is something, . . . not a mere symbol’ (CPBR, VI, 56), but did not characterize it any more precisely. Then in May 1913 he proposed that the form of all relational complexes is ‘Something is related to something’. I shall call this the ‘existential proposal’. On 27th May Russell showed Wittgenstein a ‘crucial part’ of what he had 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. It is difficult at this distance to tell exactly what the objection was (not least because Russell, our only source, did not understand it). Perhaps, though, Wittgenstein was now objecting to Russell’s existential proposal. At any rate he had indeed himself considered this, as a letter he wrote to Russell in January 1912 makes clear. In Russell’s manuscript, though, there are signs that he was uneasy with the existential proposal in any case. If the form is to occur in the analysis of my judgment, I must be acquainted with it, and yet it ‘is not a “thing”, not another constituent along with the objects that were previously related in that form’ (CPBR, VII, 98). If it were a thing, there would be an ‘endless regress’ involved in assembling it with the other constituents into a complex. Moreover, ‘as a matter of introspection, it may often be hard to detect such acquaintance’, but he insisted nonetheless that this difficulty ‘cannot be regarded as fatal, or as outweighing a logical argument of which the data and the inference seem to allow little risk of error’ (CPBR, VII, 99). The most obvious difficulty with the existential proposal, though, is that it makes acquaintance with the form dependent on a grasp of the very form in question. The definition ‘sounds circular,’ Russell

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suggested, before insisting, somewhat desperately, that ‘what is intended is not circular’ (CPBR, VII, 114). A further difficulty concerns the form’s complexity. ‘The logical nature of this fact is very peculiar,’ he conceded, since it ‘contains no constituent at all’. In a sense, it is simple, since it cannot be analyzed. At first sight, it seems to have a structure, and therefore to be not simple; but it is more correct to say that it is a structure. ‘Language’, he awkwardly ended up admitting, ‘is not well adapted for speaking of such objects.’ (Ibid.) The objection to the existential proposal which Wittgenstein included in the Notes on Logic built on this difficulty. He invited us to consider the suggestion that the proposed existential form is simple. ‘In that case,’ he asked, ‘what is the meaning of (e.g.) “∼( x, y). xRy? Can we put “not” before a name?’ (NoL, B5) The answer to this rhetorical question is of course, no. E

Permutative complexes The existential proposal was hopeless, then, but it was not the only difficulty Russell was wrestling with at the end of May. He was also trying to solve a problem with our understanding of what he called ‘permutative’ complexes, i.e. those in which the order of the constituents matters. Because he conceived of naming as akin to pointing, he thought that a name could not express a direction (CPBR, VII, 148). I can name the complex A-next-to-B, for instance, but not A-above-B, because no name is capable of encoding the difference between this and B-above-A. There is plainly another echo here of Stout’s direction problem, but Russell’s focus was now on our acquaintance with a complex rather than with how it occurs in the judging relation. Russell made an elaborate attempt to address the difficulty by reconstructing the directionality of the relation using non-directional properties as his resources. If C is the complex A-above-B, for instance, he proposed to analyse it into the conjunction of A-precedes-in-C and B-succeeds-in-C. At first sight this might not seem like an improvement: all we have done is to replace one permutative complex with two. The difference, however, is that the two complexes we are now dealing with are ‘heterogeneous’, i.e. they can only occur one way round: ‘C precedes in A’, for instance, would be obvious nonsense. On 1st June, though, Russell saw a ‘real difficulty’ (CPBR, VII, 154) with this strategy. What if my belief as to C’s structure is defective and A is not in fact above B? His analysis of my erroneous judgment that it is went via the complexes A-precedes-in-C and B-succeeds-in-C, but in the hypothesized case there are no such complexes. The whole theory of judgment had been designed to explain false judgments without requiring acquaintance with non-existent complexes, but that is exactly what the theory now did. Having noted the objection, Russell

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simply postponed it for later consideration. He never did return to it, though, or to the theory which spawned it, strongly suggesting that he realized its seriousness.

Judging a nonsense A week later, on 8th June, Russell abruptly stopped writing. Ostensibly his reason was that ‘inference (which is next) wants a lot of thought’, but the problems we have been discussing—both Wittgenstein’s objection to the multiple relation theory and his own ‘real difficulty’ with judgments about non-permutative complexes—no doubt contributed. In the previous few days his mood had alternated between pessimism—‘I have only superficially and by an act of will got over Wittgenstein’s attack’ (1st June)—and optimism—‘My work still goes swimmingly’ (6th June). Even when Russell had stopped writing, Wittgenstein was still not done with his criticisms: I can now express my objection to your theory of judgment exactly: I believe it is obvious that, from the prop[osition] ‘A judges that (say) a is in the Rel[ation] R to b’, if correctly analysed, the prop[osition] ‘aRb ∨ ∼aRb’ must follow directly without the use of any other premiss. This condition is not fulfilled by your theory. (June 1913, in CL, no. 14) In the Notes on Logic Wittgenstein expressed the point in a less technical fashion. 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.) And he said it again in the Tractatus. ‘A correct theory of judgment must show why it is impossible to judge a nonsense.’ These are evidently variant expressions of a single objection to Russell’s multiple relation theory, but there is no consensus among commentators about quite what the objection is. Before we come to what I believe to be the correct account, let us see what is wrong with some others. First, it is unlikely that the problem was with the very idea of judgment as a multiple relation; for if it were, Russell would not still have claimed in his Lowell lectures the following year that it is ‘necessary, in analysing a belief, to look for some other logical form than a two-term relation’ (OKEW , 68); nor, when he conceded in 1917 that the 1911 theory was ‘somewhat unduly simple’, would he have insisted nonetheless that a judgment is ‘a relation of a mind to several entities, namely the entities which compose what is judged’ (CPBR, VI, 154). (Wittgenstein’s own theory of judgment in the Tractatus, it is worth saying, was also a multiple relation theory, although, as we shall see, it differed in a crucial respect from Russell’s.) Nor is Wittgenstein likely to have been concerned about the difficulty mentioned earlier of making Russell’s correspondence theory of truth cohere

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with his theory of types. Although this difficulty is genuine, it is too technical to have been Wittgenstein’s, and it does not have much to do with the impossibility of judging that the table penholders the book. Another red herring is the ‘real difficulty’ with Russell’s analysis of judgments involving permutative complexes. His treatment of permutative complexes was undoubtedly un-Wittgensteinian and had little prospect of success, but its fault cannot comfortably be described as an inability to show why it is impossible to judge a nonsense. Moreover, although Wittgenstein no doubt objected to Russell’s inclusion of a name of the form as a term in the judging relation, it is unlikely (contra Pears 1989) that this was the whole of his objection: he would hardly have included in the Tractatus an objection to a theory Russell had never published; and, if he had, Russell would not have described it explicitly in his Introduction as an objection to the theory which he did publish. Perhaps Wittgenstein’s objection on 27th May was narrowly to the existential proposal. As we shall see in Chapter 42, he had by the autumn of 1913 broadened this into a general objection to any account according to which the form of a complex can be named; but the available evidence does not determine whether he had already taken this step by June. Even if he had, though, this was hardly devastating to Russell’s theory: Wittgenstein’s claim was only that the form cannot be named, not that it cannot be represented at all. Could we not respect Wittgenstein’s Fregean sensibilities by representing it with the unsaturated expression ‘xRy’? Nonetheless, although this cannot have been the whole of Wittgenstein’s objection, it is at least on the right track. What he wished to insist, I think, was that each of the terms involved should occur in the judgment with just the logical multiplicity that it has in the fact that obtains if the judgment is true. If Desdemona had loved Cassio, the occurrence of the verb loves in that fact would have had two argument-places; but in Russell’s proposed analysis of the fact that Othello believes that Desdemona loves Cassio loves must have three argument-places, one each for Othello, Desdemona and Cassio. This is recognizably a variant of the objection which, I have suggested, Wittgenstein had made on 20th May. Russell thought he had dealt with it by introducing the form into the judgment, but he had not: even if the form allowed a noun to function like a verb, it could not make it function like two different verbs at once. Wittgenstein had already made essentially this point to Russell six months earlier, in a letter he wrote just after visiting Frege. I now think that Qualities, Relations (like Love), etc. are all copulae! That means I for instance analyse a subject-predicate prop[osition], 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 there

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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. . . . All theory of types must be done away with by a theory of symbolism showing that what seem to be different kinds of things are symbolized by different kinds of symbols which cannot possibly be substituted in one another’s place. (Jan. 1913, in CL, no. 10) What Frege had evidently urged on Wittgenstein was that an unsaturated expression such as ‘x is human’ must always have its argument-place marked explicitly, thus ensuring that it is impossible to judge a nonsense. The moral Wittgenstein drew from this was that the ‘type’ of an expression depends on its meaning and cannot be conferred on it from without. Applied to the present case, the verb in ‘Desdemona loves Cassio’ is not loves but ‘x loves y’; so written, it cannot occur in a multiple relation of the kind Russell proposed, because there is no third argument-place for ‘Othello’ to fit into.

Further reading Not all philosophers agree that Russell was right to abandon his earlier conception of propositions: for a (not wholly successful) defence see Gaskin (2006, ch. 6). For more on Russell’s ‘real difficulty’, see MacBride (2013) and Levine (2013). Sellars (1974) gives a useful account of Wittgenstein’s objection to Russell’s multiple relation theory of judgment. For competing explanations, see Geach (1957) Pears (1989) and Griffin (1985).

41 FACTS

In October 1913 Wittgenstein compiled the Notes on Logic, in two parts, as a summary of the work he had been doing in Cambridge under Russell’s supervision: the first part he dictated from his working notebooks in German to a shorthand typist in Birmingham; the second was written in English in Russell’s rooms at Cambridge, partly by Wittgenstein himself and partly by dictation to Russell’s secretary, before being typed up and sent to Wittgenstein (by then in Norway) for correction. Russell was immediately convinced of the notes’ importance— ‘as good as anything that has ever been done in logic’, he told Morrell (3 Oct. 1913)—and the following February he devoted considerable labour to ‘translating and copying and classifying’ them to use in a course of logic lectures he was due to give at Harvard. The First World War began soon thereafter, and for the next three years Russell devoted himself largely to pacifist campaigning. His return to philosophy was marked by two sets of popular lectures, the first (between October and December 1917) on mathematical philosophy, the second (between January and March 1918) on logical atomism. Both sets were published, the former as a worked-out text written in prison between May and September 1918, the latter as a verbatim transcript. Both these publications, as well as his later recollections, explicitly acknowledged the influence on him of Wittgenstein’s work—not the Tractatus, which he had not yet seen, but the Notes, which ‘together with a large number of conversations, affected my thinking during the war years while he was in the Austrian army and I was, therefore, cut off from all contact with him’ (MPD, 83). The next two chapters will be devoted to describing this influence.

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Complex and fact Most significant for Russell was the distinction emphasized in the Notes between complex and fact—between, for instance, the complex A-above-B and the fact that A is above B. Complexes and facts both have internal structure—both are, in some sense, complex entities—but the kind of complexity is different. The salad on the table is a complex consisting of tuna, lettuce, cucumber, beans, anchovies and dressing: it exemplifies the fact that these ingredients are thus assembled; but it also exemplifies the fact that the dressing coats the other ingredients evenly; that the anchovies are arranged on top; etc. The correspondence between complex and fact is thus one-many. Until he met Wittgenstein, Russell had routinely ignored this distinction, treating the fact that A is above B as identical to the complex A-above-B. By June 1913 Wittgenstein had evidently persuaded him that there is a distinction, but not yet that it matters; for he still maintained (CPBR, VII, 80) that each complex exemplifies exactly one fact. On Wittgenstein’s view Russell’s notion of ‘the form of a complex’ was a misnomer: it is facts that have forms, not complexes. ‘You’, he wrote accusingly in the margin of the Notes on Logic typescript, ‘imagine every fact as a spatial complex.’ (NoL, C26) Not until his lectures at Harvard the following year did Russell place any emphasis on the distinction between complex and fact. ‘I should not call Napoleon a fact,’ he now said, ‘but I should call it a fact that he was ambitious, or that he married Josephine.’ (OKEW , 60) In 1918 he took the further step of claiming that facts rather than complexes are primary. The analysis of apparently complex things . . . can be reduced by various means to the analysis of facts which are apparently about those things. Therefore, it is with the analysis of facts that one’s consideration of the problem of complexity must begin, not by the analysis of apparently complex things. (CPBR, VIII, 171) He may even have become persuaded that his failure to distinguish between complex and fact had been part of the explanation for the error in his pre-war theory of judgment. ‘You cannot make what I should call a map-in-space of a belief.’ (CPBR, VIII, 198) Plainly, then, Wittgenstein did not first learn the contrast from Russell. Frege, on the other hand, drew it explicitly in ‘Thoughts’. ‘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.’ (CP, 354) Some commentators have gone as far as to suggest not merely that Wittgenstein got the distinction from Frege, but that he remained confused about it until much later in his career. The source of this view is an anecdote due to Geach (1976, 67–8). Wittgenstein, he said, had once told him about a conversation in which Frege, having read the Tractatus, had asked him whether a fact is bigger than its components. The story cannot be quite right as it stands, though: Wittgenstein’s meetings with Frege were all before the war. Frege did

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write to Wittgenstein after reading the Tractatus, but the question he asked was not quite the one Geach reported. You write, ‘It is essential to a thing to be able to be a part of an atomic fact.’ 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 an atomic fact. 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. (Frege to Wittgenstein, 28 June 1919) Frege’s remarks certainly point to the importance of distinguishing carefully between two radically different kinds of complexity, but much more would have to be said to show that the Tractatus is guilty of confusing them. (Superficially, at least, the Tractarian response to Frege is that the components of an atomic fact are simple and hence do not have parts.)

Negative facts The objection to the multiple relation theory described in the last chapter is no doubt sufficiently serious to have forced a pause in Russell’s work on his Theory of Knowledge book. Yet on his later telling Wittgenstein had persuaded him only that the theory was ‘somewhat unduly simple’ (CPBR, VI, 154n), not utterly wrong. Why, then, did he never resume work on the book? In October 1913, soon after first reading the Notes, Russell re-read his own manuscript and reported to Morrell that it seems to me the early part is as good as I thought at the time, but that it goes to pieces when it touches Wittgenstein’s problems, as he said at the time. I hardly thought then that he would get out as much as he has lately, and it seemed not worth waiting. Now his work has put a completely new face upon whole vast regions. (24 Oct. 1913) Russell was right: the ontology of facts Wittgenstein proposed in the Notes was indeed sufficiently radical on its own to ‘put a completely new face upon whole vast regions’. Consider, for instance, the question of negative facts. On Russell’s pre-war conception it is obvious that a complex cannot be negative: if the existence of a complex C would make the proposition p true, what would make p false would simply be its non-existence, not the existence of some other complex not-C. (There are various ways to demonstrate this, the most obvious of which is that if there were such a complex as not-C, C would be part of it and hence would exist too.) This reasoning does not apply to facts, however, and in the Notes Wittgenstein claimed (for reasons to be discussed in the next chapter) that there are both positive and negative facts. By repeating this claim in his lectures at Harvard Russell

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‘nearly produced a riot’ (CPBR, VIII, 187). What, though, is the structure of a negative fact? If it consisted of a positive fact together with negation, the earlier objection to negative complexes would extend to the current case too. It must not be supposed that the negative fact contains a constituent corresponding to the word ‘not’. It contains no more constituents than a positive fact of the correlative positive form. The difference between the two forms is ultimate and irreducible. (CPBR, VIII, 279–80) It is far from clear, though, that the position Russell found himself defending at Harvard was any more convincing. If, as he now proposed, facts come in two kinds, positive and negative, the difference between which is ‘ultimate and irreducible’, we are left without any explanation of the necessity that each fact of the first form excludes a fact of the second form from obtaining. Indeed, he could hardly complain at this objection, since he himself used it against Demos (1917), one of the rioters, who had tried to argue against negative facts by appealing to a primitive relation of incompatibility. Russell’s discussion of the issue is hampered by his failure to distinguish sufficiently between two distinct questions: whether the absence of a fact is itself a fact, or whether mere absences would suffice as truth-makers; and, if so, whether one of the two possibilities—presence or absence—is intrinsically positive and the other negative, or whether they form a wholly symmetrical pair.

General facts In the autumn of 1913 Russell noticed a further difficulty. What makes it true that all men are mortal cannot be just the mortality of Socrates, Plato, Aristotle, etc.: we also need the general fact that these are all the men there are (see OKEW , 55–6).1 This commitment to general as well as particular facts modifies Russell’s atomism substantially, for two reasons. First, it suggests that the logical and nonlogical parts of the world cannot be wholly separated. Second, it ‘affords a refutation of the older empiricists’ (OKEW , 56), because we clearly know some wholly general facts about the external world, he thought, whereas perception only ever justifies particular facts, not general ones. Perhaps the most notable feature of Russell’s commitment to general facts, though, is that it contradicted not merely the ‘older empiricists’ but Wittgenstein, who had claimed in the Notes (B36) that ‘if we form all possible atomic propositions, the world will be completely described if we declared the truth or falsity of each’. When Russell translated this remark, he added in parentheses, in a rare gesture of dissent, ‘I doubt this.’ It is striking how much of the Notes he accepted wholesale. Indeed, his attempted defence of negative facts

1 Strictly speaking, in Our Knowledge Russell only committed himself to general truths, but in his 1918 lectures he was explicit that there are general facts (CPBR, VIII, 164–5).

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(CPBR, VIII, 187–9) was so inept that one might wonder whether he believed in them simply on Wittgenstein’s say-so. He espoused general facts, on the other hand, despite Wittgenstein’s denial in the Notes that there are any. (We shall see later that Wittgenstein eventually evaded this objection by holding that Russell’s problematic ‘general facts’ are unsayable and therefore not really facts at all.)

Names and particulars Frege’s mirroring principle (Chapter 16) asserted that linguistic expressions coincide in saturatedness or unsaturatedness with what they refer to. At the heart of Wittgenstein’s work was a similar ‘harmony principle’, which requires there to be an identity of structure between a symbol and what the symbol represents. Oddly, though, he never quite stated the principle in this general form in the Notes, but made the point laboriously case by case. 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. (B77) In his lectures at Harvard Russell was more succinct. ‘The structure of the symbol must be identical with the structure of the symbolized.’ (7 Apr. 1914) ‘There is always a sort of fundamental identity between symbol and symbolized.’ (11 Apr. 1914) The ‘symbol’ which shares its structure with what it symbolizes cannot always be read off straightforwardly from a sentence’s surface structure, of course: the harmony principle presupposes a programme of analysis. Yet in order to make this analysis consistent with harmony Wittgenstein had to amend it significantly away from its Russellian origins. Russell had held that I may name any complex provided only that my relation to it is sufficiently direct. I can, for instance, give the name ‘C’ to a complex A-above-B. But this contradicts harmony, since the name is simple whereas what it names is complex. Wittgenstein proposed instead to treat ‘C’ as an incomplete symbol and analyse any judgment about C as the conjunction of the judgment that A is above B and some further judgment about A and B (see NoL, B17). Let us call the incomplete symbol ‘C’ a ‘relative’ name of the complex (cf. IMP, 174). The key point to note here is that Wittgenstein’s analysis is ontologically eliminative after the manner of Russell’s treatment of class-terms, in contrast with his non-eliminative analysis of definite descriptions. Russell’s theory described an entity by means of its (external) properties; Wittgenstein’s analysed it into its (internal) constituents. Russell’s quantifier had ranged

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over complexes as well as simples, whereas on Wittgenstein’s account complexes are logical fictions. ‘Complex objects do not exist.’ (NB, 17 June 1915) Notice, though, that Wittgenstein’s highly schematic analysis hides a wealth of difficulty. The transition from the predicate ascribed to the complex in the analysandum to that ascribed to its constituents in the analysans is far from trivial. In his convenient example (PI, §60), the fact that the broom is in the corner is analysed as the fact that the brush is in the corner, the broomstick is there, and the brush is fixed to the broomstick (PI, §60); but for the broom to be six feet long is not for its constituents all to be so, and in cases such as this it is not at all easy to say what the predicate in the analysans is. Nor did Wittgenstein explain how to deal with the case in which a single complex C exemplifies two different facts and hence gives rise to two non-equivalent analyses.

Forms and universals Frege had persuaded Wittgenstein during his Christmas 1912 visit that propositions must have two fundamentally different kinds of components which cannot be fitted together nonsensically. At that time, though, Wittgenstein had not yet changed from an ontology of complexes to an ontology of facts. Once he did so, he implemented Frege’s insight by drawing a binary distinction among the components of a fact between the constituents (content) and the manner in which they are configured (form). As Russell noted in the typescript of the Notes, ‘Components are forms and constituents.’ (NoL, C50) The form is distinguished from the other components by being responsible for the fact’s overall structure. There is therefore no need to posit, in addition to the components of a proposition, a further element, the copula, whose job it is to marshal these components into propositional form. Thus is Bradley’s regress avoided. Russell told Bradley in 1914 that through Wittgenstein’s work he ‘seem[ed] now to see answers about unities’ (30 Jan.), but the answers were in fact thoroughly Fregean. Just as Frege deduced from his mirroring principle that ‘the concept horse’, being saturated, cannot refer to a concept, so Wittgenstein deduced from his harmony principle that there is ‘no name which is the name of a form’ (NoL, B48). It is no surprise, therefore, that it proves tricky to say just what the form of a proposition such as aRb is. In ‘aRb’ ‘R’ looks like a substantive but is not one. What symbolizes in ‘aRb’ is that R occurs between a and b. Hence ‘R’ is not the indefinable in ‘aRb’. (NoL, C9) We could try saying instead that the third component is ‘xRy’, but this is not quite right either, since it is not a component. We are left having to approach the matter indirectly by saying that the form is that expression which, when combined with ‘a’ and ‘b’ in that order results in the proposition ‘aRb’. The form thus plays in Wittgenstein’s account the role that a universal (earlier called a concept) played in Russell’s, but with the important difference that the

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form, unlike Russell’s universal, cannot occur as a logical subject. In 1902 Russell had rejected Frege’s argument concerning the concept horse, on the ground that universals had to be capable of being logical subjects in order to count as entities at all, but now he meekly accepted Wittgenstein’s version of the same argument. ‘If a thing is a verb,’ Russell now said, ’it cannot occur otherwise than as a verb.’ (CPBR, VIII, 198) The verb may grammatically be made subject: e.g. we may say, ‘Killing is a relation which holds between Brutus and Caesar.’ But in such cases the grammar is misleading, and in a straightforward statement, following the rules that should guide philosophical grammar, Brutus and Caesar will appear as the subjects and killing as the verb. (IMP, 142) Russell was still ‘inclined to think that there are things that are not in time’ (CPBR, VIII, 223), and hence to divide individuals into two kinds, concrete and abstract, but he no longer called these two kinds ‘existents’ and ‘subsistents’, adopting instead the modern usage according to which existence does not entail temporality. Where, though, does this leave the form of the fact? Since it cannot be a logical subject, Russell inferred that I cannot be acquainted with it. ‘Words of which the logical meaning is universal can’, he now thought, ‘be employed correctly, without anything that could be called consciousness of universals.’ (AMi, 228) The fact is that ’acquaintance’ cannot be applied straight off except to particulars, i.e. the only two-term cognition relation of the form K

S·− →·O has its converse domain confined to particulars. Kn[owledge] by descr[iption] is kn[owledge] of a general prop[osition] of the form ( x)φx. In such cases, we cannot properly speak of acquaintance with φ xˆ ; ‘φ xˆ ’ must never be put in a subject-place, i.e. it must only occur in positions where it is doing the proper work of a function. Universals, prop[osition]s, functions, facts cannot be named, & cannot occur in subject-places; they are not ‘things’. The symbols which are concerned with them are never simple, and do not name them. E.g. redness is introduced by ‘the meaning of “x is red” whatever x may be’. This is Wittgenstein’s theory & I am sure it is right. (Letter to Broad, 10 Feb. 1914, in RA3.17C) E

Previously, Russell had held that our knowledge of atomic complexes rests ultimately on the binary relation of acquaintance, but this was yet another of the ‘vast regions’ in which his views required revision. Because he now thought that the objects of acquaintance must be simple, he had to posit a second relation, ‘perception’, which multiply relates us to a fact. Sense, he now said, gives acquaintance with particulars, and is thus a two-term relation in which the object can be named but not asserted, and is inherently incapable

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of truth or falsehood, whereas the observation of a complex fact, which may be suitably called perception, is not a two-term relation, but involves the propositional form on the object-side, and gives knowledge of a truth, not mere acquaintance with a particular. (CPBR, VIII, 6) Instead of being directly acquainted with the universal, Russell’s idea was that we derive our grasp of it indirectly, by subtraction, from our perception of a fact in which it occurs. Understanding a predicate is quite a different thing from understanding a name. . . . To understand a name you must be acquainted with the particular of which it is a name, and you must know that it is the name of that particular. You do not, that is to say, have any suggestion of the form of a proposition, whereas in understanding a predicate you do. To understand ‘red’, for instance, is to understand what is meant by saying that a thing is red. You have to bring in the form of a proposition. You do not have to know, concerning any particular ‘this’, that ‘This is red’ but you have to know what is the meaning of saying that anything is red. (CPBR, VIII, 182) In this respect, he thought, understanding a predicate differs from understanding a name. In order to understand a name for a particular, the only thing necessary is to be acquainted with that particular. . . . No further information as to the facts that are true of that particular would enable you to have a fuller understanding of the meaning of the name. (CPBR, VIII, 179) Russell’s explanation was hesitant, as Wittgenstein’s had been in the Notes, but it seems that at this stage the idea was to align the distinction between semantically complete and incomplete expressions with that between names and forms.

Further reading Potter (2009) gives a detailed account of the circumstances in which the Notes on Logic were composed. On Russell’s defence of negative facts see Rosenberg (1972).

42 LATE LOGICISM

We have seen that by the time of Principia Russell had already given up several of the claims about logic that he had defended in Principles. Now, under Wittgenstein’s influence, he gave up, or heavily modified, several more: that logic has a subject matter which is worldly, not symbolic, and includes a range of distinctive universals (known as logical constants); that logical truths are objective, not psychological; that pure mathematics is part of logic; that higher-order quantification may be explained in terms of a domain of entities, not in terms of the symbols used to express them.

The subject matter of logic Since the Principles Russell had consistently supposed that logic has as its subject matter a domain of abstract entities. In 1913 he still claimed that logical knowledge is grounded in what he called ‘logical experience’ (CPBR, VII, 97). Such words as or, not, all, some, plainly involve logical notions; and since we can use such words intelligently, we must be acquainted with the logical objects involved. But the difficulty of isolation is here very great, and I do not know what the logical objects involved really are. (CPBR, VII, 99) Wittgenstein, by contrast, had told Russell that ‘there are NO logical constants. Logic must turn out to be of a TOTALLY different kind than any other science.’ (22 June 1912, in CL, no. 2) This special status could be explained, he thought, only by holding that logic has no subject matter of its own. Perhaps, indeed, Russell was on the verge of agreeing. ‘Logical constants’, which might seem to be entities occurring in logical propositions, are really concerned with pure form, and are not actually

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constituents of the propositions in the verbal expression of which their names occur. (CPBR, VII, 98) Nonetheless, he still claimed that our knowledge of logic requires acquaintance with this ‘pure form’ (‘logical intuition’, as he also termed it). Acquaintance with logical form, whatever its ultimate analysis may be, is a primitive constituent of our experience, and is presupposed, not only in explicit knowledge of logic, but in any understanding of a proposition otherwise than by actual acquaintance with the complex whose existence it asserts. (CPBR, VII, 99) That autumn, however, he conceded (making use, he said, of ‘unpublished work by my friend Ludwig Wittgenstein’) that logical constants are not entities. Such words as or, not, if, there is, identity, greater, plus, nothing, everything, function, and so on, are not names of definite objects, like ‘John’ or ‘Jones’, but are words which require a context in order to have meaning. All of them are formal, that is to say, their occurrence indicates a certain form of proposition, not a certain constituent. ‘Logical constants’, in short, are not entities; the words expressing them are not names, and cannot significantly be made into logical subjects except when it is the words themselves, as opposed to their meanings, that are being discussed. This fact has a very important bearing on all logic and philosophy since it shows how they differ from the special sciences. (OKEW , 208) Although Russell had abandoned his commitment to propositions as logical entities when he adopted his multiple relation theory of judgment in 1909, he still conceived of the constituents of a judgment as parts of the world, not mere words. In the Theory of Knowledge manuscript he reaffirmed that a ‘purely linguistic’ definitions of a proposition as a kind of sentence ‘will not bear scrutiny’ (CPBR, VII, 105–6). In the Lowell Lectures, though, he shifted abruptly to this very terminology, calling a proposition a ‘form of words’ (OKEW , 62). Soon, too, he made a corresponding shift with the phrase ‘propositional function’, which he now used to mean ‘an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition’ (IMP, 155–6, my emphasis). ‘A propositional function in itself is nothing: it is merely a schema.’ (CPBR, VIII, 206) Although Russell now agreed with Wittgenstein in rejecting a platonistic ‘world of logic’ and in giving renewed attention to the structure of the conceptscript, their conceptions of the symbolism’s role in logic soon began to diverge. Wittgenstein had been trying to design a concept-script which makes a sign’s logical role wholly transparent. Inevitably he failed, but his letters to Russell from Norway demonstrate that he did not admit defeat until some time later. When he eventually did so, the moral he drew was that the logician’s task is to see past language to the symbol’s non-psychological essence.

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What symbolizes in a symbol, is that which is common to all the symbols which could in accordance with the rules of logic = syntactical rules for manipulation of symbols, be substituted for it. (NdM, 116) It will be for a later chapter to explain the role Wittgenstein’s belief that there are no logical constants played in shaping the Tractarian conception of logic as empty. Russell, on the other hand, took Wittgenstein’s Grundgedanke as showing the importance of questions of meaning to the logician. He thus abandoned the view he had held since the Principles that ‘meaning, in the sense in which words have meaning, is irrelevant to logic’ (PoM, §51). On the way of speaking he now adopted, logic is about signs, not about what they refer to, and is therefore to that extent psychological. ‘The theory of types is really a theory of symbols, not of things.’ (CPBR, VIII, 232) ‘A logic independent of the accidental nature of space-time becomes an idle dream.’ (CPBR, IX, 138) That Russell now thought of logic as psychological might surprise anyone tempted to locate anti-psychologism concerning logic in the conceptual core of analytic philosophy. It is worth bearing in mind, though, that he distinguished between a ‘word-proposition’—a publicly perceptible entity formed from words—and an ‘image-proposition’—a mental entity formed from images. Nowadays only the latter would be called ‘psychological’, but in Russell’s usage psychology included in its scope part of what we would now regard as linguistics. If he now held that logic is psychological, he meant by this that it is about signs, whether mental or linguistic.

The new theory of types Even before Russell had seen the Notes, Wittgenstein’s closest friend at Cambridge, David Pinsent, wrote in his diary: It seems that both Russell and old Whitehead are most enthusiastic about his recent work in Logic. It is probable that the first volume of Principia will have to be re-written, and Wittgenstein may write himself the first eleven chapters. That is a splendid triumph for him! Wittgenstein’s work in philosophy may have begun as a critique of Russellian logicism, but it was not at first a very revisionary one. Whatever was wrong with Principia, it seems not to have been the technicalities of the theory of types that were centrally at issue: the Notes, compiled shortly afterwards, envisage a ramified theory, not a simple one. Take (φ)φ!x. Then, if we describe the kind of symbols for which φ stands, the 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!x kind. (NoL, C51)

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Wittgenstein had to say something about Russell’s paradox, though. After all, his account of the variable, like Frege’s in Begriffsschrift, permits us to treat any expression occurring in a proposition as constant, and therefore licenses higher-order quantification (as, for instance, in the preceding quotation). What he sketchily proposed was an amendment of Russell’s vicious circle principle. 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. (NoL, B76) Russell had said that no proposition can depend on itself: Wittgenstein adopted this principle, but with ‘dependency’ interpreted as symbolic containment. So interpreted, the principle does not on its own solve Russell’s paradox, since it does not provide us with the criterion for telling whether a symbol contains itself. Where it perhaps improves on Russell’s vicious circle principle, however, is that it at least indicates a strategy for settling the issue: it initiates a research programme whose goal is to clarify the structure of the symbols in question. The fact that Wittgenstein’s research focused on the structure of propositions is therefore not the shift of topic away from the philosophy of mathematics that it might superficially seem: the task of rewriting the first 11 chapters of Principia would have had clarifying this structure as its central aim. The Tractatus, when it eventually arrived after the war, was perhaps not the new prolegomenon to Principia that Russell had been expecting. For current purposes, though, what is relevant is that Wittgenstein’s proposal for grounding the theory of types depended on a conception of symbols that, as we have seen, Russell showed no inclination to adopt. Without this conception, therefore, Russell was left to write himself the new Introduction to Principia that he had previously envisaged as Wittgenstein’s task. What made this new Introduction urgent was the nominalistic conception of propositional functions that Russell had now adopted. In our consideration of quantification so far I have recommended being fairly relaxed about the difference between a world-level and a broad language-level account. That, though, was on the assumption that the substituends for the relevant variables do refer. On Russell’s new view that assumption no longer applied to higher-order variables. We do not need to ask, or attempt to answer, the question ‘What is a propositional function?’ A propositional function standing all alone may be taken to be a mere schema, a mere shell, an empty receptacle for meaning, not something already significant. (IMP, 157) On the other hand, Russell could hardly restrict himself to first-order quantification, since that would have lamed his system completely. In the second edition of Principia, therefore, he accounted for second-order quantification by adopting the doctrine that a propositional function can occur only ‘through its values’. Any meaningful sentence in which it appears to be the subject is to be recast. Thus

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‘wisdom is a virtue’ might be interpreted to mean ‘all wise acts are virtuous’. He then sketched a system of ramified logic in which second-order quantification is in principle reducible to an infinite conjunction of first-order predications. The logic of general propositions concerning a given argument would, he said, ‘be intolerably complicated if we abstained from the use of variable functions, but it can hardly be said that it would be impossible’ (PM, 2nd edn, I, xxxii). The axiom of reducibility, dubious on the old interpretation, is on the new nominalistic interpretation utterly implausible. In an appendix to the second edition of Principia Russell attempted to prove mathematical induction without reducibility, but his proof was unfortunately fallacious (see Gödel 1944), and later work (Myhill 1974) has shown that (given certain plausible background assumptions) it is impossible to generate the standard theory of natural numbers in a ramified second-order theory without reducibility. Even without this error, however, Russell’s abandonment of reducibility would have curtailed logicism’s scope substantially, since it put the standard Weierstrassian theory of the real numbers out of reach. Whether the theory could be adapted so as to preserve enough of classical mathematics to be plausible is a complex technical question. Although he continued thereafter to hold that ‘mathematics and logic are identical’ (PoM, 2nd edn, v), it is difficult to see how he could do so in relation to mathematics as standardly practised given that he now excluded the axiom of reducibility from his logic.

Propositions are not names ‘It is perfectly evident as soon as you think of it’, Russell said in 1918, 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 . . . illustrates how the relation of proposition to fact is a totally different one from the relation of name to the thing named. (CPBR, VIII, 167) Once again, Russell was here reporting what he had learnt from the Notes. ‘I never had realized it’, he confessed, ‘until it was pointed out to me by a former pupil of mine, Wittgenstein.’ I have already mentioned Wittgenstein’s argument that propositions constitute a logical category distinct from names of objects. For all that I said then, however, the manner in which propositions refer to facts might be strictly parallel to, although distinct from, that in which names refer to objects. Now Wittgenstein went a step further, arguing that the two cases are radically different. ‘Names are points, propositions arrows—they have sense.’ (NoL, B23) By ‘sense’ here he evidently meant ‘direction’: p and ∼p point at the same fact—which he for a

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time referred to as their Bedeutung—but from different directions. This is what he called the ‘bipolarity’ of the proposition (NoL, C30). From this radical difference between propositions and names Wittgenstein drew consequences for the logic of propositional operators such as negation or disjunction. Russell had treated these as functions taking propositions as arguments. In that case, though, one might expect f (p) to be of higher type than p, just as f (φ xˆ ) is of higher type than φ xˆ , and that would lead to a propositional calculus of appalling complexity. Wittgenstein’s way out of this difficulty took as its starting point that ∼∼p is the same proposition as p (NoL, C12). From this it follows that negation (and hence, by extension, the other propositional connectives) do not raise the type of a proposition, and hence are not propositional functions; in the Tractatus Wittgenstein called them ‘operations’. The key feature of an operation is that, in contrast to a propositional function, it is repeatable. Wittgenstein held that ‘propositions, owing to sense, cannot have predicates or relations’ (NoL, C15). He presumably hoped thereby to dissolve the liar paradox, which requires us to ascribe to a proposition the property of being false. To avoid it, Russell had at one time proposed a hierarchy of kinds of truth—first truth, second truth, etc.—each not applicable to itself (see Chapter 35). Wittgenstein, by contrast, only needed one kind of truth, since if propositions cannot have predicates, the paradox is ill-formed. It is unclear, though, to what extent he had yet thought this through or, if he had, whether Russell followed him. Notice, though, that in order to distinguish between names and propositions Wittgenstein did not have to appeal to bipolarity. It would have been enough for his purposes to point out that names are complexes, propositions facts. The key question left unaddressed in the Notes, though, is how an operation differs from a propositional function so that the vicious circle principle applies only to the latter, not the former. In Norway, therefore, Wittgenstein continued to search for a conception of the structure of a proposition that would allow him to explain why ∼p does not ‘say anything about’, and hence is not a predicate of, p (see NdM, 116).

Bipolarity and truth Previously, Russell had held a correspondence theory according to which the truth of a judgment consists in the existence of appropriate complexes. Now he had to adjust this theory to the new ontology of facts. He did this by proposing what might be termed a ‘bi-correspondence’ theory of truth. Propositions occur in pairs [having] the same objective, but opposite meanings. . . . A proposition is true when it points towards its objective, and false when it points away from its objective. (AMi, 273) He explicitly credited this way of looking at the matter to Wittgenstein, who had indeed claimed that propositions are like arrows (NoL, B23). Nonetheless, in the

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manner in which Russell took up this idea he ignored warnings Wittgenstein had posted elsewhere in the Notes against interpreting bipolarity as a theory of truth in this fashion. The error in Russell’s theory was clearly articulated by Ramsey. Any such view either of judgment or of perception would be inadequate for a reason which, if valid, is of great importance. Let us for simplicity take the case of perception and, assuming for the sake of argument that it is infallible, consider whether ‘He perceives that the knife is to the left of the book’ can really assert a dual relation between a person and a fact. Suppose that I who make the assertion cannot myself see the knife and book, that the knife is really to the right of the book, but that through some mistake I suppose that it is on the left and that he perceives it to be on the left, so that I assert falsely ‘He perceives that the knife is to the left of the book’. Then my statement, though false, is significant, and has the same meaning as it would have if it were true; this meaning cannot therefore be that there is a dual relation between the person and something (a fact) of which ‘that the knife is to the left of the book’ is the name, because there is no such thing. (FoM, 140) Ramsey summarized his conclusion thus: ‘ “That the knife is to the left of the book”, whether it is true or false, cannot be the name of a fact.’ As an argument against Russell, this is unfortunately phrased, since he accepted Wittgenstein’s argument that a proposition is not a name, but Ramsey was right nonetheless. There are two conditions our account of meaning should meet: first, understanding the meaning of a proposition requires an understanding of what is the case if the proposition is true; second, the meaning does not depend on whether the proposition is true or false. Russell’s account did not satisfy these conditions.

Logic and necessity From 1906 onwards Russell admitted that the axiom of infinity, despite being expressible in purely logical terms, is not a logical truth. This left him with the problem of supplying some other criterion, if not logical expressibility, that distinguishes logical from non-logical truths. We have already noted his longstanding suspicion of counterfactual reasoning. As late as 1913 he maintained that possibility always marks insufficient analysis: when analysis is completed, only the actual can be relevant, for the simple reason that there is only the actual, and that the merely possible is nothing. (CPBR, VII, 27) In 1918, on the other hand, Russell understood a proposition to be logically necessary if it is ‘true in all possible worlds’ (IMP, 193). Pure logic, he now said, aims at being true, in Leibnizian phraseology, in all possible worlds, not only in this higgledy-piggledy job-lot of a world in which chance has imprisoned us. There is a certain lordliness which the logician should preserve: he must

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not condescend to derive arguments from the things he sees about him. (IMP, 192) The axiom of infinity, for instance, is not logically necessary, because there are possible worlds in which there are only finitely many individuals. This was admittedly not the first time that Russell had made use of ‘Leibnizian phraseology’: in the Problems (121–3) he appealed to possible worlds to explain our contrasting attitudes to the empirical generalization that all men are mortal and the arithmetical truth that 2 + 2 = 4. That, though, is a psychological matter—in the passage he repeatedly used the phrase ‘we feel’—and it would have been open to him to insist that there his talk of possible worlds was intended only to be psychologically persuasive, while strictly erroneous. By 1918, however, the suggestion that possible worlds might be merely a heuristic device is harder to maintain. What is odd, nonetheless, is that elsewhere in the Introduction (165) Russell repeated his earlier view that what appears to be the ascription of necessity to a proposition should properly be interpreted as saying that a certain propositional function is always true. We are thus left with the puzzle of how he could have held both views at once. Once again, the explanation may lie with Wittgenstein, although this time not the Notes, from which any discussion of modality is absent, but his dictation to Moore in Norway in April 1914, where he began to expound the idea that logical truths are tautologies, true in all possible worlds. It should be said, though, that this dictation had notably less effect on Russell than the Notes. Most of the latter he had discussed with Wittgenstein himself, whereas by the time of the former not only had the two men been apart for seven months but they had fallen out and were no longer in regular correspondence. In interpreting the dictation Russell therefore had only Moore to help him, and that, Russell suggested, was no help at all (21 June 1919, in CL, no 65). (Moore’s indignant response was that he had not helped because Russell had not asked.)

Further reading It is disputed how the theory of types in the second edition of Principia should be understood: Cocchiarella’s (1989) account is probably as good as any. Before the second edition Weyl (1918) had already attempted to develop a predicative account of the real numbers, and there has been much subsequent work in a similar vein.

43 POST-WAR JUDGMENT

After giving his two lecture courses in 1917 and 1918 Russell used his five months in prison to prepare for ‘a large projected work, Analysis of Mind’ (CPBR, VIII, 313), in which he hoped to develop new accounts of judgment and of the relationship between mind and matter.

Language and vagueness Wittgenstein’s conception of logical analysis ruled out Russell’s conception of the range of my acquaintance as a proper subclass of the range of the individual quantifier. Whether or not Russell ever fully accepted the argument, he did at least recognize that it made inter-personal communication puzzling. The meaning you attach to your words must depend on the nature of the objects you are acquainted with, and since different people are acquainted with different objects, they would not be able to talk to each other unless they attached quite different meanings to their words. (CPBR, VIII, 174) Russell then drew the only conclusion Wittgenstein’s argument seemed to allow him, namely that the language in which he conversed could not in fact be the logically perfect language of logical atomism. A logically perfect language ‘would be very largely private to one speaker’ and would therefore fail ‘to serve the purposes of daily life’ (CPBR, VIII, 176). If you were to insist on language which was unambiguous you would be unable to tell people at home what you had seen in foreign parts. It would be altogether incredibly inconvenient to have an unambiguous language, and therefore mercifully we have not got one. (CPBR, VIII, 174)

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Russell resisted a solipsism of acquaintance, then, but only by making communication seem like a kind of miracle and the logically perfect language a mistaken ideal. The fully precise propositions whose structure coincides with that of the world remain tantalizingly out of reach; and the world remains correspondingly out of reach too. When in 1919 Russell sought an account of meaning, it was not for the (relatively) precise language of Principia, but for the ordinary, and hence ambiguous, language of every day.

Rejecting the subject Before the War Russell had consistently held that in any act of sensation there is a recognizable distinction between subject and object. In his plan for the new book, on the other hand, he noted that ‘the “subject” as anything but a construction must be avoided’ (CPBR, VIII, 314). What explains his change of mind? The only reason he offered was that the distinction between act and object was ‘not empirically discoverable’. Nor did he yet explain in any detail what was wrong with the ‘most conclusive’ argument for the Self that he had offered in 1913, namely that without it we have no prospect of an account of egocentric indexical thought. Another possibility, though, is that he was influenced by his reading of the Moore dictation. The relation of ‘I believe p’ to ‘p’ can be compared to the relation of ’ “p” says (besagt) p’ to p: it is just as impossible that I should be a simple as that ‘p’ should be. On Wittgenstein’s view, then, I do not occur as a simple component in the analysis of my beliefs. If Russell did change his mind simply because of this remark, however, that would surely demonstrate a remarkable degree of deference to his student’s metaphysical authority (especially given that Wittgenstein’s explanation in the dictation was hardly transparent). Russell took it to be a consequence of abandoning the ‘pinpoint subject’ that his pre-war multiple relation theory of judgment should also be rejected. The theory of belief which I formerly advocated, namely, that it consisted in a multiple relation of the subject to the objects constituting . . . the fact that makes the belief true or false, is rendered impossible by the rejection of the subject. (CPBR, VIII, 305–6) This is puzzling because Russell had only rejected the ‘pinpoint subject’ and remained open to the possibility of constructing the subject as a complex of entities such as sensations and images. There is no immediately obvious reason why such a construction could not occur as one of the relata of a multiple relation. (Of course, if the subject were to be a construction out of judgments, then this would be circular, but Russell has certainly not given us so far any reason to suppose

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that this might be the case.) So although he was perhaps correct to say that ‘the constituents of the belief cannot, when the subject is rejected, be the same as the constituents of its “objective” ’, he had done nothing to explain why they could not be these plus something else (namely the constructed subject just mentioned). What makes his stated reason for rejecting his multiple relation theory especially odd is that, as we shall see shortly, the new theory that he adopted in its place seems to involve just such a complex subject.

Meaning Since Russell now held that the subject matter of logic is words and images, not what they stand for, he owed an account of their meanings which his earlier views had not required. The account he proposed was in two parts: he first treated the meaning of images, and then hoped to derive the meanings of words in terms of association. The former case is simpler, he thought, because ‘images resemble what they mean, whereas words, as a rule, do not’ (CPBR, VIII, 292). He filled this out somewhat by using the notion of mnemic causation which he had learnt from Semon, but it remained sketchy. Consider, for instance, the image-proposition which is involved when I judge that the window is to the left of the fire. In this case, I have a complex image, which we may analyse, for our purposes, into (a) the image of the window, (b) the image of the fire, (c) the relation that (a) is to the left of (b). The objective consists of the window and the fire with the very same relation between them. In such a case, the objective of a proposition consists of the meanings of its constituent images related (or not related, as the case may be) by the same relation as that which holds between the constituent images in the proposition. When the objective is that the same relation holds, the proposition is true; when the objective is that the same relation does not hold, the proposition is false. (CPBR, VIII, 303) It is easy to see how this might be called a ‘picture theory’ of the meaning of image-propositions. The main difficulty of the theory, Russell thought, was that it could not cope with a proposition whose meaning is negative. ‘Propositions, whether of images or words, are always themselves positive facts.’ (CPBR, VIII, 303) Surely just as serious, though, was that Russell did not say how to circumvent Berkeley’s objection (Principles, §8) that ‘an idea can be like nothing but an idea’. Russell claimed that the relation between the images in my mind is the same as that between the window and the fire, but since the latter is spatial, this entails that mental entities can be spatially related too. How? Wittgenstein also proposed a picture theory, but was careful not to suggest—and explicitly warned Russell against—the notion that propositions are spatial facts. ‘Images’, he later noted, ‘are surely no kind of picture at all.’ (PR, §49)

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Having dealt, so he thought, with image-meaning, Russell then moved on to word-meaning, for which he aspired to provide a naturalistic explanation. Such explanations are sometimes known as ‘use’ theories, but this is too vague unless it is specified in what terms the use is to be described. As Dummett (1978, 188) has noted, ‘The limit of total triviality would be reached if it were permissible to describe someone’s use of, e.g., the word “ought” by saying, “He uses it to mean ‘ought’.” ’ Nonetheless, despite its apparent banality, the view that meaning depends on use is hardly to be found in the 19th-century literature. Even Frege, who repeatedly emphasized meaning’s public character, made no explicit attempt to connect it to use. The idea can be traced to Alfred Sidgwick (cousin of the more famous Henry). ‘The meaning of a name is not something distinct from its use.’ (1892, 254) Schiller then acknowledged his ‘extensive indebtedness’ to Sidgwick’s ‘original and penetrating work’ in his Formal Logic (1912, xi), which Russell reviewed dismissively (CPBR, VI, 296–7). We may distinguish two aspects to our use of a word, acquisition (input) and manifestation (output). Naturalistic theories of meaning may be categorized according to which of these they focus on: causal theories appeal to input, pragmatist ones to output.1 Russell’s, however, appealed to both: someone understands a word, he said, ‘when (a) suitable circumstances make him use it, (b) the hearing of it causes suitable behaviour in him’ (AMi, 197). To explain a word’s meaning, therefore, both the causes and the effects of its use are relevant (AMi, 203). In 1919 Russell proposed an explicitly non-behaviouristic account according to which word-meaning depends on image-meaning. ‘In thinking of the meaning of a word, the word calls up images of the object which is its meaning.’ (CPBR, IX, 8) Moreover, he noted explicitly that although ‘ “meaning” is a relation involving causal laws’, it nonetheless ‘involves also something else which is less easy to define’. He had encountered the contrary, behaviouristic conception of meaning first in reviewing Dewey’s Essays (1916), then, while in prison, in the work of Watson (1913, 1914), a behaviourist who proposed to proceed in the opposite direction to Russell, treating the image-proposition as if it were a word-proposition spoken very quietly. At first, Russell supposed the explanation to be that Watson happened not to possess a visual imagination and had therefore mistaken ‘a personal peculiarity for a universal human characteristic’ (CPBR, VIII, 285). Nonetheless, Russell’s own view was far from satisfactory. Although he had made the meaning of word-propositions dependent on that of image-propositions, it was unclear how the dependency worked. Even if a sense

1 Do not confuse this use of the word ‘pragmatist’ to describe a theory of meaning with a later distinction within meaning between ‘pragmatic’ features, which are sensitive to context, and ‘semantic’ features, which are not. The latter usage, although widespread, is unfortunate, since it is far from obvious why the ‘pragmatic’ features should be any more closely related to action than the ‘semantic’ ones.

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could be made out in which an image-proposition resembles part of the world, this resemblance would not extend to the word-proposition we use to express it. He therefore had to admit lamely that his imagistic account was ‘inadequate’ (AMi, 278), and by 1926 he had abandoned it (CPBR, IX, 139) (without, it should be said, replacing it with anything more satisfactory).

The feeling of belief The salient feature of Russell’s pre-war multiple relation theory, in all its variants, had been that it attempted to explain judgment without going via the proposition judged. Now, though, he claimed that ‘truth and falsehood, in their formal sense, are primarily properties of propositions rather than of beliefs’ (CPBR, VIII, 306). This did not amount to a return to his 1904 binary theory, of course, because by ‘proposition’ he now meant a sentence or its mental correlate, not what the sentence expresses. He coined the now-familiar phrase ‘propositional attitude’ (CPBR, VIII, 296) to refer to a feeling directed in some way towards the proposition. (It cannot be explained simply as the feeling on its own, he explained, because one might be entertaining two propositions at once and only believing one of them.) This proposal was originally Hume’s: the standard objection against it, to which Russell’s version is also susceptible, is that there are cases, such as optical illusions, in which we might be said to feel one thing while rationally believing another. At first, Russell’s account of the feeling of belief or doubt directed towards an image-proposition was obscure, because he concentrated more on the feeling of desire and left his view of belief to be inferred by analogy. A desire, on his account, is the cause of a sequence of actions called a ‘behaviour-cycle’; the ‘purpose’ of the desire is whatever brings the cycle to quiescence. The purpose of a belief state is to help us act in ways that achieve our desires. We noted earlier a difficulty for Russell’s account of propositional meaning arising from the fact that propositions are always positive facts, even when what they represent is negative. He proposed to solve this difficulty by treating belief and disbelief as ‘different feelings towards the same content, not the same feeling towards different contents’ (CPBR, VIII, 304). It is a measure of his distance from the ‘Frege point’ that he even suggested there might be disjunctive or implicational belief feelings (AMi, 250) and hazarded that ‘it is possible to develop a behaviouristic logic, starting with the definition that two propositions are logically incompatible when they prompt bodily movements which are physically incompatible’ (CPBR, VIII, 299). Such a logic might, he thought, explain disjunction in terms of hesitation, for instance. Russell showed some attachment to this suggestion, repeating it in his later writing (IMT, 84f), and yet he also insisted that in logic we are concerned with propositions rather than beliefs, since logic is not interested in what people do in fact believe, but only in the conditions which determine the truth or falsehood of possible beliefs. (AMi, 241–2)

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He did not explain how to reconcile the evident tension between the two views. We noted in Chapter 35 Russell’s pre-war hostility to the pragmatists’ attempt to define truth in terms of success. In the 1926 edition of Encyclopaedia Britannica he rehearsed this account in markedly less antagonistic terms. Both human beings and animals act so as to achieve certain results, e.g. getting food. Sometimes they succeed, sometimes they fail—, when they succeed, their relevant beliefs are ‘true’, but when they fail, at least one is false. There will usually be several beliefs involved in a given piece of behaviour, and variations of environment will be necessary to disentangle the causal characteristics which constitute the various beliefs. This analysis is effected by language, but would be very difficult if applied to dumb animals. A sentence may be taken as a law of behaviour in any environment containing certain characteristics; it will be ‘true’ if the behaviour leads to results satisfactory to the person concerned, and otherwise it will be ‘false’. Such, at least, is the pragmatist definition of truth and falsehood. (CPBR, IX, 196) The last sentence suggests continued hesitation on Russell’s part, but in fact his own account has similar consequences. The difficulty, as Wittgenstein later noted, is that whether my expectation is satisfied need not coincide with whether the proposition believed is true. I believe Russell’s theory amounts to the following: if I give someone an order and I am happy with what he then does, then he has carried out my order. (If I wanted to eat an apple, and someone punched me in the stomach. taking away my appetite, then it was this punch that I originally wanted.) (PR, §22) The proposition desired or believed cannot be straightforwardly read off from my behaviour. I only use the terms the expectation, thought, wish, etc., that p will be the case, for processes having the multiplicity that finds expression in p, and thus only if they are articulated. . . . I only call an articulated process a thought: you could therefore say ‘only what has an articulated expression’. (Salivation—no matter how precisely measured—is not what I call expecting.) (PR, §32) For Wittgenstein, therefore, there are only two things involved in a thought’s being true, namely, the thought and the fact; for Russell there are three things, namely, the thought, the fact, and a third thing which, if it occurs, is the recognition. (PR, §21)

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On Wittgenstein’s view, therefore, Russell’s mistake once again was to have dissociated his theory of judgment from his theory of truth. Why, then, did Russell not circumvent Wittgenstein’s objection by saying that the purpose of a desire (or the object of a belief) is whatever state of affairs makes the proposition towards which the feeling is directed true? The reason is that he held non-linguistic animals to be capable of having desires and beliefs, and therefore needed an account of them that does not depend on a capacity for propositional thought. And indeed this decoupling of belief from propositional meaning is just what Wittgenstein objected to. What is a little curious, though, is that his objection is a close relative of one Russell himself had already made in 1913 against James’s account of knowledge. Suppose, for example, that I wish to be with my dog, and start towards the next street in hopes of finding him there; but on the way I accidentally fall into a coal-cellar which he has also fallen into. Although I find him, it cannot be said that I knew where he was. (CPBR, VII, 26) It is surely obvious that examples of this sort can be used just as well to point up the intentionality of belief.

Further reading Russell’s 1919 theory of judgment is notably ill served by the secondary literature: Quinton (1972) makes some worthwhile points. On the influence of Semon on Russell’s theory of word meaning see Pincock (2006). For a modern defence of belief as a kind of feeling see Cohen (1992). Lebens (2017) usefully discusses Ramsey’s argument against Russell’s bi-correspondence theory of truth. Davidson (1982) objects to ascribing beliefs to non-linguistic animals. On Russell’s move to conceiving of language as inevitably vague see Faulkner (2003, 2008) and Levine (2016).

44 NEUTRAL MONISM

There are prima facie three possible ways of denying the dualism between mind and matter: materialists hold that everything is physical; subjective idealists that everything is psychical; neutral monists that everything is of a single kind that is not determinately either physical or psychical. In this chapter we shall trace Russell’s shift after the War towards the latter view.

Adopting neutral monism Versions of ‘neutral monism’ can be traced to Spinoza and Hume, among others. Russell will no doubt have been struck by Ward’s description of the view (under this name) as ‘now in vogue among scientific men’ (1899, II, 206), but there is no sign that at this stage he was yet tempted by it. In 1912 and 1913 he explicitly argued against versions of it advanced by James and Mach (1886). His conversion from rejection to acceptance took two steps. The first step, described in the last chapter, was that in 1918 he abandoned the ‘pinpoint subject’ (CPBR, VIII, 268) and the distinction between sense datum and sensation that he took to depend on it. Quite soon, indeed, he was puzzled that he had ever held such a view. I formerly believed that my own inspection showed me the distinction between a noise and my hearing of a noise, and I am now convinced that it shows me no such thing, and never did. . . . It seems to me now . . . that the whole belief was based on theory and bias, as indeed philosophical beliefs almost always are. I was anxious to rescue the physical world from the clutches of idealism, and I thought it undeniable that there is an exclusively mental event called ‘hearing the noise’. Therefore I made the noise itself as distinct as possible from the hearing of it, in order that the noise might be physical. (CPBR, VIII, 255)

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Russell never published this passage, though: perhaps he thought better of handing his opponents quite so candid an account of how he went about forming his philosophical views. Russell now held that the mental and material worlds overlap—‘Sensations belong in equal measure to physics and to psychology’ (CPBR, X, 4)—but that there may also be purely physical entities such as unobserved events (if phenomenalism is false), and purely mental ones such as images (if behaviourism is false). In order for his position to count as neutral monism, he had to take the further step of holding that all these entities are constructible from a single base of ‘neutral stuff ’. Russell certainly found this second step towards full neutral monism attractive. My bias remains. I still wish to rescue the physical world from the idealist. But if I could rescue the so-called ‘mental’ world from him too! Then the reason for making a gulf between the mental and the physical would disappear. (CPBR, VIII, 255) Neutral monism would thus allow him to reject both materialism and idealism. If we take his views to be either materialistic or idealistic, he suggested, they will seem to involve inconsistencies, since some seem to tend in the one direction, some in the other. For example, when I say that my percepts are in my head, I shall be thought materialistic; when I say that my head consists of my percepts and other similar events, I shall be thought idealistic. Yet the former statement is a logical consequence of the latter. (AMa, 382) Notice, though, that while sufficing to explain why Russell found neutral monism attractive, this falls some way short of an argument for it. In his later writings he sometimes elided this second step, giving the impression that once we have abandoned the inference to the pinpoint subject, neutral monism follows directly. Yet when he first maintained that sensations are both mental and physical, he acknowledged explicitly that this was only a ‘first step . . . towards abolishing the dualism of mind and matter’ (CPBR, VIII, 254), and said of the second step only that it ‘is very attractive, and I have made great endeavours to believe it’ (CPBR, VIII, 289). In the first version of his lectures on the analysis of mind, given in May and June 1919, he held that images ‘obey only psychological laws’ (CPBR, IX, 479). And in the published version of the lectures he still said that ‘the American realists are partly right, though not wholly, in considering that both mind and matter are composed of a neutral stuff which, in isolation, is neither mental nor material’ (AMi, 25, my emphasis). The italicized qualification is surely a slip, however, since elsewhere in the lectures he argued for neutral monism without qualification. The earliest record we have of his adoption of this view unqualified occurs in the syllabus for the expanded version of the lecture course that he delivered later in 1919; here he referred to ‘a unified world out of which both mind and matter arise by different methods of classification’ (CPBR, IX, 484).

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The lecture he was here summarizing, entitled ‘Mind and matter’, was probably written in June 1919: at any rate a version of it certainly existed at the beginning of July, when he sent it to an acquaintance (Percy Nunn) for comments. The lecture itself has not survived, since Russell omitted it from the published version, but in the book he explicitly committed himself to the true metaphysic, in which mind and matter alike are seen to be constructed out of a neutral stuff, whose causal laws have no such duality as that of psychology, but form the basis upon which both physics and psychology are built. (AMi, 287) One thing that helped Russell to accept neutral monism may have been that he found a way to avoid the over-generation problem, namely that there are far more ‘things’ and ‘biographies’ than physics or psychology requires. What he now proposed was that the constructed entities should count as physical if they obey the laws of physics, and as mental if they obey the laws of psychology. Thus his view did not make him a panpsychist: not everything is mental, he thought, because not everything obeys psychological laws. Nor did it require him to be a physicalist: it is a further question, on which he hesitated, whether the mental objects also obey physical laws and hence are physical.

A second wave? Russell had thus adopted neutral monism in its unrestricted form by the end of June 1919. The date is of some interest, because it was, by chance, just before he first read the Tractatus. (Keynes had arranged for Wittgenstein to send him the typescript which he had with him at the Armistice from his Italian prisoner-ofwar camp by diplomatic bag. Keynes acknowledged receipt on 28th June and forwarded it to Russell straightaway.) Forty years later Russell recalled that Wittgenstein’s influence on him had come in ‘two waves’ (MPD, 112), one before the war and one after: of the first wave we have already seen ample evidence; of the second, curiously, there is hardly any. One looks almost in vain in Russell’s work for ideas from the Tractatus that are not already in either the Notes or the Moore dictation. The one exception is Wittgenstein’s claim that intentional propositions can be analysed in such a way that they do not threaten an extensional hierarchy: Russell thought this proposal sufficiently interesting to devote an appendix to the second edition of Principia to assessing it. Why did the Tractatus have so little influence on Russell? One reason, perhaps, is that he had no sympathy with Wittgenstein’s remarks on the unsayable: in the case of ethics and religion, he regarded them as the regrettable result of reading too much Kierkegaard; in the case of semantics, he held that they could be overcome by use of the distinction between object language and metalanguage—a distinction which he first made (although not using these words) in his Introduction to

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the Tractatus. Russell’s failure to take seriously the limits that unsayability places on our ability to state the theory of types may explain Wittgenstein’s scathing reaction to his Introduction to Mathematical Philosophy. I should never have believed that the stuff I dictated to Moore in Norway six years ago would have passed over you so completely without trace. In short, I’m now afraid that it might be very difficult for me to reach any understanding with you. And the small remaining hope that my manuscript might mean something to you has completely vanished. (12 June 1919, in CL, no. 64) In his dictation to Moore Wittgenstein had expressed the idea that the theory of types is in principle inexpressible, because it is a product of the expressive capacities of the symbolism—capacities which can be shown but not said. In Wittgenstein’s view this inexpressibility is absolute, but Russell persistently took it to be merely relative—capable of being overcome by moving to a metalanguage.

The inference from percepts to events Neutral monism changed Russell’s conception of the base to be used in constructing the external world. The unsensed sensibilia of 1914 were replaced (in the case of vision) with electromagnetic waves, while sense data were replaced with events in the visual cortex. In recognition of this change he dropped the term ‘sense datum’ and now called entities that are both physical and mental ‘sensations’—or, when he realized that this made them sound too mental, ‘percepts’. One difficulty with this new conception is the violence it does to our sensory vocabulary. For instance, Russell now claimed that if a neurosurgeon performs an operation, what he ‘sees’ are actually events in his own brain, not in his patient’s (OP, 146ff, AMa, 320, 383). One way to resolve this oddity would be to insist that the very same entities which are described in the vocabulary of perception as, for instance, patches of colour in my visual field may also be described using physical vocabulary, in which case they are called events in my brain: under one description, it is indeed bizarre to say that the surgeon sees events in his own brain; but not under the other. In The Analysis of Mind Russell did not describe his construction of mind and matter out of neutral stuff in any detail—perhaps that was dealt with in the lecture on ‘Mind and matter’ that he left out of the published volume—but in 1927 he published The Analysis of Matter, a ‘companion volume’ in which he revised his construction of the external world to take account of relativity theory. This required him to hold that even if my percepts are located in a sequence of momentary spaces corresponding to the specious present, the public world to be constructed from these resources is not a sequence of spaces but a single space-time. How Russell met this purely technical challenge need not detain us here. What is of more general interest is that the construction still required an inference

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from observed to unobserved entities: in 1914 it had been an inference from sense data to sensibilia; now it was from percepts to events. He thus still drew something like the earlier distinction between knowledge by acquaintance and knowledge by description, although not under these titles. (This required him, of course, to accept that we can quantify over things we are in no position to name.) Our knowledge of events that are not percepts is purely structural: we know the external (or formal) properties of the relations between them, but we know nothing about their internal natures (or contents). We infer this structural knowledge from our direct knowledge of the relations between percepts by means of what Russell called the ‘causal theory of perception’—the principle that the events that are not percepts are related in ways that are isomorphic to the relations we perceive between percepts. Russell attempted to align the distinction between internal and external properties of events with that between psychology and physics: if a thing is psychical, introspection may reveal to us its internal properties; physics, on the other hand, is concerned only with its external (i.e. structural) properties. Because percepts are subject to both psychical and physical laws, we can know both their internal and their external properties and hence ‘give physics the greater concreteness which results from our more intimate acquaintance with the subject matter of our own experience’ (AMa, 10). This conception of physics as purely structural was the target of a well-known objection by Newman (1928), who pointed out that if a set is endowed with relations having certain structural properties, then any other set of the same cardinality can be endowed with relations having the same structural properties. So the inferred knowledge of the structure of the set of events which Russell claimed really amounts to no more than knowledge of its cardinality.

Emergence Even if we can construct out of neutral stuff entities that play the roles of matter and mind, it is a further question whether their properties are constructible. Alexander (1920, II, 46–7) called non-constructible properties ‘emergent’, a terminology that is now standard. For instance, it might be that we could construct matter out of events, but that mass was not definable in these terms, in which case mass would be emergent relative to this base. Russell swithered over whether any of the properties that science attributes to matter might be emergent: in The Analysis of Matter (285–6) he claimed that what passes for emergence is mere incompleteness: the properties of a whole are in principle definable in terms of the properties of its parts and hence at most practically, not theoretically, emergent; yet in the very same year he conceded elsewhere (OP, 294–6) that although the continuous properties of the physical world can be deduced from those of events, the discontinuous properties posited by quantum mechanics are probably emergent. Moreover, he also held, with rather more confidence, that the properties of mind are emergent, because

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our knowledge of data contains features of a qualitative sort, which cannot be deduced from the merely mathematical features of the space-time events inferred from data, and yet these abstract mathematical features are all that we can legitimately infer. (OP, 295) Unless we can rule out the possibility that the properties of matter and mind are emergent relative to those of particulars, the project of construction looks inadequate. The attraction of neutral monism was supposed to be that it obviated the need for a mysterious relation between mind and matter; now the difficulty was in danger of recurring in the need for an equally mysterious relation between their properties. What is striking, indeed, about the whole of Russell’s discussion of neutral monism in the 1920s is its focus on analysing entities, not the facts of which they are constituents. His enthusiasm for Wittgenstein’s conception of analysis, which privileged the latter over the former, was evidently short-lived.

Further reading For more recent discussions of emergence see Kim (2006) and, in relation to the mind, Crane (2001). For helpful discussions of neutral monism see Lockwood (1981) and Rosenberg (2004).

45 RUSSELL’S LEGACY

Russell’s later career was only fitfully that of an academic and he did not, after Wittgenstein, have research students in the conventional sense. (The nearest would be Wrinch, who acted as his secretary for a time and contributed significantly to Jeffreys’s work on scientific method, but is remembered more for her later research in molecular biology.) So although he directly influenced Wittgenstein before the First World War, and Ramsey after it, his wider effect on 20th-century philosophy was not principally through his students.

The demise of absolute idealism Bradley, the absolute idealist who dominated British philosophy at the end of the 19th century, is nowadays hardly read. Modern anti-realists do not typically deny, as he did, that our ordinary claims about the world are straightforwardly true. For this decline the revolt that took place in 1898 was undoubtedly at least partially responsible. Russell himself was clear that it was Moore who deserved credit for it, but, as we noted earlier, what Moore wrote, while certainly a rejection of idealism, fell well short of refuting it. In Russell’s case the main reason he rejected it may well be that he found realism allowed him to make philosophical progress that he had not made without it. Absolute idealism did not vanish instantly, but simply failed to gain new adherents, so that it died at the same rate as its followers. The philosophy that replaced it had obvious connections with the British empiricism of Locke and Hume, with the result that absolute idealism came to be seen as ‘an exotic in the English scene, the product of a quite recent revolution in ways of thought due primarily to German influences’ (Warnock 1958, 9). Nonetheless, the reason it disappeared can hardly have been that it was too German: Bradley and Bosanquet were

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thoroughly British, and logical positivism was at least as much a German (or at least Austrian) import. Nor can it be wholly a matter of temperament or politics: the British idealists were not all religious, nor their opponents all atheists, for instance; and although it is true that Russell and Moore opposed the First World War when McTaggart and Ward did not, that is hardly enough to explain the influence of the former or the post-war neglect of the latter. Modern commentators sometimes present this an instance of history belonging to the victors: they accuse Russell and Moore of caricaturing Bradley’s views. The difficulty with this, though, is that Bradley did indeed say many of the things they accused him of saying. It is just that he also said the opposite: his aim was to show that the notions in question were defective and should be given up. It is a mark of the gulf between Bradley’s work and the present that most modern philosophers, if they read him at all, are puzzled not that his popularity waned but that he was ever popular in the first place. This is partly because his roundly rhetorical style alienates modern readers, but also because they are largely ignorant of the Kantian background he presupposed.

The external world programme One of the central components of Russell and Moore’s revolt against idealism was that the sense data we are directly aware of in sensation are non-mental—a view shared by others such as Alexander (1909) in Manchester and Nunn (1910) in London. In the 1920s and 30s the sense-datum theory was widely accepted. I have mentioned Russell’s tendency, under Moore’s influence, to deduce the nature of sense data from little more than first-person testimony. One strand of the programme, therefore, sought to refine the characterization of sense data by use of experimental evidence from psychology. Carnap conceived of his Aufbau explicitly as attempting to complete Russell’s programme, and took Russell’s principle of preferring construction to inference as his motto (1928, 1). The difference was that Carnap attempted to start, solipsistically, from the sense data of a single person (what he called the ‘auto-psychological base’), a project which, as we noted in Chapter 39, Russell had abandoned as implausibly ambitious. However, Carnap achieved this more ambitious aim only by making use of what is not really a construction in Russell’s sense, since he relied crucially on implicit, rather than explicit, definitions. Neither Russell’s nor Carnap’s version of the project was altogether successful, but their failure might have spurred on others to more elaborate attempts if philosophers had not begun to doubt whether there is any immediate basis on which our understanding of the world could intelligibly rest. One doubter was the later Wittgenstein, who argued that meaning, being public, cannot be explained in terms of private elements such as sense data, and questioned thereby Russell’s Cartesian assumption that the internal is epistemologically more secure than the external. It is an important question, however, whether Wittgenstein’s private

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language argument applies only to an intrinsically private base, in which case sense data would be defensible provided that they were only accidentally available to only one person. A second kind of attack, mounted most famously by Austin (1962), argued that Russellian sense data were the by-product of an erroneous assimilation of illusion to veridical perception. Even if we were to make the prior admission . . . that in the ‘abnormal’ cases we perceive sense data, we should not be obliged to extend this admission to the ‘normal’ cases too. For why on earth should it not be the case that, in some few instances, perceiving one sort of thing is exactly like perceiving another? (1962, 52) A further attack was made by Sellars (1956) on the ‘myth of the given’, the foundationalist epistemology of hard data that Russell and Carnap presupposed. Sellars argued that each of our cognitions is theory-laden, and hence cannot be treated as basic. The fact that Russell’s external world programme did not flourish may explain why the neutral monism which he advocated in the 1920s failed to attract many adherents at the time, although similar views have become popular more recently.

Logicism Principia, on which he and Whitehead laboured for almost a decade, is one of the most celebrated but least read of Russell’s works. Their project, like Frege’s in the Grundgesetze, had two parts, one formal, the other philosophical: the former aimed to show that mathematics can be captured in a single formal system; the latter that it is (conditionally, at least) part of logic. These two parts are prima facie independent: even if mathematics can be presented as a formal theory, it is a separate question whether that theory is part of logic; and it would require argument to establish that logic itself is in this sense formal. The first part of the project was dealt a significant blow in 1931 when Gödel proved that no consistent formal system proves all the truths expressible in the language of arithmetic. Notice, though, that Gödel’s theorem does not refute formalism outright, because the committed formalist need not recognize the metalinguistic notion of truth to which the theorem appeals. (In recognition of just this sort of concern, indeed, Gödel in the paper carefully stated his theorem in a more restricted form which made no explicit use of the concept of truth at all.) By the middle of the 20th century a sort of ‘informal formalism’ had become a ubiquitous part of the folklore of mathematics: it was formalism, because it held that there is a particular formal system in which all genuinely mathematical theorems can be both expressed and proved (the qualifier ‘genuinely’ being the required sop to Gödel); but it was informal, because no one showed much interest in actually carrying out the formalizations.

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Nonetheless, the formal system in which mathematicians supposed that their theorems could be presented was not Principia but first-order Zermelo-Fraenkel set theory, ZF. This was nothing to do with Gödel’s theorem: whatever the title of his paper (‘On formally undecidable propositions of Principia Mathematica and related systems’) might have suggested, his theorem is no less applicable to other formal systems than it is to Principia. Mathematicians’ preference for ZF may be traced to two features: first, ‘x ∈ y’ is meaningless in Russell’s theory if x and y are not of the correct types, whereas in ZF it is merely false; second, types in Principia are disjoint (so that ‘x ∈ y’ can be true only if the type of x is one less than that of y), whereas in ZF they are cumulative (so that the type of x has only to be less than that of y). The first of these differences is technically inconsequential, but practically important: a system in which lots of things cannot be said is less convenient in practice than one in which they can be said but are false. The second difference, on the other hand, is significant technically: in Principia every type can be obtained from the type of individuals in a finite number of steps, whereas in a cumulative theory types can be (and in ZF are) iterated into the transfinite. How far this matters in practice is moot, however, since very little of the mathematics that interests working mathematicians really depends on transfinite types. To treat Principia as a ‘formal system’ in the modern sense (according to which whether something is a proof must be mechanically verifiable) involves in any case a considerable degree of charitable reconstruction. Not only did the book confuse use with mention so frequently that Frege gave up reading it in despair (PMC, 81–4), but the rewriting rules for eliminating incomplete symbols such as class-terms were not unambiguously specified. What, though, of the second part of Russell’s project, the claim that mathematics is part of logic? Here his great difficulty was the lack of a principled criterion of logicality. Insofar as he had one, the axiom of infinity and reducibility counted as non-logical, in which case the first edition of Principia should best be regarded not as a proof of logicism but as a regressive demonstration of the need for non-logical axioms in the foundations of mathematics. (The second edition unfortunately cannot be regarded as a convincing demonstration of anything very determinate, because Russell was so unclear there about the rules of the theory of types he was proposing.) The regressive method was taken up by Gödel (1944, 1947) and hence became widespread as an account of our belief in some of the axioms of set theory (principally strong axioms of infinity such as the axiom schema of replacement or large cardinal axioms).

Naturalism Russell was a major influence not only on the Vienna Circle but on Ayer, its British disciple. The latter’s inclusion of Russell’s ‘Logical atomism’ in a widely read collection on logical positivism (1959) contributed to the tendency to regard him as more sympathetic to this cause than he was. Although he agreed with the scientific outlook of the positivists and, following Wittgenstein, with their view of

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logic, Russell was never quite the empiricist that those who treated him as Hume’s successor supposed. ‘We all in fact are unshakeably convinced’, he said in 1935, ‘that we know things which pure empiricism would deny that we can know. We must accordingly seek a theory of knowledge other than pure empiricism.’ (CPBR, X, 326) And as late as Human Knowledge (1948) he continued to hold that pure empiricism is self-refuting (because its claim that all knowledge is grounded in experience is not itself grounded in experience). Nor did he share the logical positivists’ distaste for metaphysics, regarding Wittgenstein’s attempt to limit the scope of the sayable as a failure. Importantly, though, Russell did share the positivists’ scientistic conception of philosophical method. In 1912 he told Ottoline Morrell of his long-held dream ‘to found a great school of mathematically minded philosophers’ (29 Dec.). Philosophical knowledge, he said, ‘does not differ essentially from scientific knowledge; there is no special source of wisdom which is open to philosophy but not to science’ (PP, 233). And later he affirmed even more emphatically that ‘there is one method of acquiring knowledge, the method of science’ (CPBR, IX, 91). If modern philosophers are classified by whether they conceive of their subject (as Wittgenstein did, for instance) as a kind of prolegomenon to science or (with Quine) as continuous with it, then Russell was one of the founders of the latter group. The issue here is not whether we take science seriously, but what we regard as its scope: philosophers in the former camp typically regard the scientific method as the best route to truth within its own domain, but hold nonetheless that there is another domain where it gets no purchase; those in the latter emphasize instead the unitary character of the notion of knowledge. One of the most significant battlegrounds of naturalism in 20th-century philosophy was the theory of meaning. Modern writers have not always been clear, however, about the difference between naturalism and behaviourism. Russell did not originate this, but played a part in its popularization. Particularly influential, perhaps, was Skinner’s adoption of behaviourism as a result of reading his work.

Cambridge analysis In a 1922 book review Russell distinguished between ‘the analytic type, to which most British philosophers have belonged’ and ‘the synthetic type, which has been almost exclusively dominant on the Continent’ (CPBR, IX, 406). For the remainder of the 20th century the division which he thus labelled ran deep, but it has been notoriously difficult to identify the defining characteristics of either type. At the very least the method of analysis attempts to get clear about what we mean. In the case of Moore, whom Russell was discussing in the book review, this method was to be applied to claims which common sense shows to be true but philosophers have doubted. The most celebrated example of the method of analysis was of course the theory of descriptions. Russell later claimed that this theory struck Stout, the

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then editor of Mind, as ‘so preposterous that he begged me to reconsider it and not to demand its publication as it stood’ (MPD, 83); but unfortunately he did not go on to report what Stout’s objection to it had been. Soon, though, it came to be seen as a crucial example of the method of analysis. Wittgenstein described it as ‘quite definitely correct’, Ramsey as ‘that paradigm of philosophy’ (FoM, 263n). Important as it was, though, the theory of descriptions could not on its own have sustained a whole school: what for a time held out the prospect of substantial philosophical dividends was the method of incomplete symbols of which it was an instance. Russell’s influence is visible in the work of philosophers such as Wisdom and Stebbing—sometimes known as the ‘Cambridge school of analysis’—but it was largely transmitted to them via Moore rather than by Russell himself (thus explaining, perhaps, the unscientific tilt of their work). The journal Analysis was founded in 1933 with the explicit purpose of publishing articles ‘concerned . . . with the elucidation or explanation of facts . . . the general nature of which is, by general consent, already known’. This emphasis on elucidating what we all already know was Moore’s. The Cambridge school of analysis did not last long: Stebbing died in 1943, while Wisdom’s later work grew markedly less analytical under Ryle’s influence. The real reason the school withered, though, was that it never delivered what it promised. For instance, none of its members came close to analysing ‘Britain declared war on Germany’ into a series of claims about individual people. Nonetheless, Russell’s label, ‘analytic philosophy’, stuck. Already in his 1922 book review he noted prophetically that on the question whether philosophic truth is to be arrived at by analysis or by synthesis, it is apparently impossible to find any argument which ought to appeal to both sides. Every argument begs the question by being either analytic or synthetic. (CPBR, IX, 408)

Ordinary language philosophy Assessing Moore and Russell’s influence on 20th-century philosophy is complicated by the divergence of their interests after the First World War. While Russell was pursuing his external world programme in The Analysis of Mind and The Analysis of Matter, Moore’s application of common sense tended increasingly towards a kind of clarification of everyday locutions that became known, when carried out by Austin’s followers in Oxford after the Second World War, as ‘ordinary language philosophy’. The distance Moore himself travelled away from Russell’s approach is visible in his painstaking assessment (1944) of the theory of descriptions, in the course of which he queried Russell’s analysis of ‘Scott was the author of Waverley’ on the bizarrely pedantic ground that Scott might have been the author without having written the book (if, for instance, he had dictated it).

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Much as Frege chose to write ‘On sense and reference’ as if it was about the philosophy of language, even though his real concern was primarily with logic, so also Russell presented ‘On denoting’ as if his concern were with analyzing George IV’s curiosity about the authorship of Waverley rather than with solving the logical paradoxes. It is understandable, therefore, that many readers have read Russell’s theory of descriptions, like Frege’s theory of sense, as a contribution to the philosophy of language. Taken in this way, the theory was challenged by ordinary language philosophers immediately after the Second World War, most notably Strawson (1950), who complained that it fails to give plausible analyses of various uses of definite descriptions in English. For instance, its analysis of ‘The table is covered with books’ entails, implausibly, that the sentence cannot be true unless there is precisely one table in the universe. Notice, though, that this complaint is not specific to Russell’s theory of descriptions, but applies equally to Frege’s account of quantifier phrases. If I said at the beginning of a class, ‘Everyone’s here, so I’ll start,’ I would plainly intend the domain of the quantifier to be contextually restricted to whoever I was expecting to attend, not everyone in the world. Such restrictions are a ubiquitous feature of everyday discourse; it is plainly a legitimate (and far from trivial) task for the philosopher of language to determine the rules governing them. What is less clear is whether Russell’s failure to state these rules is a good objection to his theory. In 1957 he wrote a response to Strawson’s paper in which he expressed his disenchantment with ordinary language philosophers. ‘They are persuaded’, he said, that common speech is good enough not only for daily life, but also for philosophy. I, on the contrary, am persuaded that common speech is full of vagueness and inaccuracy, and that any attempt to be precise and accurate requires modification of common speech both as regards vocabulary and as regards syntax. Everybody admits that physics and chemistry and medicine each require a language which is not that of ordinary life. I fail to see why philosophy, alone, should be forbidden to make a similar approach towards precision and accuracy. (CPBR, XI, 632–3) Strawson’s error, Russell suggested, had been to suppose that he had been engaged in analysing the state of mind of those who utter descriptions, whereas he had in fact been attempting ‘to find a more accurate and analysed thought to replace the confused thoughts which most people at most times have in their heads’ (CPBR, XI, 634). He contemptuously diagnosed the interest ordinary language philosophers took in ‘the different ways in which silly people can say silly things’ (CPBR, XI, 624) as resulting from ‘the desire to separate [philosophy] sharply from empirical science’, and hence as directly threatening his naturalistic conception of the discipline. Russell was now writing more than 50 years after ‘On denoting’, but if his recollection was correct, it demonstrates the rather theoretical character of his principle of acquaintance. In order for me to grasp an ‘accurate and analysed

310 Russell

thought’, I would have to be acquainted with all its constituents, but what about the ‘confused thoughts’ that I have in my head instead? Are they even determinate enough to have constituents in the required sense? If so, is it these I have to grasp, or is it the constituents of the precise thoughts that lie in the theoretical background? Strawson could reasonably have complained, too, that if it is wrong to regard ‘On denoting’ as a contribution to the philosophy of ordinary language, then the error was of Russell’s own making. Although the reason he was interested in the problem of denoting concepts was that he hoped it would shed light on the paradoxes, he chose not to mention them at all in ‘On denoting’, instead suggesting merely that his aim was the solution of ‘logical puzzles’. Without further guidance, how was a reader to tell which were logical puzzles, and which merely linguistic? Strawson’s specific charge against Russell’s analysis of ‘the f is g’ as the conjunction of ∃x∀y(fy ≡ x = y) and ∀x(fx ⊃ gx) was that it ignored the distinction between the content of an utterance and its presuppositions. He claimed that if ‘the f ’ is the logical subject of the analysandum, the first clause of the analysis should be thought of as a presupposition of the utterance rather than as part of its content. At first, he restricted this point to definite descriptions, but soon (1952, ch. 6) he widened it to apply to quantifier phrases as well, suggesting that an utterance of ‘Every f is g’ presupposes, but does not assert, that there is at least one f . Strawson’s diagnosis was thus that Russell had tried to analyse sentences when he should have been analysing utterances. The difference is significant in the case of a sentence containing an indexical, which expresses different propositions in different contexts of utterance. On Strawson’s view we cannot analyse the sentence ‘The present King of France is bald’ until we know when it was uttered: someone who uttered it in the reign of Louis XIV would have said something true; in the reign of Louis XV, false; nowadays, truth-valueless. There is nothing to prevent a precisely specified formal language from containing indexicals. So Russell could not now respond merely by insisting on his lack of interest in the vagaries of ordinary language. Instead, he argued that the problem he had been addressing in ‘On denoting’ did not depend on indexicality. Strawson, he said, fastens upon the egocentric word ‘present’ and does not seem able to grasp that, if for the word ‘present’ I had substituted the words ‘in 1905’, the whole of his argument would have collapsed. (CPBR, XI, 630) As Russell’s bad-tempered tone indicates, he plainly resented being given a lesson in the logic of indexicals. In fact, however, Geach (1950) had already made essentially the same argument as Strawson without appealing to indexicals: the issue still arises for mathematical sentences such as ‘The largest prime number is odd’ or scientific ones such as ‘Vulcan is a planet’, even though mathematical language is, and scientific language at least aspires to be, free of indexicals. Whether it is

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correct to distinguish between the presuppositions of an utterance and its content therefore cannot depend solely on whether it contains an indexical. Strawson certainly had some linguistic intuitions on his side: Russell’s bare claim that ‘The present King of France is bald’ is ‘plainly false’, rather than truthvalueless, is dubious. On the other hand, there are sentences for which our intuitions are more favourable to Russell’s case: most of us would say, for instance, that ‘I met the present King of France last night’ is false. One might wonder, however, how much our intuitions about what Quine (1960, 260) called ‘don’t care cases’ matter for logical purposes. If Russell was as uninterested in ordinary language as he later claimed, perhaps he would have been better not to rely on intuition to support his case at all. He could have granted Strawson’s distinction between presupposition and content, and instead maintained that his aim was to supply, for each sentence p, a sentence p0 such that (1) in circumstances in which the presupposition for the utterance of p is fulfilled, p0 is materially equivalent to p; (2) otherwise p0 is false. If p were ‘The f is g’, p0 would be the Russellian translation ∃x(∀y(fy ≡ x = y)∧gx); and if p were ‘Every f is g’, p0 would be ∃xfx∧∀x(fx⊃gx). In another well-known article Donnellan (1966) argued that Russell’s account is not so much wrong as incomplete, because it fails to distinguish between two sorts of use of definite descriptions. Suppose (to amend his example slightly) that I utter the sentence The man drinking a martini is happy,

(1)

but that the happy man I can see is in fact drinking water, not a martini. I have plainly made a mistake, but of what sort? Suppose that there is as it happens a miserable man just outside my line of sight who is drinking a martini. Which have I referred to? Russell’s account analyses what Donnellan calls the ‘attributive’ use, on which I have said something false about the man who is in fact drinking a martini. If, however, I am making a ‘referential’ use, I have said something true about the man drinking water, but erred in my description of him. Kripke (1977) subsequently suggested that this is a special case of a more general distinction between what he called the semantic reference and the speaker’s reference of a singular term. Kripke himself drew from this the moral that Donnellan’s point could not be a complaint against Russell’s theory of descriptions as such, since the phenomenon to which he drew attention is not limited to descriptions alone. This, though, was because Kripke had, for independent reasons, rejected Russell’s treatment of proper names as disguised descriptions. To anyone who accepts Russell’s treatment, Kripke’s distinction need not be regarded as substantially more general than Donnellan’s. One possible response would be to suggest that the issue Donnellan was addressing was not relevant to Russell’s project. As listeners, we bring various methods to bear, often subconsciously, when a speaker makes a mistake, with the aim of retrieving the thought which the speaker intended to express; but the rules for doing this belong to linguistics, not logic.

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Later reception Ordinary language philosophy withered after Austin’s death—his works are now hardly more likely to be read in analytic philosophy departments than Bradley’s— but its demise did not mark a return to Russellian analysis. In the 1970s and 80s Anglo-American analytic philosophy was dominated by the linguistic turn of Dummett and Davidson, whose distance from Russell may be gauged by the fact that Dummett (1993a) proposed a characterization of analytic philosophy that largely excluded him from it. Since then, Russell has been ‘historicized’—treated as if he were of interest only to historians and not to active philosophers. Despite his many changes of mind, however, the principles that underlay his work held stable: that philosophy proceeds by analysis and has as its goal making certain things precise; that ‘all our knowledge of matter is derived from perceptions, which are themselves causally dependent on effects on our bodies’ (AMa, 27); but that we also have other knowledge not so derived. This last assumption makes his work inimical to naturalists, of course, although some have curiously failed to notice that he made it. Among those who accept his assumptions, though, the reason his influence has declined is that he never found an account of perceptions that sufficiently acknowledge their complexity. The sense-datum theory cast a shadow over his thinking from which he never fully emerged.

Further reading On British philosophy in the 1930s, see Black (1938). Lugg (2005), Stevens (2006) and Morris (2015) discuss Russell’s influence on Quine’s naturalism. Baldwin (2013) discusses the influence of Russell and Moore on the Cambridge school of analysis, while Urmson (1956, ch. 10) suggests reasons for its demise. On Strawson’s notion of presupposition see Sellars (1954).

PART III

Wittgenstein

46 BIOGRAPHY

The Wittgensteins were one of the wealthiest families in Austria. Ludwig’s father, Karl, made his money in steel before becoming in retirement a noted patron of the arts in Vienna: in 1905 he commissioned Klimt to paint his daughter’s wedding portrait; the Secession art gallery was built with his money; and at the family house (known without apparent irony as the Palais Wittgenstein) he hosted musical evenings attended by Brahms, Schoenberg, Strauss and Mahler. Ludwig, the youngest in a family of eight children (a ninth died at birth), was born in 1889 and initially educated at home. From the age of 14 to 17 he attended the Oberrealschule in Linz, where, notoriously, he overlapped for one year with Hitler; but although they were the same age, they were not in the same class—Wittgenstein was a year ahead of his age in Class V, Hitler a year behind his age in Class III—and there is no persuasive evidence that they ever met. Wittgenstein first trained in engineering at a technical college in Berlin, before conducting research on jet engines in Manchester for three years. There he became interested, for reasons now obscure, in the foundations of mathematics. He read Russell’s Principles and wrote to Jourdain, a friend of Russell, with a proposed solution (now lost) to Russell’s paradox. In October 1911 he turned up at Russell’s rooms in Cambridge to announce his intention of doing research in the foundations of mathematics; he spent the next two years there with Russell as his supervisor and, soon, friend. Throughout his philosophical career it was his practice to keep a journal in which he recorded his philosophical thoughts day by day, but his pre-war journals have not survived; what little we know of his earliest philosophical views is either contained in the handful of letters he wrote to Russell or inferred from the occasional anecdotes about him with which Russell entertained his lover, Ottoline Morrell.

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Wittgenstein decided to spend the academic year 1913–14 in Norway. Just before he departed, he compiled some of the remarks from his journals into a text—his earliest surviving philosophical work—now known as the Notes on Logic. Ostensibly the compilation was for Russell’s benefit, but it also provided Wittgenstein himself with a convenient summary of his Cambridge notebooks that he could take away with him. The problems which occupied him in Norway centred on providing an account of the foundations of logic and linguistic meaning. He made what he at least believed was substantial progress during the year, but his Norwegian notebook is also lost and the only surviving record of this work, apart from a few more letters to Russell, is some notes he dictated when Moore visited him in April 1914. When the war began in July 1914, Wittgenstein was, as it happened, back home in Vienna. He volunteered almost immediately and served in the Austrian army throughout the war: as a searchlight operator on a boat; an engineer repairing military equipment (injured in an explosion at the workshop where he was working); then, from April 1916 onwards, a front-line soldier in an artillery regiment (twice decorated for bravery). During the war Wittgenstein continued to keep a philosophical journal: three of these wartime notebooks have survived. Then he began (probably towards the end of 1915, although the exact date is uncertain) compiling remarks from his notebooks and typescripts into a philosophical treatise (in German, Abhandlung). The large hardback volume in which he carried out this process is referred to as Bodleianus (because it now resides in the Bodleian Library, Oxford). The decimal numbering of the paragraphs in the Tractatus is one of the book’s best-known features, but in Bodleianus its primary purpose was the purely functional one of indicating, as he copied remarks, the order in which he wanted them to be printed. The survival of Bodleianus therefore allows us to trace the order in which he compiled the book as well as that in which he intended it to be read. The most striking feature it preserves is Wittgenstein’s original intention of ending the book not with the famous proposition 7—‘Whereof one cannot speak, thereof one must be silent’—but with the rather more prosaic claim about the general form of truth-functions at proposition 6. Not until he had filled Bodleianus as far as page 64 did he begin to add the 6s, which contain his accounts of the nature of logic, mathematics, science, ethics, and philosophy itself. It seems, then, that he did not originally intend to discuss any of these in the Tractatus, or indeed to address the peculiar difficulty implicit in reading a text which says of itself that it is nonsensical. Eventually Wittgenstein had a typescript made—the so-called Prototractatus— and then, after a further round of revisions in the summer of 1918, a final typescript, with a carbon copy. One of these (now known as the Engelmann typescript) was with him when he was taken prisoner by the allies at the Armistice; the other was sent off to a publisher. (Later another copy was typed to send to Frege.) He had considerable difficulty getting the book accepted for publication

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in post-war Austria, however, and it was only after Russell had agreed to write an Introduction that it appeared in 1922, first in a poorly produced German edition, then in a dual-language edition with an English translation on facing pages. This translation was originally attributed to Ogden, the editor of the series in which the book appeared, but was in fact done in March of that year by Ramsey, then a second-year mathematics undergraduate at Cambridge. Ogden sent Ramsey’s draft to Wittgenstein in Austria, who made detailed revisions to it. In a few places the English he recommended was a gloss rather than a literal translation: curiously, he chose not to amend the German at all, even where he admitted that it was unclear. Disparaging remarks he made to friends later in life about the translation suggest that he had by then forgotten the extent of his involvement in it. After his death (and in ignorance of some of these facts) Pears and McGuinness made a second translation—a more consistently accurate representation of Wittgenstein’s German than the first, which contains a few definite mistakes. By the time the Tractatus was published, Wittgenstein had been persuaded (in part, perhaps, by his reading of Tolstoy) of the virtue of poverty, and had given away all his inherited wealth to his siblings to become a primary school teacher in Lower Austria. This phase of his life ended abruptly in 1926 when he resigned after striking a pupil. He was charged with assault, but the case was dropped. He devoted much of the next two years to the detailed design of a house his sister was building in Vienna. During this time Schlick encouraged him back to philosophy by means of a series of meetings with members of the Vienna Circle. He returned to Cambridge in 1929 to begin a second phase in his philosophical career.

References Wittgenstein’s writings are referred to by the following abbreviations: NoL NdM PTLP TLP PR PG BB PI PPf OC CKO CL PO PPO

Notes on Logic Notes dictated to Moore Prototractatus Tractatus Logico-Philosophicus Philosophical Remarks Philosophical Grammar The Blue and Brown Books Philosophical Investigations Philosophy of Psychology—A Fragment On Certainty Letters to C. K. Ogden Cambridge Letters Philosophical Occasions Public and Private Occasions

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Decimal numbers unqualified refer to the Tractatus, of which I use the Ogden–Ramsey translation, since it was sanctioned by Wittgenstein himself. Readers using the Pears–McGuinness translation need to be aware that it differs in its translation of the following technical terms. German Tatsache Sachverhalt Sachlage

Ogden–Ramsey fact atomic fact state of affairs

Pears–McGuinness fact state of affairs situation

Further reading Wittgenstein’s philosophy is harder than that of most philosophers to disentangle from his life. McGuinness (1990) is a perceptive study of the relationship between the two that reaches as far as the publication of the Tractatus. The best singlevolume introduction to Wittgenstein’s philosophy (early and late) is Kenny (1973), although Child (2011) comes a close second. On the Tractatus Anscombe (1959) remains worth reading, as do Russell’s Introduction and Ramsey’s Critical Notice. Among collections of essays on the book McGuinness (2002) and Block (1981) stand out. Stages in the composition of the Tractatus are represented graphically at http://tractatus.lib.uiowa.edu.

47 FACTS

Although I shall not follow Wittgenstein’s order of composition slavishly, I shall concentrate first on expounding the first 64 pages of Bodleianus. These represent the Tractatus just before he took the decision (probably some time in 1916, although the date is uncertain) to change the book’s structure by extending it beyond proposition 6. Indeed, the outline of the book on the first page of Bodleianus provides us, especially if we focus on the remarks with whole numbers as labels (see Pilch 2015), with a convenient conspectus of the book’s structure. 1 2 3 4 5 6

The world is everything that is the case. What is the case, the fact, is the existence of atomic facts. The logical picture of the facts is the thought. The thought is the significant proposition. Propositions are truth-functions of elementary propositions. The general form of truth-function is [p, ξ , N(ξ )].

What stands out straightaway is that the outline ends not with the injunction to silence of the final published version but with a technical claim (to be explained in Chapter 51) about the expressive power of a certain notation. Wittgenstein’s route to this conclusion starts from the world before passing successively to what is the case, the structure of facts, the thought, the proposition, and finally the truthfunction. The central claim of the 1916 Tractatus is thus that the world may be pictured logically by means of propositions obtained from elementary propositions by recursive application of the N-operator.

Facts ‘The world is everything that is the case. . . the totality of facts, not of things.’ (1–1.1) A great deal is contained in this brief beginning. Later in the book

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Wittgenstein used ‘thing’ as a synonym for ‘object’, a word to which he gave a technical meaning (involving logical simplicity) rather distant from its ordinary language use; but here, at the beginning, he was probably still using the word to include complex entities as well as simple—alluding, that is to say, to the distinction between complex and fact. I have already mentioned the central role this distinction played in the 1913 Notes, and the extent to which adjusting to it contributed to Russell’s paralysis. What Wittgenstein now claimed was that the world consists of what is the case (facts), not what there is (things). On reading the Tractatus Frege wrote to ask Wittgenstein (3 Apr. 1920) whether proposition 1 was a definition or a substantial claim. It is a good question. The book’s interest would plainly be substantially diminished if it were a definition, and yet Wittgenstein provided no argument for it. It is doubtful whether any freestanding argument is available at this stage in the exposition to show that the world consists of facts, not complexes. Russell’s gloss in the Introduction was that to describe the world completely it is not enough to say what things there are in it; we need to say also how those things are combined together, i.e. which facts there are. But this is not the argument we are looking for, since it applies only if we read ‘thing’ to mean simple thing. The Tractatus does indeed later claim that the components of facts are simple, but we shall be able to reach that conclusion only if we grant first that the world consists of facts. And if we read ‘thing’ to include complexes, then we can describe the world completely by saying what things there are in it, since we shall thereby specify which atomic facts obtain. One partial argument for a fact-based ontology does suggest itself, but it depends on Wittgenstein’s picture theory. As we shall see in the next chapter, this theory holds that it is of the essence of the relation of representation that its relata should share a common form and hence be facts, not complexes. This would at least show that the representable part of the world consists of facts. To reach the conclusion that the whole world consists of facts, though, we need what I shall call the ‘internalist premiss’, that the world is representable. Wittgenstein did not state this premiss explicitly in the Tractatus: the nearest he got, perhaps, was in the course of his argument for solipsism, where he said that ‘the limits of my language mean the limits of my world’ (5.6); but even this falls a little way short, since what we really need is that the limits of language mean the limits of the world. I shall return later to the extent to which he nonetheless relied tacitly on the premiss. In the Tractatus Wittgenstein called the manner in which the components of a fact are combined its ‘structure’; the possibility that there should be a fact with this structure he called its ‘form’. Two facts may have the same form even though their components differ; for instance, aRb has the same form—relational form— as cSd. Notice, though, that in using the word ‘form’ in this way Wittgenstein was deviating from the Notes, where he had meant by it a privileged component of the fact responsible for determining its Tractarian form. (We shall discuss the reason for this change of terminology in Chapter 52.)

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There is thus a distinction between content (the components of a fact) and form (how those components may be configured). This is the prototype of a distinction that runs through the whole Tractatus between material (external) and formal (internal) properties: the former are those that may fail to obtain; the latter are independent of how things stand. It is internal to this shade of blue that it is lighter than that other shade (4.123), external that it is the colour of my shirt. We saw earlier that Russell struggled to explain what he meant by a relation’s being external. Wittgenstein’s conception of facts gave him the criterion that Russell’s modal actualism had disallowed: a relation is external just in case it can fail to obtain.

Atomic facts A fact may be made up of simpler sub-facts: the fact that the Smiths live at number 32 is made up of the fact that Mr Smith lives at number 32 and the distinct fact that Mrs Smith does too. Wittgenstein said little about the manner in which a fact is made up of sub-facts, but he seems to have conceived of it as no more than bare, unstructured summation. A distinctive role is then played in his theory by those facts which have no sub-facts. In this connection he used the word Sachverhalt, but it is a matter of controversy among commentators whether he meant by this word the sub-facts themselves or the complexes whose existence would constitute these sub-facts. One point in favour of the former interpretation is that Ogden and Ramsey, with his permission, translated Sachverhalt as ‘atomic fact’. Another is that when Russell asked him in 1919 to explain the difference between Tatsache and Sachverhalt, he replied: 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. This clearly implies that every Sachverhalt is a Tatsache. On the other hand, it favours the other interpretation that Wittgenstein repeatedly spoke of a fact as consisting in the obtaining (Bestehen) of Sachverhalte, not in the Sachverhalte themselves. (Indeed, Ogden and Ramsey translated Bestehen in this context not as ‘obtaining’ but as ‘existence’.) I am inclined to favour the former interpretation, and to treat Wittgenstein’s use of Bestehen as merely a response to the linguistic difficulty that the word ‘fact’ is factive, and yet in laying out his system he had occasionally to talk about facts that do not obtain as well as ones that do. I should add straightaway, though, that I do not think much hinges on this. The confusion between a complex and the fact of its existence is not as significant in the atomic case as it is in general. Part of the importance of the distinction lay in the underdetermination of the fact by the complex—one complex may exemplify several different facts—but in the case of a Sachverhalt there is only one fact for the complex to exemplify. So in this case, but not in general, the fact may be said to be the existence of the complex.

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Molecular facts There is further unclarity when we turn to the question of what other sorts of facts Wittgenstein countenanced. According to the letter to Russell just quoted, all facts are sums of atomic facts. If this is right, then he allowed neither disjunctive nor negative facts. In the case of disjunctive facts, this stance is confirmed by the text of the Tractatus. Why, though, did Wittgenstein take this stance? One possible argument against positing disjunctive facts is that they are unnecessary: once we have an inventory of all the non-disjunctive facts, the disjunctive ones add nothing. This can hardly have been Wittgenstein’s own reason, though, since by the same token we have no need of conjunctive facts: once we have an inventory of all the atomic facts, the conjunctive ones add nothing. A more plausible reason is that he might have conceived of facts as determinate, whereas a disjunctive fact would be, by its nature, indeterminate. When we turn to negative facts, matters are less clear. The letter to Russell entails that there are no negative facts, and several remarks in the Tractatus confirm this. For instance, ‘the world is determined by the facts, and by these being all the facts’: there would be no need for the second clause if negative facts were permitted, since once we had listed all the positive facts, we could add the negative fact that no other positive facts obtain. If this was Wittgenstein’s view in the Tractatus, however, he had changed his mind: in the 1913 Notes he was explicit that there are both positive and negative facts (NoL, B7). That he should have changed his mind between 1913 and 1919 is not itself especially surprising, of course; what is surprising is that he repeated this very claim in the Tractatus. ‘The existence of atomic facts we also call a positive fact, their non-existence a negative fact.’ (2.06) Moreover, he went on to say that ‘the existence and non-existence of atomic facts is the reality’ (2.06) and ‘the total reality is the world’ (2.063), from which it would follow (since the world is the totality of facts) that the nonexistence of an atomic fact is a fact. There is, as far as I can see, no prospect of making these remarks consistent with the 1919 letter. What is clear at least is that Wittgenstein could not allow there to be negations of non-atomic facts without contradicting his rejection of disjunctive facts. (The negation of the fact that p and q would be the disjunctive fact that not p or not q.) But the negation of an atomic fact is not disjunctive, since there is only one way for an atomic fact not to obtain.

Further reading On the claim Wittgenstein was confused about the distinction between complex and fact in the Tractatus see Kenny (1974) and Potter (2009, ch. 27).

48 PICTURES

Everyone knows that the Tractatus contains a ‘picture theory of the proposition’. We should distinguish, though, between the picture theory proper, which attempts to explain how any method for representing the world truly or falsely must work, and Wittgenstein’s more specific account of the propositions by means of which we represent the world in language. I shall focus on the picture theory in this chapter, and turn to his theory of propositions in the next.

Meaning According to the picture theory, the vehicle by means of which it is possible to represent, truly or falsely, how things stand is a fact. What enables it to perform this task—what makes it a ‘picturing’ fact—is that its components are paired off one-to-one with entities in the world (2.131). It then represents that these worldly entities are combined just as their proxies are in the picturing fact. What it is for a picture to be true, therefore, is that there should be two facts—picturing and pictured—with different components but the same structure. Wittgenstein did not himself call this a ‘picture theory’, but he did accept ‘picture’ as the English translation of his German Bild. Some commentators prefer the translation ‘model’, but little hangs on this, since the most likely source of misunderstanding is one to which both words are equally susceptible, namely that in ordinary usage a picture (or model) is a complex, whereas for Wittgenstein it is a kind of fact (2.141). In order to understand his theory correctly, therefore, we must distinguish carefully between the picturing fact and the complex which exemplifies it. Wittgenstein was famously inspired by a description he read early in the war of the practice of magistrates in Paris law courts of letting witnesses to a

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traffic accident describe what they had seen by positioning dolls and toy cars in the same configuration as what they said took place (NB, 29 Sep. 1914). As he had already adopted the harmony principle, which embodies the essence of the picture theory, in Cambridge more than a year earlier, however, what he derived from the description cannot have been the theory itself but only a vivid illustration of it. What the example makes clear is that in order to set up the system of representation to be used in describing the traffic accident, witness and magistrate have only one semantic task to perform, not two: they have to agree which doll is to stand for which person, which toy car for which real car, etc.; but not how to interpret their spatial arrangement. By arranging the dolls and toy cars in a certain way the witness expresses without further ado that the real people and cars were arranged in just that way. In ‘Thoughts’, Frege offered an argument against regarding propositions as pictures. ‘It might be supposed’, he suggested, 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. (CP, 352) A pictorial complex, Frege here argued, does not make a determinate claim about the world, since if I merely show it to you, you will not know which of the various facts about it I am alluding to. The example he used, a picture of Cologne Cathedral, already occurred in his 1897 draft, but there the argument against the picture theory was less specific. ‘Thoughts’ cannot be a direct response to the Tractatus—Frege sent it off for publication in 1918 before first seeing the book in typescript—but it is tempting to speculate that his pre-war conversations with Wittgenstein encouraged him to think a detailed argument against treating propositions as pictures was worthwhile. Nevertheless, although Frege was here making a good objection to a picture theory, it is not an objection to Wittgenstein’s picture theory—not, at any rate, to the one he included in the Tractatus. For the view Frege was arguing against was that a proposition is a pictorial complex, whereas Wittgenstein’s claim was that a proposition is a picturing fact. His picture theory held that what allows a picture to represent the world is an identity of form between the former and the latter. No such theory could be offered of how a pictorial complex represents a real complex, because there is in general no such thing as the form of a complex: a complex may have various forms according as it is taken to exemplify different facts. This explains the point made in the last chapter that a represented world must consist of facts, not of things. According to the picture theory representation is a relation between two facts, picturing and pictured. It could not be a relation between complexes, because complexes do not have a single form.

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Truth In the Notes there are hints, enthusiastically taken up by Russell, of a sort of bi-correspondence theory of truth, but elsewhere in the notes Wittgenstein had already moved away from this idea, thinking that it wrongly detaches the proposition from the criterion for its truth or falsity. His objection to Frege’s treatment of sentences as names of truth values was that it erroneously separated meaning from truth in just the way that naming a point on a sheet of paper can be separated from saying of the point that it is black or white. ‘That which “is true” or “is false” must already contain the verb.’ (4.111) Wittgenstein’s conception of pictures as facts was intended to capture this insight by holding that a picture is true when things are as it says they are. Commentators often describe the picture theory as a theory of meaning, but Wittgenstein thought, for the reason just stated, that any theory of meaning must be simultaneously a theory of truth. The picture theory fulfils this requirement by offering a condition which a picture has to satisfy in order to be true. Is this then a correspondence theory of truth or an identity theory? In one obvious sense it is the former: the components of the picture are not identical with, but only correspond to, the components of the world about which the picture says something. In another sense, though, it is the latter: the manner in which the components of the picture are configured does not merely correspond to, but is identical with, the manner in which the components of the world are configured if the picture is true (2.161). The former sense, in which it is a correspondence theory, is what enables it to explain false propositions (the Achilles’ heel of Moore’s early identity theory): the witness in the law court is free to lie by positioning the toy cars contrary to the facts of the actual accident. Nonetheless, as a theory of truth it is an identity theory, because what makes a picture true is the identity of form between it and what it represents. The respect in which there is correspondence—namely between the components of the picture and the components of the world—is in that regard secondary. In another noteworthy passage in ‘Thoughts’, Frege proposed an argument against a correspondence theory and in favour of an identity theory of truth. Correspondence comes in degrees, he said, but truth does not. ‘What is only half true is untrue. Truth does not admit of more and less.’ (CP, 353) A correspondence can be exact, he maintained, only if the two entities related by it do not merely correspond but coincide. Wittgenstein’s theory, it may be said, conforms to the first part of Frege’s view but not the second: it aims for exactness without coincidence.

Relative inexpressibility Every picture has a ‘form of representation’—a kind of background structure on which its meaningfulness depends: the arrangement of toy cars in the law court has spatial form, for instance; a rhythm tapped out on a table has temporal form;

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it would also be possible (though somewhat laborious) to use warmth as a form of representation—to express how hot someone is by heating a doll to the same temperature. Understanding a picture—understanding what would make it true or false—is conceptually prior to knowing which of these two possibilities obtains. The picture therefore presupposes, and cannot itself express, its own meaningfulness; for if a picture did express its own meaningfulness, it could fail to be true only by being meaningless. The picture . . . cannot represent its form of representation; it shows it forth. The picture represents its object from without (its standpoint is its form of representation), therefore the picture represents its object rightly or falsely. But the picture cannot place itself outside of its form of representation. (2.172–4) Notice, though, that the inexpressibility thus claimed is relative to the form of representation of the picture in question: there is nothing in principle to prevent us from expressing the form of representation of one picture by means of a picture of a different form. The form of the arrangement of toy cars in the Paris law court, being that of two-dimensional geometry, cannot be expressed by another twodimensional picture, but it is expressible in the language of Euclidean geometry, in which the properties of the two-dimensional plane can be axiomatized. This merely relative sort of inexpressibility is no more mysterious in itself than the fact that classical Latin has no word for a mobile phone. The form of representation that enables the picture to say what it does is not the only thing that is inexpressible according to the Tractatus. It is equally impossible to say of any component of what is pictured that it has the form it has. A picture can express only material (external) properties and relations, not formal (internal) ones. Wittgenstein’s metaphysical distinction between the components of a fact (content) and the manner in which they are configured (form) is mirrored in his semantics by a distinction between what a picture says and what it shows.

Possibility The form of a fact is the possibility of assembling its components in that manner, and the picturing fact shares its form with the pictured fact. So the possibility of assembling a picturing fact guarantees the possibility of a pictured fact in which the corresponding worldly entities are identically assembled. A picture may represent what did not happen, but not what could not happen. Notice, though, that this guarantee of possibility depends crucially on the identity of form between picture and pictured. The illusion of an impossible staircase in the Escher drawing relies on the two-dimensional drawing’s lack of a commonality of form with the three-dimensional world it seeks to depict.

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When Wittgenstein distinguished between a geometrical and a physical impossibility, he was not making an absolute point about the a priori nature of geometry but only a relative one about the nature of spatial representation. ‘We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry.’ (3.0321, my emphasis) This is because a spatial picture is subject to the laws of geometry to the same extent as what it presents. If, on the other hand, our picture were physical, it would be just as incapable of presenting a physical contradiction. There is no reason why a nonspatial picture could not represent a fact which contradicted the laws of geometry (as, for instance, in an axiomatic treatment of non-Euclidean geometry). Wittgenstein took inspiration for his picture theory from Hertz. We form for ourselves pictures or symbols of external objects; and the form which we give them is such that the necessary consequents of the pictures in thought are always the pictures of the necessary consequents in nature of the things pictured. In order that this requirement may be satisfied, there must be a certain conformity between nature and our thought. (1899, 1) It was one of Wittgenstein’s most notable traits to press an idea more resolutely than its originator. His conversion of Hertz’s ‘certain conformity’ between world and thought into an identity is an instance of this. Hertz also stressed the need for the picture to match the complexity of what it pictures. 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 connections that it can be answered; but by removing the contradictions existing between those already known, and thus perhaps by 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. (1899, 8) We fall into perplexity, that is to say, if we use pictures of greater complexity than required. Wittgenstein later said that this passage ‘seemed to him to sum up philosophy’ (PPO, 379), and considered using its last sentence as the motto for the Philosophical Investigations. The method of philosophy consists, he thought, in discerning the true complexity of the facts we are attempting to represent.

Further reading Preston (2008) has critically evaluated the idea that Hertz influenced Wittgenstein’s picture theory or his conception of philosophical method. For other expositions of the picture theory see Pears (1987-8, I, ch. 6) and Griffin (1964, chs 8– 10). On Wittgenstein’s picture theory as an identity theory of truth see Sullivan (2005a). On the identity theory independent of this Tractarian context see McDowell (1994) and Hornsby (1997).

49 PROPOSITIONS

It is said (Bartley 1974, 28) that Wittgenstein originally intended to call his book Der Satz (‘The Proposition’), and an account of the proposition is indeed central to the 1916 Tractatus. ‘In the proposition’, he said, ‘the logical picture of the facts . . . is expressed perceptibly through the senses.’ (3, 3.1) A proposition is thus a picture with two extra features: that it is logical, and that it is perceptible. Let us consider these two conditions in turn.

Logical form One pictorial form may be more general than another: three-dimensional spatial form is more general than two-dimensional; colour more general than black-andwhite. None of these forms is wholly general, though: each is suited to expressing a particular kind of fact. Wittgenstein held that there is one form—logical form— which is maximally general, in that all forms have it in common. A picture that has this form—a logical picture (2.181)—he called a ‘thought’ (3). Here, though, I shall sometimes use this last word to refer generically to the class of all such pictures. To see what Wittgenstein had in mind, consider the case in which I wish to say that someone (a witness to an accident, say) was standing beside a car. I can represent this by placing a doll, proxy for the witness, beside a toy car, proxy for the real car. What does the representational work is then the spatial relationship between the doll and the toy car. An alternative method for representing the same situation, though, would be to add to the picture a third element, namely a relational sign with two arguments places to stand for the spatial relationship of ‘standing beside’. What does the representational work in this new picture is no longer the spatial relationship between the components, but the logical device

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of placing the original two objects in the argument-places of the third, newly introduced one. Logical form has none of the special features that distinguish more specific forms. In a spatial fact, for instance, the spatial relation between the constituents contributes to the nature of the fact, whereas in a fact whose form is logical the manner of combination of the constituents is mere concatenation: they ‘hang one in another like the members of a chain’ (2.03). This thin conception of the manner of combination have two virtues, Wittgenstein thought: its insubstantiality ensures that it imposes no conditions on reality, hence explaining the generality of logic: and its immediacy ensures that there is no third thing standing between two links in the chain, and hence no danger of Bradley’s regress. ‘The meaning’, he told Ogden, ‘is that there isn’t anything third that connects the links but that the links themselves make connection with one another.’ (CKO, 23) In this respect he was of course influenced by Frege, whose explanation of ‘aRb’, for instance, was that the names ‘a’ and ‘b’ occupy the two argument-places in the unsaturated relation-word ‘xRy’; logical form makes no further contribution. ‘The logical form of the proposition must already be given by the forms of its component parts.’ (NB, 1 Nov. 1914) Wittgenstein held that every picture has logical form as well as its more specific picturing form (2.182): every spatial picture, for instance, has logical form in addition to spatial form. This is puzzling, because it suggests that a single picture might have two forms simultaneously, in apparent contradiction to the idea that in a fact, in contrast to a complex, the components have a single determinate manner of combination. His thought seems to have been that logical form is a kind of minimum necessary for expressiveness. One might think, though, that an analogy with the links of a chain is a weak foundation on which to build a theory of propositions. Wittgenstein gave no argument at this stage that there is such a maximally general form. Why should it not instead be the case that every form has particular features that limit its range of applicability, and hence render it non-logical? The reason he did not answer this question explicitly is presumably that he was already powerfully persuaded of the correctness of the Fregean insight that logic is ‘of a totally different kind than any other science’ (letter to Russell, 22 June 1912, in CL, no. 2). The generality of logic is a generality of form, not of content. What singles logic out is not that it applies to all things, but that it applies to all thoughts. If there were not a single, maximally general form, there would (absurdly, in Wittgenstein’s view) not be a single logic.

Sign and symbol The second feature a picture must have in order to be a proposition is that it be ‘expressed perceptibly through the senses’ (3.1). We cannot perceive the thought directly, because logical form is not directly perceptible. So another fact—what Wittgenstein called a propositional!sign—must act as the vehicle. One might be

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tempted to think that by ‘propositional sign’ Wittgenstein just meant ‘sentence’, except that this word, like ‘picture’, in ordinary usage means a complex, whereas Wittgenstein was explicit (3.14) that a propositional sign is a fact, i.e. it is already parsed so that its grammatical structure is revealed. A propositional sign and a proposition are both facts, but their forms are different: perceptible in the former case; purely logical in the latter. It is this difference in form that explains why we cannot simply read off the structure of the world from that of language. Propositions share their form with the parts of the world they represent; propositional signs do not. We should be alert to the ambiguity of Wittgenstein’s use of the word ‘language’ in the Tractatus, sometimes (e.g. 4.002) to mean a sign-language, whether the ordinary language of 5.5563 or the semi-formal concept-script, and sometimes (e.g. 4.001, 5.62) to mean what I am here calling ‘thought’— the symbol-language that abstracts away from the signs’ contingent, and hence logically inessential, features. (In my terms, then, what shares its structure with the world is not language but thought.) Having thus distinguished between propositional sign and proposition, Wittgenstein distinguished analogously between their meaningful constituents, which he called signs and symbols (or expressions) respectively. (The qualifier ‘meaningful’ is needed here to rule out treating, for instance, ‘es is mor’ as a sign because it occurs in ‘Socrates is mortal’.) ‘What is essential in a symbol’, Wittgenstein said, ‘is what all symbols that can serve the same purpose have in common.’ (3.341) In the Moore dictation he was slightly more explicit about the ‘purpose’ in question here. ‘What symbolizes in a symbol is that which is common to all the symbols which could in accordance with the rules of logic . . . be substituted for it.’ (NdM, 117) Wittgenstein’s notion of a symbol thus plays in the Tractatus essentially the role that conceptual content played in Begriffsschrift and sense in Grundgesetze, namely that of focusing attention on the features of a word’s content that contribute to its logical role. ‘Seeing the symbol in the sign’ consists in seeing past the contingent properties of a sign to these features. It Can Never indicate the common characteristic of two objects that we symbolize them with the same signs but by different methods of symbolizing. For the sign is arbitrary. We could therefore equally well choose two different signs and where then would be what was common in the symbolization? (3.322) In his Critical Notice Ramsey glossed this using Peirce’s terminology of type and token (in Ogden & Richards 1923, 433–4). Sign and symbol are not token and type—a sign is already a type, as Wittgenstein’s gnomic observation that ‘ “A” is the same sign as “A” ’ (3.203) was intended to make clear—but rather two ways of typing the tokens, whether according to syntactic properties (signs) or logical role (symbols). This way of putting the matter might at first seem to have

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the advantage of discouraging us from hypostasizing symbols as occupants of a Fregean ‘third realm’ intermediate between language and reality. Nonetheless, Ramsey later noted that a symbol need not actually be instantiated. ‘It cannot be any concern of ours whether anyone has actually symbolized’ it (FoM, 33). Logic should not depend on the contingency of which symbols have actually been tokened and must therefore grant, he thought, that there may be types with no token instances. Wittgenstein himself did not mention this point in the Tractatus, but it must be accepted if we are to save the book from an extreme solipsism of acquaintance. (In this connection it is worth mentioning, perhaps, that Ramsey made the point about untokened types after he had discussed the Tractatus with its author.) So although we should not regard symbols as intermediaries between language and world (as Russell, for instance, seems to have regarded Fregean senses), it would also be wrong to conceive of them as dependent on any particular human mind. (In Chapter 56 we shall draw a distinction, between empirical and non-empirical conceptions of the self, that may help to make this point clearer.) Because of the ambiguities of ordinary language, classification of sign tokens into interpreted signs cuts finer than that into signs: two occurrences of the sign ‘John’, for instance, count as different interpreted signs if they are used to refer to different Johns. On the other hand, classification into symbols is typically coarser than classification into signs: different signs will count as the same symbol if they have the same logical role. (Just what this logical role is will be for discussion in subsequent chapters.) Wittgenstein used the sign ‘=’ (much as Frege had used ‘≡’ in Begriffsschrift), to stand between signs that express the same symbol. In particular, as he explained in the Prototractatus (4.2213), he used it between propositional signs that express the same proposition. Oddly, though, he dropped this explanation from the final version of the book, despite continuing there to use it in this way (e.g. 4.0621, 5.51, 5.52). This is what we should nowadays call a ‘metalinguistic’ use of ‘=’, since it says something about signs rather than about their meanings. In logic we frequently use ‘operators’ to build up complex signs from simpler ones, but since our real interest is in symbols, not signs, we should restrict our attention to those operators that take logically equivalent signs to logically equivalent signs (and hence give rise to functions from symbols to symbols). Wittgenstein called an operator an ‘operation’ if it operates on propositions in this manner, i.e. if from p = q follows (p) = (q). The paradigm example of an operation is negation. Wittgenstein noted explicitly that ∼∼p = p, which illustrates two features of operations: they are repeatable; and applying them does not increase the complexity of the symbol. No proposition is intrinsically negative.

Absolute unsayability Wittgenstein’s account of logical form may seem disappointingly negative: logical form is merely whatever remains when all the particular features of a form of

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representation are removed. Perhaps, though, this is not Wittgenstein’s fault but the inevitable consequence of the form’s generality. That a picture cannot depict its own form of representation is a case of merely relative unsayability: the spatial form on which my picture of Cologne Cathedral depends might be expressible in a more general form of representation such as that of formal geometry. If the picture is a proposition, however, its picturing form is logical, and so there is no more general mode available in which to describe the features that make it expressive. ‘To be able to represent the logical form, we should have to be able to put ourselves with the propositions outside logic, that is outside the world.’ (4.12) Logical form is therefore unsayable not just relatively (because we lack the right words) but absolutely. Just as the features of spatial language can be expressed only in a non-spatial language, so expressing the features of logical language would require a non-logical language; but there is no such thing. Notice, though, that Wittgenstein here ran together two issues. Even if he was right that I cannot say informatively what the logical properties of the world are (since for that I would need a non-logical language), it does not follow that I cannot do so uninformatively. He did not, that is to say, explain the difference between logical truths, which can be expressed but are uninformative, and structural features of language, which cannot be expressed at all but only shown. Relative unsayability is, we have already noted, unsurprising: that a fact should be unsayable in some particular language is no reason to think that there is anything particularly problematic about it. Nor does absolute unsayability on its own pose any deep problem. It becomes more troubling, however, when combined with Wittgenstein’s internalist premiss that the world is representable, since this entails that whatever it is that we are failing to express is not worldly. The puzzle, then, is to understand what it could be instead. Nowadays, logicians tend to suppose that what cannot be said in the object language can be unproblematically said in the metalanguage, i.e. that the difficulty is merely one of relative unsayability, and this is the proposal that Russell made in his Introduction to the Tractatus. If the language in question is logical, however, Wittgenstein’s view was that retreating to a metalanguage will be of no avail, since what we are attempting to say is not even capable of truth or falsity.

Wittgenstein’s context principle Wittgenstein did not draw the sign/symbol distinction in the Notes. At that time he still hoped to devise a ‘logically perfect’ notation which made the logical properties of a sign transparent. By the time of the Moore dictation, however, his search for a logically perfect notation had stalled and he began instead to stress the importance of seeing past the ‘particular scratches’ to the symbolism’s ‘logical properties’ (NdM, 112). Nonetheless, with no label for the distinction he was compelled to speak opaquely of ‘seeing the sign in the sign’ (NB, 23 Oct. 1914). Not until he had reached page 54 of Bodleianus did he introduce a separate word

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for the part of the sign that is relevant to logic. Even then, his immediate purpose in doing so was only to make a point about the ambiguity of ordinary language, namely that one sign might represent different symbols in different occurrences. Not until later, it seems, did he realize that even if he had, per impossibile, found a logically perfect notation, there would still have been a role for the distinction, as a reminder that whereas the identity of structure between symbol and world is necessary, that between sign and world would still be contingent. There are two main differences between a propositional sign and a proposition. The first is a difference in form: the form of a propositional sign might be spatial if written, temporal if spoken; the form of a proposition is logical. The second difference is that a propositional sign is uninterpreted: it consists in the fact that its component signs stand in some (spatial, temporal, etc.) relationship, but these signs are not yet matched up with the world. One might therefore try to distinguish correspondingly two stages in the transition from sign to symbol: first, we pair off the components of the sign with parts of the world; then we use the ‘method of projection’ to transform this into a proposition. Such a distinction between sign and interpreted sign might seem natural, but Wittgenstein did not draw it, and in the Tractatus his use of the word ‘sign’ oscillates uncomfortably between the two. The explanation for the conflation lies in his changing attitude to semantic incompleteness. We have already noted that Russell was persuaded by his reading of the Notes that names are semantically complete, forms not. In his dictation to Moore Wittgenstein repeatedly tried to make use of descriptions of the syntax of interpreted signs as a way of circumventing the inexpressibility of the structure of the world. What made this possible, however, was that at this stage he still envisaged that each simple sign could be interpreted (i.e. allocated a meaning) on its own. ‘a has a meaning independently of φa,’ he said. (NdM, 117) Contrast this with the Tractatus, where in a deliberate echo of Frege’s context principle he held that ‘an expression has meaning only in a proposition’ (3.314), i.e. that all sub-sentential expressions are semantically incomplete. Why this shift? If there is a harmony of structure between thought and world, as the picture theory requires, we should expect metaphysical claims to have semantic correlates. So here. Wittgenstein’s context principle is the internalist premiss transposed into a linguistic key. Confirmation of this reading may be found in a remark he made in the 1930s. The world does not consist of a catalogue of things and facts about them (like the catalogue of a show). . . . What the world is is given by description and not by a list of objects. So words have no sense except in propositions, and the proposition is the unit of language. (L30, 119) Just as in Grundlagen the context principle gave Frege’s realism an internalist flavour, so in the Tractatus it gives linguistic expression to Wittgenstein’s internalism.

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Wittgenstein’s adoption of the context principle explains, then, why the notion of an interpreted sign makes no appearance in the Tractatus. The context principle collapses the two semantic stages identified above into one by entailing that ‘assigning a meaning’ to a sign is the very same process as specifying the contribution the sign makes to determining the senses of propositions in which it occurs.

Further reading On the connection between Wittgenstein’s doctrine of inexpressibility and Frege’s paradox of the concept horse see Geach (1976). On the sign/symbol distinction and psychologism see Diamond (2006).

50 SENSE

Since Wittgenstein shared Frege’s view that logic is concerned with truth, he also shared Frege’s understanding of the sense of a propositional sign as consisting in its truth-conditions. Frege’s account had suffered, however, from a lack of precision about when two sets of truth-conditions should count as equivalent. Wittgenstein aimed to rectify this defect.

Truth-possibilities Wittgenstein called a proposition ‘elementary’ if it says that an atomic fact obtains. The world is described completely by saying which atomic facts obtain and which do not; this may be done, he claimed, by specifying which elementary propositions are true and which false (4.26). Although he did not say so explicitly, it follows that for every possible atomic fact there must be a corresponding elementary proposition, true if the fact obtains, false otherwise. This may seem implausible, especially if the number of these is uncountable, but it underlines Ramsey’s point about the need for logic to allow symbols that have not been and perhaps never will be instantiated. Wittgenstein called an ascription of truth or falsity to each elementary proposition a truth-possibility (4.3). If there are n elementary propositions and Kn truthpossibilities, then Kn = 2n , since there are two possible ascriptions—‘true’ and ‘false’—for each elementary proposition. Oddly, though, Wittgenstein did not quite say this, but offered instead (4.27) the more complicated formula

Kn =

n   X n ν=0

ν

.

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Why? It would be strange indeed if he knew enough about binomial coefficients to work out this formula without also being aware that, by a trivial application of the binomial theorem, n   X n = 2n ; ν ν=0 but it is hard to think of any other reason for him to have given the more complicated formula, unless he just thought it looked more impressive. To describe the world completely we would have to specify a single truthpossibility, but in practice a proposition falls short of this standard, and determines not one truth-possibility but a whole class: its sense consists in the claim that the world is described by one of the truth-possibilities in this class. So the senses are wholly determined by the truth-possibilities, which in turn are wholly determined by the elementary propositions. If there are n elementary propositions (and hence Kn = 2n truth-possibilities), then the number Ln of senses is n

Ln = 2Kn = 22 . Again, however, the formula Wittgenstein actually gave for Ln (4.42) makes an unnecessary detour via binomial coefficients. Wittgenstein once told an acquaintance (see Lokhorst 1992, 24–5) that in thinking through the ideas in the book he had found it helpful to suppose that the number of elementary propositions is finite, so that all the truth-possibilities can be expressed in a finite truth-table. Nonetheless, this is a simplification: in the Tractatus he left it open whether or not the number of elementary propositions is actually finite. If instead the number were an infinite cardinal κ, the number κ of senses would become Lκ = 22 , which is uncountable, and so we cannot expect to be able to construct a single recursive language in which all of them are expressible. Notice, though, that this is not a matter of absolute inexpressibility: no one recursive language can express every sense, but this does not entail that there is any particular sense which no language can express.

Tautology and contradiction Two senses are worth singling out as special cases: the class containing no truthpossibilities; and the class containing all. The former (which Wittgenstein called ‘contradiction’) places a condition on the world so severe that it cannot be satisfied; the latter (‘tautology’) places no condition at all. The picture theory, which was already under strain when it was required to express disjunctive and negative facts, breaks down completely in such a case. If the witness in the Paris law court wanted to tell the magistrate that his car either was or wasn’t in front of the other car, nothing he did with the toy cars would achieve his aim: the best he could offer would be an eloquent shrug. Properly speaking, therefore, no picture (and a fortiori no proposition) can express either of these two special cases. So if there

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are n elementary propositions, the number of distinct non-trivial senses capable n of expression by means of genuine propositions is 22 − 2. Wittgenstein did not always adhere strictly to this terminology, however; occasionally he extended to tautology and contradiction the courtesy title of propositions, even though they cannot really be pictured. Although tautology and contradiction are not strictly propositions with sense, the propositional signs which express them are formed by assembling meaningful component signs, in a similar fashion to the construction of signs that do express genuine senses; the difference is that now the components cancel each other out. Wittgenstein called such signs ‘senseless’ (sinnlos) and contrasted them with the ‘nonsensical’ (unsinnig) ones that result when we use a sign in a context in which it has no meaning. To see the difference, consider the following: It’s raining; It’s raining or sunny; It’s raining or sunny or overcast; It’s raining or sunny or overcast or hailing; . . . It’s raining or it isn’t. This is a sequence of propositions saying progressively less about the weather, until the last—a tautology—says nothing at all. A tautology is thus a limiting case which can be approached by way of senseful propositions. By contrast, ‘˙‘‘a” refers to a’ does not express a sense, even an empty one, as can be seen by noting that it cannot be approximated by senseful sentences in the same manner.

Independence A ‘truth-possibility’, we have said, is an assignment of truth and falsity to elementary propositions. Which of these assignments are genuinely ways the world might be? Wittgenstein held that all are. Or, to state the matter in terms of the world rather than of language, Atomic facts are independent of one another. From the existence or nonexistence of an atomic fact we cannot infer the existence or non-existence of another. (2.061–2) (Particularly eye-catching is the possibility that every elementary proposition is false, and hence no atomic fact obtains at all.) Despite stating various versions of this independence claim in the Tractatus, Wittgenstein never gave an argument for it. The nearest he came was to suggest that in analysing we can use independence as a criterion of whether we have reached the elementary, because it is a ‘sign of an elementary proposition, that no elementary proposition can contradict it’ (4.211). So if p entails q but not conversely, this merely shows that p is not elementary but decomposable as the conjunction of q and a further component r. However, this begs the question whether there are any propositions that are elementary in this sense. What if r in turn entails, but is not entailed by, a further proposition s? ‘It is obvious’, Wittgenstein claimed, ‘that in the analysis of propositions we must come to elementary propositions, which consist of names in immediate combination.’ (4.221) In truth, though, this is not obvious at all: it is not obvious that the

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propositions q, r and s which we obtain from p by analysis need be successively simpler, nor, therefore, that this process of analysis will terminate. It is easy to see, though, why Wittgenstein wanted elementary propositions to be independent. He distinguished two categories of propositions: contingent propositions, which express ways the world might be; and senseless propositions, which are tautologies or contradictions. If elementary propositions were not independent, there would be a third category consisting of propositions that are senseful but whose senses contain truth-possibilities that are not ways the world could really be. These would in effect be synthetic a priori propositions (and their negations), and we would be owed an account of what it is—not the world, but not logic either—that makes them true or false. Wittgenstein’s view, by contrast, was that what makes each atomic fact possible—its form—is wholly internal to it. The independence of atomic facts was therefore an aspect of the generality of logical form. For atomic facts to be incompatible would be for the logical form of reality to constitute a constraint on what is possible. Part of the difficulty with assessing this independence claim, though, lies in saying quite what is at stake in making it. Wittgenstein did not hypostasize possible worlds; the only world is the real world—everything that is the case, not everything that might be. (He was not a modal realist, despite what some commentators have claimed.) So it is difficult to articulate what it means to claim that a truth-possibility is ‘genuine’. It is striking how different his approach to this question was from Russell’s. For Russell, the analysis of propositions so as to exhibit their underlying forms is subject at every stage to the limits of our current knowledge. There might be other aspects of the world, not encoded in our current analysis (and perhaps not directly knowable by us), that constrain the possibilities of combinations of objects. Once he had extricated himself from the belief that all relations are external, Russell treated whether there are such constraints as a matter for investigation. ‘Perhaps’, he granted, ‘one atomic fact may sometimes be capable of being inferred from another, though this seems very doubtful.’ (OKEW , 62) Wittgenstein, by contrast, simply insisted that there must be a complete analysis: for possibilities of occurrence of atomic facts in any analysis to be dependent on one another would be for there to be another aspect of the world not represented by those possibilities, and hence for the analysis to be incomplete. So persuaded was he of this, indeed, that he dismissed apparent counterexamples as showing only that the facts in question were not atomic (6.3751). For instance, if a point in my visual field is red, it cannot also be green. Therefore, he thought, colour ascriptions are not atomic but can be further analysed. Yet he did not say what the analysis is. Nor, it seems, did he give any serious consideration to the idea that colour incompatibilities might be due to laws of nature rather than to the essence of the objects involved, in which case there would indeed be no logical incompatibility between red and green (see FoM, 280).

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The general form of proposition Wittgenstein structured the 1916 Tractatus so as to make the determination of the general form of the proposition its central concern, but the preliminary account of this that he offered at 4.5—‘Such and such is the case’—is strikingly modest. Even the phrase ‘is the case’ is included only to meet the demands of ordinary grammar: as he later noted (PI, §134), he might just as well have said, as indeed he did say a few paragraphs later, that ‘the general proposition form is a variable’ (4.53). There are two things that might be meant by a ‘theory of meaning’: an explanation of what, in general, grasp of the meaning of a sentence amounts to; or a set of rules determining what meaning each propositional sign of our language has. Nowadays, it is usually the latter that is meant, but the picture theory, as described thus far, is at most the former. Perhaps Wittgenstein thought that in relation to the wholly general notion of a language nothing more informative can be said. At any rate, although he offered a theory of meaning of the second kind later in the book, it was one that applied only to his concept-script, not to ordinary language. It is far from clear what a theory of meaning for the latter in this more ambitious sense would be like. Nonetheless, Wittgenstein’s statement of the general form of proposition, although modest, is not vacuous: he was at least making the substantive claim that there is a single variable under which all propositions fall. The task of the theory of meaning is to explain what it is for a proposition to be true, and, he thought, ‘you must do it once for all’ (NdM, 113). Presumably his thought was that if there were a multiplicity of forms of proposition, our grasp of the world would fragment correspondingly into a multiplicity of incommensurable conceptual schemes. (Of course, he was not the first to express the need for such a unitary conception. Kant, for instance, called it the ‘transcendental unity of apperception’.) Wittgenstein’s difficulty was that he was not yet in a position to say informatively what this single variable is. For the time being, at least, the general form of proposition remained merely a desideratum placed on his account by philosophical considerations; its satisfaction still required technical work to support. In fact, finding a single variable to encapsulate this general form proved to be the most troublesome task he faced in the 1916 Tractatus. There must be such a variable, but there seemed to him to be reasons internal to logic why there cannot. In the next chapter we shall examine the extent to which he succeeded in resolving this tension.

Further reading Pears (1981) discusses Wittgenstein’s reasons for wanting elementary propositions to be independent of one another. On a world where no atomic facts obtain see Page (1997). Sullivan (2004a) discusses Wittgenstein’s desire for a single form of proposition.

51 WITTGENSTEIN’S CONCEPT-SCRIPT

The sense of a proposition is a specification of which truth-possibilities make it true, and which false. In the finite case, in which there are n elementary propositions p1 , . . . , pn , every such sense is expressible as f (p1 , . . . , pn ) for some n-ary n truth-function f . (Since there are 22 such truth-functions, this agrees with our 2 earlier calculation.) There are, for instance, 16(=22 ) binary truth-functions, each specifiable by means of a truth-table in the now familiar fashion: p T F T F

q T T F F

p⊃q T T F T

If we agreed a canonical ordering of the 2n lines of the truth-table, we could use the Ts and Fs in the right-hand column as a notation for the truth-function: p⊃q, for instance, could be written (TTFT)(p, q). If there are infinitely many elementary propositions, every sense is still a truthfunction of them, but perhaps now an infinite one (i.e. one taking an infinite ordered list of truth values as input and returning a single truth value as output). The number of such functions is uncountable, and so we cannot hope to devise a finite notation in which every one of them is expressible. For instance, the T-F notation just considered becomes an infinite sequence of Ts and Fs. The best we can hope for is to devise finite notations for some salient countable subclass of the possible senses.

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The N-operation The components of a proposition represent components of the world, and the proposition as a whole represents a way they may be configured. But how does it do this? The simplest case is that of an elementary proposition, and here the picture theory seems to be straightforwardly applicable to the propositional sign, in which simple signs stand next to each other, and what is required for it to be true is just that the corresponding objects should stand in immediate combination, i.e. that the atomic fact corresponding to the elementary proposition should obtain. Consider next the case of a ‘molecular’ (i.e. logically complex) proposition. Now the propositional sign pictures how things stand only indirectly, by identifying the corresponding sense. One notation for doing this is that of truth-tables, but even in the finite case this is horribly unwieldy, and in the infinite case (which, let us recall, Wittgenstein did not rule out a priori) it is not just unwieldy but impossible (at least for us), since we cannot write down infinite truth-tables. In Norway in the autumn of 1913 Wittgenstein devised a second method, using diagrams rather than tables to represent the truth-possibilities; and for a time he seems to have hoped that this would supply a kind of graphical decision procedure for logic, representing in a perspicuous form the truth-functional structure of any proposition. Soon, though, he gave up on this ambition, omitting the diagram method from the 1916 Tractatus (although he added a brief mention of it to the final version of the book during the last round of revisions). Instead, he settled on a method that, unlike the two just mentioned, is linear. The method depends on devising a notation for picking out a class of propositions whose senses are supposed already to be fixed. Wittgenstein called such a notation a ‘propositional variable’, and used ‘ξ ’ as his generic instance of it. We then use ‘N(ξ )’ to express the joint denial of the propositions in the class thus picked out, i.e. the proposition which is true just in case all the propositions in the class ξ are false. This N-operator is an operation in the sense mentioned earlier, except that because it takes as its argument not a single proposition but a class of them, it is in effect multigrade. Its expressive power therefore depends on what propositional variables are available in our language.

Propositional variables Wittgenstein’s account of propositional variables has two features worth noting, one restrictive and the other permissive. The restrictive feature is that the specification of the range of propositions picked out by a propositional variable should be conducted at the level of symbols only, and not say anything about the objects (3.331). He did not explain the reason for this restriction, but presumably he wanted to avoid the kind of self-reference that leads to paradox. The permissive feature of the account is that the description of the propositions is, Wittgenstein said, ‘unessential’ (5.501). This suggests that any method of describing a class

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of propositions which obeys the restriction just mentioned—that the description should take place at the level of symbols—will give rise to a legitimate propositional variable. The vagueness implicit in this criterion shows how far Wittgenstein was from conceiving of his task as that of specifying a formal system in the modern sense. Given his permissive attitude, there is little prospect of obtaining a precise characterization of the signs which are to count as legitimate variables. In a sense, indeed, this permissive attitude towards methods for forming propositional variables was dictated by his understanding of what language is—an understanding which he shared with Frege. The nub of Frege’s objection to formalism (see Gg, II, §§86–137) was that a formal language is not really a language at all but a meaningless game. As soon as we learn how to read the signs of such a language as meaningful, which of them make sense is determined entirely by what sense there is to be made, not by the formal rules. In the Tractatus Wittgenstein listed three specific ways of defining a propositional variable: ‘1. Direct enumeration. . . . 2. Giving a function ‘fx’ whose values for all values of ‘x’ are the propositions to be described. 3. Giving a formal law, according to which those propositions are constructed.’ (5.501) He did not say these three were the only ways, though, and therefore implicitly left open the possibility that there may be other legitimate notations that extend the expressive power of the system still further.

Direct enumeration The first of Wittgenstein’s three methods of picking out classes of propositions, direct enumeration, can be dealt with swiftly. Here the sign is simply a list of the propositional signs expressing the propositions to be picked out: if there is only one proposition p, the variable is just the sign for this proposition, and N(p) is its negation; if there are two, p and q, then (p, q) expresses the class containing just them, and N(p, q) is true just in case both are false; and so on. The origin of the N-operation lies in work by Sheffer (1913) showing that a single binary connective p|q (nowadays called the ‘Sheffer stroke’ in his honour) is adequate to express all the finite truth-functions of propositional logic. Wittgenstein saw this work in the summer of 1913, when Sheffer sent a copy of his paper to Russell. The N-operation generalizes the Sheffer stroke, since when applied to a sign enumerating two propositions p and q, its effect is just that of the stroke. Since the stroke is adequate to express all finite truth-functions, the N-operation together with direct enumeration is adequate to express them too. Although Wittgenstein explicitly treated direct enumeration as a propositional variable, this is hard to square with his claim that ‘every variable represents a constant form that all its values possess’ (4.1271). At any rate he did not explain what common form two unrelated propositions p and q might have that would make them both values of the variable (p, q).

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Propositional functions If there are only finitely many elementary propositions, direct enumeration is the only kind of propositional variable we need. If there are infinitely many, on the other hand, some truth-functions are not expressible by this means, and we need another method. What Wittgenstein proposed was to pick out a class of propositions by giving a function (by which he meant a propositional function) whose values are the propositions in question. When we combine the N-operation with a variable obtained from a propositional function, we obtain a symbol for a quantified proposition. To explain this, we need a notation, which Wittgenstein did not give, for the variable that picks out fa, fb, fc, etc. Here I shall use x˜ fx for this. (Russell’s notation f xˆ has the disadvantage that it does not mark scope.) With this notation N(˜xfx) is the joint denial of the propositions fa, fb, fc, etc.; its negation N(N(˜xfx)) is the proposition nowadays written ∃xfx. In order for this method to cope with multiple generality we need to be able to form a further propositional variable from the result of this process. Thus from aRb we form the variable y˜ aRy, and hence the proposition N(˜yaRy). We then form the variable x˜ N(˜yxRy), and hence the proposition N(˜xN(˜yxRy)) (in modern notation, ∀x∃y xRy). The account just sketched is due to Geach (1981). What Geach did not address, however, was how this account affects our understanding of the structure of propositional symbols. We must not think of N as obliterating the structure of the variable to which it is applied, because we have to be able to recognize the occurrence of ‘a’ in N(˜yaRy) so as to be able to form from it the variable x˜ N(˜yxRy). The fact that Wittgenstein did not mention this point makes one wonder how carefully he ever considered it. On the other hand, we cannot suppose that applications of the N-operation (or indeed of operations in general) obliterate altogether the structure of the symbols to which they are applied; for that would remove any hope of relating the structure of complex propositions to that of elementary ones via the N-operation. And if not via the N-operation, then how? The conclusion to draw must be that what contributes to the structure of a proposition of the form N(ξ ) is only the structure of the propositional variable ξ . The N-operation itself has to be thought of as transparent, not obliterating structure, but not adding to it either. Wittgenstein’s account of the semantics of quantification operates narrowly at the level of symbols, not of signs. If, for instance, ‘a’, ‘b’ and ‘c’ were the only names in the language, then ∀xfx would be the same symbol as fa ∧ fb ∧ fc. This requires every object in the domain of quantification to have a name, irrespective of whether this name actually has any token instances. In other words, Wittgenstein’s is, in the terminology I introduced earlier, a narrow account: he did not use the device of adding an auxiliary name to the language; the propositions which a propositional variable picks out are propositions in a single language. This highlights Ramsey’s point, noted earlier, that logic should ignore whether a symbol is actually instantiated.

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This account is plainly influenced by Begriffsschrift §9, where Frege arrived at the notion of a variable by identifying a component of a proposition that is replaceable. The difference is only that Wittgenstein’s account emphasizes the role of the whole expression rather than of the replaceable part, thus making ‘constants prominent’ (5.522) and hence diminishing the temptation to think that we owe an account of how the signs ‘x’ and ‘y’ function independent of their occurrence in propositions. In this Wittgenstein was guided by what might be called a ‘variable context principle’, namely that ‘an expression presupposes the forms of all the propositions in which it can occur’ (3.311).

Formal series The third kind of propositional variable mentioned by Wittgenstein is a formal series—an infinite sequence of propositions specified recursively. Nowadays the method of defining a class by recursion is a commonplace, but when Wittgenstein was writing, it was only beginning to be widely used. The method has two components: first, we specify a base class; then we provide a function which may be applied to its members to obtain further members. Wittgenstein used [A, ξ , f (ξ )] as a compact notation to express what we would now call the closure of A under f , i.e. A is the base class and f is the function which may be applied to an arbitrary class ξ to obtain further members. Anyone familiar with Principia, where this is a defined notion, might be surprised to find Wittgenstein treating it separately. However, he claimed that Russell’s way of defining the ancestral, and hence of generating a variable of Wittgenstein’s third kind, was illegitimate. If we want to express in logical symbolism the general proposition ‘b is a successor of a’ we need for this an expression for the general term of the formal series: aRb, ( x) aRx xRb, ( x,y) aRx xRy yRb, etc. E

E

The general term of a formal series can only be expressed by a variable, for the concept symbolized by ‘term of this formal series’ is a formal concept. (This Frege and Russell overlooked; the way in which they express general propositions like the above is, therefore, false; it contains a vicious circle.) (4.1273) Wittgenstein’s objection seems to have been to the impredicativity of the definition of the ancestral. The ‘vicious circle’ consists in the fact that the ancestral is defined by quantifying over a class of relations of which it is itself a member. When this point arose earlier (in Chapter 7), I quoted Russell’s argument that the impredicativity is unproblematic, because quantifying over a domain does not require

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an individual grasp of each of its members. On Wittgenstein’s narrowly substitutional understanding of quantification, however, this resolution of the difficulty is not available.

Wittgenstein’s vicious circle principle In the Notes Wittgenstein proposed to solve the paradoxes by means of a variant of Russell’s vicious circle principle according to which no symbol can contain itself. While he was working with Russell at Cambridge, he still hoped to devise a logically transparent language in which the ‘symbol of the proposition’ would somehow be visible to the naked eye. He recognized that questions about the nature of judgment ‘cannot be solved without a correct apprehension of the form of the proposition’ (NoL, B55), but in fact his solution to Russell’s paradox was at least as much dependent on such an apprehension. Even when this holy grail eluded him, the need remained urgent of providing a criterion to settle when one symbol is contained in another. If ∼∼p is the same proposition as, and hence does not ‘contain’, p, then similar reasoning will presumably show that N(p1 , . . . , pn ) does not ‘contain’ p1 , . . . , pn . The notion of containment here in play therefore cannot be at all the one that operates at the level of the propositional signs of Wittgenstein’s concept-script. This problem of settling when one symbol ‘occurs’ in another remained one of his central concerns while he was in Norway. There are internal relations between one proposition and another; but a proposition cannot have to another the internal relation which a name has to the proposition of which it is a constituent, and which ought to be meant by saying that it ‘occurs’ in it. In this sense one proposition can’t ‘occur’ in another. (NdM, 116) As so often, though, he here made a negative observation when a positive one was plainly required. We may grant readily enough, perhaps, that two propositions cannot be related to each other as a name is to a proposition in which it occurs; but what we need to know if we are to apply his vicious circle principle is what relation propositions can have to each other if not that one. Wittgenstein’s vicious circle principle shows at a minimum that ‘types stand in hierarchies’ (NB, 26 Apr. 1916), i.e. that at least the simple theory of types applies to them (3.333). What, though, of the ramified theory? Whether the vicious circle principle requires the hierarchy to be ramified depends on whether the symbols for instantiations of a variable can be said to ‘occur in’ a propositional function that makes use of that variable. How, then, did Wittgenstein resolve this question? At first, he simply postponed it. When he began Bodleianus, he left the formula for the general form of truth-function blank, to be filled in later (Pilch 2015). Remarkably, then, he seems to have started compiling the book without really knowing how it ended. This gives further support to the suggestion I made in the last chapter that he knew there had to be a single general

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form, even if he did not yet see how to secure it. It seems to have been quite late in the composition of the book that he finally decided he did not need a ramified theory of types. Only then could he add to it the final claim that the general form of the truth-function is [p, ξ , N(ξ )], i.e. that a truth-function is whatever can be obtained recursively from elementary propositions by a finite number of iterations of the N-operation. Quite when Wittgenstein did this is uncertain, and even then he seems to have remained far from clear about whether he had really eliminated, or only fudged, the ramification. In January 1917 he still thought there was a ‘sense in which there is a hierarchy of propositions’. At that stage, then, he seems to have envisaged treating the sign ‘N’ in the general form of proposition as typically ambiguous. He even suggested, somewhat optimistically, that it does not matter at all whether the given fundamental operations, through which all propositions are supposed to arise, change the logical level of the propositions. (NB, 21 Nov. 1916) Even as late as page 86 of Bodleianus, well after the 1916 Tractatus was finished, he wrote of finding ‘the right level’ at which to approach the world, only deleting this later. What Wittgenstein was groping for, but had by the end of 1916 evidently not yet achieved, was the certainty that all the paradoxes are either solved by the simple theory of types or inexpressible in the concept-script. Even so, he can hardly be said to have explained adequately in the book why this is the case, and it was left to Ramsey, some years later, to make good on the claim (see Chapter 67). Proposition 6 was of course Wittgenstein’s second attempt in the book at describing the general form of proposition. The first attempt amounted to no more than the demand that there should be a single thing that grasping a proposition consists in. How well did proposition 6 satisfy this demand? There are two parts to this. Is there a single method for generating propositions from elementary propositions? And is there a single way of grasping elementary propositions? The first of these questions we have seen Wittgenstein answering hesitantly in the affirmative by means of the recursive application of the N-operation. What of the second? How are we to understand a variable p that ranges over all the elementary propositions that stand at the base of the recursion? This question will be the subject of the next chapter.

Further reading Sullivan (2004b) discusses the difficulty involved in the notion of a variable ranging over all elementary propositions. Sullivan (2000) discusses the sense in which p is contained in ∼p and the relation of this question to the solution of the paradoxes.

52 OBJECTS

Shortly before he first met Wittgenstein in 1911 Russell proposed ‘un atomisme logique’ (CPBR, VI, 412), according to which logical analysis, in which the theory of descriptions played a central role, might reveal the simple objects out of which the world is composed. Wittgenstein inherited at least the outline of this project—the theory of descriptions was, he told Russell in 1913, ‘quite definitely correct’—although by then, as we saw in Chapter 41, his method of analysis already differed from Russell’s. Perhaps while they were working together in Cambridge, Wittgenstein even supposed in Russellian fashion that the simple objects were those with which we could in principle be acquainted; but by the time he wrote the Tractatus he had come to his own conception of their nature—a conception in which modality, not epistemology, played the dominant role.

Simplicity and elementary propositions ‘Russell’s merit’, Wittgenstein said, ‘is to have shown that the apparent logical form of the proposition need not be its real form.’ (4.0031) As we saw in Chapter 41, they shared a conception of analysis as exhibiting what appear to be names as really incomplete symbols, but differed on the details: a Russellian description picks something out by means of its external properties; a Wittgensteinian relative name picks it out by means of its internal properties. Wittgenstein analysed a proposition about a complex into the conjunction of the statement that the complex exists and a statement about its constituents. One consequence is that there are two ways in which the proposition may be false, depending on whether it is the former or the latter that fails. Thus ‘The broom is in the corner’ may be false either because the broom’s constituent parts have not been

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correctly assembled; or because they are not in the corner (see PI, §60). (Here ‘the broom’ is a relative name of the complex consisting of the brush attached to the broom-handle.) For the proposition to have a sense, however, it must be capable of having only one sort of falsity, and hence cannot contain any relative names. ‘The postulate of the possibility of the simple signs is the postulate of the determinateness of the sense.’ (3.23) (This should not, of course, be taken to mean that he held there to be such a thing as an ‘indeterminate sense’: as we saw in the last chapter, he did not.) An elementary proposition cannot contain relative names (since if it did it would be analysable as above and hence not elementary). In the Tractatus Wittgenstein used the word ‘name’ only for absolute names. Here I shall respect the sign/symbol distinction by using ‘name’ for the symbol, ‘simple sign’ for the corresponding sign. (Wittgenstein himself was not always so careful.) The proposition ‘A exists’ makes sense only if ‘A’ is a relative name. The mistake of treating existence as a genuine property arises, he thought, because ordinary language deals at the relative level, whereas at the absolute level existence is a formal concept and hence cannot be expressed but only shown. Wittgenstein’s analysis permits us to say that the entities to which we give absolute names are simple in the sense of having no (sayable) internal properties. He called these simple entities ‘objects’, and for the relation between a name and its reference he used the same verb ‘bedeuten’ as Frege (translated by Ogden and Ramsey as ‘mean’). Objects, then, are fixed elements—hinges, so to speak— around which the possibilities turn. What varies between worlds is not the objects (which are by definition constant between different possible worlds), but which of them are combined into atomic facts. As he later explained, ‘What I once called “objects”, simples, were simply what I could refer to without running the risk of their possible non-existence.’ (PR, 72) Object is therefore a formal concept: I cannot grasp a name without grasping it as the name of an object. It follows that for Wittgenstein there was a significant difference between restricted and unrestricted quantification. We can formalize ‘All philosophers are wise’ by ∀x(Px ⊃ Wx), but we cannot analogously formalize ‘All objects are extended’ by ∀x(Ox⊃Ex), because that would be to treat objecthood illegitimately as if it were a material, not a formal, property. Just as each object has a form, so the name whose referent the object is has a form, which consists in its possibilities of combination with other names to form elementary propositions; in more homely terms, it is the name’s grammatical category (noun, verb, adjective, etc.). The picture theory tells us that this exactly matches the form of its meaning. A name’s form, just like an object’s, is internal to it. 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! (19 Aug. 1919)

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Wittgenstein thus objected (for reasons that we shall discuss later) to what he took to be Russell’s conception of grammatical categories as containers, comprehensible independently of their instances (4.12721). Objects have both external (material) and internal (formal) properties. The external properties of an object consist in the obtaining of atomic facts in which it occurs. Its internal properties are further divisible into two kinds, form and content (2.025): an object’s form consists in ‘the possibilities of its occurrence in atomic facts’ (2.0141); its content consists in its being the object it is rather than some other object of the same form (2.0233). Objects thus have an internal nature, and so Wittgenstein’s description of them as ‘the substance of the world’ (2.021) conforms to the sense in which I have been using the term. The internal properties of an object are unsayable, but it does not follow that they are unknowable, and indeed Wittgenstein explicitly said that to know an object I must know all its internal properties (2.01231).

The argument for substance If I analyse a fact into simpler constituent facts, what guarantee is there that I ever reach atomic facts? Why should there not be infinite sequences of ever simpler facts? In one place in the Tractatus Wittgenstein held it to be ‘obvious that in the analysis of propositions we must come to elementary propositions, which consist of names in immediate combination’ (4.221), but elsewhere he attempted an argument—known as the ‘argument for substance’—for the existence not of atomic facts themselves, but of the simple objects that compose them. In fact, what he proposed is perhaps best thought of as two variant arguments towards a single conclusion. Here is the first. If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true. It would then be impossible to form a picture of the world (true or false). (2.0211–2.0212) This is phrased as a kind of regress argument, but it relies heavily on the conception of logical analysis previously sketched, according to which any proposition containing a relative name ‘C’ can be split into the supposition that C exists and a claim about C’s constituents, and hence that it has a determinate sense only if the supposition is true. How, then, are we ever to express a proposition with a determinate sense, i.e. ‘form a picture of the world (true or false)’? Only if there are absolute, Tractarian names, Wittgenstein suggested, can the process even get started. We are in some danger now, though, of entering an explanatory circle. It is not obvious why the regress should be thought vicious unless we give the notion of sense the specific technical meaning it has in the Tractatus, namely that the sense of a proposition consists in a division of all the ways the world could be into two classes such that the proposition is true if the way the world is lies in one of the

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classes, false if it lies in the other. This notion of sense would be incoherent in the case of a proposition whose sense depended on the truth of another proposition, since that other proposition could be false (even if in fact it is not), and in that case the first proposition would not have a sense at all and hence would not divide all possible worlds into two classes as required. But the notion of sense in the Tractatus has particular features which belong to the same circle of ideas as the notion of substance Wittgenstein was here trying to justify. Shorn of these features it is not immediately clear why sense should not be determinate despite depending on how the world is. Consider now the second variant of the argument. It is clear that however different from the real one an imagined world may be, it must have something—a form—in common with the real world. This fixed form consists of the objects. (2.022–3) Wittgenstein here claimed that picturing a possibility requires resources that only the actual world can supply. As he later explained, Supposing we asked: ‘How can one imagine what does not exist?’ The answer seems to be: ‘If we do, we imagine non-existent combinations of existing elements.’ A centaur doesn’t exist, but a man’s head and torso and arms and a horse’s legs do exist. ‘But can’t we imagine an object utterly different from any one which exists?’—We should be inclined to answer: ‘No; the elements, individuals, must exist. If redness, roundness and sweetness did not exist, we could not imagine them. (BB, 31) We can, that is to say, make nothing of the notion of an imagined world which does not share its constituents with the real world, since we have no resources with which to begin imagining it. It is incoherent to conceive of some aspects of the world as varying without correspondingly conceiving of others as fixed, since our notion of variability is of something’s varying relative to something else; it does not simply vary on its own. If everything varied, we should lose our grip on the very idea of variation. One obvious objection to both variants of the argument, as I have presented them, is that they commit a quantifier-shift fallacy: if for each proposition there is something simple which that proposition presupposes, it does not follow that there is something—a substance—which all propositions presuppose. This need not show not that my account is incorrect, but only that Wittgenstein had for some reason dismissed the notion that what is simple might depend on context in this way. He certainly had not failed to see the issue: a long passage in his notebook for June 1915 discusses just this point. Frustratingly, though, the notebook breaks off in mid-flow, and the following notebook, in which he might have explained his reasons, is missing.

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Components In the Tractatus, then, an atomic fact is composed of objects, each of which has a form, an internal property which determines its possibilities of combination. This is not the terminology of the 1913 Notes. There the form of a fact was the component responsible for determining its structure. The first sign we have of the change in terminology came in 1915, when Wittgenstein wrote in his notebook: ‘Relations and properties, etc. are objects too.’ (NB, 16 June 1915) By ‘object’ here he meant whatever is simple in experience—‘the concept This’. At this stage, then, his thought need only have been that if objects are whatever is necessarily existent, universals deserve that title just as much as particulars: I cannot represent the possibility of the non-existence of spatial relations, for instance, any more than I can represent the possibility of the non-existence of spatial positions. Modal metaphysics, then, give us no reason to draw a binary distinction among the components of a fact. Might there not nonetheless be some other purely logical reason to draw a binary distinction among them? One such distinction, that between semantically complete and incomplete, Wittgenstein had given up when he adopted the context principle, since this entailed that all objects are semantically incomplete. In the Notes, however, he had also adhered to another distinction, that between constituents and forms. As just noted, what he there called the ‘form’ of a fact was the component responsible for the fact’s structure. It is this feature, he thought, that makes the form hard to express: any explanation of it would be tantamount to an explanation of the form of the proposition, contrary to the doctrine of inexpressibility. Why, though, should one component take all the responsibility for the fact’s structure in this way? In the Tractatus Wittgenstein dropped the conception of the form as a separable component of a fact. Instead, he proposed that each component of a fact should be taken to have a form—an internal property consisting in its possibilities of combination with other components to make up atomic facts. On this new view no one component need be held uniquely responsible for a fact’s structure; the forms of ‘Socrates’ and ‘is mortal’ are jointly responsible for the structure of ‘Socrates is mortal’. As a consequence he now dropped the distinction in the Notes between components and constituents. In effect, Frege’s distinction between saturated and unsaturated components had attributed such a form only to the latter. On Wittgenstein’s new view Frege had mislocated the paradox of the concept horse: what cannot be named is not one or other component of a fact, but the manner in which these components are combined. Wittgenstein continued as before, though, to hold that the forms of the components suffice to explain the form of the whole; logical form is not a further substantive element.

Forms of object Suppose it is granted a priori that there are simple objects. What are their forms? Malcolm’s much later recollection was that Wittgenstein took this to be an empirical issue.

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I asked Wittgenstein whether, when he wrote the Tractatus, he had ever decided upon anything as an example of a ‘simple object’. His reply was that at that time his thought had been that he was a logician; and that it was not his business, as a logician, to try to decide whether this thing or that was a simple thing or a complex thing, that being a purely empirical matter! (1962, 86) How can this be? What we learn from experience is that such and such is the case, not that something exists. So which forms there are is not experienced directly, but must be somehow exhibited in our experience. Russell’s conception of the atoms out of which he sought to construct the external world was essentially epistemological: they were whatever reflection showed to be indubitable. Even if while he was at Cambridge Wittgenstein’s conception of atoms was as phenomenological as Russell’s, he certainly moved away from this soon thereafter. In the Prototractatus he held that space and time are forms of object. What would the objects be in that case? They could not be regions of space and intervals of time, since these obviously are (and are represented by us to be) complex. So he seems to have thought that points in space and moments of time are objects. But did he mean private or public? In the Prototractatus he seems to have intended a contrast between the two. ‘In the same way’, he added, ‘colour (being coloured) is a form of visual objects.’ (PTLP, 2.0252) But he later deleted this contrast and claimed instead that ‘space, time and colour (colouredness) are forms of objects’ (2.0251), thus suggesting that he no longer wished to contrast private and public. In the 1930s, moreover, he gave ‘a point in visual space’ as an example of a Tractarian object. (L30, 120, my emphasis) Some commentators have gone so far as to claim that in the Tractatus Wittgenstein was wholly agnostic about what the objects might be. Perrin (2007, 19), for instance, says that he was ‘astonishingly silent’ about their nature. And Ramsey at one time represented Wittgenstein as holding that about the forms ‘we know and can know nothing whatever’ (FoM, 133). In light of 2.0251 this claim is plainly a little too strong, and Ramsey later (135) replaced it with the weaker claim that we do not in fact know all the forms. By saying that space and time are forms of object, Wittgenstein of course invited comparison with the Transcendental Aesthetic, where Kant held these to be the pure forms of intuition. Where they differed was that Wittgenstein made no attempt to suggest, far less argue, that these were all the forms of object. Empirical reality may be ‘limited by the totality of objects’ (5.5561), but, unlike Kant, he left it open just what form empirical reality has. Wittgenstein further encouraged the Kantian comparison by presenting the argument for substance as transcendental in character: from the premiss that we are capable of talking determinately about the world he deduced a necessary feature of that world, namely that it is made up of simple objects. In both versions of the argument, therefore, he implicitly assumed the internalist premiss that the

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world is representable. In the first version, this is apparent from the fact that the argument cannot even be stated solely at the level of the world: his claim was that a world with no substance would be a world about which I could not begin to think. And in the second version he asked us to consider an imagined (by which he meant represented) world different from the actual one. To represent such a world I need some material to work with, and the only material available to me is what is present in the actual world. This argument gets no purchase on a world that does not share with the real one a common scheme of representation. If we dropped that assumption, perhaps we could, for instance, suppose there to be a world which varied continuously like a fluid: that would allow us to retain our conception of variation as relative, but not commit us to simple objects around which the variation hinges. Wittgenstein’s atomism thus depends essentially on his conception of thought as standing in a harmonious representing relation to the world.

Thought and world What is the order of priority between thought and world in Wittgenstein’s enquiry? His first outline of propositions 1–6 of the Tractatus seems to make the answer very clear: he begins with the world, and then moves via thoughts about the world to the means necessary to represent these thoughts in language. This order of exposition might lead one to suppose, therefore, that Wittgenstein took himself to be deducing the structure that any language must have if it is to succeed in representing an independently given world, On this ‘world-first’ (realist) view, it might be regarded as a considerable achievement that thought should latch onto the world closely enough to be able to represent how things are. If so, then it is reasonable that the explanation for it should be a feature of the world’s structure, namely that the facts which make it up decompose into simple, necessarily existent objects. Contrast this with a ‘thought-first’ (idealist) view, according to which the structure of the world is merely that which thought imposes on it; on this view thought’s fit with the world is no achievement. The overall layout of the book makes it seem, then, as if the world-first reading is straightforwardly the right one. Yet the details of its execution do not. The most obvious example of this that we encountered previously was that the only way we could find to motivate proposition 1 already presupposed that the world was representable in thought. In this chapter we have seen another example, namely that the argument for substance cannot be explained wholly at the worldly level but requires Wittgenstein’s notion of sense for its exposition. Such cases as these suggest that both the world-first and the thought-first views are mistaken. The book does not deduce the structure of thought from that of the world, but nor does it do the reverse. Rather, it deduces both simultaneously. The world which the opening sentence of the book announces as its subject is a world that is already represented in thought.

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Further reading For other accounts of the argument for substance see Noonan (1975), Ludwig (1976), Proops (2004) and Morris (2017). Pears (1987-8, I, ch. 5) discusses whether Wittgenstein held that objects are material points, sense data, or something else. Anscombe (1965) offers an early discussion of the different notions of incompleteness in the Tractatus. On Wittgenstein’s interpretation of the paradox of the concept horse see Geach (1976). On object as a formal concept see Long (1982). For a world-first reading of the book see Pears (1987-8); for a thoughtfirst reading see McGuinness (1981) and Ishiguro (2001). Johnston (2017) gives a version of the neutral reading. For further discussion see Levvis (1998).

53 IDENTITY

It is an immediate consequence of the Tractarian conception of objects that there can be no material relation of identity between them: since object is a formal, not a material, concept, identity is, correspondingly, a formal, not a material, relation (otherwise ‘x = x’ would express objecthood). Another way to the same conclusion would be to note that what varies between possible worlds is only how the objects are combined, not what they are.

Wittgenstein’s argument Despite the availability of a straightforward argument that identity is not a material relation, the argument for this conclusion that Wittgenstein included in the Tractatus itself is much more oblique. Identity is patently not a relation between objects. This becomes very clear if one considers, e.g., the proposition: ‘(x) fx ⊃ x = a’. What this proposition says, is simply that only a satisfies the function f , and not that only such things satisfy the function f as have a certain relation to a. One could of course say that in fact only a has this relation to a, but in order to express this we would need the identity sign itself. (5.5301) It is hard to see why this argument should convince anyone who did not already believe its conclusion. To Wittgenstein it may have seemed obvious that if saying ‘a is f ’ does not involve identity, then saying ‘only a is f ’ does not either.1 If the only objects are a, b and c, for instance, then ‘only a is f ’ is just ∼fb ∧ ∼fc. 1 By ‘only’ here Wittgenstein means ‘nothing other than’.

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Russell, however, would surely have protested at this analysis, because it omits the information that a, b and c are the only objects, and for that we would need identity once again: ∀x(x = a ∨ x = b ∨ x = c). (He had a similar objection to analysing ‘everything is f ’ as fa ∧ fb ∧ fc.) Wittgenstein countered this by insisting that the extra information Russell wanted to add can only be shown, not said. But Russell’s response to that would have been that the reason it cannot be said is just that there is no such relation as identity, which is precisely the point at issue.

Doing without identity Since there is no material relation of identity, Wittgenstein’s concept-script contain no identity sign. Instead, he stipulated that no object should be attached to two different simple signs. Notice, though, that the feasibility of this stipulation depends crucially on his atomism. In a non-atomistic world we might inadvertently introduce two different simple signs (e.g. ‘Aphla’ and ‘Ateb’), not realizing that we were naming the same thing twice, but Tractarian objects (unlike mountains) do not have multiple aspects and hence cannot trick us into inadvertently naming them twice. At this point it is instructive to compare Wittgenstein’s reasons for the sign/symbol distinction with Frege’s for the word/sense distinction. Frege’s reasons always came down to the case of an identity sentence: we need to recognize a level of sense, he thought, in order to allow for the fact that we might innocently give two different names to the same mountain when seen from different angles. For Wittgenstein, though, the point was not specific to identity sentences: the sign/symbol distinction arises once we recognize that although the form of thought is necessarily that of the world, the form of language is not. So much for names. What about variables? Russell’s theory of descriptions makes use of the identity sign in the analysans: so it looks as though we need it in our concept-script after all. Wittgenstein’s solution was to propose a new interpretation of the quantifier, for which I shall here use ‘∃0 ’ to distinguish it from the usual one. He proposed to read ∃0 x, y f (x, y) as meaning that there exist distinct objects x and y such that f (x, y), i.e. what would normally be written ∃x, y(f (x, y)∧x 6= y). With this notation in place, Russell’s theory of descriptions becomes expressible once more: f (the g) =df ∃0 x(fx ∧ gx) ∧ ∼∃0 x, y(gx ∧ gy). In Wittgenstein’s notation we can express ‘There are at least n f s’ as ∃0 x1 . . . xn (fx1 ∧ · · · ∧ fxn ). If we apply this in the case of a ‘tautological’ propositional function, i.e. a function that is true of every object, such as Tx =Df gx ∨ ∼gx, we seem to obtain a proposition pn expressing that there are at least n objects. In that case, though, pn ∧ ∼pn+1 says that there are exactly n objects, contradicting our earlier claim

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that this is nonsensical. What has gone wrong? To resolve this puzzle, we need to remind ourselves that Wittgenstein’s convention that the domains of quantified variables are disjoint is only a notational device, not part of the concept-script proper. To find out whether a sign makes sense, the test is always to see whether it can be re-expressed in the basic system. If we apply that test in the case where the only objects are a, b and c, we obtain p1 = Ta ∨ Tb ∨ Tc; p2 = (Ta ∧ Tb) ∨ (Tb ∧ Tc) ∨ (Tc ∧ Ta); p3 = Ta ∧ Tb ∧ Tc; p4 = ? Each of p1 , p2 and p3 converts straightforwardly into a sign expressing tautology, but p4 resists conversion altogether and therefore remains a mere arrangement of signs. More generally, if the number of objects is n, then the signs p1 , . . . , pn express tautology, but pn+1 , pn+2 , . . . fail to express anything. Wittgenstein did not explain this point in the Tractatus itself, but Ramsey evidently raised it with him in September 1923; for next to the remark that we cannot say ‘There are 100 objects’ (4.1272) he wrote in the margin of Ramsey’s copy, ‘The proposition “there are n things such that . . . ” presupposes for its significance, what we try to assert by saying “there are n things”.’ (See Lewy 1967) If there are no senseful propositions expressing identity or non-identity of objects, then ‘a = a’ is never senseful for any name ‘a’. Why, though, is it not a tautology? The reason is that a tautology is obtained by putting together meaningful components in such a way that what they express is empty, whereas what we have here is a sign in which one of the components—the sign of identity ‘=’—does not pick out a material relation at all. By the same token any attempt to say how many objects there are results in nonsense, since for there to be such a proposition, the number of objects would have to vary between possible worlds. In order that our language should not be ambiguous we need to ensure that a sign does not express different symbols in different occurrences. In the case of a simple sign, this amounts to the requirement that there should be a single object which is its referent in all occurrences. We have noted already that Wittgenstein used the ‘=’ sign to stand between signs that express the same symbol. In the case of simple signs this amounts to saying that they mean the same object (4.241). So here once again his account resembles Frege’s in Begriffsschrift. The difference is that Frege held identity sentences to be an ineliminable part of the symbolism proper, whereas for Wittgenstein they are ‘only expedients in presentation’ (4.242), i.e. part of what we would nowadays call the metalanguage, and say something about signs, not about the world. Frege’s later objection to such an account of identity was that it got the subject matter wrong, making ‘Hesperus=Phosphorus’ seem to be about language, when really it is about astronomy. What enabled Wittgenstein to resist

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this objection was that on his account identity sentences will only ever tell us about language. The kind of content that concerned Frege could not arise in an atomistic world, where simple objects cannot have different aspects.

Identity and classes According to Wittgenstein, there is no such relation as identity. Yet in Principia Russell defined identity by x = y =df ∀φ(φ!x ≡ φ!y). How so? Wittgenstein’s explanation was that the relation Russell had defined was not identity. Russell’s definition 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.) (5.5302) Suppose, for instance, that a and b are distinct objects of the same form. By the independence of elementary propositions it is possible that they should have all their elementary properties in common, i.e. that for each true elementary proposition φ(a) involving a the corresponding proposition φ(b) involving b is also true; and vice versa. But if this holds for every elementary property, it will hold for every property, because Wittgenstein’s logic is truth-functional. So it is possible for a and b to have all the same properties, in which case Russell’s definition would wrongly pronounce them identical. ‘The theory of classes’, Wittgenstein maintained, is altogether superfluous in mathematics. This is connected with the fact that the generality which we need in mathematics is not the accidental one. (6.031) In the Tractatus Wittgenstein did not group this remark with his discussion of identity, but he evidently conceived of them as related, because in 1923 he wrote the word ‘identity’ next to it in Ramsey’s copy (see Lewy 1967). The allusion is to a problem for Whitehead and Russell’s project of basing arithmetic on logic. In Principia they deduced arithmetic from the theory of classes, which in turn they derived from logic by treating class-terms as incomplete symbols. On this treatment a class-term such as ‘{x : φx}’ is accidental in the sense that the class for which it goes proxy in the theory has as its members whichever objects happen to have the property φ: these are different in different possible worlds. Their account of arithmetic made use only of these accidental class-terms, and so the theorems they proved would depend on how the world happened to be. To see this, consider their account of the natural numbers, in which the number 1 is defined as the class of all one-membered classes, and 2 as the class of all twomembered classes. In a world in which any three objects share all their properties,

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no accidental class-term would pick out a one-membered or two-membered class. According to the definitions in Principia, therefore, the numbers 1 and 2 would both come out as empty, i.e. in such a world 1 = 2. In order that arithmetic should be independent of the contingent facts of the world, then, we need what I shall call essential class-terms—terms for classes whose members do not vary between worlds. We need, for instance, an expression ‘{a, b}’ to play the role of the class whose members in every world are the objects a and b. If there were a relation of identity, we could of course synthesize this by means of the incomplete symbol ‘{x : x = a ∨ x = b}’, but without such a relation we are stymied. Modality once again plays a vital role in Wittgenstein’s argument here. His objection to Principia was only that it might get arithmetic wrong, not that it did. He took it as obvious that ‘the generality which we need in mathematics is not the accidental one’, but it would be open to a determined defender of Principia simply to deny this. The denial would come at a heavy price, however, since the theorems of mathematics would no longer be logical truths. We would on this account have at best a transcendental argument for the necessity of mathematics, conditional on the possibility of distinguishing between the objects of the actual world. This is a price which Russell seems, in 1919 at least, to have been willing to pay.

Further reading White (1978) offers an elegant explanation of what is at stake in Wittgenstein’s rejection of identity as a relation. We shall discuss in a later chapter Ramsey’s attempt to subvert Wittgenstein’s attack by rehabilitating identity (and hence logicism).

54 SOLIPSISM

The discussion of solipsism in the 1916 Tractatus was notably briefer than that in the 5.6s of the final version. The limits of my language mean the limits of my world. This remark provides a key to the question, to what extent solipsism is a truth. In fact what solipsism means is quite correct, only it cannot be said, but it shows itself. That the world is my world shows itself in the fact that the limits of the language (the language which I understand) mean the limits of my world. The thinking, presenting subject; there is no such thing. Moreover, the passage was originally positioned so as to be subsidiary to the discussion of identity: Wittgenstein only later renumbered it to form a separate section. In this chapter I shall be discussing the passage in its original form and location.

The argument for solipsism Let us begin with Wittgenstein’s claim that language is my language—‘the language which I understand’. In the 1922 edition Ogden and Ramsey mistranslated ‘die Sprache, die allein ich verstehe’ as ‘the language which only I understand’ and hence created the impression that Wittgenstein countenanced some sort of private language. What I have quoted is the correction that Wittgenstein adopted in the 1933 reprinting, although this is still needlessly obscure. The translation Russell gave in the Introduction—‘the only language I understand’— is better. What is at any rate clear is that by ‘language’ here Wittgenstein did

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not mean German or English—if he had, he would hardly have emphasized its uniqueness—but language as a whole, abstracted from the contingent features of any particular human language. He meant, in short, the symbol-language of thought. If there is in this sense only one language that I understand, what, though, about a language I do not understand? In the notebooks Wittgenstein briefly wondered whether language is the only language (29 May 1915), but he did not answer his own question and in the Tracatus he simply ignored it. In order to understand what led Wittgenstein to solipsism, let us first contrast his view with Russell’s. Russell’s conception of analysis aimed to explain how we can refer to something for which we do not have a name. This is possible, he thought, because the variable ranges over a domain that is more extensive than that of things with names. His account left room for multiple points of view by allowing that your understanding of the variable is the same as mine, even though your stock of names is different. Wittgenstein’s method of analysis, by contrast, was ontologically eliminative, and his narrow symbol-level account of quantification ruled out this two-part conception of the domain of quantification, leaving no room for the Russellian account of different perspectives. Just as each name in my language means a worldly object, so, one might say, the limits of my language mean the limits of my world. Here Wittgenstein used bedeuten—the same verb that Frege used to express the relation between a proper name and the object it refers to. When he said that what solipsism means is quite correct, on the other hand, he used meinen, which might be translated as ‘intends’. What solipsism intends is that the limits of my language mean the limits of my world. In a conventional logical system such as Russell’s, this would be expressible—if ‘a’, ‘b’ and ‘c’ were the only names in the language, it would be ∀x(x = a ∨ x = b ∨ x = c)—but in Wittgenstein’s system it is inexpressible, because there is no such relation as identity. So much for Russell. What of Frege? In his 1914 letter to Jourdain he argued (PMC, 80) that the notion of sense was required in order to explain how my conception of the mountain I know as Aphla leaves room for your distinct perspective within which you know it as Ateb. Tractarian objects, however, cannot have other aspects to them, unlike mountains. Wittgenstein’s atomism leaves no room for the notion that someone else’s language might provide a way of representing the world from a different point of view. So the representable world just is my world. Combining this with his internalist premiss that the world is a representable world, we thus arrive at the solipsist’s conclusion that the world is my world.

The thinking subject Let us move on now to Wittgenstein’s rejection of the thinking subject. Whatever it was that he took himself to be rejecting was not the self that he could see in

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the mirror. (I shall return to this empirical self in Chapter 56.) He would of course have come across arguments purporting to demonstrate the existence of a thinking subject, Descartes’ cogito being only the best known. Remarks he added to the book later suggest that what he took himself to be arguing against here was a subject located in the world. This is a comparatively unimportant issue in itself, since it depends on little more than how we choose to use the word ‘world’: we could, if we wished, exclude the self from the world by fiat. What matters rather more is that if the thinking subject he was arguing against were in the world, it would have to be simple, since only a simple entity could be knowable a priori in the manner Descartes had claimed. What Wittgenstein was arguing against, therefore, was the notion that the world should be conceived of as represented from a single viewpoint that I call ‘mine’. On the same day as his discussion of solipsism he wrote a paragraph that might at first seem to be Wittgenstein’s argument against such a viewpoint. In the book ‘The world I found’ I should also have to report on my body and say which members are subject to my will, etc. For this is a way of isolating the subject, or rather of showing that in an important sense there is no such thing as the subject; for it would be the one thing that could not come into this book. (NB, 23 May 1915) We should be cautious, though, about placing much weight on this paragraph. Wittgenstein did eventually add a variant of it to the Tractatus, positioning it so as to act as a commentary on his rejection of the thinking subject, but the fact that he did not do so until later (page 76 of Bodleianus) suggests that he did not regard it as crucial. Moreover, the paragraph does little to persuade anyone not already convinced of the conclusion, because it is more a restatement of the claim than an argument for it: the paragraph does not explain why the book ‘The world I found’ could not mention the thinking subject. Might Wittgenstein’s denial of the thinking subject then be what its position in the 1916 Tractatus makes it out to be, namely a more or less straightforward corollary of the solipsism to which it is appended? If there were a thinking subject, the fact that it represented the world would consist in a relationship (or perhaps a series of relationships) between it and other objects. We could therefore conceive of its being duplicated, so that there would now be another thinking subject distinct from the first, bearing all the same relations to the rest of the world as the original one. Yet this would plainly no longer count as a solipsistic world. If the argument for solipsism is correct, therefore, it cannot have a thinking subject capable of duplication as its focus. The proponent of the thinking subject takes every proposition p to be silently accompanied by ‘I think’, and hence really of the form ‘I think that p’; but this ‘I’ has no identity conditions and therefore drops out as an idle wheel. Strawson (1966, 163–9) detected in Kant’s paralogisms a rather similar argument against the Cartesian thinking subject, but Wittgenstein need not have been inspired by so exotic a source; for the argument just

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sketched is really just the mirror image of one we previously attributed to Russell (Chapter 29), when he argued that Reality is an idle wheel in Bradley’s claim that the true form of p is ‘Reality is such that p’. Another way to express Wittgenstein’s view might be this. According to the picture theory, the symbol-language of thought lies so close up against the world as to make the task thinking performs, of representing the ways in which the world may be configured, wholly transparent. There is no role for the thinking subject to play as indexing the perspective from which propositions represent the world, because there is no sense to be made of the notion of different perspectives on a single object. The Tractarian world is indeed a represented world, but not one that is represented from one perspective rather than another. Wittgenstein’s solipsist may gesture at the world and say, ‘All this is mine,’ but not, ‘All this is mine alone.’ Whatever the sense is in which the world is mine is not one in which it is even intelligible for it to be anyone else’s. By now, though, the reader may well feel a sense of deflation at the turn our discussion has taken. Wittgenstein’s initially eye-catching assertion that the world is mine has been so devalued by the dissolution of the self who claims this ownership that his solipsism seems to collapse into no more than an idiosyncratic restatement of the internalist premiss that the world is a represented world. In Chapter 58 we shall see how he later began to wonder whether to nuance his dismissal of the thinking subject.

Further reading The correct interpretation of the passage discussed in this chapter is controversial. For a different exegesis of Wittgenstein’s solipsism see Hintikka (1958). Further useful discussion is to be found in Schulte (1995), Diamond (2000) and McGuinness (2001).

55 ORDINARY LANGUAGE

The argument for solipsism in the last chapter assumed that my language is of the logically perfect, atomistic language sort posited in the Tractatus. Perhaps, then, I could shrug the argument off merely by denying that my language is atomistic. To stymie this move Wittgenstein needed to insist that my language, superficial appearances to the contrary notwithstanding, can indeed express the same thoughts as the atomistic one, and is therefore susceptible to the same conclusion. He needed to claim, that is to say, that ‘all propositions of our colloquial language are actually, just as they are, logically completely in order’ (5.5563).

Vagueness We saw in Chapter 43 how Russell, under the pressure of Wittgenstein’s argument for solipsism, came to think that the vagueness of ordinary language is what makes communication possible. At first Wittgenstein was tempted by a similar idea. Could it be possible that the sentences in ordinary use have, as it were, only an incomplete sense (quite apart from their truth or falsehood), and that the propositions in physics, as it were, approach the stage where a proposition really has a complete sense? . . . Is it or is it not possible to talk of a proposition’s having a more or less sharp sense? Without quite answering this question, he went on to observe that ‘it seems clear that what we MEAN must always be sharp’ (NB, 20 June 1915). This is at the end of the second surviving notebook, and there is no clear indication there of how he resolved the issue. In the 1916 Tractatus, however, he came down firmly in favour of the view that the sentences of ordinary language do express complete, determinate senses.

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All propositions of our colloquial language are actually, just as they are, logically completely in order. That simple thing which we ought to give here is not a model of the truth but the complete truth itself. (Our problems are not abstract but perhaps the most concrete that there are.) (5.5563) Even if the point of including this in the book is reasonably clear, the remark itself is puzzling, and it is rendered no less puzzling by the gloss Wittgenstein later put on it. The prop[osition]s of our ordinary language are not in any way logically less correct or less exact or more confused than prop[osition]s written down, say, in Russell[’]s symbolism or any other ‘Begriffsschrift’. (Only it is easier for us to gather their logical form when they are expressed in an appropriate symbolism.) (Letter to Ogden, 10 May 1922) What did he mean by ‘propositions’ here? If he was using the word in its technical meaning (propositions with a Tractarian sense), the remark is disappointingly anodyne: of course those are completely in order. On the other hand, Wittgenstein does seem sometimes to have thought (influenced, perhaps, by Levin’s experience talking to his farm workers) that colloquial language restricts itself to making factual claims about the world around us; in that case what he meant here was merely that these are logically in complete order. Even in the case of Tolstoian peasants, however, this is implausible: they often make statements—belief claims, for instance, or equations of elementary arithmetic—which the Tractatus classifies as nonsense. (Later, we shall encounter other examples such as ethics and religion.) At most, then, Wittgenstein might be entitled to claim that some propositions of our colloquial language are logically ‘completely in order’, not that all are. What is clear, at least, is that Russell was wrong, in the Introduction, to attribute to Wittgenstein the view that ‘the whole function of language is to have meaning, and it only fulfils this function in proportion as it approaches to the ideal language which we postulate’ (CPBR, IX, 101). This was indeed Russell’s view, and may have been Wittgenstein’s before the war, but by the time of the Tractatus he had rejected it. Russell held that a sentence may come close to making sense; Wittgenstein, that it either makes sense or it does not. Only then would it follows that his solipsistic conclusion applies to me.

Austerity Proposition 5.5563 provides textual authority for ascribing to Wittgenstein the view that when we mundanely engage in fact-stating discourse, we make sense in not just the ordinary but the technical Tractarian sense. Contraposing, therefore, sentences that the Tractatus characterizes as nonsensical do not make ordinary, factual claims. What, then, do they say? The answer, ‘Nothing’, has come to be known as the ‘austere’ conception of Tractarian nonsense, in contrast to the

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inaustere conception according to which such sentences may gesture towards what is ineffably so. In the Introduction Russell canvassed an extreme inaustere view according to which what I cannot say in my current language can be said in some larger language. It is worth stressing, though, that he offered this as an alternative to the account of nonsense in the Tractatus, not an interpretation of it. Wittgenstein’s own view, by contrast, was that logic’s generality consists in its utilization of a form of representation from which no retreat to a metalanguage is possible. It is hard, in any case, to see how enlarging the language could address the difficulty he saw in nonsensical claims, namely that they do not carve the possibilities as to how the world can be into two classes. If we reject Russell’s extreme inausterity, ought we to adopt extreme austerity instead? One frequently suggested reason for doing so is that there cannot be degrees of nonsensicality: once we have recognized that something does not make sense, we have no choice but to put it in the same category as ‘fribble frabble’. As a claim about the ordinary use of the word ‘nonsense’, however, this is questionable: we do treat nonsense as if it comes in degrees, as the frequency with which we put intensifiers such as ‘sheer’ or ‘utter’ before the word suggests. Moreover, we can sometimes help someone out of a confusion by deliberately participating in it to some extent. Only if we regard the confusion as extreme (whether culturally or intellectually) do we refuse to engage with it at all. Defenders of extreme austerity must maintain, nonetheless, that in so participating we talk complete nonsense, its similarity to sense being an illusion with only a psychological, not a semantic, explanation. Such readers, although they deny that there are degrees of nonsense, need not resist the notion that there are kinds of nonsense. There may be various things to be said, for instance, about the effect that hearing some kinds of nonsense has on us. Such explanations would count as inaustere only to the extent that they attributed semantic content to the sounds in question. What should we make of sentences that obey the rules of ordinary grammar but fail to make Tractarian sense? The context principle diagnoses the failure as arising when at least one constituent of a propositional sign has not been given a reference in the relevant context. In such a case the picture theory breaks down: there is no picture, because one of the required constituents is missing. Might some other, weaker notion of sense be applicable here? Frustratingly, the 1916 Tractatus does not say.

Philosophy The natural sciences, Wittgenstein held, consist of the true contingent propositions (4.112). Philosophy, by contrast, does not consist of propositions at all and is ‘not a theory but an activity’, a critique of language (though not, he added helpfully, in Mauthner’s sense). Wittgenstein took as his model the project of philosophical analysis in which Russell had ‘shown that the apparent logical

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form of the proposition need not be its real form’ (4.0031). Russell’s analysis of ‘the present King of France is bald’, for instance, shows how to understand this sentence without positing a mysterious, non-existent King. In other cases, Russell’s method may show that there is no proposition being expressed at all. In these cases, too, philosophy can supply clarity, by indicating to us the limits of what is thinkable. Philosophers, Wittgenstein thought, have spent much of their time trying to answer questions ‘of the same kind as the question whether the Good is more or less identical than the Beautiful’ (4.003). They would be better to concentrate instead on showing these questions to be nonsensical. Notice, though, that although logical analysis is a legitimate weapon in the philosopher’s armoury, he did not think that this was a matter of achieving a complete analysis. There is some doubt about whether he thought such an analysis is even feasible, but at any rate he did not think it had yet been carried out, nor that clarity depended on achieving it. The remarks about the nature of philosophy in the 4s make no mention of the reflexive paradox that the Tractatus is by its own lights largely nonsense. Indeed, it is not as clear as some commentators suppose that Wittgenstein intended these remarks to apply to the book itself. If philosophy aims to clarify what we are saying—whether by revealing its structure or by showing that we have not really said anything at all—one can only be struck by how poor an exemplar of the genre the Tractatus is. Much of the book is taken up with an extended exposition of the relationship between thought and world, an exposition of precisely the sort that philosophy, we are told, must eschew. Only once this was in place did Wittgenstein make any attempt to discuss the limits of language; and even then, he fell well short of the specificity one might wish for. Rarely did he practise philosophy as he himself had characterized it. Perhaps, then, he regarded his book more as a prolegomenon to philosophy than as a free-standing contribution to it, and hence took it to be exempt from his own characterization. The understanding of philosophy as an activity is a consequence of the picture theory, not a route to it.

Further reading The literature on the austere conception of nonsense is extensive: begin with Diamond (1981). On the relation of Wittgenstein’s view to Frege’s see Geach (1976) and Shieh (2015). On connections with modernism see Ware (2013).

56 MINDS

As Russell later recollected it, the nub of Wittgenstein’s 1913 criticism of his multiple relation theory of judgment was that if I say that A believes that p, I in particular express the proposition p, and to do this I must use its verb as a verb. Russell’s multiple relation theory made this impossible, since it ensured that the verb occurred with different valencies in p and in ‘A believes that p’. Let us examine how Wittgenstein’s own theory of judgment circumvented this difficulty.

Judgment In saying ‘A believes that p’, Wittgenstein thought, we pick out the fact in A’s mind that consists in his believing that p and express what it is that this fact says (namely that p). ‘It is clear’, he claimed, that ‘A believes that p’, ‘A thinks p’, ‘A says p’, are of the form ‘ “p” says p’: and here we have no co-ordination of a fact and an object, but a coordination of facts by means of a co-ordination of their objects. (5.542) However ‘clear’ Wittgenstein himself may have found this, commentators have been puzzled about which two facts are coordinated in ‘ “p” says that p’. The only answer that makes the passage intelligible, I think, is that the first fact is an interpreted propositional sign (or the psychical correlate thereof), the second a proposition: the first could not be an uninterpreted propositional sign, because that does not yet say anything; the second could not be a worldly fact, since if A’s belief is false, there is no such fact. Consider, for instance, Othello’s belief that Desdemona loves Cassio, and suppose for convenience that Desdemona, Cassio and loving are simple. There

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occur in Othello’s mind tokens of simple signs which he interprets as referring to Desdemona, Cassio and the loving relation. For him to have the belief that brings about his tragic downfall is for these tokens to be configured as a mental picture which says that Desdemona loves Cassio and is accompanied by a feeling of belief. (If Othello had doubted whether Desdemona loves Cassio, the mental picture would have been accompanied by a feeling of doubt; and similarly for other propositional attitudes.) Wittgenstein’s theory of judgment thus depends on the picture theory. A judgment consists in the coordination of two facts, an interpreted propositional sign and a proposition, by means of a coordination of their components: it is because the signs in Othello’s mind are interpreted as referring to their worldly correlates that the mental picture says that Desdemona loves Cassio. For Wittgenstein, as for Russell, judgment is a multiple relation. The difference is that in Wittgenstein’s analysis Othello himself does not occur: this is what permits loving to occur in Othello’s belief with its correct multiplicity. I may make a judgment in writing by means of a propositional sign; or I may make it in thought by means of a psychical fact. Wittgenstein said much less about the latter than the former. We saw in Chapter 43 that by the time the finished book reached England in the summer of 1919, Russell had written his own account of these psychical facts. It is therefore no surprise that he pressed Wittgenstein to say more about their constituents. Wittgenstein replied that they are not words but ‘psychical constituents that have the same sort of relation to reality as words’ (CL, no. 68). What occurs in my mind is thus like a token of a propositional sign, except that its constituents are psychical, not physical. One might be tempted to put this by saying that psychology is parallel to, but distinct from, linguistics: the former studies psychical signs, the latter propositional signs. Wittgenstein, though, made no such contrast, and seems not at this stage to have regarded the difference between the two kinds—that only the latter are directly perceptible by the senses—as interesting or important. That ‘p’ says that p is, according to Wittgenstein, inexpressible in the language: he called it a ‘pseudo-proposition’. Nowadays we call it a ‘semantic’ claim, but Wittgenstein did not use that word. (Russell treated it as falling within the scope of epistemology.) Because it is not just relatively but absolutely inexpressible, the now familiar move to the metalanguage—first proposed by Russell in his Introduction to the Tractatus—is of no help. By using the sign ‘p’ I show what ‘ “p” says that p’ tries, but fails, to say.

The empirical subject Wittgenstein’s theory of judgment envisages analysing ascriptions of beliefs to other thinkers. How is that consistent with his solipsism? To explain this we need to distinguish two different uses of the word ‘I’: one refers to an empirical being that exists contingently and is viewable in the mirror; the other refers to a putative entity knowable a priori. If we call these the ‘empirical’ and the ‘thinking’ subject

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respectively, then the selves that are involved in belief ascriptions are empirical, existing contingently and therefore not the kind of thing that might be entitled to the solipsist’s a priori claim of ownership of the world. However necessary this distinction is, though, it has to be conceded that Wittgenstein took some time to make it explicitly himself. Even as late as August 1916 his notebook shows him uncertain of how to apply the distinction. This theory of judgment occupies a singular position in the books’s composition: it is in the Moore dictation of 1914, but Wittgenstein did not include it in the 1916 Tractatus at all; and when he did eventually add it, he presented it as if its role were merely to block a potential counterexample to his claim that every proposition is a truth-function of elementary propositions (since ‘A believes that p’ is evidently not a truth-function of p). So why did he omit the theory from the 1916 Tractatus? It might of course be a simple mistake, but it might also indicate his hesitancy concerning the relationship between the psychical facts that make up the mind and the perceptible propositional signs that make up language. Not until the autumn of 1916 did he exclaim, as if it was a discovery, that ‘thinking is a kind of language’ (NB, 12 Sep. 1916). The most notable feature of Wittgenstein’s analysis of ‘A believes that p’ is that A does not occur in it: A is merely the location of the psychical fact whose meaning what it does constitutes the belief. So empirical subjects are not Tractarian objects capable of being related in judgment to other objects, but (presumably) complexes of psychical facts. Accounts of the mind of this sort are usually called ‘Humean’, but to say any more about what Wittgenstein intended would be speculative, since he showed no interest whatever in elaborating. That, he thought, would be a task for the psychologist, not for him. I suggested earlier that Wittgenstein’s account of belief might have influenced Russell when he rejected the ‘pinpoint subject’. Could it also have led Wittgenstein to reject the thinking subject? Probably not, given the history just related. The account leads to a conception of the empirical subject as complex, but says nothing directly about the thinking subject. Even so, one might still try to argue that it says something indirectly, in the following way. The empirical subject is a complex of mental signs expressing various propositions and accompanied by various feelings. The thinking subject, if there were such a thing, would be like this except that its language would already be that of thought, and so the semantic task involved in propositional attitudes—that of using a propositional sign ‘p’ to say that p—would evaporate. Even if there is an argument of this shape, though, I doubt whether it was Wittgenstein’s; for it depends on a distinction between empirical and thinking subjects that even in the autumn of 1916 he was still struggling to clarify. Russell distinguished between a solipsism of acquaintance, which attempts to construct matter from my own sense data, and a phenomenalism of acquaintance, which enlarges the constructional base to include the sense data of other people. The difficulty he then faced was that the constructional base of each person’s

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language consisted of sense data with which that person was acquainted. Since these classes were disjoint, it became puzzling how we succeed in communication with one another. For Wittgenstein, on the other hand, naming is linked not to acquaintance but to simplicity. It is in principle possible that you and I should both grasp simple signs which refer to the same simple object and hence are instances of the same Tractarian name. In the 1916 notebook Wittgenstein treated solipsism as an extreme version of idealism. ‘Idealism’, he said, ‘singles men out from the world as unique, solipsism singles me alone out.’ (NB, 15 Oct. 1916) By the time of the final Tractatus, however, the mention of idealism as an intermediate step on the route to solipsism had been excised. Now we can see why. The 1916 remark applied to the empirical level, at which Russell’s phenomenalism might indeed be said to single out humans, and his solipsism to single out him alone. At the level of symbols, though, there is only one language, namely the language of thought, and so the distinction between idealism and solipsism, between ‘us’ and ‘me’, dissolves. In the midst of his discussion of solipsism, Wittgenstein wrote in his notebook that ‘there really is only one world-soul, which I for preference call my soul and as which alone I conceive what I call the souls of others’ (NB, 23 May 1915). There really is, he might just as well have said, only one language, which I for preference call my language. The souls of others can be thought of as expressing propositions in this one common language. Wittgenstein’s way of expressing the point carries rather more mystical connotations, however. Weininger, for instance, talked about the world-soul. And the remark shows an affinity with Emerson (an author whom Wittgenstein read in the early part of the war). ‘There is’, Emerson said, ‘one mind common to all individual men.’ (1979, 3) ’Of the universal mind each individual man is one more incarnation.’ (Ibid., 4) It is ‘that Over-Soul, within which every man’s particular being is contained and made one with all other’ (ibid., 160).

Logic and psychology Wittgenstein’s account of belief may be applied to a tautology just as to a proposition with sense. So ‘ “p” is a tautology’ is, like ‘ “p” says that p’, a nonsensical pseudo-proposition. The difference is that in the case of a tautology, what the propositional sign expresses is empty (and hence, strictly speaking, not a proposition at all). If we can read—i.e. see the symbol in—the sign, we can already recognize whether it expresses a tautology or not. Someone who knows that p is a tautology knows nothing. The non-trivial part of the process consists in the reading. In a logic exam, for instance, students might be asked to determine whether various propositional signs are tautologies by recognizing what these signs say. There is therefore a significant difference in the use of the ‘=’ sign, according as the signs it stands between are simple or not: when it stands between simple signs, it can be reduced to triviality by adopting Wittgenstein’s convention that there should only be one simple sign in the concept-script for each object; when

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it stands between propositional signs, on the other hand, no concept-script can render it trivial, because a method of deciding whether two propositional signs express the same sense would in particular solve the Entscheidungsproblem. In Wittgenstein’s account the point of insisting on the distinction between sign and symbol was to secure the objectivity of logic. Since the empirical subject—‘the human soul of which psychology treats’ (5.641)—is a complex of psychical facts, of no direct relevance to logic, Frege was wrong to make the assertion sign central to his treatment of logic. Frege’s sign ‘ ’ is logically altogether meaningless; in Frege (and Russell) it only shows that these authors hold as true the propositions marked in this way. ‘ ’ belongs therefore to the propositions no more than does the number of the proposition. A proposition cannot possibly assert of itself that it is true. (4.442) If the only purpose of the sign ‘ ’ is to mark the psychological fact that whoever uses it judges that a certain proposition is true, it has no place in a logic book.

Theory of knowledge The solipsist holds that the only things in the world are those my language has names for. But which are these? Perhaps when they were arguing about matter in 1912 (see Chapter 41), Wittgenstein still accepted Russell’s view that naming is akin to pointing, so that they are just the objects of my own acquaintance. This ‘solipsism of acquaintance’, according to which ‘no one exists but himself ’, seems rebarbative to common sense, requiring for its refutation no more than Christine Ladd Franklin’s quip to Russell (in a letter of 21 Aug. 1912) that she was surprised the view was not more popular. Soon, however, Wittgenstein changed his conception in several respects. He rejected Russell’s epistemologically based conception of the atoms as objects of acquaintance and replaced it with a modally based conception of them as whatever is necessarily existent. Also, the sign/symbol distinction allowed him to conceive of a name as a certain kind of symbol, irrespective of whether I have uttered any token of it. This freed him from the implausible claim that I have to conceive of the world as consisting of those things which I have actually named with simple signs. The language spoken of in the 5.6s, whose limits mean the limits of the world, is intended to abstract from such contingencies. His account thus remained solipsistic, but it was no longer a solipsism of acquaintance. Moreover, Wittgenstein’s adoption of the context principle transformed naming into a task to be performed at the level of whole propositions, not individual names. This shift does not in itself disqualify acquaintance from playing any role in the process. To give a sign ‘a’ meaning, it is enough to find a unique description of a in terms of its material properties, e.g. as the φ. There are three points to note about this, however. First, doing this

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does not make ‘a’ merely an abbreviation for ‘the φ’: the former is rigid, the latter not. Second, this method might work for a single simple sign, but it could not be used to give meaning to all the signs simultaneously. Third, the method works only if a can be uniquely described, but, as we noted earlier, the independence of elementary propositions guarantees that if a0 is another object of the same type as a, then there is a possible world in which a and a0 are indistinguishable. On Wittgenstein’s account, interpreting a sign, and hence turning it into a symbol, is a psychological task. He offered two descriptions of this task, one more public in character than the other. The first, more public description was that ‘to recognize the symbol in the sign we must consider the significant use’ (3.326). How a sign symbolizes—which symbol it expresses—is thus a matter of its ‘logical syntactic application’ (3.327). The second, less public answer was that ‘the method of projection is the thinking of the sense of the proposition’ (3.11). In his later writings Wittgenstein sometimes appealed to a metaphor according to which the sign is an ‘utterly dead and trivial thing’ (BB, 4) brought to life—turned into a symbol—by our thinking what it means. It is controversial, though, how much of a contrast Wittgenstein intended between his two accounts of the symbol.

Further reading The account of Wittgenstein’s theory of judgment offered here is indebted to Kenny (1981). For an attempt to link the account of the mind to Wittgenstein’s solipsism see O’Brien (1996). On Wittgenstein’s criticisms of Frege’s assertion sign see Proops (1997). Putnam (2008) makes some suggestive remarks about ‘I’ and ‘we’.

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The 6.1s constitute a late addition to the 1916 Tractatus—the first, modest extension of the book beyond its originally intended conclusion concerning the general form of a proposition.

The propositions of logic ‘The propositions of logic are tautologies.’ They therefore say nothing (6.1–6.11). We knew already, of course, that tautologies say nothing. What Wittgenstein now added was the claim that they are the whole of logic—every logically necessary proposition is a tautology—and hence that logic is in this sense empty. This further claim could be deduced by a straightforward squeezing argument from the independence of elementary propositions, since this entails that the only non-senseless propositions are the contingent ones. When the Tractatus was finished, what united the 6s was a concern with the nonsensical (unsinnig). One might at first think, therefore, that by then the discussion of logic was mislocated. Are the propositions of logic not senseless (sinnlos) rather than nonsensical?1 The explanation is that although the propositions of logic are senseless, Wittgenstein also used the word ‘logic’ to describe not the propositions themselves but what in his later writings he called ‘metalogic’, i.e. the attempt to delimit logic by saying which are senseless. Metalogic consists of pseudo-propositions, which show not which atomic facts obtain, but which are possible.

1 The Ogden-Ramsey translation incorrectly says that tautologies are ‘not senseless’ (4.4611): the German is ‘nicht unsinnig’.

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One advantage Wittgenstein claimed for his conception of the propositions of logic as empty was that it solved at a stroke the problem of the epistemology of logic—a problem which, he thought, Frege had signally failed to solve. Frege had held both that logic is not psychological and that it is capable of delivering non-trivial truths (most prominently, those of arithmetic). In deriving these nontrivial truths we begin from particular logical truths—the so-called ‘basic laws’— that we take to be self-evident, but Wittgenstein complained that self-evidence is a psychological matter with no role to play in the epistemology of logic. ‘It is remarkable’, he said, ‘that so exact a thinker as Frege should have appealed to the degree of self-evidence as the criterion of a logical proposition.’ (6.1271) Wittgenstein solved—or rather dissolved—the problem of the epistemology of logic by holding that in logic there is nothing to know. ‘I know, e.g. nothing about the weather, when I know that it rains or does not rain.’ (4.461) It was not only Frege who had suffered from the lack of a criterion for logical truth. In the Principles Russell characterized the logical truths as being those whose generality is limited only by their concern with the ‘eight or nine’ logical constants. In the course of writing Principia, however, he found that the development of mathematics required him to make several assumptions expressible in pure logic—reducibility, infinity, the multiplicative axiom—whose status as logical truths is questionable. Nor did the issue arise only for abstruse propositions of the theory of types, as Ramsey’s famous example—‘Any two things differ in at least thirty ways’ (FoM, 4)—shows. In the ‘big book’ itself Russell did not so much address this problem as postpone it, but in October 1912 he tried again. ‘Logic is the study of the forms of complexes,’ he now suggested. ‘A complex is logical if it remains a complex whatever substitutions may be effected in it.’ (CPBR, VI, 55) The trouble with this characterization, though, was that it did not seem very different from the one he had attempted a decade earlier. ‘Wittgenstein has been here arguing logic,’ he wrote to Ottoline Morrell after he had been working on the problem for a couple of weeks (13 Oct. 1912). ‘It is difficult but I feel I must have another go at it.’ A couple of days later he gave up. There is no sign that he yet understood Frege’s insight that what is special about logic is not merely its generality. Logic is concerned with the predicate ‘true’ in a quite special way, . . . analogous to that in which physics has to do with the predicates ‘heavy’ and ‘warm’ or chemistry with the predicates ‘acid’ or ‘alkaline’. (PW , 128) Other sciences have truth as their goal; only logic has truth as its subject matter. Wittgenstein expressed this Fregean demand with elegant simplicity in a remarkable letter to Russell of June 1912. ‘Logic’, he said, ‘must turn out to be of a totally different kind than any other science.’ Yet if his motivation was Fregean, his solution was not. His recipe for guaranteeing to logic this special status was to insist that ‘there are no logical constants’, since only thus could it be ensured that logic has no subject matter. Wittgenstein’s ‘fundamental thought’ may thus be

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conceived of as based on an essentially Fregean insight while nonetheless opposing another of Frege’s doctrines. A formalization of logic has two parts, a language in which to express thoughts, and a system of derivation for deducing which thoughts so expressed are logical truths. In Chapter 51 we discussed how Wittgenstein dealt with the first of these. In the 6.1s he dealt summarily with the second, by insisting that he did not need it. The reason is that he had given a semantic account of when a propositional sign expresses tautology. This sufficed, he thought, since it provided, at least in principle, a method of characterizing the propositions of logic all at once. ‘It is possible, even in the old logic,’ he said, ‘to give at the outset a description of all “true” logical propositions.’ (6.125) By ‘the old logic’ is usually meant mediaeval logic prior to the rediscovery of Aristotle’s logic in the 13th century, but Wittgenstein here commandeered the phrase with casual chutzpah to refer to all logic before the Tractatus (i.e. including Frege’s Begriffsschrift). He hoped that what was possible but tricky in the old logic would be trivial in his: the formal properties of a sign would reveal themselves ‘by mere inspection’ (6.122). While he was in Norway in 1913–14, he spent some time trying to devise such an ‘adequate’, i.e. logically transparent, notation, using diagrams in which propositional signs were connected to ‘poles’ so as to exhibit their truth-conditions explicitly. His method works, somewhat creakily, for a finite truth-function, but he never got it to work properly in the infinite case. The reason is the undecidability theorem (Church 1936, Turing 1936), which shows that there is no mechanical method for determining whether an arbitrary propositional sign in a language capable of representing multiple generality expresses a tautology. Wittgenstein did not know this, but his failure in Norway to devise a transparent notation in which whether a sign expresses tautology would be immediately visible must at least have persuaded him that to ‘recognize the symbol in the sign’ (3.326) is a non-trivial task.

Logic as transcendental There was a stage in the book’s composition, before the rest of the 6s had been added, when it ended by claiming that ‘logic is transcendental’ (6.131). The Kantian resonance was no doubt deliberate. Kant distinguished between the transcendental—what is contained in the structure of experience—and the transcendent—what lies beyond experience. Logic, according to Wittgenstein, is therefore ‘before the How, not before the What’ (5.552). It is not a selfsubsistent structure into which the objects that make up the worldly facts must be fitted, but rather the structure which these objects instantiate. Every attempt to represent how things stand comes ‘with logic built into it’ (Nagel 1997, 61). Wittgenstein no doubt took himself here to be disagreeing with Russell, who had conceived of logic as having a subject matter of its own and hence as being in principle comprehensible independently of the world. ‘If there were a logic, even if there were no world, how then could there be a logic, since there is a world?’ (5.5521) An even more extreme later instance of the view Wittgenstein

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opposed was Carnap, whose Logical Syntax of Language presented a ‘logic-first’ view according to which any formal language has logic built into it from the start; the empirical vocabulary is whatever is left over once logic has done its work. Wittgenstein’s account was precisely the other way round: it cannot be logic that constrains the possibilities open to the world, because logic arises only once our scheme for representing these possibilities is in place. Commentators sometimes ask whether the Tractatus commits Wittgenstein to transcendental idealism. Before we answer this question, let us remind ourselves how transcendental idealism arises in the Critique. Kant distinguished two sorts of logic, formal and transcendental. Formal logic, he thought, is so trivial as to require no justification. Transcendental logic, by contrast, applies the categories— a priori concepts which underlie the most general conception we have of what an object is. The transcendental deduction is required in order to show that these concepts are genuinely applicable to experience. In the course of this deduction, however, we come to see these concepts as conditions that anything must satisfy if objectively valid judgments concerning it are to be possible. The categories thus function as constraints that something might—and that things-in-themselves do—fail to satisfy. Hence transcendental idealism. How, then, does Wittgenstein’s logic fare in comparison? On his account the generality of logic is secured by the generality of logical form—a generality which involves no categories and hence, he hoped, imposes no genuine constraints on the world’s structure. He offered no transcendental deduction of the applicability of logical form in representing the world, because he thought none was required. Remember, though, that before reaching the transcendental deduction of the categories Kant had already offered a route to transcendental idealism in the Aesthetic, namely that pure intuition presents space and time to us as structures prior to the particular points in space and moments in time that we experience as subject to these structures. Does the Tractatus allow anything like this second route to transcendental idealism? It does at least share with the Critique a conception of space and time as forms of object. Where it differs, though, is in rejecting Kant’s conception of these as prior to the objects that possess them. This explains the urgency with which Wittgenstein insisted against Russell that form is internal. Russell, by conceiving of logical types as containers, made it seem a coherent possibility that there might be objects (things-in-themselves, as one might call them) which belonged in none of them.

Further reading For further discussion see Moore (2013).

58 THE METAPHYSICAL SUBJECT

In the 1916 Tractatus Wittgenstein’s discussion of solipsism ended with a curt rejection of the ‘subject that thinks or entertains ideas’. Later, though, he added further remarks about the subject, drawn from notebook entries for August and early September 1916 and numbered so as to follow on from, and hence presumably form a kind of commentary on, the earlier rejection. The subject does not belong to the world, but it is a limit of the world. [2 Aug.] Where in the world is a metaphysical subject to be found? [4 Aug.] You say that this case is altogether like that of the eye and the field of sight. But you do not really see the eye. [4 Aug.] And from nothing in the field of sight can it be concluded that it is seen by an eye. [4 Aug.] For the field of sight has not a form like this: [12 Aug.]

Eye

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

g

This is connected with the fact that no part of our experience is also a priori. Everything we see could also be otherwise.

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Everything we can describe at all could also be otherwise. [12 Aug.] Here we see that solipsism strictly carried out coincides with pure realism. The I in solipsism shrinks to an extensionless point and there remains the reality co-ordinated with it. [2 Sep.] There is therefore really a sense in which in philosophy we can talk of a non-psychological I. [11 Aug.] The I occurs in philosophy through the fact that the ‘world is my world’. [12 Aug.] (5.632–5.641) It will be convenient to consider this difficult passage alongside one further remark which Wittgenstein added later. The philosophical I is not the man, not the human body, or the human soul of which psychology treats, but the metaphysical subject, the limit—not a part of the world. [2 Sep.] Our task in this chapter will be to understand the sense, if any, in which philosophy can talk of this ‘non-psychological’ or ‘philosophical’ I.

The eye and the visual field The world, according to Wittgenstein’s earlier discussion of solipsism, is my world. What this boils down to is that the components of language are multiply related by the representing relation to the components of the world. Wittgenstein now suggested that we should conceive of this multiple relation as somehow constituting a new, less substantial, ‘metaphysical’ subject. If this is not to be straightforwardly inconsistent with his earlier rejection of the thinking subject, however, it is of course vital that what was objectionable about the latter should not infect the former. There are places where Wittgenstein talks as though what matters is that the metaphysical subject is not in the world. ‘Where in the world’, he asks rhetorically, ‘is a metaphysical subject to be found?’ As I have already noted, however, it can hardly be the crucial innovation merely to locate the metaphysical subject outside the world. What matters, rather, is that it should not be conceived as a single entity and used, as various philosophers have sought to use it, as a binding agent to give unity to my representation of the world. That unity, on Wittgenstein’s account, is already supplied by the general form of proposition, which generates a single structure within which all my representations can be located. Wittgenstein seems to have regarded this conception of the world as seen from a single point as a sort of transcendental paralogism—a persistent illusion to which we are inevitably tempted. He attempted to make the putative error vivid by appealing to an analogy (already mentioned in his notebook in July 1916) between my relationship to the world and the eye’s to the visual field. The analogy is not original to Wittgenstein but occurs repeatedly in The World

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as Will and Representation, where Schopenhauer says, for instance, that ‘the I or ego is the dark point in consciousness, just as on the retina the precise point of entry of the optic nerve is blind, . . . and the eye sees everything except itself ’ (II, 491). The analogy is not, in any case, particularly surprising: philosophers have often resorted to visual metaphors—‘points of view’, ‘perspectives’, etc.— to discuss different ways of representing the world. What is important, rather, is the link the analogy suggests between conceiving of the self as an entity outside the world and conceiving of the world as limited. Wittgenstein’s point was that the picture of the eye and the visual field makes two complementary errors: it represents the visual field as circumscribed by a boundary; and it explicitly marks the field’s viewpoint. The analogy thus presents these two features as inextricably linked: we represent the visual field as limited only insofar as we represent it as seen by an eye. In the same way, the suggestion goes, I am presented with two complementary temptations, variant expressions of a single mistake: that of conceiving of myself as an entity; and that of seeing my world as merely part of something more all-encompassing. One might also draw an analogy with two ways of watching a play: the first is that of the theatre critic, aware of his place in the audience at a distance from the action taking place on stage, and hence aware of the limits which the proscenium arch imposes on the action and of the fact that the actors come out of character as soon as they step offstage; the second is that of the uncritical viewer who immerses himself in what is happening, oblivious that he is watching a play at all, and hence unaware of the stage’s limits or of any contrast between it and the wider world. Schopenhauer, indeed, deployed just this theatrical simile, although, rather oddly, he identified what I have been calling the ‘critical’ perspective with a child’s. In childhood, he suggested, ‘life looks like the scenery in a theatre, as you view it from a distance’, whereas in old age, so he thought, ‘it is like the same scenery when you come up quite close to it’ (1995a, ch. 5).

The a priori order ‘Everything we see could also be otherwise.’ Why remind us of this now? Was it not central to Wittgenstein’s atomism that what we see—facts, not things—is always capable of being otherwise? The point of reminding us now that ‘no part of our experience is also a priori’ was presumably to help us to interpret the analogy with the eye and the visual field correctly—to realize that the world does not have a form like the diagram he drew on 12th August any more than the visual field does. This is, he wanted to stress, a diagram of how our representation of the world is not. It would thus be a mistake to conceive of our relationship with the world as like that of the critic who sees what happens on stage as a consequence of contingent directorial decisions. If we adopted that attitude to the real world, we should correspondingly see as contingent what ought rather to be viewed as inevitable.

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All analogies are imperfect, of course. One might wonder, for instance, how far the analogy with lines of perspective can be carried. However, Wittgenstein proposed the alternative of regarding the self as a limit of the world. Of course, this cannot be taken too literally either: plainly there is an element of metaphor in calling the self a limit. The metaphor is presumably suggested in part by the analogy with the visual field already mentioned, and like that analogy it has two purposes: first, that we conceive of the self only to the extent that we conceive of the world as limited; second, that we may somehow become aware of the limits of the world even though they are not given to us directly in experience. The subject may thus be thought of as a limit (i.e. boundary) of the world, but it is also a limit in the slightly different sense that the incorrect conception comes in degrees: the closer to the action on stage the critic becomes, the less he is aware of its limitations. The correct conception, in which the critic disappears and all that is left is the action of the play, may be thought of as a limit in the sense in which the word is used in mathematics—a value to which the incorrect conception can be made to approximate arbitrarily closely. As Wittgenstein put it, ‘The self of solipsism shrinks to a point without extension, and there remains the reality co-ordinated with it.’ (5.64) Features of the mode of depiction correspond to structural features of what it depicts, so that limitations of the mode of depiction entail limitations of the world. As Wittgenstein recognized straight away, though, such limitations would problematically re-introduce the thinking subject. ‘In spite of this,’ he therefore had to remind himself, ‘it is true that I do not see the subject. It is true that the knowing subject is not in the world, that there is no knowing subject.’ (NB, 20 Oct. 1916) Why does the a priori structure of the world not contradict this rejection of a knowing subject? The explanation can only be that my world is not limited in the way that my visual field is limited. If I used a spatial language, for instance, I would be limited to describing what is spatial. For Wittgenstein’s denial of the thinking subject to survive, he had to insist that the mode of depiction is logical and hence maximally general.

Talking of the I ‘In philosophy we can talk of a non-psychological I.’ (5.641) This is a puzzling remark. In the notebook entry from which it is derived Wittgenstein said that there ‘can and must’ be such talk, but the surrounding text supplies at best a reason why there must, not an explanation of how there can. ‘The I makes its appearance in philosophy through the world’s being my world.’ How does this come about? When the solipsist says, ‘The world is my world,’ we are perhaps owed some sense of who is claiming ownership, but what Wittgenstein had said over the previous few days served only to emphasize the illusory nature of the self in question by reminding us of his earlier emphatic rejection of the thinking subject.

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Isn’t the thinking subject in the last resort mere superstition? . . . The thinking subject is surely mere illusion. . . . The I is not an object.. . . I objectively confront every object. But not the I. (4–11 Aug. 1916) The problem becomes even starker, indeed, when we recall that the world’s being my world is just what the solipsist cannot express. The ‘sense in which in philosophy we can talk of a non-psychological I’ is therefore far from straightforward, since whatever we say about it will strictly speaking be nonsense. Notice, though, that when he transferred the remark to Bodleianus, Wittgenstein positioned it next to the one from a couple of weeks later about how the I of solipsism ‘shrinks to an extensionless point’. Perhaps, then, the metaphor of the eye and the visual field suggests a way in which we could indeed talk about a non-psychological I if only the world were limited. Perhaps, that is to say, Wittgenstein now proposed to regard the correct conception as a limiting case of the incorrect one. What he meant by ‘talking of the I’ here was not a matter of what is ‘sayable’ in his technical sense. The only self that can be spoken about in my language is the (philosophically uninteresting) empirical one studied by psychology. The I makes its appearance in philosophy in the context of a claim—that the world is my world—which is paradigmatically unsayable. So the sense in which there can be talk in philosophy of the non-psychological I is at best indirect, namely that in the course of representing the world as I do, I make manifest that it is, in this weak sense, my world. ‘The subject—we want to say—does not drop out of the experience but is so much involved in it that it cannot be described.’ (PG, 156) The sense in which it is philosophy that talks of the I is therefore recognizably that in which Wittgenstein had already (4.112) described it as an activity of clarification (and in which, I have suggested, the Tractatus itself is not a work of philosophy). It serves to ‘limit the unthinkable from within’, and hence, he supposed, gradually to fill in the structure of my representation of the world, to understand what it is for the world to be mine, or—what he took to amount to the same thing—to come to an understanding of the metaphysical subject. Wittgenstein thus distinguished implicitly between two levels of enquiry. The study of the signs we humans actually use to represent what we want to express is a matter for psychology (if the signs are mental) or linguistics (if physical), but in either case it belongs to the natural sciences. What belongs to philosophy is the study of the structure of the symbols which these signs express—a study which on Wittgenstein’s view will show the nature of the metaphysical subject. The remarks on the metaphysical subject that we have been considering in this chapter were added to the Tractatus as part of a set of revisions—begun no earlier than the autumn of 1916, but the exact date is uncertain—which extended the book beyond the 6.1s so as to include sections on arithmetic, science, ethics and philosophy. In the Critique Kant held that these subjects all supply us with synthetic a priori knowledge. If this were right, it would refute Wittgenstein’s claim that the only necessity is logical necessity. Soon, therefore, Wittgenstein

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began to add remarks attempting to show that each of Kant’s apparent counterexamples is really nonsensical. We shall turn to these remarks in succeeding chapters.

Further reading The distinction between limits and limitations is discussed by Moore (2003). The large literature on the Kantian reading of the Tractatus goes back to Stenius (1960, ch. 12). Hacker (2013, ch. 2) makes a somewhat different attempt to establish that the Tractatus does not contain transcendental arguments in Kant’s sense.

59 ARITHMETIC

We have seen that Wittgenstein rejected Russell’s attempts to ground mathematics in logic via the theory of classes. The only part of the subject for which he offered any positive account was elementary arithmetic: he seems to have assumed (implausibly) that this was the only part of mathematics which makes contact with reality.

Numbers Wittgenstein’s account of arithmetic is inspired by the treatment of numbers in terms of numerically definite quantifiers that Frege considered (but rejected) in the Grundlagen. That treatment, when rewritten in Wittgenstein’s notation, involves defining ∃≥n x fx =df ∃0 x1 , . . . , xn (fx1 ∧ · · · ∧ fxn ) ∃n x fx =df ∃≥n x fx ∧ ∼∃≥n+1 x fx, from which can be derived an account of the use of numerals in counting things. For Wittgenstein’s purposes, however, this account is too specific: it cannot explain how numbers can also be used to count not just things but the terms of other formal series, such as the one we discussed earlier. aR1 b =df aRb aR2 b =df ∃x(aRx ∧ xRb) .. .

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Wittgenstein therefore generalized Frege’s treatment to make numerals indices of the repetition of any operation. If  is any operation and p is any proposition, he defined 0 (p) =df p ν+1 (p) =df (ν (p)) The numerals 0, 0 + 1, 0 + 1 + 1, . . . (normally abbreviated 0, 1, 2, . . . ) thus play the role, when occurring in exponent position, of coding the number of times the operation  has been applied to the proposition p. One might expect that Wittgenstein would then define the arithmetical functions by recursion, e.g. µ+0 (p) =df µ (p) µ+(ν+1) (p) =df µ+ν ((p)) Instead, however, he proposed more compact definitions of addition and multiplication that exploit the direct links between these functions and composition.1 µ+ν (p) =df µ (ν (p)) µν (p) =df (µ )ν (p) He did not say whether he preferred these merely because they are more compact or because he had some objection to the method of recursive definition. If the latter, it would of course make a significant difference to his system, since it is far from obvious how to give correspondingly compact definitions of other arithmetical functions such as exponentiation. With the definitions of addition and multiplication just given we can prove explicit numerical equations. For example, 2+2 (p) = 2 (2 (p)) = ((2 (p))) = (2+1 (p)) = (2+1)+1 (p) = 3+1 (p) = 4 (p). In general, an equation µ = ν is to be interpreted to mean that µ (p) = ν (p), i.e. that ‘µ (p) ≡ ν (p)’ is a tautology. For this to be extensionally correct, we must plainly mean to be asserting this for every operation . (If it were sufficient that it hold for some operation , the case ∼∼p = p would show that 2 = 0.) But we cannot achieve this effect by genuine quantification, because that would involve quantifying into the pseudo-proposition that ‘µ (p) ≡ ν (p)’ is a tautology, 1 More accurately, Wittgenstein stated the definition of multiplication and left the reader to work out the corresponding one for addition.

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So if the account is to come out right, we shall have to regard the letters ‘’ and ‘p’ as schematic. Wittgenstein thus treated each arithmetical equation as abbreviating not a tautology but a claim that all the signs of a certain form express tautologies. He evidently intended to stress the parallels between metalogic and arithmetic, both of which contain pseudo-propositions that do not themselves express thoughts about the world, but are used to license inferences between genuine propositions that do express thoughts. Indeed in real life a mathematical proposition is never what we want. Rather, we make use of mathematical propositions only in inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (6.211) Wittgenstein made his account of arithmetic subsidiary to his statement of the general form of proposition in order to indicate their common source. The general form of proposition is recursive: propositions are obtained from elementary propositions by iterations of the N-operation. Since all operations apply to propositions, they are ultimately expressible in terms of the N-operation. Numbers are merely a notation for expressing their repeated application. So our understanding of numbers in general is based ultimately on our understanding of the general form of proposition. None of this, of course, amounts to an explanation of the validity of mathematical induction. It does, however, link induction to the grasp of language as a whole that we have when we understand the general form of proposition. Wittgenstein saw this grasp as playing the same role in his theory that the ‘limitlessness in the progression of intuition’ (A25) plays in Kant’s. To the question whether we need intuition for the solution of mathematical problems it must be answered that language itself here supplies the necessary intuition. The process of calculation brings about just this intuition. (6.233)

Dependence on infinity Does Wittgenstein’s account ensure that the numbers are all distinct? If ‘µ’ and ‘ν’ are distinct numerals, can we conclude that µ 6= ν? In order to prove the corresponding result in their system, Whitehead and Russell needed to assume an axiom of infinity asserting that there are infinitely many individuals, but the axiom of infinity cannot play the same role in Wittgenstein’s account. The infinitude of the world (if it is indeed infinite) shows itself in the language we use to describe it and cannot be expressed by any one proposition in the language. Suppose for a moment that there are only finitely many elementary propositions. Then there are only finitely many propositional senses, and consequently the set of all operations, i.e. functions from propositional senses to propositional

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senses, is finite too. A straightforward group-theoretic argument then shows that there must then be numbers M, N with M < N such that for every operation  and every proposition p we have M (p) = N (p). But this is just what we said was meant by M = N. Consequently there are no natural numbers other than 0, 1, 2, . . . , N − 1. Not only that, but subtraction cannot be defined in Wittgenstein’s system; for if it could, we would have N −M = 0, and so N−M (p) = p for every operation  and proposition p; in particular, if we apply this in the case in which (p) = p0 for all p, where p0 is some fixed proposition, we would obtain p = p0 for all p, i.e. all propositions would have the same sense. Contradiction. Although reality is limited by how many elementary propositions there are, this is not a limit we can express in our language. The possibilities we can express are possibilities as to the ways the objects could be configured, not possibilities as to how many objects there are. So if the form of arithmetic is dependent on how many objects there are, that does not directly contradict Wittgenstein’s injunction that ‘the generality that is required in mathematics is not accidental generality’. The reason the dependence of arithmetic on the number of objects nevertheless seems embarrassing to his account is that there is plainly another sense in which it is possible that the number of elementary propositions should be different from what it is, namely that logic allows for a different number even if language as a whole does not. Language draws the limits of the world, the limits of what is possible. But we can divide our grasp of language into two parts, the elementary propositions on the one hand and the general form of proposition on the other, and corresponding to this division we can detect two levels of possibility that language as a whole allows for. ‘Mathematics’, Wittgenstein said, ‘is a logical method.’ (6.2) A consequence of this is that mathematics should not limit the world any more than logic does. If mathematics is to lie at the same level of generality as logic, it must be necessary not only with respect to the level of contingency that our language expresses, namely contingency as to which arrangements of objects into states of affairs actually occur, but also with respect to the higher level inexpressible in the language itself but represented by the general form of proposition, namely contingency as to what elementary propositions there are. So if Wittgenstein’s account of arithmetic was to be satisfactory, he did indeed need to show that our language is one that represents there to be infinitely many elementary propositions.

Further reading Frascolla (1997) interprets Wittgenstein’s account of arithmetic at the level of signs, not symbols, and hence as avoiding the dependence on the axiom of infinity of which I have accused it.

60 SCIENCE

The natural sciences, Wittgenstein said, consist in ‘the totality of true propositions’ (4.11). Indeed, since tautologies are not strictly speaking propositions, he might just as well have said ‘contingent truths’; these constitute, in effect, a complete description of the world. As an account of the practice of science, though, this would surely be inept: scientists do not simply accumulate facts: they formulate general laws expressing regularities. This leads to a difficulty. As Wittgenstein reminds us at the beginning of the section of the Tractatus on science, ‘Logical research means the investigation of all regularity. And outside of logic, all is accident.’ (6.3) Hence there are no regularities in the world of the non-logical sort that scientists purport to express. How so?

The mesh Wittgenstein’s account of science depends on the observation that scientists show no interest whatever in the sort of logical analysis that would be needed to determine the structure of the elementary propositions; nor do they seek to find out their truth values one by one. The language they use may not be quite that of everyday discourse, but they share the feature of operating at a level well above that of elementary propositions, and it is only at this level that scientists attempt to state laws—contingently true generalizations expressing regularities. Wittgenstein suggested that the language they use may show something about the world (6.341–2). He used the analogy of describing a black-and-white picture by laying a mesh over its surface. We obtain an approximate description of the picture by saying of each square in the mesh whether it is black or white (as is familiar nowadays in the process of pixilation). Which shape of mesh we use influences how accurate the representation is. Wittgenstein suggested (5.511) that

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the Tractarian language of elementary propositions corresponds to the case of an infinitely fine mesh. Perhaps this is an exaggeration, since the process of analysis comes to an end eventually, but it at least makes clear the contrast he intended between the atomistic language of elementary propositions and the macro-level language of science: the logical analysis of propositions is generally too complicated to be carried out. The elementary propositions are, as it were, invisible to the naked eye. So the scientific laws we actually adopt will attempt to approximate the microscopic reality at the macroscopic level. One example which makes salient the role played by the shape of the mesh is the law of induction, that the unobserved resembles the observed in some respect. What is delicate is determining the respect in which we claim resemblance. For instance, if we have repeatedly observed that objects of some kind have a certain colour, there is an instance of the law which says that any object of that kind can be expected to have the same colour. But whether this turns out to be true will depend on which colour language (which shape of mesh) we use. Anyone who applied the law of induction using Goodman’s alternative colour vocabulary— something is grue if it is green and first observed before time t0 , or if it is blue and first observed after t0 , and it is bleen if it is blue and first observed before t0 , or if it is green and first observed later—would be shown to be wrong soon after t0 , when their belief that all emeralds are grue turned out false. Something about the world is shown by the fact that some shapes of mesh are better than others—our ordinary colour language better than the gruesome alternative, for example.

Principles Laws are contingently true generalizations and therefore pose no threat to the claim that the only necessity is logical necessity. In science, however, there are also some claims—here I shall call them ‘principles’—that seem not to make contingent claims about the world. Wittgenstein’s explanation of these is that they are not really laws but forms of laws—organizing criteria for determining the shape of the generalizations we are willing to consider as putative laws. As an illustration, consider the principle of least action (more accurately, stationary action), first formulated by Maupertuis in the 18th century. This states that a system will evolve in such a way that its action is a stationary point, where what is meant by ‘action’ varies according to the case. This general principle is not itself a law, but various physical laws may be derived from it: in optics it gives us Fermat’s law, that light takes the quickest path between two points; when it is applied to the motion of a particle, it gives us Hamilton’s Law, that the path followed by the particle minimizes its Lagrangian (the difference between its kinetic and potential energy); in Hertz’s Principles of Mechanics the principle is used to derive a mechanical principle called the law of least curvature; the principle has also been fruitfully applied in relativity theory. On Wittgenstein’s account, least action is an organizing principle, but the laws we obtain from it are empirical generalizations,

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and hence falsifiable. The important point is that the principle does not dictate what the instance will be: that depends on the definition of action for the case in question. If we found a counterexample to some specific law derived from it, that would not falsify the principle but merely show that the relevant notion of action for the case in question was different. Another example is the principle of sufficient reason, which states that every event has a cause. This is not itself a law. (What would it be for an event to have no cause? At any rate, we could hardly expect to establish this.) There is, Wittgenstein held, no ‘causal nexus’ compelling one event to occur because another occurs. Among empirical generalizations, however, there are some that have a causal form, and the principle of sufficient reason expresses our preference for such laws. It ‘treat[s] of the network and not of what the network described’ (6.35). Nonetheless, Wittgenstein’s account of these principles obscures the essential question, which is why there are any true laws of the sorts they stipulate. If we scattered black marks randomly on a sheet of paper and then found when we laid a net over it that this revealed a pattern enabling us to give a neat finite description of the marks, that would surely deserve explanation. Yet in the scientific case it is not merely that Wittgenstein offers no explanation, but that his account seems to deny us the very tools we would need if we wanted to construct one.

The independence of the will ‘The world is independent of my will.’ (6.373) Wittgenstein presented this claim, towards the end of the section of the book on science, as if it were a consequence of the general claim that the only necessity is logical necessity. Since there is no ‘logical connection between will and world, which would guarantee’ that everything we wished should happen, it follows that there is no connection. The obvious rejoinder, though, is that even if there is no logical compulsion about the matter, it is a well-confirmed empirical generalization that some parts of the world (my limbs, for instance) do very often conform to my will. Wittgenstein’s reason for holding this to be irrelevant was in the end an ethical one. The independence claim is a transitional remark that looks forward to the 6.4s on ethics, where it emerges that my will does not belong to the world, and hence cannot participate in the sort of empirical generalization just envisaged. Wittgenstein positioned the claim in the Tractatus directly after a remark (drawn from his notebook for May 1916) about the modern ‘illusion’ that ‘the so-called laws of nature are the explanations of natural phenomena’. This juxtaposition invites us to note that in both cases we are tempted to seek explanations that go beyond the facts themselves. Once we recognize that ‘in the world everything happens as it does happen’, Wittgenstein hoped that we might give up the temptation to seek such explanations. Since there is no necessity except logical necessity, a ‘so-called law of nature’ is merely a way of uniting in a single form the description of a large

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number of contingent facts, and hence cannot be explanatory: where chance is concerned, there is nothing to explain. At this point Wittgenstein drew an interesting contrast between science and religion: the ancients, he suggested, invoked religion as a way of expressing that some phenomenon does not have a natural explanation, whereas we moderns are guilty of continuing to look for natural explanations even where none are to be had. Even if he were right to say that invoking religion is clearer than appealing to science, though, it certainly would not follow that it is more honest. The great theistic religions, although they present God as a single explanatory terminus, plainly do intend to offer at least a partial explanation of how things stand. Wittgenstein himself was uninterested in this conception of religion; but his real point, somewhat obscured by his reference to God or Fate, was that the most honest stance of all would be to agree that nothing counts as an explanation of why the world is as it is. That we should not expect religion to offer such an explanation is merely an instance of that general point. As a matter of history, though, it is surely dubious whether the ancients were really clearer about this than we are. Wittgenstein thought it significant that they tended to appeal to religious explanation at just the point where ordinary explanation came to an end, but it is a separate question whether their doing so constituted, as he claimed, an implicit recognition that no genuine explanation is possible. Nonetheless, this understanding of religion persisted in Wittgenstein’s later thinking. Schlick says that in theological ethics there used to be two interpretations of the essence of the good: according to the shallower interpretation the good is good because it is what God wants; according to the profounder interpretation God wants the good because it is good. I think that the first interpretation is the profounder one: what God commands, that is good. For it cuts off the way to any explanation ‘why’ it is good, while the second interpretation is the shallow, rationalistic one, which proceeds ‘as if ’ you could give reasons for what is good. The first conception says clearly that the essence of the good has nothing to do with facts and hence cannot be explained by any proposition. If there is any proposition expressing precisely what I mean, it is the proposition, ‘What God commands, that is good.’ (17 Dec. 1930, in Waismann 1979, 115) Wittgenstein thus continued to hold that any mention of God is a way of signalling that we have moved out of the space of reasons: to say that something is what God commands is to grant that it is good while cutting off any possibility of explaining why.

Further reading Proctor (1959) and McGuinness (1969) discuss Wittgenstein’s account of science in more detail.

61 ETHICS

The 6.4s on ethics are Wittgenstein’s response to a problem he posed in July 1916. What do I know about God and the purpose of life? I know that this world exists. That I am placed in it like my eye in its visual field. That something about it is problematic, which we call its meaning. This meaning does not lie in it but outside of it. That life is the world. That my will penetrates the world. That my will is good or evil. Therefore that good and evil are somehow connected with the meaning of the world. The meaning of life, i.e. the meaning of the world, we can call God. And connect with this the comparison of God to a father. To pray is to think about the meaning of life. It is worth recalling the context in which Wittgenstein wrote these remarks. The Brusilov Offensive, one of the bloodiest battles in human history, was launched in June 1916: throughout that month he was in the thick of the fighting and made no entries in his notebook at all. The passage just quoted was the first philosophical entry in his notebook when he resumed in July. The timing can hardly be coincidental. When he said in the Notes on Logic that ‘Philosophy consists of logic and metaphysics’, he evidently meant to exclude ethics. In the first part of the war he made ethical or religious remarks only on the left-hand (personal) page of his notebook. It thus amounted to a significant shift that he should now discuss the meaning (i.e. significance) of life on the right-hand (philosophical) page. ‘My

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work,’ he said, ‘has widened out from the foundations of logic to the nature of the world.’ (NB, 2 Aug 1916, amended)

Moral nonsense The 6.4s begin by claiming that ‘all propositions are of equal value’. Why? Wittgenstein took it as obvious that there can be no ethical propositions in the Tractarian sense; for if there were, their truth or falsity would be contingent, and this would be absurd, since value must transcend the happenstance of how objects are configured to form facts. ‘In the world everything is as it is and happens as it does happen. In it there is no value—and if there were, it would be of no value.’ (6.41) If, for instance, killing babies happened to be wrong, there would be other possible worlds in which it was not wrong, in which case the real issue would be which world had got its ethics right; the problem would merely have been pushed up a level. Remember, though, that the only categories Wittgenstein’s system allowed for were propositions with sense, tautologies and contradictions, and nonsense. Even if our claim about the wrongness of killing babies tells us nothing about how the objects are combined into atomic facts, it is plainly not a tautology—an empty limiting case of contentful propositions in the manner of ‘Either it’s raining or it isn’t’. So by a process of elimination it is nonsensical. ‘Propositions cannot express what is higher.’ (6.42) ‘Ethics cannot be expressed.’ (6.421) This is not quite enough, though, to give us the conclusion that facts have no value in themselves. Might their value not be an internal, and hence inexpressible, property of them. To rule this out, Wittgenstein considered the case of a lifeless world (NB, 2 Aug. 1916), in which he took it as obvious that no fact would have any value. But internal properties are independent of how things happen to stand. So no fact in the actual world has any value either. If Wittgenstein had been a moral nihilist, he could then have concluded that all propositions are of equal value, namely none; yet there is no evidence in the notebook that he ever even contemplated this response. Even before the war, when he denied that ethics is part of philosophy, this was a comment on the scope of philosophy, not a denial of the existence of value. The challenge, he thought, is to explain how, not whether, the world is ultimately valuable. In that case, though, why suppose that, however value is bestowed on the world, all propositions will end up being valued the same? Wittgenstein offered no argument for this, but seems to have been powerfully gripped by the notion that ethics contemplates the world as a whole rather than its individual parts. One might try to motivate this view by recalling the sort of things that constitute Tractarian facts. If a typical elementary proposition says that there is colour at some point in space at some moment in time, it is hard to see why my contemplation of the world, however conducted, might lead me to value one such proposition over another. Let us recall, though, the lesson we drew from our discussion of the law of induction

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in the last chapter, namely that a world which at the level of the elementary propositions has no discernible regularity might at another level of description (i.e. with the right shape of mesh) appear to be regular. In much the same way we might suspect that evaluative terms are emergent, i.e. they get a grip on the world only at higher level of description and are not practically reducible to elementary propositions. One plausible source for Wittgenstein’s view that individual facts are not in themselves good or bad is Russell. Good and bad . . . are the reflections of our own emotions on other things, not part of the substance of things as they are in themselves. And therefore an impartial contemplation, freed from all preoccupation with Self, will not judge things good or bad, although it is very easily combined with that feeling of universal love which leads the mystic to say that the whole world is good. (CPBR, VIII, 46) Russell here made the same slide as Wittgenstein from saying that good and bad ‘are not part of the substance of things as they are in themselves’ to the conclusion that ‘an impartial contemplation’ will withhold judgment from particular facts entirely.

The willing subject The problem Wittgenstein set himself in July 1916 was to explain how the world as a whole could have a value that does not reside in the individual facts that make it up. His solution, in outline, was that value lies not in the world itself but in how it is viewed. The world of the happy differs from that of the unhappy, although the facts are the same, because the happy view it in the right spirit. (Here the word glücklich might be better translated ‘fortunate’ or even ‘blessed’ instead of ‘happy’.) This is only an outline, however, and it immediately invites the further question, who is it who is doing the viewing? In the 1916 notebook Wittgenstein was at first tempted to answer this question by conceiving of another entity standing outside the world which he called the willing subject. But how does willing differ from mere wanting? He attempted to understand this by analogy with someone who is paralysed, but he made little headway because his claim that the world is independent of my will made the willing subject seem too like a passive spectator on the passing show. He therefore began instead to think of the will as accompanying worldly events so as to convert them into acts and hence give them moral significance. This led him to the view (in which more than a hint of Schopenhauer’s influence is detectable) that every worldly fact—the motion of distant stars just as much as of my hand as I write this sentence—is (unsayably) accompanied by willing, so as to give it moral significance. If we conceive of willing as thus giving value to worldly events, however, this cannot be in the ‘popular sense’, in which ‘there are things that I do, and other

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things not done by me’ (NB, 4 Nov. 1916). What Wittgenstein needed at this point was a distinction between two senses of ‘will’, empirical and metaphysical. Curiously, though, he drew this distinction explicitly only in the very last stage of revisions to the Tractatus, inserted by hand in the final typescript. ‘Of the will as the bearer of the ethical we cannot speak. And the will as a phenomenon is only of interest to psychology.’ (6.423) I have said that Wittgenstein was ‘at first’ tempted to explain value by means of the willing subject. It is controversial what role (if any) the willing subject still played in the finished Tractatus, where it is not mentioned explicitly at all. Some hold that he conceived of the metaphysical subject, somewhat obscurely, as playing this ethical role. If so, then he will have intended his metaphor of the self of solipsism shrinking to an extensionless point (5.64) to have an ethical as well as a semantic interpretation.

The lecture on ethics If all propositions are of equal value, it follows that there is no moral point to doing one thing rather than another. So ethics as Wittgenstein understood it concerns not what I do but how I do it. The good life consists not in doing one thing rather than another but in ensuring that whatever I do is done in the right spirit. Some commentators find this radical dissociation of ethics from facts implausible. Kelly (1995, 570), for instance, dismisses as ‘absurd’ the notion that empirical facts are irrelevant to ethics and infers that this could not really be what he meant. To understand how it could, a useful resource is the ‘Lecture on ethics’. This dates from November 1929, ten years after the Tractatus, but Wittgenstein had been away from philosophy for almost all that time, and his thinking was therefore still recognizably Tractarian. The one significant addition to what he said in the Tractatus was that he now made explicit a distinction between absolute and relative value that had previously only been implicit. An act has relative value if it is a good way to achieve some goal, but this is no reason to attribute absolute value to it unless we have reason to think that the goal is in turn absolutely valuable. Drinking water is a good way to assuage a thirst, but this does not make drinking absolutely good unless assuaging thirst is already absolutely good. Wittgenstein took it to be obvious that no considerations of relative value could ever lead us outside the circle of contingent facts so as to arrive at absolute value. Our feeling that statements of value are not nonsensical is to be diagnosed as a confusion between the two sorts. The claim that killing babies is wrong makes sense as a statement of relative value (and is obviously true relative to the criteria we all accept in making everyday moral judgments); but this gives us no reason to hold that it is wrong absolutely (i.e. in a way that transcends these contingent, everyday criteria). When a moral statement appears to make sense to us, Wittgenstein’s explanation would thus always be that what makes sense is really only some claim of relative, not absolute, value.

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This distinction between relative and absolute value is reminiscent of Kant’s distinction between hypothetical and categorical, but Wittgenstein pressed it more resolutely. Kant conceived of the categorical imperative as a tool guiding human action, whereas for Wittgenstein considerations of absolute value can never recommend one action over another. Another obvious precursor was Hume’s claim in the Treatise (bk III, pt 1, §1) that the dependence of an ‘ought’ on an ‘is’ is ‘altogether inconceivable’. Perhaps the most interesting, though, was Schopenhauer’s argument against Kantian ethics, that only by invoking God could we ever justify the notion of an absolute goal to the attainment of which our moral rules are subservient. Without such an—in Schopenhauer’s view illegitimate—appeal to theology, rules could only be relative to our interests and hence not genuinely moral. Instead, he claimed, the moral worth of an action should be assessed not by the end to which it is directed but by the motivation with which it was willed (1995b, 145). It is worth noting, though, how much further Wittgenstein took this line of thinking. Schopenhauer still supposed that something of an action’s motivation could be inferred from its external (i.e. empirical) character, whereas Wittgenstein held that there can be no route from the external character of the action to an assessment of its moral worth. On this view goodness is, as Levin says in Anna Karenina (pt 8, ch. 12), ‘beyond the chain of cause and effect’. To Wittgenstein, the inexpressibility of absolute value seemed wholly compelling. I at once see clearly, as it were in a flash of light, not only that no description that I can think of would do to describe what I mean by absolute value, but that I would reject every significant description that anybody could possibly suggest, ab initio, on the ground of its significance. That is to say, I see now that these nonsensical expressions were not nonsensical because I had not yet found the correct expressions, but that their nonsensicality was their very essence. . . . Ethics so far as it springs from the desire to say something about the ultimate meaning of life, the absolute good, the absolute valuable, can be no science. What it says does not add to our knowledge in any sense. (PO, 44) If we begin from the system of the Tractatus, perhaps this is right. What is less clear is whether it would seem so compelling if we took a different starting point—if, for instance, our description of the world made use of thick moral concepts from the outset.

Religion Wittgenstein delivered the ‘Lecture on ethics’ in November 1929 to the Heretics, a Cambridge society for the discussion of religion, and he made it clear in the lecture that he regarded religious language as just another vehicle for attempting to express our sense of value. In fact, this had long been his view. As his sister reported in 1917,

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Ludwig says religion and ethics are absolutely inseparable. . . . In Ludwig’s opinion anyone who poses . . . the question about the meaning and purpose of life comes to concepts such as God, godly, etc. (In Prokop 2003, 94) If Wittgenstein took religious language to be just another way of talking about ethics, that may explain why he said so little about it in the Tractatus: he simply left us to perform the translation between ethics and religion for ourselves. For instance, his reasons for holding that the world is independent of my will would show just as well that it is independent of God’s will too. ‘How the world is, is completely indifferent for what is higher. God does not reveal himself in the world.’ (6.432) God cannot by an exercise of will bring about a change to the worldly facts—a miracle, for instance—any more than I can (and anyway if all propositions are of equal value, He has no reason to try). It need hardly be said that this is not the conception of religion espoused by traditional Western Christianity, which emphasizes God’s intervention in the world and makes at least one miracle, the bodily resurrection of Christ, crucial to faith. Wittgenstein’s own upbringing was loosely Catholic—he was confirmed as an adolescent—but a kind of religious fatalism is a common theme of much of his reading while he was composing the Tractatus. For instance, he bought a copy of Tolstoy’s The Gospels in Brief in Tarnov early in the war—it was, he claimed, the only book in the shop—and read it repeatedly thereafter. Not only is this a notably fatalistic retelling of the gospels, but it strikingly omits all the miracles. This is not, Tolstoy explained, because they did not occur, but because he considered them irrelevant: they have usually been assigned the epistemological role of providing us with evidence of God’s existence, but they fail in this, he thought, because those who do not already believe in God will not believe they occurred. It is surely a rather small step from there to wondering whether any religious role remains for them.

Further reading Levy (2009) usefully fills out what Wittgenstein’s view of ethics might amount to. On the relation between ‘is’ and ‘ought’ see Atkinson & Montefiore (1958). Garver (1971) suggests a Spinozistic account of Wittgenstein’s religious views. Williams (1955) discusses whether anything worthwhile in religion might remain once God’s intervention in the world is abandoned. On the willing subject in the Tractatus see Tejedor (2013).

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Even if all factual questions were answered, Wittgenstein said, we would still have a feeling that there was another question, not precisely formulable in our language, that had still not been answered (6.52). There is an echo here of Boltzmann, who spoke in one of his popular lectures of ‘the feeling that it is an insoluble riddle how I am alive at all, that a world can exist, and why it is just thus and not otherwise’ (1905, 343). From this Wittgenstein drew the moral that philosophy, to the extent that it has pretensions to solve this riddle rather than to clarify our thinking, is nonsensical and should be abandoned.

The riddle In Chapter 55 we discussed the conception articulated in the 4.1s of the 1916 Tractatus of philosophy’s role in determining the range of possible thoughts. What is new in the 6.5s is the (patently inaustere) affirmation that there is something that lies outside this range. Here, much more than in the earlier parts of the book, Wittgenstein deliberately courted paradox by encouraging us to entertain a view that he has previously dismissed as nonsensical. In affirming the existence of the mystical, Wittgenstein identified it with the nonsensical; but he also identified it with one particular instance of the nonsensical, namely the existence of the world, which he took to be the paradigm of the miraculous. (In the ‘Lecture on ethics’ he again referred to this as a miracle.) The riddle may thus be posed in various ways: as the problem of the meaning of life; of the existence of the world; or, more generally, of understanding what is strictly nonsensical. Wittgenstein called our awareness of the riddle, under any of its forms, the ‘mystical feeling’.

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The religious resonance of the word ‘mystical’ was no doubt deliberate. Although the world’s existence is not in itself an ethical matter, it can somehow be converted into one by the attitude we take towards it. Even if we grant a connection between the semantic inexpressible and the moral, however, it is far from easy to say what it is. We cannot, after all, treat the connection as inevitable, given that the world in itself has no value. Our account must leave room, that is to say, for finding the world valuable to be a task. To see the world as a limited whole is, Wittgenstein said, to see it ‘from an eternal perspective’. In the 1916 notebook he used Spinoza’s phrase sub specie aeternitatis, which he probably picked up either from his reading of Schopenhauer or in conversation with Russell, who much admired Spinoza’s ethical writings and later used the same phrase in his own writing (CPBR, XII, 102). In the notebook Wittgenstein equated viewing the world in this way with the good life (NB, 7 Oct. 1916), and most commentators therefore assume without further ado that in the Tractatus he was similarly recommending the perspective of eternity as something towards which we should strive. Perhaps it should give us pause, though, that in the Tractatus he altered the phrase to sub specie aeterni (changing the genitive noun into an adjective). This slightly different phrase originated with Hegel, who used it to refer to an abstract way of thinking that aspires to objectivity; but Wittgenstein most likely picked it up from Kierkegaard, who subjected Hegel’s use of it to relentless criticism in the Concluding Unscientific Postscript (1978–98, XII.1, 81, 171, 192, 217, 227, 301, 306, 307, 308, 309, 329, 362, 533), and insisted that the perspective it picks out is the prerogative of God alone. It is therefore worth considering whether perhaps in the Tractatus Wittgenstein was advising against seeing things sub specie aeterni. This would accord much better not just with Kierkegaard but with other possible influences, which emphasize the virtue of living in the moment, rather than timelessly. When Levin helps his workers bring in the harvest, his scythe cuts better the less he thinks about it. Tolstoy recommends a life wholly embedded in the world, in deliberate contrast with the frustration Levin feels when he attempts to engage in abstract argument—sub specie aeterni, perhaps—with his half-brother Koznyshev, the noted intellectual. That Kierkegaard and Tolstoy disavowed a view is, of course, insufficient in itself to prove that Wittgenstein did so too; it is surely hard, though, to see how he could have been straightforwardly recommending us to adopt a perspective on the world sub specie aeterni, since on his view there is no such thing: if there were such a perspective, it would be one from which one could say ‘this and this there is in the world, that there is not’ (5.61), and from which, therefore, the riddle would be answerable after all. Perhaps, then, the view sub specie aeterni and the role of the willing subject in enabling it are best thought of once again as transcendental illusions to which we inevitably succumb but which we should nonetheless strive, however futilely, to reject.

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Quietism ‘Whereof one cannot speak thereof one must be silent.’ Commentators have sometimes treated the famous closing remark of the Tractatus as if it had philosophy as its only target, but why suppose that he intended it to apply any less to nonphilosophical attempts to express the inexpressible—to religious discourse, for example? The quietism which the remark recommends is not the same as austerity: its appeal is most obvious in relation to the austere conception of nonsense, perhaps, but even someone who conceived of Tractarian nonsense as an attempt to gesture inausterely towards an insight tantalizingly out of reach of precise enunciation might hold that the correct response is to abandon the attempt. Only Russell’s extreme inausterity unarguably entails that this response is too pessimistic, since it holds out the prospect that once our language has been enlarged appropriately, the attempt to say what is currently unsayable may be successful. The connection between quietism and austerity is rather that quietism is an alternative response to the tension which the dispute between austere and inaustere accounts generates: if we give up the desire to answer the question, perhaps we may also give up the desire to delineate the contours of what we previously thought we wanted to know. If it is to be satisfying, however, quietism must also be therapeutic: it cannot rest content with showing us that the question was not well-posed, and thereby convincing us to abandon the attempt to find an answer; it must also diagnose what drove us to ask the question in the first place, since only then will mere silence seem an adequate response. That is not to deny that there may be cases where the former leads to the latter—in the process of realizing that a question is nonsensical I may come to understand the delusion I was under when I asked it—but it does suggest that this is by no means inevitable: as Kant noted, some delusions are so persistent that recognizing them as delusions is not enough to immunize us from them. The notion that there is a sort of question we cannot even ask is certainly not original to the Tractatus. At its weakest, it may amount to no more than a sense of the inexpressibility of certain emotions. In Ernst und Falk, for instance, Lessing says that there is a secret at the heart of freemasonry which one can know but not say. In War and Peace Natasha feels that what she and Princess Maria have lived through at Prince Andrew’s deathbed ‘could not be expressed in words’. At the end of Anna Karenina Levin is about to try to tell his wife about the religious insight he has just reached but pulls back at the last moment, realizing that it is ‘not to be put into words’ (pt 8, ch. 19, 853): is this because verbalization would cheapen the insight or because Levin fears it would evaporate in the attempt? It is a commonplace of Buddhism that someone who had attained nirvana could not be expected to express the insight thus acquired. The Old Testament instructs us, ‘Let thy words be few.’ (Ecclesiastes, 5:2) The New Testament tells us that the peace of God ‘passeth all understanding’ (Philippians, 4:7). And Engelmann in his memoir emphasized Wittgenstein’s ‘wordless faith’ (1967, 135).

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The reason we cannot say what can only be shown is not that we lack the words, but that what we are trying to express does not have the form of a proposition—it is not, properly speaking, a thought—and so adding more words to our language will not help. On any remotely plausible conception of Tractarian nonsense, therefore, the fact that most of the book is by its own lights nonsense poses a ‘reflexive paradox’—as Wittgenstein eventually acknowledged in 6.54. The metaphor of the ladder which he there deployed is not original with him: he could have found it in Schopenhauer (1966, II, 80), Mauthner (1901-2, I, 2), or Sextus Empiricus. Nor is the idea of discarding the means after we have achieved the end: the founder of Zen Buddhism, for instance, spoke of ‘throwing away the raft when it has taken one to the further shore’. (Note, incidentally, the stress Buddhism places on the journey towards enlightenment, rather than on reaching the destination.) The interpretation of 6.54 is hotly contested by commentators. Some debate whether ‘überwinden’ is best translated ‘surmount’ (Ogden–Ramsey), ‘transcend’ (Pears–McGuinness), or ‘overcome’ (Goldfarb 2011). Others worry that it was a mistake for Wittgenstein to say that propositions can be nonsensical at all: should he not more correctly have said that his ‘propositional signs’ were elucidatory? Perhaps it is the brevity of 6.54 that has made it such an appealing subject for scholarly attention: the textual evidence it supplies is slight enough to give commentators’ interpretative imaginations free rein. For Wittgenstein the purpose of the ladder metaphor was no doubt partly ethical. He did, after all, tell a friend that the point of the book was ethical. To Russell, though, he stressed the semantic aspect, and it is in relation to this that the reflexive paradox is most troubling.

The dialectical reading Some commentators respond to the paradox by, in effect, making light of it— urging the reader to leave it unresolved. I shall call this the ‘dialectical’ reading, since it places the reader, once the ladder has been discarded, in a state of perpetual dialectical instability, leaving ‘inconsistency at the heart of the Tractatus’ (Williams 2004). This reading cuts across the battle line between austere and inaustere conceptions: one could in principle adopt it whatever one’s conception of nonsense; at most, it might make the book seem more or less paradoxical depending on how austerely nonsensical we take its sentences to be. The particular paradox of a book saying that it is itself nonsense is one that Wittgenstein evidently enjoyed. Moreover, some of the authors he read celebrated paradox. Schopenhauer, for instance, insisted that his philosophy had a contradiction at its heart, and Kierkegaard (1985, 82) made great play with the ‘monstrous paradox’ of religious belief, whereby Christian faith is not merely unsupported by, but actively repugnant to, reason. In another presentiment of Wittgenstein, he regarded silence as paradoxical when it is a response to an ethical dilemma

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(1985, 114). Even Russell, so concerned to eliminate paradox from his logical system, sometimes embraced it willingly in his more popular writings: consider the relish with which he characterized mathematics as ‘the subject in which we never know what we are talking about, nor whether what we are saying is true’ (CPBR, III, 366). Note, however, that although Wittgenstein advised his readers to throw away his propositions after recognizing them as nonsensical, he did not, in contrast to Kierkegaard, explicitly revoke them. Indeed, when Kierkegaard (or rather his alter ego, Johannes Climacus) revoked his propositions at the end of the Concluding Unscientific Postscript, he observed that saying something and then revoking it is not the same as never saying it at all. One of the aims of the Tractatus, plainly, is to persuade us that some things we previously took to make sense do not. The dialectical reading is correct to note that there is something inherently unstable about this. Perhaps we should expect moments of confusion en route to clarification. In that case, though, Wittgenstein’s metaphor seems inept because it presents throwing away the ladder as a one-off affair. (Mauthner, by contrast, described a gradual process in which I demolish the ladder rung-by-rung as I climb.) It would be a very obtuse reader who did not feel the ladder beginning to give way long before 6.54. There is thus some truth in the dialectical reading as a description of how we do in fact respond to the text. If, after reading it to the end, we start again from the beginning, we are once again tempted to see before us a series of contentful claims, assessable for truth or falsity, about how things are: that the world is the totality of fact, not of things, for instance; or that propositions share their form with the part of the world that makes them true or false. This need not in itself be of any great significance, however, unless our concern is with explaining the psychological attraction of metaphysics. More interesting is the normative question of whether a stable, ladder-free conception of the claims apparently advanced in the book is possible at all, despite our psychological tendency to lapse into dialectical instability. In the end, though, the flaw in the dialectical reading is that it is hard to understand as anything other than a comment on our psychological tendencies. If it is right that certain claims made in the early part of the book are nonsensical, it cannot also be right that they make sense, unless they are somehow systematically ambiguous. Yet where is the ambiguity?

The resolute reading The project of working out the consequences of the reflexive paradox given an austere conception of nonsense has become known as the resolute reading of the Tractatus. The 1916 version of the book contains no hint of this project, but presents itself straightforwardly as proposing solutions to various problems in metaphysics and logic. Resolute readers typically claim, therefore, at most that Wittgenstein was resolute by the time he finished writing the book. They are consequently unimpressed by evidence that he later said things of the form, ‘I

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used to think that p’, where p is a claim in the Tractatus: he did indeed think that p while he was writing the book, they say, but he had rejected it as nonsensical by the time he finished. This interpretation is perhaps a little strained, though, because some of the remarks about what he used to think were recorded in the 1930s by his Cambridge students, who were trying to understand the Tractatus but presumably did not know the details of how it was composed, and therefore had no way to guess that he was, somewhat oddly, telling them only about his views in 1916 and not about how to understand the published book. According to the resolute reading, the finished book is so far distant from the 1916 Tractatus that it seeks not to endorse, but to exhibit the incoherence of, the picture theory which was the earlier version’s central proposal. The resolute reading therefore has the following striking feature. In order for us to accept the austere conception, we needed an argument that Tractarian nonsense is ordinary nonsense, but any such argument will at once be applicable to the passage in the Tractatus where the technical notion is defined, whereupon we realize that this passage is not merely technical but ordinary nonsense. How, in that case, are we now to understand it? At its most radical, the resolute reading concludes that, since the theory advanced in the book will turn out to be nonsense, it does not much matter what the theory is. Conant and Diamond, for instance, tell us that it makes little difference whether the account given of the supposed theory be one that rests on an appeal to verifiability, bipolarity, logical syntax, or some other putative respect in which ‘philosophical propositions’ are to be identified as nonsensical because of having been put together in some special kind of logically or conceptually illegitimate manner. All such accounts will qualify equally as instances of an irresolute reading, if they are committed to ascribing to the Tractatus a theory which its author must endorse and rely upon (if he is to be able to prosecute his program of philosophical critique) and yet which he must also regard as nonsense (if he thinks through the commitments of his own theory). (2004, 48) On their view, therefore, the book’s rejection of semantic theory as nonsensical is largely independent of the form which that theory takes. This might be plausible, though, only if there were a route to the self-applicability of the notion of nonsense that did not depend on the details of the account. I am not aware of any such route. Wittgenstein’s argument depends crucially, for instance, on the assumption that our language uses the logical mode of depiction: only then will there be a guarantee that what is inexpressible in it is absolutely, not just relatively, inexpressible. Goldfarb (1997) has argued, less radically, that Wittgenstein needed to develop his account of sense in careful detail in order to show that it is the best that can be done. Only when this account undermines itself fatally do we come to realize that there can be no such thing. Even in this less radical version, therefore, all

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that is left once we have thrown away the ladder is a set of linguistic practices which we previously conceived of as making sense, but which now we simply get on with. The ‘right method of philosophy’ is ‘whenever someone wished to say something metaphysical, to demonstrate to him that he had given no meaning to certain signs in his propositions’ (6.53). The tools available to us for achieving this aim will be strictly limited, however. There will be little we can do except to join the aspiring metaphysician in his mistaken enterprise pro tempore in the hope that we can thereby encourage him to give it up. The continuity between this and Wittgenstein’s later philosophy is of course striking, but so is the perplexity to which it gives rise. If someone declines to give up using certain signs, it is hard to see how, without the ladder, we can convict him of error, since now we cannot sensibly accuse him of failing to make sense. This is not the place to discuss how resolute the later Wittgenstein was in his scepticism towards meaning. We need only note how hard it is to keep any grip on the Tractarian metaphysics once we have resolutely discarded the framework used to present it. Why, though, is the Tractatus even as long as it is? Once the picture theory is in place, are we not in a position to discard the ladder straightaway? The much slower ascent of it which Wittgenstein’s text imposes on us seems quite unnecessary. Whatever justification the 6s of the Tractatus may have as an aid to living with the nonsensicality of so much that we previously took to make sense, no such justification could be offered for the solutions to an assortment of philosophical problems that he chose to give along the way. Why, for instance, did he bother giving a solution to Kant’s problem of incongruent counterparts (a solution which, it is worth noting, he did not add until page 99 of Bodleianus, some pages after recommending us to throw away the ladder)? If clearing up such puzzles does not contribute to the aim of showing that they are nonsensical, it is hard to see what point is served, on the resolute conception, by clearing them up at all. When we come to realize that the problems are nonsense, we shall realize that their solutions are nonsense too, and the satisfaction we briefly felt at having cleared them up will sound distinctly hollow. It hardly helps at this point to mention once again the Buddhist trope of the journey as an ineliminable part of the process of enlightenment. The inclusion in the book of answers to particular philosophical questions is explicable by the resolute reader only if working through these answers might be a useful part of the therapy—might help us abandon the temptation to ask the questions in the first place—but why think that? The resolute therapist should be concerned not with the details of the beliefs the patient will be giving up, but only with the features that show them to be nonsensical. Perhaps Wittgenstein included in the Tractatus remarks of which he was fond, even though they no longer quite expressed his view, but the resolute reading ascribes to him a nostalgic attachment to his earlier sayings that borders on the manic. Any attempt to describe the structure of the world is, by the lights of the Tractatus, nonsensical. On an inaustere understanding, this nonsensicality shows

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that the world towards which our thinking is directed has a structure which, although determinate, is systematically elusive, so that our attempts to put its features into words inevitably fail. Some resolute readers argue, by contrast, that the world has nothing that can count as a structure at all. Since an unstructured world could not play the role demanded of it by the realist, of being that in virtue of which our thoughts are true or false, realism drops away as untenable: resolution leads to anti-realism when strictly thought through. This line of reasoning is unconvincing, however: resolution leads no more directly to anti-realism in any of its usual senses than to realism, since thought mirrors the world, and so its structure is no less elusive. To the extent that resolution leads us to abandon the notion of a determinately structured world, it should lead us to abandon such a notion of thought as well (see Sullivan 2002). This position certainly merits the epithet ‘resolute’, since if we jettison both thought and world, all that is left to talk about is language, but is it even coherent? If prior to throwing thought overboard we had succeeded in shaping a logically perfect language to act as its proxy, matters would perhaps not be altogether hopeless. The strictures of reflexivity do not apply to the syntax of language, and the argument just considered, however resolutely understood, would therefore give us no reason to question the determinacy of that. Even if by the time he finished the book Wittgenstein had not abandoned such a goal, he certainly had not yet achieved it. So what this version of resolution leaves us with is not a logically perfect language but its imperfect, everyday counterpart. It thus makes the author of the Tractatus a proponent not of anti-realism but of ordinary language philosophy. The difficulty with this interpretation (apart from its historical implausibility) is that it renders opaque what is now to guide our critique, if not the world or a logically perfect language.

The frame To explain how to read the book, we might try dividing it into two parts. The outer part, sometimes called the ‘frame’ (Conant 1992, Diamond 2000, 159) consists of (senseful) instructions for reading the inner (nonsensical) part. It is not altogether clear, however, which sentences belong to which part. We can perhaps agree that the frame includes the instructions to the reader to throw away the ladder once he has climbed up it—at least if we ignore Conant’s puzzling suggestion (1989, 337) that we should throw this remark away as well. What, though, of the passages in the main body of the text in which Wittgenstein tells us that philosophy ‘is not one of the natural sciences’ (4.111); that it ‘is not a theory but an activity’; that it ‘should make clear and delimit sharply the thoughts which otherwise are, as it were, opaque and blurred’ (4.112); and that it is full of ‘the most fundamental confusions’ (3.324)? Should these be thrown away or not? Conant has even gone as far as to speculate that it is up to us to determine the scope of the frame for ourselves according to our own purposes in reading the book.

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There can be no fixed answer to the question what kind of work a given remark within the text accomplishes. It will depend on the kind of sense a reader of the text will (be tempted to) make of it. (2000, n. 102) Proops (2001, 380) has argued that it is a weakness in Conant’s reading if the contours of the frame are uncertain. This is because Conant has stressed the frame’s role as a set of instructions for reading the book. If he has to grant that Wittgenstein left those instructions wholly unclear—left it unclear, indeed, even which sentences constitute the instructions—he makes Wittgenstein seem implausibly clumsy. Suppose, though, that we focus not on the author’s intentions but on the book’s philosophical coherence. Even then, the extent of the frame remains problematic, because the notion of sense is one of those that according to the picture theory is inexpressible. In Chapter 49 we decomposed the task of making sense into two stages: first, discern a propositional sign, i.e. a syntactic fact consisting of some arrangement of signs; second, connect the components of this syntactic fact to components of the world in such a way that the whole sign becomes a symbol. Nonsense arises, Wittgenstein held, when the second stage fails: if a possible proposition has no sense, ‘this can only be because we have given no meaning to some of its constituent parts’ (5.4733). The first stage of the task is expressible—grammar books state the rules for arranging words to form grammatically correct signs—but the second is not. The most we can do is to say whether or not the sign has a meaning, much as someone who knows no German might be able to report which of the words in a purported German sentence have meaning by looking them up in a word list (such as the one Wittgenstein himself later published as a spelling aid for schoolchildren). According to the resolute reading, therefore, if I report that some sentence is or is not senseful, I must be doing no more than reporting whether or not the words it contains occur on some list. But whose? If we compiled the list before reading the Tractatus, while in the grip of a mistaken conception of meaning, it will have to be thrown away as part of the ladder; but no criterion for drawing up a new list is available. This difficulty is not so severe for irresolute readers, since although they cannot express the notion of sense any more than resolute readers can, they nonetheless believe that there is (unsayably) such a notion and that we can recognize whether an expression makes sense or not.

The preface If, as some contend, Wittgenstein became resolute only late in the composition of the book, his preface will be an especially relevant text, since its position at the end of Bodleianus suggests that he wrote it not long before he had the final typescript prepared in the summer of 1918. There are undeniably sentences in the preface which seem irresolute. The book, Wittgenstein tells us, ‘will perhaps only be understood by those who have themselves already thought the thoughts that are expressed in it—or similar thoughts’. In it, moreover, ‘thoughts are expressed’

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whose ‘truth’ (the emphasis is his) seems to him to be ‘unassailable and definitive’. He might of course have meant merely that those sentences that express thoughts (namely those in the frame) are true, but such a weak claim would surely be starkly at odds with the manifest chutzpah of the rest of the preface. It is surely compelling to suppose that he was here referring not just to the frame but to the bulk of the text. Resolute readers offer various responses to this: Kremer suggests that when in the preface Wittgenstein used the word ‘truth’ he meant it in a Biblical sense to mean a ‘way to be followed’ (2001, 61), whereas Floyd suggests that the preface is ‘ironic’ (1998, 87) and Wittgenstein is ‘seducing us’ into reading his remarks irresolutely in order that the eventual hijacking will be all the more shocking when it comes. Both suggestions strike me as implausible: the first, because it stretches to breaking point the words Wittgenstein actually used; the second, because the irony Floyd detects is so nearly invisible in the text. In support of her reading one might, it is true, try to invoke Kierkegaard, who so often advanced claims in an evidently ironic manner with the intention of deconstructing them from within. Yet the parallel seems lame: Kierkegaard elaborately signposted his intention to play with the notion of authorship, deliberately giving his writings ‘the appearance of chance and caprice’ (1967–78, V, 320) so as to deceive the casual reader, since ‘only the difficult inspires the noble-hearted’ (1967–78, I, 303); he published under pseudonyms; and he expatiated at length in the Concluding Unscientific Postscript on the authorial trick he had played, for the benefit of anyone who had still not got the joke (such as it is). Wittgenstein did none of these things, and the discomfort he later expressed about Kierkegaard’s habit of teasing his reader should surely warn us further against detecting irony in the Tractatus (see Schönbaumsfeld 2007, 27). ‘There is something in me that condemns this teasing,’ he wrote. ‘The idea that someone uses a trick to get me to do something is unpleasant.’ (PPO, 131) It is worth noting, too, that Kierkegaard intended his pseudonymous authorship merely to postpone, not permanently to suppress, interest in his biography as an interpretative key. ‘The time will come,’ he confidently announced (1967–78, V, 419), ‘when not only my writings but my whole life, the intriguing secret of the whole machinery, will be studied and studied.’ To Wittgenstein, on the other hand, such a notion was plainly abhorrent.

Further reading The best discussion of Wittgenstein’s mysticism is by McGuinness (1966). For an introduction to the resolute reading see Goldfarb (1997) and White (2011). The literature on the resolute reading is now large. For a representative sample see the collections edited by Crary & Read (2000) and McCarthy & Stidd (2001). The reading was largely originated by Diamond (1991), who remains its most consistently interesting proponent. Evidence of Wittgenstein’s own irresolution is marshalled by Hacker (2000, 2001) and Proops (2001).

63 THE LEGACY OF THE TRACTATUS

The extent to which the Tractatus influenced the logical positivists has often been over-stated. They certainly read it carefully, discussing it line by line at meetings of the Circle, and Schlick in particular regarded it as a kind of bible. Yet their distinctive view that Kant’s synthetic a priori judgments are really linguistic conventions stemmed not from the Tractatus but from Schlick’s Allgemeine Erkenntnislehre (1918), and their hostility to metaphysics owed more to Neurath than to Wittgenstein. When they used the verification principle to justify this hostility by arguing that the claims of metaphysics (along with those of theology and ethics) are meaningless, because nothing would count as a verification or a refutation of them (see Carnap 1932), their source cannot have been the Tractatus, from which this principle is absent. Moreover, they treated the question of sayability more formally than did Wittgenstein, and therefore tended to regard Tarski’s recursive definition of truth as a sort of refutation of the saying/showing distinction. One reason that Wittgenstein did not influence the Vienna Circle more was their irreconcilable difference in attitude. ‘All of us in the Circle’, Carnap recalled, ‘had a lively interest in science and mathematics. In contrast to this, Wittgenstein seemed to look upon these fields with an attitude of indifference and sometimes even with contempt.’ (1963, 29) Wittgenstein had no sympathy with their scientific approach to philosophy. ‘Philosophers’, he warned, ‘are irresistibly tempted to ask and answer in the way science does. This tendency . . . leads the philosopher into complete darkness.’ (BB, 18) Another factor limiting his influence, though, was more practical: his ideas soon began to change, and the positivists risked his contempt if they made use of the old ones he now regarded as mistaken; but he resented any attempt to use the new ones before he had published them himself (which he never did). In 1932, for instance, he complained to Schlick that Carnap had plagiarized him by hypothesizing that

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everything we say about the world is in principle translatable into a physicalist language. Carnap was never forgiven. In Britain, too, the influence of the Tractatus was soon limited by the growing suspicion that its author was in the process of disowning it. Copies of the Blue and Brown Books began to circulate in the late 1930s, and the publication of the Philosophical Investigations in 1953 led many to regard the Tractatus as obsolete. In fact, however, the popular conception of two Wittgensteins was always too simplistic—he did, after all, originally ask that the two books should appear together in a single volume—and the most important strand of influence of the earlier book, on which I shall here focus, was on its author’s own later writings.

The independence of elementary propositions We earlier counted it a virtue of the picture theory that the possibilities of combination for the toy cars in the picture coincide with those for the real cars on the road. It seemed then to be merely a defect of ordinary language that it should allow us to express arrangements which the structure of space prevents. Perhaps, though, we should not have dismissed this so casually. The spatial mode of depiction allowed us to show, but not to say, that there could never be an experience of the other man’s car being both in front of and behind mine; when we switch to the logical mode, on the other hand, this a priori insight, which in the spatial mode of depiction was unsayable, seems to become a sayable law of geometry. Similar issues arise with the a priori insight concerning time that if t is after t0 and t0 is after t00 , then t is after t00 ; or with the insight concerning colour that a point in the visual field cannot be both red and green. Wittgenstein reflected on all these examples in the autumn of 1916. On 12 October 1916, for instance, he insisted that ‘having only one direction is a logical property of time’, but the reason he gave was only that it is in the logical nature of an event that it cannot be repeated: he did not offer an analysis that demonstrated why this should be so. Similarly, he considered the colour exclusion problem on 16 August 1916, and again on 8 January 1917. His solution was to grant that it is sayable that a point is both red and green, but to insist that the correct analysis will show that this is a tautology after all (and hence not a synthetic a priori truth). Once again, though, he did not, either here or in the corresponding remarks in the Tractatus (6.3751), go into the details of how colour expressions are to be analysed so as to exhibit the ‘logical structure of colour’. In his 1929 article, ‘Some remarks on logical form’, Wittgenstein addressed again the question he had dodged in 1916. Let pn be the statement that a certain patch of light has intensity n. Suppose we analysed p1 as E(b). What about p2 ? We cannot analyse it as E(b) ∧ E(b), since that is logically equivalent to E(b), but nor can we analyse it as E(b) ∧ E(b0 ), since we then would not know which of E(b) and E(b0 ) was p1 . He concluded that no analysis was possible and that his original

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supposition which led him to attempt it, namely that elementary propositions are logically independent of one another, was false. (It is curious, though, that it should have been the visual cases that persuaded him of this, since it is surely even less plausible that space and time should have a wholly logical structure than that colour should.) Wittgenstein here attempted to suggest that accepting a priori entailments between elementary propositions is only a minor adjustment to the Tractarian account rather than a wholesale rejection. One atomic fact, he suggested, may exclude another in much the way that one person sitting in an armchair excludes another. It is surely remarkable, though, that he treated this so casually. The obvious rejoinder is that what prevents the two of use from sitting in the same chair is properties of space. To allow that this places a priori conditions on what it is possible for me to experience can hardly count as a ‘minor adjustment’, since it leads directly to the transcendental idealism that in the Tractatus he aimed to avoid. In an attempt to save himself from this, Wittgenstein adopted the view that the sentences in question do not express genuine propositions but only rules of grammar: it is part of the grammar of colour that something red is not also green; and part of the grammar of time that if t is before t0 then t0 is not before t. Nonetheless, this meant that he was now admitting two kinds of necessity, logical and grammatical, and one might well think that this was just as bad.

The shape of space Although Wittgenstein probably derived his Tractarian analogy between the self and the eye from Schopenhauer, the two authors drew different morals from it. Schopenhauer, despite his fondness for comparing the ‘pure subject of knowing’ with the ‘eternal eye of the world’ (e.g. 1966, I, 186, II, 371), was not greatly interested in using it as a route to deducing the existence of the subject, since he held (somewhat obscurely) that we have a more direct means of knowing it via the will. Wittgenstein, on the other hand, began in the autumn of 1916 to wonder whether the visual analogy offered a route to a transcendental argument for the existence of a viewpoint from the existence of the view. He wondered, that is to say, whether this might supply the (somewhat indirect) sense in which ‘in philosophy we can talk of a non-psychological I’, namely that it might make itself manifest through our activity of representing the world. It is clear that my visual space is constituted differently in length from breadth. The situation is not simply that I everywhere notice where I see anything, but I also always find myself at a particular point of my visual space, so my visual space has as it were a shape. (NB, 20 Oct. 1916) Hence correlatively, he suggested, ‘I always find myself at a particular point of my visual space’. In the classical theory of perspective the lines in a picture can be

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used to deduce the point of view (the ‘station point’). In the same way the visual field has information encoded in it which enables us to deduce the point from which it is viewed—what Wittgenstein would later call the ‘geometrical eye’ (BB, 63–4). By analogy, therefore, we might conjecture that language somehow has the perspective from which it represents the world encoded in it. The visual analogy thus suggests a route by which we might attempt to deduce from structural features of the world the existence of a self representing it. This is a temptation to which in the Blue Book (63–4) Wittgenstein showed clear signs of being attracted. In 1916, however, he was more circumspect and recognized the weakness of the visual analogy, namely that the idea of a perspective on the world carries with it the possibility, incompatible with his atomism, of other perspectives on the same world. The ‘shape’ of visual space, which allows us to deduce its point of view, also leads us to conceive of the things we see as having hidden aspects which other points of view might reveal. Atomism, by contrast, requires us to conceive of the world as represented in such a way that no inference of a point of view is possible—none, at least, that leaves room for other points of view alongside it. Although perspective is a (no doubt welcome) feature of human vision, it is not a necessary feature of any sort of representation. Consider, for instance, a radar screen on which the position of the radar antenna was not shown. In order to deduce something about the representing subject from the representation, we would have to show that the structural features of the representation are relevantly like those of the visual field (i.e. perspectival) rather than like those of the radar screen. Yet the structural features in question are only those which a proposition has merely by virtue of being a logical picture, i.e. those which any representation will inevitably have. We have been given no reason to think that these features are in the relevant sense perspectival.

Atomism Another problem traceable to the wartime notebooks but left unresolved in the Tractatus concerns the notion of simplicity on which Wittgenstein’s atomism depended. If I say ‘My watch is on the table’ I seem to express something quite determinate, even though I do not mention (and may not even have the least notion of) the watch’s internal make-up. In June 1915 Wittgenstein gestured briefly towards the idea that simplicity might be context-sensitive, but his discussion breaks off at the end of the surviving notebook, leaving us with no clue as to how (or indeed whether) he resolved it. Yet he included in the Tractatus the argument for substance, which claims (unconvincingly, as we have seen) to demonstrate the existence of a non-context-sensitive substratum of simple objects. He queried that claim once again in the early 1930s, and in the Investigations he explicitly repudiated it, adopting instead the idea he had floated but not pursued in 1915, namely that simplicity is context-relative.

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Notice, though, that even if we abandon the doctrine of a fixed substance invariant between different propositions, we can still hold that each proposition depends for its sensefulness on a distinction between what that proposition assumes to be fixed and what it conceives of as varying. In his later work Wittgenstein retained his earlier insight that understanding is distinct from truth, and it is this that underlies the part of the argument for substance that survives when the quantifier-shift fallacy is eliminated, namely that nothing I say can express all its own presuppositions. This is the essence of the appeal of the direct-reference theory of names as against the descriptivist one. (If it did not seem too much like a contradiction in terms, one might call this an argument for relative substance.) In his later work (OC, §§96–9) Wittgenstein used the metaphor of the shifting riverbed in an attempt to capture this. There is a sense in which the proposition about my watch is incomplete, namely that by treating the watch as simple I do not represent its internal make-up. In the wartime notebooks Wittgenstein also considered another kind of incomplete proposition, namely one that is disjunctive or negative, but he did not then satisfactorily resolve the problem this generates, and in 1929 he returned to it. I can see, for instance, that there is no one in the room, but it seems implausible to analyse this as meaning that Carnap is not in the room, Schlick is not in the room, etc. Wittgenstein was now finally willing to grant—what Russell had long pressed—that a generalization may say something that goes beyond the conjunction of its instances (see PG, 268–9). This led him back to the distinction between restricted and unrestricted quantification. Only the former, he now held, is equivalent to a logical product or logical sum. To represent the latter, Wittgenstein proposed to use a free variable, so that φx would say that something or other is φ. He thus in effect divided the language into two parts, a ‘primary language’ consisting of truth-functions of elementary propositions and a ‘secondary language’ consisting of incomplete pictures corresponding to generalizations. (We shall see in Chapter 72 how Ramsey developed this idea.) This puts further pressure on the argument for substance, since its central premiss was that each proposition has a single, well-defined negation. This argument, even in its relativized form, would not apply to the incomplete propositions in the outer part of language that Wittgenstein was now willing to countenance. (A variable has no substance.) Allowing incomplete propositions also undermines the argument for solipsism in the 1916 Tractatus, and in the 1930s Wittgenstein no longer advocated it.

The picture theory In the Investigations (§22) Wittgenstein repeated his criticism of Frege’s way of decomposing an assertion into an assertion sign and a complex name of a thought. In his discussion Wittgenstein called a Fregean thought a ‘proposition-radical’ or ‘assumption’ (Annahme). It is worth noting, though, that Wittgenstein’s objection

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was not to the whole notion of a force/content distinction, but only to Frege’s implementation of it, according to which the ‘assumption’ (i.e. content) is a complex, not a fact. In contrast to the Tractatus, however, Wittgenstein later emphasized the disparate nature of language: ‘language-game’ is a family-resemblance concept with no single unifying feature. Even if the picture theory could be adapted to deal with other forces such as the interrogative and imperative, therefore, it is unlikely to be able to get much purchase on the ‘countless’ other uses of language to which he increasingly drew attention (PI, §23), not all of which are analysable as the attachment of a force operator to a proposition. So the picture theory is at the very least incomplete, and the more kinds of speech act we struggle to encompass in our account, the more the ‘crystalline purity’ of logic will come under threat. It is a commonplace of the secondary literature, however, to represent the later Wittgenstein as holding that the picture theory is not merely incomplete but wholly mistaken. Wittgenstein was insisting that a proposition and that which it describes must have the same ‘logical form’, the same ‘logical multiplicity’. Sraffa made a gesture, familiar to Neapolitans as meaning something like disgust or contempt, of brushing the underneath of his chin with an outward sweep of the finger-tips of one hand. And he asked: ‘What is the logical form of that?’ (Malcolm 1962, 69) Malcolm offered the following gloss on this. Sraffa’s example produced in Wittgenstein the feeling that there was an absurdity in the insistence that a proposition and what it describes must have the same form. This broke the hold on him of the conception that a proposition must literally be a ‘picture’ of the reality it describes. It is doubtful, though, whether Wittgenstein ever repudiated the picture theory in the way that Malcolm suggested. He certainly did reject the notion that a picture is a sort of mediating element standing between the sentence and the fact, but that would amount to a correction to the Tractatus only if we read that work, implausibly, as hypostasizing propositions as free-standing elements of a Fregean third realm. If there is no single language but only a patchwork of overlapping languagegames, it follows in particular that there is no single form—logical form—which all representations share. So the Tractarian argument against the existence of a metaperspective from which to articulate the relationship between language and world falls away. The rule-following considerations constitute Wittgenstein’s attempt at a new argument for the same conclusion. In the Tractatus the general form of proposition played a sort of unifying role, akin to that of the transcendental unity of apperception in Kant. Once Wittgenstein had abandoned the notion that there is a single language, it became

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problematic once more what explains the unity of my representation of the world. In the Investigations Wittgenstein introduced the notion of our ‘form of life’ to play this role. Although it is obscure just what he meant by this, he repeatedly appealed in explaining it to notions that seem intrinsically communal and hence unable to fulfil the required role. His difficulty was that that role was inherently transcendental, but that acknowledging this would have compelled him to recognize the transcendental idealism to which it leads.

Philosophy One of the most obvious continuities between early and late Wittgenstein is his therapeutic aim of persuading us to stop asking certain questions. Some commentators try to use this to bolster the austere reading of nonsense in the Tractatus, even though the pressure towards inausterity is stronger in the late work than in the early, because the notion of a patchwork of language-games naturally throws up the possibility that something unsayable in one language-game may be sayable in another. The later Wittgenstein continued to hold that ‘philosophy is not a theory but an activity’ (4.112), its purpose being to free us from mistaken conceptions. In two respects, however, his later view was more radical. The first is that he placed on the practice of therapy the constraint that the philosopher should only say what everyone believes. It is questionable, though, whether he obeyed this constraint. I noted earlier that his characterization of philosophy applies only imperfectly to the Tractatus itself, and the same can be said of his later work. The Investigations contains many claims which are certainly not what we all believe. The second change is that Wittgenstein came to think that the process of therapy should not involve arguments. His view became increasingly radicalized as a sort of particularism, according to which there is no pattern capable of being extracted as a philosophical generalization. This is one of his most troublesome doctrines, because it can seem to be an attempt to make him immune to philosophical criticism. In the wrong hands, indeed, it might risk turning his later philosophy into a sort of cult. In the thirty years after Wittgenstein’s death interest focused on attempts at understanding his later work. The new interest in the Tractatus that the resolute reading generated was unfortunate not because the reading is wrong but because it led to even less attention being given to the apparently positive but ultimately nonsensical claims made in the core of the book. What, then, should modern philosophers learn from the Tractatus before they throw it away? Even before Wittgenstein had added the puzzling and problematic remarks about the metaphysical subject, the 1916 Tractatus was already flawed, because logical atomism is flawed. That should not blind us, though, to the insights the book contains. Although the picture theory of the proposition is not quite right as it stands, it is closer to being right than any other account of

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meaning and truth for empirical declarative sentences that has yet been offered. Although Wittgenstein’s attempt to block the ascent to the metalanguage outright was illegitimate, he was right to object to the ‘side-on’ conception of the relationship between language and world. And he was right, too, to dismiss as incoherent the persistent attempts of philosophers to conceive of our representation of the world as undertaken from one viewpoint among many.

Further reading For a reading of the Tractatus in logical positivist terms see Stebbing (1933). Kenny (1974) discusses how the Tractatus influenced Wittgenstein’s later work. On the notion of simplicity deployed in the argument for substance see Sullivan (2003). On the continuity in Wittgenstein’s attitude to solipsism see Bell (1996). The view that the later Wittgenstein did not so much give up as amend the picture theory is advanced by Kenny (1973) and opposed by Hacker (1981), although the latter’s opposition seems heavily dependent on interpreting the phrase ‘picture theory’ in a way that Kenny probably did not intend. On Wittgenstein’s later particularism see Pears (1994).

PART IV

Ramsey

64 BIOGRAPHY

Frank Ramsey was born and raised in Cambridge, where his father was a hardworking but not especially distinguished maths don at Magdalene College and his mother, who had read history at Oxford, a noted campaigner for women’s suffrage. The house near the top of Castle Hill where he grew up (built on land which his father had, remarkably, bought from his own college while serving as its Bursar) is now part of a Dominican priory. Ramsey won a scholarship to Winchester College (one of England’s leading public schools), where he excelled at mathematics as well as winning a German prize. (The later claim of Ivor Richards, an English don at his father’s college who knew him, that he mastered the language ‘in almost hardly over a week’ is an unhelpful exaggeration.) His father was a Congregationalist, his mother an Anglican. His younger brother, Michael, became a priest and was Archbishop of Canterbury in the 1960s, but Ramsey himself, although confirmed at 13, soon became, and thereafter remained, a militant atheist. In 1920 he went to Trinity College Cambridge to read mathematics at the young age of 17; most of his colleagues would have been at least two years older, those returning from war service older still. At the end of his first year there he went on a walking holiday in the Austrian Alps, during which a member of the party was killed by a fall. (He did not himself witness the accident and indeed continued with the holiday afterwards.) At Cambridge Ramsey’s interest in philosophy was soon recognized. In his second year he was invited to give a talk to the Moral Sciences Club on Russell’s theory of judgment and to review Keynes’s book on probability for a Cambridge periodical. Keynes described him as ‘certainly far and away the most brilliant undergraduate who has appeared for many years in the border-country between Philosophy and Mathematics’ (letter to Broad, 31 Jan. 1922, in Coates 1996,

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138). That year he was also invited by Ogden to make the first draft of the English translation of the Tractatus: he dictated it at the university typing office, and Ogden then liaised with Wittgenstein about amendments before publication. Ramsey graduated in 1923 as what was then called a B* wrangler (i.e. with a First in Part II(A) and a distinction in Part II(B) of the Mathematical Tripos). That August he wrote a Critical Notice of the Tractatus remarkable not only for the astuteness of its criticisms of the book but for the extent to which it articulated issues that dominated his own philosophical work thereafter. The following month he spent a fortnight with Wittgenstein (by then a primary school teacher in Lower Austria) during which they discussed the book in detail. His copy, containing markings of the passages he wanted to discuss as well as annotations by Wittgenstein, has survived (see Lewy 1967). He wrote to his mother during the visit. He is prepared to give 4 or 5 hours a day to explaining his book. I have had two days and got through 7 (+ incidental forward references) out of 80 pages. And when the book is done I shall try to pump him for ideas for its further development which I shall attempt. He says he himself will do nothing more, not because he is bored, but because his mind is no longer flexible. . . . I think he exaggerates his own verbal inspiration, it is much more careful than I supposed but I think it reflects the way the ideas came to him which might not be the same with another man. (CKO, 78) Soon after beginning graduate work Ramsey won a scholarship, which he used to return to Austria for six months. Ostensibly, this was in order to work further with Wittgenstein, but in fact they only met a couple of times, and he was really there to be psycho-analysed by Theodor Reik, a disciple of Freud. He claimed that the purpose of this was to cure himself of an unhealthy obsession with a married woman, but that is hardly likely to have been the whole story. The following winter Ramsey wrote ‘The foundations of mathematics’ as an entry for an essay competition open to Cambridge mathematics graduates. It did not win the prize, but was published the following year nonetheless. By then he had been made a Fellow of King’s College, largely at Keynes’s instigation. He was remarkably young for such an appointment, but it was a teaching post, not a research fellowship, and the load was substantial: he typically supervised undergraduates 15 hours a week during term. Ramsey had met Lettice Baker when they were both undergraduates: she read the Moral Sciences Tripos at Newnham, graduating in 1921. She returned to Cambridge in 1924 to work in the psychology department, and they met again at a Moral Sciences Club meeting that November. They married the following August (in deference, it seems, to his mother’s conventional mores) and soon had two daughters. They treated it as an open marriage by mutual agreement, and both had extra-marital affairs, although Ramsey was shocked to discover that he minded his wife’s affairs much more than she minded his.

Biography 421

In 1925 Cambridge University was in the midst of major reforms to its employment structure. Until then, university lectureships had been a random assortment with disparate duties and rates of pay. An act of parliament created a more coherent structure, as a result of which a number of new university lectureships became available. To strengthen his candidacy, Ramsey wrote up his paper on ‘Universals’ (rather hurriedly, if he is to be believed) for publication. In July 1927 Ramsey and Moore took part in a symposium on ‘Facts and propositions’ at the Joint Session, held that year at Bedford College London. The paper Ramsey wrote for this occasion is frequently cited nowadays both for its endorsement of the thesis concerning the transparency of truth and for the sympathy towards pragmatism which it displays—a sympathy he credited in the paper to Russell. In his remaining years he tried to develop some of the ideas in this paper into a book on truth, but what remained on his death was far from finished. In August 1927 Ramsey’s father crashed his car, killing Ramsey’s mother, who was a passenger. It was Frank who formally identified her body. Although Ramsey’s appointment was in the Mathematics Faculty, the bulk of his publications were in philosophy. One notable exception was his work on the Entscheidungsproblem, the problem of finding a mechanical method of deciding, for each sentence of pure logic, whether it is logically valid. For monadic first-order logic the problem is straightforwardly soluble by the Venn diagram method, but for polyadic logic it was shown to be unsolvable in general by Church and Turing in 1936. Ramsey provided a solution for the case of sentences whose normal form contains no universal quantifiers. In the course of proving the correctness of his method, he instigated a branch of combinatorics now known as Ramsey theory. Ramsey also had an interest in mathematical economics, on which with Keynes’s encouragement he published two remarkably innovative papers. The first of these (1927) asked at what rate we should tax goods in order to do least damage to the economy. He argued that if the economy is efficient, so that the proportions of consumption of the various kinds of goods are already optimal, we should tax them in such a way as to keep these proportions constant. This entails, counter-intuitively, that we should tax essentials at a higher rate than luxuries, since it requires a greater disincentive to reduce our consumption of them. Ramsey’s second paper in economics asked at what rate we should save in order to achieve the maximum obtainable rate of enjoyment, which he called ‘bliss’. The problem was that ‘the more we save the sooner we shall reach bliss, but the less enjoyment we shall have now, and we have to set the one against the other’ (1928, 545). His solution—on the simplifying assumptions of a constant population, no technical progress and no discounting of utility—was that the rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment falls short of the maximum possible rate of enjoyment. (1928, 543)

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The practical significance of this is that ‘the rate of saving which the rule requires is greatly in excess of that which anyone would normally suggest’ (1928, 548). Part of the explanation for the disparity is that the simplifying assumption of no technical progress is false: in practice, our future income is likely to be ‘obtainable with less sacrifice than at present’, which is a reason to save less and enjoy more now. Keynes described this paper as ‘one of the most remarkable contributions to mathematical economics ever made’ (FoM, x), and it eventually gave birth to a sizable branch of the subject. In January 1929 Wittgenstein returned to Cambridge and for the rest of that year he and Ramsey met frequently. Previously Ramsey had been hostile to intuitionism, calling it a ‘Bolshevik menace’ (FoM, 56), but now he became increasingly interested in a finitist or intuitionist attitude to mathematics. In November 1929, Ramsey caught an infection, originally diagnosed as jaundice and treated at home in Cambridge. When he had not recovered by January, he was transferred by ambulance to Guy’s Hospital in London, where his wife’s uncle was a senior surgeon, for an operation. By then, however, his liver and kidneys were severely compromised, and he died a few days later, just short of his 27th birthday. His body was cremated at Golder’s Green crematorium; he is commemorated on the headstone of his mother’s grave in the Ascension Burial Ground, Cambridge. The corpus of Ramsey’s published work spans little more than five years. Even so, we shall not here discuss the whole of it, but only the part that is directly philosophical.

References Editions of Ramsey’s works are cited using the following abbreviations: FoM Foundations of Mathematics and other Essays, ed. R. B. Braithwaite N Notes on Philosophy, Probability and Mathematics, ed. M. C. Galavotti OT On Truth, ed. Nicholas Rescher & Ulrich Majer Two other editions of Ramsey’s philosophical papers (both edited by Mellor) unfortunately omit his Critical Notice on the Tractatus.

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Ramsey is often quoted as saying that there is ‘no separate problem of truth’ (FoM, 142). He made the remark in 1927, but the view he was then expressing may be traced to talks he gave to Cambridge audiences as a second year undergraduate.

Propositions In November 1921 Ramsey gave a talk to the Moral Sciences Club (a Cambridge University philosophy society) arguing that Russell had been wrong, in his multiple relation theory of judgment, to treat propositions as incomplete symbols. If we analysed judging that p and that q as relations between the components of p and of q respectively, the analysis of judging that p ⊃ q would, he thought, be hopelessly complicated, since it would differ according to the various structures p and q might have. Ramsey took it to be obvious, indeed, that we need to be able to speak of the object of a belief, because we need to be able to give a sense to the claim that A’s belief has the same object as B’s, even if A and B express their beliefs in different sentences—perhaps even different languages. In our analysis of a belief, that is to say, we shall have to distinguish between what he called ‘mental’ and ‘objective’ factors (FoM, 138). What he regarded as controversial was only what the objective factors are, not whether there are any. In the 1921 talk he called a belief ’s object or objects its ‘reference’; later he called this ‘propositional reference’ (OT, 7). The role of propositional reference in his account is thus similar to that of thoughts in Frege’s or propositions (prior to 1909) in Russell’s. Indeed Ramsey expressed some hesitancy over whether just to call the reference a ‘proposition’: that he decided not to should not, however, be interpreted as evidence that he denied the need to recognize objective factors in belief.

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Although he rejected Russell’s account of propositions as incomplete symbols that disappear from the analysis of the judgments in which they apparently occur, Ramsey agreed with him that the analysis of judgments involves a multiple relation. His own view, as expressed in the talk, was that the ‘propositional reference’ of a belief is a property multiply relating the belief to the entities it is about. What, then, of propositions that no one currently believes? Presumably he would have said that the property exists even if it has no instances, thus avoiding the awkwardness of Russell’s appeal to propositions coming into existence whenever the logician wants to mention them. (Compare Ramsey’s insistence that it cannot be the logician’s concern whether anyone has named an object.) Had Ramsey read the Tractatus by the time he gave this talk? If so, it can only have been very recently. The only copy then in Cambridge was an offprint of the page-proofs which Ogden had received in early November, less than four weeks earlier. His father’s later recollection was that Ramsey had been lent the Tractatus to read—presumably, therefore, Ogden’s offprint—and it ‘interested him greatly’. Perhaps this was what led Ogden to involve him in the translation. At any event, the view Ramsey expressed in the talk, that belief involves a multiple relation between the things the belief is about, is similar, in essence if not in detail, to the Tractarian view. The difference between the two lies in the direction of explanation: Wittgenstein explained the belief in terms of the proposition, Ramsey the other way round.

Truth ‘The most certain thing about truth’, Ramsey suggested at the end of the talk, ‘is that “p is true” and “p”, if not identical, are equivalent.’ (N, 103) He did not here define what he meant by ‘equivalent’, but he was probably using the word, like Johnson (1921-4, I, ch. 4), to mean sameness of logical content. At any rate, he did not intend to suggest—and a year later he explicitly denied—that there is no difference in tone between ‘p is true’ and ‘p’. In a paper he read to the Apostles (another Cambridge discussion society) the following April he suggested that ‘p is true is merely a different verbal form for p’ (2007, 385). Having thus enunciated the transparency schema as an undergraduate, he then adhered to it throughout his career, repeating it in both his 1927 paper and his late manuscript On Truth. He did not originate it, of course: Frege stated it several times, as we have noted, but Ramsey probably got it from Johnson, who discussed it in his Logic (1921-4, I, 52). The transparency schema entails that there is a significant sense in which merely using the word ‘true’ cannot, in Blackburn’s neat phrase, ‘raise the temperature’ (2005, 63). Whether the schema entails deflationism about truth is much less certain. Frege, for instance, deduced from it the converse, inflationary moral that truth is implicated in every thought. Ramsey, on the other hand, did not think the schema amounted to a ‘theory’ of truth at all. Even his first recorded

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statement of it in 1921 was followed by a discussion of three different theories of truth that might be consistent with it. And in his 1923 Critical Notice of the Tractatus he suggested that if we can say what it is for a proposition token to have a certain sense, we incidentally solve the problem of truth, or rather, it is already evident that there is no such problem. For if a thought or proposition token ‘p’ says p, then it is called true if p, false if ∼p. We can say that it is true if its sense agrees with reality, or if the state of affairs which it represents is the actual one, but these forms only express the above definition in other words. (FoM, 275) Ramsey was here making the Tractarian point, which Russell had been slow to grasp, that a theory of meaning must be simultaneously a theory of truth. If we explain what it is for a judgment to be a judgment that p, we thereby explain the condition for it to be true. When Ramsey said in 1927 that there is no ‘separate problem of truth’ (FoM, 142), he meant no problem separate from that of explaining the meanings of sentences. The transparency schema can be at most a constraint on theorizing, he thought, not a theory on its own. By mentioning truth, then, we do not raise the temperature, but nor do we lower it. The transparency schema does not commit us to conceiving of truth as ‘internal’—explicable without reference to the world—and indeed Ramsey in various places explicitly affirmed an externalist understanding of it. In particular, he endorsed Russell’s refutation of the coherence theory of truth (OT, 95). ‘The beliefs of a man suffering from persecution mania may rival in coherence those of many sane men but that does not make them true.’ (OT, 94) And he rejected in even stronger terms the pragmatist account of truth in terms of usefulness. Whatever the complete definition may be, it must preserve the evident connection between truth and reference that a belief ‘that p’ is true if and only if p. We may deride this as trivial formalism, but we cannot contradict it without absurdity, and we should value the necessity of conforming to it as a check on our deeper investigations; for the disregard for this check is one of the main sources of errors for the pragmatists. . . . If the essential relation of truth to propositional reference is overlooked, philosophers are liable to give a vague and inaccurate definition of truth, in terms not of propositional reference but of the supposedly more primitive notions of utility and habits of action, and then use this definition either to establish a false relation between truth and propositional reference, or to deny that there is any close relation between them, and in either case falling into patent absurdity. (OT, 92–3) As an instance, Ramsey criticized James for holding that a religious belief may be true because it is useful to us to believe it (OT, 91–2), but the point is not restricted to religious beliefs. Invalids, for instance, may find it useful (in the sense

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of promoting well-being) to hold false beliefs about their illnesses. There may even be cases in which false beliefs are useful epistemically—as a route to true ones. One could of course try to deflect such examples by arguing that the supposed wellbeing promoted by a false belief is fake: well-being is not genuine—religious faith not genuinely redemptive—if it is based on falsehood. But that move, whatever plausibility it might have on independent (e.g. religious) grounds, is of no use to the pragmatist here, since it appeals to just the distinction between truth and falsity that is in dispute. There has been an unfortunate tendency in the secondary literature to equate externalism about truth with the correspondence theory. Ramsey was in fact content that his own view should be categorized as a correspondence theory— true propositions, he was happy to say, correspond with the facts—but the casual tone in which he said this suggests that he did not regard it as an especially significant claim. He was at any rate commendably free of the superstition that ‘factstating’ talk raises the metaphysical temperature any more than the transparency schema does: such talk, he said, is not an analysis of truth but merely ‘a cumbrous periphrasis’ (OT, 18).

Expressing and describing The transparency schema licenses us to eliminate the predicate ‘is true’ from contexts in which it is predicated of a declarative sentence, and hence to regard it in these contexts as an incomplete symbol in Russell’s sense. There are, however, two other sorts of context, description and quantification, from which its elimination is less straightforward. Consider first the case in which truth is predicated not of a declarative sentence but of a description such as ‘what John just said’. In order to deal with this case, Ramsey drew a distinction between expressing a sense and describing it. ‘I use “relative names”,’ he said in the Critical Notice, ‘to include “p”, the expression for a given sense p; in contrast to a description of that sense, such as “what I said”.’ (FoM, 280–1) In ‘Othello believes that Desdemona loves Cassio’, for instance, what Othello believes is expressed; in ‘Othello believes what Iago wants him to believe’ it is only described. Ramsey’s distinction is important, but other logicians were curiously slow to draw it. Kneale (1962, ch. 11), for instance, attempted to draw a distinction between ‘expressing’ and ‘designating’ a thought, but then bungled it by classifying ‘that Socrates is mortal’ as a designation, not an expression, hence collapsing the distinction into the familiar one between a sentence and a noun phrase. The latter distinction is no doubt important for the grammar of ordinary language, but on Ramsey’s view it is the former that matters in logic. The prefix ‘that’, which converts a sentence into a noun phrase, is cancelled by the suffix ‘is true’, whereas in the case of a description there is no prefix to cancel; but this, he thought, is merely a parochial feature of English. What matters to logic is that the prefix

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‘that’ does not opacify the thought; the noun phrase ‘that Socrates is mortal’ is as transparent an expression of it as the sentence ‘Socrates is mortal’. We have already encountered Wittgenstein’s argument in the Tractatus (based on the disanalogy with naming a point on a sheet of paper) against assimilating sentences to names, on the ground that the former represents transparently the structure of what it says in a way that the latter need not. Ramsey in effect relocated this point. What matters, on his view, is that a descriptive term is always parasitic on an expressive one (see OT, 23). By saying ‘What John just said is true’ I express a thought only if John already did so. Since ‘what John just said’ is a definite description, this case is reducible, by the usual Russellian analysis, to that of quantification into sentence position such as ‘John has said something true’. In his 1922 paper to the Apostles Ramsey proposed to analyse this as ‘There are terms and a relation, such that he has asserted the relation to hold between the terms and such that the relation does in fact hold between the terms’ (2007, 385). The difficulty he saw with this analysis, however, was that it worked only for an atomic relational proposition, whereas the sentence to be analysed did not specify the form of what was said. The reason he restricted himself in this way to propositions of one particular form was probably that at that time he still adhered to the ramified theory of types of Principia, which does not permit quantification over all propositions. In his paper he downplayed this difficulty—oddly, given that he had raised a rather similar objection against Russell’s multiple relation theory of judgment. By the time he revisited the problem in 1927 (see FoM, 143), however, he had (for independent reasons to be discussed in Chapter 67) abandoned the restriction on quantification, and could therefore now analyse ‘John has said something true’ straightforwardly as ∃p(John has said p and p). Recall our explanation (Chapter 5) of quantification into name position using anaphoric pronouns. To do the same for quantification into sentence positions, we would need anaphoric ‘prosentences’, as Ramsey (OT, 10) called them. What makes this awkward is that English has rather few of these: he claimed that the only ones are ‘yes’ and ‘no’, but in fact there are some others, such as ‘amen’ or the ‘so’ in ‘So you say’. More often, English uses locutions such as ‘That is true’, but on his view the occurrence here of the word ‘true’ is merely a linguistic quirk, not a genuine reference to the concept of truth. And whether we can express our account of quantification in ordinary language is in any case incidental to the logical point at issue here. Later discussion of Ramsey’s account has been distorted by the common view (e.g. Quine 1970, 66–7) that referential quantification makes sense only in name positions, and hence that the only sense to be made of quantifying in sentence positions is substitutional. Why think that? Certainly the restriction to name positions does not follow from the account in Begriffsschrift §9, which requires only that the part of a sentence to be removed should make a distinguishable

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contribution to determining the conceptual content of the whole. Quine’s objection arose from his suspicion of the propositions to which sentences are supposed to refer—a suspicion based principally on the failure to supply them with sufficiently determinate identity conditions. Wittgenstein could have responded to this by pointing to the identity conditions supplied by his notion of sense. For Ramsey, though, the important point is not so much whether the entities to be quantified over have identity conditions, but whether they form a determinate range; and by the time of ‘Facts and propositions’, at least, he thought that this task had been sufficiently dealt with by the Tractarian notion of sense. The purpose of Ramsey’s discussion of describing a proposition, as opposed to expressing it, was to replicate without overt use of the truth predicate such locutions as ‘What John just said is true’. That is not to say, however, that we should unrestrictedly adopt this device in our formalism; if we do, the liar paradox beckons. This paradox is often said to show that no language can consistently contain its own truth predicate, but on the view we are now considering this mislocates the problem. It is not the truth predicate that leads to paradox but the ability to construct a proposition that contains a description of itself.

Reverse semantics In the Critical Notice Ramsey noted a gap in the Tractarian account of belief. Wittgenstein had said that ‘A believes that p’ has the form ‘ “p”’ says p’, in which two facts are coordinated by means of a coordination of their objects. What this compressed formulation conveniently obscured, however, was that the objects to be coordinated are incompletely specified. If I say, ‘Othello believes that Desdemona loves Cassio,’ I do not thereby indicate which propositional sign Othello used to formulate his belief. So Wittgenstein’s analysis really contains an implicit quantification over the signs Othello might have used. In order to complete it, therefore, we need a reverse semantics—a procedure which, given a proposition p, generates all the propositional signs which say that p. Having raised this problem, Ramsey sketched a solution for the specific case of negation. A sign expressing the negation of ‘q’ is, he said, ‘ “∼q or ‘∼∼∼q’ or ‘∼q ∨ ∼q’ or any of the other symbols constructed according to a definite rule’. But which rule? He admitted that it ‘may be doubted whether it is possible to formulate this rule, as it seems to presuppose the whole of symbolic logic’. Indeed so: in the case of a tautology, for instance, the ‘definite rule’ would have to enumerate all the tautological propositional signs in the language. Ramsey went on to say that ‘in Mr Wittgenstein’s notation with T’s and F’s there would be no difficulty’ (FoM, 278). No difficulty, that is, except that the notation is usable only in the finite case. In the infinite case, things are much trickier, at least if we require the ‘rule’ in question to be mechanical: for a first-order language there is indeed such a rule, but this fact depends on Gödel’s completeness theorem, not proved until 1929; and for the higher-order language of the Tractatus there is none.

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The problem of reverse semantics is thus quite difficult enough when restricted to a single formal language; but in fact it is even more general than that. If I merely ascribe a certain belief to you, I do not thereby specify (and need not even understand) the language in which you formulated it. So the implicit quantifier in Wittgensteins analysis of ‘A believes that p’ has to range over all the languages A might have used to say that p. Characterizing this class is plainly a tall order. In the Critical Notice Ramsey briefly mentioned Russell’s idea (AMi, 250) that the more complicated cases might involve ‘special belief feelings’. Logical constants might then be significant as substitutes for these feelings, which would form the basis of a universal logical symbolism of human thought. (FoM, 279) Four years later, in ‘Facts and propositions’, Ramsey expanded a little on this Russellian idea, suggesting that believing not-p is equivalent to disbelieving p. This neatly dodges the difficulty of representing negation, since whether it is p or not-p that we believe, in both cases what we represent is the same: what differs is only the accompanying feeling, belief in the first case, disbelief in the second. Even so, it leaves unaddressed ‘the central difficulty of the subject’ (FoM, 147), which is to explain how believing not-p is equivalent to disbelieving p. Ramsey here followed Russell’s suggestion that this should be done in terms of the causal properties of the mental states in question. ‘Feeling belief towards the words “notp” and feeling disbelief towards the words “p” ’, he said, ‘have then in common certain causal properties.’ If then I say about someone whose language I do not know ‘he is believing that not-aRb,’ I mean that there is occurring in his mind such a combination of a feeling and words as expresses the attitude of believing not-aRb, i.e., has certain causal properties, which can in this simple case be specified as those belonging to the combination of a feeling of disbelief and names for a, R, and b, or, in the case of one who uses the English language, to the combination of a feeling of belief, names for a, R, and b, and an odd number of ‘not’ ’s. Besides this, we can say that the causal properties are connected with a, R, and b in such a way that the only things which can have them must be composed of names for a, R, and b. (This is the doctrine that the meaning of a sentence must result from the meaning of the words in it.) (FoM, 148–9) This account was, as Ramsey acknowledged, ‘very vague and undeveloped’, and it would take a lot more to turn it into a convincing explanation of the psychology of meaning. For instance, the restriction he made to the English language in the last part of his treatment was far from innocent: a Frenchman’s belief that it is not raining and mine may share many of their causal properties, but not all, and it is by no means easy to say which they must share in order to count as a belief in the same proposition.

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Ramsey said in ‘Facts and propositions’ that his ‘pragmatism’ was ‘derived from Mr Russell’, by which he no doubt meant the account of meaning in The Analysis of Mind that we discussed in Chapter 43. Russell had in fact distinguished between pragmatist (output) and causal (input) components, but Ramsey showed little interest, either here or elsewhere, in this terminological distinction, calling both components ‘pragmatist’. It is ‘the essence of pragmatism’, he said, that the meaning of a sentence is to be defined by reference to the actions to which asserting it would lead, or, more vaguely still, by its possible causes and effects. (FoM, 155) Ramsey held, then, a pragmatist theory of meaning, but he also held that there is no separate problem of truth. Some modern commentators have supposed that from these two views a pragmatist account of truth follows straightforwardly, and have therefore been unable to explain why he explicitly rejected this. Their mistake has been to minimize the significance of Ramsey’s non-psychologistic conception of the range of propositional reference. Which propositional references there can be is fixed by which possible senses there are, and this in turn is wholly determined, according to the Tractatus, by the range of elementary propositions. There is no separate problem of truth because the Tractarian account of what senses there are will already have accounted for it. Where pragmatism has a role, on Ramsey’s view, is only in explaining which mental states are beliefs in which propositional references—‘what propositional signs have what sense’, as he put it in the Critical Notice (FoM, 277).

Further reading The word ‘prosentence’ was also coined independently by Brentano (1930, 76) in German and, much later, by Prior (1967, 229). Quine’s insistence that quantification be into sentence position is discussed by Prior (1971, 34–9) and Richard (1996). Grover (1972) discusses the extent to which natural language contains prosentences. Sullivan (2005b) discusses the non-pragmatist component of Ramsey’s account of meaning. Misak (2016, 173) expresses the view (which I do not share) that Ramsey’s attribution of his pragmatism to Russell is ‘astounding’.

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Knowledge, Ramsey said, is true belief which is ‘not a fluke’ (N, 127). This view later became popular under the name ‘reliabilism’ (coined by Peirce).

Reliabilism ‘I have always said’, Ramsey remarked in 1929, ‘that a belief was knowledge if it was (i) true, (ii) certain, (iii) obtained by a reliable process’ (FoM, 258). Russell had already expressed some sympathy with reliabilism in The Analysis of Mind (260–1), but had rejected it on the ground that mere sensitivity to the facts cannot explain our preference for truth over falsity. A thermometer which went down for warm weather and up for cold might be just as accurate as the usual kind; and a person who always believes falsely is just as sensitive an instrument as a person who always believes truly. The observable and practical difference between them would be that the one who always believed falsely would quickly come to a bad end. . . . Accuracy of response to stimulus does not alone show knowledge, but must be reinforced by appropriateness, i.e. suitability for realizing one’s purpose. (AMi, 261) This, though, is at most an argument against replacing truth with reliability, not an objection to Ramsey’s proposal that both are required for knowledge. According to Ramsey, though, Russell’s main error was to suppose ‘that because we are sometimes mistaken, in no case can we know with certainty’ (N, 105). Inferential knowledge is fallible—e.g. when I make a mistake in a calculation—but so are judgments derived from perception, as Russell conceded. How, though, should Russell explain Stumpf ’s example of a row of colour patches

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of which any two adjacent ones seem to be the same colour but those at either end are plainly different colours? Surely, Ramsey thought, this example shows that even my acquaintance with sense data could be erroneous (OT, 60), and hence that Russell’s Cartesian project of finding ‘hard data’ which ‘resist the solvent influence of critical reflection’ (OKEW , 77–8) was doomed to fail. If a man doubts his memory or his perception we cannot prove to him that they are trustworthy; to ask for such a thing to be proved is to cry for the moon. (FoM, 197) Theory of knowledge, Ramsey thought, must, much more than formal logic, concern itself with human frailty: not only can we not know everything, but any of our knowledge might be mistaken. ‘We sometimes make mistakes, and it’s no use pretending we don’t.’ (OT, 62) On Ramsey’s account, we are reasonable if we form our beliefs by a process which as a matter of fact works. The mention of whether a method ‘works’ might suggest a pragmatist slant, and he himself caused some confusion among commentators (e.g. Sahlin 1990, 50–1) by describing his own view as ‘a kind of pragmatism’ (FoM, 197). It deserves emphasis, therefore, that whether methods ‘work’ is not, for Ramsey, a matter of whether deploying them brings me what I want, but rather of whether ‘the opinions they lead to are for the most part true, or more often true than those which alternative habits would lead to’. (FoM, 197-8) A true belief need not amount to knowledge, of course, and the theory of knowledge has been dominated by the search for the further property required in order to make it so. Since Armstrong (1973), it has been common to classify accounts as internalist or externalist depending on whether or not this further property is accessible to the believer on suitable reflection. On this taxonomy Ramsey’s reliabilism is externalist, because whether a process is reliable depends not just on me but on the world. I might adopt processes that are from my own perspective justified, but they would not result in knowledge unless the world cooperated so that they actually worked. In fact, though, Ramsey never formulated his reliabilism in enough detail to be convincing. The most immediate difficulty concerns the notion of ‘process’, which carries with it the notion of repetition. His definition seems therefore to rule out cases, in which the method of knowledge-acquisition is unrepeated or even unrepeatable. Moreover, an act of belief-acquisition typically does not carry the process of which it is an instance (its ‘reference class’) on its face. The problem here is not merely that you need not be aware of the process by which you acquired a piece of knowledge (although that is certainly true), but that it may simply be indeterminate. Any one case of knowledge acquisition might be an instance of two different processes, one reliable, the other not. Suppose, for example, that learning from a schoolteacher is reliable, but not from a drunk. Is something knowledge, then, if you were told it by a drunken schoolteacher?

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Induction In a paper he read to the Apostles in Autumn 1922 Ramsey used his reliabilist conception of knowledge to argue that the principle of induction can give us knowledge, because it has in fact proved a reliable method in the past. A type of inference is reasonable or unreasonable according to the relative frequencies with which it leads to truth and falsehood. Induction is reasonable because it produces predictions which are generally verified, not because of any logical relation between its premiss and conclusion. On this view we should establish by induction that induction was reasonable, and induction being reasonable this would be a reasonable argument. (OT, 123) It should be conceded straightaway that Ramsey’s reliabilist argument can at most explain why it is reasonable to believe the principle of induction, given its reliability, but provides us with no guidance on how the principle should be formulated so as to be reliable, e.g. by ruling out induction on Goodman’s (1955, ch. 3) ‘gruesome’ properties. It should also be granted that since Ramsey’s argument depends on the fact that induction is reliable, it is unlikely to impress the internalist, who demands a justification for induction. ‘What is wanted’, Russell complained, is not the mere historical record of past regularities, but some justification, however, inconclusive, of our expectation that these regularities will continue. (CPBR, X, 113) Ramsey held that Russell’s demand was as unreasonable as wanting a proof of the validity of memory, and hence that Russell was once again crying for the moon. The familiar objection to arguments for the principle of induction is that they are circular. Ramsey, however, explicitly embraced the circularity. Braithwaite (a friend, as well as editor of his posthumous work) later suggested (1953, ch. 8) that although circularity is vicious in an argument for a single proposition, it need not be so in an argument for a rule. What is clear, at any rate, is that the information that the general validity of a rule follows from one instance of it may well be nontrivial. How great an epistemic advance it represents to us depends on whether we feel any more certain of the instance than of the rule. Of course, a rulecircular argument may be wrong—e.g. there is a counter-inductive argument for the rule of counter-induction—but that shows only that we should not accept the conclusion unless we accept the premisses. The externalist’s point is that we should distinguish whether an argument is correct from whether it represents an epistemic advance. Ramsey was plainly aware that his argument for induction would not convince someone who genuinely doubts its conclusion; his point was that the argument may be correct nonetheless.

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Human logic Formal logic, in the sense which Ramsey inherited the term, is concerned only with which sets of propositions are consistent, not with which are true. In order to emphasize this, he referred to it—probably following Mill (1882, bk 2, ch. 3, §9), although the terminology originated with Kant—as the ‘logic of consistency’, and contrasted it with the science of acquiring true beliefs, which he called the ‘logic of truth’. He also called these formal and human logic respectively; or (in a nod to Hegel) the lesser and the larger logic (FoM, 186). Nowadays, of course, we use ‘logic’ for the former, ‘epistemology’ for the latter, but for Ramsey that word encompassed what would now be called semantics. ‘By epistemic’, he said, ‘I mean connected with the relation of sign to thing signified, which includes the relation of a thinker or thought to its object.’ (N, 85) So when he claimed that ‘logic as the term is ordinarily used consists largely of psychology’ (OT, 4), he did not mean by this that formal logic is largely psychology. In Chapter 8 we noted Frege’s distinction between science, which has truth as its goal, and logic (i.e. formal logic), which has truth as its subject matter. Although Frege took the latter to be maximally general, he did not think that this feature characterizes it. Ramsey agreed, making the point with the neat example, ‘Any two things differ in at least thirty ways’. This, he suggested, ‘is a completely general proposition, it could be expressed as an implication involving only logical constants and variables, and it may well be true’ (FoM, 4); but it is not a truth of formal logic. Human logic (epistemology), on the other hand, ‘can only be distinguished from the natural sciences by the greater generality of its problems’ (FoM, 198). There is thus a sense in which it is characterized by its maximal generality. On Ramsey’s account, knowledge’s concern with truth distinguishes human logic from formal logic, but also from descriptive psychology.

Further reading Baldwin (2002) describes Russell’s reluctant engagement with the externalist account of knowledge from The Analysis of Mind onwards. Feldman (1985) discusses the reference-class problem for reliabilism. The modern literature in epistemology contains much discussion of the arid question whether justification is internal or external: it seems preferable to recognize that in ordinary language the word is ambiguous between the two. On the general problem of rule-circularity see Boghossian (2000). For objections to the externalist account of knowledge see Bonjour (1980). Variants of Ramsey’s reliabilist account of induction have been advanced by Braithwaite (1953), Mellor (1991, ch. 15), Papineau (1992) and others.

67 THE FOUNDATIONS OF MATHEMATICS I: TYPES

In 1913 Russell had expressed the hope that Wittgenstein would revise the early parts of Principia Mathematica; but in the event the parts of the Tractatus that are devoted to the theory of types are too sketchy to constitute a determinate description of a formal system and the book gives no clear argument against the ramified theory of Principia. In ‘The foundations of mathematics’—published in 1925, but written at the end of the previous year, soon after his return from six months in Austria, as an (unsuccessful) entry for a Cambridge graduate dissertation prize—Ramsey aimed to fill this gap by revising the theory of types on Tractarian principles.

Ramsey’s simple hierarchy Our account of the formal system of the Tractatus came in two stages: first, we described the Tractarian notion of a sense as a class of truth-possibilities of elementary propositions; then, we described notations for expressing these senses, recognizing as we did so that if there are infinitely many elementary propositions, we cannot expect these notations to express all of them. Ramsey’s account of the hierarchy of propositional functions runs parallel to this. The base class now consists not of the elementary propositions, but of what he called ‘atomic propositional functions’, i.e. propositional functions obtained from elementary propositions by replacing one or more names with variables. Russell’s ramified hierarchy, let us recall, proceeded step by step, first taking the class of what he called ‘predicative functions’—functions obtainable from the atomic functions using the symbolic means available in the formal system. Ramsey, by contrast, proposed to ‘disregard how we could construct [i.e. express] them, and to determine them by a description of their senses’. With this in mind he proceeded immediately to

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the class of all truth-functions (finite or infinite) of the atomic functions, ignoring whether they are expressible using the symbolic resources available. He called these functions ‘predicative’ (FoM, 39), but this invites confusion with Russell’s earlier use of the word, and so I shall here call them ‘predicating’. This notion thus abstracts completely from how propositional functions are to be given finite expression. If there are infinitely many individuals, there are uncountably many first-level predicating functions, and so in any particular formal language many of them will not be expressible: we shall ‘include many functions which we have no way of constructing’ (FoM, 39). Ramsey defended this on the ground that the possibility of indefinable classes ‘is an essential part of the extensional attitude of modern mathematics’ (FoM, 23). The classes of male and female angels may be infinite and equal in number, so that it would be possible to pair off completely the male with the female, without there being any real relation such as marriage correlating them. (Ibid.) So far, though, the only reason Ramsey has given us for preferring a ramified to a simple hierarchy is the regressive one that it accords better with the ‘extensional attitude of modern mathematics’. To supply a more principled argument, he appealed to Wittgenstein’s sign/symbol distinction, according to which two signs express the same symbol just in case they play the same logical role. In particular, two propositional signs express the same proposition just in case they have the same sense. By analogy, Ramsey said, two propositional function signs should be said to express the same propositional function (i.e. the same symbol) just in case they express the same sense whenever they are completed by the same names. Notice, though, that whereas a propositional function’s level is a property of the symbol, its order is a property only of the sign: two propositional function signs of different orders may express the same propositional function. The order of a propositional function is therefore like the numerator of a rational number (FoM, 47): it makes no sense until we have specified the sign being used to express it. It follows that order cannot be a logical feature, and so propositional functions should be typed only according to level, not order. This obviates the need for the problematic axiom of reducibility. The impredicative constructions for which Russell needed the axiom are straightforwardly valid in the simple theory. What, though, of the paradoxes? Ramsey classified them in two groups: ‘Group A’ (what I shall here call ‘logical’ paradoxes) consists of those, such as Russell’s paradox and the Burali-Forti paradox, that can be expressed in logical terms; ‘Group B’ (‘semantic’ paradoxes) consists of those, such as the liar or Richard’s paradox, that ‘contain some reference to thought, language or symbolism’ (FoM, 20). In a Tractarian language all such notions are inexpressible, and so the ramification Russell imposed in order to avoid them is unnecessary; the simple theory of types, on the other hand, already avoids the first group. Ramsey disowned the credit for drawing this distinction among the paradoxes: Peano

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(1906) had observed that Richard’s paradox belongs not to mathematics but to linguistics; and Chwistek (1922, 241) had noted that if we rejected paradoxes constructed by ‘Richard’s method’, the simple theory of types would suffice. It might seem from what has just been said as if there were, puzzlingly, two distinct arguments in play in Ramsey’s defence of the simple theory of types, one derived from the sign/symbol distinction, the other from the inexpressibility of the semantic paradoxes in the logical formalism. In fact, though, these are different aspects of a single argument, because how finely symbols are individuated in the first argument depends on what sorts of inferences are permitted in the second. If we could express semantic notions such as meaning in the object language, symbols would perforce be individuated finely enough to require a ramified hierarchy. When Russell adopted the ramified theory of types in Principia, he was of course aware of the problematic status of the axiom of reducibility. Why, then, did he not adopt Ramsey’s solution of consigning the semantic paradoxes to the metalanguage? Since we have distinguished two components in Ramsey’s argument for the simple theory of types, however, the question concerning Russell may be asked separately about each component. In relation to the first component, one might respond by noting that Russell did not have Wittgenstein’s sign/symbol distinction. Indeed, when he was persuaded by the Notes on Logic to drop his previous realism concerning the subject matter of logic, he switched directly to a consideration of signs and hence to a weakly psychologistic conception of logic. Nonetheless, this response, although correct, would be superficial. Even if he had had the sign/symbol distinction, as long as he held that propositional attitude statements are logically expressible, he would have been obliged to individuate symbols sufficiently finely to require a ramified theory. Ramsey’s argument thus depended crucially on Wittgenstein’s claim that propositional attitudes are not a counterexample to the truth-functional character of logic. In relation to Ramsey’s second argument, one might respond by noting that Russell had not yet drawn the distinction between object language and metalanguage on which the argument depends. Although such a distinction is implicit in Frege’s writings, the first place where it is drawn explicitly is in the Introduction which Russell wrote for the Tractatus in 1921. Once again, though, this response would be superficial. The problem for Russell was not so much that he did not have the distinction, but that his universalist conception of logic disbarred him from exploiting it.

Saying the unsayable The simple theory of types is, Ramsey said, ‘a natural consequence of the logical theories of Mr Wittgenstein’ (FoM, 33). Nonetheless, Ramsey did disagree with the Tractatus in one important respect, namely his attitude to the unsayable. Already in his Critical Notice he denied that sentences attempting to express the internal properties of an object are mystically ineffable. It is, he said, ‘possible to give reasons why these sentences are nonsense and a general account of their origin and apparent significance, which have no mystical implications’

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(FoM, 280). His famous quip that ‘you can’t say what you can’t say, and you can’t whistle it either’ (FoM, 238) was directed at one particular instance of the Tractarian unsayable—which objects there are—but his published work from the Critical Notice onwards displays a continuing interest in saying, inausterely, what Wittgenstein claimed could only be shown. In this he was no doubt influenced by Russell’s suggestion in his Introduction to the Tractatus that what cannot be said in the original language can be said in a larger language. The important point is that the latter cannot be another Tractarian language. The relevance of Ramsey’s attitude to the unsayable to his account of the theory of types arises in relation to Wittgenstein’s explanation of Russell’s paradox, which was that in ‘F(F(f xˆ ))’ the outer and inner instances of the sign ‘F’ symbolize differently. Because Wittgenstein held that how a sign symbolizes is absolutely, not just relatively, inexpressible, he felt no obligation to say any more about it. Ramsey, on the other hand, aimed to bring precision to this question, and thereby to determine the proper range of the higher-order quantifiers. In the case of a first-order variable, the range consists of some type of objects, and this is, as he put it, ‘an objective totality which there is no getting away from’ (FoM, 36). The second-level variable in his theory of types has to range over all first-level predicating propositional functions, and the issue is whether this is likewise an objective totality. The ramified theory is two-dimensional: it types functions first according to the types of their real variables, then of their apparent variables. Ramsey’s theory is also two-dimensional, but the second dimension is now a hierarchy of meanings (inexpressible in the object language) of the word ‘meaning’. The signs that express a single propositional function are stratified according to the kind of meaning they have: some have first-level meaning; others second-level meaning; and so on.

A transcendental argument Reducibility was not the only problematic axiom in Principia: Russell had to concede that the axiom of infinity, too, is not a law of logic. In the Tractatus Wittgenstein was neutral on whether there are infinitely many objects. This creates a severe difficulty for mathematics, much of which depends on the axiom of infinity. Perhaps he realized this and was not troubled by it. What he seems not to have realized, though, is that if there are only finitely many possible atomic facts, it is not just higher mathematics but elementary arithmetic that is in difficulties. This is because in that case we cannot establish inequalities. If Wittgenstein did not see this difficulty, though, Ramsey certainly did. He mentioned it in his Critical Notice of the Tractatus and spent some time thereafter on the problem of justifying appeals to infinity in mathematics. One of his earliest attempts was an ingenious transcendental argument. In order to understand it, we need to recall a feature of Wittgenstein’s account that we mentioned in

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Chapter 53. Whereas the usual notation ∃x1 , . . . , xn f (x1 , . . . , xn ) expresses the disjunction of all values of the propositional function f (ˆx1 , . . . , xˆ n ), Wittgenstein’s, which we wrote as ∃0 x1 , . . . , xn f (x1 , . . . , xn ), expresses the disjunction only of those values for which x1 , . . . , xn are distinct. Now this makes sense only if there are at least n objects. If there are fewer than n, the sign will attempt nonsensically to express the disjunction of an empty class of propositions. Ramsey utilized this observation in the following way. Let T(ˆx1 , . . . , xˆ n ) be a propositional function which expresses tautology for every one of its values. (For instance, we could let T(x) = g(x) ∨ ∼g(x) for some propositional function g(ˆx), and then let T(x1 , . . . , xn ) = T(x1 ) ∧ · · · ∧ T(xn ).) Now put pn =df ∃0 x1 , . . . , xn T(x1 , . . . , xn ). If the number of objects is N, then p1 , p2 , . . . , pN are tautologies, whereas pN+1 , pN+2 , . . . are nonsense, i.e. we cannot say intelligibly (even if falsely) that there are more than N objects. Yet we can intelligibly say this. We deduce, therefore, that there must be infinitely many objects. This is, as Ramsey briskly observed (N, 179), a ‘Kantian’—by which he meant transcendental—argument: from the mere intelligibility of saying that there are infinitely many empirical objects, we deduce that there are infinitely many Tractarian objects. He himself never published the argument, however, for reasons that will emerge in the next chapter.

Further reading Whether Russell had good reasons to ramify the hierarchy of types is discussed by Goldfarb (1989). Whether he adhered to a universalist conception of logic is discussed by Proops (2007). Priest (1994) questions the significance of Ramsey’s division of the paradoxes into two categories. On Ramsey’s transcendental argument see Potter (2005).

68 THE FOUNDATIONS OF MATHEMATICS II: LOGICISM

The part of ‘The foundations of mathematics’ discussed thus far mandates a significant simplification of the theory of types in Principia, but leaves untouched Wittgenstein’s argument against logicism (Chapter 53). This concerned two kinds of class-expressions, accidental (e.g. ‘{x : φx}’) and essential (e.g. ‘{a, b}’), which give rise to two distinct formal hierarchies. Wittgenstein’s complaint was that the theory of types gives us only the former, whereas the latter, which is the one needed for the logicist reduction of arithmetic, is at most an empty formalism and not part of logic. The second part of Ramsey’s article attempted to extend the logical formalism so as to show, on the contrary, that the essential hierarchy is indeed part of logic.

Propositional functions in extension The essential notion of class is not part of logic, Wittgenstein thought, because identity is not: identity is not a material relation, and hence cannot be expressed in the formal language. Russell’s attempted definition, x = y =df ∀φ(φ!x ≡ φ!y), picks out the relation that holds between objects just in case they share all their predicative properties. This is a perfectly respectable relation, perhaps, but not identity, since there is a truth-possibility in which two distinct objects have all the same predicative properties. Ramsey even took it to be possible for the actual world to contain indistinguishable objects. He could not give an example, of course, since this would require him to name them, but this is no odder, he said, than knowing there are two people on earth with the same number of hairs on their heads without

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knowing who they are. We can quantify over them without actually having chosen signs to name each of them. This suggested to him that although identity cannot occur as a material relation, that does not rule out its occurrence within the scope of a quantifier. To explain how this might be possible, he resurrected an idea from the Principles, in which Russell briefly considered two different conceptions of a propositional function: first, as what is obtained from a proposition by replacing some meaningful component of it with a variable; second, as a function in the mathematical sense taking a proposition as its value. Ramsey’s notion of a predicating function discussed in the last chapter is recognizably a descendant of the first conception, generalized so as to remove the dependence on finite expressibility; what he called a ‘propositional function in extension’ (PFE) is a descendant of the second. A PFE, that is to say, is a function in the mathematical sense, but one which takes an object as its argument and delivers a proposition as its value. I shall follow Ramsey in indicating a PFE with a subscript e, e.g. φe . To take his own example, we might specify a function so that φe (Socrates) =df Queen Anne is dead φe (Plato) =df Einstein is a great man We thus ‘drop altogether the notion that φa says about a what φb says about b’ (FoM, 52). Once we have the notion of a PFE, we can then define identity: x =e y =df ∀φ(φe (x) ≡ φe (y)). I have used the subscripted sign ‘=e ’ in order to indicate that as thus defined identity is not a predicating function but a PFE. ‘Its value is tautology when x and y have the same value, contradiction when x, y have different values.’ (FoM, 53) The crucial difference from Russell’s definition of identity is that now the variable φ ranges over all PFEs, not just predicating ones. As a consequence, Ramsey’s definition is extensionally correct under all truth-possibilities, since even if x and y shared all their predicating properties, there would be a PFE which sent x to a true proposition and y to a false one. In the case where the domain is finite we can define any PFE explicitly by means of a table. If the only objects were a, b and c, for instance, we could define a PFE as follows: fe a =df p fe b =df q fe c =df r Indeed the single PFE of identity would be sufficient, since we could use it to explicitly define fe x =df (x =e a ∧ p) ∨ (x =e b ∧ q) ∨ (x =e c ∧ r)

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In the infinite case, on the other hand, this method of defining a PFE by enumeration is not available, and so we shall not be able to give explicit examples of non-predicating PFE. The point is that we can make use of a variable that ranges over all PFEs of a particular level. Once we have PFEs, we can as a by-product retrieve the essential notion of class too: thus {a, b}, for instance, is obtained as {x : x =e a ∨ x =e b}. More generally, any essential class is obtained as {x : φe x} for some suitable φe . The effect of adding PFEs to our formal language, therefore, is to subsume the essential class hierarchy within logic. (And since every predicating function can be treated as a PFE, the accidental class hierarchy is subsumed as well.)

Wittgenstein’s objections When he read Ramsey’s paper, Wittgenstein raised no objection to the simple hierarchy of predicating functions, but he was immediately suspicious of PFEs, not only dictating a letter to Ramsey containing objections to them, but returning to the matter repeatedly in the next few years. Wittgenstein’s initial suspicion was based on a mistaken supposition that Ramsey was arguing for a genuine material relation of identity. Ramsey had to re-assure him that he accepted the Tractarian rejection of such a relation. His intention was to rehabilitate identity only within the scope of a quantifier. He intended, that is to say, to add to the ways in which a propositional variable may be specified. His claim was only that a symbol containing a variable φe that ranges over PFEs is an ‘intelligible notation’ (FoM, 53), because it is a coherent way of picking out a class of propositions. Once this misunderstanding had been ironed out, Wittgenstein’s suspicion remained, but he had some difficulty in formulating just what he objected to. One obvious objection which he did not raise is that on Ramsey’s account the expression ‘φe (Socrates)’ is a name, whereas in the way he used it what he required for grammaticality was a sentence. This was not merely a casual error on Ramsey’s part, but an instance of his previously noted view that the difference is merely a parochial feature of ordinary language and of no logical significance. More to the point, though, is that ‘φe (Socrates)’ is, in the terminology of Chapter 65, a descriptive name, not an expressive one. This was what Wittgenstein was objecting to when he likened the difference between a predicating function and a PFE to that between a mirror and a painting. Ramsey’s theory, he said, makes the mistake that would be made by someone who said that you could use a painting as a mirror as well, even if only for a single posture. If we say this we overlook that what is essential to a mirror is precisely that you can infer from it the posture of a body in front of it, whereas in the case of the painting you have to know that the postures tally before you can construe the picture as a mirror image. (PG, 317)

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Ramsey’s response to this objection, however, would have been to grant that a descriptive name does not on its own express a proposition. After all, he was not claiming to be able by this means to enlarge the class of possible senses, but only to find new ways of picking out classes of propositions for the purposes of quantification. Wittgenstein also attempted what seems at first to be a different objection arising from the fact that PFEs are given as functions. Consider, for instance, the PFE fe purportedly defined earlier by means of a table. The signs ‘fe a’, ‘fe b’, ‘fe c’, Wittgenstein objected, are no more function and argument than the words ‘Co(rn)’, ‘Co(al)’ and ‘Co(at)’ are. (Here it makes no difference whether or not the ‘arguments’ ‘rn’, ‘al’, ‘it’ are used elsewhere as words.) (PG, 317) Is this objection valid? If we conceive of the definition as simply a way of attaching labels to the three propositions p, q and r, then it is. The occurrences of the letters ‘a’, ‘b’ and ‘c’ in these names is merely a contingent feature of the signs we have picked: they are, in the terminology of the Tracatus (5.02), affixes to a name rather than arguments to a propositional function. Surely, though, this is not the correct way to regard the definition. We make it clear in advance that what we are doing is not merely labelling the propositions but defining a mathematical function. If we admit this notion at all, it is hard to see how it can be illegitimate to define one in the finite case by means of a table of arguments and values. Indeed Ramsey could have pointed out that Wittgenstein himself had in effect done the same thing in the Tractatus when he regarded the direct enumeration (p, q, r) as a legitimate way of specifying a propositional variable. What, though, of the infinite case? Here, it is true, the idea of defining a function by means of a table involves a significant degree of idealization. Once again, though, Ramsey could have pointed out that the logic of the Tractatus is based in its entirety on the idea that the infinite case resembles the finite. Indeed, anyone who wanted to impose a more constructivist attitude to logic would already have objected to the first part of Ramsey’s article, where he replaced Russell’s elementary functions (finite truth-functions of atomic functions) with his own predicating functions (arbitrary truth-functions of the same). What Wittgenstein’s objection to PFEs amounted to was the complaint that the variable ‘x’ in ‘fe xˆ ’ does not occur in the manner of Begriffsschrift §9, and hence that this is not a case of ‘function and argument’ in Frege’s sense. Indeed not, but Ramsey did not claim that it was. His claim was that a PFE is a kind of symbol that collects propositions together for the purpose of quantification in cases where predicating functions have proved inadequate.

The demise of the transcendental argument One intriguing side-effect of Ramsey’s acceptance of PFEs was the collapse of his transcendental argument for the axiom of infinity. That argument, let us recall,

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relied on its being the case that if there were only N objects, then pn , which attempts to say that there are at least n objects, would be a tautology for n ≤ N and nonsense for n > N; so it would be impossible to say (even falsely) that there are more than N objects. Using PFEs, though, Ramsey could re-define pn as ∃x1 ∃x2 . . . ∃xn (x1 6=e x2 ∧ x1 6=e x3 ∧ · · · ∧ xn−1 6=e xn ); and for n > N this becomes not nonsense but a contradiction. If PFEs are accepted, then, the axiom of infinity is either a tautology or a contradiction, but we cannot tell which. Ramsey was therefore forced to attempt a regressive justification for it, namely that if we do not take it to be a primitive proposition, then ‘all analysis is self-contradictory and meaningless’ (FoM, 61). (In fact, he over-stated the point: if the axiom of infinity were a contradiction, not a tautology, then the propositions of higher mathematics would no doubt be almost all contradictory, but he should not have said that they would be meaningless.) This is a serious difficulty for the applicability of higher mathematics. It is not sufficient that the axiom of infinity should be consistent, since the applicability of mathematical analysis in physics ‘seems to require its mathematics to be true and not merely to follow from a possibly false hypothesis’ (FoM, 80–81). Ramsey’s logicism was therefore hostage to a hypothesis about the number of things in the world.

Further reading Ramsey’s notion of a PFE is criticized by Sullivan (1995) and Trueman (2011).

69 UNIVERSALS

Frege’s logic drew a binary distinction between saturated and unsaturated entities; Wittgenstein’s did not. In the Tractatus the components of an elementary proposition were all called names indifferently, and the corresponding constituents of an atomic fact were all called objects. Yet Wittgenstein offered no explanation for this departure from Frege’s views. Ramsey’s paper, ‘Universals’, is devoted to supplying the missing argument.

Incompleteness What concerned Ramsey was only whether logic should recognize a binary distinction, not whether metaphysics or psychology should. He offered no criticism, for instance, of the distinction between entities existing in one place and those existing in more than one; this distinction, precisely because it applies only to the spatio-temporal, could be of relevance only to physics, not to logic. Nor did he rule out that there might be a difference between two faculties—‘perception’ and ‘conception’, let us say—by means of which we become acquainted with entities; but this would be relevant only to psychology, not to logic. Ramsey’s principal target was Frege’s supposition that in an atomic sentence such as ‘Socrates is wise’ the decomposition into the function ‘ˆx is wise’ and the argument ‘Socrates’ reflects the true structure of the thought expressed. This analysis, Ramsey claimed, ‘has nothing to do with the logical nature of Socrates or wisdom, but is a matter entirely for grammarians’ (FoM, 116). It is ‘as clear as anything can be in philosophy’ that the sentence ‘wisdom is a characteristic of Socrates’ expresses exactly the same proposition, even though it would be analysed into ‘wisdom’ and ‘ˆq is a characteristic of Socrates’. Frege’s distinction reflects no

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fundamental difference, but is merely ‘a matter either of literary style, or of the point of view from which we approach the fact’ (FoM, 116). The distinction Ramsey was arguing against was originally Frege’s, then, but oddly Ramsey contrived not to mention Frege in the article at all, citing instead Russell’s 1918 lectures on logical atomism. The omission is particularly egregious because his central claim that ‘Socrates is wise’ and ‘wisdom is Socratic’ express the same proposition tacitly exploited Frege’s notion of conceptual content. On Ramsey’s view, Frege (and, following him, Russell) had wrongly assimilated two quite different things: first, the function-argument method of extracting a (complex) predicate from a sentence; second, the subject-predicate structure of an atomic sentence such as ‘Socrates is wise’. On the one hand, he thought, the former has no ontological implications: functions ‘do not mean single objects as names do, but have meaning in a more complicated way derived from the meanings of the propositions that are their values’ (FoM, 37). On the other hand, the latter carries no implication of functional incompleteness: wisdom is not in itself functionally incomplete any more than Socrates is. From an atomic proposiˆ and f xˆ ; but tion fa we may extract symmetrically two propositional functions, φa the two constituents of the proposition, f and a, are not themselves functionally incomplete.

Narrow and wide ranges The fact remains, though, that the decomposition of ‘Socrates is wise’ into ‘Socrates’ and ‘ˆx is wise’ seems overwhelmingly more natural to us than the— in Ramsey’s view equally defensible—decomposition into ‘wisdom’ and ‘ˆq is Socratic’. Why so? Ramsey’s diagnosis appealed to an apparent asymmetry previously noted by Johnson between particulars and universals (‘substantives’ and ‘adjectives’, as he called them). Johnson had argued that we may properly construct a compound adjective out of ‘simple’ adjectives, . . . yet the nature of any term functioning as a substantive is such that it is impossible to construct a genuine compound substantive. (1921-4, II, 61) On Johnson’s view, then, the compound ‘rational and animated’ is as genuine an adjective as its components, ‘rational’ and ‘animated’ (a fact confirmed by the fact that we have a single word ‘human’ to express it), whereas ‘Plato and Socrates’ remains resolutely compound and hence not a genuine substantive. Ramsey reexpressed this as a matter of scope, and hence as Russell’s distinction between complete and incomplete symbols in another guise. The scope ambiguity in ‘Plato and Socrates love someone’ shows, he thought, that ‘Plato and Socrates’ is incomplete (in Russell’s terminology) or compound (in Johnson’s); this explains our unwillingness to suppose that there is in reality a disjunctive entity denoted by it. An incomplete symbol is one for which we can distinguish two ranges of proposition: ‘the narrower range will be that in which the [symbol] has primary

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occurrence, the wider range that in which it has any sort of occurrence primary or secondary’ (FoM, 127). This explains the ‘felt difference’ between Socrates and wisdom. In the case of wisdom we can distinguish two ranges: the narrow range consists of all the values of the propositional function ‘ˆx is wise’, e.g. ‘Plato is wise’, ‘Aristotle is wise’, etc.; the wide range consists of the values of the secondˆ x is wise), and thus also includes such propositions as ‘Everyone order function Φ(ˆ is wise’ and ‘Aristotle is wise unless Plato is foolish’. In the case of Socrates, on the other hand, there seems to be only a single range consisting of the values of φ(Socrates), i.e. all propositions whatever that ascribe a property to Socrates. On Ramsey’s view, however, this was where Johnson was mistaken. There indeed seems to be only a single range, but we could draw an exactly analogous distinction between narrow and wide ranges for Socrates if we wanted to: the narrow variable would range only over propositions of the form ‘Socrates is q’, where q is a simple property; the wide range would include all propositions ascribing a property, simple or complex, to Socrates. Symmetry would then be restored. The ‘felt difference’ between ‘Socrates’ and ‘wise’ was an illusion created by the fact that we do not normally regard the distinction between simple and complex properties as important. Once we see past this, we can recognize that complex adjectives, just as much as complex substantives, are incomplete symbols that can be eliminated contextually. Several distinguished commentators (e.g. Anscombe 1965, Dummett 1973, 61ff) have attempted to counter Ramsey’s argument by pointing out that an adjective can be negated but a substantive cannot. Thus ‘wise’ has a negation ‘unwise’ such that ∀x(x is unwise ≡ ∼(x is wise)), but there is, they claim, no ‘anti-Socrates’ such that ∀φ(φ(anti-Socrates) ≡ ∼φ(Socrates)). In fact, though, these commentators have failed to notice that Ramsey had already noted this argument and responded to it. The two cases they compared are not really alike: in the first, x has the narrow range; in the second, φ has the wider. If the variable in the first case had the wider range, there would not in general be an adjective satisfying the required equivalence; and if, conversely, we restricted the variable in the second case to range only over simple properties, then substantives would have contradictories (albeit complex ones). The felt difference between substantives and adjectives in relation to negation is once again evidence only of our subjective interests, not of the ultimate structure of the world.

Complex universals Ramsey thus argued that apparent reference to compound universals, such as ‘being wise unless Plato is foolish’, should be eliminated. However, he also considered a second case, which he described merely as ‘simpler’, although in fact

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its character is different. This second case is that of apparent reference to a complex universal, where the complexity is internal to an atomic, not a compound, proposition. Suppose, for instance, that R is a binary relation, and we define fx =df xRb, gy =df aRy. One might think that f and g were complex universals, in which case we would have found the sought-after asymmetry, given that there is no analogous way of defining a logically complex particular. Ramsey disagreed. We have certainly defined two symbols ‘fx’ and ‘gy’, but that is quite a different matter from holding that f and g are entities in their own right. If we admitted such entities as these, we would be confronted by an ‘incomprehensible trinity, as senseless as that of theology’ (FoM, 118), consisting of three distinct analyses—aRb, fa and gb—of a single fact. The correct view, Ramsey held, is that ‘f ’ and ‘g’ are incomplete symbols, not names, and hence carry no ontological implications. Here, then, is Ramsey’s diagnosis of that ‘great muddle the theory of universals’. The logician calls the result of replacing names of individuals by variables ‘functions’, irrespective of whether the constant part of the function is a name or an incomplete symbol, because this does not make any difference to the class which the function defines. The failure to make this distinction has led to these functional symbols, some of which are names and some incomplete, being treated all alike as names of incomplete objects or properties. (FoM, 134)

Unigrade and multigrade terms Ramsey considered one further way, derived from Russell (PM, 2nd edn, I, xv), of distinguishing logically between particulars and universals, namely to observe that the constituents of atomic propositions split into two kinds, unigrade and multigrade: the former are signs for universals, the latter for particulars. ‘Socrates’, for instance, is multigrade, since it may occur either with a predicate or with a relation-sign, whereas ‘taller than’ is unigrade, since it requires exactly two signs to complete it. Ramsey responded to this by denying Russell’s claim to know what the forms of atomic propositions might be. ‘Of all philosophers,’ he said, ‘Wittgenstein alone has seen through this muddle and declared that about the forms of atomic propositions we can know nothing whatever.’ (FoM, 134) Ramsey gave neither a textual reference nor an explanation for this attribution, which overstates what Wittgenstein ‘declared’. Indeed he was explicit in the Tractatus that space, time and colouredness are forms of object. What he declared to be impossible was only

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knowing that we had found all the forms, not that we had found some of them. A year later Ramsey withdrew the stronger claim, having realized that in order to respond to Russell all he needed to deny was that it is in fact known what sorts of atomic propositions there are. From this it follows that Russell’s view cannot be positively asserted; and there is no strong presumption in its favour, for I think that the argument of my article establishes that nothing of the sort can be known a priori. (FoM, 135) Russell’s claim is anyway not as obvious as he seems to have supposed. Ordinary language contains many multigrade predicates, and he gave no argument why these should be eliminable in the final analysis. By then Russell had, as we have seen, adopted an essentially Fregean analysis according to which each predicate contains a fixed number of singular argument-places, and on such a conception no argument would seem necessary. In the present context, however, the validity of that conception is just what is in dispute, and so appealing to it would be question-begging.

Further reading Ramsey’s arguments are assessed by Simons (1991), MacBride (2005) and Hale (2006). Oliver & Smiley (2004) discuss multigrade predication in detail. On the criticism of Frege see Long (1969).

70 DEGREES OF BELIEF

In 1926 Ramsey wrote a paper on the theory of partial belief, part of which he read to the Moral Sciences Club. The paper has become famous for what is barely more than a passing remark, in which he originated the method of so-called ‘Dutch book’ arguments in decision theory.

Logical probabilities Ramsey’s interest in partial belief can be traced to his first philosophical publication, a review of Keynes’s Treatise on Probability (1921) which appeared in the Cambridge Review in 1922. Dominant at that time, in Cambridge at least, was the view of Johnson (1921-4, I, xxxix–xl) and Keynes that there is a logical relation of probability between propositions, and hence that probability is part of logic. Keynes held that sometimes, but not always, this relation is expressible numerically as a real number P(p|q), and in these cases the real numbers obey the laws of probability theory, which therefore becomes subsumed within logic. In his review, Ramsey accused Keynes of confusing the objectivity of a claim with the objectivity of our ground for it, but he did not express hostility towards the very idea of logical probability relations. He suggested that Keynes’s argument for non-numerical probability relations was really just an argument that we do not know what the value is, but he did not object to the part of Keynes’s theory which deals with cases in which the probability relations are numerical. Keynes based his treatment of probability relations on the ‘principle of indifference’, the claim that there are basic classes of propositions which, prior to any more specific evidence, are of equal probability. Thus ascriptions of a finite range of determinates that fall under a single determinable (e.g. colour) might be such a basic class. In his review Ramsey noted that such an account ‘can be applied to dice, coins and cards, but not to such cases as the position of a point on a line, in

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which the number of possible absolute determinates (e.g., points on the line) is infinite’ (1922, 4). This suggests a significant restriction on the scope of Keynes’s theory, but does not refute it altogether. In a talk he gave as an undergraduate Ramsey acquiesced readily enough in Wittgenstein’s rather similar Tractarian principle of indifference, according to which every elementary proposition has probability 12 of being true. This view is highly implausible, however: there are many propositions to which there seems no rational reason to allocate any particular probability whatever; giving them the probability 12 is hardly a solution. By 1926, however, Ramsey was prepared to doubt whether there are the logical probability relations which Keynes, Johnson or Wittgenstein had proposed. In support of this doubt he reported that he did not perceive them—a report which, like most such first-person reports, was hardly likely to persuade his opponent. Aware, perhaps, of its inadequacy as an argument, he then offered the rather more pertinent observation that anyone who tries to decide by Mr Keynes’ methods what are the proper alternatives to regard as equally probable in molecular mechanics . . . will soon be convinced that it is a matter of physics rather than pure logic. (FoM, 189) Ramsey’s objection was thus only to the interpretation of the probability calculus in terms of logical relations, not to an interpretation in terms of physical propensities or long-run frequencies.

Betting In his 1926 paper Ramsey rejected Keynes’s notion of logical probabilities, but nonetheless hoped to show that formal logic has some normative role in constraining partial belief, just as it constrains full belief. Formal logic does not tell us whether to believe p or not-p, but only not to believe both of them. And in something like the same way, he thought, if we partially believe p to degree α, it tells us to partially believe not-p to degree 1 − α. To see why, it will be convenient to begin by simply supposing that there is such a notion as ‘believing p to degree α’ and ask how this is to be measured. (We shall return to the problem of how to define it later.) Ramsey quickly dismissed the idea that α might be thought of as the intensity of the feeling of belief: for there are many beliefs of which we are quite certain without feeling strongly about the matter; and of course someone’s report of their degree of belief in a proposition may not be accurate even if it is sincere. He therefore proposed to use instead the ‘old established’ (FoM, 172) idea that your degree of belief in a proposition p can be measured (approximately, at least) by offering you a bet on the occurrence of p. This idea goes back at least to Venn (1866, ch. 6) if not before, although Ramsey may have been made aware of it when he read a collection of essays by Peirce (1923, ch. 3) in January 1924. Betting odds are

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conventionally expressed as ratios of the form M : N, where £N is the stake you pay to the bookmaker and £(M + N) is the amount you get back if you win, but for mathematical purposes the fraction α = N/(M + N), known as the ‘betting ratio’, is more convenient. So if you bet £N at a ratio of α, you end up better off by £N(1/α − 1) if you win, worse off by £N if not. Ramsey’s starting point was the idea that your acceptance of the bet might be taken as evidence that your degree of belief in p is greater than α. There are several difficulties with this idea, however. If α is close to 1, you may decline to bet not through lack of certainty in the outcome but because even if you win, your gain will be too small to be worth the trouble; and if I try to make the bet interesting to you by increasing the stake, there is the opposite problem that you may be unable to afford it. On the other hand, if α is close to 0 (as in most lotteries, for example), a significant part of your stake may be a payment for the thrill of betting rather than an expression of your degree of belief in the outcome. For these reasons betting may be a poor way of measuring degrees of belief close to 0 or 1. A further difficulty arises if a certain quantum of money is important to you. If, for instance, you have only 90p in your pocket but the bus fare home is £1, you might bet on a coin toss at 1:9 if those are the only odds the bookmaker at the bus stop is willing to offer you, not because you believe the coin is biased but because the bet gives you an even chance of being able to take the bus rather than face a long walk home. Remember, too, that some people will always refuse to bet for religious reasons. This is an obstacle to defining degrees of belief in terms of bets: we plainly do not want to say that principled non-betters have no degrees of belief at all. Admittedly insurance is also a kind of betting, more tolerated by most religions, but this tolerance does not help us much with the current problem, since it extends only to events affecting policyholders adversely, whereas they plainly have partial beliefs concerning many other events too. (Negative insurance, where you lose if something pleasant happens to you, would of course be possible in principle, but there seems to be less demand for it.) Some of the literature on degrees of belief tries to get round the difficulty by hypothesizing that you are compelled to bet, but this ignores the obvious problem that the very compulsion might radically alter the degrees of belief we are attempting to measure. For all these reasons, then, the method of betting odds is, as Ramsey himself recognized, at best an imperfect and approximate way of measuring a subject’s degree of belief: working the theory out in detail would, he said, ‘be rather like working out to seven places of decimals a result only valid to two’ (FoM, 180).

Synchronic Dutch books What degree of belief you have in a proposition is a fact of your psychology, and measuring it is therefore a problem for psychology. Ramsey suggested, however, that the possibility of what are now known as Dutch books places constraints not

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on what your degrees of belief are, but on what they need to be to avoid the risk of financial loss. A set of bets or ‘book’ is said to be ‘Dutch’—the term was introduced into the probability literature by Lehman (1955, 251)—if one or other party is sure to lose in any event. Dutch books are readily available in practice. On a major sporting event with only two possible outcomes, odds will typically be offered such that anyone who bets both ways in certain proportions is sure to lose: this is the bookmaker’s profit margin. As long as no one person does this, it does not show that any one person is being inconsistent, of course, but only that betters differ in their degrees of belief towards the same events. If you object to betting for religious reasons, you can obtain a Dutch book by buying both a life insurance policy, which pays out when you die, and an annuity, which pays out while you don’t. (Such Dutch books have sometimes been used as a way of avoiding inheritance tax, the tax saved more than compensating for the loss.) A similar idea is involved in the practice of arbitrage whereby stock market traders seek to profit from inconsistent derivative prices. Ramsey’s idea was to show that if you wish to avoid Dutch books, you need to ensure that your degrees of belief obey the laws of probability theory. Suppose, for instance, that I offer you a betting ratio of α on p and β on ∼p. You place bets at these odds of £α and £β respectively. You are then sure to get back £1 either way: on the first bet if p occurs; on the second if not. Hence if α + β > 1, you are sure to lose, but if α + β < 1, then I am. For neither party to be acting inconsistently, we must have α + β = 1, i.e. P(p) + P(∼p) = 1. Similarly, let α and β be the betting ratios I offer you on disjoint events p and q respectively, and γ the ratio you offer me on p ∨ q. Suppose that you place with me bets of £α and £β on p and q respectively, and I place with you £γ on p ∨ q. If either p or q occurs, you will pay me £1, but I will have to pay it back to settle the bet you placed with me; if neither p nor q occurs, neither of us has to pay out. So your overall gain on these bets will be £(α + β − γ ) (the difference in the original stakes) come what may: if α + β < γ , you are sure to lose; if α + β > γ , then I am. As before, therefore, consistency requires that α + β = γ . So P(p) + P(q) = P(p ∨ q). These arguments demonstrate that if your bets contravene the laws of probability theory, you are vulnerable to a Dutch book. Ramsey also stated, but did not prove, the converse theorem (Lehman 1955, Kemeny 1955) that if your bets are conform to these laws, no Dutch book can be made against you. This does not show, though, that if your partial beliefs obey the laws of probability, your life will go better, or even that you will win more bets. That depends on how many Dutch bookmakers you encounter. Even if the odds on offer are inconsistent, the book will not be Dutch unless the stakes are in the right proportions, whereas in practice bookmakers let punters choose their stakes (within wide limits). And of course whether someone’s bets pay off is an entirely different matter from whether they are consistent: you could bet in perfect consistency and yet always lose. This

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is what places probability theory within the logic of consistency, not the logic of truth.

Diachronic Dutch books A Dutch book argument may also be used to show that the odds you should in consistency accept on a conditional bet on p given q (i.e. a bet which is void if q does not occur) is P(p|q) = P(p ∧ q)/P(q). Your odds should therefore in consistency accord with this rule if you promise in advance to bet if q occurs, since that is in effect just a conditional bet. And in a similar manner you will be susceptible to a Dutch book if you do not overtly promise, but nonetheless somehow reveal your future betting intentions to the bookmaker. What, though, if you have neither made any binding promises nor revealed your future intentions? It is clear what you should do in such circumstances if you wish to act consistently with your past self. ‘Obviously’, Ramsey said, if p is the fact observed, my degree of belief in q after the observation should be equal to my degree of belief in q given p before. . . . When my degrees of belief change in this way we can say that they have been changed consistently by my observation. It is a quite separate question, though, whether you ought to revise your degrees of belief in this manner. Diachronic consistency is quite different from the synchronic consistency considered earlier, and it is hard to see what Dutch book could be made against you in these circumstances. As Ramsey noted later in the article, therefore, the degree of someone’s belief in q given p is not the same as the degree to which he would believe q, if he believed p for certain; for knowledge of q might for psychological reasons profoundly alter his whole system of beliefs. (FoM, 180) That observation, though, is only descriptive. What we really want to know is whether rationality demands diachronic consistency. Putting all your faith in Bayesian updating in this manner would absolve you altogether from exercising judgment and place all the weight instead on the reliability of your initial degrees of belief before the process of updating begins. Given that it lacks even a Dutch book argument to recommend it, why do it? This highlights an underlying difficulty in choosing optimal strategies for the future, namely that when the time comes, you may have changed your mind. Ulysses had to decide whether to stay at the helm as he sailed towards the Sirens and risk being lured by their call to jump to his doom, or tie himself to the mast to avoid this fate but risk losing control of his ship. The best strategy would have been to stay at the helm and then not jump: only his psychological frailty rendered that impracticable (see van Fraassen 1995). The problem, however, is

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that this is not a general rule. As Keynes is supposed to have said (but probably didn’t), ‘When my information changes, I change my mind. What do you do?’

Degrees of belief and frequencies So far, we have been supposing that degrees of belief are there to be measured, but we have postponed the question of what they are. Ramsey held that a degree of belief is a causal property of a mental state, which he defined in terms of the frequency with which the proposition partially believed turn out to be true. Supposing goods to be additive, belief of degree mn is the sort of belief which leads to the action which would be best if repeated n times in m of which the proposition is true. (FoM, 188) This lax formulation masks a difficulty, however. A proposition is true or false simpliciter: it cannot have different truth values on different occasions. What Ramsey meant (and what he actually said just before the lax formulation just quoted) was that we have to replace the proposition with a propositional function whose variable indexes the occasion on which the action occurs. A partial belief cannot in general be connected uniquely with any actual frequency, for the connection is always made by taking the proposition in question as an instance of a propositional function. What propositional function we choose is to some extent arbitrary and the corresponding frequency will vary considerably with our choice. (FoM, 188) This, though, represents a major shift: previously we had been discussing only the odds at which we place a one-off bet; now we discover that Ramsey’s definition of degrees of belief applies only to a sequence of bets on instances of a propositional function. If, for instance, I keep betting on the toss of a fair coin at odds less than 12 , in the long run I will almost surely lose. So ‘the very idea of partial belief involves reference to a hypothetical or ideal frequency’ (FoM, 188). The difficulty with this is that it seriously undermines the objectivity of the proposed criterion. How likely am I to die in the next decade, for instance? I could perhaps consult various actuarial tables, giving death rates for men of my age born in Scotland, for non-smokers, for Cambridge philosophy professors, for Oxford graduates, etc. Which of these frequencies ought my degree of belief to match? One is tempted to reply: all of them. What I am after, that is to say, is the life expectancy of male, Scottish-born, non-smoking, Oxford-educated Cambridge philosophy professors. The problem is that there has only ever been one of those, and when he dies, it will be too late for me to adjust my beliefs. So on Ramsey’s account my current degree of belief in my own survival is left in limbo until I choose the propositional function (nowadays known as the ‘reference class’) of which to regard it as an instance. A life insurance company will assess the terms it offers me by assigning me to some reference class, and whether it makes a profit on

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the policies it writes on that class will depend on the long-run life expectancy of its members. But I care about the life expectancy of non-smoking Cambridge philosophers only insofar as it is an accurate indicator of mine. The decision I confront is only whether to insure one life, and it seems odd to say that my belief about how likely I am to survive can have no determinate degree until I have placed myself in some particular reference class. What if I am in the intersection of two different reference classes and know the long-run frequencies for both of them? To which of them should my degree of belief conform? Ramsey’s somewhat downbeat answer was that ‘I can but put the evidence before me, and let it act on my mind’ (FoM, 202).

Partial belief and desire By referring earlier to the constraints imposed by a desire to avoid Dutch books as those of ‘consistency’ I was following Ramsey, who argued that these constraints are analogous to those which conventional formal logic imposes on full beliefs. This argument was weakened as it progressed, however, by the deficiencies of the method of measuring partial beliefs by means of betting odds. In offering bets, he suggested, we may measure things other than the subject’s degree of belief, ‘partly because of the diminishing marginal utility of money, partly because the person may have a special eagerness or reluctance to bet’ (FoM, 172). To minimize this difficulty, he proposed instead to pose the argument in terms of bets on any goods the agent desires. When phrased in this way, the argument cannot be dodged simply by declining to bet. All that is required is that we should be willing to make choices between outcomes, some of which we desire more than others, and no religion outlaws that, since it is what we have to do in order to count as having a life at all. All our lives we are in a sense betting. Whenever we go to the station we are betting that a train will really run, and if we had not a sufficient degree of belief in this we should decline the bet and stay at home. (FoM, 183) At the centre of Ramsey’s paper was the technical project of proving a representation theorem to the effect that if your degrees of belief and utilities obey certain assumptions, they may be assigned real number values. He took the basic idea here from the classical theory of measurement. What made the problem more difficult in his case, however, was that it now had two dimensions of variation rather than one: he had to measure both your degree of belief and your utility function. The trick he used to tackle this problem was to suppose that there are some propositions towards the occurrence or non-occurrence of which you are indifferent. (Ramsey called these, somewhat tendentiously, ‘ethically neutral’ propositions.) His idea was that he could start the process of measurement by assigning a degree of belief 21 to any ethically neutral proposition on which you are indifferent to betting one way or the other for the same stake. Using such a proposition as

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a reference point, he could then extend the scheme of measurement to other utilities and degrees of belief by comparison. Notice, though, that if we bet on utils rather than on money, the Dutch book argument does not deliver an unconditional conclusion, but shows only that the degrees of belief should obey the axioms of probability theory if the utilities are additive. The earlier example with the bus fare shows that the utility of money is not additive; and so we need an argument to show that there is some other commodity whose utility to us is additive. Why should that argument be any easier to come by than the argument that degrees of belief are additive (see Schick 1986)? Notice too that whether an outcome is desirable is a subjective matter. (Suppose that you offer me a Dutch book in which stakes and winnings are transacted in apples: this will suit me very well if my orchard has a glut of apples I want to be rid of.) This underlines the point that Ramsey’s representation theorem does not offer a way of measuring degrees of belief simpliciter, but only of linking them to the strength of desires, which may be just as hard to determine.

Further reading Gillies (2000) expounds the reference class problem in more detail. Kyburg (1978) and Maher (1993, §4.6) discuss the limitations of Dutch books in determining degrees of belief. van Fraassen (1984) discusses diachronic Dutch book arguments. Mellor (2005) discusses the boundary between descriptive psychology and human logic from a slightly different perspective. Eriksson & Hájek (2007) discuss the problem of defining degrees of belief. On Bayesian updating see Earman (1992, ch. 6).

71 FACTS AND PROPOSITIONS

A year after ‘Truth and probability’ Ramsey wrote ‘Facts and propositions’ for a symposium at the Joint Session, with Moore as his respondent. In the modern literature this paper is often said to be one of Ramsey’s greatest achievements, but his own assessment of it was much more modest. The paper, he said, aimed merely ‘to fill up a gap’ in Wittgenstein’s system (FoM, 155): although Wittgenstein offered an account of belief (that ‘A believes that p’ has the same form as ‘ “p’ says that p’) he did not explain which propositional signs express which propositions— he did not supply a ‘theory of meaning’, on one understanding of that ambiguous phrase. Moreover, Ramsey’s attempt to fill up this gap in Wittgenstein’s account did not go much further in 1927 than it had in the Critical Notice of 1923. When Braithwaite’s posthumous edition of Ramsey’s philosophical papers appeared in 1931, Russell’s review of it was (unsurprisingly in the circumstances) laudatory in tone. The one notable exception was that he dismissed ‘Facts and propositions’ as containing little that was original. Russell was thus agreeing with Ramsey’s own assessment in the paper itself. Everything that I have said is due to [Mr. Wittgenstein], except the parts which have a pragmatist tendency. . . . My pragmatism is derived from Mr. Russell; and is, of course, very vague and undeveloped. (FoM, 155) In this chapter we shall focus on one respect in which Ramsey’s modesty was misplaced, namely an argument which may shed light on his previous paper, ‘Truth and probability’.

Chicken beliefs In ‘Facts and propositions’ Ramsey made one significant point that is not to be found already in his earlier work; but he made it, as was often his way, so swiftly

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that many commentators (Russell, perhaps, among them) have missed its significance. It concerns Russell’s attempt to provide an account of belief that would apply to non-linguistic animals as well as to humans. According to this account, an animal’s dispositional state is ‘true’ just in case the state leads to results it finds satisfactory. In his article Ramsey paused briefly to consider such an account. It is, for instance, possible to say that a chicken believes a certain sort of caterpillar to be poisonous, and mean by that merely that it abstains from eating such caterpillars on account of unpleasant experiences connected with them. (FoM, 144) If we spoke in this way, we would then owe an analysis of how the mental factor in the belief—the chicken’s dispositions to adopt certain behaviour—is related to the objective factor—the toxicity of the caterpillars. Russell had proposed that pragmatism may be able to offer such an analysis, namely that the chicken’s actions were such as to be useful if, and only if, the caterpillars were actually poisonous. Thus any set of actions for whose utility p is a necessary and sufficient condition might be called a belief that p, and so would be true if p, i.e. if they are useful. (Ibid.) The kind of belief for which this account can be offered is now known, following Ramsey’s example, as a ‘chicken-belief ’. An agent is said to have a chicken-belief that p just in case the agent acts in ways that are useful to it if and only if p. Having introduced Russell’s account, however, Ramsey went on straight away to say that ‘without wishing to depreciate the importance of this kind of belief ’, he proposed to focus instead on those beliefs which are expressed in words, or possibly images or other symbols, consciously asserted or denied; for those beliefs, in my view are the most proper subject for logical criticism. (FoM, 144) Why did he think that chicken-beliefs are not a ‘proper subject for logical criticism’? His argument is contained in a highly compressed footnote. A chickenbelief that p is a set of actions that is useful if and only if p. So it is useful to believe that p just in case it is useful to do things which are useful if and only if p. But this last is equivalent to p. So the proposal entails that it is useful for a chicken to believe a proposition just in case the proposition is true. Whether or not this is the right thing to say about the beliefs of chickens, it is plainly the wrong thing to say about ours, since it is sometimes useful to hold false ones. A chicken-belief that p is beholden to an external standard, namely whether p, but it is assessed only relative to the usefulness of the chicken’s actions, and usefulness is not an absolute matter but relative to the goal being pursued. Neither chickens nor humans have their purposes inscribed on their breasts, and whether an action is useful to you depends on your goals. If I criticized one of your chicken-beliefs on the ground that your actions turned out not to be useful to you, it would be

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open to you to claim that I had mistaken your intentions and that your actions had indeed been useful relative to your purposes. The reason that a chickenbelief cannot be a ‘proper subject for logical criticism’ is that any assessment of it must depend on what is useful to the chicken, and that is not a matter on which formal logic can be expected to take a stance. When it comes to chickens, this point is perhaps obscured by our tendency to assume that we know their goals without having to ask. In articulating these goals we either impose some teleological conception of what constitutes flourishing for a chicken or import an evolutionary explanation. A human belief that is expressed propositionally, by contrast, can be criticized—at least by anyone who, like Ramsey, has rejected the pragmatist account of truth—simply for being false, irrespective of its usefulness.

Partial belief and chickens Let us return now to ‘Truth and probability’. It is surely worthy of note that Ramsey did not himself publish this paper. Why not? Its defect, he later said, was that it took partial belief as a psychological phenomenon to be defined and measured by a psychologist. But this sort of psychology goes a very little way and would be quite unacceptable in a developed science. (FoM, 256) There are perhaps two parts to his complaint: on the one hand, he seems to have thought that the analysis he offered was superficial when regarded as descriptive psychology; on the other, he seems to have objected to treating partial belief as a ‘psychological phenomenon’ at all. For the second of these, if not the first, Ramsey’s objection to chicken-beliefs provides us with a possible explanation. He had, let us recall, defined belief of degree mn to be ‘the sort of belief which leads to the action which would be best if repeated n times in m of which the proposition is true’. (FoM, 188) So a degree of belief 1 is one that leads to the action which would be best if the proposition were true; but this is just the definition of a chicken-belief. So if chicken-beliefs are not a ‘proper subject for logical criticism’, partial beliefs are not either. This is not to say that his definition of partial belief is wrong in itself, of course. In ‘Truth and probability’, however, Ramsey claimed (somewhat obscurely, it should be said) that the laws of probability theory are not merely an approximate description of the degrees of belief that we in fact adopt, but belong to the logic of consistency and hence to that extent provide a normative constraint on how we ought to form our degrees of belief. ‘There is’, he said, ‘a theory of consistency in partial beliefs just as much as of consistency in certain beliefs.’ (FoM, 186) It is this last claim to have situated his account of partial belief within formal logic that is threatened by the objection to chicken-beliefs he expressed in ‘Facts and propositions’. Ramsey’s mistake was his claim that ‘all our lives we are in a sense betting’ (FoM, 183). This remark is too casual to bear the weight he placed on it. All our

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lives we are no doubt making choices about action whose consequences matter to us, but that is some way short of betting. The key difference between betting on a horse and acting on the belief that a train will run is that in the former case we must agree in advance what will count as winning, whereas in the latter the desire we are trying to satisfy may be (and usually is) left wholly unsaid. You may go to the station intending to catch a train which has in fact been cancelled, but be satisfied with the replacement bus service that runs in its stead. To avoid this ambiguity, I would have to get you to agree to bet on articulated (and hence propositional) outcomes. Already in 1926 Ramsey had at least an inkling of a related point. He claimed that his account was approximately true as a description of how we in fact act, assuming ‘the theory that we act in the way we think most likely to realize the objects of our desires’. This theory, he admitted, ‘cannot be made adequate to all the facts’—presumably he had in mind the Freudian view that our conscious desires are often hijacked by the subconscious, leading us to do what is not in our own conscious best interests—but it is nonetheless ‘a useful approximation to the truth particularly in the case of our self-conscious or professional life’ (FoM, 173). Once we generalize the theory to encompass all our actions rather than just our betting strategies, therefore, it belongs not to formal logic but to psychology. That is not to condemn it, of course, but only to highlight an inadequacy in Ramsey’s paper in the form in which he wrote it in 1926, namely that it does not identify with any clarity the point at which this boundary is crossed. In notes written in 1929, which seem to derive from a conversation with Wittgenstein, Ramsey observed that ‘what is wrong with my probability is its externality’ (N, 277). The difficulty we have just been discussing is indeed a close relative of the objection Wittgenstein made in the Philosophical Remarks to Russell’s account of belief. As we saw in Chapter 43, Wittgenstein complained that Russell made whether an action satisfies my wish external to how the wish is articulated (PR, §21), and therefore failed to acknowledge sufficiently the categorial distinction between chicken-beliefs and propositionally articulated thoughts. Ramsey’s account of degrees of belief similarly failed to arrive at the required articulation.

Further reading Bermudez (2003, ch. 4) is unusual in discussing chicken-beliefs in the setting to which Ramsey thought them suited, namely that of ascribing beliefs to nonlinguistic animals; for criticisms see Beck (2013). On the prospects for an evolutionary explanation of propositional representation see Putnam (1992, ch. 2).

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Wittgenstein returned to Cambridge in January 1929, and he and Ramsey met frequently to discuss philosophy during that year. Some of Ramsey’s notes from that time explore issues in the philosophy of science, the common theme of which is the attempt, inadequately explained in the Tractatus, to express worldly regularities by means of scientific laws. It should be stressed, though, that many of these late manuscripts were plainly experimental in character, and it is difficult to say with any confidence in what direction he might have taken his ideas if he had lived. This includes the book manuscript, On Truth, of which Braithwaite said, ‘He was profoundly dissatisfied with it.’ (FoM, xiv) Braithwaite also reported that ‘in 1929 Ramsey was converted to a finitist view which rejects the existence of any actual infinite aggregate’ (FoM, xii). The surviving papers confirm his interest in finitism, but unfortunately do not reveal the reasons for his conversion or any interesting consequences of it.

Laws Central to Ramsey’s late work was the distinction between what Johnson (19214, III, ch. 1, §3) called ‘universals of fact’ and ‘universals of law’. If we accept a Tractarian account of the former as (in effect) conjunctions, the problem is then to explain the latter—what Ramsey called ‘variable hypotheticals’ (FoM, 237). There is something right, he thought, about Johnson’s idea that a universal of law is simply more general than a universal of fact. ‘All the practical man wants to know’, he said, ‘is that all people who take arsenic die, not that there is a causal implication.’ The problem is to explain the notion of scope that is in play here, given the Tractarian framework which does not allow for variation in the size

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of the domain. Braithwaite (1927) proposed that universals of law are those the ground of our knowledge of which is non-demonstrative; but Ramsey rejected this, because some universals of fact are believed on non-demonstrative grounds (e.g. ‘Everyone there was asleep’, believed on the ground of testimony) and some universals of law are believed on demonstrative grounds (e.g. ‘Whenever this balloon was filled with hydrogen and let go, it rose’, believed on the ground of observing all its instances). In the Tractatus Wittgenstein at least distinguished between generalizations that are accidentally and necessarily true, but since he allowed only one kind of necessity, he could make no room in his formalism for non-logical laws (e.g. laws of physics). He attempted to explain such laws as patterns unsayably exhibited by the true accidental generalizations, thus rendering obscure the application of his view to explicit cases. He was similarly obscure in his account of colour vocabulary. Once it is accepted that the incompatibility of ‘This is red’ and ‘This is green’ is not a tautology, the question of its status becomes urgent. Since it is evidently implausible to insist that the incompatibility is unsayable, he was driven to recognize a category of sentences—grammatical rules—that have a use and yet do not belong to the primary system. The first product of his return to Cambridge in 1929 was the paper ‘Some remarks on logical form’, in which he argued for the need for such rules. In response to these difficulties Ramsey began to explore the idea that in addition to the language described in the Tractatus (the ‘primary system’) there is another (the ‘secondary system’) consisting of general principles with which we ‘meet the future’ (FoM, 241). He conceived of the primary system as somehow grounded in experience—he did not say exactly how—so that its scope would depend on the perceptual faculties of its owner: for a blind man, for instance, colour-words would belong to the secondary system; if he were operated on so as to give him sight, they would become part of his primary system (FoM, 261). This was a precursor of the distinction, widely used by the logical positivists, between observational and theoretical vocabularies. It is by no means clear, however, whether the distinction is as clear-cut as they liked to think (see Putnam 1962).

The Ramsey test Consider now a conditional that is not straightforwardly truth-functional, because it belongs to the secondary system. Ramsey made (in a footnote, typically enough) a proposal for interpreting this. If two people are arguing ‘If p will q?’ and both are in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q. (FoM, 247) Later logicians have extracted from this remark the ‘Ramsey test’, according to which one should believe ‘If p, q’ to the same extent that one believes q given p.

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Notice, though, that this need not be the extent to which one would believe q if one did become certain that p, because coming to believe p might alter one’s system of belief (FoM, 180); rather is it the extent to which one should now promise to bet on q given p (if one is inclined to bet at all). In effect, this account of conditionals turns a variable hypothetical into a promise: to say ‘All φs are ψs’ is to promise, if you meet a φ, to regard it as a ψ. Variable hypotheticals therefore do not have truth-conditions any more than promises do, but that is not to say that they cannot be refuted by the evidence. Even if we promised to treat every φ as a ψ, being presented with a φ that is manifestly not a ψ would compel us to abandon our promise. Our aim should presumably be to choose a secondary system whose consequences in the primary system match the known facts as far as practicable. The ideal, of course, would be to choose a system that matches all the facts, not just the known ones, but it is not given to ordinary mortals to achieve this. It is worth noting the pragmatist spirit of this account of the secondary system, but also Ramsey’s explicit statement that he did not intend it to apply to the primary system. This is Peirce’s notion of truth as what everyone will believe in the end; it does not apply to the truthful statement of matters of fact, but to the ‘true scientific system’. (FoM, 253) It should be noted that at this point Ramsey’s account was (in the terminology of Chapter 55) inaustere. He aimed to give the sentences of the secondary system an explicable linguistic role, even though they are in Tractarian terms unsayable. We saw earlier that the austere conception of nonsense followed from Wittgenstein’s claim that ‘all propositions of our colloquial language are actually, just as they are, logically completely in order’ (5.5563). It is therefore no surprise to find Ramsey dismissing this remark as ‘absolutely false’ (N, 277), a ‘typical piece of scholasticism’ which treats ‘what is vague as if it were precise’ (FoM, 269). Ramsey also differed from Wittgenstein, it should be said, in his attitude to the ethical unsayable. His militant atheism disinclined him to treat what cannot be said as nonetheless ineffably profound. He never felt the attraction of viewing the world sub specie aeternitatis. Rather, he said, his picture of the world was drawn in perspective and not like a model to scale. The foreground is occupied by human beings and the stars are all as small as three-penny bits. I don’t really believe in astronomy, except as a complicated description of part of the course of human and possibly animal sensation. I apply my perspective not merely to space but also to time. In time the world will cool and everything will die; but that is a long time off still and its present value at compound discount is almost nothing. (FoM, 249)

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Causation As we saw in Chapter 39, Russell had argued that causation has no place in modern physics, because physics is time-symmetric, causation not. Its role in ordinary life, he suggested, is due to the asymmetry of human memory: we remember past events but not future ones. Ramsey amended this proposal by holding that the relevant notion is agency, not human memory. Causal laws, he claimed, are variable hypotheticals that we use in deliberating. Their temporal asymmetry is explained by the fact that when deliberating we treat the past as fixed and our own proposed action as ‘an ultimate and the only ultimate contingency’ (FoM, 250). The past, we think, is settled; if this means more than that it is past, it might mean that it is settled for us, that nothing now could change our opinion of it, that any present event is irrelevant to the probability for us of any past event. But that is plainly untrue. What is true is this, that any possible present volition of ours is (for us) irrelevant to any past event. To another (or to ourselves in the future) it can serve as a sign of the past, but to us now what we do affects only the probability of the future. (FoM, 250) Ramsey thus aspired not to eliminate causation from science, as Russell had done, but rather more modestly to classify it as belonging to human, not to formal, logic. The principal difficulty with this, however, is that Ramsey did not explain why past events are irrelevant to our current deliberations. We are no doubt in the habit of thinking the past is settled, but he seemed to leave it open that one could resist this habit by a sheer act of will. Would someone who did so be able to affect the past? Presumably not. Perhaps there is something right about the idea that as a matter of psychological necessity we have to deliberate as if the past is fixed; perhaps, too, we have to deliberate as if our actions are not wholly determined by the laws of physics. The crucial question, though, is whether we are right to do so. Some philosophers have argued that there is nothing incoherent about supposing that our actions might sometimes affect the past. For instance, Newcomb’s paradox is held to demonstrate this for the case in which someone else is able to predict with high accuracy what our actions will be.

Theories A further problem in interpreting sentences of the secondary system is that they may use theoretical terms such as ‘electron’, ‘mass’ or ‘magnetic field’. Given that we have, presumably, nothing akin to Russellian acquaintance with these terms, some kind of semantic holism seems appropriate for them: they acquire their meaning from the role they play in the theory in which they occur, and this in turn acquires its justification from its consequences in the primary system. Put more formally, the secondary system contains various propositional functions together with axioms and a dictionary. We aim to select these in such a way that the consequences in the primary system are true.

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But this way of presenting the matter does not really capture the holism just mentioned. In order to understand the theory, it seems as if we need to understand each of the propositional functions mentioned in it, and hence to grasp already the words involved in expressing it. But where do we look for an understanding of words like ‘electron’ independent of the axioms of the theory? Ramsey proposed to deal with this problem by replacing the theoretical terms with existentially quantified variables. Consider for simplicity a theory with a single axiom T(τ1 , . . . , τm , φ1 , . . . , φn ), where τ1 , . . . , τm are terms of the secondary, scientific language and φ1 , . . . , φn terms of the primary language. The Ramsey sentence of T is then the sentence R(T) =df ∃X1 . . . ∃Xn T(X1 , . . . , Xm , φ1 , . . . , φn ). Instrumentally, R(T) and T play interchangeable roles, because their logical consequences in the primary language are the same; but R(T) contains no terms of the secondary language. One neat consequence is that if I use a theoretical term differently from you, we need not be disagreeing: our terms are just variables inside the scope of different quantifiers. Disagreement between theories arises only in relation to their consequences in the primary language. Notice, though, that those who attempt to couple Ramsey’s proposal with a realist understanding of the quantifier in question depart from his intention. He regarded the secondary system as fictitious and held that realism concerning it is ‘foolish’. ‘ “There is such a quality as mass” is nonsense unless it means merely to affirm the consequences of a mechanical theory.’ (FoM, 261) On his view, therefore, the point of replacing the theoretical terms of a theory with variables is not only to substitute the general for the specific, but to enable a deductivist reading of it. Our understanding of the theory would then be akin to our understanding of a fiction. The sentences of the theory are not, therefore, strictly propositions by themselves just as the different sentences in a story beginning ‘Once upon a time’ have not complete meanings and so are not propositions by themselves. (FoM, 231) The notion of a Ramsey sentence was rediscovered some years later by Carnap, who noted that the theory T is equivalent to the conjunction of the theory’s Ramsey sentence, R(T), and what is nowadays called its Carnap sentence, namely R(T) ⊃ T. Carnap’s idea was that the Ramsey sentence encapsulates the factual part of the theory (since it has the same observational consequences), whereas the Carnap sentence encapsulates the analytic part (since it has only the nonobservational consequences). Following its advocacy by Lewis (1970) the method of replacing a theory with its Ramsey sentence became popular with philosophers of science, who were attracted by its prospects for dissolving apparent disagreement between alternative theories with the same empirical consequences, and for distinguishing between

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intra-theoretic reasoning (taking place inside the scope of a quantifier) and intertheoretic disputes (concerning which such theory to adopt). One particularly noteworthy application (Lewis 1972) has been to the functionalist theory of the mind, which characterizes mental states in terms of their causes and effects. The main consequence of explaining this in terms of the theory’s Ramsey sentence is to impose a holistic conception according to which each mental state is to be understood only in terms of its relations to all the other states within the scope of the quantifier.

Further reading Ramsey supposed that our actions cannot affect the past, only the future. Dummett (1964) argues on the contrary that there is no incoherence in the notion of bringing about the past, at least in the case of past events we have no knowledge of. The immediate consequences of our actions lie in the future, but the consequences may be correlated with past events. For further discussion see Ahmed (2014, ch. 8). The Ramsey sentence method has been much discussed: see, for instance, Bohnert (1968) and Koslow (2006). Ketland (2004) argues that the Newman objection trivializes Ramsey sentence structuralism. On Ramsey’s attitude to the unsayable in 1929 see Diamond (2011) and Methven (2016). I have been unable to detect the textual basis for Misak’s (2017, 17–18) claim that Ramsey advocated a global pragmatism concerning truth. Holton & Price (2003) argue that restricting pragmatism to the secondary system is unstable.

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One place we might expect to find Ramsey’s influence, given the closeness of their interactions in 1929, is in Wittgenstein’s later work, but in fact this influence is notably muted. Wittgenstein certainly shared Ramsey’s interest in finitism and, at first, in the distinction the primary and the secondary systems, as is visible in the Philosophical Remarks. Yet after Ramsey’s death his work on the philosophy of arithmetic stalled, because he locked himself into conceiving of the meaning of an arithmetical generalization as its proof, hence making it opaque both how we could understand a conjecture in advance of proving it and how we could have two different proofs of the same theorem. Ramsey’s concern that ‘lack of knowledge and facility would prove a serious handicap’ (in PO, 48) to his work on the foundations of mathematics was prophetic. Wittgenstein’s work on the philosophy of mind and language, on the other hand, rapidly moved away from the kind of pragmatism Ramsey had been exploring. When we look elsewhere for Ramsey’s influence, though, the difficulty is often to distinguish genuine effect from independent rediscovery: Prior (1967, 229) rediscovered his use of prosentences in explaining sentential quantification; various authors rediscovered his reliabilist account of knowledge; de Finetti rediscovered the Dutch book argument. Davidson (1999, 32), who rediscovered Ramsey’s trick of using ‘ethically neutral’ propositions to measure degrees of belief, called this the ‘Ramsey effect’—discovering something and then finding that he got there first.

Truth and meaning Some of Ramsey’s ideas have been ignored, to the detriment of progress. On truth, by contrast, he has suffered the converse fate of having his authority cited

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in favour of two views—deflationism and pragmatism—which he did not hold (and in the latter case explicitly disowned). One of his earliest influences was on the view of truth taken by the logical positivists, both in the Vienna Circle and in the British variant that followed Ayer’s publication of Language, Truth and Logic. Ayer referred to Ramsey in his discussion, but claimed that there is not merely no separate problem, but ‘no problem of truth as it is ordinarily conceived’ (1936, 89). It was widely believed among positivists that the notion of truth, being metaphysical, should be avoided in polite company, and the deflationary conception provided the means to do this. It was because of this suspicion of the very notion that Tarski’s formal definition of truth made such a splash. He showed that a rigorous definition of truth for a formal language is possible, provided only that the predicate is not supposed to belong to the language in question, but only to a more inclusive metalanguage. Tarski thus adopted a hierarchy of languages of the sort first proposed by Russell in his Introduction to the Tractatus and then adopted by Ramsey to resolve the semantic paradoxes. Ramsey’s view that there is no separate problem of truth also had an influence on other Oxford philosophers such as Strawson. His account of truth, although needlessly complicated, at least avoided any straightforward deflation. The transparency schema, he said, is ‘true, but inadequate; . . . right in what it asserts, and wrong in what it suggests’ (1949, 83–4). Several authors (e.g. Whyte 1990, Mellor 2012) have proposed variants of the idea that a belief is true just in case acting in accordance with it is successful. Some proponents of this approach quote Ramsey’s remarks about chicken-beliefs, but ignore that he mentioned them only to set them aside. Moreover, as Ramsey noted, false beliefs are sometimes successful. Blackburn (2005) recognizes this difficulty, but hopes nonetheless that it can be overcome. One attempted response is to widen the range of circumstances in which the belief has to be successful to count as true, but this is tricky: we have all had moments when we acted appropriately on the basis of true beliefs and yet were still unsuccessful in achieving our desires. Another response is to try to regulate the notion of success that is in play here, but this too is tricky: it is difficult to see what objective notion of success we could appeal to that did not beg the question at hand. Much of what has gone under the name ‘pragmatism’ in recent decades has consisted in attempts to deflate not just truth, but other semantic notions such as meaning, judgment or propositions to naturalistically acceptable proportions. One inheritor of this approach was Sellars. More recently, Blackburn has attempted something similar in relation to ethics (although his inspiration comes more from Hume than from Ramsey). Recently, there have been rather more questionable attempts to cite Ramsey in support of a global deflation of semantics. The main difficulty for this project is that it is much easier to demonstrate conceptual connections between semantic concepts than to eliminate them. The moral was well stated by Strawson (a philosopher who was himself significantly influenced by Ramsey).

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Whatever may be believed, doubted, hypothesized, suspected, supposed, affirmed, denied, declared, alleged . . . is an intensional abstract entity, but nonetheless an item of a kind such as we constantly think of and refer to whenever we think of, or comment on, what someone has said or someone has written. (2011, 194–5)

Logicism Ramsey’s paper on ‘The foundations of mathematics’ was soon influential, both for its distinction between logical and semantic paradoxes and for its advocacy of a simple rather than a ramified theory of types. As a foundation for mathematics, however, the theory of types eventually lost out to set theory. One reason was that although Ramsey removed the ramification, he left in place Russell’s requirement that types be disjoint, and hence that they cannot be iterated into the transfinite. Another reason, no doubt, was notational simplicity: set theory places few restrictions on mathematical practice, apart from remembering to avoid sets that are ‘too large’; type theory, even if we omit the type subscripts from variables, tends to make its technical restrictions more visible. A further reason was suspicion of higher-order logic. Type theory is not logic, it was said, because it is not ontologically innocent; because its canons of acceptable reasoning cannot be codified as a recursive list of rules; or, more vaguely, because it is ‘set theory in sheep’s clothing’ (Quine 1970, 66). What, though, of the second part of Ramsey’s paper, on propositional functions in extension? The fact that Wittgenstein repeatedly attempted to express what was wrong with this notion suggests that he thought it mattered. The underlying reason was his opposition to Ramsey’s claim—dubbed ‘Tractarian logicism’ by Methven (2015)—that mathematics consists of ‘more complicated tautologies’. A mathematical sentence, Wittgenstein thought, is not on a par with a sentence expressing a proposition of logic, but rather with a sentence expressing a metalogical fact that something is a tautology. Overall, Tractarian logicists are left with an uncomfortable choice: they can accept that Ramsey’s notion of a PFE is genuinely logical, which leaves the axiom of infinity having only regressive support; or they can accept Wittgenstein’s complaint that PFEs are mere labels without the internal structure necessary for quantification, in which case there is a transcendental argument for the axiom of infinity but mathematics ceases to have the necessity we previously supposed. Except for Wittgenstein, however, philosophers of mathematics largely ignored Ramsey’s notion of propositional function in extension. Perhaps they did not grasp Wittgenstein’s earlier criticism of Russell’s definition of identity, and hence were not in a position to appreciate the significance of Ramsey’s proposal.

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Universals The reception of ‘Universals’ has a particularly sorry history. Some commentators have used against it arguments that Ramsey already countered in the paper itself. Others have treated it as if it were a piece of metaphysics, despite his insistence that his concern was only with a putative logical distinction, not with a metaphysical one. Still others have failed to recognize the paper’s dependence on its Tractarian background. Given, though, that the argument of the paper stands or falls with the notion of an incomplete symbol as it is used in the Tractatus, a more reasonable objection would be not that the paper is wrong but that it is of no interest to anyone who does not accept that Tractarian assumption. One might complain, that is to say, that Ramsey merely replaces an essentially Fregean distinction between saturated and unsaturated expressions with a Russellian one between complete and incomplete symbols. That complaint has some merit, perhaps, but it should not be allowed to obscure the importance of the underlying point, namely that Frege had confused the distinction between function and argument that he announced in Begriffsschrift §9 with the quite different distinction between noun and verb.

Further reading The influence of Ramsey on modern pragmatism is discussed by several of the contributors to Misak & Price (2017). The connections between iterative set theory and the simple theory of types are explained by Quine (1969, chs 11 and 12), although the ideas are already contained in Gödel’s 1933 lecture, ‘The present situation in the foundations of mathematics’ (1986–2003, III, 45–53). McGuinness (2006) is perceptive on the nature of Ramsey’s relationship with, and the extent of his influence on, Wittgenstein; for an alternative view see Misak (2016). Stern (1995) discusses the details of Wittgenstein’s views in the early 1930s.

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INDEX

abstraction principle: 86, 87, 89, 90, 118, 119, 122, 144, 198, 199 accidents: see qualities Ackermann, Wilhelm: 25 Ackrill, John: 64 acquaintance, Russellian: 205, 211, 224, 226, 227, 233, 251–257, 261, 262, 265, 267–270, 280, 281, 283, 290, 291, 301, 309, 331, 370–372, 432, 465 actualism, modal: 26, 189, 191, 229, 321 aggregate: 61, 71, 76, 77, 196 Ahmed, Arif: 467 Alexander, Samuel: 301, 304 Allison, Henry: 187 analytic and synthetic: 15, 46, 70–72, 74–75, 82, 84, 86–90, 98, 131, 188, 189 ancestral: 48–51, 85, 87, 88, 246, 344 Anscombe, Elizabeth: 98, 139, 318, 354, 447 Antonelli, Aldo: 132 Aquinas, Thomas: 13, 61, 180 Aristotle: 11–15, 19, 20, 30, 39, 41, 61, 141, 376 Armstrong, David: 432 assertion act of: 31, 34, 133, 141 Russellian: 208, 214

assertion sign: 372, 373, 412 Atkinson, R. F.: 397 attribute: see substance austere conception of nonsense: 365–367, 400–404, 414, 438, 464 Austin, John: 13, 142, 305 auxiliary name: 40, 131, 343 thought: 19, 54, 55, 115 Ayer, Alfred: 141, 150, 187, 262, 306, 427, 469 Baker, Alan: 75 Baldwin, Thomas: 105, 187, 312, 434 Barnes, Jonathan: 30 Bartley, William: 328 Barwise, Jon: 100 basic law: 25–27, 35, 42–43, 47, 56–58, 75, 118, 121, 123, 127, 132, 375 Basic Law V: 8, 87, 110, 119, 121, 122, 144 Baumann, Julius: 70 Beaney, Michael: 4, 202 Beck, Jacob: 461 Beethoven, Ludwig van: 152 behaviourism: 293, 294, 307 Bell, David: 4, 99, 415

494 Index

Beltrami, Eugenio: 124 Benacerraf, Paul: 90 Bentham, Jeremy: 225 Bergmann, Gustav: 142 Berkeley, George: 138, 151, 170, 244, 251, 266, 292 Bermudez, José: 461 Biermann, Otto: 170 bipolarity: 287–288 Black, Max: 312 Blackburn, Simon: 31, 227, 424, 469 Blackwell, Kenneth: 264 Blanchette, Patricia: 43, 117, 132 Blanshard, Brand: 230 Boer, Stephen: 250 Boghossian, Paul: 434 Bohnert, Herbert: 467 Boltzmann, Ludwig: 398 Bolzano, Bernard: 3, 17, 171 Bonevac, Daniel: 20 Bonjour, Laurence: 434 Boole, George: 19–20, 174 Boolos, George: 20, 42, 51, 83, 123 Bostock, David: 212, 257, 264 Bourbaki, Nicholas: 38 Bradley, F. H.: 191, 221, 228, 363 regress argument: 191 Bradley, Francis: 151–154, 165 Braithwaite, Richard: 433, 434, 463 Brentano, Franz: 430 Brouwer, Egbertus: 143 Burge, Tyler: 58, 99, 105, 117 Butler, Christopher: 234 Byrd, Michael: 205 calculus ratiocinator: 14 Candlish, Stewart: 154, 187, 234 Cantor, Georg: 128, 171, 175, 217 theorem: 122, 213 Carnap, Rudolf: 74, 163, 304, 408 Cartwright, Nancy: 259 Cartwright, Richard: 212, 227, 234 Cassin, Chrystine: 227 categorical logic: see logic, categorical Cauchy, Augustin-Louis: 170 characteristica universalis: 14 chicken beliefs: 458–461 Child, William: 318 Chisholm, Roderick: 143

choice, axiom of: see multiplicative axiom Church, Alonzo: 39, 100, 217, 240, 267, 376 Chwistek, Leon: 239, 437 class: 196, 197, 199, 205, 216, 226 empty: 196, 200 class-concept: 197, 209, 214 Cocchiarella, Nino: 289 Cohen, Jonathan: 296 Coleridge, Samuel Taylor: 152 complex and fact: 275–276, 279, 281, 287, 320–324, 329, 330, 413 compositionality principle: 35, 41, 66 Conant, James: 403, 405, 406 concept-word: 41, 59–61, 63, 66, 68, 69, 78, 106–110, 112, 118, 131 conditional proof: 25, 26, 31, 126 connotative and denotative: 94–96, 153 content stroke: 24–25, 54, 55, 114, 115 content, conceptual: 27–28, 34, 39, 44, 46, 59, 67, 92, 94, 207, 330, 428, 446 context principle: 65–69, 82, 88, 120, 142, 226, 332–334, 344, 351, 366, 372 for reference: 120, 144 contextual definition: 66, 69, 85, 90, 221, 225, 226, 238, 239, 247 Copi, Irving: 240, 242 Couturat, Louis: 188 Craig, Edward: 69, 163 Crane, Tim: 187, 302 Crysippus: 10, 11 Currie, Gregory: 31, 110 curvature of a manifold: 156, 157, 159–161 Czolbe, Heinrich: 18 Davidson, Donald: 142, 296, 468 Dawes Hicks, George: 259 de Finetti, Bruno: 468 de Morgan, Augustus: 14, 32 de re and de dicto: 96, 222, 223 Dedekind, Richard: 49, 83, 88, 90, 128, 170, 198, 202, 217, 241 correspondence with Weber: 90 deductivism: 125–127, 202, 466 deflationism: 103, 424, 469 Demopoulos, William: 250

Index 495

Demos, Raphael: 277 Dennett, Daniel: 143 denotation, occurrence as: 215 denoting concept: 207–212, 218, 220, 255, 256, 310 Descartes, René: 67, 80, 81, 139, 141, 253, 304, 362, 432 Dewey, John: 293 dialectical reading: 401–402 Diamond, Cora: 334, 363, 367, 405, 407, 467 Dickie, Imogen: 47 difference, qualitative: 192 Dini, Ulisse: 170 direction principle: 86, 87, 144 diversity, numerical: 192 Donnellan, Keith: 311 Dreben, Burton: 56 Dudman, Victor: 30, 110 Dugac, Pierre: 173 Dummett, Michael: 4, 9, 27, 31, 36, 46, 54, 58, 62, 65, 69, 70, 79, 91, 94, 98–100, 103–105, 110, 123, 137, 139, 142, 220, 293, 312, 447, 467 Dutch book: 450, 456, 457, 468 diachronic: 454–455, 457 synchronic: 452–454 Earman, John: 457 Eliot, T. S.: 148 emergent properties: 301–302, 394 Emerson, Ralph Waldo: 371 empiricism: 17–19, 52–53, 77, 81, 116, 138, 152, 157, 205, 249, 277, 303, 307 Engelmann, Paul: 400 equinumerosity: 85, 86, 89, 118, 119, 198, 199, 205 Erdmann, Benno: 18 Eriksson, Lina: 457 Euclid of Megara: 10 Evans, Gareth: 22, 43, 99, 104 Ewing, A. C.: 163 existent and subsistent: 79, 180, 182, 183, 185, 187, 211, 221, 223, 224, 244, 280 existential proposal: 13, 269–270, 272 externality, form of: 159, 162, 184

fact atomic: 276, 318–322, 327, 335, 337–339, 341, 348, 349, 351, 374, 393, 410, 438, 445 general: 277–278 molecular: 322 negative: 55, 276–277, 281, 322, 336 Faulkner, Nadine: 296 Feldman, Richard: 434 Ferreira, Phillip: 154 Field, Hartry: 82 Fine, Kit: 31 Fischer, Kuno: 7, 21 Floyd, Juliet: 407 force and content: 31, 133, 413 form logical: 130, 131, 225, 271, 283, 328–329, 331, 332, 338, 347, 351, 365, 367, 377, 413 of representation: 325, 326, 332, 366 formal series: 344–345, 384 formalism: 57, 143, 175, 305, 342, 425 axiomatic: 124, 125, 127 game: 70–71, 78, 143 term: 78, 143 Fraassen, Bas van: 454, 457 Fraenkel, Abraham: 74 Frascolla, Pasquale: 387 Frege point: 23, 31, 34, 37, 133, 183, 265, 266, 294 Frege, Alfred: 8 Frege, Gottlob: 1 ‘17 key sentences’: 26, 45, 52, 54, 56–58, 77, 102, 115 ‘On the aim of the conceptual notation’: 34 ‘On the aim of the conceptual notation’: 22, 24, 30, 34 ‘The argument for my stricter canons of definitions’: 113 Begriffsschrift: 3 §2: 23, 30; §3: 23; §4: 22; §5: 33; §6: 35; §7: 34; §8: 46, 50; §9: 28; §11: 37, 40, 42; §13: 30; §23: 50; §24: 66; Preface: 21, 22, 32, 141

496 Index

‘Boole’s logical calculus’: 38, 39, 41, 42, 47, 62 ‘Comments on sense and meaning’: 44, 97, 107, 375 ‘Compound thoughts’: 102, 136 ‘On concept and object’: 106–110, 136, 138 correspondence with Hilbert: 125, 127; Husserl: 28, 108, 141; Jourdain: 35, 135, 136, 306, 361; Liebmann: 127, 129; Marty: 59, 62, 84; Russell: 122, 253 ‘A critical elucidation of some points in E. Schröder’: 40, 61, 77 ‘Dialogue with Pünjer’: 45, 61 ‘On Euclidean geometry’: 126 ‘On the foundations of geometry (1903)’: 26, 109, 110 ‘On the foundations of geometry (1906)’: 25, 126, 131, 132 ‘Function and concept’: 110, 114 Die Grundlagen der Arithmetik §3: 26, 56; §10: 73; §14: 56, 57, 74, 127; §17: 53; §24: 81; §26: 81; §29: 61; §47: 60; §51: 63; §53: 60, 83; §54: 77; §57: 61, 79; §59: 68; §60: 66, 68; §63: 86; §64: 87; §66: 63; §68: 89, 118; §88: 15, 39, 50; §90: 111; §95: 129; §105: 98; §106: 64 Grundgesetze der Arithmetik §1: 112; §10: 120; §23: 113; §24: 113; §32: 103, 120; §66: 128; §§86–137: 342; §91: 71; §97: 120; §99: 30; §147: 119; Foreword: 18, 112, 116, 121 Habilitation: 26, 49 ‘Introduction to logic’: 98, 109, 112, 134 ‘Logic’: 24, 26, 53, 54 ‘Logic’ (1897): 53, 83, 93, 94, 102, 103, 116, 137 ‘Logic in mathematics’: 27, 103, 123, 126 ‘My basic logical insights’: 103 ‘Negation’: 135, 136

‘A new attempt at a foundation for arithmetic’: 26, 123 ‘Notes for Ludwig Darmstaedter’: 123, 134, 135 ‘On Schoenfliess’: 122 ‘On sense and reference’: 92 review of H. Cohen: 53, 79 ‘On the scientific justification of a conceptual notation’: 21, 22, 40, 67 ‘On sense and reference’: 8, 45, 92, 94, 95, 105, 134, 136, 142, 203, 210, 309 ‘Thoughts’: 24, 103, 135, 138, 139, 275, 324, 325 ‘What may I regard as the result of my work?’: 133 French, Robert: 163 Frost-Arnold, Greg: 4 Gabriel, Gottfried: 77, 83 Garver, Newton: 397 Gaskin, Richard: 99, 273 Geach, Peter: 13, 20, 23, 29, 31, 97, 139, 143, 212, 227, 273, 275, 310, 334, 343, 354, 367 geometry descriptive: 161–163 Frege’s views on: 55–57, 74, 86, 116, 124–132 metric: 155–159 neutral: 156–158, 162, 174 Gillies, Donald: 457 Gödel, Kurt: 100, 130, 143, 286, 306, 471 Goldfarb, Warren: 234, 242, 401, 403, 407, 439 Goodman, Nelson: 82, 433 Gray’s Elegy argument: 218–220, 253, 256 Grayling, Anthony: 187 Grice, Paul: 31, 231 Griffin, James: 327 Griffin, Nicholas: 154, 163, 167, 273 Grossmann, Reinhardt: 107 Grover, Dorothy: 430 Hacker, Peter: 383, 407, 415 Hale, Bob: 83, 91, 144, 449 Hamilton, William: 252

Index 497

Hanks, Peter: 31 Hanna, Robert: 144 Hardy, G. H.: 173 Hare, Richard: 31 harmony principle: 278, 279, 324, 333 Hawtrey, Ralph: 234 Heck, Richard: 51, 91, 117 Hegel, Georg: 151, 152, 154, 164, 165, 173, 184, 192, 214, 229, 399, 434 Heine, Eduard: 78 Helmholtz, Hermann: 156, 157 Hempel, Carl: 230 Herbart, Johann Friedrich: 17–19, 23, 53, 76, 77, 83, 183 hereditary property: 49–50 Hertz, Heinrich: 327, 389 heterologicality: see paradox of heterologicality Heymans, Gerard: 157 hierarchy, truth-functional: 240, 358, 437 Hilbert, David: 25, 129, 130 Hintikka, Jaako: 56, 117, 142, 363 Holton, Richard: 467 horizontal: 114, 115 Hornsby, Jennifer: 327 Hoüel, Jules: 124 Hovens, Frans: 58 Humberstone, Lloyd: 194 Humboldt, Alexander von: 22 Hume’s Principle: 85–90, 119, 124, 127, 144 Hume, David: 17, 86, 152, 244, 252, 258, 259, 294, 297, 303, 307, 370, 396, 469 Husserl, Edmund: 18 Hylton, Peter: 154, 205, 206 hypothetical logic: see logic, hypothetical hypothetical, variable: 462, 464, 465 idealism: 17–19, 138–139, 173, 183–184, 186, 209, 232, 297, 298, 353, 371 absolute: 101, 151–154, 164, 165, 177, 179, 180, 184, 185, 192, 199, 225, 228, 230, 303–304 subjective: 251, 297 transcendental: 377, 410, 414

impredicative: 50, 87, 119, 122, 144, 240, 246, 436 incomplete symbol: 141, 225–227, 238, 239, 247, 266, 267, 278, 306, 308, 347, 358, 359, 423, 424, 426, 446–448, 471 incomplete, semantically: 65, 225, 333, 351 indexicals: 137–139, 256, 291, 310, 311 induction enumerative: 73, 389, 393, 433–434 mathematical: 123, 286, 386 infinitesimals: 66, 79, 169–171 infinity, axiom of: 83, 226, 241–243, 250, 288, 289, 306, 375, 386, 387, 438, 443, 444, 470 internal and external relations: 190–192, 194, 197, 244, 301, 326, 338, 349 internalism on judgment: 24 on knowledge: 432–434, 459, 461 on meaning: 96–98, 115 Irvine, Andrew: 250 Ishiguro, Hide: 354 Jackson, Frank: 257 James, William: 230, 231, 234, 254, 256, 296, 297, 425 Jevons, William Stanley: 19 Joachim, Harold: 228, 234 Johnson, W. E.: 221, 424, 446, 450, 462 Johnston, Colin: 354 Jordan, Camille: 171 Jourdain, Philip: 216 judgment stroke: 22–25, 30, 35, 112, 114 theory of existential: 152–154, 179–181, 185, 225; multiple relation: 265–273; Russell’s post-war: 290–296; Tractarian: 368–369 Julius Caesar problem: 85, 88–91, 120, 127, 128, 144, 199 Kant, Immanuel: 60, 396 correspondence with Eberhard: 163 Critique of Pure Reason: 7, 15–17, 25, 26, 34, 46, 50, 52–54, 56,

498 Index

70–72, 74, 75, 84, 96, 113, 142, 155–157, 163, 184–185, 187, 188, 193, 204, 205, 229, 304, 339, 352, 362, 376, 377, 382, 383, 386, 396, 400, 408, 413, 434, 439 Logic: 15 ‘The one possible basis for a proof of the existence of God’: 13 Prolegomena: 404 Kanterian, Edward: 9, 23, 107 Kaplan, David: 137 Keen, C. N.: 154 Kelly, John: 395 Kemeny, John: 453 Kenny, Anthony: 9, 139, 318, 322, 373, 415 Kerry, Benno: 51, 89, 130 Ketland, Jeffrey: 467 Keynes, John Maynard: 450 Kierkegaard, Søren: 399, 401, 402, 407 Kim, Jaegwon: 302 Klement, Kevin: 123, 217, 227, 246 Kneale, Martha: 9, 20, 36 Kneale, William: 9, 20, 36, 426 Koslow, Arnold: 467 Kreiser, Lothar: 9 Kremer, Michael: 31, 407 Kripke, Saul: 98, 99, 105, 311 Kuratowski, Kazimierz: 197 Kyburg, Henry: 457 Lakatos, Imre: 250 Landini, Gregory: 227, 246 Langford, Cooper: 82 language-games: 413, 414 Lebens, Samuel: 296 Lehman, Sherman: 453 Leibniz, Gottfried: 19, 22, 46, 147, 171, 188–194, 288, 289 correspondence with Clarke: 158, 193 New Essays: 14, 26, 53, 71 Leng, Mary: 230 Levine, James: 91, 200, 212, 227, 273, 296 Levvis, Gary: 354 Levy, David: 397 Lewis, David: 83, 143, 466, 467

Lewy, Casimir: 358, 420 Liebmann, Otto: 18 limitation of size: 216, 217 linguistic turn: 142, 312 Locke, John: 14, 67, 68, 77, 81, 138, 152, 181, 303 Lockean model of meaning: 67–69, 93, 138, 139 Lockwood, Michael: 64, 302 logic categorical: 11, 15, 19 human: 434, 457 hypothetical: 19 polyadic: 39, 75, 142, 144, 177, 421 transcendental: 15–18, 20, 75, 184, 377 logically perfect language: 203, 290, 291, 332, 333, 364, 405 logicism: 8, 49, 50, 74, 75, 84, 89, 122, 130, 148, 177, 201–206, 214, 243–250, 282–289, 306, 359, 440, 444, 470 neo-Fregean: 144 Tractarian: 470 logocentric predicament: 57, 58, 116 Long, Peter: 354, 449 Lotze, Hermann: 7, 18, 19, 26, 52, 54, 55, 115, 152, 177 Lowe, Jonathan: 68 Ludwig, Jan: 354 Lugg, Andrew: 312 Łukasiewicz, Jan: 35 MacBride, Fraser: 110, 273, 449 Mach, Ernst: 297 Maddy, Penelope: 248 Maher, Patrick: 457 Makin, Gideon: 227 Malcolm, Norman: 352, 413 material conditional: 3, 10, 19, 25, 32, 34, 35, 207, 221 paradoxes of: 34 Mates, Benson: 194 Mauthner, Fritz: 401 Mayo-Wilson, Conor: 250 McDowell, John: 97, 99, 327 McGuinness, Brian: 318, 354, 363, 391, 407, 471 McLeod, Stephen: 64

Index 499

McTaggart, John: 164–167 meaning occurrence as: 215, 218 theory of causal: 293, 430; picture: 323, 327; pragmatist: 293, 430 Meinong, Alexius: 232 Mellor, Hugh: 434, 457, 469 Mendelsohn, Richard: 47 Menzies, Peter: 259 Mertens’ conjecture: 73 metalanguage: 30, 167, 208, 299, 300, 332, 357, 366, 369, 415, 437, 469 metalogic: 56–58, 132, 204, 244, 374, 386, 470 Methven, Steven: 467, 470 Mill, John Stuart: 18, 46, 71, 96, 153, 434 mirroring principle, Fregean: 60, 109, 110, 278, 279 Misak, Cheryl: 430, 467, 471 Moleschott, Jacob: 18 Moltmann, Friederike: 83 monism, neutral: see neutral monism Monk, Ray: 150 Moore, Adrian: 377, 383 Moore, G. E.: 143, 168, 177, 179–187, 191, 204, 205, 209, 226, 228, 229, 231, 234, 251–253, 256, 289, 300, 303, 304, 307, 308, 312, 316, 325, 421, 458 ‘Identity’: 192, 193 ‘The nature of judgment’: 180, 182, 183, 186, 187, 189, 191 review of Russell’s Essay on the Foundations of Geometry: 184 ‘Russell’s theory of descriptions’: 308 Some Main Problems of Philosophy: 251 ‘The subject matter of psychology’: 252 Morrell, Ottoline: 148, 168, 315 Morris, Michael: 354 multigrade: 267, 268, 341, 448–449 multiplicative axiom: 241–243, 375 Munch, Fritz: 94 Myhill, John: 215, 286

N-operation: 341–386 Nagel, Thomas: 58, 376 naturalism: 142, 293, 306–307, 309, 312, 469 Neale, Stephen: 227 Netz, Reviel: 20 neutral monism: 297–302, 305 Newman, Max: 301 Nidditch, Peter: 144 nonsense: 394 moral: 393 Noonan, Harold: 9, 110, 227, 354 numerically definite quantifier: 85, 88, 384 Nunn, Percy: 304 O’Brien, Lucy: 373 O’Neill, Onora: 163 object logical: 83, 90, 122, 144, 198, 282 metalogical: 130 occurrence, primary and secondary: 222–223, 226, 447 Oliver, Alex: 31, 200, 449 ordered pair, definition of: 196, 197 orders of truth: 234 Owens, Joseph: 99 Page, James: 339 Papineau, David: 434 paradox Berry’s: 237 Burali-Forti’s: 216, 217, 436 Grelling’s: 237 König’s: 237 of heterologicality: 237, 239 of material conditional: 34 of relativity: 166–167, 176–177, 192–193 of the infinitely divisible: 169 of the infinitely large: 171–172 of the liar: 228, 233, 234, 236, 237, 266, 287, 428, 436 Richard’s: 436 Russell’s: 40, 128, 129, 131, 144, 213–438 Russell-Myhill: 215, 217, 235, 236 Parsons, Charles: 51 Parsons, Terence: 187

500 Index

particular: see universal Peacock, George: 175 Peano, Giuseppe: 196, 201, 203, 208, 237, 437 Pears, David: 150, 257, 272, 273, 327, 339, 354, 415 Pedriali, Walter: 31 Peirce, Charles: 13, 33, 49, 177, 196, 330, 431, 451, 464 permanence of forms, principle of: 175 permutative complexes: 270–272 Perrin, Denis: 352 Perry, John: 139 PFE: see propositional function in extension phenomenalism: 261, 370, 371 Philo of Megara: 10 Pilch, Martin: 319, 345 Pincock, Christopher: 296 Plato: 64, 82 plenum: 193 plurality: 77, 248 Poincaré, Henri: 124, 125, 129, 162, 163, 245 Popper, Karl: 143 Potter, Michael: 4, 91, 281, 322, 439 pragmatism: 432, 459, 468, 469, 471 concerning meaning: 293, 421, 430, 458 concerning truth: 228, 230–231, 234, 295, 425, 426, 430, 460, 464, 467, 469 Preston, John: 327 presupposition: 310–312, 349, 412 Preti, Consuelo: 187 Price, Huw: 259, 467 Priest, Graham: 167, 439 primary and secondary occurrence: see occurrence, primary and secondary system: see system, primary and secondary Prior, Arthur: 430, 468 Proctor, George: 391 Proops, Ian: 110, 206, 354, 373, 406, 407, 439 proposition elementary: 240, 319, 335–337, 346–349, 370, 386–389, 393,

394, 409–451; independence of: 337–410 general form of: 339, 346, 374, 379, 386, 387, 413 propositional function: 174, 202, 207, 208, 212, 214–216, 226; hierarchy of: 235; in extension: 440–444, 470; predicative: 236, 238–240, 435, 436; reducible: 216, 217, 239 reference: 423–425, 430 sign: 329–331, 333, 335, 337, 339, 341, 342, 345, 366, 368–372, 376, 406, 428, 430, 436, 458 pseudo-axiom: 126–129 psychologism: 16–18, 23, 24, 26, 52–53, 65, 67, 74, 77, 94, 95, 98, 116, 138, 141, 144, 152, 183–185, 212, 243, 283, 284, 289, 334, 375, 430, 437, 460 pure general logic: 15–17, 75 Putnam, Hilary: 163, 373, 461, 463 qualities and accidents: 14, 159, 181, 182, 192, 193 quantificational semantics content-level: 40, 41 language-level: 40, 112 narrow and broad: 40, 41, 51, 120, 343, 345, 361 reference-level: 112, 113 symbol-level: 361 quasi-quote: 112 Quine, Willard: 10, 20, 38, 64, 82, 111, 112, 122, 141, 142, 240, 311, 427, 470, 471 Quinton, Anthony: 296 Ramsey (née Baker), Lettice: 420 Ramsey test: 463–464 Ramsey, Frank: 1, 2, 173, 231, 296, 303, 308, 317, 318, 321, 330, 331, 335, 343, 346, 348, 357–360, 374, 401, 412 ‘Causal qualities’: 463, 466 Critical notice of the Tractatus: 338, 425, 426, 428–430, 438

Index 501

‘Facts and propositions’: 224, 288, 423, 425, 427, 429, 430, 458–461 ‘The foundations of mathematics’: 239, 331, 375, 422, 434–444, 446 ‘General propositions and causality’: 462–465 ‘The infinite’: 439 ‘Knowledge’: 431 ‘Mathematical logic’: 444 ‘Mr Keynes on probability’: 451 ‘The nature of propositions’: 424 On Truth: 230, 423, 425–427, 432–434 ‘Philosophy’: 464 ‘Probability and partial belief ’: 460 ‘Theories’: 466 ‘Truth and probability’: 432, 434, 450–457, 460, 461, 464 ‘Truth and simplicity’: 424, 427 ‘Universals’: 352, 445–449 real property, Kantian: 13 realism, naive: 260 recognition-statement: 64 reducibility, axiom of: 238–240, 243, 247, 248, 286, 306, 375, 436, 437 redundancy schema: see transparency schema reference principle: 100, 103, 104 regressive method: 158, 186, 198, 242, 248–250, 306, 436, 444, 470 Rein, Andrew: 58 relation in intension: 197, 244 relative name: 347–349, 426 relativity, paradox of: see paradox of relativity reliabilism: 431–434, 468 Resnik, Michael: 132 resolute reading: 402–407, 414 Richard, Mark: 430 Richards, Joan: 173 Ricketts, Thomas: 58, 132 Riemann, Bernhard: 156 Robbins, Peter: 154 Rorty, Richard: 234 Rosenberg, Jay: 281, 302 Royce, Josiah: 251

Russell (née Pearsall Smith), Alys: 147–149, 155 Russell, Bertrand ‘The a priori in geometry’: 158–160, 166 Analysis of Mathematical Reasoning: 176, 179–181 The Analysis of Matter: 298, 300, 301, 312 The Analysis of Mind: 280, 287, 293, 294, 298, 299, 429, 431 ‘Analytic and synthetic philosophers’: 307, 308 ‘Analytic realism’: 243, 254, 347 ‘Are Euclid’s axioms empirical?’: 167 Autobiography: 147, 148, 195, 264 ‘On the axioms of the infinite and of the transfinite’: 242–244 ‘The basis of realism’: 191 ‘The classification of relations’: 191, 192 correspondence with Frege: 253; Joachim: 164; Jourdain: 226, 244 ‘On denoting’: 148, 211, 218–227, 256, 309, 310 Essay on the Foundations of Geometry: 155–163 The Fundamental Ideas and Axioms of Mathematics: 192 Human Knowledge: 307 Inquiry into Meaning and Truth: 189, 294 Introduction to Mathematical Philosophy: 83, 278, 280, 283, 285, 288, 289 Introduction to the Tractatus: 365 ‘Is position in time absolute or relative?’: 211 ‘Is position in time and space absolute or relative?’: 193 ‘Knowledge by acquaintance and knowledge by description’: 253, 255, 267, 271, 276 ‘Leibniz’s doctrine of substance’: 189 ‘The limits of empiricism’: 307

502 Index

‘The logic of geometry’: 157, 159, 160 ‘On the logic of relations’: 198 ‘Mathematical logic as based on the theory of types’: 237–239, 247 ‘On matter’: 259, 262 ‘On the meaning and denotation of phrases’: 211 ‘The meaning of “meaning” ’ (1920): 307 ‘The meaning of meaning’ (1926): 284, 294 ‘Meinong’s theory of complexes and assumptions’: 232 ‘Mr Strawson on referring’: 309, 310 My Philosophical Development: 161, 170, 172, 180, 182, 190, 196, 206, 247, 274, 299, 308 ‘Mysticism and logic’: 394 ‘The nature of sense data’: 252, 259, 261 ‘On the nature of truth’: 229, 234, 266 ‘The nature of truth’: 229, 231 ‘On the nature of truth and falsehood’: 232, 233 ‘On propositions’: 292 Our Knowledge of the External World: 249, 263, 271, 275, 277, 283, 338, 432 Outline of Philosophy: 300–302 ‘The paradox of the liar’: 241, 265 ‘The paradoxes of logic’: 241, 246, 248 ‘Philosophical analysis’: 309 ‘The philosophical importance of mathematical logic’: 250 The Philosophy of Leibniz: 171, 181, 188–194 ‘The philosophy of logical atomism’: 211, 225, 233, 250, 263, 275, 277, 278, 280, 281, 283, 284, 286, 290 ‘Pragmatism’: 231 Principia Mathematica: 148, 195, 197, 225, 226, 234–244, 246–249, 260, 266, 267, 282, 284, 285,

291, 305, 306, 344, 358, 359, 375, 427, 435, 437, 438, 440; 2nd edn: 286, 299, 448; 2nd edition: 197, 285, 286; 2nd edn: 289 The Principles of Mathematics §1: 202; §3: 206; §6: 204; §10: 205; §11: 203; §12: 204; §15: 207; §17: 204; §27: 197; §41: 207; §46: 181; §48: 182; §51: 203, 212, 284; §56: 209, 212; §58: 196, 209; §59: 210; §64: 210; §72: 209, 211; §73: 196, 210; §85: 214; §104: 214; §105: 214; §141: 211, 212; §150: 199; §242: 193; §357: 176; §427: 182; §430: 185, 190; §433: 204; §482: 29, 208; §496: 51; §500: 216; early drafts: 171, 192, 198, 204, 214; Introduction to 2nd edn: 202, 286; Preface: 185, 186, 205, 255 The Problems of Philosophy: 251–260, 262, 267, 268, 289, 307 ‘On propositions’: 277, 291–294, 298 ‘On quantity and allied conceptions’: 176 ‘Recent work on the principles of mathematics’: 172, 199, 201, 202, 205, 206, 402 ‘The regressive method of discovering the premises of mathematics’: 248, 249 ‘The relation of sense data to physics’: 281 ‘On the relations of number and quantity’: 199 ‘On the relations of universals and particulars’: 244 ‘A reply to Dr Schiller’: 266 review of Heymans: 157 review of James: 256 review of Ramsey, The Foundations of Mathematics: 433 ‘On some difficulties in the theory of transfinite numbers and order types’: 216, 217, 235

Index 503

‘On some difficulties of continuous quantity’: 169, 171 ‘Some explanations in reply to Mr Bradley’: 191 ‘On the substitutional theory of classes and relations’: 225, 247 Theory of Knowledge: 246, 252, 254–257, 266, 268–270, 275, 282, 283, 288, 296 ‘Theory of knowledge’ (1926): 295 ‘The theory of logical types’: 232–234, 248, 266 ‘The ultimate constituents of matter’: 263 undergraduate essays: 154, 157 ‘What is logic?’: 269, 375 ‘William James’s conception of truth’: 231 Russell, Frank: 147 Ryle, Gilbert: 187, 189 Sachse, Leo: 7, 77, 83 Sahlin, Nils-Eric: 432 Sainsbury, Mark: 227, 257 Salmon, Nathan: 99 saturated: see unsaturated Savage, Wade: 257 Scarrow, David: 227 Schaffer, Jonathan: 154 schematic letter: 30, 41, 42 Schick, Frederic: 457 Schiller, F. C. S.: 230, 293 Schlegel, Friedrich: 152 Schlick, Moritz: 408 Schönbaumsfeld, Genia: 407 Schopenhauer, Arthur: 380, 396, 401, 410 Schröder, Ernst: 32, 177, 196 Schroeder, Mark: 31 Schulte, Joachim: 363 scope wide and narrow: 222 Searle, John: 98, 99, 105 self-reproductive: 217 self-subsistent: 79, 85, 90 Sellars, Wilfrid: 143, 273, 305, 312 semantics, reverse: 428–430 sense Fregean: 92–99 Tractarian: 335–339

sense data: 148, 241, 251, 252, 254–256, 259–263, 297, 300, 304, 305, 312, 354, 370, 371, 432 sensibilia (singular: sensibile): 261–264, 300, 301 Sextus Empiricus: 10, 401 Shapiro, Stewart: 123 Sheffer, Henry: 57, 241, 342 Shieh, Sanford: 58, 367 Sidgwick, Alfred: 293 Sidgwick, Henry: 3 sign interpreted: 331, 333, 334 and symbol: 329–331, 372 Simons, Peter: 449 Skiba, Lukas: 105 Skorupski, John: 20 slingshot: 100, 101, 105 Smiley, Timothy: 200, 449 Smith, Nicholas: 30 solipsism: 82, 138, 139, 143, 261, 304, 320, 360–365, 369–373, 378, 379, 381, 382, 395, 412, 415 of acquaintance: 261, 291, 331, 370, 372 Sommers, Fred: 64 sortal concept: 77, 78, 86 space public and private: 163–264 relativity of: 159, 160, 168 Spadoni, Carl: 154 Spinoza, Baruch: 297, 397, 399 Stanley, Jason: 58 Stebbing, Susan: 2, 211, 226, 415 Stenius, Erik: 383 Stepanians, Markus: 105 Stern, David: 471 Stern, Robert: 154, 234 Stevens, Graham: 246, 312 Stirling, James: 151 Stolz, Otto: 171 Stone, Alison: 154 Stout, George: 153 Strawson, Peter: 77, 98, 163, 309, 310, 362, 469, 470 subject empirical: 331, 369–372 metaphysical: 378–383, 395, 414 pinpoint: 291, 297, 298, 370

504 Index

thinking: 361–363, 370, 379, 381, 382 willing: 394–395, 397, 399 subjectivism: 184, 187 substance and attribute: 13, 14, 79, 158, 181, 182, 188, 190, 349 Tractarian argument for: 349–350, 352–354, 411, 412, 415 Sullivan, David: 83 Sullivan, Peter: 58, 64, 91, 117, 327, 346, 405, 415, 430, 444 supposition, mediaeval: 12–13, 20, 209 synthetic: see analytic system, primary and secondary: 412, 463–468 Tappenden, James: 26, 69 Tarski, Alfred: 18, 30, 234 Taschek, William: 117 tautology: 336–337, 357, 371, 374, 376, 393, 409, 428, 439, 444, 470 Taylor, Gerald: 257 Tejedor, Chon: 397 Teller, Paul: 194 thing and concept: 181–183, 223 Thomae, Johannes: 71 thought Fregean: 97, 98, 101–105, 109, 114, 115, 120, 130–136 mock: 103, 104, 115 Tractarian: 319, 328, 329 Tiergarten programme: 164–167 Tolley, Clinton: 20 tone: 27–28, 31, 94, 102, 142, 207, 424 Torretti, Roberto: 21, 163 transcendental argument: 16, 159, 163, 184, 187, 359, 383, 410, 438–439, 443–444, 470 deduction: 16, 17, 25, 155, 163, 185, 377 logic: 15–18, 20, 75, 184, 377 vs transcendent: 16 transparency schema: 103, 104, 115, 421, 424–427, 469 treadmill argument: 102, 180 Trendelenburg, Adolf: 22 Trueman, Robert: 110, 444

truth, theory of bi-correspondence: 287, 296, 325 coherence: 228–231, 234, 249, 425 correspondence: 102, 105, 180, 181, 183, 231–234, 271, 287, 324, 325, 426 deflationary: 103, 424, 469 externalist: 426 identity: 152, 228, 231–234, 325, 327 picture: 323–327 pragmatist: 230–231, 234, 295, 425, 426, 430, 460, 467 primitivist: 231–233 truth-possibility: 335–338, 440 Turing, Alan: 39, 376 types, theory of early Russellian: 214–216, 234 Fregean: 59, 78, 107, 111–117 ramified: 235–241, 244–246, 249, 266, 284, 286, 345, 346, 427, 435–438, 470 simple: 215, 235–238, 240, 242, 245, 345, 346, 435–438, 442, 470, 471 substitutional: 226 typical ambiguity: 236, 239, 241, 346 universal and particular: 14, 82, 108, 223, 244, 246, 250, 254–255, 279–281, 351, 445–449 unsaturated: 59, 62, 63, 104, 106–110, 112, 134, 135, 208, 220, 272, 273, 278, 279, 329, 351, 445, 471 Urmson, James: 312 Urquhart, Alasdair: 227 vagueness: 290–291, 309, 364–365 value-range: 8, 111, 118–122 van Heijenoort, Jean: 56, 58 variable propositional: 341–344, 442, 443 real and apparent: 208, 236–238, 245, 438 Venn, John: 32, 177, 451 vicious circle principle: 245–246, 249, 285, 287, 345–346 von Neumann, John: 217

Index 505

Walker, Ralph: 234 Ward, James: 153, 297 Ware, Ben: 367 Warnock, Geoffrey: 303 Watson, John: 293 Wehmeier, Kai: 9 Weidemann, Hermann: 61 Weierstrass, Karl: 170–173, 198, 242, 286 Weir, Alan: 123 Welty, Ivan: 31 Weyl, Hermann: 289 White, Roger: 359, 407 Whitehead, Alfred: 155, 162, 172, 260, 284 falling out with Russell: 264 Universal Algebra: 174–178 work on Principia: 148, 214, 215, 222, 228, 235, 236, 241–244, 246, 267, 305, 358, 386 Whyte, Jamie: 469 Wiener, Norbert: 197 Williams, Bernard: 397 Williams, Michael: 401 Wilson, Mark: 90 Windelband, Wilhelm: 18 Wisdom, John: 2 Wittgenstein, Ludwig Blue Book: 68, 350, 373, 408, 411 Bodleianus: 316, 319, 332, 345, 346, 362, 382, 404, 406 Notebooks 29 Sep. 1914: 324; 23 Oct. 1914: 332; 1 Nov. 1914: 329; 23 May 1915: 362, 371; 16 June 1915: 351; 17 June 1915: 279; 20 June 1915: 364; 26 Apr. 1916: 345; 2 Aug. 1916: 393; 12 Sep. 1916: 370; 7 Oct. 1916: 399; 15 Oct. 1916: 371; 20 Oct. 1916: 381, 410; 4 Nov. 1916: 395; 21 Nov. 1916: 346 Notes on Logic B5: 270; B7: 322; B17: 278; B23: 286, 287; B48: 279; B55: 345; B76: 285; C9: 279; C12: 287; C15: 287; C26: 275; C30: 287; C50: 279; C51: 284

On Certainty: 412 Philosophical Grammar: 382, 412, 442, 443 Philosophical Investigations: 279, 339, 348, 413 Philosophical Remarks: 292, 295, 348, 461 Prototractatus: 352 Tractatus 1: 319; 1.1: 319; 2.01231: 349; 2.0141: 349; 2.021: 349; 2.0211–2.0212: 349; 2.022–3: 350; 2.0233: 349; 2.025: 349; 2.0251: 352; 2.03: 329; 2.06: 322; 2.061–2: 337; 2.063: 322; 2.131: 323; 2.141: 323; 2.161: 325; 2.172–4: 326; 2.181: 328; 2.182: 329; 3: 328; 3.0321: 327; 3.1: 328, 329; 3.11: 373; 3.14: 330; 3.203: 330; 3.23: 348; 3.311: 344; 3.314: 333; 3.322: 330; 3.324: 405; 3.326: 373, 376; 3.327: 373; 3.331: 341; 3.333: 345; 3.341: 330; 4.001: 330; 4.002: 330; 4.003: 367; 4.0031: 225, 347, 367; 4.063: 55; 4.11: 388; 4.111: 325, 405; 4.112: 366, 382, 405, 414; 4.12: 332; 4.123: 321; 4.1271: 342; 4.1272: 357; 4.12721: 349; 4.1273: 344; 4.211: 337; 4.221: 337, 349; 4.241: 357; 4.242: 357; 4.26: 335; 4.27: 335; 4.3: 335; 4.42: 336; 4.442: 372; 4.461: 375; 4.4611: 374; 4.53: 339; 5.02: 443; 5.4733: 406; 5.501: 341, 342; 5.511: 388; 5.522: 344; 5.5301: 355; 5.5302: 358; 5.542: 368; 5.5422: 115; 5.552: 376; 5.5521: 376; 5.5561: 352; 5.5563: 364, 365, 464; 5.6: 320; 5.61: 399; 5.62: 330; 5.632–5.641: 379; 5.64: 381, 395; 5.641: 372, 381; 6.1–6.11: 374; 6.122: 376; 6.125: 376; 6.1271: 375; 6.131: 376; 6.2: 387; 6.211:

506 Index

386; 6.233: 386; 6.3: 388; 6.341–2: 388; 6.35: 390; 6.373: 390; 6.3751: 338, 409; 6.41: 393; 6.42: 393; 6.421: 393; 6.423: 395; 6.432: 397; 6.52: 398; 6.53: 404; 3: 328

Wollheim, Richard: 154 Wordsworth, William: 152 Wright, Crispin: 144 Zermelo, Ernst: 248 Zermelo-Fraenkel set theory: 247, 306