The Development of Bertrand Russell's Philosophy 9781317853312, 9780415295451

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Muirhead Library of Philosophy

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

MUIRHEAD

Muirhead Library of Philosophy

20TH CENTURY PHILOSOPHY In 22 Volumes I

III

Contemporary American Philosophy (Vol I) Contemporary American Philosophy (Vol II) G E Moore

IV

Ludwig Wittgenstein

V VI VII VIII IX X XI

Philosophy in America Contemporary Philosophy in Australia A History of Philosophy (Vol I) A History of Philosophy (Vol II) A History of Philosophy (Vol III) Ideas The Development of Bertrand Russell's Philosophy Contemporary British Philosophy (Vol III) Contemporary British Philosophy (Vol IV) A Hundred Years of British Philosophy Lectures on Philosophy Commonplace Book Philosophical Papers Some Main Problems of Philosophy Bernard Bosanquet and His Friends Contemporary British Philosophy (Vol I) Contemporary British Philosophy (Vol II) Bertrand Russell Memorial Volume

II

XII XIII XIV XV XVI XVII XVIII XIX XX XXI XXII

Adams & Montague Adams & Montague Ambrose & Lazerowitz Ambrose & Lazerowitz Black Brown & Rollins Erdmann Erdmann Erdmann Husser! Jager Lewis Lewis Metz Moore Moore Moore Moore Muirhead Muirhead Muirhead Roberts

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

RONALD JAGER

~~~J~:n~~~up LONDON AND NEW YORK

First published 1994 by George Allen & Unwin Ltd Published 2013 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN 7ll Third Avenue, N ew York, NY 10017, USA Routledge is an imprint of the Taylor & Francis Group, an informa business © 1972 George Allen & Unwin Ltd All rights reserved. No part of this book may be reprinted or reproduced or utilized 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. The publishers have made every effort to contact authors/copyright holders of the works reprinted in the Muirhead Library 0/ Philosophy. This has not been possible in every case, however, and we would welcome correspondence from those individuals/companies we have been unable to trace. These reprints are taken from original copies of each book. In many cases the condition of these originals is not perfect. The publisher has gone to great lengths to ensure the quality of these reprints, but wishes to point out that certain characteristics of the original copies will, of necessity, be apparent in reprints thereof. British Library Cataloguing in Publication Data A CIP catalogue record for this book is available from the British Library

The Development of Bertrand Russell's Philosophy ISBN 0-415-29545-9 20th Century Philosophy: 22 Volumes ISBN 0-415-29526-2 Muirhead Library of Philosophy: 95 Volumes ISBN 0-415-27S97-X

ISBN 978-0-415-29545-1 (hbk)

MUIRHEAD LIBRARY OF PHILOSOPHY

An admirable statement of the aims of the Library of Philosophy was provided by the first editor, the late Professor J. H. Muirhead, in his description of the original programme printed in Erdmann's History of Philosophy under the date 1890. This was slightly modified in subsequent volumes to take the form of the following statement: 'The Muirhead Library of Philosophy was designed as a contribution to the History of Modern Philosophy under the heads: first of Different Schools of Thought-Sensationalist, Realist, Idealist, Intuitivist; secondly of different Subjects-Psychology, Ethics, Aesthetics, Political Philosophy, Theology. While much had been done in England in tracing the course of evolution in nature, history, economics, morals and religion, little had been done in tracing the development of thought on these subjects. Yet "the evolution of opinion is part of the whole evolution." 'By the co-operation of different writers in carrying out this plan it was hoped that a thoroughness and completeness of treatment, otherwise unattainable, might be secured. It was believed also that from writers mainly British and American fuller consideration of English Philosophy than it had hitherto received might be looked for. In the earlier series of books containing, among others Bosanquet's History of Aesthetic, Pfleiderer's Rational Theology since Kant, Albee's History of English Utilitarianism, Bonar's Philosophy and Political Economy, Brett's History of Psychology, Ritchie's Natural Rights, these objects were to a large extent effected. 'In the meantime original work of a high order was being produced both in England and America by such writers as Bradley, Stout, Bertrand Russell, Baldwin, Urban, Montague, and others, and a new interest in foreign works, German, French and Italian, which had either become classical or were attracting public attention, had developed. The scope of the Library thus became extended into something more international, and it is entering on the fifth decade of its existence in the hope that it may contribute to that mutual understanding between countries which is so pressing a need of the present time.' The need which Professor Muirhead stressed is no less pressing today, and few will deny that philosophy has much to do with enabling us to meet it, although no one, least of all Muirhead

himself, would regard that as the sole, or even the main, object of philosophy. As Professor Muirhead continues to lend the distinction of his name to the Library of Philosophy it seemed not inappropriate to allow him to recall us to these aims in his own words. The emphasis on the history of thought also seemed to me very timely: and the number of important works promised for the Library in the very near future augur well for the continued fulfilment, in this and other ways, of the expectations of the original editor. H. D. LEWIS

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

BY

RONALD JAGER

LONDON: GEORGE ALLEN & UNWIN LTD NEW YORK: HUMANITIES PRESS INC

First published in 1972 This book is copyright under the Berne Convention. All rights are reserved. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright Act, 1956, no part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, electrical, chemical, mechanical, optical, photocopying, recording or otherwise, without the prior permission of the copyright owner. Enquiries should be addressed to the publishers.

©

George Allen & Unwin Ltd 1972 BRITISH ISBN USA SBN

0 04 192028 7 391-00176-0

Printed in Great Britain in 11 point Baskerville type by Clarke, Doble & Brendon Ltd Plymouth

FOR GRACE

szne qua non

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Philosophy, though unable to tell us with certainty what is the true answer to the doubts which it raises, is able to suggest many possibilities which enlarge our thoughts and free them from the tyranny of custom. Thus, while diminishing our feeling of certainty as to what things are, it greatly increases our knowledge as to what they may be; it removes the somewhat arrogant dogmatism of those who have never travelled into the region of liberating doubt, and it keeps alive our sense of wonder by showing familiar things in an unfamiliar aspect. Russell, The Problems of Philosophy (1912)

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PREFACE

It may be well to admit at the outset that this book attempts not what is difficult but what is impossible. What it attempts is a critical account of Russell's philosophy-just that-without supposing that every reader is himself a philosopher at the beginning, though he may be at the end. It is written for those who know of Russell's philosophy and wish to know about it, for those who know about it, and wish to know it. For me, it was a mixture of high purpose and indiscretion that prompted the project, stubbornness that kept it going, and intimations of coherence in Russell's philosophy that kept it interesting. But of course Russell's work is too massive, too prolonged, too much a chaos of lucidity to focus in one volume. But I did not find that a reason for not trying it. The book was named after it was written, after I had read what I wrote and learned what I thought. What I think now is that I have emphasized only those continuities, and not nearly all of them, that are of philosophical, and not merely scholarly interest. What I also think is that more of Russell's philosophy, even his very technical work, can be made accessible to the general reader who cares to concentrate a bit, than is usually supposed. For example, this book does not presuppose anything more than a nodding acquaintance with modern logic: I have endeavoured to teach the reader what he needs to know in the places where it is relevant. The book might have been called an 'Introduction' to Russell's philosophy, but for the solid fact that five hundred pages is too much introduction to anything-for who then would have the hardihood for the real thing? There are several collections of essays on Russell's philosophy, and there are a few individual books on aspects of his work. Fewer than ten books, including biographies, and fewer than five good ones. And no comprehensive one, done from one point of view. But I have not tried to 'fill a gap in the literature,' if only because I think gaps tend to take care of themselves much too well as it is. The secondary literature in the professional 15

16

PREFACE

journals on ideas that derive from Russell is, however, enormous beyond mastery; it is almost at one with the development of analytical philosophy in this century. It is the Russell who is a central figure in this development that I have tried to portray, the effortlessly fertile supplier of topics for philosophy, and of ways of discussing them. I have resolutely attempted to think things through from Russell's point of view, even at the expense of travelling some familiar territory, even at the expense of lingering in some duller stretches that others have hurried past. Though I have dealt with men and ideas that influenced him, I have not made very many references to the writings of other philosophers on Russell, except those major figures whom Russell actually engaged in discussion. I have no wish to imply disparagement of the legions of others: Russell has stimulated some of the finest philosophy that has been written; and, starting with Moore and Wittgenstein, he diverted to philosophy some of the best minds of his age. But I had to choose between writing my own book and writing a scholarly treatise, embellished with acknowledgements, refutations, and dusty debates. This is not to be disparaged either, but the professional journals accommodate it quite nicely. So this is meant to be a book, not a debate and not a collection of essays; it has, I hope, a certain obstinate integrity-at least that is a quality I have often sought in Russell and sometimes found. The purpose and order of the book are made as clear as I can make them in the introductory chapter. What that chapter says the long way is this: Russell's philosophy can be grasped best conceptually if it is grasped first imaginatively. That is the way his thought developed, and that is the way this book is written; that is why I have not hesitated to describe or allude to the panoply of his thought even where I could not tarry to discuss its operation. Between the first and last draft of that chapter there occurred: the depositing of Russell's papers at the Mills Memorial Library of McMaster University in Hamilton, Ontario, Canada, the publication of Russell's three-volume Autobiography, the death of Russell himself at age ninety-seven in February 1970. Russell's name continues to be one of the most evocative in the language, and his ideas will be in circulation for as long as we can imagine.

17

PREFACE

I should like to tender the usual thanks and absolutions. Each of the following read some ancestor of at least one of the ensuing chapters, and some stalwart ones read more: Robert Brumbaugh, Michael Dunn, Frederic Fitch, Robert Fogelin, Karsten Harries, Ted Honderich, Stephan Korner, Frederick Oscanyan, John E. Smith, Casimir Lewy, Rulon Wells, Nicholas Wolterstorff, Peter Wright. In addition, four persons were more especially helpful, not only by reading portions of the manuscript but in other ways as well: Professors John Findlay, H. D. Lewis, Gilbert Ryle, and George Schrader. Thanks are also due to Kenneth Blackwell at the Russell Archives, to Pat Slatter and Nancy Kaplan at the typewriter, to Richard Olsen; and most of all, at every point, to my wife Grace. All these, as critics, colleagues and friends have been most helpful. Even where they have found the alleged structure of my thought inscrutable, they have sometimes made suggestions for smoothing the facade of my gargoyled prose. They are to be congratulated, warmly thanked, and absolved. Yale University

R.J.

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CONTENTS

page 15

Preface

1 INTRODUCTION: THE CHARACTER OF R U SSE L L ' S PHI LOS 0 PH Y Overview of Russell and His Work 1. Russell's books 2. The philosophic everyman 3. The place of philosophy 4. Philosophy intellectualized B. Design of Russell's Philosophy and of this Book 5. Chronology 6. Realist, Atomist, Neutral Monist 7. Two major themes c. Beginnings of Analytical Philosophy 8. Moore and metaphysics 9. Cambridge influences 10. The Principles of Mathematics

A.

2 THE EARLY METAPHYSICS PART

25

35

45

52

I

Being and Relations 1. Existence, being, subsistence 2. Immutable 'terms' 3. Internal and external features 4. The reality of relations B. Properties and External Relations 5. The reducibility of relations 6. Properties and relations: summary 7. Later concessions c. Contingent and Necessary Properties 8. Aristotelian realism 9. 'Necessary' and 'intrinsic' 10. Diversity and difference 11. Do differences differ? 12. The drift to atomism

A.

25

53

69

76

20

CONTENTS PART II D.

E.

Propositions and Their Constituents 13. Propositions as unities 14. 'Wisdom' as universal and particular 15. ' ... is wise' as a function Propositions: Their Truth and Their Unity 16. The 'truth' of propositions 17. What unifies propositions? 18. Realism in retrospect

3 THE THEORY OF LOGIC A. "Philosophical Logic" 1. The Nature of logic 2. The Indefinables of logic 3. Three primitives B. The Propositional Calculus: Truth Functions 4. The nature of truth functions 5. The axioms of Principia 6. The status of rules 7. The 'paradoxes of material implication' 8. The confusions of 'formal implication' 9. 'Inference' and Principia c. The Predicate Calculus: Quantification 10. Quantifiers 11. Propositional functions 12. The universal quantifier 13. 'Some' and 'a' 14. Russell's three quantifiers 15. The 'realistic' context D. Set Theory and Paradoxes 16. Frege and Russell on 'the paradox' 17. The shape of the paradox 18. The theory of types 19. The vicious-circle principle 20. Systematic ambiguity 21. Metaphysical types 22. Logic and metaphysics 4

THE PHILOSOPHY OF MATHEMATICS A. The Poetry and Essence of Mathematics 1. Mathematics and poetry 2. Principia

B7

99

111 112

11B

137

154

179 181

CONTENTS

3. Russell's critique of Peano 4. Retarding influences in mathematical philosophy B. Logicism 5. Classes and numbers 6. Defining the numbers 7. Summary 8. How to picture numbers c. Counting and Reasoning 9. How to count 10. Natural numbers 11. Mathematical reasoning D. Critical Perspectives 12. von Neumann's numbers 13. Are the definitions circular? 14. The nature of logic 15. Reinterpreting Principia 16. Principia, Kant, and mysticism 5

6

ATOMISM: THEORIES OF LANGUAGE A. The Theory of Descriptions and Its Consequences 1. The influence of language 2. The theory itself 3. Definite and indefinite descriptions B. Reference, Metaphysics, and Existence 4. The uses of definite reference 5. 'Means,' 'denotes' and 'incomplete symbols' 6. Logically proper names 7. The improprieties of proper names 8. Proper names as ghosts of 'terms' 9. Quantifiers as deputy proper names 10. The departure of 'existence' 11. Ontological simples c. Acquaintance and Constructions: Anticipations 12. Implications of "On Denoting" 13. Hume and Kant and Russell 14. The reductive vision ATOMISM: THEORIES OF KNOWLEDGE A. Acquaintance, Particulars, and Universals 1. The acquaintance principle as a truism

21

193

201

209

224 225

236

262

270 272

22

CONTENTS

2. What is acquaintance? 3. Acquaintance with universals 4. Acquaintance with particulars B.

Sense Data and Constructions 5. The nature of sense data 6. Sense data: 'at' and 'from' 7. Perspectival ism 8. The construction of time 9. The nature of constructions

c. The Atomist World Picture 10. The atomic landscape 11. Atoms as objects of acquaintance 12. Does the system cohere? 13. Ideal languages 14. Simples 7

NEUTRAL MONISM: MIND AND MATTER A.

The Neutrality of Sensations 1. Kinds of neutral monism 2. Identity theories 3. Sensations as neutral entities 4. Sensation and perception 5. What are sensations? 6. Perceptual and bodily sensations

The Construction of Mind 7. Images and perception 8. Images and having images 9. The ghost of the Cheshire Cat 10. Mental activity 11. The mind of the moralist c. Perception and the External World 12. The philosophy and science of perception 13. A 'casual' theory of perception? 14. Sensations as ingredients, as effects 15. Casual laws and casual dependence D. Beyond Neutral Monism. 16. The development of neutral monism 17. Percepts: two kinds 18. Percept2 as a bastard notion 19. Percept 1 as a legitimate notion 20. Seeing our own brains B.

285

302

320 321

335

351

363

8

CONTENTS

23

NEUTRAL MONISM: THE PRIVATE AND THE PUBLIC WORLD

377

A.

The Experiences of Others 1. Privacy and structure in knowledge 2. Physics and ontology in competition 3. My knowledge of others' experiences 4. Compresent qualities 5. Grounds and verifiers again

379

B.

Perspectival Privacy and Egocentricity 6. 'Here-now' as the point of privacy 7. 'This' and egocentricity 8. 'This' and logical privacy 9. Abstract structure and sensory content

391

c. Solipsism and Privacy 10. Solipsism without a subject 11. The escape from solipsism 12. No 'Self' at the centre D.

9

Structure and Knowledge 13. The 'Postulates of Scientific Inference' 14. The logic of the postulates 15. Knowledge of structure, not of content 16. Russell, Wittgenstein, and 'private language' 17. The style of Russell's mind

404

410

POLITICS AND EDUCATION

424

Philosophy and Politics 1. "A Free Man's Worship" 2. Freedom, philosophy, and politics B. Psychology and Politics 3. The roots of pacifism 4. Thwarted instincts 5. National hegemonies 6. Inner tension and outer anarchy c. War and Politics 7. Peace through disarmament in 1936? 8. Rights and wrongs of his pacifism 9. Common Sense and Nuclear War 10. The escape from individualism D. Education and Freedom 11. Beacon Hill 12. The Individual and the Citizen

426

A.

431

441

450

24 10

CONTENTS

ETHICS AND RELIGION PART I

461

ETHICS

A.

Objectivism 1. Shades of three theories 2. Moore and Russell's intuitionism 3. Two kinds of goodness

463

R.

Subjectivism 4. Santayana and the slide to subjectivism 5. Is Russell consistent?

471

c. Intimations of Naturalism 6. Passion and Reason 7. The escape from subjectivity

479

PART II RELIGION

D.

Polemics 8. The break in Russell's outlook 9. Combating religion

484

E.

The Worship of the Free Man 10. Early religious sentiments 11. "A Free Man's Worship" 12. "The Essence of Religion" 13. "Mysticism and Logic" 14. Religion joins science

487

F.

The Final Synthesis 15. Is the religious subjectivism coherent? 16. Unity with a world reconstructed 17. Religion and Philosophy

499

Works Cited Index

509 517

CHAPTER 1

INTRODUCTION: THE CHARACTER OF RUSSELL'S PHILOSOPHY

A.

0 V E R V lEW 0 F

R U SSE L LAN D

HIS

W 0 RK

1 Bertrand Russell is the great intellectual adventurer of the century. This book is a critical account of his voyage, and it tries to convey something of the spirit of his odyssey and the wonder of his productivity. It is appropriate to begin by pausing briefly at the spectacle of his immense reputation, the volume of his publications, and the length and breadth of those activities on which the reputation rests. He was intermittently associated with Trinity College, Cambridge, for about eighty years: from the time he went up in 1890 till his death in 1970. Of contemporary philosophers he lived longer, wrote more books, exerted more influence, started more discussions, was known to a wider audience than any other. He was a commanding presence both in the Anglo-Saxon philosophical world and on the larger intellectual scene of Europe, as a public figure, storm centre, foe of orthodoxy, friend of mankind, from the 1890s through the 1960s. These seventy-plus years comprise one sort of record unequalled by any other philosopher in history. Russell writes extremely easily, extremely well, and doubtless too much. He has said, not quite truly perhaps, that he never rewrites anything; and there is evidence that he seldom rereads what he didn't rewrite. Granted this, it is surprising that he repeats and contradicts himself so little. Only a very few philosophers are a persistent source of new ideas decade after decade, and Russell is pre-eminent in this group. His 1948 book, Human Knowledge, is sometimes cited as his 'last serious work in philosophy proper'; but it is notable that it is separated by precisely half a century from the lectures on Leibniz which became his second serious work in philosophy proper, A Critical Exposition of the Philosophy of Leibniz. Knowledge is Russell's theme, and there is no record of any other philosopher who has had new and 25

26

THE DEVELOP,MENT OF BERTRAND RUSSELL'S PHILOSOPHY

important things to say on epistemology for a continuous period of fifty years. "I came to philosophy through mathematics," he wrote in retrospect, "or rather through the wish to find some reason to believe in the truth of mathematics." He stayed to do the same for metaphysics, logic, ethics. Yet he is always in and out of the subject. Some of his works might be called occasional pieces, if they had been less impressive. For example: at twenty-three he worked in Germany for a few months and then wrote what was for a time an important book on that country, German Social Democracy (1896, reprinted in 1966); at fifty he visited the scene of the Russian revolution and wrote a perceptive and unorthodox, because unfavourable, book on that, The Practice and Theory of Bolshevism (1920); he went to China and wrote a book on that, The Problem of China (1922); he was divorced and wrote a book on that subject, Marriage and Morals (1929); he wrote a book on education, On Education Especially in Early Childhood (1926), started his own school and then wrote another, wiser, book, Education and the Social Order (1932). He fought, unsuccessfully, against wars, and wrote books on that subject. Some of his technical philosophy came about in the same kind of way: he was asked at the last minute to give a course of lectures on Leibniz, and with only this accident for excuse he wrote an enduring classic, A Critical Exposition of the Philosophy of Leibniz (1900); he was invited to give the Lowell Lectures in Boston, and, pressed for time, he dictated (without interruption, he said) what became Our Knowledge of the External World (1914); he was sent to prison for anti-war activity, and there, working a few hours a day for only a few months of his term, he wrote Introduction to Mathematical Philosophy (1919). These books are all in print today, still maintain their relevance; some, particularly the last, being of unsurpassed cogency and elegance. Not all his books came about in this casual way, not certainly The Principles of Mathematics (1903), nor his most distinguished work, the three-volume Principia Mathematica (1910-13), coauthored with A. N. Whitehead during the first decade of the century. During the composition of this latter work Russell took out time for public campaigning, unsuccessfully, for Parliament, women's suffrage, and revised taxation laws; and for philosophical campaigning, largely successful, against pragmatism and

INTRODUCTION

27

idealism. Several of his most important books are also collections of philosophical essays, the most significant being Philosophical Essays (1910, reprinted 1966,) Mysticism and Logic (1918). Some of his books were avowedly written for money at a time when he needed it, especially the popular science, The ABC of Atoms (1923), The ABC of Relativity (1925); some were written for editors or publishers, the most successful being what he called his "shilling shocker," The Problems of Philosophy (1912); some were written for fun, Sceptical Essays (1928), In Praise of Idleness (1935), Unpopular Essays (1950). All abound in lucid intelligence, insight, and wit. In 1953 he published the first of two volumes of short stories, Satan in the Suburbs, declaring that after devoting his first eighty years to philosophy he planned to devote the next eighty to other forms of fiction. Twenty titles have just been mentioned -less than half of his books. The catalogues of paperback books in print today list more than thirty of his titles. Few of these facts, or the pages more that could be adduced, are of any philosophical interest, though a sociologist may someday attend to them with profit. Someone may speculate that the evidence, namely those long shelves of books, is most easily explained by the hypothesis that "Bertrand Russell" has always been a front for a committee of opinionated experts, the membership changing as the committeemen die of old age, or are discredited for youthful follies. But the hypothesis is wrong, at least concerning the first sixty-five years of Russell's published work, which is all that is to be considered here. There is enough material prior to the publication in 1963 of his Unarmed Victory, the book that marked the presence of an alien influence, to give a context for rationalizing the kind of order I wish to impose upon the tumult of his ideas. I therefore want to make clear, or at least audible, the philosophic tone of voice adopted here.

2 Russell is the philosophical Everyman. If it is an intellectual idea, he has probably tried it; if it is a philosophical hope or distress, Russell has shared it; if a technical novelty, he has worried and exploited it; a noble theme, he has enlivened and varied it; an error, he has been tempted by it. To study the reactions of ideas in the laboratory of his mind is to~ see in sharp light - what is true of every man, but easily missed by the intro-

28

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

spective gaze - how variously volatile mere ideas are. While some central ones are stable elements resting quietly upon the evidence, and most of these enlighten, others of his ideas are more swiftly active: we see them seduce and coerce, blind and bemuse; and some of them are simply banners, and some are simply weapons. He is the philosophic Everyman in technique as well. The lure of the part and the challenge of the whole - these are equal and opposite intellectual inclinations, the holy passions of philosophy. They are equally, and very intensely, present in Russell's pages, the catalysts of all his thought. Pulverizing ideas may be his special forte, but locating the residue in some larger vision is his compensating impulse. His books, collectively, comprise a large slice of the autobiography of the modem philosophical temper. There is the notion, attractive and wrong in my view, that Russell is essentially an ingenious logical technician with an indecently large embroidery of miscellaneous opinions on diverse other subjects. I intend not to combat this idea, but to ignore it. I am ignoring it wherever I explore the connections, very many, between his technical and non-technical work; and I am ignoring it too wherever I linger to elicit the point of view from which Russell's ideas are compelling. I care about this opinion, and discard it, partly because I care about the unity of his philosophy; and I care about that partly because I care about Russell's conception of the philosophical vocation. The conclusion of this line of thought is probably lodged somewhere in the lofty proposition: we can become disciplined by reflecting on Russell's arguments but we can be helped to become wise only by reflecting on his enterprise. But such solemnity can wait, at least until we have sketched a first impression of what his philosophical project is. Viewed in the largest possible dimensions, Russell's enterprise is a relentless exploration of a certain hypothesis. To formulate the hypothesis and ascribe it to him is, too obviously, a massive oversimplification. Before venturing it, I must acknowledge that Russell never described his work in just one way, for he has never systematically tried to get his social and moral theories under the same rubric with his logic, epistemology, and metaphysics. He would, therefore, be suspicious of this sweeping summary of what he was up to. But we ought to try it anyway, if only because we can be

INTRODUCTION

29

more detached than he. Here is the Russellian hypothesis, dominating sixty years and sixty books. Philosophical scrutiny begins at the subjective pole, where certainty lies, and works outward in criticism, reconstruction, tentative hypothesizing, toward the public world, and then beyond to the structure of systematized science. The axis of his thought runs from immediate perception to scientific objectivity. "I have throughout been anxious to discover how much we can be said to know and with what degree of certainty or doubtfulness," he wrote in My Philosophical Development (1959, p. 11). The hypothesis which, I have said, his entire philosophy elaborates is relevant to logic as well as ethics, to the philosophy of mathematics as well as social theory. Since this may be surprising, it had better be explained. Whether the subjective starting points - which it may take some philosophical work to identify and arrive at - are the intuited indefinabIes of logic, or whether they are sense data, or whether they are the stuff of our souls (loves and fears, impulses and feelings), or whether individual freedom itself, the presumption of the hypothesis, to restate the point, is that one starts subjectively, gathers his certitudes, fortifies himself with logic, and works outwardly toward objectivity, that is, through "the metaphysics of the stone age to which common sense is due," to science or to social theory. That is the characteristic Russellian line of thought; it is subjectivism in method, but not necessarily in doctrine. Philosophy, by this hypothesis, is seen to be a purification of common sense by an elaborate process of intuition, postulation, bootstrapping, and logic chopping. It follows from this, and is true in any case, that a social or ethical theory which orients itself around a collective as a primary moral entity is alien to his outlook, and any which orients itself around the individual is, so far, congenial to it. Likewise, any epistemological theory with its point of departure in common sense or ordinary language is alien, and any that assumes that this public area is precisely what is problematic is, so far, congenial. Similar things hold for metaphysical theories that yield autonomy straight off to physical objects without justifying their place in a scheme that leads from the privacy of perception to the publicity of science. In such a vast context perhaps can best be seen Russell's lifelong opposition to Communism, his correlative emphasis upon the autonomous individual; his early

30

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

opposition to idealisms and pragmatisms, seen as differently astray in their subtle subjectivisms; and, latterly, his opposition to 'linguistic philosophy,' seen as dogmatically ascribing autonomy to the wrong area. At different times in his career Russell concentrated his attention on the line that gazes outward from subjectivity in very different ways, often in indifference to the larger picture, often in exclusive attention to the logical theories that sustained it. For this reason, and because of the extremely general character of the hypothesis in any case, it has been identified only for orientation, and cannot serve as an outline for exposition. There is more to Russell's philosophy than any summary can cover, and as much interest in the detail as in the design. 3 Under the microscope Russell's philosophy is technical, piecemeal, intricate, frequently cross-purposed. In the telescope it is a continuous zigzag movement, a veritable armada of ideas; it tacks into contrary winds, it lists with a ballast of undiscarded half-forgotten doctrines. So I shall sometimes trouble to explain how he gets from there to here, sometimes postponing the question of whether either the zig or the zag is quite true. There is a good reason for this; or at least one that belongs to my conception of philosophical investigation. One thing this book wants to say, and to say it if possible by showing it, is this: it is a very valuable thing in philosophy to recapture for an idea, even an unsuccessful one, the source of its attraction. Until this is attempted - as, for example, Russell himself did in his book on Leibniz - we have no clear understanding of its power either to illumine or mislead. This is more than a matter of charting the lineage of his ideas. In the endeavour to articulate the attractiveness of an idea, which may be other than its truth, and which may be tangled in a mass of fact and falsity, is the one firm place where philosophy and the history of philosophy overlap. Beguiling falsehoods are as much philosophy's business as are its banal truths. So too are the vaguer things that enliven and sustain them both: preconceptions, expectations, ideologies, and the permanent insights and temptations they embody when greatly handled by great minds. Sometimes, in short, the fact that Russell's argument is good, or bad, may be easier to see, but may teach us less, than an inquiry as to why it seemed persuasive, and why it seemed to matter. This

INTRODUCTION

31

has everything to do with philosophy, and nothing to do with filial piety. For all of that, we may, near the centennial of Russell's birth, be at just the wrong point to gain the right view: he is perhaps no longer contemporary nor yet historical. But there would be no merit in anxiety here, and no help for it. Sometimes, doubtless, a shadow falls between the noonday brightness of contemporaneity and the later lustre of historical importance. Professor Ryle has said that Russell cannot now be quite rightly seen until all remaining philosophers who have personally known him have passed from the stage. Still, books have begun to appear nearly annually on his work, and few professional journals are long free of references to him - though this is perhaps more true of Wittgenstein. It may be that Russell's style of thought, his central accent on the importance of formal logic, his concern for the first and fewest principles at the foundation of any inquiry, will soon again super1ede the influence of Wittgenstein among professionals. But for a larger audience, unattuned to fast-fading fashions, Russell continues, as always, to stand for philosophical pioneering. This book therefore does not so much aspire to draw him from professional shadows, if such there be, as to rescue him, for as large a philosophically inclined audience as possible, from becoming merely a distant and indefinable shape on the landscape of our experience. For reasons like these, direct retreat to mere historical reportage is inadequate; it is inviting, though, just because Russell has interacted with his contemporaries enough to suggest that the history of the period could be written in tolerable completenesswith notable gaps - by correcting the relation of Russell's views to those of his fellow philosophers. Here specialists of many sorts may wonder why I spend so little space correcting Russell's understanding of, for example, Frege, Wittgenstein, Cantor, Peano, Meinong, Bradley, Leibniz, Moore, Whitehead, Dewey, James, Carnap, Strawson; or theirs of him. It is to be hoped that the question comes in approximately that form; for then the answer is apparent. This is a book about Russell. At the same time I do not intend to be an uncritical reporter of Russell's dialectic; only a sympathetic one. No one should live long in the thickets of Russell's philosophy without an axe to grind, and no philosopher can. So this has to be a book in philosophy as well.

32

THE DEVELOP,MENT OF BERTRAND RUSSELL'S PHILOSOPHY

4 Though I have tried so far, somewhat failingly, to keep solemnity at bay, I may now succumb to it. For I said that Russell is the philosophic Everyman. This notion is a trifle pretentious, but also in need of a touchy and important qualification. Russell's philosophy, viewed as a whole and within the entire context of what philosophy has been, lacks -lacks what? An indefinable quality of thought and concern, such as we find in Spinoza (whom Russell, incidentally, enormously admired) and feel inclined to call passion. There is Pascal's I'esprit geometrique, but what of Ie coeur? Does the philosophy as a whole lack what Russell the man does not lack: humaneness, warmth, spiritual depth and finesse, passion? At any rate, all its refinements are intellectualized. With Russell, and even more in Moore, philosophy becomes austere and professional. Rationality reaches one of its new peaks of refinement, but so doing it tends to extinguish the sensibility of, say, Santayana, the rugged humanity of William James; it lacks altogether the personal passion of Kierkegaard, the inwardness of Sartre. There are books on practically everything - but none on aesthetics. Even Russell's moral theory, whether objective and rationalistic as at first, or subjective and emotion-based as at the last, is, like his religious belief and disbelief, urbane, almost wry. Compare Russell and Nietzsche on religion: though both are called atheists, there is a world of difference between the articulate and donnish agnosticism of Russell and the stammering anguished unbelief of Nietzsche. There are explanations for this: what it means, and why it is. In the early days of his career, Russell, who was perceived by his friends as somewhat puritanical and ascetic, squeezed the passion from his thought only to discharge it in his celebrated wit, or, more ponderously, to enshrine it in a halo of romantic despaira moving but disconnected super-imposition upon his technical work. One half of this disjunction is specifically and revealingly evident in, for example, the capital-lettered cadences of such early essays as "On History" and "A Free Man's Worship." The final paragraphs of these essays are representative; the first is schmaltz, but the second, which is not, differs only slightly in tone. Here paraphrase is impermissible: "Year by year, comrades die, hopes prove vain, ideals fade; the enchanted land of youth grows more remote, the road of life

INTRODUCTION

33

more wearisome; the burden of the world increases, until the labour and the pain become almost too heavy to be borne; joy fades from the weary nations of the earth, and the tyranny of the future saps men's vital force; all that we love is waning, waning from the dying world. But the past, ever devouring the transient offspring of the present, lives by the universal death; steadily, irresistibly, it adds new trophies to its silent temple, which all the ages build; every great deed, every splendid life, every achievement and every heroic failure, is there enshrined. On the banks of the river of Time, the sad procession of human generations is marching slowly to the grave; in the quiet country of the Past, the march is ended, the tired wanderers rest, and all their weeping is hushed." ("On History," 1904, reprinted in the 1966 ed. of Philosophical Essays.) "Brief and powerless is Man's life; on him and all his race the slow, sure doom falls pitiless and dark. Blind to good and evil, reckless of destruction, omnipotent matter rolls on its relentless way; for Man, condemned today to lose his dearest, tomorrow himself to pass through the gate of darkness, it remains only to cherish, ere yet the blow falls, the lofty thoughts that ennoble his little day; disdaining the coward terrors of the slave of Fate, to worship at the shrine that his own hands have built; undismayed by the empire of chance, to preserve a mind free from the wanton tyranny that rules his outward life; proudly defiant of the irresistible forces that tolerate, for a moment, his knowledge and his condemnation, to sustain alone, a weary but unyielding Atlas, the world that his own ideals have fashioned despite the trampling march of unconscious power." ("A Free Man's Worship," 1903, reprinted in Mysticism and Logic, hereafter cited as M&L.) At about the same time Russell wrote a rhapsodic essay "The Study of Mathematics" (1902), full of "lofty thoughts that ennoble," in which the timeless purity of mathematics is commended as the ultimate response of the universe to the human hunger for perfection. Was the passion mobilized in early years into the service of this stirring but impersonal romanticismwhile the intellect researched the logical foundations of mathematics - poured in later years, during and after the Great War, into moral and political activism? If so, this did not heal the breach created by the early divorce of thought and feeling, a cleavage B

34

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

which became both conscious and resolute. It is, for example, no accident that what Russell tells us about the analysis of matter in The Analysis of Matter (1927) excludes the rhythms of the "trampling march of unconscious power"; and it is no accident either that what he tells us about the analysis of mind in The Analysis of Mind (1921) omits the insights into the workings of mind in Marriage and Morals (1929). I shall say further, and more developed, things about all this in later chapters of this book, particularly the last one. For the present I am remarking that Russell simply took it as a fundamental tenet that the intrusion of passion upon philosophical thought was a trespass. But it is a curiously restricted view of what this would have involved. He supposed, or affected to suppose, that it would always take the pernicious, emotional form of diverting logical arguments toward merely personally satisfying conclusionsitself a narrow and intellectualized view of what passionate thought amounts to. Russell, though he has explained himself as thoroughly as any major philosopher in history, has left the resolution of these matters, if there is one, to the sympathy and prowess of his interpreters. For now it is enough to remark that most of Russell's official philosophy does not grow up directly out of the experience of life; only his journalism does. His philosophy grows out of the exercise of intellect, out of his marvellously imaginative thought about the nature of the universe. He did not, for example, start talking about sense data because, every day, he perceived them all about him: he did so because a theoretical conception of knowledge seemed to require them. In none of this was Russell nearly so extreme as Moore, who said that he did not think that the world or science would have suggested any philosophical problems to him at all but for the queer things other philosophers, including Russell, had said about them. The world and science suggested legions of philosophical problems to Russell. But - to extend now the point about passionate thought to philosophy itself - it was not often to be Russell's way, as it was Wittgenstein's, so to philosophize as to embody within the essence of his philosophy a suspicion about philosophy's own validity. The cool address of the intellect to the ambiguities of a concrete question, the invention of a novel logical stratagem - such was Russell's usual antidote to those symptoms of self-obsession so

INTRODUCTION

35

prominently featured in twentieth-century philosophy. For all his writing on philosophical method, it was not Russell's intention to make philosophy a problem unto itself, only to redirect it. B.

DESIGN OF

RUSSELL'S THIS

PHILOSOPHY AND

OF

BOOK

5 So far I have been concerned only with the shape and texture of Russell's philosophy and a way of viewing it - but not the stuff itself, not even with principles or a plan for coming to terms with it. It is time to come to that. One stringent principle, derived from what has just been said, is this: to try to forget for a few hundred pages that Russell has been for so many so much more than a technical analytical philosopher, to put aside the moral energy, personal and social, that prompted so much of his work and at least half of his writing. All this, with justified lapses, is consigned to the last two chapters of the book. Accordingly, the initial account of Russell's work given in this chapter requires to be supplemented by the account I could only give later in the book, particularly that attempted at the close of each of the last three chapters. Next, the organization and sequence of chapters tries to acconwdate the needs of the newcomer who may need a map to avoid a traffic jam of ideas in Russell's books. Philosophies are like cities, they grow largely unplanned; intermittf1tnt re-zoning demolishes obsolete commitments and relocates some structures intact. Russell is famous for his merciless way with his own past: he said for example that his new theory of 'existence' removed "a slum of metaphysical debris," some of it his own. The mapmaker often obscures the growth order in the interest of logical order, unless he is lucky. But in my book we ought to be able to have it both ways with Russell. It is as if, surprisingly, his philosophy grew in a more or less logical way. Anyhow, the topics which here give title to the chapters are connected in his thought by sequence and by doctrine, and they are also a list of major subdivisions of philosophy itself. Russell's attention to these subdivisions could hardly have been as orderly and sequential as this might suggest; but though there will be plenty of glancing forward and backward within each chapter, the chronology of his development is kept intact through the first eight chapters. The last two run parallel with the rest.

36

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

Finally, the exposition throughout recognizes an important threefold division, developmental and doctrinal, within Russell's work. These three phases represent, as it were, the successive ways in which Russell perceived the entire philosophical landscape, and which therefore supported and shaped the architecture of his ideas. I mean by his realist phase primarily his work to 1912 (basically, the first four chapters), by his atomist phase his work to the nineteen twenties (the next two chapters), and by his neutral monist phase (the seventh and eighth chapters) the remainder of his philosophical work. Since this threefold distinction is more an instrument of exposition than of controversy, we can also sometimes ignore or override it, as Russell did. It would be convenient to say simply that Russell's philosophy has a beginning, a middle, and an end; it does, but not all of it comes in that order. However, most students of Russell will agree that these distinctions accord with definite shifts of emphasis and preconception in his work, shifts that can often be traced to specific pressures. Having postulated that much of Russell's work does fall into a fairly natural order, we can go on to incorporate the idea, somewhat more debatable, that each of these phases of his thought has its own philosophical centre of gravity. This fact can be simplified and pinpointed now for later explanation: in the realist pha,;e he is preoccupied with the metaphysical doctrine of external relations, in the atomist phase with the notion of logical form, in the neutral monist phase with the ideals of science. 6 I just intimated that the atmosphere of each phase of his thought was dominated respectively by metaphysics, by logic, by science. It may be helpful now to make a very brief and sweeping survey of the landscape which is to be thus triangulated. We can arrive then at some generalizations about the slope and direction of Russell's thought. Realist. To get the feel of Russell's early philosophy, which was devised explicitly as support for a philosophy of mathematics, it is necessary to be struck, as he was, by such a question as this: What is the ultimate explanation of the difference between "Caesar died" and "The death of Caesar"; and also to have some ideas on what sort of answer might suffice, what sort of further questions might come up. For example, what kinds of

INTRODUCTION

37

entities are involved, in what kinds of combinations; why is one a proposition and the other not? In 1900, in his book on Leibniz, Russell wrote: "That all sound philosophy should begin with an analysis of propositions is a truth too evident, perhaps, to demand a proof" (p.8). He was to offer indirect 'proof' as time went on, though he was to change his conception of analysis even as he was to change his conception of propositions. Curiously, it is impossible to state just what sort of thing Russell at first took a proposition to be. He never managed to make it entirely cleara very unsatisfactory situation indeed - but we can make this tolerable by citing some of the things he repeatedly emphasized. Propositions are not sentences, for then they would be composed of words; they are not judgments, for then they would be mental; they are not subjective, for then they would not be present to different knowers. Propositions are objective, the primary objects of cognition. Propositions are connected by logical relations, they can imply and contradict each other. They are the truths and falsehoods of the world. They are, or may be, the premises or conclusions of arguments. Finally, they are composed of terms, whose nature and combinations it is the peculiar business of philosophy to scrutinize. Already, this is a lot to swallow straight off, but the important point for now is that in his early realist phase Russell took all this as the merest philosophical common sense. It was as if he did not expect anyone but woolly-minded idealist philosophers to dispute any of it. So we must first try to take it in an innocent way. If we do, we get this very general description of the realist stage of Russell's thought. First and foremost, the main polemical target is usually some form of philosophical idealism with, as Russell saw it, its two cardinal doctrines: that the knowing process affects the known object, and 'the axiom of internal relations' (roughly: that it is its necessary relatedness to everything else that gives anything its identity). Secondly, he looks upon anything as a philosophical datum only insofar as it is, or may be, an element in an objective proposition: epistemology is the study of the mind's relation to such elements; metaphysics studies the relations among them; logic studies the forms of propositions and the relations between classes of them; ethics studies a special subset of the elements, chiefly goodness and obligation. Thirdly, language is the transparent vehicle for traffic between minds and

38

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

propositions. The entire conception is emphatically Cartesian in its boldness and simplicity of viewpoint, in its confident rationalism, and in its vague analogy to mathematics. Put otherwise, Russell is a pluralist, and all the world's unities are propositional. Atomist. Roughly speaking, metaphysical realism was to have provided a foundation for Russell's philosophy of logic and mathematics; equally roughly, the atomism of the next stage was a speculative consequence of that philosophy of mathematics, a consequence that in turn undermined a good bit of the foundations. Metaphysical realism preceded his pivotal work with Whitehead on Principia Mathematica (1910), logical atomism succeeded it. It was in 1914 with the publication of Our Knowledge of the External World that Russell first used the term 'logical atomism' to describe his work. The book contains a famous chapter, characteristic of this phase of Russell's thought, entitled "Logic as the Essence of Philosophy," in which he dilates on the importance of logical analysis if philosophy is to make progress. Throughout his atomist phase it is not idealism, but philosophical scepticism that becomes the main polemical target. A characteristic statement occurs at the beginning of his 1918 lectures, "The Philosophy of Logical Atomism" (hereafter "Lectures"). "The kind of philosophy that I wish to advocate, which I call Logical Atomism, is one which has forced itself upon me in the course of thinking about the philosophy of mathematics. . . . The reason that I call my doctrine logical atomism is because the atoms that I wish to arrive at as the sort of last residue in analysis are logical atoms and not physical atoms. Some of them will be what I call 'particulars' - such things as little patches of colour or sounds, momentary things - and some of them will be predicates or relations and so on. The point is that the atom I wish to arrive at is the atom of logical analysis, not the atom of physical analysis" (reprinted in Logic and Knowledge, 1956, pp. 178-9, hereafter L&K). An interesting sidelight is the fact that the shift to atomism in metaphysics, which was not as abrupt as this survey suggests, coincided with his relinquishing what he called "the belief in the objectivity of good and evil," since he believed that his early ethical views had suffered a decisive refutation in Santayana's Winds of Doctrine. It coincided

INTRODUCTION

39

also with his first detailed attacks on the problems of perception, and his concern to base epistemology on what he and Moore were calling 'sense data' - the immediate objects of perceptual awareness. Today and for a decade past, the magnificent and profound artificialities of logical atomism are widely thought to be Russell's most distinguished philosophical work.

Neutral Monist. I here use the term, somewhat inaccurately, to designate a phase of Russell's work, as well as a pervasive thesis. The doctrine itself, first presented in 1921, concerns the idea that mind and matter are not ultimate kinds of entities but 'logical constructions' from metaphysically neutral stuff. It is the scientifically orientated frame of mind, anticipated in "Mysticism and Logic," in which this typical and central doctrine is held that is of present interest: "a truly scientific philosophy will be more humble, more piecemeal, more arduous, offering less glitter of outward mirage to flatter fallacious hopes" (M&L, p. 32). Russell reports that he "began to think it probable that philosophy had erred in adopting heroic remedies for intellectual difficulties, and that solutions were to be found merely by greater care and accuracy" (L&K, p. 324). The doctrinal source of this emerging conviction is plain to see in many typical passages: "What are we to take as data in philosophy? What shall we regard as having the greatest likelihood of being true, and what as proper to be rejected if it conflicts with other evidence? It seems to me that science has a much greater likelihood of being true in the main than any philosophy hitherto advanced (I do not, of course, except my own). In science there are many matters about which people are agreed; in philosophy there are none. Therefore, although each proposition in a science may be false, and it is practically certain that there are some that are false, yet we shall be wise to build our philosophy upon science, because the risk of error in philosophy is pretty sure to be greater than in science" (L&K, p. 339). What it is to build philosophy upon science is itself a perplexing philosophical question. The idea would not have tempted the early Russell; the task was more recommended than explained

40

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

by the later Russell. His most characteristic instance was the casual theory of perception that inspired neutral monism. There is in this third and later phase very much favourable comment about the nature of science, its methods and its modesty, and the qualities of mind its pursuit supposedly inculcates. Though Russell did manage to avoid many of the pitfalls of loose favourable talk about science, he did not avoid all of them. One of the latter is the tendency, in Russell and others, to think of a new and promising ism (say, behaviourism) as science, until its most sanguine claims are shown inadequate, and then to call it philosophy. In general this holds: he was at first inspired by the possible contributions of philosophy to science; he was later inspired by the prospect of important contributions by science to philosophy. Thus, in later books he re-engaged the perplexing problem as to how we escape the privacy of our own individual experience. Correspondingly, the problem for the philosophy of science, as he saw it, was to get the actual method of scientific inquiry stated and clarified in a way that could be of use in philosophy. It was this which led him once more to the classical problem of probability and induction, and culminated in his exposition of 'the postulates of scientific inference' which concludes his 1948 Human Knowledge. Earlier, I ventured to characterize the whole of Russell's philosophy as an escape from subjectivity. I have now brought a part of that idea into more usable form: identifying three phases highlights certain doctrinal overtones of a project that was quite differently conceived at different times. Very much the metaphysician at the outset, Russell moved, from 1905 onward, always in the direction of minimizing metaphysical commitments, by turning attention to questions of method, knowledge, and language; and for this he had a well developed rationale, having invented a number of technical devices for avoiding unpalatable assumptions. Throughout there is this striking historical symmetry: the Cartesian confidence of his realist phase, the Leibnizian vision of his atomist phase, the Humean empiricism of his later phase; for it is true that Russell more than any other recent philosopher has lived through, in his work, a modern version of the classical systems of the seventeenth and eighteenth centuries. To stress sequential phases in Russell's work is a convenient,

INTRODUCTION

41

though artificial, way to do justice to some facts at the expense of humbler ones. The humbler ones contain all the reasons and arguments, the very stuff of Russell's philosophy. Not only have we yet to be presented with an argument, we have yet even to describe one. 7 There are two very broad classes of arguments that need to be described, so broad indeed, and so fundamental, that they may be called, at the outset, more simply 'themes.' The first theme, explicitly present, concerns the twin roles of analysis and synthesis in his philosophy; the second theme, only implicitly present, concerns the reciprocity between the technical and the philosophical in his work. These themes are prominent in this book because they are essential to his philosophical outlook. Consider each of them in turn. (a) Analysis and Synthesis. Russell wrote in 1924 "the business of philosophy, as I conceive it, is essentially that of logical analysis, followed by logical synthesis" (L&K, p. 34). The most notable form of analysis at his hands was not definition as much as the elimination of surplus meaning; analysis typically works from what is given and certain and vague, to something perhaps less certain but certainly more precise, more logically definite. Analysis, for example, tentatively reduces knowledge to sense perception, time to instants, mind to mental experience, numbers to classes, and the like, in order to see what must be retained and why. The most notable form of synthesis was not speculative system-building but what he called logical reconstruction from analytic components. Thus analysis and synthesis, each conceived and executed in a very special way, are two sides of one doctrinal and methodological coin. Together they constitute the most prominent and pervasive theme in his philosophy, source of his most characteristic and most controversial arguments. The more readily grasped of the two is the doctrine of logical constructions (a special form of 'synthesis'), and, further simplified for convenient summary now, it may be stated as follows. Statements that seem to refer to puzzling entities are to be reformulated (according to well-articulated criteria) so that this apparent reference is eliminated. Russell states the general rule this way: "whenever possible substitute constructions out of known entities,

42

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

for inferences to unknown entities." It is, quite deliberately, an updated positive version of Occam's razor, which stated: "entities are not to be multiplied without necessity." What are the puzzling inferred entities to which reference is to be avoided? Russell's view is that reference can be avoided, and for philosophical purposes should be avoided, to a good many things we normally take for granted. First, it was only reference to logical oddities, square circles, classes of non-self-membered classes; then it was reference to fictitious entities, golden mountains, present kings of France, that was to be circumvented. But the systematic development of the restating techniques led to eliminating references to numbers (in favour of classes), classes (in favour of properties), matter (in favour of sense data), minds (in favour of sensations). Russell apparently knew the temptation of what he once called the 'ratiocinator's fallacy' - which explains each thing by showing that it is really something else. Anyway, all such entities were to be considered not as the direct referents of our thought or statements, but as items logically constructed out of the more immediate material. Analysis, accordingly, was commonly thought of as the process of thought required to yield those indubitable building blocks of logical construction. In the early buoyant days Russell put it this way: "The discussion of indefinables - which forms the chief part of philosophical logic - is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple. Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them" (Preface, The Principles of Mathematics, hereafter Principles). When Russell said in "Logical Atomism" that "the most important part [of philosophy] consists in criticizing and clarifying notions which are apt to be regarded as fundamental and accepted uncritically" (L&K, p. 34), he is not merely issuing general cautionary notes, but is summarizing entire volumes of his own more technical work in analysis. Analysis and construction therefore comprise a central and explicit theme throughout Russell's philosophizing, a methodological theme surviving many shifts of doctrine, giving an

INTRODUCTION

43

identifying style to his thought, suggesting certain characteristic kinds of argument, and discouraging others. (b) Logic and Philosophy. I mentioned a second theme which I called rather vaguely the reciprocity of the technical and the philosophical in Russell's thought. What I mean is this. For Russell each technical device within logic or the foundations of mathematics characteristically has its more general philosophical counterpart; and conversely, every genuinely philosophical insight or achievement has its technical expression, consequence, or analogue. This bald thesis about the formation and structure of his thought needs plenty of qualification, for Russell has himself never put it so concisely. Even so, it is, or so I shall argue, the heart of a very important feature of his work, a doctrine more allusive than direct, operative rather more than discussed. This logical and philosophical interplay is easily traceable to the influence of Leibniz, but I shall not trace it. The theory of descriptions, a technical trick which was unusually fertile in his thought, and the theory of types, another symbolic artifice with many an ontological overtone, are but the two most notable illustrations of this pervasive feature, this propensity to find a larger moral in every logical insight. Indeed, if a single disposition of mind is to be identified as the wellspring of Russell's enormous creativity, it is the habit of counterpointing his every score. I am not now commenting on the justification for this inclination, a vast subject, but only highlighting the fact. The fact is present at every important turn, and it is what binds together what otherwise appear to be disparate methodologies. There is, for example, his casting of the (metaphysical) doctrine of internal relations in the form of the (logical) doctrine that every proposition is subject-predicate in form. There is - to take another random example - his way of arriving at what had been Hume's view that the human Ego amounted to a bundle of sensations and nothing more. Let us illustrate briefly with this notion. Russell is considering defining 'persons' and he says (my interpolations) : "We proceed here just in the same way as when we are defining numbers. We first define what is meant by saying that two classes 'have the same number,' and then define what a number

44

THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

is. The person who has a given experience x will be the class of all those experiences which are 'experiences of the same person' as the one who experiences x . ... Therefore we shall say that a person is a certain series of experiences [d. a number is a certain class of classes]. We shall not deny that there may be a metaphysical ego [d. we shall not deny that there may be metaphysical beings called numbers]. We shall say merely that it is a question that does not concern us. . . . What we know is this string of experiences that makes up a person [d. what we know is this class of classes that makes up a number]" ("Lectures,"

L&K, p. 277).

What I call the reciprocity of the technical and the philosophical in Russell's work is certainly not something that came about unconsciously, but neither did he ever raise the whole matter with full and direct explicitness as a subject for critical scrutiny. Rather more typical was, for example, his way of charting the route to logical atomism by saying that it is a view "which has forced itself upon me in the course of thinking about the philosophy of mathematics, although I should find it hard to say exactly how far there is a definite logical connection between the two" (L&K, p. 178). Well, it is hard to see just what that linkage is; moreover, the 'logical connection' between his technical and more general philosophical views was a different sort of connection in different places. In the longer quotation above the connection is only a speculative parallel between persons and numbers; in the case of the metaphysics of atomic particulars the connection with the logical variables of quantification was quite different; different still from this were his ontological reflections on the logical theories of types and descriptions; and the search for the postulates of scientific inference is still otherwise related to its counterpart. The idea of axioms for a deductive system; and so on. Yet these and many other instances are variations on what is really one methodological theme, present throughout Russell's work. It is possible to get some idea of how important and how controversial is the mode of thought which supports this theme by reflecting on the casual way in which it issued in certain very severely conceived maxims. "I always wish to get on in philosophy with the smallest possible apparatus, partly because it diminishes the risk of error, because it is not

45

INTRODUCTION

necessary to deny the entities you do not assert" (L&K, pp. 221-2). There are doubtless many reasons for wanting to get on in philosophy with the smallest apparatus, and many reasons against it; but the point here is that Russell has merely transformed an ideal from a formal system of logic into a philosophical principle, and has then cast about among his current philosophical preoccupations for the handiest and briefest rationale for the newly borrowed principle. It is a very consequential and audacious procedure. In section 6 above I commented on the surface texture of three stages of Russell's philosophy: realism, logical atomism, neutral monism. In section 7 I tried to identify the limbs and lay bare the skeleton and sinews. To return momentarily to a procedural matter: what has been said may be focused in the promise that, in addition to the philosophical subtopics that entitle the chapters, there are just two kinds of principles of organization in this book: (1) the three chronological phases of Russell's work, and (2) the two doctrinal themes. These two themes, (a) the method of analysis and synthesis (construction) on the one hand, and (b) the reciprocity of the technical and the more generally philosophical on the other hand, are themselves intimately connected, and one of our final goals will be to put this fact on display by putting it in context. When it is sufficiently on display we will be appreciating something of the uniqueness and mobility of Russell's mind. C.

BE GINNING S

OF ANALYTICAL

PHILO SOP HY

8 We have worked our way round to the beginning. Even the most creative philosopher's work does not spring full blown into an alien context. Certainly not Russell's. Let us elicit some of the crosscurrents of influence that stirred the beginnings of analytical philosophy, a rare moment in the history of philosophy - robust, confused, and endlessly fertile. In 1902, Russell, just thirty years old, is prefacing his fourth book, The Principles of Mathematics: "On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted

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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. 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 J believe to be otherwise insuperable. The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics." Let us momentarily ignore what this summary passage says to consider what it conveys. What it conveys is representative of several things at once: of Russell's characteristic generosity; of the metaphysical impulse within his work on the foundations of mathematics; of the inter-connectedness between his ideas and those of his contemporaries; of the sense of liberation rather than mere curiosity with which Russell greeted philosophical insight. In just one way is the passage untypical of Russell: what it says is cryptic and obscure. Yet once we have read sufficiently in and around Russell to have figured out that the "non-existential nature of propositions" means just that propositions are logical entities, objects of thought, which are not relative to the mind as are ideas, nor to time and place as are sentences; and once we have discerned that 'existents' are those objects of thought which occupy space and time and are contrasted with 'entities' which, though including existents, include also thinkables not related to space or time, like universals, numbers, and relations; and once we have seen that the clause about relations was a flag-waving farewell to the philosopher Bradley and the doctrine that all relations are internal - then the passage is easily seen to be the elegant testimony to the simplified metaphysical realism that it is. We may only wonder why Russell, who had already written an ingenious book on geometry and an important book on Leibniz, should have bothered to credit Moore and to highlight these simple, sober, and entirely unedifying opinions. The reason is that Russell is announcing a kind of British philosophical home-

INTRODUCTION

47

coming, having already completed his second philosophical turning. 9 He had done his first serious thinking, before and during his student days at Cambridge, in the philosophical tradition of J. S. Mill. McTaggart then beguiled him and Moore into what he later regarded as a youthful aberration, and became thus the first of a series of colleagues to exercise decisive influence. Despite an unexcelled creativity of his own, Russell had an equally astonishing capacity for learning from others. Repeatedly, he seems in fact simply to have adopted whole clusters of ideas from others, and then quickly stirred them into some novel mixture of his own. Before 1920 he fell successively under the influence of McTaggart and Bradley, Moore, Whitehead, Wittgenstein, and thereafter J. B. Watson and the logical positivists.* The casualty of the first-mentioned encounter was the empiricism of his early Cambridge years. He wrote in "My Mental Development": "The set in which I lived was very much influenced by McTaggart, whose wit recommended his Hegelian philosophy. He taught me to consider British empiricism 'crude,' and I was willing to believe that Hegel (and in a lesser degree Kant) had a profoundity not to be found in Locke, Berkeley, and Hume, or in my former pope, Mill ... [Stout] and McTaggart between them caused me to become a Hegelian; I remember the precise moment, one day in 1894, as I was walking along Trinity Lane, when I saw in a flash (or thought I saw) that the ontological argument is valid. I had gone out to buy a tin of tobacco; on my way back, I suddenly threw it up in the air, and exclaimed as I caught it: 'Great Scott, the ontological argument is sound.' I read Bradley at this time with avidity, and admired him more than any other recent philosopher" (The Philosophy of Bertrand Russell, ed. P. A. Schilpp, p. 10, hereafter Schilpp).

* If this were, as it is not, an essay on the philosophers :who most influenced Russell one would include, in addition to these personal acquaintances, Peano and Frege amongst logicians; Meinong, Mach, James, Santayana, and Carnap amongst other contemporaries; and Leibniz, Hume, Kant, and Mill amongst predecessors. Plato and Aristotle, except in the most general sense, have had very little effect on his thought; Aquinas, and medieval philosophy generally, are a closed book. First Descartes and later Berkeley are philosophers whose themes are very congenial to Russell, but their direct influence is very small. Hegel, after the initial falling away, was his first scapegoat, linguistic philosophers his second: these bear, respectively, the opprobrium of making philosophy muddled and making it trivial.

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In this heady atmosphere Russell first became a philosopher. "There was a curious pleasure," he writes in somewhat prejudiced retrospect, "in making oneself believe that time and space are unreal, that matter is an illusion, that the world really consists of nothing but mind" (Portraits from Memory, 1956, p. 22). In that remote period he published a number of reviews; an article which, six decades later, he called "unmitigated rubbish" ("On the Relations of Number and Quantity," Mind, 1897); his first philosophical book, An Essay on the Foundations of Geometry (1897), also since dismissed as "somewhat foolish"; and accumulated a large number of notes for a work on the foundations of the sciences, some samplings of which he published, as curiosities, in 1959. The rubbish article has in fact an impressive cogency, though it and some of the rest, including the notes, are, in the manner of young admirers of Hegel even today, festooned with periodic 'dialectical transitions,' entirely abstract anxieties about 'abstraction,' and casual discoveries of 'contradictions. ' Enter G. E. Moore. "He also had had a Hegelian period, but it was briefer than mine. He took the lead in rebellion, and I followed, with a sense of emancipation. Bradley argued that everything common sense believes in is mere appearance; we reverted to the opposite extreme, and thought that everything is real that common sense, uninfluenced by philosophy or theology, supposes real. With a sense of escaping from prison, we allowed ourselves to think that grass is green, that the sun and stars would exist if no one was aware of them, and also that there is a pluralistic timeless world of Platonic ideas. The world which had been thin and logical, suddenly became rich and varied and solid" (Schilpp, p. 12). "Thin and logicaL" This, if we can believe it, is Russell's description of the Hegelian world as edited by Bradley and McTaggart, and as contrasted with the rich and varied one of his new realism. A fairly widespread opinion has it that it is Russell's philosophical world that is thin and logical. This paradox may be mitigated by a few examples. Whereas idealism had persuaded Russell that, for instance, belief in the reality of space and time was logically insupportable he now came not merely to discount that, but actively to believe that points of space and instants of time were, like particles of matter, among the meta-

INTRODUCTION

49

physical 'existents' of which we spoke earlier. It was just here that Russell came later to feel the pressure of Whitehead's views. For it was Whitehead's 'construction' of points, instants, and particles as sets of events, each of finite extent, that Russell adopted wholesale within the next few decades, subsequently taking the method of constructions as his model with respect to all the objects of perception. Here, with the concepts of space, time, and matter, is one clear lineal descent from idealism through realism to atomism. Among the other elements giving variety and solidity to this early world of Russell were numbers. Whereas he had understood Hegel as having taught that mathematics was not quite true, he later reported himself as having reacted to the point where he "imagined all the numbers sitting in a row in the Platonic heaven." Then once more a large scale vision was adopted from someone else. "I thought of mathematics with reverence, and suffered when Wittgenstein led me to regard it as nothing but tautologies" (Schilpp, p. 19). Again the direct line is apparent, from idealism to realism to atomism. That universe, so rich and varied and solid, into which Moore had led him from out of the shadow of Bradley's Absolute, was destined to become thin and logical again.

10 The Principles of Mathematics, the book whose prefacing tribute to G. E. Moore launched this survey, is a formidable work of over five hundred pages, and we shall have to find a strategy for coming to terms with it. "T he Principles of Mathematics (1903) first made it perfectly clear that a new force had entered British philosophy. The rigorous philosophical examination of logico-mathematical ideas was a genuine novelty, and there was an atmosphere of intellectual adventure about the whole book which stamped it as an achievement of the first order. ... Russell there tries to formulate the logical principles and methods which, so he thinks, must be involved in any construction of mathematics. No work since Aristotle's time has had so striking an effect upon the logic ordinarily taught at universities" (Passmore, A Hundred Years of Philosophy, p. 218).

The "atmosphere of intellectual adventure" of which Passmore rightly spoke identifies one quality of the book. Also, for

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sheer quantity and novelty of argument, for historical penetration and for the frequency of its many spasms of ingenuity, it is unmatched by any of Russell's other books. It begins with a prose version of the philosophical foundations (metaphysical as Russell thought, being influenced by Moore) of mathematics; develops in passing a set of new logical theories about denoting, relations, and implication; proceeds to examine the nature of a multitude of mathematical entities and expressions; comments at great critical length on the work of his predecessors, especially Dedekind and Cantor; moves on with a flurry of arguments to space and time, to matter and motion; devotes entire chapters to Newton, Kant, Euclid; closes with an appendix on Frege and an early version of the theory of types. The book awed and worried Bradley, at whose philosophy it was partly directed ("I am incompetent utterly to sit in judgment on Mr Russell's great work," he wrote in Mind of April 1910), intimidated Moore (who wrote a long and unpublished review), entranced Josiah Royce at Harvard (who also wrote a long and unpublished review of it), stirred C. D. Broad to become a philosopher, prompted Whitehead to join in writing the projected second volume, which became Principia M athematica, and created the subject matter from which Wittgenstein drew and withdrew in his Tractatus. Yet of Russell's major books it is today perhaps the least appreciated. Why? Partly because the book is so vast and diverse; partly because its logical parts are largely superseded by Principia Mathematica (1910-13); partly because Russell explicitly abandoned much of its underlying metaphysics; partly because its exposition is often careless and confused, Russell thinking out loud;* partly because philosophers influenced by

* In 1968 the printer's manuscript of Principles, in Russell's longhand, was deposited in the Library of McMaster University, Hamilton, Ontario, Canada. It is not difficult to track down there some reasons for a number of anomalies in the published text. Quite obviously, pages from an earlier draft were spliced into the final manuscript, sometimes in a rather hurried way. It is readily apparent, too, that most of these earlier pages express an outlook even more resolutely realistic, in the sense described earlier. Several independent discussions and some footnoted references to Frege were also added at the proof stage. Last of all was added the appendix on Frege with its scattered retractions. Russell worked on the manuscript for almost three years, and it is apparent that the development of his own thought, the invasion of Peano and Frege, and perhaps the fact that he never had a completed typescript to give him perspective on his own work, marred the unity of the result.

INTRODUCTION

51

Wittgenstein and the later Russell, and whose knowledge of history of recent philosophy is spotty, have been content to view it mainly negatively, as a sourcebook of convenient targets; and partly because the philosophy of science, to which several hundred of its pages are devoted, scarcely had an academic audience until decades later when the book was dated or even superseded by Russell's own Analysis of Matter (1927). However that may be, Principles, lingering strands of idealism and all, is the backdrop for the entire development of Russell's philosophy and is therefore indispensable as an early stopping point in our exposition. There is no point whatever in considering Principles as a whole; so I shall simply use it in successive stages, considering first its underlying metaphysics and logic. Ensuing chapters will excavate, for their own purposes, other ideas from this imposing and confusing book. There may be compensation in the reflection that to have one's book thus butchered for ends entirely different from those for which it was written is homage paid only to classics. In any case, liberated from present responsibility to any topic but the one at hand, which is metaphysics, we can plunge in and plunder the book at will.

CHAPTER 2

THE EARLY METAPHYSICS

The world of being is unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of metaphysical systems, and all who love perfection more than life. The world of existence is fleeting, vague, without sharp boundaries, without any clear plan or arrangement, but it contains all thoughts and feelings, all the data of sense, and all physical objects, everything that can do either good or harm, everything that makes any difference to the value of life and the world. According to our temperaments, we shall prefer the contemplation of the one or of the other. The one we do not prefer, will probably seem to us a pale shadow of the one we prefer, and hardly worthy to be regarded as in any sense real. But the truth is that both have the same claim on our impartial attention, both are real, and both are important to the metaphysician. Russell, The Problems of Philosophy (1912)

CHAPTER 2

THE EARLY METAPHYSICS

This chapter explores the main lines of Russell's metaphysical ideas throughout his realist period, that is, until about 1912. His arguments against idealist philosophers and his attempts to provide a reasoned and objective basis for logic and the philosophy of mathematics are his central concerns; what he called "the doctrine of external relations" is at this stage his most pervasive idea. However, from the perspective of more than half a century what is of equal philosophical importance are these things: the perennial interest of the realistic hypotheses he adopted; the topics connected with meaning, reference, and analysis he opened up; the general shift of philosophical interest and technique toward formal logic that he helped to bring about. Part I of this chapter is concerned with properties, relations, universals, and necessity. Part II is concerned with particulars, propositions, truth, and analysis. The chapter begins and ends with the idea of being - or 'subsistence' as he often called it: metaphysical considerations bring it in and logical considerations begin to usher it out, a metaphor for Russell's development. PART I A.

BEING AND RELATIONS

1 Russell is amongst the most quotable of philosophers. Passages such as the following have enticed laymen and bemused philosophers even more than Russell intended: "Being is that which belongs to every conceivable term, to every possible object of thought-in short to everything that can possibly occur in any proposition, true or false, and to all such propositions themselves .... 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 53

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them. Thus being is a general attribute of everything, and to mention anything is to show that it is. "Existence, on the contrary, is the prerogative of some only amongst beings. To exist is to have a specific relation to existence -a relation, by the way, which existence itself does not have .... This distinction is essential, if we are ever to deny the existence of anything. For what does not exist must be something, or it would be meaningless to deny its existence; and hence we need the concept of being, as that which belongs even to the nonexistent" (Principles, pp. 449-50). The conviction that reality was thus zoned formed a vital part of Russell's thought for almost two decades. Though he came naturally to this idea and abandoned it in the atomist years with reluctance and without direct refutation, he was aided and abetted a good deal by the study of Meinong, whose metaphysical work he had deeply admired, and by Moore, whose conclusions he was perhaps too reluctant to dispute. The larger region of being mentioned above long housed the relations, numbers, universals and possible objects of thought that were Russell's continuous concern throughout his realistic phase. In fact, Seinsvergessenheit was one of Russell's themes before it was Heidegger's. In 1903 he wrote: "misled by neglect of being, people have supposed that what does not exist is nothing" (Principles, pp. 450-1). But of course only the term and not the intent is the same. Let us adopt the terminology that Russell often (not always) used whereby Being is the largest category and it consists of existents and subsistents, the latter consisting of beings which do not exist.* Russell's basic intuition, which was not his alone, that the realm of objects of thought has to be subdivided, is, on the

* This will do for Russell, but not for Meinong, whose Aussersein pertains to things which neither exist nor subsist. Russell's term 'subsistence,' though not wholly new to the philosophical lexicon, was explicitly derived by him from 'bestehen' in Meinong ("Meinong's Theory of Complexes and Assumptions," Mind, 1904, p. 531 - hereafter cited as "Meinong's Theory"), who thus designated objects, such as universals, "of a higher order." Perhaps 'supersistence' is the idea intended, rather than subsistence, but Russell would have found this language a bit too elevated. He settled for the faint paradox, not in Meinong, that objects of a higher order subsist, and covered his tracks by using 'subsistence' in a much more elastic sense than Meinong's 'bestehen.'

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face of it, a plausible one. It is easy to feel a need for some general format for all those terms with which we wish to have intellectual commerce, even those which don't exist. We can as easily make a list of and talk about things which don't exist as of things which do; and no one but the most sophisticated philosopher has trouble understanding such lists. All the words on both lists can be meaningful; we can discuss which list 'shadows,' 'tickles,' and 'beauty' should go on; and we know something of what is evolved in putting 'numerals' on the list of existents and 'numbers' on the list of subsistents. There will be arguments about many candidates, sometimes culminating in agreement, and there will be innumerable candidates assignable to the lists with no argument at all. Perhaps the arguable cases will start exhibiting common features not possessed by the easy ones. Still, sooner or later problems about the distinction of existent from subsistent beings have to be faced: does 'a black cbquixw' belong on either list?; exactly how and where is the zoning line to be drawn and what are its criteria? Russell was profoundly puzzled by this last question. Only he did not say he was puzzled. Sometimes, he just drew the line in different ways. For example, in the passage (1903) which begins this chapter, being is a generic category which indudes existence, but in the passage (1912) which serves as preface to this chapter being excludes existence. It will not do to say that this is the difference between Being and being. It is important to recognize that what Russell is working with is not merely a plausible hunch that some distinction of existence from subsistence is needed, plus a nagging problem about how to make the distinction. There is a droplet of argument in each of the paragraphs quoted above, and by applying to them a little logical energy it is easy to generate a metaphysical cloud. The first point above is that we need not only existent but also subsistent beings to accommodate the things which we can make propositions about; and his second point, a specific application of this, is that we need to appeal implicitly to subsistence in order to attribute or deny existence to anything. Consider denial: we want to say, The cities of gold which the Conquistadors sought did not exist. But what did not exist? What are we talking about? The cities of gold: these cities were sought, they are being mentioned, referred to, thought about. Russell rightly recognizes that seeking, talking about, or denying the existence of these cities

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differs profoundly from seeking, talking about, or denying the existence of just nothing at all. Hence, in the reality scale these cities of gold are more than nothing but less than exi9tent. They are subsistent. This line of thought, which was also shared by Moore, is very venerable and very seductive. Anything whatever that can be mentioned has a kind of shadow reality, subsistence at least: it is a different and further question whether it also has existence. Russell saw that difficulties about how to draw the distinction between existence and subsistence did not themselves dispose of the idea that some such distinction was needed. Three preferred ways of drawing the distinction are present in what he wrote. The first is to say that subsistence is appealed to when existence is attributed to anything: 'A exists' entails that A's existence subsists. The second is to say that subsistence is whatever is appealed to in the denial of existence: the golden cities (subsistent) do not exist. (This line of thought will be taken up in detail, later in this chapter, in the discussion of truth and propositions.) A third way - different from· the first two, though Russell did not attend to the difference, partly because he just took it over from Mooreis to consign everything with spatial or temporal relations to existence, and all the remainder of the things that can be mentioned to subsistence. "We tend to ascribe existence to whatever is intimately related to particular parts of space and time; but ... except space and time themselves, only those objects exist which have to particular parts of space and time the special relation of occupying them. On a question of this kind argument seems scarcely possible" ("Meinong's Theory," p. 211). The coincidence of these three criteria makes the entire situation unruly, because it threatens to generate three or more kinds of subsistent beings, different and yet overlapping. Here new problems fester. Russell, unlike Meinong, has no patience for the needed distinctions: for example, he does not even distinguish being and subsistence. Even if all subsistents were easy to talk about (which they are not) only some of them (e.g. numbers, classes, relations) are easy to argue about. Suppose I say that the fat elf feeding at the trough has six wings and you say it has four; are there facts to settle the matter? If occupying space and time is the criterion of existence, do an elf's wings occupy space and take time? For that matter, do images and shadows

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occupy space, or are they only spatially located? We had better not ask about the shadow of an elf. Some subsistents might seem more amenable since coerced by facts: numbers for example. But how shall we then distinguish familiar numbers (the even prime) from the impossible ones (the primes divisible by six), or from irrational numbers, or from the number of the elf's wings? And, if classes belong to being, what of the class of female husbands or elf husbands, and what is the difference between them? Meinong, who encouraged Russell's intuitions, boldly re-zoned being, by allowing Aussersein to accommodate new beings, even impossible ones. This can appear to be helpful- for a time. For Russell's idea that subsistence is needed to deny existence raises the question, How might you deny subsistence? First, Russell courageously believed that you could not do it. "To mention anything is to show that it is," he had said (Principles, p. 449). Indeed he held that "in some sense nothing is something" (p. 73), anticipating Heidegger again. Russell is actually echoing Bradley's dictum: "every denial must have a positive ground." If this does not satisfy we may consider the other option: perhaps we can say that there is no subsistent golden city by appeal to a sub-subsistent city which does not subsist. Similar doubt falls on the more innocent looking parallel circumstance that Russell had wanted to accept: when 'A exists' is true then A's existence (or the existence of A) subsists. But if it is the subsistence of A's existence that makes 'A exists' true, one wonders what might have made it false - the non-subsistence, or perhaps the subsubsistence, of A's existence? Either way, one's intuitions are bound to rebel at the entire idea that you cannot affirm something without affirming something else, and that you cannot deny something, even an obscure something, without affirming it as a still more obscure something. It is too obvious that Russell would be making the theory of subsistence successful, and by making it successful make it incredible. These exasperating complications were always for Russell only annoying side issues. Meinong had faced them head-on, but Russell did not take himself to be primarily fighting metaphysical battles: he had wanted merely to take over some ideas for his own purposes. But once he had pulled being inside his own camp he had a Trojan horse on his hands, for being disgorged

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Meinongian satraps and sub-subsistents so smoothly that Russell feared the gates would be opened again to subjectivism. In 1904 he battled the issue to a standstill on the question of false propositions (see end of this chapter), and in 1905 he started to clear the ground for a drastically new approach to the whole issue. One thing he wanted, at a minimum, was a way of denying both existence and subsistence to something at one stroke. He saw that the appeal to subsistence to deny existence was made compelling only in virtue of the misleading subject-predicate structure of the way the problem was posed - as if every grammatical subject had to stand for some being and every grammatical predicate had to assign or deny some feature to it. In this way existence too is a kind of property which, say, the elves and the golden cities do not happen to have. But if there is another analysis of 'The golden cities do not exist' which avoids these suspicious suggestions of grammar, that argument for subsistence disappears. This new analysis was in fact a new theory about descriptive phrases and by implication about existence, and it changed Russell's entire outlook after 1905. (Discussed in detail in Chapter 5). Essentially what this new theory said was that a denial of existence could be effected merely by holding of a given description that it had no application. Since it was to be a theory not about non-existent things but about non-applying descriptions, no appeal to subsistents comes in. But this new theory of existence did not immediately nullify the entire impulse that led to distinguishing being and existence, in particular it had nothing directly to do with the idea of confining existents to spatial or temporal entities, and assigning to subsistence those other genuine entities whose reality one wanted to acknowledge not, now, because the affirmation or denial of existence required them but because they just seemed to be real, though outside space and time. Numbers, relations, universals, minds, classes - surely these are real and yet of a different order of reality from the things to be seen and felt in the empirical world. These things Russell continued to call beings, not subsistents, and he continued to distinguish them from existents for almost a decade after he had discarded those subsistents that the denial of existence had seemed to require. Thus in 1912, in the gallant words quoted at the head of this chapter, he plumped for the full fledged reality of both being and existence.

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But this did not last either. The new theory of existence which first destroyed subsistents was next applied to the remaining beings. Since this takes us beyond our present story we can simply survey the situation from afar. The original dignity (as distinguished from argument) for the category of non-existent being had derived very largely from the dignity of numbers, its most distinguished kind of guest. And numbers, disguised as classes by new theories of logic, had already by 1.910 led the parade of beings across the gulf into the redrawn boundaries of existence: classes, he concluded, were comprised by their members, and their members were existents. Here was a model for other beings. By the 1920s Russell had emptied being of all its hostages, naturalizing some as existents, reconstructing surrogates out of existent materials for others (e.g. minds), exterminating still others, and exposing some (e.g. logical constants) as misleading shadows cast across the gulf by passing words. The details of these campaigns largely coincide with the development of Russell's philosophy for two decades. This hasty exposition of one feature of Russell's outlook during the realist phase and beyond gives us a context for considering more exactly that item which, even more than numbers, encouraged the cultivation of being, and outlived it. This is the matter of relations - an absolutely central idea for the whole of Russell's philosophy. Relations, we shall see, are never more crucial than when making their often implicit transition from being to existence.

2 Russell called all beings and existents, all objects of thought, 'terms'; all could be subjects of relations, all could "occur in any true or false proposition" : "A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned is sure to be a term" (Principles, p. 43). One can easily admire the abstract simplicity of this conception, but it quickly takes on a portentous character. "Every term, to begin with, is a logical subject: it is, for example, the subject of the proposition that itself is one. Again every term is immutable and indestructible. What a term is, it is, and no change can be conceived in it which would not destroy its identity and make it another term" (p. 44). "The notion that a term can be modified arises from neglect to observe the eternal self-identity of all terms and all logical

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concepts, which alone form the constituents of propositions" (p. 448, my italics). We shall have to try to improve the propaganda-to-argument ratio in these passages: for it seems that if terms embrace everything and terms are all immutable, then nothing changes, nothing is ever modified. Something had better be wrong with this. What Russell means is that terms, especially existents, take on and abandon relations to other terms. "What is called modification consists merely in having at one time, but not at another, some specific relation to some other specific term; but the term which sometimes has and sometimes has not the relation in question must be unchanged, otherwise it would not be that term which had ceased to have the relation" (Principles, pp. 448-9). This saves appearances, or appears to. Consider: the grass (an existent) which was fresh and green and now is brown has dropped its former relations to freshness and greenness (beings) and acquired a new relation (a being) to brownness (a being). What of the grass itself? Russell does not directly raise this question, nor any recognizable version of it. But we can see what he is implying: he is implying not merely that it remains the same grass (having taken on new relations); he clearly seems to be implying that the grass is unchanged and arguing that this must be true. The difference, which is very vast and crucial, between something's being the same thing (though changed) and something's being unchanged simpliciter, is not allowed for by Russell. Actually, he has another argument in addition to the one quoted above, and here too his reasoning, like Parmenides', is entirely a priori: "To be modified by the relation could only be to have some other predicate, and hence we should be led into an endless regress. In short, no relation ever modifies either of its terms" (p. 448, my italics.) This last sentence is a succinct summary of the "doctrine of external relations"; it is the first and polemical reason why relations were so important to him. But neither of the italicized 'arguments' for immutable terms in the preceding paragraphs is convincing as it stands. On the surface, at least, the first is circular and the second is question begging. This can be explained. Russell is not so much arguing against a metaphysics that allows for change as presenting his view of terms and propositions, and simply taking the con-

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sequences: change is unintelligible, unless it is understood in a wholly relational sense. Terms embrace literally everything whatever, and they are the changeless constituents of propositions: we shall keep returning to this idea, for it has many attractive features. It is important to spell it out graphically because the doctrine of relations that we are to examine is one initially formulated, as it were, largely in indifference to the empirical world. In fact, without saying so, Russell was thinking of terms in propositions after the manner of numbers in equations. We can be more precise: insofar as Russell was looking at the world at all, he was looking at it from the point of view of the terms of his propositions. What he saw may be expressed in a way, though it was not his way, that makes a good deal of initial sense: substances do not themselves change, only their states do, by taking on and abandoning relations with other things that do not themselves change. (Such a view, of permanent substances and changing states, was employed by Russell in his exposition of Leibniz's philosophy, though he had disapproved of this conception of substance.) Now to see what Russell is saying we merely identify these substances with the terms of his propositions. Even in this abstract sketch a very important point comes into focus: philosophically speaking the structure of the world is given by the structure of propositions, for these are one and the same. This is the Leibnizian idea that Wittgenstein picked up from Russell and embedded in his Tractatus: there, 'objects' (Russell's 'terms') formed the immutable and external "substance of the world"; their configuration changed, but they did not. It is because this crucial idea, obscure as it is, is so central to Russell that the notion of relation, as the source of the unity of a proposition, is vital- and this is the second and positive reason why relations are important. Though Russell seems to be comInitted to the inference 'things do not change, therefore there is no change' he is saved implicitly by a relational understanding of what might have been called changing states or configurations. 3 The amount of attention which Russell, and Moore with him, gave to relations in the early days of analytical philosophy is less appreciated now than it was formerly. One reason for this is that it is widely presumed that, on the fundamental issues at least, they simply won the battle against Bradley and the idealists.

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But for Russell it was many battles at once. Summing up his first twenty-five years of thought on the subject, he wrote in "Logical Atomism", "The question of relations is one of the most important that arise in philosophy, as most other issues turn on it: monism and pluralism; the question of whether anything is wholly true except the whole of truth, or wholly real except the whole of reality; idealism and realism, in some of their forms; perhaps the very existence of philosophy as a subject distinct from science and possessing a method of its own" (L&K, p. 333). This, especially the last remark, is rather expansive; but we can move in slowly. Russell's view was in the first instance a negative response to a doctrine that he credited Bradley with having distilled from Hegel. "I called this 'the doctrine of internal relations,' and 1 called my view 'the doctrine of external relations.' The doctrine of internal relations held that every relation between two terms expresses, primarily, intrinsic properties of the two terms and, in ultimate analysis, a property of the whole which the two compose" (My Philosophical Development, p. 54, hereafter MP D). Let us enlarge this ascription just a bit, in order to get a feel for Bradley's picture of the world - at least as Russell seems to have remembered it; for there is no evidence that he really studied Bradley after he came to reject him. (Relevant passages from Bradley are: Chapters 3, 13, and Appendix of Appearance and Reality.) We have to move smoothly from platitude to paradox in somewhat the following fashion. Everything is apparently somehow related to everything else, and thus the so-called relational facts - such as that A is to the left of B, that B is heavier than C, that B is known to D - are really facts about A, B, C, and D; facts that report elements of their respective natures. Each thing is thus involved in the nature of everything else. Hence it is not really relational facts, nor even independent things over against each other, which is the ultimate truth, but only the interpenetration of everything, making in effect one comprehensive unity. For the same reason, even an idea, conscious awareness of anything, as an ostensible relation, submits to this interrelatedness, being therefore involved in the unity; conversely, the fact of the unity's being presented to consciousness is also a fact about consciousness. Therefore it must be, and so is, this all-comprehensive conscious unity which is the ultimate destiny

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of any consistent attempt to think out the full implications of any single fact. The deeper Reality to which Appearance (things, relations, ideas) points thus absorbs relatedness and individuality. . . . This is a condensed picture of the path to the Absolute. Bradley knew only too well that propaganda on these themes could confound even where it could not convince. This picture does not encourage an ultimate distinction between the properties of a thing, which are its very own (e.g. this pencil's colour), and relations, which it sustains to others (e.g. the pencil's being on the desk). Bradley was inclined to think of relations as properties, and Russell was inclined to think of properties by means of relations ("the so-called predicates of a term are mostly derived from relations to other terms" -Principles, p. 471). Now there is a vast difference between discoursing about the features of a thing on a general level in a way that does not distinguish properties from relations, and having a theory that one can be expressed in terms of the other. We must consider the respective theories, but it will be helpful to get our bearings by sticking first to the general level, speaking simply of 'features,' without prejudice to whether they are relations or properties; the question is only whether they are internal, and what it amounts to to say that they are or are not. This is the level from which most philosophical arguments on this topic take their departure. It is as if, to use a very crude physical model, we are to think of external features as attached to their bearers with hooks or glue, you can detach them without damage; internal or necessary features are dovetailed and screwed, you cannot tear them out without wrecking the bearer. The notion that some features at least are not internal to their terms makes an undeniable appeal to common sense. We are naturally disposed to think that some features of a thing belong less essentially to it than some others. The colour of this black pen and its lying on my desk seems relatively indifferent; the same pen could be painted blue tomorrow, and be in my hand. Its being a writing instrument does not seem indifferent; that belongs to its very nature and is an internal feature. Similarly, De Gaulle's being a man, being French, and being rational, seem internal features of him; his being over six feet tall, once President of France, and a father seem external. Yellow's being lighter than red seems internal to yellow; but its being present in this

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pencil seems external. Apparently, common sense and metaphysical realism generally are on the side of saying that most of the things that occupy the world of space and time and perhaps some subsistent beings as well will be found to have both accidental or external features and essential, internal, or necessary features. The internalists (e.g. Bradley, on some interpretations) say all features are internal; the externalists (e.g. Russell, in his inclinations) say none are; sober realism (e.g. common sense) says some are and some are not. The internalist can challenge his opponents to present criteria in terms of which features are to be more than haphazardly identified as internal or external. The challenge is extremely hard to meet satisfactorily.* On the other hand, internalists can also be challenged to state just exactly what internal difference it is supposed to make to the nature of, say (almost any example will do), this pen, that it yesterday had the feature of being-in-my-pocket, and now has the feature of being-in-myhand. This challenge is also most difficult to meet without resorting to sheer repetition.t What is bedazzling, and philosophical, about the topic on this general plane is that it can easily seem impossible to get a debate going - and then promptly seem impossible to get it stopped. The idealist affirmation, as usually interpreted in the idiom of 'internal relations,' is an exceedingly sweeping view. In insisting that all of a thing's features are 'essential' or 'internal,' or 'belong to the nature of the terms,' or 'express an intrinsic property of

* A general indication of several lines of search for the required criterion is this: some resort to semantics (internal features correspond to properties picked out by the predicates of analytic, or by necessary, or by a priori, propositions); some resort to intuition (there can be no general criteria, but we can often tell in particular cases what is what); some resort to pro pertyism (the criterion can be made, not in terms of the natures of things, but in terms of two kinds of properties: those that are always internal to whatever bears them and those that never are). Failing on these lines, some realists resort to complete externalism (all features are more or less external; it is merely a matter of degree). t Attempts can be made: some internalists resort to physics (the gravitational field is different, thus . . . ); some resort to mysticism (nothing is really completely independent of anything, so, . . . ); some resort to hierarchies of natural kinds ('man,' 'animal,' 'organism,' are internally related going upward, not down). Failing on these lines, some resort to complete internalism (all relational properties are more or less internal; it is significantly a matter of degree). See section 9-12 below for my improvement on this debate.

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the term' they have invited equally sweeping retorts. But it would be entirely inaccurate to leave the impression that the issue, some form of which has become, largely as a result of Russell's work, a permanent aspect of analytic philosophy, involves merely an exchange of rhetoric whose outcome affects nothing - though too easily and too often it can involve that. 4 Internalists keep comfortable company with idealists; externalists keep company with atomists. When he entered the debate Russell was in motion from the first to the second. Earlier we mentioned and then ignored a distinction among features between properties (which belong to a thing) and relations (which connect it with others), a distinction that puts us in the path of Russell's views. What is a relation? Russell does not answer this question directly, nor feel a need to. He casually accepted the suggestions of the language: for graphic examples he looked to prepositions - 'between,' 'north of,' 'behind' - each of which indicated a relation; for profounder examples he looked to verbs, especially transitive verbs. 'John teases Mary behind the water cooler': here are indicated three terms and two relations; this Russell takes to be self-evident. Sometimes he said it was obvious that all verbs indicated relations. So we can let the answer to our blunt question grow in stages, as Russell did. His first and fundamental thought is merely a vivid concern for the reality and ultimacy of relations: they had to be authentic bits, as he said, of "the furniture of the universe," not disguisable as anything else and particularly not as properties. This aim, so simple and direct, had a specific occasion and a specific raison d'etre. The raison d'etre was that the metaphysical reality and autonomy of relations was, or seemed to be, presupposed by the use which Russell made of relational ideas in logic and in his philosophy of mathematics. The occasion was the denial by Bradley and other idealists of the autonomy of relations, in particular (but not only), the well-known argument that the very notion of a relation involved an infinite regress of further relations to relate the relation to its terms, to relate the relating relations to the terms, etc. One reply to this argument of Bradley's (deriving from Cook Wilson), namely, that there is no such need for a tie between the relation and its terms if we admit that it is always a particular instance of a relation (not the universal relation as such) which c

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relates terms, was unavailable to Russell, because he believed that relations had no instances. "Relations do not have instances, but are strictly the same in all propositions in which they occur" (Principles, p. 52). So the first point (to be discussed further later) is that relations are universals without particulars. Do they exist, or merely subsist? For many years Russell did not say. Perhaps this was because subsistence seemed too little honorific for them; perhaps it was because some relations, spatial and temporal relations, seemed to be bound up with his ideas about 'existence' as distinguished from 'being.' But when we consider examples this becomes very obscure: behind is spatial, but not in the way a table is; earlier than is temporal though not as an event is. Is it that some relations, those of space and time, confer existence on other things but lack it themselves? Russell did not discuss any of these questions. In The Problems of Philosophy (1912) he just said that relations had being, not existence. What mattered, first to last, was their reality, their autonomy, their irreducibility; and these features showed up most dramatically in the fact that they had certain logical properties. The general angle of Russell's vision here, as distinct from any argument, may be indicated as follows. Consider: 'If plums are larger than grapes and apples are larger than plums, then apples are larger than grapes.' No one doubts it; yet how could we be content with such arguments if 'larger than' did not indicate a real relation, something for the argument to rest upon? But there would be no such thing as the relation larger than if being-Iargerthan-grapes is merely one more feature of apples, like being sweet and being light. If there is no such affair as larger than, if this thing has no being at all, the argument seems to lack a needed support, for it is not the nature of apples and grapes just as such that makes the argument go. What does make the argument go? Very roughly, 'logical form' was Russell's later answer; the being of a relation was his present answer. Again, here is an arithmetical law '(a+b)+c = a+(b+c).' But how can this be accepted if it is not acknowledged that '+' indicates something, namely the manner in which a, b, c, are related? But conjunction has little chance of being a relation if we take '+ b' to be just one more feature of 'a' - whatever that could possibly mean. So of course conjunction is a genuine relation in its own right. If taken as argument, these remarks are like Swiss cheese. But

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they are not yet arguments; they are only logical and mathematical slants on a Russellian world "rich and varied." They are attractive remarks in a world where a realm of being is taken seriously. They begin to look like arguments only when conjoined with Russell's theory that relations have no particular instances and with the fact that such relations as larger than and conjunction can themselves be organized in terms of certain laws which hold true of them, and not of any non-relations. Larger than for example was just shown (though not proved) to be transitive and asymmetrical; and the arithmetical law just given in effect states that conjunction is associative. When such laws are themselves made the target of proofs it is not easy to deny that relations are real, that they have some autonomous status, perhaps the status of being. But neither must it be thought, as Russell later acknowledged he too easily did, that this alone gives a clear meaning to admitting their reality. * So this kind of approach, encouraged by realistic metaphysics, to the defence of relations leaves us uneasy. We shall be better positioned to understand the spirit of Russell's inquiry, and the kind of achievement a genuine knockdown argument against Bradley would be, if we intensify the uneasiness by taking a specific example of a putative relation and noticing how Russell gave expression to his worries and inclinations. Consider and: "What is meant by A and B? Does this mean anything more than the juxtaposition of A with B? That is, does it contain any element over and above that of A and that of B? Is and a separate concept, which occurs besides A, B? To either answer there are objections. In the first place, and, we might suppose, cannot be a new concept, for if it were, it would have to be some kind of relation between A and B; A and B would then be a

* More than fifty years later Russell still toyed with this question: "For my part, I think it as certain as anything can be that there are relational facts such as 'A is earlier than B.' But does it follow that there is an object of which the name is 'earlier'?" Clearly, he had once thought so, but now doubts have crept in. "But if we try to descry some entity denoted by these relation-words and capable of some shadowy kind of subsistence outside the complex in which it is embodied, it is not at all clear that we can succeed." (MP D, pp. 172-3). Throughout his realist phase he had thought he could succeed. The shift of thought, as Russell saw it, is that from the primacy of universals to the primacy of particulars.

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proposition, or at least a propositional concept, and would be one, not two. Moreover, if there are two concepts, there are two, and no third mediating concept seems necessary to make them two. Thus and would seem meaningless. But it is difficult to maintain this theory. To begin with, it seems rash to hold that any word is meaningless. When we use the word and, we do not seem to be uttering mere idle breath, but some idea seems to correspond to the word. . . . Thus it seems best to regard and as expressing a definite unique kind of combination, not a relation, and not combining A and B into a whole, which would be one" (Principles, p. 71). Russell's question is: Is there an and in the zone of being? The answer, reluctant and dubious, is No. (This passage, as the manuscript shows, is from the earliest drafts of Principles and is in some respects inconsonant with the ensuing paragraph in the published text. See footnote above, p. 50.) If and were a genuine relation it would either collect and collapse its terms or it would add an extra item to them. Still, as a quasi-relation it is not to be reduced to a property. So he settles for the idea that 'and' expresses something "definite" and "unique." Not a very happy solution. Russell does not consider what sort of metaphysics you might get if you worked not in English but in Latin: ' ... et A, et B.' We shall not learn very much from or about Russell if we succumb to the temptation to lazily sweep this queer worry and its messy solution under the nearest rug. It has to be seen that when Russell talks about the reality of external relations this is as much a general orientation for metaphysics as a specific tenet for logic. And there is no a priori reason why this orientation does not at a given point supply relevant questions and useful answers. But if, nevertheless, Russell has got us into fairyland, the lesson to be extracted for the future is that he is resolutely trying to probe every conceivable area of inquiry with his theory of relations, just as Bradley had done with his theory. This is the most important critical point that this chapter has to make. It is a first and dramatic instance of Russell's lifelong penchant to find, if possible, a logical and philosophical side for every dogma. As for just turning away in dismay from this stir about and, there is to be found encouragement from Russell himself. In

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1937, in the Introduction to the second edition of Principles, he wrote: "Not even the most ardent Platonist would suppose that the perfect 'or' is laid up in heaven, and that the 'ors' here on earth are imperfect copies of the celestial archetype." His own ardour had cooled to the point where he no longer even recalled this ancient speculation about 'and.' With the advent of logical atomism his theory of relations had been drastically reined in. The interesting question is what effect it had, early and late, upon his general philosophical outlook. Weare not merely pursuing historical curiosities, though we are also doing that. There is something of a different order in the quoted passage that requires to be noted. It is the deep and uncritical conviction - to which an exception is painfully allowed - that every word or phrase requires, as a condition of its being meaningful, its corresponding object. The idea adumbrated for 'and,' namely, of admitting that some words contribute to the meaning of their sentences not in isolation but only in context, was destined to become the principle of his theory of descriptions, one of the most fertile ideas in all of Russell's writings. He put it succinctly by saying that his old problem "arose from the belief that, if a word means something, there must be some thing that it means" (MPD, p. 63) - a belief altogether too convenient for his doctrine of relations. The opposed idea, toward which Russell is looking, that one should inquire about the meaning of a word only in the context of a proposition, is, on its face, not at all startling. Everything depends upon what one does with it in his semantic theory. In fact Frege had long since laid down just this idea as one of the three fundamental principles of his great work, Die Grundlagen der Arithmetik (1884). But Russell had not yet read Frege when he wrote this passage, and he had forgotten that the same point had been made, less systematically than Frege, by Mill in The System of Logic. He had not yet begun to think of semantics as anything other than a reflection of ontology. B.

PRO PERTIE SAND

EXTE RN AL

RE LATION S

5 It is time to see that Russell has what the foregoing pages stand in need of: a knockdown argument to show that some relations cannot be reduced to properties; that relations, in his

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words, are therefore 'real.' Consider the relation 'A is greater than B' which, Russell argues, cannot be reduced simply to the conjunction of A possessing a property 'being greater than B' and B possessing a property 'being less than A.' Of his several reasons, two may be cited. (1) 'Being greater than B' is the putative description of a property which essentially contains reference to B for its identification. That is, there is no way to pick out that property without a direct, or possibly indirect, reference to B. What the property is, therefore, depends as much upon B as upon its bearer, which is A. Yet it is a property of A, intelligible only in reference to B, "and this is merely a cumbrous way of describing a relation," says Russell. What he means is that even if we say the original relation has been subverted by being transferred into a relational property we cannot assign this property to anything, for example A (which we must do to capture the original meaning), without thereby reintroducing the relation between A and B. The same considerations hold if we start with 'being less than A.' (2) Another argument is this. Our original relation was asymmetrical (that is: if A is greater than B then it is not the case that B is greater than A - a matter of logic). How shall this be registered in the presumed reduction to properties? We shall have to see to it that the properties fashioned, namely, 'being less than A' (for B), and 'being greater than B' (for A), show the original asymmetry, and this will have to be done by giving them each an additional further property but not in terms of a relation, since relations were to be avoided. For example, B (which by hypothesis has the property 'being less than A') must have the property designed to exclude the possibility of also having the property 'being greater than A.' But this property can be defined only with reference to A, introducing the impasse described above, or alternatively, by introducing more properties to specify which asymmetry it is supposed to accomplish ... and so on, and on, until we tire of tugging at our bootstraps. "This kind of infinite process is undoubtedly objectionable, since its sole object is to explain the meaning of a certain proposition, and yet none of its steps bring it any nearer to that meaning" (Principles, p. 223). The argument can be generalized to other kinds of relations. The heart of the matter for both arguments is not at all recondite, for it is just this: the required properties cannot be identified and assigned without reintroducing relations,

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and relations have their own kind of characteristics (e.g. asymmetry, transitivity) which cannot, short of infinity, be exhibited by any display of relations as properties. Bradley, to be sure, had general reservations, scarcely acknowledged by Russell, about reducing relations to properties, or relational propositions to subject-predicate propositions; but he used these only to cast aspersions on "the relational way of thinking." Russell took the specific example of asymmetrical relations and used it to defend the relational way of thinking. He had detailed and conclusive reasons to show that the idea of order, which is essential for understanding the serial character of the natural numbers, as well as many connected mathematical ideas! could be defined only by appeal to asymmetrical relations. "Broadly speaking, we may say that, if we wished as far as possible to dispense with relational propositions and replace them by such as ascribed predicates to subjects, we could succeed in this so long as we confined ourselves to symmetrical relations . . . but it is formally impossible when the relations are asymmetrical. ... Asymmetrical relations are, we may say, the most characteristically relational of relations, and the most important to the philosopher who wishes to study the ultimate logical nature of relations." (The clearest expositions of this are in Chapters IV and V of I ntroduction to Mathematical Philosophy, 1919 - from which this passage comes, p. 44-5; and Chapter II of Our Knowledge of the External World, 1914). In addition there was something else he was later to make much of. The fallacious reduction of relations went hand in glove with another venerable and false idea: "a great deal of traditional philosophy depends upon the assumption that every proposition really is of the subject-predicate form, and that is certainly not the case. That theory dominates a great part of traditional metaphysics [including part of Russell's own] and the old idea of substance and a good deal of the theory of the Absolute" ("Lectures," L&K, p. 207). Very well, a relation is not a property; not a property of its respective terms; not (as Bradley would have argued in rejoinder) a more sophisticated property of the whole that comprised the terms. Some relations at least, the most characteristic ones in fact, in being irreducible to features of things, are external to things, and are beings in their own right. In 1924 Russell wrote (L&K, p. 335) that this was the whole of what he meant by his

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doctrine of external relations, and that is why it was important to emphasize the defence of relations per se. But this gives a much too narrowed view of the actual debate. For, abstractly considered, some relations could be real, in Russell's sense of being irreducible to properties, and also internal in some important but yet unexplained sense. Russell gives no reason at all for denying in general that a relation ever could 'enter into the nature of its terms' (the idealist way of putting it); and such a reason would not be the same as the reason for holding that a relation is not reducible to properties of its terms. Wittgenstein, for example, who would have agreed with all of Russell's arguments on the reality of relations, nevertheless held the view, not at all implausible, that "the order of the number-series is not governed by an external relation but by an internal relation" (4.1252). It is a crucial point that Russell never discussed, but should have; why not, will be explained in the next section. An example of an internal relation from Moore (Some Main Problems of Philosophy) may illustrate that the omission is of general significance. Consider the relation part of, and also this rectangle which is a part of this larger square. So, 'having this rectangle as a part' gives us an essential or necessary property of this square, whilst 'being a part of this square' gives us a nonessential or contingent property of this rectangle. It appears that in this sense the relation is part of enters into the nature of one of the terms (the square) but not the other. But this does not impair the reality of the relation; indeed on its most natural construction, is part of is asymmetrical, and hence the very paradigm of a relation. Russell gives us no reason why this relation is not (1) both internal in one of the senses he attributed to Bradley's use of this word (viz., that it yields a necessary property of a term) and (2) external in one of the senses he appropriated from that word. The point might be made more emphatic by considering 'equality' - assuming this gives us a relation. Suppose the surface of circle A is equal to the surface of square B. This can be translated without apparent loss into property ascriptions: 'equal in surface to B' is a property of A, and 'equal in surface to A' is a property of B. Does this mean that equality of surface is an internal relation; is it then not a real, autonomous relation? There are other problems here, but one thing is certain: we have not yet made clear just what is going on when Russell describes

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this as a dispute about "the axiom of internal relations" as opposed to "the axiom of external relations." He did not make it clear himself. 6 What has been said so far shows that Russell wins one important round in the debate with Bradley, though his way of drawing the moral is seriously defective. We must therefore try to secure a larger perspective on the entire debate. Bradley's Principles of Logic, Russell's numerous critical passages in The Principles of Mathematics, Joachim's book The Nature of Truth, Russell's essays on it, especially "The Monistic Theory of Truth" (1906), Russell's and Bradley's exchanges, public and private, several works of Stout, McTaggart's powerful Hegelianism at Cambridge, Moore's 1911 lectures published as Some Main Problems of Philosophy, adjacent comment by figures as remote as Josiah Royce in America and Cook Wilson at Oxford - these are the principal materials. Altogether it is one of philosophy's most extraordinarily confusing pieces of dialectic, and on it for a time the fate of realism and idealism seemed to pivot. What needs to be said here about Russell can best be said in three overlapping stages. (a) It is apparent that in propagating the doctrine of external relations Russell was primarily affirming the irreducibility rather than the externality of relations - so this dispute was confined to the connections between properties and relations. Both Russell and Bradley managed to combine, or confuse, this relatively narrow debate with the much larger issue, old as Aristotle, whether all features of a thing (be they properties or relations) were in an important sense internal or necessary to a thing. Now Russell called Bradley's affirmative answer to this question the 'axiom of internal relations.' For himself he denied that any relations were internal, and he also kept slipping his own view of properties into the argument without warning ("the so-called predicates of a term are mostly derived from relations to other terms"-Principles, p. 471). This is an extremely confusing situation, but we must see how the argument has been loaded in order to understand what is going on when Russell simply assumes that the affirmation of the axiom of external relations amounted to the denial of the axiom of internal relations. In a way this was true, but misleading, because Russell's arguments

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for and against these axioms kept getting interference from assumptions about properties. Indeed, throughout these discussions, arguments about reducing properties to relations (Russell) and relations to properties (Bradley) are being mixed with arguments about the internality of properties and/or relations. We can clear the air somewhat by recognizing one extreme form of opposition between Russell and Bradley. Bradley held that all relations were ultimately to be understood if at all not only as properties but as internal properties; Russell held that all properties were to be understood not only as relations but as external relations. Doubtless it was partly a result of the controversy that enlivened those days, especially with one another, that each adopted these extremes. (b) Bradley's position assumed that if those features which appeared to be relations were reducible to properties, then that was as good as saying that they were internal or necessary properties of the thing. What is interesting is that Russell, without so much as a by-your-Ieave, went along with this crucial assumption. In fact, we know that he long had no theory of nonnecessary or contingent properties as such at all. Properties, he said, could be defined as terms in external relation. This is part of what is involved in the idea, already discussed, that Russell, with the metaphysician's sublime indifference to the empirical world, had argued against the very possibility of the terms of propositions being mutable. Accordingly, he was easily led to suppose that the way to prevent relations from melting wholly into their terms was to show that some of them at least were not properties at all. In this latter, Russell was entirely successful, but his motivation was unsteady just because his idea of a property was, inadvertantly, the idea of a necessary property. In the long run, the quick assumption he let slip by - supposing that properties, if allowed, would become necessary - was as important as anything else. (c) It is now clear that the debate is somewhat misnamed as being about internal relations, for it was as much about properties. Furthermore, each had a very tendentious assumption about properties. Russell held that properties could be defined by means of relations; Bradley held that any property was ipso facto a necessary property. That is why Russell could record his disagreement by affirming the 'axiom of external relations' - mean-

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ing thereby that relations are not generally reducible to properties, implying that no properties are necessary, and assuming that properties are reducible to relations. He could attribute to Bradley the 'axiom of internal relations' - meaning that according to Bradley properties (which he, Russell, thought of as relations) were internal, necessary. These details are admittedly tedious. Russell is developing his own ideas and confusing his memories of Leibniz and Bradley, while trying to sharpen his conception with inadequate terms. For example, he defends relations with the notion of series in mathematics, yet his polemics are such that he is blocked off from inquiring whether the relations of the number series are internal or external. Too many arguments are decked out in slogans. At any rate I have now explained just what these slogans amounted to; fortunately, with this out of the way, it will be easier to enter into the relevant part of Russell's positive ideas. 7 Like Russell, I have discussed relations without indicating very exactly what they are. This was no oversight. It might be supposed that the idea of an internal relation that Russell is opposing, a relation which 'enters into' the nature of its terms, 'modifies' its tenns, is 'internal' to its terms, and so on, could be discussed in the following way: we consider examples like 'Fire melts ice' and wonder if the relation melts affects the term ice. Nothing so straightforwardly empirical is on Russell's mind, because nothing so straightforward is on Bradley's mind, and Russell is accepting his problems from Bradley. Despite his saying that all verbs depict relations, the only polemical examples Russell used (though he generalized from them freely) were relations with clean logical properties: symmetry, transitivity, difference, and the like. The two important polemical points he made against idealists, that not all relations were reducible to properties, and that not all properties were necessary (this latter was not his way of putting it) were not themselves nullified by his abstract address to the problem, but we shall see that a good deal of his own positive theory was affected by it. In 1959, a half century after this controversy, Russell wrote that it was plausible to regard some relations as internal, and he gives as an example "X loves Y" (MPD, p. 54). The idea, I suppose, is that being in love transforms one. The point of view

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of this concession is entirely incompatible with his platonic stance during those early contentious years. He had certainly not then thought that one could look and see whether a relation 'affected' the nature of its terms, and if so, concede that it was internal. It had not been that kind of debate. He had thought then only of the logical features of relations, and took them as universals without particulars. One may agree, perhaps in platonic moods, that loves as a universal does not 'affect' the X and Y involved (just as one may say that the number two does not 'affect' a pair of twins); but the fact of being loved may 'affect' Y, and it is this banal fact that he had slighted. But how attending to it would show that loves is an internal relation is something he left for his readers to figure out. Just now we have other things to do. C.

CONTINGENT AND

NE CE S SARY PRO P ERTlE S

8 We need perspective on Russell's own emerging ideas. In particular we can understand his arguments with idealists by looking at them through conceptions he developed a few years later. So we skip ahead momentarily to 1911, which is after his main controversies about relations have subsided. The crucial paper is "On the Relations of Universals and Particulars" (reprinted in Logic and Knowledge). Here we pick up a set of ideas which can be applied retrospectively to the earlier controversies. Russell did not do this, and there is no way of knowing whether he would approve, but we need to articulate and use, now and later, that view which bridged Russell's shift from the early platonic realism to his atomism. This view, though he did not hold it for long, has the advantage of being plausible, perhaps true. One reason why we need it now is quite simple: we have repeatedly noticed that Russell's alienation, metaphysically speaking, from the empirical world of space and time is putting things out of focus. He came only slowly to realize this and never was explicit about it. It is in this 1911 paper, a presidential address to the Aristotelian Society, that he takes up the question of space and time in a new way. He there for the first time clearly defends particulars - but they are not yet the particulars of his atomism. Briefly, they are such particulars as might be indicated by expressions like these: 'The yellow of this pencil,' 'the wisdom of

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Socrates,' 'the shape of his nose,' 'the tone of that bell,' where colours, shapes, wisdom and tones are abstract universals instantiated as particulars in empirical objects and there individuated as particulars by spatial or temporal criteria. I shall refer to this view and the semantic trappings to be extrapolated from it as aristotelian realism, thus distinguishing it from both the platonic realism of the first phase - which is to be criticized by it - and the atomism of Russell's middle phase. It is a convenient accident that it was an Aristotelian Society address, but it is a fact that he in no way attributed the view to Aristotle. Yet this Russellian particular is one which students of Aristotle, especially of the Categories (la20ff.), will recognize as those aristotelian qualities which are not "said of" a subject but are "in" a subject, though "not as a part," and which, according to Aristotle, are "numerically one," "particular," and "cannot exist separately." Since I intend to make a good deal of use, now and later, of Russell's new conception of particulars it may be well to quote his statement. He summarizes what I call 'platonic realism' (his early view) and what I call 'aristotelian realism' (his new view) as follows: "Of the above two views, the first, which does not introduce particulars, dispenses altogether with predication as a fundamental relation: according to this view, when we say 'this thing is white,' the fundamental fact is that whiteness exists here. According to the other view, which admits particulars, what exists here is something of which whiteness is a predicate - not, as for common sense, the thing with many other qualities, but an instance of whiteness, a particular of which whiteness is the only predicate except shape and brightness and whatever else is necessarily connected with whiteness" (L&K, p. 111). This distinction of universals from particulars is one thing we need. A second thing we need, something which Russell also was explicit about for the first time in this essay, is the notion of an intrinsic property, such that it is at least conceivable that two things may have the same intrinsic properties and yet, because they occupy different places, be numerically diverse (p. 118). The third thing we need is something Russell was not explicit about. It is the distinction among intrinsic properties between those which are necessary and those which are contingent. The way to explain this is to take what Russell said about intrinsic properties and develop it in such a way that it can be used as a

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parameter against which to measure the necessary-contingent distinction. The latter distinction is important but suppressed in Russell, though it is very near the surface in 1911. For metaphysical realism generally to take seriously the notions of space and time is to take seriously the notion of change - through time and in space. To do that, in the jargon of this context, is at a minimum to take seriously the notion of a contingent property which can come and go without destroying its bearer, as distinguished from a necessary property. It is also to take seriously something like the notion of sameness or identity as distinguished from similarity in some or other respect. We have, in short, to connect the necessary-contingent distinction with the samenesssimilarity distinction. (For the time being we can forget about relations, speaking only of properties, without prejudice to any of Russell's animus against reducing relations to properties. We need only admit, what is obvious, and what was repeatedly emphasized by Moore but not by Russell, that, to put it crudely, relations entail properties: 'A is greater than B' entails 'A has the property of being-greater-than-B.') The idea of a necessary property has a history of self.~ conscious controversy. The idea of an intrinsic property has a history of blurred assumptions, with no known lineage, and no clarity. For Bradley, and such of his followers as Russell was most concerned with, Joachim and McTaggart especially, both 'intrinsic' and 'necessary' collapsed into 'internal,' and Russell's own use prior to 1911 tended therefore to be indiscriminate. Moore was inclined to be more careful, but he too sometimes let 'intrinsic' overlap in use if not in meaning with 'necessary.' We can and must therefore take liberties on Russell's behalf and, without violence to any clear tradition, simply give to 'intrinsic' the precise but somewhat unusual use which he implied for it in 1911. I now turn to this. 9 The fundamental distinction between 'necessary' and 'intrinsic' shows up most importantly in the way these terms are logically connected, respectively, with 'sameness' and 'similarity.' A necessary property, of which the opposite is a contingent property, may be characterized initially as a property which is essential to a thing's being the thing it is, whose removal is incompatible with its remaining the same thing. To take some

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plausible if not incontrovertible examples: being-greater-thanfive is a necessary property of the number ten; being-darker-thanyellow is a necessary property of black, as being-a-carnivore is of a hyena, as being-spatially-extended is of a coloured object, as being-a-rectangle is of a square.* The important point is the connection between something's being a necessary property and the self-sameness or self-identity of a thing, that is, with something's being or not being the same thing. Part of the utility of the idea derives from the need to speak of the continuity of one and the same thing through time and through change. By contrast, the notion of an intrinsic property (the opposite is an extrinsic property) is linked with the idea of a plurality of things being similar. The connecting idea here is expressible by a formula which Moore usually put as follows: "If A has intrinsic property p then anything exactly similar has property p also." (This formula is not a tautology, nor an analytic sentence, but it may be taken as expressing, by means of an a priori proposition, what it is for something to be an intrinsic property.) For example, this page is white, and anything exactly similar to this page is white also. At the limit, two things may have all the same intrinsic properties, and they would thus

* Two connected forms of recent empiricism deny that we can pick out genuine necessary properties in this way. The first and most sweeping form holds that the attempt derives merely from a misinterpretation of analytic truths, a matter of meaning being read as a matter of fact. A variant and a refinement of this holds that there are necessary properties only in the trivial case in which there is a conceptual tie between the way the property and the property's bearer are referred to. A blatant example, supposedly paradigmatic, is this: 'Being born at sea' is a necessary property of the 'first dog born at sea,' but only because both are described that way, not because of the internal nature of the dog. And so for all other cases of necessary properties: they are functions of modes of reference and description. The official way of propagating these doctrines is to say: all de re modalities are reducible to de dicto modalities. This is attractive, often important, in some cases certainly true. But it is vastly complicated in its detail, not to be accepted or dismissed lightly. But to suppose uncritically that this is the whole truth about necessary properties is to make it impossible to understand Bradley, difficult to understand Russell, and entirely unlikely that the historic uses of the concept of necessity will be appreciated. Nevertheless, Russell himself has provided on the whole as much impetus for this set of views as he has for the views in the text I am assigning to him. The 1911 view was in fact an interlude between the inclination of the realist phase to treat all properties as necessary and the inclination of the atomist phase to treat all properties as contingent. All the more reason to develop it.

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be exactly similar. But they would not thereby be the same thing. Hence sameness of intrinsic properties add up as a limit to exact similarity of things but not to sameness of thing. This shows a way in which the use of the concept of similarity (of thing) presupposes the use of sameness (of property). Intrinsic properties thus considered may include necessary properties, but what is more important, they may also include nonnecessary or contingent properties. Whiteness is a contingent intrinsic property of this page. Being greater than nine is a necessary intrinsic property of the number twelve. The major thing to see is the definitional links between, on the one hand, necessary properties and sameness of thing, and on the other, intrinsic properties and exact similarity. The notion of a necessary property is thus bound to a notion of sameness, and is most naturally illustrated by the adjacent idea of the continuity of one and the same thing through time and through change by loss of contingent properties; the notion of an intrinsic property is bound to a notion of similarity which itself has to do primarily with different things, partially or wholly similar, irrespective of time. Hence the logical liaisons of sameness and exact similarity reach out in entirely different directions. This is shown too by the fact that it is not an exhaustive distinction (some properties, some relational ones, for example, may be neither necessary nor intrinsic but extrinsic); and it is not an exclusive distinction (for some intrinsic properties are necessary, and perhaps with universals they all are); and it is also not merely a distinction between one class and a more inclusive class. Rather, it is a distinction between the logical aspects of different kinds of properties that is oriented toward the different logics of the concepts of sameness and similarity. I want now to urge several things: that these ideas are clearer than Russell's jargon of external and internal relations; that they can be made to cover that ground too; that they are implicit in his conception as of 1911; that they are indispensable for exposing the logical foundations of his realism. So there is nothing here - except emphasis - which Russell need have disagreed with in 1911. It suitably provides for going on to speak of the distinction which then interested him, viz. properties correlated with universals (e.g. Greenness) and with particulars (e.g. the green of this leaf). Moreover there is nothing here offensive to common

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sense. To a surprising extent the family of concepts that gathers about the notion of identity may be comprehended in the terms laid down. Such troublesome terms as 'nature' and 'difference' also fall into place conveniently. 'Different' is precisely ambiguous in most uses with respect to 'not the same thing' / 'not similar'; and the term 'nature' in most of those philosophical uses that have to do with the nature of a thing or concept bestrides our two basic ideas: things may have the same nature without being the same thing (e.g. two black sheets of paper); on the other hand things may also have the same nature and be the same thing, which happens just in case all the shared intrinsic properties are also necessary properties (e.g. the number two and the even prime, and so perhaps for the properties of concepts generally). We can now return to the robust realism of Russell's early period, putting this intervening version of aristotelian realism to work: first to compare the diagnosis it gives of some arguments for internal relations with the diagnosis Russell actually gave; and next to compare the account it suggests of particulars and universals with the one Russell first gave.

10 Russell said he did not find many reasons advanced in favour of what he identified as the axiom of internal relations, and he frequently accuses its proponents of merely assuming it (Philosophical Essays, 1910, Chap. VI). This line of attack is unfairly harsh, requiring for the axiom a kind of self-denial. To expect a list of arguments to be advanced on behalf of a specific doctrine, which is what he did for his axiom of external relations, is already to conceive it as a kind of independent entity whose support can be nicely discriminated from the fabric of presuppositions and consequences into which it is woven - as if its proof were external to it. This sort of thing the doctrine of internal relations is not and was never supposed to be: the axiom is itself internally related to its philosophical context. However that may be, he does ferret out a brace of specific arguments and one of these may be considered here. This argument of his opponents Russell first phrases for them as follows: " ... there is the fact that, if two terms have a certain relation, they cannot but have it, and if they did not have it they would be different; which seems to show that there is something in the

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terms themselves which leads to their being related as they are. ['Different' is the crucial word here. He promptly tries another formulation of their argument in which the pivotal word is 'other'] .... We may say 'If A and B are related in a certain way, then anything not so related must be other than A and B, [sic] hence, etc.' But [Russell goes on] this only proves that what is not related as A and B are must be numerically diverse from A or B; it will not prove difference of adjectives [properties], unless we assume the axiom of internal relations" (pp. 143-4). One senses that what is said here is vaguely right, until one sees that it is precisely wrong. For that, the distinction between sameness and similarity helps. If we return to the argument that Russell last put in quotation marks and insert in the place marked 'sic' the words 'or be A and B at a different time,' we can see that it is just this omission that the argument trades on; and it trades on it by its word "other" which gestures towards the words just inserted without actually taking their consequences. As modified with the inserted words the presumed argument is rendered entirely impotent, a harmless truism proving nothing. Russell does not catch this, does not, that is to say, catch the fact that "other" is being used ambiguously with respect to 'not the same things as' j'not exactly similar to.' Instead Russell grants that what is not related as A and B must be numerically diverse from them, which is certainly incompatible with what we have just said (assuming, as we must, an equivalence of "numerically diverse" and "not the same"). Russell in fact is conceding that a diversity of relational properties may yield terms which are not the same numerically but which are exactly similar, i.e. "without difference of adjectives." I suggest that the view proposed in aristotelian realism gives the far more plausible converse: diversity of relational properties may yield terms which are the same, but which (being at different times) are not exactly similar. This gives us a grip on the rather special meaning which Russell is appending to his doctrine of external relations: it is a conception that is hostile to the notion of continuity and self-identity of things through time and change. Just this was indeed, as we saw at the very beginning, a feature of his general metaphysical outlook, sending massive repercussions in many directions. Russell inaugurated the battles over external relations on a plane of

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argument such that, had the empirical world entirely disappeared, his arguments would have been unaffected. Return to the quotation's beginning in the previous paragraph. The crucial word (last line p. 81) is 'different,' and though there are other complications here, 'different' at least hides just the ambiguity that 'other' did. If 'different' means 'not the same terms' then it is not an argument but an entirely unargued assertion of the so-called axiom of internal relations; if 'different' means 'having different intrinsic relations or properties (while being the same terms), then it is an entirely harmless truism. As an argument, therefore, it absolutely depends upon trying to have both meanings at once, without taking the consequences of either: an unsupported assertion and a truism are merged and disguised as an argument. This seems to be the right analysis of the argument which Russell attributes to his opponents; but it is emphatically not Russell's diagnosis. It is a diagnosis sponsored by aristotelian realism and closed to platoniC realism. 11 Here is the place to explore this latter contrast further by considering the first and most abstract of Russell's arguments for the view that relations do not have particular instances. He shows this, or tries to, by generalizing from an argument about the relation of Difference (diversity, in the foregoing sense). If relations had instances then, in particular, different differences would differ: "And it seems plain that, even if differences did differ, they would still have to have something in common. But the most general way in which two terms can have something in common is by both having a given relation to a given term. Hence if no two pairs of terms can have the same relation, it follows that no two terms can have anything in common, and hence different differences will not be in any definable sense instances of difference. I conclude, then, that the relation affirmed between A and B in the proposition 'A differs from B' is the general relation of difference, and is precisely and numerically the same as the relation affirmed between C and D in 'C differs from D.' And this doctrine must be held, for the same reasons, to be true of all other relations; relations do not have instances, but are strictly

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the same in all propositions in which they occur" (Principles, pp.

51-2). This is the sort of snappy proof that abounds in Russell's work. The argument is fast and valid, and just as we expect, the tug is in the direction of Plato's world. There is a commonsense analogue to the philosophical idea that relations have no instances, and Russell was probably influenced by this: The order 'give me an instance of betweenness' seems rather more queer than 'give me an instance of redness.' But there is a problem about the premises of the quoted arguments. Since we are already familiar with Russell's principle that what we call similarity (sameness of property) and what he here calls having "something in common," is to be analysed as "having a given relation to a given term," the aristotelian-realist is bound to rise to the bait of Russell's second sentence as follows. Russell holds that if two things are similar or have something in common this entails that each of them bears a relation, the same relation, to a third term; but it may be that they bear instead different but similar relations to the third term: in the present case, the different differences, that is, the particular cases of difference, would not be the same, but they would be similar. How is this similarity of differs! (e.g. 'a cow differs from a horse') and differs 2 (e.g. 'cheese differs from crackers') to be explained? Russell would say more: only by both having the same relation, R, to the universal Difference; so if R cannot have instances there is no reason whatever to think Difference has. Aristotelian-Russell replies that having similar relations, r! and r 2 , to Difference is enough. Platonic-Russell then says that they were not "in any definable sense instances." AristotelianRussell says that it is arbitrary to require that the sense in which particulars are instances be "definable"; it is enough that it can be explained - which would require of course reference to the properties of the particulars. Whitehead is said to have said that "Russell is an entire Platonic dialogue in himself" - and so he is. To the uninitiated this may look like sheer verbalism. Yet, profoundly different metaphysical schemes are involved. Russell's platonism, where relations have no instances, has only universals and no particulars. The particulars of aristotelian realism are to be understood in terms of the sameness/similarity schema, and therefore too they are to be thought of as having various intrinsic

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properties and external relations. Indeed, the latter's very premise that non-sameness and non-similarity are two kinds of difference militates against Russell's basic argument that difference has no instances. Russell could have replied to this last point (though he did not) that relations may have species but no specimens: Difference, for example, has non-sameness and non-similarity. This would have been a first step toward aristotelian realism. Russell did not, in 1911 when he hovered temporarily over aristotelian realism, apply its doctrines to his earlier views as I am doing, nor in the earlier years did he argue against the view I am exploiting. The point is to make it evident that he might have done both. In Part II of this chapter, on propositions, these metaphysical arguments, between platonic-Russell and aristotelian-Russell, will be notably sharpened and recast in a more logical form. For the really driving idea here, that properties are expressible as relations, is both an ontological and a purely logical notion, and will return as a central idea in the logical atomism of both Russell and Wittgenstein. There is, however, on this metaphysical level, something bizarre in the consequence just discussed, that relations have no instances, that, for example, differences do not differ. Can we make sense of the idea that specimen differences do not have their own peculiarities? Surely it will not do to hold, as this appears to require, that the difference between horses and cows is the same as the difference between crackers and cheese. And to resort to the truism that after all they are both differences is to be rather more fond of abstract thinking than of facts. ("The world of being is unchangeable, rigid, exact, delightful to the mathematician, the logician, the builder of metaphysical systems, and all who love perfection more than life.") Consequently, it appears that Russell's crucial idea (second sentence in the long quotation above) that having a common property is having a "given relation to a given term" may have to be relinquished for ontological purposes in favour of having similar particular properties, a move urged by aristotelian realism. Curiously, Russell has a more complicated way out.*

* This other way is important for understanding the direction of Russell's thought: why, for example, he leapfrogged so curiously from realism to atomism with only a pause at the intervening aristotelian realism, when it was precisely the latter's new emphasis on spatial and temporal relations that gave him the format he needed. Briefly the situation is this. In Principia Mathematica he gave himself a variety of reasons for extending to properties

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Let us notice that the earlier way of interpreting properties as relations fits in with his argument that relations have no instances to give a platonic conception of all qualities. The redness of this book consists in its having a relation to redness; that red book bears the same relation to redness. There is nothing such as the particular case of red presented by this book. Russell drew just this conclusion in 1904: "there seems no such entity [the aristotelian particular] as the blackness of the table: there is blackness, and the table, and the proposition 'the table is black.' When the table is black, 'the blackness of the table' is merely another expression for the proposition 'the table is black' ("Meinong's Theory," p. 520). This is opposed to his later aristotelian realism precisely in that the latter takes 'the blackness of this table' as a particular, a particular enduring through time, present in space, and independent of perception, a particular similar in appropriate ways to the blackness of that table, both tables exemplifying the universal blackness. But then aristotelian realism needs a theory about its universals, and Russell anticipated that this might be a problem. His new way, mentioned in the footnote above, will rescue 'blackness' from Plato's heaven and will turn it into a class, and therewith the blackness of the tables into cases of membership in the class of black things. He took it that this meant that he no longer had to eschew the particular blackness of this table, or to affirm it. He was then free (in his atomist phase) to affirm something else in its place, viz. the-blackI-perceive.

12 We have been canvassing a situation in avowedly metaphysical terms, just as Russell did. The arguments tend to be abstract, sometimes soft, sometimes to proceed from arbitrary generally the privilege which was first reserved for 'number-properties,' that is, having a property is to be explained in terms of being a member of a class, and having a common property in terms of being members of the same class. He came to this view in stages. For a time he held both that to have a property was to have an external relation to another term, and that for some properties to have that property was to be a member of a class. But for logical reasons not now relevant he found it increasingly convenient to move to the view that in all cases the external relation in question was 'membership' and the term in question was the class. The new problem was then to define the class in question, and this he did, for classes of empirical properties, on the analogy of the definition of numbers - giving him what he sought, explicit links between metaphysics and mathematics.

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premises, or from tendentious assumptions about language; and, in the passages above, they have been merely described so as to sketch in the picture. But all this gives a distorted view, even as it facilitates exposition and reflects one aspect of Russell's thought, namely, its shift from platonic realism to aristotelian realism to atomism. We have not got to the heart of any matter, for this is essentially an affair of logical theory. This means concretely that we have to rethink things from the point of view of propositions. Let us see where in general we are and where we are going, and then turn to propositions. We are in a position to see at least that Russell's slowly growing and finally (in the atomist phase) dominant interest in particulars derived not from lasting sympathy for aristotelian realism but from independent sources. One general motive for this, coming from logic, has just been mentioned, viz. to think of universals as classes of particulars. A second motive, an epistemological one, is to be found in Russell's growing concern, first stimulated by Meinong, for the question of the status of particular objects of immediate sensory experience. It is all very well to talk about redness, squareness, etc., as objects of knowledge (when that was the form of his problem), but it came to seem intolerable to him to think that we see the universal redness; and it came to seem both obvious and important that in perceptual experience (when that was his problem) what we see is the particular redness of this, to wit, this red. From this point of view, then, in some sense there had better be particulars, be they ever so humble and ephemeral. To understand all this is to grasp the precise points at which a platonic metaphysics dominated by timeless terms and relations will, at the behest of logic and epistemology, yield its prominence to an atomistic metaphysics of transient particulars, logical skeletons, and universals. For a metaphysics that takes itself with great seriousness and yet casts its lot with the career of its logic, delicate changes transform timeless terms into passing particulars, Plato's Forms into Hume's impressions. PART II D.

PROPOSITIONS

AND

THEIR

CONSTITUENTS

13 Russell's early work on Leibniz had convinced him that almost the whole of the latter's philosophy had followed from his

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doctrine of propositions. This was not only a novel achievement on Leibniz's part; it was a novel insight on Russell's part as well. It was a fact too which was profoundly influential. Hence it is time to look more deeply into the fact that Russell, in the same mood, has cast his early metaphysics in the mould of terms and propositions. Bradley registered the following complaint, which was quoted and considered by Russell in "Logical Atomism" (1924). "Mr.. Russell's main position has remained to myself incomprehensible. On the one side I am led to think that he defends a strict pluralism, for which nothing is admissible beyond simple terms and external relations. On the other side Mr. Russell seems to assert emphatically, and to use throughout, ideas which such a pluralism surely must repudiate. He throughout stands upon unities which are complex and which cannot be analysed into terms and relations. These two positions to my mind are irreconcilable, since the second, as I understand it, contradicts the first flatly" (L&K, p. 336). For a philosopher of Bradley's stamp, who prized coherence above all, to fall into this incoherence was to sin against the light. Russell thought the criticism entirely unjust and politely said so. Yet if we allow for a certain vagueness, we must admit that Bradley has put his finger on the crucial spot: the ambivalence in which the uncritical notion of proposition is central - between complex unities and simple terms externally related is .a fundamental one for Russell; and even more fundamental in the early phase from which Bradley doubtless took his cue, than in the atomist phase from which Russell marshalled his rejoinder. Great philosophers often speak past each other, often, alas, in the very simple way of unknowingly addressing different topics. Since Russell's ambivalence is centred upon the notion of proposition, to consider this notion in some detail will also serve as a reminder that the preceding arguments too all revolve within the framework of this doctrine. In barest outline the situation for Russell's pluralism is this. It is the world that is responsible for diversity; it is the proposition that is responsible for unity. We said at the beginning and need now only summarize the fact that there are apparently obvious aspects of the doctrine:

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propositions are the truths and falsehoods of the world, they are neither 'mental' nor 'verbal,' they are what we know, believe, etc., they are independent of our apprehension of them and are composed of terms, whether beings or existents. Russell said that he had borrowed this idea of propositions from Moore - which was true, though the preoccupation with propositions goes back to Russell's work on Leibniz. The source several times cited by Russell is a very curious paper by Moore, "On the Nature of Judgment" (Mind, 1899), often identified as a pivotal point in the swing from idealism. Only Moore's main thesis is relevant. Trying to fight his way out of idealism he took the word 'concept,' pruned it of subjective, mental and psychological associations, and said that the world was composed of nothing but propositions, and that propositions were complex unities composed of concepts! Russell was in no way horrified: he absorbed the argument, welcomed the conclusion, and adopted the terminology of 'terms' as the worldly 'constituents' of propositions. Thenceforth, until for a complex of reasons he demoted propositions to something more like sentences in his atomism period, whatever there was in the world of being or existence could actually 'enter into,' be a constituent of, a proposition. Keynes writes in his Two Memoirs that "Moore had a nightmare once in which he could not distinguish propositions from tables." If so, Russell dreamed the same dream, and wakened more slowly. In 1910 Moore concluded a long argument by saying that "there are no propositions at all," a move which did not deprive the world of its furniture, because Moore had taken the precaution to reconstruct everything out of sense data, a project in which Russell promptly joined. A way to get an intuitive grip on Russell's idea is to take rather literally the idea that a proposition is what is propounded or proposed. Here, then, is a reconstruction of a theme throughout Principles and of the argument in "Meinong's Theory" (p. 521). Discussing the wisdom of my going to London, I conclude by saying 'Still, the proposition seems to me unwise.' What are we talking about? What is it that 'unwise' applies to? Certainly not a sentence (it is not the words 'I am going to London' that are unwise); certainly not a fact (there is no fact there). It is rather the complex entity my-going-to-London that is held to be unwise. That complex is precisely what the proposition is. And of course

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my-going-to-London does not exist: so the proposition is a subsistent being. "I have accepted from him [Moore] the nonexistential nature of propositions," said Russell. And here nothing seems to prevent us from analysing the proposition into its three components: I, the relation am going to, and London. There is nothing false or obviously fallacious in this; but it is eccentric and evasive. Eccentric in its starting point because the terms of appraisal of propositions, like 'wise,' 'irrelevant,' may not in general supply a sound clue as to what sorts of entities propositions are; and proposals in general are future oriented as propositions are not. Run through the argument again, changing the tenses to past, and putting 'undertaking' for the 'proposition.' Everything goes through smoothly, which shows how little the argument shows. Surely we do not want to divide the world into an ontology of undertakings, which exist because completed, and propositions, which as yet only subsist. Finally, the conclusion that London is a constituent ('literally a constituent,' the facile critic says) of the proposition seems rather courageous. But Russell has made converts, and many philosophers besides him and Moore have retorted, Why not? What they, or Russell at any rate, have in mind may be shown by arguing our way across the gap between this idea of constituents and the idea that we already know Russell was deeply committed to, viz. that propositional terms are timeless entities, free of change. Propositions are repeatable. If so, and if they are composed of terms, the same terms (not merely the words - that is indifferent) must recur. We can say the same thing about two individuals with these sentences: 'Caesar was a statesman,' 'George III was a statesman.' So 'statesman' indicates a term, the same term both times, related now to one man, now to another (cf. Principles, p. 84). When Gibbon said 'Caesar was a statesman' he expressed the same proposition as we did, with the same terms; and so might Sallust have, though he would have used different words. If we can say today and tomorrow again, what Gibbon or Sallust said, make the same proposition, ascribe the same predicates to the same subjects, whether the words are the same or different, then we must be presupposing that the terms in question are independent of time, that they are permanently there, being or existent, free of change. The difference among the propositions, viz. that statesman is related now to Caesar and now to George

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III is, says Russell, just a difference of external relations, precisely as the difference between my expressing the proposition and Gibbon's doing so is a difference of external relations. This elliptical argument, which I have cooked up from Russell's stock ingredients, shows how he was prepared to invoke his doctrine of external relations at every possible point. Again something is right and something is eccentric. Russell has got hold of the sound idea that something permanent must render possible our making the proposition that Gibbon made. He is surely right that it cannot be merely language, nor even the meaning of 'statesman' and the other words, for these two could change (perhaps have), and we could still produce Gibbon's proposition. But must the changeless element be the term? We can say this, but only if we are prepared to tie the idea of a term to something like 'the meaning of what Gibbon said' - which thing seems as obscure as it seems fixed. Russell could have said just this, and doubtless believed it. But here we need a theory of meaning, and since he had no notion of meaning other than of objects arranged externally, he would come back to his constituent terms. What Russell lacks is a sharp focus on the questions which Wittgenstein's Tractatus created in this vicinity: for example, how are stable terms related to the 'definiteness of sense' of propositions? We have made some headway into what is inviting in Russell's central conception, but it is yet far from clear. One reason why it is and must remain unclear is that there is going to be no decisive argument to show that everything that philosophy can deal with must be a constituent in a proposition. Russell was inclined rather toward proclamations: "I should have thought the subject of a proposition was a constituent of a complex in the fundamental sense from which all others are derivative" '("Review of A. Meinong's tJber die Stellung," Mind, 1907, p. 532). The reason he did not think he needed an argument is not merely that he had simply accepted this idea from Moore, nor yet that a variety of different considerations like these suggested and to come convinced him that it would work. It was primarily because of a certain frame of mind, inspired by his idealistic forebears, of this sort. We may think we can get to things or to the facts pure, but any fact we get will immediately present itself in propositional dress; in knowledge claims we cannot get past making propositions. (cf. We cannot get to the world bare~ but only to

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our perceptions of the world - a later and analogous temptation.) Consider how much idealist philosophy revolves around this hoary. banality: to know an object is only to know a known object; you cannot, as it were, get to the object and know it in its pure state, as it is in itself, apart from the knowing situation. Russell never looked twice at this argument, so convinced was he that an apparently opposite banality was the simple and entire truth: knowing never makes any difference to the object known. If he had looked twice, he might have seen that he was himself struggling within a mirror image of that idealist net: you can never in knowledge, or belief, or supposition get past the proposition. * So he is found to be assuming that of course whatever is known or knowable in the world bears the limitation that it is so only insofar as a proposition might transmit it, insofar as it is subject, predicate, relation or some term in a proposition. It does not matter, just yet, that whatever is clear enough to be true in this idea is also true enough to be empty. It does not need to be clear to be taken as significant. Propositions are the limits of the world. When Wittgenstein gave Russell's idea a linguistic re-expression in his Tractatus it was recognized that it contained a seed of solipsism. But if propositions - which for Russell in this period were at once the limits of our world and the limits of our knowledge - contain the objects of a realistic metaphysics, solipsism is avoided, as Russell saw. His proposition was a philosophical beachhead just above the wash of idealism and just short of the plains of subjectivism. (Did he come to see, accordingly, that it was built upon the sand?) Nevertheless, the philosophical outlook that ensues from all these reflections can be made compelling, the more so since Russell often took it in

* The idealists talked of judgments - acts of mind; Russell talked of propositions - objects of mind. This gave him objects independent of the knower -- what he demanded above alI. RusselI made use of the concept 'judgment' but always as a generic term for the different possible attitudes of mind toward propositions. He never meant to use 'judgment' (though his practice was somewhat slipshod just because of his reading) as the idealists did, to combine his notion of judgment and proposition. But the shared view, that knowables are constituents of propositions (RusselI) and of judgments (idealists) was unquestioned by both. Incidentally, a good bit of the pattern of this thought was to be repeated by Russell, four decades later (see Chapter 8), but with perception replacing propositions as the ultimate format. He came, in fact, to speak of the experience and structure of percepts in accents rather like those here employed for the subject and structure of propositions.

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a sufficiently general sense so that the alternatives seemed always to partake of some idealistic sophistry. Besides, a quite liberal and quite undefined commitment to the ideas of external relations and non-existent being seemed, in their very generality, powerfully helpful at crisis points.

14 "Complexes, as soon as we examine them, are seen to be always products of propositions" ("Meinong's Theory," p. 346). We must move on from the metaphorical seductiveness of this notion of a proposition to find its logical heart. Throughout Principles Russell argued for an interesting and pregnant idea which he expressed by saying "every term can be made a logical subject." I shall call it 'the doctrine of movable terms.' This important doctrine I want to consider first from Russell's (a) platonic view and then from (b) his aristotelian view. (a) One thing implied in the above doctrine is that within propositions the terms indicated by a noun and an adjective of the same root are the same term. "The fact is, as we shall see, that human and humanity denote precisely the same concept, these words being employed respectively according to the kind of relation in which this concept stands to the other constituents" (Principles, p. 42). This means that the same propositional terms are involved, though differently referred to in, for example, "Socrates is wise" and "Wisdom belongs to Socrates." The fact - if it is a fact* - that we can manreuvre the same term into different positions, and into different propositions, argues that it

* Russell disputed with Frege on this point, understanding him to hold that the attempt to move, say, human from predicate to subject place as our examples do, gets us only the name of the entity in the latter place. Russell replied, in effect, that this was contradictory since to say of anything that it cannot be made a logical subject is to make it a logical subject (Principles, p. 510). He did not think that Frege's important doctrine that a function cannot be an object - which in Russell's idiom says that human cannot be the same term as humanity - goes any deeper than, for example, to say that you cannot make a predicate into a subject for then it will no longer be a predicate. There are deep issues here which we cannot now explore, but they will return before chapter's end, for it will be seen that Russell is rejecting not the utility but the finality of Frege's distinction between objects and functions. Incidentally, we shall see (Chapter 5) that in his atomist period Russell rejected his present doctrine, insisting upon a categorical difference between subjects and predicates. Still later, in the neutral monist period (Chapter 8), he partially returned to the present view, analysing most grammatical predicates as logical subjects.

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must be a universal that is being dealt with. Is it not the nature of the universal to occur and re-occur, to be known and to be knowable in various contexts? What is involved in his treating it as a universal is that Socrates' wisdom or humanity is one with Aristotle's wisdom or humanity, and with everybody's. Russell, like Frege before him, is not here distinguishing a qualityuniversal (wisdom) from a kind-universal (humanity). On the face of it, it is more plausible to hold that Socrates' humanity is one with Aristotle's than to hold that his wisdom is; and platonism was encouraged by the penchant for taking kind-universals as examples. But the point for now is that there is nothing about the wisdom of Socrates that links it in any specially intimate way with Socrates, since the same thing may be linked to Aristotle, or to us. That is why Russell can say that Socrates is externally related to wisdom; that is also why one may think nothing is impaired by analysis, why one can break the proposition into Socrates, wisdom and some relation, just as a superficial glance would have suggested. Furthermore, in virtue of his general doctrine that every term can be made a logical subject Russell has a theory for verbs parallel to the one considered above: "The concept which occurs in the verbal noun is the very same as that which occurs as verb" (Principles, p. 48). The above considerations would also apply here; and there are other linguistic arguments. * This, then, is the platonism at the heart of his propositions.

* In this view about verbs - that their corresponding universals have no instances, that verbs and verbal nouns denote the same concepts - Russell is apparently exploiting facts like these: Suppose Sammy runs and I also run; then in the fundamental sense we are said to do the same thingRussell would say we have the same relation to running. To ask whether Sammy's running and my running are numerically one and the same is to ask a spurious question, since, for Russell, the beginning and end of the matter is that we are doing the same thing. Notice secondly that in reporting the matter we switched from the verb 'runs' to the verbal noun 'running': apparently we can report facts or theories about the relation indicated by 'runs' by talking about the relation indicated by 'running.' Yet everything seemed in order, and is. Discourse with verbs can introduce the same data as discourse with verbal nouns, can describe the same parts of the world. What Russell characteristically does is to seize upon such facts in order to explain them by means of a doctrine of external relations to which he can address other a priori arguments as well- that relations have no instances, that a verb and its verbal nouns always register the same relation, etc. But the upshot was that Russell is giving metaphysical explanations for

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(b) Let us now try to consider the matter from the view of particulars. We start again with the general point that, like wise and wisdom, "human and humanity denote precisely the same concept" and try to work it around to the aristotelian conclusion. The examples must be changed slightly, for a reason that will become apparent. (1) Socrates is wise (2) The wisdom of Socrates is not in dispute. We are now to interpret these examples (they are different propositions) as each involving the same particular: not the universal wisdom as above, but the particular, viz. Socrates' wisdom. Any theory of universals can be held in abeyance, recognizing only that Aristotle's wisdom would have to be another particular. Apparently our particular is such that it can be identified only by identifying it as Socrates': it is just Socrates' wisdom and nothing else that we are talking about. It is as if its pertaining to Socrates is a necessary or internal property of this particular. Alternatively said, our particular, viz. the wisdom of Socrates, cannot without destruction of its very nature be analysed into wisdom and Socrates: it cannot be analysed at all, since it is one unitary particular. We can no more detach Socrates, even in thought, from the-wisdom-of-Socrates than we can detach the shape of a thing from that thing, or detach the idea of 'London' from the idea of my 'plan-to-go-to-London.' Such detachments would only be destructive of the entity analysed. This, then, in contrast with platonic-Russell is aristotelian-Russell. I have just tried to fit two theories to Russell's doctrine of movable terms. He took the former platonic way until 1911, and then in "On the Relations of Universals and Particulars" (L&K) he came over to something equivalent to the aristotelian way. The former way is simpler: there are only terms and external linguistic facts for which he later gave logical explanations. He came to see that in logic we can discard many worries about the character of objects in propositions since, for purposes of deduction, it is often all one whether the predicate letter in our symbolism goes proxy for a verb, verbal noun, or adjective, because they are one and all going to be thought of after the manner of functions. This itself is not going to presuppose any firm commitment on the question of universals and particulars, or relations, whatever it may suggest for further theory.

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relations; it makes analysis intuitively based and intuitively acceptable; it fits in exactly with his current doctrine that relations have no particular instances; it postulates no unbreakable unities. There are important reasons why the aristotelian way is not at first open to Russell, and why he did not stay with it later. The first reason, and for a time this alone would have been enough for Russell, is that the aristotelian particular, a unitary complex (here we touch on Bradley's criticism) appeals to a kind of 'connection' between Socrates and wisdom that is not an external relation. We might say that the question of relations cannot come up in this form, but Russell clearly thought it could. A second point is this: in the examples we did warp the topic out of shape by supposing that the movable term was not that indicated by 'wise' and 'wisdom' (as would have paralleled the platonic analysis) but by 'the wisdom of Socrates' which, we said, was the same in denotation as 'Socrates' wisdom.' Now Russell can flatly deny that one of our propositions, viz. "Socrates is wise" involves this entity at all, and deny therefore that we have considered, as we pretended to do, a case of the kind he was talking about, where a predicate was made into a logical subject. He can say that our examples, which are not parallel with his examples, are malign. The truth is that they were meant to be subversive. It is also true that "Socrates is wise" does not contain as constituent (nor is it about, nor contain a reference to) Socrates' wisdom. But it does contain something else, to be considered shortly. However, something that is very fundamental about Russell's thought now comes out, the principal reason why he could and did take the aristotelian way for a time, and did not retain it. It is fundamentally because he was there (in 1911) concerned not with the analysis of the constituents of propositions, as he was before and after, but simply with, to make it banal, the nature of things in the world. He then took as his problem the question whether there is a "fundamental division" of objects into universals and particulars, deciding that there was. The reason why aristotelian particulars do not naturally emerge from a Russellian analysis of propositions is just that they are not movable propositional terms in Russell's sense - which is precisely what is involved in Aristotle's doctrine (Categories la20ff.) that though wisdom is "predicated of" Socrates'-wisdom (which is just to

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say that Socrates'-wisdom is wisdom), and though wisdom is "in" man by virtue of the fact that Socrates' -wisdbm is "in" Socrates (and he is a man), yet it is nonsense to "predicate" Socrates'wisdom (= the-wisdom-of-Socrates) of Socrates. Socrates'-wisdom is not, and cannot be made into, a propositional term in Russell's sense.

15 Weare not yet done with this argument. Aristotelianism was a resting point - not only in specific doctrine but in the very conception of the philosophical task - between platonism and atomism. The logical core of this fact can be laid bare by returning to our examples. Consider that it might be said that we can find in the examples one and the same movable term after all. The term is indicated by ". . . is wise," in (1) and "the wisdom of ... " in (2). These are expressions for what Frege had long since called "functions," and which he made essential to the analysis of propositions. So here an entirely new analysis breaks on the scene, incompatible with platonism, different from aristotelianism. Yet it can now become plausible again to say that we are dealing with a universal, for there seems nothing obviously wrong with Frege's implied belief, which is also Aristotle's, that these expressions depict one and the same movable term. The question, from Russell's direction, is, Which term? Here it is no good saying it is the platonic wisdom, and it is also no good saying it is the aristotelian Socrates' wisdom. (Though it is very likely that it was reflection on this Fregean analysis that opened to Russell the aristotelian possibility of admitting unanalysable particulars.) Yet the function appears to provide, and does provide, an extremely important piece of machinery for the analysis of propositions. Only in his atomism period was Russell willing to accept this logical analysis as being fundamental. Only then did he make the notion of the 'logical form' of a proposition (rather than, as here, ontological autonomy) essential for understanding universals (see Chapter 5). For the present he emphatically denied that any grammatically fragmentary expression like those for Frege's functions could be indicative of the final analytical constituents of the proposition. This is a crucially important part of his outlook. He did not deny the logical power of the idea, which was central to his entire conception, but ontologically he said that Frege's function which D

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"could not become a subject" was a "non-entity." He has basically two reasons for this: one a very technical one, roughly, to the effect that there are cases where we cannot keep track of the function when certain combinations of terms fill its gaps; and another to the effect that the function as such "is apt to have no sort of unity, but to fall apart into a set of disjointed terms," because it has a gap in it, and because different kinds of expressions - 'is wise,' 'the wisdom of .. .' - can pick it out (Principles, p. 509). (There is nothing whatever in Russell's philosophy to allow 'is' and 'of' to depict the same relation.) He means that the function does not have the sort of unity that is had by the class of all those terms (in our example, all the wise things) that it is a function of; nor does it enjoy the sort of unity possessed by the class of true propositions it is abstracted from; and this is correct. Russell thought it important because of his general commitment to the importance of classes and because he supposed that an analysis of propositions, to be sound and acceptable, must resolve into the final unities of the world, must exhibit the proper objects of which the world consists. Logical analysis of propositions must meet ontological criteria. So the Frege analysis does not give the kind of answer Russell believes he needs. . . . With this, I think, we touch bottom: this is the deepest theme in Russell's realism. It ought to be clear now - what a mere reading of the texts does not make very plain - why the aristotelian particular, though it tempted, could not detain him. One arrives readily at that entity only if attending directly to the empirical world (as he was in 1911), where change and contingency prevail; or, more liberally, if attending to the analysis of things, unaffiicted by undefined worries about internal relations; not if searching for the metaphysical components of propositions (as Russell was first), nor if drawing the ontological conclusions of a logical theory (as he did later). A metaphysical analysis which accepts a universal wisdom as basic, and a logical analysis which accepts a function 'the wisdom of .. .' as basic, are both different from an aristotelian outlook, but in diverse ways: the latter can accept the logical analysis for the proposition on the one hand and also accept a unitary particular, the wisdom-of-Socrates, as a metaphysical entity.

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Though it may be said that Russell needs this kind of particular, we shall see in ensuing chapters that both in his atomism and in his neutral monism he always held the idea off at arm's length. A general reason for this reticence about unitary particulars, such as the-wisdom-of-Socrates, will be discovered, next chapter, in his logical theory. We shall see that eventually the same kinds of questions are involved in his discussion of the identification of sense data (Chapter 6), of the individuation of experiences and perceptions (Chapter 7), and of the notion of privacy (Chapter 8). E.

PROPOSITIONS:

THEIR TRUTH AND

THEIR UNITY

16 In 1908 Russell called "the nature of truth" "the fundamental question of philosophy" (Philosophical Essays, 1910, p. 114). We must chart the roots of this conviction in his own philosophy, but there is a polemical context as well. One reason why he said this was that the idealist coherence theory of truth was part and parcel of the doctrine of internal relations, which he had been warring against; in addition, he was only recently confronted with the pragmatist theory of truth, which he found equally implausible. Both these schools - the dominant ones in the English-speaking world, besides his own - had adherents fond of claiming that with their doctrine of truth their philosophy would stand or fall. And so they fell. Russell and Moore were principal combatants in both cases. Russell's Philosophical Essays contains what is arguably his best polemical writing; it is as remarkable as it was unfortunate that the book was out of print for fifty years - until 1966. But Russell had his own reasons for putting the problem of truth at the centre: he had trouble fitting the notion into his own philosophy. These problems go back ten years to his book on Leibniz (1900). Throughout that book, more influenced by Leibniz than he knew, he is thinking of truth as a property intrinsic to true propositions, which in virtue of his not distinguishing 'intrinsic' from 'necessary' led him, though it was not the only thing that led him, to make truth a necessary property of true propositions. This view, explicit in 1900, helps to explain what we have seen long since: the absence of any clear notion of contingency other than

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that implied in external relations. The general problem is so persuasive and so entwined with his metaphysics that we cannot avoid considering some aspects of it. In Principles he states flatly: "there seems to be no true proposition of which there is any sense in saying that it might have been false" (p. 454). If a proposition consists of its constituents in a certain combination, as Russell supposed, then the identification of a true proposition consists in these constituents being recognized to be in the appropriate combination. On this view it does, in a curious way, make no sense to say of a true proposition that it might have been false. It may seem that it would make sense in that it might be supposed that this would mean only that these constituents might not have been in that combination. But for them not to be in that combination is for the proposition, which is their being in that combination, simply not to obtain at all. To imagine that combination of constituents away is to imagine the proposition away: nothing remains of the proposition to be false. It is a consistent but perplexing view. It is interesting right here because it raises the problem of the analysis of false propositions. Thus in Russell's next publication, the distinguished series of essays on Meinong ("Meinong's Theory of Complexes and Assumptions," Mind, 1904), he keeps circling back to this puzzle, trying to draw a bead on the presumed constituents of a false proposition. In Principles he had hoped that we could get by with saying that the constituents of the false proposition were just the constituents of the correlative true proposition, plus negation. A philosopher like Russell, who merely happened upon the problem, can perhaps brazen it out with that, or postpone the matter - unlike a Plato who deliberately courted the problem because he knew that it hid something tempting and wrong. Two sorts of false propositions especially puzzled him in the Meinong essays: (a) There is the false proposition made by my saying of this brown table: 'this table is black.' (b) There is the different kind of case where, say, 'A exists' is false. Russell quickly convinces himself (pp. 519-22), following Meinong, that there is, quite certainly, a transcendent 'object' of some sort - a fact, Objectiv, or proposition - whose structure and analysis are crucial for both these kinds of falsehoods. Let us consider them in turn.

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(a) As to the first problem: we cannot long be content with the idea that the false proposition is analysed by the corresponding true proposition plus negation, if only because this obscures the distinction between 'this table is not brown' and 'this table is black.' The problem for Russell's constituent ontology is that if the latter proposition is false (since the table is really brown) it must contain some constituent in its predicate place. The question is, What? Russell can say that though it contains no existent blackness it yet contains a subsistent blackness: the falsehood derives from the fact that it contains a predicate from the wrong ontological level (pp. 516ff). Russell does not need to be told about the artificiality of this, and of the further artificiality in having to say that we detected its falsehood by noting that the proposition contained a merely subsistent blackness. It is here, in fact that he is developing his first suspicions about the capacity of subsistent being to rescue foundering ideas, for he actually begs off, saying that the expression 'the blackness of the table' is "misleading." Then he appears to change the subject, by mixing the subsistence-existence puzzle for the blackness in question with the issue of its universality or particularity (p. 520). Recall (above p. 86) that Russell holds that there does not exist an aristotelian particular, the-blackness-of-this-table, even when the table is black, and, he now appears to think, such a particular does not even subsist. What subsists is only the platonic blackness, which, as it happens, is not instantiated in this brown table. The idea that is influencing him is that a particular blackness (were there such a thing) could only be identified and individuated by the table, thus already granting it some specific status, whether subsistent or existent. That is why he says that the difficulty of admitting the particular at all is "to see what it is that is denied" when it is "said not to subsist; and this difficulty seems fatal to the view in question" (p. 521). The view in question is the admission of particulars and he is right in sensing that there is a special difficulty in denying the subsistence of particulars, and hence in providing for false propositions about them, since the subsistent particular denied would, unlike the universal, have to have been antecedently individuated by its bearer. What Russell has come upon is a special case of the very important puzzle to the effect that once you have identified and individuated an entity you need some special manreuvre to deny subsistence to it. The way

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out of this, he discovered only later, was to say: not that we identify something and then deny existence or subsistence to it, but rather that the identifying expression identifies nothing (see Chapter 5). The total upshot for the present argument, however, is this: instead of answering his own question about the subsistent constituents of the false proposition, he has diverted the argument so as to use subsistence to attack particulars, and so preveut them from giving an unwanted answer to his question. Speaking of the false proposition, he says: "it is very difficult to see what the proposition is" (p. 521). And so it is. Russell is very aware that, if not how, his conclusion is unsatisfactory. One reason it is unsatisfactory, beyond the subsistence theory, is that he is careless about the distinction and connection between identifying a proposition by means of its constituents and raising the question of its truth or falsity. Suppose we wish to consider the identification of a proposition without considering or knowing whether it is true or false. We wish to know, for example, merely whether two candidates are the same proposition. Abstractly, the problem calls simply for considering whether they have the same constituents in the same combination. So, for Russell, if I say 'I am unhappy' and you say of me 'He is unhappy,' the same constituents are at issue and compose the same proposition. The language of 'constituents,' despite its frequent and appalling problems, makes short work, as is certainly advantageous, of any worries about the verbal difference, or any - real but irrelevant - differences in the meaning or sense of 'I' and 'you.' By getting the same complex of constituents we get the same proposition. This is sound enough as far as it goes, but it does not go very far; and it calls for support from what Russell has no clear inkling of - a distinction like Frege's between the sense and reference of expressions. It is in such a context - where sense-reference distinctions are suppressed, and where determining the constituents of a proposition is already determining whether it is a true or false - that we must understand a certain famous statement of his. He suddenly just waved his hand and remarked: "It may be said and this is, I believe, the correct view - that there is no problem at all in truth and falsehood; that some propositions are true and some false, just as some roses are red and some white" (p. 523, my

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italics).* It is an appealing boldness, cutting the exasperating Gordian knot, but it did not keep him from taking seriously the next problem about false propositions. (b) Our second example raised this question: What is the corresponding fact when, say, 'A exists' is false. We can reconstruct his thought (p. 523) as follows. First, there is the tempting and simplifying idea that a fact is only a true proposition, in this case expressed by , "A exists" is false.' But he acknowledges that this would be a dodge. He then remarks that whereas when A exists is true not only do we have the existence of A but we also have the subsistence of A's existence; so when it is false, what do we have the subsistence of? He knows that the answer should be: the subsistence of A's non-existence. But Russell is not stalwart enough to face such a monster, and what he suggests instead is the converse: viz. that "something does not subsist which would subsist if the proposition were true" (p. 523). What would this be - a creature whose failure to subsist accounts for the falsehood of 'A exists' in the way the subsistence of A's existence does account for the truth of 'A exists' when it is true? The answer is : it is the existence of A that fails to subsist. So now we have two affairs to answer to the falsehood of 'A exists' : (1) the subsistence of A's non-existence, (2) the non-subsistence of A's existence. These answers, though here somewhat more succinct than Russell made them, are in fact equivalent; moreover, they are logically arrived at, and in some sense 'correct.' They have the additional property of being preposterous. Russell retreats again: "it is hard to regard A's non-existence, when true, as a fact in quite the same sense in which A's existence would be a fact if it were true" (p. 523). On this note he concludes the entire series of Meinong papers by saying that "the analogy with red and white roses seems, in the end, to express the matter as nearly as possible. What is truth, and what falsehood, we must merely apprehend, for both seem incapable of analysis" (p. 524). Our preference for

* In Principles Russell said "now the truth of a proposition consists in a certain relation to truth" (p. 450). But if this relation is really external then once you have the proposition you have everything essential to its truthwhich is just what we are criticizing. On the other hand, his inclination to regard all truth as necessary makes truth an internal relation of true propositions. The famous quotation above is partly the result of trying to balance these incompatible (and individually implausible) positions against each other.

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truth, he suggests, must be based upon "an ultimate ethical proposition: It is good to believe true propositions, and bad to believe false ones." He does not say whether it is good to prefer red roses to white ones. After this, subsistence was never quite the same for Russell. His arguments were putting it out of business. Yet the swift retreat to intuition was not wholly warranted. We cannot blithely say that what Russell concluded in triumph is for us a reductio ad absurdum. The trouble was not just the admission of subsistence, but the muddled tendency to think of subsistence as a special kind of feature, like existence but opposite to it. Briefly, he should have tied subsistence to facts, not things. When A does not exist, we have the fact, not the subsistence, of A's nonexistence. (The preceding paragraph can be rewritten along these lines, with quite innocuous results.) The point is important, and complicated, because facts, as Russell later saw - and as Meinong certainly emphasized about Objectiven which subsist ('bestehen') - can contain existents, whereas Russell's subsistents can only be contrasted with existents. This is the precise point that Russell missed, and it is just to this degree that his discussion deviates from the doctrinal line that otherwise runs, through him, from Meinong's Objectiv to Wittgenstein's Tatsache. Both play host to existents and have themselves a different status. That Russell is partly on to this point is shown by the fact that he did not make the blunder of supposing that just as the failure of existence led to the subsistence of something, so the failure of subsistence would lead straightway to the sub-subsistence of something. But he did, manifestly, make the mistake of supposing that the failure of his subsistence analysis to produce a plausible theory of falsehood entailed the failure of a parallel existence account of truth. That is why the correspondence theory of truth at which he arrived about 1910 seemed to him to be not only a triumph over pragmatism and idealism, but also over the bold and implausible roses theory of truth that he jumped to when subsistence collapsed beneath him. One thing that Russell needed, of course, was just what the theory of descriptions gave him (see Chapter 5): a compendious method for denying existence without having to stand in the quicksand of subsistence - for denying both existence and subsistence at one stroke. Another thing he needed was a distinction

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between the criteria of a proposition and the criteria of its truth, thus giving him room for the idea of contingent truth. A third thing needed was a distinction between the sense and reference of expressions. These omissions are all connected with the general circumstance that the language of constituents creates the impression that there are, somewhere, direct and clear answers to questions about the nature and working of propositions; and in fact he gives only indirect and unclear answers. They are indirect and unclear because the language of constituents keeps company with everything that is facile and vague in appeals to their external relations, and everything that is tempting and unproductive in appeals to their subsistence or being.

17 We can take a last and somewhat hurried glance at a large field of problems that interested and worried Russell. Something, he often remarked, of the unity of a proposition is threatened when it is analysed into its components. Human and humanity may be the same propositional constituents, as he said, but their grammatical difference seems impossible to dismiss - or explain. 'Socrates is humanity' is false or nonsense, and there ought to be a reason for that. 'Caesar died' and 'the death of Caesar' (his examples) are the same and yet different. A sentence is not just a list, and a proposition is not just a stack of objects. It is at least a unified stack, and the meaning seems affected when the bits are unstacked, or restacked in another way. Perhaps analysis is falsification after all. Thus the worry becomes more and more abstract, and just for that reason more and more insoluble. * Russell countered with a maxim, one he has always maintained: analysis is not falsification even though we do not retain

* The phenomenon of asserting propositions struck him, rightly, as crucial. He saw that it is connected with truth at least in the sense that to assert a proposition is effectively to assert that it is true, and that a proposition can be unasserted just as it can be untrue. Accordingly, interesting questions both parallel and intersecting those about the connection of truth and propositions come up. However, the difference between propositions asserted and merely considered, between propositions in action and in analysis, between propositions as unities and as lists, between what is or can be true and what can't, between operating relations and ones merely consideredthese differences, all too much neglected by his predecessors, Russell knew, presented themselves to him rather too much as ideally open to one kind of explanation. And why not? - if "different differences do not differ."

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everything we start with - otherwise there would be no point in analysing. From the first he acknowledged that there are many varieties, purposes, and criteria in philosophical analysis, and no definition of it except its practice, and that a given proposition need not be thought of as always admitting only one sort of analysis. Later he was inclined to retract this last concession. Still a certain model has gripped him and' it is this which is leading him to the particular worry that "when analysis has destroyed the unity, no enumeration of constituents will restore the proposition" (Principles, p. 50). We can see that this is going to be an insoluble problem, because it is on the wrong level and Russell has the wrong equipment for dealing with it. Let us try to be brief even at the price of being brutal. One dilemma is this, writ large throughout Principles. To begin with, verbs are admittedly crucial in the unity of propositions, and verbs depict relations. Is the verb in the sentence then indicative of an internal relation, organically tying the parts together, accounting for its unity? But it is doubtful that there are any internal relations, and anyway not all verbs depict internal relations. So do they depict external relations? How then could they possibly generate the unity of the proposition? Well, perhaps there is a difference between the universal relation, static, which analysis elicits~ and the particular case of it which activates the proposition. But no, he had repeatedly argued that relations had no instances. From here there is no exit. * On a related point, Russell sometimes seems to realize, but much too obliquely, that in 'Socrates eats' or 'Socrates sleeps,' even if eats and sleeps are universal constituents only externally related to Socrates, they cannot be merely externally related to their proposition: for to supplant them with say talks and walks, also presumably externally related to Socrates, would produce entirely different propositions. Russell takes it that a term is movable within a proposition (cf. 'eating is done by Socrates'),

* He found an exit for another dilemma, the one about 'Caesar died' and 'The death of Caesar,' which he said depicted the same entities in different relations. Aristotelian-Russell saw that he did not have to admit that 'the death of Caesar' gives what is given by 'Caesar died,' but only what is given by 'Caesar's dying,' which is a particular, involves no proposition, and brings in no more worries about the function of a universal, death, both within and outside a proposition.

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and external relations facilitate this; but he does not consider the fundamental question of what is involved from the point of view of propositional identity if a movable term is replaced. And this, unfortunately, does not come up because he does not bring the discussions of internal and external relations into the discussion of propositions - except in terms of reducing one kind of proposition to another. Another vital point comes in here. Russell made extensive and, for logic, very fundamental use of what he calls "propositional functions" - briefly what is obtained by deleting a term from a proposition. Thus: from 'Socrates is a man' to ' ... is a man.' It can be rightly wondered why this does not figure more prominently in the attempt to provide a format for the understanding of propositional unity. It is not merely that the idea is more essential for his logic than for his metaphysics - which is true. And it is not merely that 'functions' do not submit very well to a preoccupation with external relations - which is also true. So far as our present purposes go Russell held that the propositional function was, as it were, but a stage in the analysis: propositional functions make sense only because they relate to, and for some logical purposes are equivalent to, classes. A function is not an authentic constituent, and it is with constituents that his ontology is concerned: "propositions may have a certain constancy of form, expressed in the fact that they are instances of a given propositional function, without its being possible to analyse the propositions into a constant and a variable factor" (Principles, p. 85). There are deep and intricate problems here, which for the time being we shall have to elide, in order to get to this point: In a sense, for Russell, only the propositions, classes of propositions and classes of propositional functions are real (Principles, p. 508). Recall here his objections to Frege's functions as too abstract to be real, just because they could not be subjects. The unity of an individual proposition was not to be explained by the external membership of one of its admittedly abstract and arguably 'unreal' components in some class. If one wonders why not - as a start, at least, in the right direction - there is the answer, distressingly vague in this application, that this would violate the doctrine of external relations. This is probably true. But the reply has to be that to rescue any fundamental idea of logical reform, to regard logical analysis, where generality is of

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the essence, as in any sense basically revelatory, we shall have to violate some versions of this doctrine: for example, the notion that we could maintain the identity of a proposition by replacing some of its merely externally related constituents. Russell is not yet on to this - or, rather, he is on to it but unwilling to take its consequences. We touch here on what was said before about his response to Frege's functions. If these cannot be assigned the final role in the analysis of propositions for the reasons there given, then there is no reason to look to them for the final account of the propositions' unity; and why should we suppose, anyway, that a 'fragment' (Russell's other equally tendentious word was 'rump') could explain a unity? The foregoing paragraph is a reconstruction of Russell's line of thought, throughout Principles, and especially in its appendixes. It shows that what he has come upon, though he at first turned resolutely away from it, is that it will never do to isolate propositions and look inward at their constituents to find' the secret of their unity. The unity of a proposition, which overlaps with its capacity to register a truth about the world, can only be understood by considering such banal affairs as its context, and such abstract things as its logical form: for example, those several kinds of relations between a proposition's constituents and its own identity. These are ideas which are simple and subversive, and which lay in his future.

18 Glancing back across this entire field of argument makes it evident that there is profound truth in Bradley's quoted (p. 88 above) criticism: something very deep in Russell's thought calls for "simple terms and external relations," something else calls for "complex unities" (propositions, functions, particulars, for examples), just as Bradley has said. It is not however that Russell won the battles while Bradley won the war. It is rather that Russell won the battles, and by doing so he redrew the map, even though he could not immediately occupy all the territory, until he was joined by Wittgenstein. In the meantime, he and Whitehead built the imposing edifice of Principia M athematica on a cleared corner of the battlefield. If we can see that Bradley was right in his critique of Russell's methods, we can also see what Bradley did not see, though Russell by the time of his reply had seen it. It is all too abstract

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a business: metaphysical tools are doing the work of syntax and logic; the suggestions of language are being heeded too literally. In sum, a doctrine of relations, though polemically victorious over an entire school of philosophy, but whose central terms and range of application are too undefined, is intersecting a doctrine of propositions whose limits are too uncritical, and whose application is too cumbersome. For a decade and a half, including the whole period of Russell's work on Principia Alathematica, he had eyed the world from the point of view of relations; at the end of this time, coinciding with the emergence of his new philosophy of logical atomism, he was beginning to look at the world from the point of view of sensory particulars, and from the austere demands imposed by his new slogan, first propounded in 1914: "logic is the essence of philosophy." It was a very fundamental shift of preoccupation, orientation, and specific doctrines - in that order - though the manner of its coming cannot be seen simply from what has so far been said. It is a remarkable fact that all the topics we have touched, and many we have not, remained in his attention, in (.ften novel ways. What also prevails is a resolute pursuit of analysis and that search for ultimate principles which he identified as the philosophical impulse. Whether metaphysics is to dictate to logic, or logic to metaphysics may appear a hard and bitter choice. Russell's way with this option was to adopt each in turn. There is much in the realist picture that is perennially tempting and plausible, and never more plausible than when it is serving as the basis for something else, for example, a philosophy of mathematics or ethics which is taken to be more important than the metaphysics - which is just what it was for Russell. Years later he spoke approvingly of the "realistic bias" which he was inclined to place at the basis of any sound logic. It was by means of his realism, its preoccupation rather than its doctrines, that Russell, with powerful support from Moore and later of Wittgenstein, accomplished a vast and historic shift in the inclinations of Anglo-Saxon philosophers: to look for philosophical models in the direction of logic and mathematics instead of, say, only to psychology and literature. Though he demonstrated by inadvertance how hard it is to argue one's way out of certain

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idealist traps, he broke successfully with that framework of subjective judgments and the "block universe" that he joined with William James in combating; and so doing raised just the issues that succeeding generations have found most important: propositions and meaning, truth and reference, relations and logical form. Thus he supplied the style of mind and the principal intellectual energy - though not at first the answers - for the creation of a new area of philosophical inquiry, and indeedwhat remains to be discussed - an entire new logical theory.

CHAPTER 3

THE THEORY OF LOGIC

The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds must be abandoned as beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncracies, but must command the assent of all who are competent to form an opinion. Russell, Our Knowledge of the External World (1914)

CHAPTER 3

THE THEORY OF LOGIC

A.

" PHI LOS 0 PHI CAL

LOG Ie"

1 This chapter will have to introduce a minor blast of technicalities. To get at what is most interesting, namely the philosophical situating of logic as Russell saw it, and the connection of logic and mathematics as a philosophical idea, requires a fairly orderly basis. So we cannot get along without talking about such items as these: primitive terms, class, axiom, implication, quantification, truth functions, denoting. Not, however, for its own sake, which is taken care of well enough by the professionals, but to indicate the kind of logical traditions which Russell and his cohorts inherited, and transformed. It would be a mistake to think that there is an uncontroversial way to do this: The nature of logic and its place among the sciences has been debated since Aristotle, and was itself an exploding question at the time Russell began. Suppose it is now asked bluntly, What is logic? Is it, as the layman is inclined to suppose, and as Russell sometimes declares, simply the study of inference? Some philosophers will reply: Of course. Some will reply: Of course not. These latter may have a more intriguing answer: Logic is the superscience, the science of sciences, it deals with the laws of the laws of nature, it is the most general and abstract discipline there is or could be. The best place to focus whatever light Russell has for this complex debate will come after his particular views have been aired piecemeal. (See Section D, Chapter 4). There is a double reason for allowing the large and barely digestible question - What is logic? - to simmer on the back burner: First, logic has itself changed drastically, in technique and theory, since Russell began working in the subject, and, secondly, his avowed interest at first was in the philosophy of mathematics, and what he had to say about logic was a means to that end. This second point will be artificially ignored until the next chapter. 112

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So Russell said what he had to say in passing: "Symbolic Logic is essentially concerned with inference in general, and is distinguished from various special branches of mathematics mainly by its generality. . . . [Logic investigates] the general rules by which inferences are made, and it requires a classification of relations or propositions only insofar as these general rules introduce particular notions ... the logical constants. The number of indefinable logical constants is not great: it appears, in fact, to be eight or nine" (Principles, p. 11). For many yearsuntil at least 1914 when he had completed most of his technical contributions to the subject - Russell found himself with little motive for trying to develop his conception of what sort of discipline logic ought to be. Like Aristotle and Leibniz he was too busy pioneering to say just what he was doing. Like them too, and like Bradley, who was looking over his shoulder, he made no sharp division between logic and the rest of philosophy; his logic appeals to metaphysics even as it reflects the process of inference, and tries to give a foundation for mathematics. In the public mind the syllogism has dominated the field of logic, just as, to a lesser extent, it has dominated the actual history of the subject since Aristotle. Russell defines his work by its opposition to this tradition, though by 1900 he was not alone in this. As for mathematics he points out that the traditional treatment of deduction was inadequate to mathematical deduction (see next chapter); and as for logic, it had considered deduction too narrowly: "It is from the recognition of asyllogistic inferences that modern Symbolic Logic, from Leibniz onward, has derived the motive to progress" (Principles, p. 10). One common way of dating the beginning of modern logic is by reference to Frege: his Begriffschrift (1879) is a natural landmark, because therein the devices known as quantification theory (about which we shall have much to say) first saw the light. It was Russell who first tried to direct wider attention to Frege, but this occurred (1903) only after Russell was himself well established and providing distractions of his own. Another way is to date modern logic, as Russell himself does, from George Boole, a reason being that it was he who first made important use of what Russell called propositional functions. There are numerous other reasons for thinking of Frege and Boole, or of others, such

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as Peano or Cantor, as Russell's principal nineteenth-century forebears. But it happens that Russell began largely independently of these direct influences, even though he did his early thinking in an atmosphere, the late nineteenth century, thoroughly saturated with the idea of an axiomatic exposition of mathematics. Though he was eventually to do more than anyone to focus professional attention upon logic, upon the importance of his predecessors, and to attract brilliant minds to the study of the subject, he began his logical reflections in a curious isolation. One reason for this has already been mentioned; he was concerned with what he then called 'symbolic logic' only as the foundation of mathematics. Thus it was to Georges Cantor, the German mathematician who laid the basis for what is known as set theory, that he was looking for light, more than to logicians. Russell reports somewhat dolefully in the Introduction to the second edition of Principles (1937) that "logic is traditionally associated with philosophy and Aristotle, so that mathematicians felt it to be none of their businessl and those who considered themselves logicians resented being asked to master a new and rather difficult mathematical technique." A connected and more important point is that the two logicians he knew best, who were most widely known at the time, viz. J. S. Mill (1806-73) and F. H. Bradley (1846-1924), fell from grace on just this point: their logic, each for its own reasons, did not allow mathematics to be true, he said, and that was enough to condemn it. (This idea will be picked up in the next chapter.) Later, Russell implied that if he had been reading Peano and Frege at the time he was in fact turning from Mill to Bradley, and from Bradley to Leibniz, and from Leibniz to his own ruminations, he would have made more progress. Perhaps he would only have made different progress, for Russell easily underestimates the highly individual trajectory of his own mind. His borrowings have been substantial, but they have always been as much atmospheric, so to speak~ as material, and it is at least doubtful that he would have seen the point of Frege's logical inquiries before he had begun to feel pangs in the same place. What is true is that a certain philosophical picture of how things hang together, one that derives largely from his own reflections and from Moore (in particular from his ideas on propositions and relations), is very much in evidence as his theories on logic are shaping up.

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2 The first, the most important and the most laboured section of Principles is entitled "The Indefinables of Mathematics." Russell wants to know what are the principal ideas, as few as possible, that must be appealed to, to explain the essential concepts of mathematics. This was the kind of philosophical inquiry that he had missed in Cantor. That these ideas will be logical in kind just is the thesis of the book. "The discussion of indefinables - which forms the chief part of philosophical logic - is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple" (Principles, Preface). That is the uniquely Russell way of putting it, though he was not alone in believing it. The logical terms he is seeking are as real as pineapples, and if they are indefinable they are realer yetthat is the spirit of the campaign. The enemy is SUbjectivism in all its forms, and the strategy is to find a firm objective basis for a rigorous reconstruction. (Moore was simultaneously sounding what was meant to be, and was taken as, an equally triumphant note with his dictum that goodness was intuitively knowable and indefinable, and the objective basis of all ethics.) In the first chapter of Principles six terms are opted for which cannot be defined by means of anything else, which are logical entities, and which are essential for mathematics. There is considerable static in this area of Russell's thought, and we cannot do more than indicate the main lines. Much might be made out of his shifting from discourse about indefinables (roughly: terms not capable of definition in any system) to primitives (roughly: terms indefinable in a chosen system but perhaps not in another). Something might also be made of the evident haphazardness of the list of indefinables in Principles. It is clear that he was, at least at the beginning, inspired by the idea that there must be some ultimate fixed and finite list of terms indefinable and primitive for any complete exposition of the logic required for mathematics. To locate and explain them, including the way they are employed in the definitions of mathematical terms and in the strategies of mathematical reasoning, is to expound "the

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principles of mathematics," or so he believed.* It is simultaneously to engage in the philosophy of logic, though he did not say quite this. This bold enterprise, indeed the very idea of a philosophical foundation in logic for mathematics, though it had been precisely Frege's idea too, was a new emphasis in the Anglo-Saxon world at the time. It is of course vaguely reminiscent of 'first principles' in Aristotle, and also of 'a priori concepts' in Kant; indeed, the scrutiny of presuppositions for the primary ideas is a venerable philosophical exercise, and in Russell it was directly inspired also by Leibniz's dream of a 'universal calculus' of thought. We can salvage something but not the whole of Russell's vision; he himself had abandoned part of it (namely, the importance of the search for the indefinables) by the time of Principia Mathematica (1910-13, hereafter PM). In any case the idea of a system of logic has to be flashed out somewhat first. The point for now is that it would do Russell's best thought an injustice to put much weight on the fact - a very evident fact it is - that the indefinables of Principles are unsatisfactory in both conception and detail. We just said, for example, that the list was haphazard: in the first presentation (Principles, p. 11) it contains material implication, formal implication, membership, such that, relation, truth - but it does not contain, for example, negation, proposition, class, propositional function, denoting, assertion - all of which in certain respects are treated, within the book, as on a par with the others. We cannot really get the hang of what his logic is up to by studying his list of terms nor what he says about them. (The manuscript of Principles shows the traces of vacillation. The first section, "The Indefinables of Mathematics," was originally entitled "The Variable," and there is evidence that Russell sometimes thought that variable was the only, or the ultimate, indefinable; though it is not in any clear

* Moore's Principia Ethica, also published in 1903, was devoted to the same enterprise for value theory: the fundamental terms (good, duty, beauty - as it turned out) and the few basic principles for the entire discipline were what Moore wanted to elucidate. Retrospectively, it is easy to see that these two volumes and the year 1903 mark the major shift in English philosophical thought, even though very little of either book has survived intact. Russell wrote a favourable review of Moore's book, and later he wrote a series of essays adopting and elaborating Moore's themes, "The Elements of Ethics." (See Chapter 10, below). Moore wrote a long review of Russell's book, though he never published it.

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sense a purely logical idea.) So something upon which Russell laid a very great deal of unclear stress - will not here be taken as seriously as he did, but it must be taken very seriously as indicative of the nature of his theoretical framework. 3 We can in fact cut the pie in a fairly routine way, extracting just what we need from the first three indefinables on Russell's list, thus to link his logic to the logic that grew out of it. A primitive term will be viewed, non-ideologically, as the characteristic idea around which the theory and problems cluster for a certain level of the developed logic. For our purposes there are just three levels of logic to consider, and the required terms are Russell's first three. Thus (1) Material implication is the key idea (technically neither indefinable nor, usually, primitive) for what Russell called the propositional calculus: the logic which treats of propositions as wholes and expounds the system of their interconnections (disjunction, con junction, negation and, of course, material implication). (2) Formal implication, misnamed because misunderstood by Russell, is nevertheless the idea that is central to what he called the predicate calculus; the idea is really that of universal quantification, which will be explained in detail later. (3) Class (or, as Russell divides things, membership in a class) is the idea central to what has become known as set theory ('class calculus' to Russell). For the nonce we simply call these 'levels' of logic, because they form a certain hierarchy, as Frege and Russell were the first to make explicit. The first level deals with propositions as units, the second deals with the internal structures of propositions, the third deals with classes, be they of propositions, terms, or anything else. This deployment of material will organize the remainder of this chapter. Nothing has just now been said that Russell need disagree with, but he would long have disagreed with the implication that we are merely identifying our data and arranging them in an orderly fashion instead of getting to the indefinable metaphysical core of the universe.*

* The above arrangement corresponds, near enough, to that of PM. However in Principles, seven years earlier, when he said "Symbolic Logic consists of three parts" (p. 11) Russell was actually combining the first two above into one level, and dividing the third into two, classes and relations. This suggests that at first Russell saw a larger cleavage between relations

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PROPOSITIONAL CALCULUS: TRUTH FUNCTIONS

4 The first wave of obscure terminology and name dropping has now passed. We can slide smoothly on to the lowest level as Russell and subsequent logicians see it: the propositional calculus, or more generally, the theory of truth functions. The goal is to understand a large part of Russell's outlook by understanding two things, the propositional calculus of Principia M athematica and the idea of material implication. We begin with some fairly obvious things about the theory of truth functions. This may be viewed as an idealization and simplification of certain pervasive features of the logical hookup among propositions as employed in ordinary thought and discourse. One thinks of propositions, in the indicative, as being atomic or molecular, the latter being composed of the former by being joined by connectives such as 'and,' 'or,' etc. The idea is then that the truth or falsity (truth value) of a molecular proposition depends exclusively upon the truth value of its atomic propositions. The way in which it depends is determined itself exclusively by the assigned meaning of the connectives. (a) Chicago is a city and Michigan is west of New York. (b) Some lions roar or Germany lost the war. Both molecular propositions are true because: in (a) each of its atomic propositions is true, and in (b) at least one of the disjoined atomic propositions is true; and these conditions coincide with the expected meanings of 'and' and 'or.' Moreover - and this is the heart of the matter - any true atomic proposition in (a) or (b) may be replaced with any other true proposition without affecting the truth value of (a) or (b); the same goes for replacing a false atomic proposition by any other false one. This is an example of what it means to say molecular propositions are truth functional- a claim which is quite unexciting except when one and classes than he did between the propositional and the predicate calculae. And this is precisely true. In PM (p. 200) but not in Principles relations are defined by means of classes; in PM (pp. 127ff.) but not in Principles the predicate calculus has its own identifying indefinables. The shift in organization represents both technical refinements (to be considered later) and, more broadly, a reversal of priorities from metaphysics to logic.

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of the connectives is 'if . . . then.' Accordingly, the theory of truth functions takes two kinds of things as its data: (1) an array of simple propositions each of which must be regarded as either true or false but never both, and (2) an inventory of propositional connectives - and, or, if ... then, if and only if, plus negation. (All definable in terms of incompatibility: 'not--or not .. .'; or of joint denial: 'neither--nor. . . .') These connectives, though staples of ordinary discourse, have to be sharpened up a bit to prune ambiguities, since formal logic cannot tolerate imprecision. (e.g. and is thought of atemporally, or is taken as and/or, and if ... then, which is material implication, has to be rather drastically streamlined, as we shall see.) In short, the theory is this: it is the truth value of the atoms and the meaning of the connectives which alone determine the truth value of the molecular propositions; as (a) and (b) illustrate, no other connection of meaning between the atoms is relevant. Of course, not all compound propositions are truth functional. For example, the truth value of 'I believe that it is raining' (which can of course be true whether the component 'it is raining' is true or false) is for several reasons not a function simply of the truth values of 'I believe' and 'it is raining.' Therefore, 'that' is not here a truth functional connective. Therefore, too, as Russell emphasized, not all propositional truth can be organized by the theory of truth functions. A truth functional compound (molecular proposition) retains its truth value (whether true or false) if any other atoms with the same truth values are substituted. That is the first point. The second point is that some compounds have a form such that they are always true, and retain their truth value no matter what atoms of whatever truth value are substituted (same for same) within them. Such are called logical truths or tautologies. Thus: (a) If grass is green or grass is green then grass is green is a logical truth because of its form, because of the role of 'or' and 'if . . . then,' because it remains true if we replace each occurrence of 'grass is green' with 'grass is purple' or 'lambs are pink' or any other sentence, of whatever truth value. It thus corresponds to the empty schema (i.e. has the form of) (1) If Por Pthen p

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A slightly more complex example is (b) If grass is purple then grass is blue or grass is purple corresponding to the empty schema: (2) If q then p or q Logical truths are such that the systematic replacing of their atoms by others (same for same) always yields a molecular proposition with the truth value of true. Logical truths, tautologies, convey no information, affirm nothing, deny nothing; they cannot be denied without self-contradiction. They may be immensely complicated and not wear their tautological character on their faces as do the above examples, but if they are genuine tautologies a technique is always available for showing that they are indeed true simply in virtue of the form which the connectives give them. 5 The technique of truth functions is an extension and purification of one segment of ordinary rational thought, applied logic. The schematic structure that underlies it is the system of logical truths, pure propositional logic. One goal of the propositional calculus is now already in view: given the definitions of the connectives, we formulate an initial set of tautologies (primitive axioms), from which other tautologies can be derived by means of systematic rules. Such a description brings to mind the image of a total system which could include, theoretically, all possible logical truths of this kind, an infinite number. This is called the propositional calculus. It meets a minimum standard of being a formal system: it has definitions, postulates, rules, and theorems. Numerous different systems of the propositional calculus are constructible, each covering the same logical truths from differing starting points. Such systems, of immense pedagogical value, are, unlike the applied logic of truth functions, of little practical value. (However, many decades after such systems were developed by Russell and Whitehead, they have proved indispensable for devising intricate electrical circuits, which circuits are used to build computers, which in turn can devise different logical systems, which can help build better computers, which can ....) But they form the basis of any attempt to present the nature of mathematics in terms of logic. They have this role

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not because they have any direct connection with mathematics, but because the rules they systematize and the logical connectives they employ are used by the next two levels of logic, the predicate calculus and the class calculus (quantification and set theory), which do have direct connection with mathematical entities, definitions and procedures. For the propositional calculus of Principia M athematica, Russell and Whitehead, always striving to minimize the basis, selected five axioms plus a rule of inference whose informal statement, the first and most honoured primitive proposition of Principia (p. 13) was this: "Anything implied by a true premiss is true" (called PPl). The axioms were the following logical truths. The first two have already been explained above and the others are explainable in terms of these.

1. 2. 3. 4. 5.

(p v p) > p q> (pv q) (P v q) > (q v P) [pv(qvr)] > [qv(pvr)] (q v r) > [(p v q) > (p v r)]

A few very brief reflections are called for. Why just these premises? "All that is affirmed concerning the premises is (1) that they are true, (2) that they are sufficient for the theory of deduction, (3) that we do not know how to diminish their number" (PM, p. 90). (Their number was diminished to one, much more complicated, premise by Nicod in 1917; and these five to four by Bernays in 1926.) How did Russell and Whitehead know they are true? Well, how did we know that (1) and (2) were true, logically true, a few pages back? It is self-evident. (In 1918 Russell said that they are tautologies: "For the moment, I do not know how to define tautology . . . in spite of feeling thoroughly familiar with the characteristic of which a definition is wanted" (Intro. to Mathematical Philosophy, p. 205).) How did they know they are consistent with one another? Because they did not in fact turn up a contradiction among the theorems. How did they know that they were "sufficient for the theory of deduction," i.e. could produce all truth functional tautologies? Merely because

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they did not, by adding to the axioms, manage to produce more theorems. Such questions are in fact, as it turned out, of great theoretical interest; and such answers ought to seem much too cavalier, for they are. They require and receive far more rigorous attention now than in 1910. We shall report on them in their proper place at the end of the next chapter. 6 Recall that Ppl was "Anything implied by a true premiss is true." This idea as the fundamental notion in logic appears in just this role already in Moore's paper ten years before, "Necessity" (Mind, 1900). For decades Russell was puzzled, rightly so, about the precise relation of this rule to the rest of the system. One aspect of his problem, still intriguing, was memorably presented in an essay by Lewis Carroll, "What the Tortoise Said to Achilles" (Mind, 1895) and is briefly this: A rule cannot be simply added to the axioms as one more premise for deriving the new conclusions, for that would require another rule (rule 2 ) to show that rule1 was the required premise for a particular conclusion, and that would require a new rule3 etc., so that an infinity of rule-premises is required for any conclusion. Thus, on the one hand, no theorem can be derived simply from premises, some other kind of principle to legitimize the procedure is required. But on the other hand, says Russell, if two propositions do imply a third, then the implication holds: if the first two are true the third is, no matter what else is the case. (For example, suppose our premises are 'p' and 'p implies q,' which together seem to imply 'q.' The question is: do we also need another rule or premise, viz. "'p' and 'p implies q' implies 'q' "?) Russell felt this problem as a dilemma. The way out seemed to be to emphasize the distinction of a rule from a premise; but this appears to him to be too easy a way. That is why he inserted the First Primitive Proposition, Ppl above. "Anything implied by a true premiss is true." In Principia M athematica Russell said that this is the principle which justifies and underlies inference. He also said "we cannot express the principle symbolically" (PM, p. 94), and referred to a discussion in Principles seven years earlier which had concluded that the curious status of the principle "points to a certain failure of formalism in general" (p. 34). What is the general nature of this problem that Russell repeatedly identified but did not resolve? It

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is sometimes supposed that he did not sufficiently appreciate that rules have to be thought of as comments about the axioms and theorems. But it is clear (PM, pp. 98, 106) that he is well on to this point. His problem is other than that and twofold. First, his notion is that in logic, though the axioms of the system are to some extent arbitrary, a rule has to be justified; and the ultimate justification is just Ppl. And this principle itself has no symbolic expression; moreover he takes it as a synthetic a priori proposition. This line of thought has not proved to be very popular among Russell's followers. But the idea that a rule stands in need of justification if the calculus is to be in any sense a map of inference cannot be lightly dismissed. Russell's idea was made to look suspicious by his appeal to a synthetic a priori principle that cannot itself be incorporated within the system. What has symbolic expression is the particular rule, phrased as a premise above, an instance of Ppl, which he uses in his deductions, viz. modus ponens: "From 'p' and 'p>q,' deduce 'q'." Secondly, Russell rightly sees that any particular application of this rule (or, indeed, any rule) "cannot itself be erected into a general rule, since the application required is particular, and no general rule can explicitly include a particular application" (PM, p. 98; d. also Principles, p. 41). So what Russell is saying is that two crucial and connected aspects of deduction cannot be reduced to formal or symbolic statement: (1) the justification of the rules by the first primitive proposition, and (2) the particular application of a rule to deduce a particular proposition. These problems are connected - indeed Russell did not clearly separate them at all. We can say that the rule of modus ponens is an instance of the first primitive proposition, and that, in an allied but different sense, a given proof is an instance of this rule. But that this is so in a concrete case is something we can at best see or declare; we cannot demonstrate it without falling into an infinite regress. So formal logic gives us the philosophical question of what it is for a purported proof to be justified by a rule or for a rule to be justified by a principle. Indeed, what is a rule of inference? Russell raised this question, and left it for Wittgenstein to pursue. Does it point, as Russell thought, to a "failure of formalism in general," to a "respect in which formalism breaks down" (Principles, p. 41)? It depends on what your expectations are for 'success.' Russell's realism had initially injected an

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optimism both vague and attractive: formal logic should be an exact model of the logical structure of valid thought. Perhaps there is no such thing. At the very least, Russell has put his finger precisely upon the sore spots; there is something curiously ineffable in formal logic. His appeal to the symbolically inexpressible was picked up by Wittgenstein in the Tractatus where it underlies the central distinction between what can be said and what cannot be said but only shown. Wittgenstein's doctrine that all the propositions of logic are tautologies seems to have amounted at his hands to the complete rejection of the distinction between rules and premises, and a rejection of higher logical principles, as attempts to say the unsayable. "We see from the two propositions themselves that 'q' follows from 'P>q'P'" (Tractatus, 6.1221). And again: "Clearly the laws of logic cannot in their turn be subject to laws of logic" (6.123). For the early Wittgenstein, unlike Russell, the very symbolism of logic embodied within itself its own 'rules' of inference (d. Tractatus, 5.514, 6.1223, and 6.1264). So what Russell finds to be a failure of formalism - roughly, its inability to articulate its own measure of success in its own terms - is for Wittgenstein an occasion for narrowing the limits of logic, and thus the limits of the thinkable. "wovon man nicht sprechen kann, damber muss man schweigen." Russell, following Wittgenstein's advice, was thereafter silent on the point (with one exception to be mentioned shortly); but curiously, Wittgenstein himself was not: his later critical and influential reflections on the phrase "the steps are determined by the rule" in the posthumous Remarks on the Foundations of Mathematics reopened Russell's question in a new way, by mustering a new attack on the ideal that had initially prompted Russell. In his own and different way, not to be discussed here, Wittgenstein came to agree that the inscrutable concept of a rule pointed to "a failure of formalism" to represent the logical structure of thought. Russell anticipated a different line of thought. In his Introduction to Wittgenstein's Tractatus he wrote, with that author's disapproval, of "some such possibility as this: that every language has, as Mr Wittgenstein says, a structure concerning which, in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of

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languages there may be no limit" (p. xxii). What prompted Russell here were not only problems about the status of rules for the propositional calculus, but also what he was calling "the theory of types" - which will be discussed later in this chapter. Hierarchies of language have become central devices in the work of some of Russell's most distinguished followers, most notably Carnap and Tarski. 7 What Russell called, unwisely but now irretrievably, 'material implication,' (symbolized '>' in the axioms above) is a cornerstone on which the entire edifice of his logic is built. It is now, and consequently, so familiar that an explanation of the idea may be scarcely necessary. It is the technical idea corresponding, sometimes, to 'if ... then' of ordinary discourse; or, more frequently to 'never--without ... .' Certainly, no worthwhile logic text now in print and in use fails to treat it, though less than a century ago perhaps no logic text treated it. The idea is not at all new, however, and was not invented by Russell, though its power was first exhibited by Frege, Whitehead, and Russell. The logical properties of material implication (under variuus names) were studied by both ancient and medieval logicians, though before Frege never in the axiomatic way we have been describing. It was Russell's work, anticipated in part by Frege, helped by Whitehead, and refined by Wittgenstein, which made this connective basic to the whole of logic, by making it the central item in the propositional calculus. Russell did not present it in an uncontroversial way. Material implication is the relation between any propositions p and q such that the molecular compound 'p> q' (read: 'p materially implies q,' or 'if p then q') is regarded as false only in case p is true and q is false, otherwise 'p > q' is regarded as true. According to this definition any such conditional proposition is true in case its antecedent is false, or in case its consequent is true, or in case both antecedent and consequent are true, or in case both antecedent and consequent are false. Thus the following compound propositions (it is a question whether they should be called 'propositions') are each regarded as true: 'If New York is in France then China is in Asia/ 'If witches ride horses then cows are bovine,' and 'If Caesar was a Greek then Napoleon was a German.' (It has to be remembered of course that we are

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here speaking merely of truth, not 'logical truth' in the sense explained above.) To the uninitiated, this definition is implausible and baffling in the extreme, and all but unintelligible. To the initiated, intelligibility is the one thing it has, and plausibility, as judged by untrained common sense, is just irrelevant. Those familiar with the elegant symbolic constructions that can be made with this device and other devices defined with its help, can quickly develop impatience with anyone's having scruples about founding a logic on so 'illogica1' a notion. This initial boo jhurrah stage can be quickly transcended: newcomers are taught to use the notation and advised to ask questions afterwards. Afterwards, the questions, now pretty much ritualized, centre upon the so-called 'paradoxes of material implication.' That a false proposition implies anything at all, it is said, is paradoxical, as is the related notion that a true proposition is implied by anything at all. Implication, as normally understood, requires that there be some connection between the antecedent and the consequent; propositions whose meanings are in no way connected such as 'Chicago is a city,' 'the Pacific is an ocean,' cannot imply one another. Yet for Russell each is implied by the other just by virtue of its being true, and each is also implied by 'cockroaches think in Greek' just because this is false. If implication (objective) is what validates inference (subjective) then, it is affirmed, implication is not truth functional; it is 'meaning-functional.' One is then tempted - it is a proclivity of logic textbook writers, and Russell in Principles takes the lead in this - to mollify the qualms by urging that though material implication is not quite the familiar sort of implication it is, as it were, the logical essence of it. And so it is sometimes said: the point about material implication is just that it allows everything except travelling by sanctioned steps from true premises to false conclusions. That one thing only is debarred, and surely it is the most important thing for logic to proscribe. How we get truth is not important for logic; preserving it is. What has to be prevented is getting falsity from truth, hence the special role of material implication. The paradoxes are a small price to pay in unfamiliarity. This is not a very enlightening dispute. The force of the charge of paradox depends upon ignoring the systematic utility of defining an operation in a certain way, no matter how arbitrarily; and

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the defence misguidedly supposes that we get nearer the heart of implication (nearer the explanation of inference) by shearing off several of its properties, and leaving others. But the 'paradoxes' are only paradoxes so long as one sustains an artificial and vague notion of what is going on. If we take Russell's words at face value ("a false proposition implies every proposition"), they are not paradoxical at all. They are false. And if we coax his words into a favourable light ('one clearly definable and immensely useful sense that may, in a formal calculus, be given to "implication," though not the ordinary one, is .. .') then there is nothing mysterious at all. Within the propositional calculus there are no paradoxes. It is therefore confusing and entirely unnecessary to speak of the 'paradoxes of material implication.' Russell is, however, directly responsible for having generated this tempest by supporting an all-chosen terminology with confused arguments.* We shall look at this shortly, but before doing so let us notice that he also has good support for his procedure. The first good argument is that at least part of what is false or paradoxical in his view may be harmless. The license to the effect that a true proposition is implied by anything at all is of no practical use and of no logical harm: if we know a proposition is true we don't have to infer it, and if we don't know that it is true we don't know that we are licensed to infer it. He also believed - a second and more controversial idea - that the stratagem making a false proposition imply anything was also harmless, because inference only proceeded from truth to truth: I can't infer 'Caesar was a general' from the falsehood 'Caesar was a Greek,' but nevertheless I can say 'Jf Caesar was a Greek then Caesar was a general' (i.e. 'Caesar was a Greek' materially implies 'Caesar was a general') if I want to define my terms that way, that is, if I want to call the entire compound true just because the antecedent is false. One way to accommodate this important point is to resolve to speak not of inferring a truth

* Moreover, there is a genuine paradox of material implication, unmentioned by Russell, but very Russellian in spirit, noticed centuries ago by Pseudo-Scotus. This paradox is that the definition of material implication is open to the counter example: 'If grass is green then this material implication is false.' Reflection on this bizarre example shows that whether we take the consequent as true or false we end up with an example that would contradict the definition. Whether this is more than a curiosity is open to dispute, but we shall not dispute it.

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from a falsehood, but of the 'truth of a conditional' with a false antecedent. What this means is not that 'p materially implies q' is about p and q and a relation between them, but that it is about the entire complex composed of propositions p and q. What it says of this complex entity is that it is never the case that p is true when q is false, never the first part of the complex without the second part. This way of phrasing it shows that he is not really offering a definition, and in particular not a definition of implication. Material implication, he said, is an indefinable. But - and this is the third point - if Russell had been quite clear on this he would not have been so inclined (as he was in Principles but not in Principia) to refer to material implication as an indefinable. He would have said that it was defined in terms of 'truth' and 'proposition,' the truth of a conditional by virtue of a false antecedent or true consequent. More likely he would have said: material implication relates any propositions p and q such that, since at least one must be true or at least one must be false, either the true one is materially implied by the other, or the false one materially implies the other. Material implication is truth functional. Lastly, however defectively Russell explained material implication, the real strength of the idea lies in the fact that in the propositional calculus nothing illegitimate can be deduced with its help. 8 We have disarmed one set of criticisms but only by raising and not answering the question of how material implication relates to valid inference. This is a question that interested Russell more than it has some of his successors. Let us be clear about how this discussion relates to what has already been said. In Section 6 above we posed the questions about the justification and application of a particular rule, agreeing that Russell was on to something. In Section 7 we put the 'paradoxes' of material implication in perspective. Now we must raise the question whether a particular inference rule that reflects the idea of material implication is the sort of rule Russell thought it was. The data for this problem are the following. By speaking loosely, we say that whether one proposition materially implies another depends solely on the truth status of the propositions; our intuitions then repel this idea by insisting that there is some important idea of 'implication' between propositions which does not depend at all upon the

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truth value of the propositions. Russell edged somewhat evasively into this situation and hoped, mistakenly, for a compromise. Let us consider what he actually said: "In the discussion of inference it is common to permit the intrusion of a psychological element .... But it is plain that when we validly infer one proposition from another, we do so in virtue of a relation [material implication] which holds between the two propositions whether we perceive it or not" - this against the idealists. He also said in the next paragraph that "any true proposition is implied by every proposition" (Principles, pp. 33, 34) - this against all familiar preconceptions. He reports that he has been "led to conclusions which do not by any means agree with what is commonly held concerning implication, for we found that any false proposition implies every proposition ... " (Principles, pp. 33-4). There is therefore something odd in what Russell is doing, and he knows it. To say that material implication is a convenient device (with some of the properties of implication) for the axioms of a formal system like that of Principia Mathematica is one thing; to say it is the only inferential relation which underlies deductive inference, and hence the rules of the system, may be quite another. We come here to several major issues in Russell's theory of logic. Recall that Russell's essential undertaking is to show that all of mathematics can be deduced from purely logical premises. But if in the course of the exposition he tells us that the deduction or inference rests only upon material implication, it may appear that the deduction of mathematics from logic is entirely pointless: it would appear that if mathematics is true it can be deduced from (i.e. is materially implied by) anything whatever, true or false! What then is the point of making such heavy weather over deducing it from logic? We might as well deduce it from numismatics or phrenology. After all, 'I have square bumps on my head' materially implies '2 plus 5 = 7.' Russell does not consider the general threat - the threat that using his logic as an escort into the foundations of mathematics can appear suicidal for his own programme. One reason why he did not is doubtless that he took with total seriousness his claim to be providing the logical basis for our belief in mathematics. It is imagined that we are not logically entitled to believe that mathematics is true until its connection with the logical E

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foundation had been exposed; so in that sense we could not know until afterwards that anything in mathematics was materially implied by something else. Notice that to say this is simply to apply the earlier point (p. 127 above) about the harmlessness of one of the 'paradoxes' of material implication. This reflection may keep at bay the general worry about Russell's entire enterprise, but it does not help to show that nothing more than material implication is required as the basis of valid inference. It is this latter question that Russell does address, referring in Principia to his earlier discussion in Principles. What Russell does, Whitehead following him in this, is to try to relieve his scruples about material implications by introducing a new notion, formal implication. This notion, though it is prominent in both Principles and in Principia is rather confused, and students of Russell have been reluctant to sort it out. They have gone along with one technical aspect of the idea - and there can be no quarrel with this - but important philosophical complications have been thereby ignored. The fact is that formal implication is precisely a fusion of three perfectly sound ideas, confused by Russell's philosophical preconceptions. Russell was pioneering in unexplored territory; it is not surprising that we can see the cleared terrain more easily than he could. To explain all this would be difficult and tedious, but some things can be said here, and they will go far toward making Russell's outlook clear. What Russell often said was that "formal implication asserts a class of [material] implications" (Principles, p. 38); and the central point to see is that in saying this he does not distinguish the ways in which inference may appeal to (a) a general rule, (b) universal quantification, (c) a connection among meaningscalling all these formal implications. This obscures all the relevant issues; but we can sort them out in stages. (a) Russell wrote: "It would certainly not be commonly maintained that '2 2 = 4' can be deduced from 'Socrates is a man,' or that both are implied by 'Socrates is a triangle.' But the reluctance to admit such implications is chiefly due, I think, to preoccupation with formal implication, which is a much more familiar notion, and is really before the minds, as a rule, even where material implication is what is explicitly mentioned" (Principles, p. 34; my italics).

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Here we must tamper and interpret. What he means, the context makes clear, is that a perfectly general rule, say modus ponens, serves as warrant for any particular deduction of 'q' from 'p' and 'p > q.' The rule in effect says that for any propositions, p, q, so related, the deduction holds. He refers to this, very misleadingly, as "a particular instance of some formal implication" (Principles, p. 34). ("Formal implication ... is involved in all of the rules of inference" - p. 40.) But he also uses 'formal implication' in a quite different way, where the generality applies not to propositions, as does an inferential rule, but simply to terms. (b) The illustration of this second sense of formal implication goes as follows: (1) 'Socrates is a man' materially implies 'Socrates is mortal.' Now, says Russell, " 'Socrates' may be varied" for (2) 'x is a man' materially implies 'x is a mortal,' for all values of x. And this latter is equivalent to (3) 'x is a man' formally implies 'x is mortal.' In symbols: (x) (Px > Qx). The equivalence of (2) and (3) here serves as Russell's definition of formal implication in this second sense. He then raises the obvious problem: "But, it may be asked, how comes it that Socrates may be varied in the proposition 'Socrates is a man implies Socrates is mortal' ? In virtue of the fact that true propositions are implied by all others, we have 'Socrates is a man implies Socrates is a philosopher'; but in this proposition, alas, the variability of Socrates is sadly restricted. This seems to show that formal implication involves something over and above the relation of implication, and that some additional relation must hold where a term can be varied" (Principles, pp. 38-9). The idea on which Russell has a very unsteady grip is that 'is a man' and 'is mortal' can be related in the same kind of way that 'p' and 'q' are when 'p materially implies q' is true; that is, never the first of the pair without the second. He calls this a 'formal implication.' But he loses sight of the fact that so to relate them (to assert a class of material implications) is only to make a general assertion, sometimes true, sometimes false. In no case does it indicate an additional relation in formal implication. True,

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what he calls the 'variability' of the term 'Socrates' bespeaks a kind of generality, at least in this example. But what does it have to do with implication? Not much. It has no more to do with implication than it has to do with conjunction or disjunction. If this is not evident in itself it can easily be made so by expressing either (2) or (3) in its equivalent form where the symbol for material implication does not appear. (4) For no x is it the case that x is a man and not a mortal. [In symbols: ---(3x) (Px· r-'Qx)] The point so far is this: there is nothing wrong in Russell's logical strategy, but there is nothing right in his explanation. His example, badly chosen, turns upon the facts, first, that 'All men are mortal' is a general proposition, and, second, that it is misleading just because it can be interpreted as a necessary proposition like 'All men are male,' and not merely an empirical proposition like 'All men are born free.' In the passage above Russell senses that something is amiss, and this is why he postulates an additional feature in the idea of formal implication. We come thus to the third idea to be separated out, after which we can tie up the strings. (c) What is the "something" besides material implication and besides formal implication in the sense of (a) and (b) that is sometimes involved? Russell concludes that in these cases there must be a special relation holding between the non-variable segments, the predicates ('is a man,' 'is morta1'), of material implications. He does not see that this notion - call it a 'meaning connection' - is tailor-made for his particular example, and has no general validity at all for formal implication in sense (b). Russell is partly misled by his own bad example. Yet he repeats the example and the confused argument in Principia Mathematica (p. 20). So what are we to say? Clearly we have three entirely different logical notions, all of them funnelled into the idea of formal implication. We have (a) the generality of a rule for propositions, (b) the generality of a term within a given proposition, that is, the assertion of something for all values of a variable, (c) a connection among meanings. There is no hope of defending Russell's belief that these are one and the same logical indefinable. Why did Russell suppose they were and what effect did this have? Three comments are enough here.

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First, let us confine the initial sense of formal implication, (a) above, to the area where it belongs. The idea there is just the idea of a rule, and specifically the rule Russell has in mind is the rule of modus ponens. We have already discussed the problems connected with justifying and applying this rule; and the fact that its status in these respects was somewhat unclear was doubtless partly responsible for Russell's confusing it with (b) and (c). Indisputably, this rule can be connected with the process of inference, and one theory about its nature, namely Russell's, is that this rule, stated in terms of material implication, is, merely in virtue of its unrestricted generality, sufficient for the propositional calculus:>:< Russell's theory about the status of this rule may be disputed, but it cannot really be disputed that this rule has nothing to do with what he calls formal implication in sense (b). Secondly, there is a familiar and important distinction between general propositions which just happen to be empirically true ('all my brothers are married') and those which are necessarily true ('all my brothers are siblings'). Yet Russell's logic, by silence and assumption, denies ultimacy to any such distinction; and it may be recalled from the previous chapter that this position is attributable to a larger system of ideas in his metaphysics. So the old problems are still hurting. One effect of this was Russell's choice of example, 'Ail men are mortal,' a rare case that straddles the distinction, and thus confuses the proper idea of formal implication, which is sense (b), with (a) and (c). Thirdly, Russell evidently has a bad logical conscience about material implication. It is clear that material implication and truth functions generally do not record logical relations holding in virtue of the meanings of propositions or predicates; but our intuitions assure us that some logical connections do hold in virtue of these. Russell, presumably having the same intuitions, is willy-nilly found trying to reinstate these connections of meaning, and reassert their relevance to inference under the now ambiguous rubric of formal implication. (See above quotation from pp. 38-9 of Principles.)

* There are places where Russell is less clear than this suggests. Modus ponens shapes up as the Rule of Detachment (PM, p. 95) without being called a formal implication. And there are parallel remarks on pp. 34-5 of Principles. That the operation known as "detachment" just is the result of modus ponens (stated in terms of material implication) was made plausible to Russell partIy by the presiding status he had assigned to PpZ, "Anything implied by a true premiss is true."

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It is important to emphasize that the propositional basis of Russell's logic is in fact entirely truth functional, that the intrusion of this obscurely postulated 'relation' among predicates is entirely alien to the basic thrust of the theory. The upshot is that we have to reject completely the notion that formal implication is a special kind of implication, whose presiding presence mitigates the novelty of material implication. To summarize: formal implication in the sense of (a) is just the general rule of modus ponens; in the nuclear sense of (b), though misnamed, is a very important and very different idea; it belongs to the next level of logic, the predicate calculus, and has nothing specifically to do with implication; it is the idea of universal quantification.* Formal implication in the sense of (c) above, has no place in Russell's logic; it looks to a different sort of logic entirely, namely modal logic. 9 Where have we now arrived? A swift survey of the scene from the level of the propositional calculus would now look approximately like this. Russell says boldly: "The relation in virtue of which it is possible for us validly to infer is what I call material implication" (Principles, p. 33). There is a hint of bluster in this. Indeed, in the next but one sentence he says: "The relation holds, in fact, when it does hold, without any reference to the truth or falsehood of the propositions involved." In one sense, Russell believed, or at least hoped, that material implication was a stronger connective than it is - which is not to deny that in another sense he knew perfectly well just what sort of connective it is. But only much later, in Introduction to M athematical Philosophy (1919), did he clarify - if not entirely vindicate - his intended meaning: essentially that the use of material implication in inference does not appeal "to the truth or falsehood of the propositions involved." He recognized in Principles that for systematic purposes material implication had to be sup-

* There is just one place where the right light breaks sharply through Russell's discussion. He says suddenly: "It is important to realize that, according to the above analysis of formal implication, the notion of every term is indefinable and ultimate" (Principles, p. 40). Indeed it is. Here is the truth of the matter in shining contradiction to his making formal implication itself an indefinable even while loading it with extra meanings. Here in Russell was born an idea (an idea long since born and buried in Frege and Peirce) that grew up to be called the universal quantifier.

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plemented, somehow, with the idea of universality; but there was no way to install in his system the more venerable logical notion of necessity. Yet, discussing inference in Principles, he brought in, in italics, the neighbouring notion of therefore (pp. 14, 35) and contrasted it with material implication; and though he wrongly linked this with formal implication (because both were confused with the general rule modus ponens) he at least indicates that he is thinking that modus ponens involves a stronger connective than material implication. He reverts, however, to his claim that this "cannot be expressed symbolically." His notion of therefore in Principles was thus ancestor to the first primitive proposition, Ppl, of Principia. In a word, then - and with this I let this vexed subject drop - Russell both fathered and repudiated the later discussion of entailment and modal logic. More generally, Russell believed, or hoped, that his calculus would exhibit the essence, or at least the foundations, of logical structure, both as it is appealed to in valid reasoning and as it provides the basis for mathematics. To build, with precisely defined terms, a calculus which at innumerable points has formal analogies with valid reasoning, is quite isomorphic with it at other points, and systematically departs from it for good reasons and i:o. various ways at other points - this is one sort of thing, and that is a fair description of the propositional calculus of Principia and similar systems; but it is not exactly the image of things Russell has in mind. His formal logic is supposed also to be a correct map of the exact logical structure of valid thought and discourse, and part of the first volume of Principia is an exact rebuilding of the main lines of that structure, brick by brick. This tension between the abstract ideal and the concrete achievement is not surprising. Russell's work transpired at historical crossroads: the ideal of logic in the traditional sense as a canonizing of the forms of valid argument was giving ground to that ideal of logic in the newer sense which is exhibited in axiomatic systems. It was Principia M athematica that gave to this newer view a major impetus even while its authors did not altogether relinquish the older view. An alternative, somewhat oversimplified, way to describe the situation is this. By resting upon truth functions, universality, and classes (instead of upon meaning, necessity, and properties) Russell's logic aspires to be 'extensional'; and then the natural

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thing to say is that it has turned out that less of the field of logic than he expected can be accounted for on this basis. For just this reason his work inspired by reaction the development of modal logic, the first such systems (by C. 1. Lewis) appearing within the first decade after the publication of Principia M athematica. Such logics presuppose a distinction between the universal and the necessary, and they take as their basis neither material nor formal implication, but rather the relation which is called variously 'strict implication' (Lewis) or 'entailment'; which relation is thought, surely correctly, to be a much closer approximation to the relation which justifies valid inferences than is material implication. Modal logic (and intensional, meaningbased logics generally) has none of the elegance or simplicity or range of Russell's logic, and it is not without puzzles and paradoxes of its own. Even its central term, 'entailment,' though intuitively obvious, is, quite unlike Russell's 'material implication,' peculiarly resistent to satisfactory definition, and its formal manipulation is a topic of unceasing controversy. For there is yet, two millennia after Aristotle and a half century after Whitehead and Russell, no systematic and universally acceptable account of this connection: 'p follows from q'; no agreement, therefore, on what the formal pattern supporting correct inference IS.

Nothing said here must be thought to reflect adversely on the deductive power and depth of the propositional calculus of Principia, nor on the role, within the calculus, of material implication. But it does affect the interpretation which Russell gives to his procedures and results: his system is more narrow and limited than he imagined, less penetrating as a diagnostic tool for valid reasoning, but systematically both deeper and purer than anyone might have anticipated beforehand, and, what was of greater importance for Russell, adequate for axiomatizing the first level of the logic of mathematics. A brief, if slightly pejorative, way of putting all this in the perspective of Russell's early metaphysics is this. If, as he believed, no properties or relations are internal, the logic which reflects reality is free to be built upon conjunctions, disjunctions, and generalizations, not upon necessary bonds and connections of meaning. Russell's extensional logic, built on material implication, is precisely the counterpart of his metaphysics of external relations: it rests upon the

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coincidences and not the internal connections of things; it cares more for truth than for meaning, and more for a generality than for necessity. C.

THE

PREDICATE

CALCULUS:

QUANTIFICATION

10 It was mentioned earlier that one commonly dates the beginning of modern logic at 1879, the year Frege presented quantification theory, the narrow base of which is essentially along the lines sketched below, familiar in all logic texts today. The expression 'quantification theory' is now normally used not merely in the narrow sense to refer to the way the quantifiers are used to diagnose propositions, but also in the sense of formal systems which, incorporating the strategies of the propositional calculus, have rules, axioms, proofs, theorems and the rest. In this chapter, however, I use the term in the narrow sense, also in the historical remarks, and consider the formalized sense only in the next chapter. The narrow sense is more important for philosophy, the broader and official sense is of course more important for formal logic. This conforms to the policy of trying to avoid material that has become standardized in texts, except such as is important for our philosophical purposes. In any case, most of what follows has more right to be called 'theory' than do the 'results' of the textbooks. Frege's technical breakthrough passed largely unnoticed at the time, curiously victimized by the clarity and novelty of the attending theory, and remained virtually so for almost twenty-five years until discovered, too late, by Russell. Later the larger world realized, what some few had immediately known, that another lonely genius, the American Charles S. Peirce (who introduced the word 'quantification'), had made the same breakthrough at about the same time as Frege. Ironically, Russell came to understand and appreciate Frege's work on this head only after he himself had made a similar breakthrough, from a different direction, and now for the third time. But Russell's method of arriving at what was then Frege's and Peirce's way (and to a limited extent also Peano's way) of using quantifiers in the analysis of propositions was quite different from theirs, more attentive to ordinary use, more metaphysically encumbered, contained insights theirs obscured, and is more difficult to describe

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in detail.* We shall attempt to excavate in three layers Russell's way of arriving at what came to be called quantification theory. But before proceeding to this task, it is wise to indicate exactly our point of arrival by summarizing, textbookwise, the technique and explanation of quantifiers. This may be done briefly as follows: Quantification theory is interested not in what propositions are about, but in how they work. How they do work is largely determined by how certain prefixes, or surrogates for them, combine with other fragments to form propositions. These prefixes may be thought of as 'operators' on propositional functions. The most important of these are 'some' and 'every,' and these prefixes are in turn considered interdefinable by means of negation. Roughly, this way: 'every F .. .' is equivalent to 'not some F is not ... .' This can be rendered more perspicuous by paraphrasing respectively into the jargon 'each' and 'there is,' and then adding 'such that,' employing connectives from the propositional calculus as needed. The following examples (which, notice, employ the > of material implication) are thus seen to be equivalent, both being canonical renderings of 'all F's are G's.' (1) (x) (Fx > Gx) i.e. 'Each x is such that, if it is F then it is G.' (2) ---(3x) ---(Fx > Gx) i.e. 'There is no x that is not such that, if it is F then it is G.' Alternative readings - 'for any x' and 'for some x' - are less clear but obviate the 'such that.' (Here is another point over which Russell wavered, wondering whether 'such that' was an

* The early history of quantification theory remains to be written. The exact knowledge which logicians, in the 1890s, had of each others' ideas is not well understood. Russell, for example, in the Preface of Principles, credits W. E. Johnson with "many useful hints," but he goes unmentioned in the rest of the book. Johnson's papers in Mind in the 1890s (read by Russell?) actually employ quantification in more modern form than Russell's, and that is partIy why in ensuing pages we probe Russell's philosophical preconceptions to explain his particular (denoting) version of quantification. Also, it is not well known just what kind of circulation was enjoyed by the important volume Johns Hopkins Studies in Logic (1884) by Peirce and others. Peano's and Frege's exchanges, and Russell's knowledge or understanding of them is also an important and unexamined part of the story. The monumental study by W. and M. Kneale, The Development of Logic, throws no light on these historical questions.

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indefinable.) The symbols for the prefixes, (x) and (3x), minus the negatives, are called 'quantifiers,' respectively universal and existential. The quantifiers are said to 'bind' the variables (the appearances of x) within the propositional function (viz. Fx> Gx), and, so bound, the variables then work within the whole proposition to make cross references somewhat in the manner of pronouns. (Russell spoke here of changing a 'real variable' to an 'apparent variable.') Thus in (1) 'it' refers back to the 'each x.' In both cases the quantifier is also thought of as effecting the primary logical link between language and reality. (This is what will be crucial for Russell's particular variation.) Briefly, the quantifiers pick out the topic of discourse; and the predicates, here F and G, discourse about them. When there is more than one topic of discourse the order in which they are picked out may be important, and may be reflected in the arrangement of quantifiers. This iS1 then, when suitably elaborated and extended, the system of ideas employed in all Russell's logical writings after 1905. Though the use of the quantifiers in making deductions is their practical value, I shall here be primarily concerned with what is involved in their use to plot the meaning of propositions. Propositions are pictured above as analysable into fragments of two sorts: prefix and remainder. The remainder Russell called a propositional function. As to the prefix, the questions are: What do they range over, stand for, refer to - if anything? What of propositions with several such prefixes? Accordingly, the three ideas (discussed in Sections 11, 12, and 13, immediately below) which we have to get hold of in order to understand the theory are: (a) propositional functions, (b) the range of variables, (c) plurality of variables - each a fairly comprehensive story in its own right, and each containing ideas relevant to Russell's later philosophical theories. 11 'A propositional function is anything containing one or more variables which becomes a proposition when the variable is replaced by a constant.' We met this idea before, in Chapter 2, where we pretended that it was quite a simple notion, though its role in Russell's metaphysics was a bit obscure. It is obscure just because it refers to Russell's propositions which, to summarize gently, were problematic. But we can ride on the illustrations

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as Russell did. Thus 'x is a man' can become by appropriate replacements 'John is a man' or 'Smith is a man,' 'somebody is a man,' etc. Obviously things become more complicated when we link several propositional functions by one of the truth functional connectives or when we wish to replace the variables by some denoting phrase other than a name. But the notion of a propositional function seems more or less clear at the outset: "a kind of schematic representation of any proposition of a certain type" (Principles, p. 29). Perhaps it is best to face, and then dismiss, the inclination to dwell on the question of the status to assign to this entity, the propositional function. Propositions for Russell, we remember, are metaphysical entities, so a propositional function looks rather like a proposition with a hole in it, a fragment of the real world. This will be pretty hard to accept just like that unless we find some innocent version of it. For the present we can acknowledge, and try to avoid, the tension between a metaphysical and a linguistic entity that seems to be built into the very conception of a propositional function, though it must be admitted that this ambivalence generated a certain confusion in Russell's writing for many years.* The way to keep some of the problems at bay, Russell divined, though he did not say it, was to think of the .

* From the point of view of his realistic metaphysics, one can see how problems may creep in because of the kind of company, linguistic and otherwise, propositional functions will keep. Associated with each is a variety of kinds of entities, identical or connected in some other intimate way. For example, grammatical predicates (which can be true of just the same subjects without being the same predicate; e.g. 'is a man,' 'is a featherless biped'). Also for each there will be a class of objects composed of just those entities which can fill up the variable gap to produce true propositions. Also there will be for each function a property possessed by each member of the class of objects just cited. Another entity, seemingly different from the foregoing, is the class of true propositions which each propositional function is associated with. And so on. The early chapters of Principles agonize a great deal over these intimate relationships. An adequate account of propositional functions has to come to terms with all these allied entities, making clear what is involved in the temptation to identify it with any or all of them. Russell was aware of all this; and he knows that his views, through the years, have not been consistent on these topics, and perhaps cannot be made so. But his inclination, in distinction from his achievement, is clear: it is to transpose ontological questions into logical questions by identifying with a propositional function any entity of which it is not demonstrable that this is impossible. By the time of the "Introduction to the Second Edition" of PM (1925), predicates, classes, relations, properties had

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functions for present logical purposes as far as possible along linguistic lines. The first level of logic, propositional calculus, deals with propositions as units intact. It seeks to find a systematic way of organizing all the relations holding externally among propositions and groups of propositions in virtue of the propositional connectives, 'and,' 'not,' 'or,' 'if-then,' 'materially equivalent.' It is thus a more circumscribed and more intricately developed region of study than, for example, classical syllogistic, which derives from Aristotle. The second level of logic according to the Russell system, the predicate calculus, which includes syllogistic as a kind of minor subdivision, is devoted to systematizing the logical connections among propositions that hold also in virtue of their internal structure. And this is where propositional functions come in, for they provide the most primitive schema for the internal analysis of propositions. A propositional function is a proposition with one or more gaps in it, we said. The essence of quantification theory, and of what Russell was calling at this time "the theory of denoting," is the devising of a systematic way of so filling the gaps that the result can express any and all of the possible propositions of the predicate calculus.

12 'All men are mortal.' This is a universal affirmative proposition. Are all such propositions really categorical, or are they conditional ('For anything whatever, if it is a man then it is mortal')? This has often been a bone of contention among philosophers, and was never more actively gnawed than at the turn of the century when every writer on logical topics, idealist, pragmatist, realist, mathematicist, managed to get in his licks. Quantification has recast this problem, and the advantage of understanding these propositions as universally quantified conditionals (by means of a quantifier and material implication) is acknowledged on all sides. Russell himself - who took his first logical problems from Bradley - had defended the categorical interpretation in his 1900 book on Leibniz and thus gave himself all in their various ways capitulated. Less pejoratively put: the merit of the idea of a propositional function, as Russell saw it, was that it provided a vehicle, essential to logic, for abstracting from ontological questions about the nature or existence of classes, propositions, properties, relations, and the rest. He devised in this connection his important 'principle of abstraction' which will be taken up later.

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reason not only to reject but to argue against the categorical view when he came round to the conditional view. It will be valuable to pick up some pieces of just one argument of Russell's, for it illuminates the idea behind the universal quantifier. Russell says: "'x is a man implies x is a mortal for all values of x.' This proposition is equivalent to 'all men are mortal. .. .' " Our question is, How is the topic of discussion picked out in this analysis? and his answer is, "It may be doubted, to begin with, whether 'x is a man implies x is a mortal' is to be regarded as asserted strictly of all possible

terms, or only of such terms as are men. Peano, though he is not explicit, appears to hold the latter view. But in this case, the hypothesis ceases to be significant ... " (Principles, p. 36). Russell's argument is this: If 'all men are mortal' is to be analysed as a conditional (the latter being represented by material implication), it has to employ a variable x, asserting something of all values of this x. But then the x cannot only range over (i.e. take as possible values) men, for that would make nonsense of the residue. The residue, which is a propositional function, and which is equivalent to 'is a mortal when a man' was to be asserted of x; but, says Russell, if x refers only to men the whole thing 'ceases to be significant.' What he means can be put this way: 'each man is such that if he is a man then he is mortal' is redundant, so that cannot be the correct logical account of the matter. Hence we have to say: 'each x is such that if it is a man it is mortal.' And this is Russell's conclusion: "We must, therefore, allow our x ... to take all values without exception ..." (p. 38). The reason Russell pauses to argue for this view is, first, that he was inclined to suppose that there has to be one correct analysis; and second, that this analysis, though it did not originate with him, can seem farfetched, even if today familiarity has bred consent: in 'all men are mortal' the original subject has disappeared, a variable has been inserted, and two predicates ('is a man,' 'is a mortal') are affixed to the variable. All this cuts against the grain of much pre-twentieth century textbook doctrine in logic. Whereas that traditional doctrine usually took 'all men are mortal' to involve or be about the class composed of all men

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(and about another class composed of just some mortals), Russell's interpretation, like Frege's, discards all this in favour of the use of a propositional function and a variable of unrestricted range, with assertions predicated of it and linked truth-functionally. The 'subject' is replaced by a variable and two predicates. This is a crucial element in the idea of quantification theory, for this is the universal quantifier in action. Fully elaborated it means that there is no room for such questions, staples of textbooks for centuries, as the following: In the propositions 'all men are mortal,' 'some cows are animals,' 'no dogs are reptiles,' what do the phrases 'all men,' 'some cows,' 'no dogs,' respectively stand for, or what do they refer to, or name, or denote? Is 'no dogs' about (name, refer to, denote) just dogs, or all dogs, or no dogs? Such questions, and many more, with their varying degrees of intrinsic plausibility and nonsense are in principle swept away by the device of regarding erstwhile universal categorical propositions as universally quantified conditionals. The variable x in the analysis is of unlimited range; it does not stand for, name, or denote anything at all, but is a device for cross-reference among predications (e.g. 'is a man,' 'is morta!'). It is more like a pronoun than anything, but it is really like nothing else at all. The proposition as a whole is not thought of as being subject-predicate in form. In the previous chapter we saw Russell deplore the subject-predicate form as having pernicious influence in metaphysics; here it has pernicious influence in logic; and a similar point will be made in each of the next two chapters. Three brief comments are called for. First, we just now said that certain questions were 'in principle swept away.' For Russell in 1903 these questions were not in fact swept away - with consequences to be investigated shortly. The reason they were not is that certain metaphysical assumptions which go naturally with his realism had a deep hold on his mind, even though this was incompatible with his conception of propositional functions. Second, it is being assumed here, and defended by Russell, that if this interpretation holds for universal propositions, an analogous account (propositional function plus quantified variable) is to be given of the other standard propositional forms deriving from the Aristotelian tradition. For example, 'Some men are mortal' becomes " 'x is a man and x is a mortal' is true for at least one value of x." There are ensuing problems for the

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aristotelian Square of Opposition which Russell did not tarry over, but which have been a sub-topic of active discussion to the present time. Third, we have just been explaining what Russell called formal implication (in the sense of (b) disentangled in Section 8 above), which we have argued was only confusingly said to be a kind of implication; it belonged, we said, to the predicate calculus. The correct picture of things may be seen in such of Russell's statements as this: "according to the above analysis of formal implication, the notion of every term is indefinable and ultimate" (Principles, p. 40). 'Every' was not yet made interdefinable with 'some' - as it was first for Russell in 1905 (see Chapter 5).

13 The traditional formal logic associated with Aristotle gets by with a large number of very gross distinctions: it does not systematically discriminate among 'at least one,' 'some,' 'a,' 'most,' 'a few'; neither does it logically separate 'each,' 'any,' 'all,' 'every.' Where syllogistic is thought of as the principal formalization of argument, this rather clumsy way with fine distinctions will seem not to matter. But in other contexts it will. Problems resulting from this were recognized and discussed by medieval logicians, but no systematic doctrine emerged, and indeed most of their ingenious insights were lost from the fifteenth to the twentieth century. Consider an example of this: 'any' does not always make the same contribution to the meaning of the proposition containing it as does 'every.' Though there are many cases where the difference is merely rhetorical, there are others where it is logical, and which an adequate theory should accommodate. 'Aristotelian' logic with its famous doctrine of 'distribution of terms' (by giving distributed subject terms to all propositions prefixed by 'any' and 'every' and 'all') cannot evaluate a fallacious inference such as this: 'Any starter can win the race, so every starter can be a winner.' An attentive reader unstudied in logic may discern that whereas the premise says in effect that of each starter it is the case that he may win the race, the conclusion says in effect that it may turn out that every starter wins; and to analyse the blunder in this is to show that it makes a difference how you treat the 'any'; what, in grammatical terms, you take it to be 'modifying.' A rather subtle point, it is of enormous consequence for logic because it is - as has finally

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emerged - just one fragment of an entire logical framework, capable of precise systemization and extension to unimaginably wide contexts. The logical notion involved here is what Russell called "the scope" of the quantifier. Let us turn to that. Something of what was wanted at the turn of the century when Russell, Peirce, Frege and others got independently interested in these questions, may be suggested by this example. Consider the ambiguous proposition: I.

A paid crook voted for every Democrat

which leaves it unclear whether reference is to just one crook, or to one for each Democrat. What was wanted was an analysis of the ambiguity involved, an account of the terms involved, and an account of the logical relations between the different propositions involved. What follows is a summary of Russell's eventual diagnosis. It sorts out the ambiguity, turns it into symbols, and then back into alternative forms of the stiffened English: (1) Some paid crook voted for every Democrat, i.e. : la (3x) (y) [Cx· (Dy

> Vxy)], i.e.:

1b Some x is such that, for any y: or There is an x for any y such that:

J x is a crook and if y is a

Democrat, then x voted for y.

(2) Every Democrat was voted for by some paid crook, i.e. : 2a (y) (3x) [Dy

>

(Cx· Vxy)], i.e.:

2b Any y is such that, for some x: ) If y is a Democrat then x is a crook and voted or For any y there is an x such that: for y.

As we already know, the quantification layout of this example discards, by its use of variables, the questions about the subject terms; and it is designed to show that as a matter of logic (2) follows from (1), but not (l) from (2). Just one crook is alluded to in (1), and any number of them in (2). This diagnosis, which is the one that would be followed by Russell when he had fully articulated the theory, is not ad hoc, but employs ideas of much wider utility. The essential point here, however, is that two

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quantifiers (operators) are used and it is their order (what Russell called "scope") that is technically important. So much for logical technique; but Russell's explanation of this analysis (which he called "the theory of denotation") did not in 1903 coincide with the explanation that would be given today in a standard textbook. Considering his actual explanation will highlight the connection between Russell's logic and his epistemology, and also make clear the fact that he is arriving by an importantly novel route at what was to become the standard technique. There are two novel features in Russell's explanation, one metaphysical and one logical. The former is along these lines: he tends to expound the features of such terms as 'some x,' 'an x,' 'every x' in terms of what entities or classes of entities are "denoted" by these expressions. He is thinking in the old realistic way of each expression standing for an entity; in the case of these denoting concepts (quantifiers) he says the entities represented are not in the propositions, only the concepts themselves are. This is a vivid example of what in the previous chapter I called giving metaphysical explanations for logical points: more on this later. His second novel feature is logical, and though it too is expressed metaphysically, I shall not do so. The essential point is this: Russell's theory of denotation (1903) may be characterized as one which envisages, in place of the now standard two quantifiers (universal and existential), three independent operators. His three operators correspond respectively to the English terms all (i.e. 'each and every'), a (i.e. 'some or other'), and some (i.e. 'some particular'). Simple illustrations showing what problems seem to have arrested Russell's attention are the following: (3) I read every book (each book). (4) I need a book (some book or other). (5) I saw a book (some particular book). Now the difference between the role of 'a' in (4) and (5) may be further brought out by considering the 'truth conditions' of each, that is, the statement of which minimum conditions must obtain in order for each to be true. The truth conditions of (5) are just that there be at least one particular book seen by me (i.e. of some specific book, it is true that I saw it); for (4) we may, for example, suppose that I lack a doorstop, and the truth condition is just

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that some book or other is needed (i.e. it is not true of some specific book that I need it). Thus there is an ambiguity in the word 'a' which the parenthetical expressions above distinguish. Now return to proposition (I) above ('A paid crook voted for every Democrat'), where it is immediately evident that its article 'a' has a similar ambiguity, and this ambiguity is diagnosed in (1) and (2) in terms of two different roles of the word 'some.' Conclusion: 'some' can be ambiguous in just the way 'a' can be. In fact we could replace 'a' in (4) and (5) with 'some,' and in (1) and (2) replace 'some' with 'a' and nothing would be disrupted. In (1) the meaning is 'some particular' and the truth condition holds that just one crook did all that voting; in (2) the meaning is 'some or other' and the truth conditions may be met with one or with many different crooks voting. That is why (2) follows from (1), but not the reverse. This shows again that 'a' and 'some' have parallel ambiguities: (1) coincides with (5), and (2) with (4). Next conclusion: There are two logically different meanings and two words, but both words bear both meanings; moreover, any confusion deriving from this is intensified as (1) and (2) illustrate, by the presence of other prefixes, such as 'any' and 'every.' These then are our data. Russell seized upon these seemingly trivial facts 1 noticed their general importance and gave them in his theory a profundity one would not have expected. The resolution in quantification theory of the ambiguity just described is, as (1) and (2) show, by means of what is called the scope of the quantifier (essentially: the order of the prefixes), and the analysis turns on the question of which quantifier appears within the area governed by the other, for just this is what assigns the appropriate meanings to 'some.' Roughly, this amounts to the difference between Something is such that anything . Anything is such that something . . . And the context brings out the difference: in the first case 'something' amounts to 'some one particular thing'; in the second case 'something' amounts to 'something or other.' The question of scope, or order, is of crucial importance for

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the systematic development of quantification theory. It relates in this way to the theory of Russell's here being examined: when an existential quantifier appears to the left of a universal quantifier it naturally makes a specific (some particular) reference, thus (1) and (la); it was this meaning of 'some' or 'a' which Russell reserved for 'some.' When the existential quantifier appears to the right of a universal quantifier it naturally makes an unspecific (some or other) reference, thus (2) and (2a). This meaning Russell reserved for 'a.' These restrictions were not arbitrary. I have been saying that 'a' and 'some' can have the same ambiguity; Russell saw that these words often record a distinction, recognized but hitherto unsystematized. He illustrates cogently as follows: "This brings us finally to some X, the constant disjunction. This denotes just one term . . . but the term it denotes may be any term of the class. Thus 'some moment does not follow any moment' would mean that there was a first moment in time, while 'a moment precedes any moment' means the exact opposite, namely, that every moment has predecessors" (Principles, p. 59). Symbolically expressed (assuming that moments are not simultaneous), with 'x' and 'y' denoting different moments, and 'P' as 'precedes,' we have logical distinction that matches the prose distinction laid out above: (3x) (y) (xPy) (y) (3x) (xPy) None of this helps in the analysis of (4) and (5), which we said had the same ambiguity, this time on 'a.' Since they do not have two quantifiers in their analysis, we cannot sketch the difference in terms of quantifiers appearing within each other's scopes, as we could with (1) and (2) and with Russell's example. I shall not now discuss the ways in which present-day logic handles (4) and (5) for there is not yet agreement either on technical strategy or rationale, even with examples so innocent looking as these. Instead we can now state briefly what Russell did with examples like (4) and (5) and then turn this method back upon (1) and (2). His method has a certain simplicity and naturalness which is not matched by any other theory.

14

We have said that he had in effect three quantifiers

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instead of two: the first is the universal quantifier (based on 'each'); the second is what we will call the variable quantifier (based on 'some or other' or on 'a'); the third is what we will call the constant quantifier (based on 'some particular'). Thus the existential quantifier of today's logic books first appeared in Russell split in twain, as constant and variable. It is easy to see how this very elementary bit of machinery is put to work. Wherever the words 'some' or 'a' have the meaning of 'some or other,' as in (2) and (4), the variable quantifier is used; and where they have the meaning 'some particular' as in (1) and (5) the constant quantifier is used. Thus it turns out that when in modern quantification theory the existential quantifier precedes a universal quantifier (e.g. la), the constant quantifier is called for by Russell's system, and when it succeeds a universal quantifier (e.g. 2a) a variable quantifier is called for in Russell's system. This is a precise, though abstract, statement of how Russell's system, by trying to stick closely to the suggestion of English idiom, fits in with modern quantifiers. But why did Russell keep to his three quantifiers? The answer is that he had an ingenious explanation for the difference between the constant and the variable quantifiers, an explanation more extensive than anything in present day quantification theory. Let us look at this. Russell saw that contexts with 'some' and 'a' were analysable (i.e. their truth conditions were stateable) in terms of disjunctions. Thus 'a man came into the room' has as its truth conditions, 'al or a2 or a3 or ... came into the room' where al a2 a3 ... are the names of all the men. He then argued (Principles, pp. 56-60) that there is in general a logical difference between a disjunction of names and a disjunction of propositions. More precisely: between a disjunction of names replacing the variables in a propositional function, and a disjunction of propositional functions with each variable replaced by one of these names. He called these disjunctions respectively the 'constant disjunction' and the 'variable disjunction' (pp. 58-9). (It was from these terms of Russell that I fashioned the constant and variable quantifier terminology.) He employed this difference as a logical explanation of the difference he assigned to 'some' and 'a.' This is debatable as a general principle. Consider the propositional function 'John loves x.' Logicians today, as opposed to Russell, do not find a distinction between

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(6) John loves (Mary or Susan or Judy or ...) [variable disjunction] (7) (John loves Mary) or (John loves Susan) or (John loves Judy) or ... [constant disjunction]. Now for Russell, (6) states the truth conditions of 'John loves a lady,' and (7) states the truth conditions of 'John loves some lady.' What are we to make of this? The orthodox response is that it is needless subtlety, that there is no difference here. But Russell's point is more general than this. Nothing prevents him from granting that there may be equivalent results at times without obliterating a general logical distinction. Thus consider the sentences (8) If anybody finds this interesting, I'll be gratified. (9) If somebody finds this interesting, I'll be gratified. What is the logical difference between (8) and (9)? Obviously none. They are equivalent. Yet we cannot conclude from this that there is no difference in meaning between 'anybody' and 'somebody,' no difference in the logical diagnosis of (8) and (9). Rather we insist, rightly, upon a different meaning for 'somebody' and 'anybody' and explain the equivalence of (8) and (9) by explaining how that difference is absorbed by the rest of the sentence, a fairly complicated but perfectly rigorous tale. We fashion our logical tools in terms of the examples where the difference is clear and cogent, extending it to the others as necessary. This example, though not quite parallel to Russell's point about 'some' and 'a,' is enough to encourage patient attention to his explanations. Let us then apply his diagnosis to our earlier examples. For (4) and (5) we take bl b2 bs ••• bn as names of the world's books. Then these propositions become (4)' 1 need (b l b2 or bs . . • or bn) [variable disjunction] (5)' (I saw bl ) or (I saw b2 ) or (I saw b s) ••• (I saw bn) [constant disjunction]. And indeed a difference between what may be called 'the object of my seeing' and 'the object of my need' is exhibited by Russell's schema. Clearly, we could not retain the original meaning of (4)

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by expressing it in the schema of (5)'; my need for a book is just not that specific. To be precise: if (4) is true, there is yet no particular book in the world of which it is the case that I need it; but if (5) is true, there is in the world a particular book of which it is the case that I saw it. So something is right about Russell's method. We may now return to (1) and (2) (p. 145 above) and see that Russell has a precise explanation of the difference between them. (To do this I shall adapt and improve some suggestions, very unsympathetic to Russell, from Mr Geach's Reference and Generality.) Taking the names of the crooks to be C1 C2 etc. and the names of the Democrats to be d 1 d 2 etc., (1) and (2) become

(1)' [(Cl voted for d1 ) and (Cl voted for d 2) and ... (Cl voted for dn)] or [(C2 voted for d 1) and (c 2 voted for d 2) and ... (C2 voted for dn)] or [(Cn voted for dn)] etc. (2)' (c 1 or C2 or ... Cn voted for d 1 ) and (Cl. or C2 or ... Cn voted for d 2) .•. and (Cl or C2 or ... Cn voted for ds) etc. And here there is no doubt whatever that Russell's schema captures precisely just the logical difference that is wanted. (2)' follows from (1)', but (1)' does not follow from (2)'; for in (1)' some particular crook, we don't know who (it may have been Cl C2 or Cn), did all the voting, while (2)' allows it that several conspired, each perhaps to vote for a different man. What Russell calls the constant and variable disjunctions are, for some examples at any rate, an adequate technique for explaining the difference between what we have called respectively his constant quantifier and variable quantifier. His diagnosis of (1) and (2) is not to be thought of as incompatible in its logical results with that given in (la) and (2a) by quantification theory, but Russell seems to have continued to think of it as giving a clearer picture of the rationale of one and the same diagnosis than does quantification theory. It turns out, though we shall not here demonstrate it, that in terms of his three quantifiers Russell can capture all the required distinctions among the various combinations of the prefixes 'each,' 'every,' 'some,' 'a,' 'any' ... a most fascinating and profound achievement. The details are contained essentially in the chapter "Denoting" in Principles, in some of the most obscure and penetrating prose that Russell ever wrote. Our general con-

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elusion is that Russell manages to achieve, in this three-quantifier denoting theory, a diagnosis which is at crucial points technically equivalent to that achieved by the two quantifiers which he later adopted. What he missed in this first attempt, devised before he had read Frege or Peirce, was the fact that his 'a' and 'some' could, for purposes of technical manipulation, be inter-defined just when they were flanked by another prefix such as 'any' or 'every.' What he gained by the omission was a convenient device, denied to present standard uses of the two quantifiers, for dealing with examples like (4) and (5). He gained something else as well. For even though he adopted the now standard two quantifiers for manipulation after 1905, it is clear that he never wholly abandoned the explanation given above. * What we have just been through is both logic and history of logic - a small selection from many fundamental topics on which Russell's ideas, though most confusingly expounded, were exciting and innovative. It is worth noticing that the philosophical genesis of his technical ideas often remained linked to the interpretation and use he was later to make of them. This was true of the cluster of ideas associated with material implication; it is also true of the quantifiers. In his logical atomism, for example, reflection on the matured theory of quantification began to influence very directly his metaphysical theses. We can locate one continuing theme here by posing a final question about the techniques just discussed.

15 Why did Russell not seize upon the fact that his 'constant' and 'variable' quantifiers could be managed as one and the same operator when flanked by the universal quantifier, differing only in placement with respect to that universal quantifier?

* It will have been observed that the analysis given in (1), and (2)' are in terms of conjunctions and disjunctions. However we recall that these notions are the staple of the propositional calculus: the predicate calculus seems to be reducible to the propositional calculus. So at least Wittgenstein seems to have believed, but not Russell. Several obstacles appear. One is that though the universal quantifier may be understood as an indefinitely long conjunction and the existential quantifier as an indefinitely long disjunction, there will be special difficulties when the conjuncts stretch to infinity. Another problem for Russell is that (2), contains a disjunction not of propositions but of names, an idea alien to the propositional calculus. Therefore just insofar as he is to retain this special feature of his denotion theory he cannot reduce its ideas to those of propositional calculus.

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The answer is twofold. The first is an epistemological reason: consider what is implied in his explanations for 'some' and 'a,' namely, for his constant and variable quantifiers, as given above. The essence of these explanations is that different ways of denoting can be expressed as different kinds of naming; that is, items in various combinations can be picked out by various arrangements of names. The fundamental idea here, viz. that the contact between proposition and reality is ultimately funnelled through the narrow channel of names and uniquely named objects, is the slim and significant basis on which entire volumes of Russell's later epistemology are to rest. There is a connected and second reason, later abandoned as too 'platonistic,, why he could not have accepted an explanation of denoting in terms simply of the two operators of quantification. To put it glibly: instead of going to the context, like a logician, Russell goes to the object, like a metaphysician. He was in fact (inconsistently with his own theory of the range of variables described earlier) often thinking of the prefixes, in the unanalysed propositions, as picking out corresponding and differing 'unities' connected with the respective propositions. 'Some F,' 'an F,' 'any F,' 'every F' - each of these denoted a different sort of combined object, and it was, he said, these "very paradoxical objects" that were ultimately at issue in his denoting theory. It is not difficult to see in this emphasis the unspoken but firm hold that his realistic metaphysics had on him even as he was working out the fundamentals of a quantification theory inconsistent with it. The twofold explanation just given will be rendered graphic and not entirely implausible by a final look at (1)' and (2)' above. There we find that each item involved combinations: denoting, he believes, is a highly sophisticated kind of naming, and this, on the level of a rationale, is precisely wherein it differs from most later interpretations of quantification. But not from his later interpretation. And the fact that within each parenthesis different combinations of names and symbols appear is correlated with the fact that, as Russell said, correspondingly different objects occur in the respective propositions, "and it is with this very paradoxical object that propositions are concerned in which the corresponding concept is used as denoting" (Principles, p. 62). These "objects," the ones picked out by the combinations of 'names' which appear in the various parentheses and brackets, were

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paradoxical to Russell precisely because, as he said, they were effected "without relations." Relations, we recall from Chapter 2, had tended to explain almost everything problematic; but if 'any x' and 'every x' denote different "objects," and if 'some x' and 'an x' also denote different "objects," what could possibly be the point of trying· to explain this, he remarks, by saying that the sets of terms of these complex objects sustain different relations? It could have no point at all; the outlook which helped to sponsor the logic is inadequate to the logic. Just here the metaphysics of external relations seems to him irrelevant - and is irrelevant - and to require drastic reorganization under the new guiding idea of the logical form of the proposition. Weare now on the way from realism to atomism. This plurality of paradoxical objects is going to be abandoned entirely, but the idea that quantifying is a more refined kind of naming will survive to influence further developments. D. -SET THEORY AND

PARADOXES

16 The present chapter emphasizes certain philosophical ideas in logical theory, specifically Russell's particular variations on themes central to the historical development of logic. Though a vast amount of material falls aside by virtue of this policy of hewing to the centre, the principal issues stand out. Thus: the ramifications of material implication comprise the special topic of the propositional calculus; the emergence of the quantifiers is the special topic of the predicate calculus; and now what is generally known as the Russell paradox is the special topic of the third level of logic, the class calculus. This paradox, which Russell came upon while studying Cantor one memorable day in 1901, has a beautiful simplicity. It derives simply from asking whether the class of non-self-membered classes is a member of itself or not. For if it is, it follows logically that it isn't; and if it isn't, it follows logically that it is. We have to see why this matters, why it opens a new act in the drama of logical development. But first, to the staging. It would be too easy, and quite mistaken, to create the impression that Russell the logician was mainly a lonely pioneer - Moore supplying appropriate metaphysical animus, Whitehead taking up the symbolic slack, no one else paying much heed. In

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fact, the turn of the century was a period of extraordinary logical ferment, not only in the foundations of mathematics but in many areas of logic proper. Russell himself was engaged in continuous dialectical interchange, both at home and abroad. In particular, the Russell paradox was just one - doubtless the most interesting of a number of paradoxes that logicians turned up at this time. In the "Introduction" to Principia (pp. 60ff.), Russell surveys seven such additional paradoxes and argues that they admit of similar solutions. If Russell and Whitehead were working in the mainstream, consolidating, advancing, dramatizing the work of Frege, Peano, Cantor, Boole and many others, this was not true of Frege, part of whose earlier work was the first victim of Russell's paradox. The story of the professionals' neglect, and their disdain for Frege's pioneering work, his frustrating struggle for recognition, his discovery by Russell, Russell's famous finding of the fatal flaw in Frege's system, Russell's own subsequent attempts to repair the damage - here is one of the more moving chapters of intellectual history. In 1919 Russell wrote of Frege's first book Begriffsschrift (1879), which contained the first system of quantification theory: "in spite of the great value of this work, I was, I believe, the first person who ever read it - more than twenty years after its publication" (Intro. to Mathematical Philosophy, p. 25n.). In the Introduction to his Grundgesetze, Frege, explaining the delay in publishing what he had promised in that first book fourteen years before, wrote plaintively of the "discouragement that overcame me at times because of the cool reception or more accurately the lack of reception - accorded by mathematicians. . . . And so at times I turned to other subjects. But I could not keep the results of my thinking, which seemed valuable to me myself, locked up in my desk for long, and the labour already expended kept requiring new labour so as not to be in vain." Frege's is a pure case of what Kierkegaard called the distinctive mark of genius: "it goes straight up against the wind." And there may have been another sort of genius in the prophetic clairvoyance with which Frege summarized his prospects: "My only remaining hope is that someone may have enough confidence in the matter beforehand to expect in the intellectual

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profit a sufficient reward, and that he will make public the outcome of his careful examination. Not that only a laudatory review could satisfy me; on the contrary, I should far prefer an attack that is thoroughly well-informed than a commendation in general terms not touching the root of the matter." [This and the preceding quotation are from the Introduction (p. 8) to Frege's Grundgesetze, 1893, translated by M. Furth and published as The Basic Laws of Arithmetic.] This was in 1893. Russell was then in his second year of mathematical studies at Cambridge, but neither he nor his teachers, one of whom was Whitehead, had ever heard of Frege. Another decade of neglect was to be Frege's lot. Then in June of 1902 came the first sign of that dramatic fulfilment of "my only remaining hope." It was Russell who, because he was working independently in the very same problems, had enough "confidence in the matter beforehand" to master Frege's difficult symbolism and semantics, and sufficient insight to discern that Frege had made progress against all the contrary winds of doctrine. Russell also saw that something was amiss and in the letter that he sent to Frege there was, far more precisely than Frege could even have dreamed "a commendation in general terms" and "an attack that is thoroughly well-informed" and a kindly promise to "make public the outcome of his careful examination" of Frege's work. Portions of Russell's letter, written in German and first made public in 1967, are as follows: "For a year and a half I have been acquainted with your Grundgesetze der Arithmetik, but it is only now that I have been able to find the time for the thorough study I intended to make of your work. I find myself in complete agreement with you in all essentials. . . . With regard to many particular questions, I find in your work discussions, distinctions, and definitions that one seeks in vain in the works of other logicians.... There is just one point where I have encountered a difficulty. You state (p. 17) that a function, too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be

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predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection does not form a totality. "I am on the point of finishing a book on the principles of mathematics and in it I should like to discuss your work very thoroughly.... "The exact treatment of logic in fundamental questions, where symbols fail, has remained very much behind; in your works I find the best I know of our time, and therefore I have permitted myself to express my deep respect to you. It is very regrettable that you have not come to publish the second volume of your Crundgesetze; I hope that this will still be done." [pp. 124-5. This and succeeding quotations are to be found in From Frege to Codel, 1967, ed. Jean van Heijenoort.] The second volume of Frege's work to which Russell referred was already at the printer, and Frege hastily composed an appendix in which he proposed a modification of his rules (unsatisfactory, as it turned out) that might prevent the contradiction which Russell described. His prompt letter of reply to the young Englishman of whose existence he had not previously been aware, but whose name was thereafter destined to be inseparably linked with his own, contained these lines: "Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic. . .. It is all the more serious since, with the loss of my Rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic, seem to vanish. . . . In any case your discovery is very remarkable and will perhaps result in a great advance in logic, unwelcome as it may seem at first glance" (pp. 127-8). Reflecting on this exchange just sixty years later, from the hither side of what Frege had foreseen as that "great advance in logic," Russell wrote in 1962 :

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"As I think about acts of integrity and grace, I realize that there is nothing in my knowledge to compare with Frege's dedication to truth. His entire life's work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known" (p. 127).

17 Suppose there is a school for logical scandal where barbers are trained to shave all those people and only those people who do not shave themselves. Will the barbers shave themselves? They do only if they do not, and do not only if they do. This is the kind of paradox Russell was fond of using in popular exposition to illustrate the problem about non-self-membered classes. (He did not say whether the barbers in question used Occam's razor.) Let us get a closer look at the Russell paradox (it is actually a contradiction), the avoidance of which has been, since 1900, a minimum condition for any system of set theory. It is important to see that the paradox is significant mainly insofar as it threatens to erupt within a formal system. The system itself is designed, as we shall see in the next chapter, to define mathematical entities, the natural numbers for example. Numbers are going to be defined as certain classes of classes. A systematic treatment of classes (meaning, since Cantor, an axiomatic treatment) obliges us to determine which classes there are (as Russell would put it), or at least to decide which ones shall be admitted (as it would today be put). The determination is made by providing for axioms which will in turn identify the appropriate classes. A class is made up of its members. Accordingly, it is the sort of thing that can be indicated at will simply by specifying the characteristic - any characteristic would seem to do - which each of the members of the class is to have. Thus we can speak of the class of things with red hair, the class of things weighing over two grams, the class of things that can't count, and so on. Here 'with red hair,' 'weighing over two grams,' etc., are defining characteristics. Call a given class x and its defining characteristic Q. A system of

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set theory with any claim to attention would seem to require an axiom which canonizes the intuitive ideas just expressed, viz. that a given class may be composed of just those objects specified by whatever value we give to Q. With these preliminaries, we can describe (not give) a formal exposition of the paradox - inexact, but adequate for exhibiting the tender spots in the reasoning involved. The fundamental relation, taken as primitive, and symbolized by 'e' is 'member of.' The system, we just said, would want an axiom of the following very general sort : (3x) (y) rye

X

= Q]. (Call this a 'Formula.')

An axiom formulated in that way would then be saying that there is a class so composed that whatever is a member of that class answers to defining characteristic Q (whatever specific value we give to Q), and also that everything admitted by characteristic Q is a member. Succinctly, the axiom would tell us that we can have a class x which contains all and only the ys admitted by Q. SO it is the values, i.e. the substitutions, for Q that guard the gates of entry to classitude. Among the classes there will naturally be classes of classes. There will even presumably be a class consisting of all the classes that there are. So this latter class would then contain itself as a member. Other classes will also contain themselves as members; for example, the class of classes described in this book will be (because it was just described in this book) a member of itself. But more frequently classes will not be members of themselves. So let us concentrate on all those classes which are not members of themselves, and formulate by means of its defining characteristic the class that contains just these. The defining characteristic Q of this class will be 'is a class which is not a member of itself.' For example, any class y will be a member just in case Q holds, which in this case is '--, (ye y).' Applying the formula suggested above then means that we have: Axiom I (3x) (y) [(y ex) = --' (y e y)] which means that there is a class x so composed that whatever class (say, y) is a member of that class has the defining characteristic that it is not a member of itself, and also that

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every class y with this characteristic of not being a member of itself is a member of our class x. Thus we have a class, called x, which contains all and only those classes not members of themselves. Let us now fix upon this class x which we have formed by this particular substitution for Q. We know what it contains, viz. the ys which are not members of themselves; but let us try to consider it further. We may ask: Does it, our class x, perhaps have the defining characteristic that all its y-type members have, namely, the characteristic of not being a member itself? Clearly, it does or it does not. Suppose it does. In that case x too would be a y; more exactly, it would be (according to the previous paragraph) a non-self-membered y, for (as we have just supposed) it is not a member of itself. But it was just those - the non-selfmembered ys - which, also according to the previous paragraph, comprised the members of x. So x, as that kind of y, is a member of x, namely of itself. But we just started the chain of reasoning by supposing that x was not a member of itself. So we have proved that if x is not a member of itself (our premise) then it is a member of itself (our conclusion). That is, we have just shown that our system, by formalizing this argument, can produce the theorem: (1) (3x) [r-' (x Ex) > (x EX)] We can get a different but equally awkward result if we start with the opposite situation. Let us suppose, contrary to the previous attempt, that x does not have the defining characteristic had by the ys which compose it. That is, x is not among those classes which are not members of themselves. That is, x is a member of itself. But what, according to our axiom, are the members of x? They are only and precisely the non-selfmembered ys. So x, by being a member of itself, is one of those ys; that is, as that kind of y it is not a member of itself. So we have just shown that if x is a member of itself it is not a member of itself. This means that we have just shown that there can also be produced from Axiom I an alternative theorem: (2) (3x) [(x Ex) > r-' (x EX)] Putting our two theorems together, by a standard rule, we get the further theorem (3) (3x) [(x Ex) - r-' (x EX)]

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which is a contradiction. It says that there is a class which is a member of itself if and only if it is not a member of itself. Russell says that "Poincare, who disliked mathematical logic and had accused it of being sterile, exclaimed with glee, 'it is no longer sterile, it begets contradiction'" (MPD, p. 76). Poincare was right. As for himself, Russell wrote more than a half century later, "I felt about the contradictions as an earnest Catholic must feel about wicked Popes" (MPD, p. 212). To spread the reasoning out, when we could have merely taken the short way by noting that (3) is a substitution instance of our formula, is to show how a variety of qualms may find a foothold; and suggest different responses. There are points in the reasoning where we may be tempted to stop the argument, but is the demurrer based on anything more than the fact that the conclusion is offensive? Can a class be indicated by the same characteristic that defines its members? Is a class of non-selfmembered classes a class at all? Can classes, as classes, be members of classes or do they forfeit classhood by becoming members? Is it true that each class either is or is not a member of itself? These are plausible questions. But for the logic of set theory they have to be trained on the point where the trouble lies. The trouble is that Q has failed us: it lets in some radical classes that promptly destroy everything. Hence the answers to the questions have to be in terms of reforms for Q - otherwise, like Zeno who disparaged metaphysics because thinking about motion led to contradictions, we shall have to jettison set theory because thinking about classes leads to contradictions. Something has got to be wrong with our intuitive starting point, that we can pick out any class at will just by specifying the defining characteristic, for that allows us to take Q as 'non-self-membership' and that way paradox lies. For more than a year Russell brooded on this situation with a near-finished manuscript of Principles on his desk. "The contradiction" it is called throughout the book; and other similar paradoxes, like the barbers, are spun out in an effort to make the problem clearer by making it bigger. The appendix proposes a line of thought which he subsequently developed into his 'theory of types' as a solution. In the chapter of the book devoted to the problem he made numerous and substantial changes and revisions after it had been sent to the printer (the manuscripts show this) and also in the final proof copies. F

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18 More than fifty years later Russell reported on his experience as follows: "When The Principles of Mathematics was finished, I settled down to a resolute attempt to find a solution of the paradoxes. I felt this as almost a personal challenge and I would, if necessary, have spent the whole of the rest of my life in an attempt to meet it. But for two reasons I found this exceedingly disagreeable. In the first place, the whole problem struck me as trivial and I hated having to concentrate attention upon something that did not seem intrinsically interesting. In the second place, try as I would, I could make no progress. Throughout 1903 and 1904, my work was almost wholly devoted to this matter, but without any vestige of success ... "While I was looking for a solution, it seemed to me that there were three requisites if the solution was to be wholly satisfying. The first of these, which was absolutely imperative, was that the contradictions should disappear. The second, which was highly desirable, though not logically compulsive, was that the solution should leave intact as much of mathematics as possible. The third, which is difficult to state precisely, was that the solution should, on reflection, appeal to what may be called, 'logical common sense' - i.e. that it should seem, in the end, just what one ought to have expected all along" (MPD, pp. 79-80). There were ancient paradoxes reminiscent of Russell's, the best known (the Liar Paradox) being the one about Epimenides, the Cretan, who said that all Cretans are liars (in which case he is lying when he says he is lying, in which case ...), and Russell promptly concluded that these and similar paradoxes must spring from a common source. He came later, under the influence of F. P. Ramsey, to sympathize with the now common view, that there are two importantly different kinds of paradoxes: those which like the Liar are more or less linguistic (semantic paradoxes) and do not affect the basis of formal reasoning; and those which like the non-self-membered classes ('logical' paradoxes) require some systematic legislation for their avoidance. But it is important to see that Russell's own ideal was for some general strategy that would explain the known paradoxes, even while being especially tailored to his own.

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Russell's solution to the paradoxes, which is called the theory of types, has been controversial in one way; his justification of that solution, designed to meet the demands of "logical common sense," has been controversial in another. His rather freewheeling general approach to the problem has stirred up important investigations in what are now three widely separate areas; set theory proper, semantic paradoxes, category topics. This is the familiar reciprocity between technical and philosophical issues that distinguishes Russell's work. The theory of types may be briefly summarized this way. We are to think of the universe as composed of objects stratified on different levels. On the lowest level there will be individuals; on the next level, classes of individuals; on the next level, classes of classes of individuals, and so on, a different type of object for every level. (Though Russell first began to think in terms of logical hierarchies while still committed to thinking of being as divided into existence and subsistence, that metaphysical idea played no visible part in the theory of types; neither, for example, does another favourite metaphysical idea, also surveyed in the previous chapter, external relations - vivid examples of how his logic puts his metaphysics out of business.) We now think of classes as having as members only things from the next lowest type. Type n + 1 for example will have classes whose members will be of type n. Thus no class either is or is not a member of itself (just as, says Russell, virtue neither is nor is not triangular), and it will be regarded not as false but as meaningless to say that a class is a member of itself. In the axioms, replacements for the Q in our formula of the sort 'tf; e cp' will be meaningful only if cp and tf; are of consecutive types; '(x e x)' for instance will be rejected as nonsense. There are many further complications in the formulation and technical details of this theory - for the trick is to get the axioms to say exactly what is wanted and nothing else - but let us first consider what is involved in the programme. The idea that a class cannot be a member or a non-member of itself restores to dignity an intuition of common sense. Perhaps we had to be nudged into believing they could in the first place. So why plant weeds only to uproot them? Answer: to learn to recognize logical weeds, the soil they sprout in, and to develop techniques for removing them. Russell's attempt to weed the

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paradoxes by erecting a hierarchy remains one of his most distinguished and controversial ideas, controversial chiefly because of the costs in complication. Here are three examples: (1) For a variety of technical reasons it is desirable in set theory to have a universal class to which everything belongs, a null class to which nothing belongs, and a complementary class for any given class, to which all non-members of that class belong. But these niceties are precluded by the theory of types, for it decrees that all members of any class must be of uniform type. Hence new and complicated devices must be found for admitting whole series of universal, null, and complementary classes, one for each type - inelegance at best. (2) According to Russell's definition of numbers (next chapter), numbers are classes of classes; but if each class contains only members of one type, there will be different classes of classes and thus different definitions of each of the natural numbers for each type level. Something, therefore, has to be done to reduce all this to a manageable and plausible minimum. (3) Moreover, if numbers are classes of classes, some of these classes will have infinitely many members. But if, say, a type two class is to have infinitely many members, there have to be an infinity of type one entities, i.e. individuals. And what grounds have we for supposing that there are? The question matters because we had wanted to use this logic to explain arithmetic and we have just now put in jeopardy everything that depends on the idea that every number has a successor different from itself. Difficulties of these kinds were not unknown to Russell, who naturally took steps to patch things up. In particular, for the last two problems he introduced special axioms (the axiom of reducibility, and the axiom of infinity) to heal the wounds which type theory made. And it is partially the patchwork, ad hoc character of these devices that has repelled some scholars, and impelled others to find paths around type theory. The upshot is that new set theories with and without a theory of types have shot up, and after more than sixty years no single theory holds the field alone. In fact the situation strongly suggests, though this is not proven, that perhaps there is no such thing as the correct set theory, any more than there is such a thing as the correct version of the propositional calculus. The analogy is imperfect, however, since all legitimate versions of the proposi-

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tional calculus are consistent with one anther, and cover precisely the same truths, which is not the case with set theories. (We speak of the propositional calculus, since one body of truths is organized, though in different ways; but less commonly of the class calculus, as Russell did, since there is no one body of truths organized by the different set theories.) Sets or classes were first taken by Russell to be, like numbers, perfectly objective entities, independent as the stars, and if so the truth about them ought to be as final as the truth about astronomy. The point is debatable, and debated. So described, is appears that Russell's conception of the nature of classes is of a piece with his realistic metaphysics. This is essentially true despite the important fact - to be discussed laterthat in Principia he had provided himself with a symbolic way of redefining expressions which appealed to classes so as not to assert that the contemplated classes existed. For his outlook remained orientated to the idea that whatever the truth about set theory, it ought to be ultimately represented in one theory incompatible with all others. The trick was merely to find out what the truth was, meanwhile to remain uncommitted on the existence of classes.

19 Russell's three criteria - removing the contradictions, saving as much of mathematics as possible, appealing to "logical common sense" - are only moderately well met. The first fares very well, the second is very costly in complication, and the third needs more support. As to that, the key idea is one which Russell took over from Poincare and called "the vicious-circle principle." Violations of this principle, which consisted in the misuse of self-referential expressions, were at the root of all the fallacies. His argument, a bit opaque in the following crucial passage, shows that he is looking for a single solution for all the paradoxes. Weare easily seduced, he says, into thinking "that a collection of objects may contain members which can only be defined by means of the collection as a whole. Thus, for example, the collection of propositions will be supposed to contain a proposition stating that 'all propositions are either true or false.' It would seem, however, that such a statement could not

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be legitimate unless 'all propositions' referred to some already definite collection, which it cannot do if new propositions are created by statements about 'all propositions.' . . . the principle which enables us to avoid illegitimate totalities may be stated as follows: 'Whatever involves all of a collection must not be one of the collection'; . . . We shall call this the 'vicious-circle principle' ... " (PM, p. 37). It is obvious how this principle applies to 'the class of all classes not members of themselves' (not because of the phrase 'all classes' but because of the phrase 'members of themselves'). But it also applies to the set of paradoxes, distinguished by F. P. Ramsey as 'semantic paradoxes,' for example, The Liar. In this one says 'I am lying' (or, 'what I am saying is false') with the paradoxical consequence that if he is lying he isn't, and if he isn't, he is. (Or, if what he says is true, it is false, and if it is false, it is true.) The illegitimate totality here consists in the collection 'what-I-am-saying' which includes the saying of 'what I am saying,' a collection also implied in the 'I am lying' version. Thus it is relatively easy to make a preliminary analysis of this paradox too in terms of the vicious-circle principle. In these and five or six other paradoxes which he discusses, Russell shows "there is a common characteristic which we may describe as selfreference ... , something is said about all cases of some kind, and from what is said a new case seems to be generated, which both is and is not of the same kind as the cases of which all were concerned in what was said. But this is the characteristic of illegitimate totalities as we defined them in stating the vicious-circle principle" (PM, pp. 61-2). A tempting question here is this: what about the penultimate sentence quoted from Russell in the previous paragraph? ("Whatever involves all of a collection must not be one of the collection.") Is this part of the theory, and thus about itself, or not? Can the theory say about itself that it cannot refer to itself, or not? If it can't say it, then can it so refer, and then can it itself violate the vicious-circle principle? Pressure here can make paradoxes spring from the theory itself, but we need not pursue this. We said that Russell's idea is that quite different paradoxes were to be soluble as mere special cases of the vicious-circle principle. Evidently the principle rules out a great deal; expressions as different as the following being proscribed as mean-

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ingless when they enter propositions unqualified: 'all words,' 'all theories about theories,' 'all propositions,' 'this proposition,' 'the words used in this sentence,' 'classes members of themselves,' 'the number of letters in the first word of this sentence.' Accordingly, philosophers have criticized Russell for putting all the philosophical weight on this principle. It is not only that it disguises significant differences among the paradoxes, as he came to agree, but it has certain other inconveniences which parallel some of the technical difficulties of type theory mentioned earlier. One such inconvenience, interesting in itself, is this. Consider

P: All propositions are false. Clearly, if P is true, it promptly follows that P, as one of the propositions involved, is false. If on the other hand P is false, nothing odd follows -;- it follows only that some propositions are true. So if P is true, it is false; and if it is false, it is false. So it is false. Why adopt a theory that makes it meaningless? Another objection, first formulated by Paul Weiss and exploited by Frederic Fitch and others, is this: if all self-reference is ruled out, the theory of types may rule itself out. (See previous paragraph.) For as a theory about all set theories it is a theory about itself, which is illegitimate. Another objection, connected with the previous one, deriving from Ramsey and exploited by Quine and others, holds that Russell's principle implies, very implausibly, that there would be something improper, say, in singling out an individual as the typical Oxford man on the basis of the Oxford averages including his own. There are answers, all of them debatable, to these and other complaints. It is evident however that Russell is right in holding that self-reference, in some sense, is the root of the trouble; but it is almost as evident that there is more to the story. Some sorts of self-reference seem permissible, and the different kinds of selfreference badly need discrimination. It only adds to the headaches to be reminded that all these questions, and more, also have technical counterparts. But enough is enough.

20 Changing accents, we may remark that there is a quite different kind of reason why Russell, at least in Principia, confined himself, beyond the axiomatic restrictions, to just one general philosophical explanation for the paradoxes. This will be mentioned briefly because, though it has gone largely

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unnoticed, it was important to him, and affected further developments. Two of these will be cited. The idea in question is the doctrine of 'systematic ambiguity,' and it was worked out in some detail in Principia. By this phrase Russell was wont to describe two sorts of things: a battery of symbolic conventions, expressed and implied, to ensure that values of the correct logical type were always given to the appropriate variables in the axiomatic development of set theory; and an associated semantic doctrine. A thesis that derives from Aristotle holds that 'being' is a word with a different sense for each of the categories: roughly, that to say of a cat that it 'is' or 'exists' is to use these words in a different sense from that in saying for example that red 'exists' and again different from that in saying hope 'exists,' etc. Russell canonizes this idea for a host of crucial terms and organizes it around the theory of types. He concludes that "the words 'true' and 'false' have many different meanings according to the kind of proposition to which they are applied" (PM, p. 42, my italics), that is, the type of entities referred to in them. In a flurry of arguments in Principia Russell also applied this conclusion, the doctrine of systematic ambiguity, to a variety of other concepts: 'function,' 'property,' 'relation,' etc. For example, 'property' has a different meaning in 'x is a property of A' from that in 'x is a property of B' when A and B are of different types. This way of dealing with truth and meaning seems alarming, but it must be understood that in 1910 he was, naturally and plausibly, far more self-conscious about a theory of truth and a theory of types than about a theory of meaning, and he was therefore more prepared than philosophers are today to do violence to an intuition about meaning in order to do justice to a theory of truth. The violence he does, to be precise, is to slip almost inadvertently into an explanation of the meaning of 'true' (d. 'Socrates is mortal' is true, and 'red is a colour' is true) in terms of kinds of indicated fact. But how, indeed, are we to determine when we have different uses of the same meaning of 'truth,' and when we have genuinely different meanings? (A question to be gone into in Section 21, below.) Perhaps such a question seemed to Russell too fastidious to discuss; but it certainly needs thorough airing before we can be forced to his conclusion. We have here another case where Russell has his eye

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on the technical logic - where it does make sense to describe his type restrictions on variables as making certain of them systematically ambigious - and is then reaching for larger philosophical consequences. His general conclusion, that a word acquires a different meaning when it is applied to a different type, is not rendered more plausible by the fact that types can be infinite. In the years that followed Principia Russell had a great deal more to say about allied kinds of ambiguity, and indeed he took more radical steps yet in his epistemology. So the first longer range consequence is that he made it a thesis (see Chapter 6, below) that most words of philosophical importance, particularly perceptual words, had different meanings, not just different uses, for different people, because they pointed to different private experiences, different ultimate sources of verification. The doctrine of the 'systematic ambiguity of truth and falsehood,' born in the most austere regions of mathematical logic, was the first step in the line of thought that culminated in the problem of epistemological privacy. There was something very vague (to be corrected shortly) about what was just said about systematic ambiguity: it can seem at once preposterous, harmless, and gratuitous to hold such a thesis. These features did not keep it from being influential. But Russell also sharpened up the idea in a second and very different way, by propounding a general criterion of type difference. We must now examine this.

21 In Russell's theory of types, logical, semantic, linguistic, metaphysical, set-theoretic issues all tended to be treated in one fell swoop, and have been going their separate ways in the literature ever since. For some years Russell's central impulse, beyond the logic of the theory, lay dormant. It was briefly raised to prominence again in the Schilpp volume on Russell's philosophy (1944) and has more recently been taken up by several philosophers. It concerns the general definition of type, rather metaphysically conceived, which Russell extracted from his logical theory. In one of his more popular expositions, written in 1924, Russell said "The definition of a logical type is as follows: A and B are of the same logical type if, and only if, given any fact of which A

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is a constituent, there is a corresponding fact which has B as a constituent, which either results by substituting B for A, or is a negation of what so results ("Logical Atomism," in L&K,

p.332).

It is of course not a definition of logical type but a criterion of 'being the same logical type.' Anyway, the idea is easily illustrated: consider 'A is thinking of bananas.' We can replace 'A' with 'John' or another human name, but not with 'the equator,' for there is no such fact as 'the equator is thinking of bananas,' nor such a fact as 'the equator is not thinking of bananas.' By his criterion then 'John' and 'the equator' belong to different types. This is what is supposed to lie behind the circumstance, long recognized, that while it may be true or false to say 'this field is triangular,' it is senseless to say 'virtue is triangular'; sensible to say 'Fido is in bed,' but not to say 'Saturday is in bed'; and so on, with comparable explanations for the absurdity of: 'ideas have three dimensions,' 'concepts are pink,' 'houses are larger than blue,' etc. (Let us call all such outlaw expressions 'Category paradoxes.') In daily life Category paradoxes are normally avoided easily by an intuition apparently trained to unarticulated rules. Like the Class and Liar paradoxes they consist, very generally, in applying a predicate of an inapplicable type to a subject. To understand what motivates Russell's particular regimentation here it must be recalled that the theory of types, fashioned for classes, extends also to relations, and thereby also to properties. To explain how this was symbolically effected would be a diversion into technicalities, but it is not difficult to illustrate one general point, which that part of the theory aims at. A relation can only relate terms of like type and then only if it is of the right type. 'Saturday' and 'bed,' for example, are not of the same type. Of such, presumably, are the reasons why Russell felt free to extrapolate from the logic of the theory of types to the general criterion above. An important question remains whether the criterion quoted is more appropriate to identifying paradoxes, or for systematically explaining antecedently recognized paradoxes. Prima facie, the vicious-circle principle and the problem of self-reference are irrelevant to the Category paradoxes. Perhaps they really are irrelevant, but Russell did not think so. What

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connects them in his mind is the doctrine of systematic ambiguity. For diagnostic purposes, he lumped the Liar and Class paradoxes together, and by implication lumped the Category paradoxes with them; but while Ramsey persuaded him later to make the first distinction, nobody interested him in the second. Nevertheless the problem set by him here is neither the nonformal analysis of Semantic paradoxes nor the formal proscription of Logical paradoxes, both of them self-referential; it is a strategy for identifying non-self-referential Category paradoxes outside a formal system - so as to guard ourselves from injecting them into the system unawares, and prevent their corrupting our less formal systems of metaphysics, epistemology, or ethics. What, after all, is our ultimate support for the hunch, appealed to earlier, that 'the equator thinks of bananas' is neither a fact nor the negation of one? Russell is disconcertingly vague here. So let us scrutinize the criterion. We can give it a more convenient expression by coining a term, 'factual framework,' which is just the notion abstracted from the "corresponding fact" that Russell's criterion appeals to. With him, we shall pretend that we know what facts are, know how to identify them. A factual framework, affirmative or negative as the case requires, goes to make up a fact whenever the name of whatever the fact is about is fitted into its gap. ( ' - - is thinking of bananas' is a factual framework.) With this notion we can express Russell's criterion of type sameness exactly: A and B are of the same type if and only if there is no factual framework that fits A and not B (i.e. if and only if both A and B fit all the same frameworks).

but more conveniently in the equivalent way: A and B are of different type if and only if there is a framework that fits A but not B.

As we saw, ' - - is thinking of bananas' is a factual framework that fits 'John' but not 'the equator,' so they are of different type. But now consider the framework ' - - runs through Brazil.' Evidently this fits both 'John' and 'the equator.' So 'John' and 'the equator' have this peculiarity: though by Russell's criterion they differ in type just because they do not fit all

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the same frameworks, they yet do fit some of the same frameworks.* Now consider a new term 'riddle' which fits the frame ' - - solved by me'; and another term 'house' which fits the frame ' - - is painted red.' But neither of these frames fits the other term (I can solve a riddle but not paint it red, and 1 can paint a house red but not a riddle red). Therefore, some terms are of different types (house, riddle) by Russell's definition, and in addition, each has a frame the other does not fit. And this is the wrinkle within Russell's type differences that his criterion does not accommodate; it is also a wrinkle which his doctrine of systematic ambiguity alluded to but was too vague to say anything about. We can see one reason why his definition does not catch it: it is because his general definition of type (though he did not say this) is precisely fitted to the requirements for type theory in the logical system of Principia, where the aim is just to provide a logical restriction for avoiding the Russell paradox. Something different is possible and is more useful for the Category paradoxes - which shows that he has really changed the subject, from logic to metaphysics. And this calls for a revised general criterion of type. A good candidate for our present need is this: A and B are of different type if there is a framework that fits A and not B and another framework that fits B and not A. The criterion is different from those contemplated on Russell's behalf in the recent footnote. The italicized words mark the addition in terms of which this criterion is different from Russell's. Apparently, this criterion would put 'John' and 'the equator' in the same type; or rather, a proof that it can separate them would depend upon ingenuity in coming up with examples. This emphasizes an important difference between the criteria: Russell's original, geared to avoiding the Russell paradox, makes it rather easy for the sake of the Category paradoxes to establish type

* It is easiest to get oriented here by imagining at this point what I think Russell imagined, namely, that abstractly one might proceed in category type theory in anyone of these three ways: to define sameness of type either (1) in terms of fitting all, or (2) in terms of fitting only some,- of the same frameworks; difference in terms of fitting only some (equivalent to (1)), or (3) in terms of fitting none, of the same frameworks. There would have to be, and are, reasons for choosing among these approaches; but I shall not pursue it except to say that anyone would also have to be justified in the face of the claims of the others.

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difference - only one recalcitrant framework is needed; the revised criterion makes it rather hard to prove type differencetwo very special frameworks are needed. Ironically, the austerity of the revised criterion for the Category paradox is just what is superfluous for the Russell paradox: no system needs two contradictions to discredit it; but a term by the revised criterion needs two frameworks to type it. The requirements for metaphysics or semantics (whatever this subject is) are different from those for logic - not an idea Russell would have liked. The revised criterion is more restrictive, suggesting what it does not prove: that we need fewer types than Russell permits. (Translation: less systematic ambiguity.) But what of that? Russell might ask: why should it be harder to pass muster, to be of a different type? One answer, too general, is: we may as well stay in tune with the long-standing metaphysical conviction that entities ought to be assumed innocent of type varieties until proven guilty - no point in multiplying types indiscriminately if the whole idea, outside of pure logic, goes against the grain. If, as Russell often reminds us a la Occam, entities should not be multiplied beyond necessity, neither should types of entities. A second answer, much better, is this: Russell's criterion, by allowing too much explains too little. We are trying to pin down the truth behind the absurdity of, for example, 'virtue is triangular.' Russell says that the theory of types, by his criterion, gives the answer;* the rejoinder is that it gives too general an answer, because it has too general a notion of type. Not too general for classes, but too general for virtue. A third reason, more complicated, reaches even farther. In making explicit reference to two frameworks the revised criterion thereby makes the whole notion of type resolutely relative to context and adjacent type. This is in the spirit if not the letter of Russell. His original theory of types was geared to his somewhat ambivalent notion of propositional function (built on the metaphysical conception of proposition); type restrictions were thought of as classifying entities

* The answer he gives appeals to the notion that, say, virtue and New Hampshire are of different type just because the latter but not the former fits the frame, ' - - is triangular.' The revised criterion requires an additional framework, say ' - - is a matter of the heart' which fits virtue but not New Hampshire. New Hampshire can be triangular but not a matter of the heart, and virtue can be a matter of the heart but not triangulara more precise reason for Russell's conclusion.

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by impositions on variables. This made him suggest, what was alien in his clearest thought, that hierarchies of types were really autonomous ontological facts, instead of, as he later made explicit, and as the revised criterion urges, systems of relations among expressions - but yet reflective of ontological facts. By revising Russell's criterion we have attempted both to prune some of the vagueness from his conception of systematic ambiguity and simultaneously to urge that there is rather less ambiguity than he implies.

22 Our revised definition helps to bring out something deeper about the endeavour to specify what a type is. The problem is how factual frameworks are themselves to be identified, how we are to be justified in holding that a given framework is really one and not, ambiguously, two. This is a still more sharpened version of the 'systematic ambiguity' question. For Russell the type difference of 'John' and 'the equator' rests simply upon the unsuitability of the latter but not the former to the framework ' - - thinks of bananas'; for the revised definition, on the other hand, their presumed sameness of type rests upon the absence of full proof to the contrary. But what sort of proof is to be acceptable? Russell might say, appealing to the doctrine of systematic ambiguity, that ' - - runs through Brazil' is itself systematically ambiguous just because it can accommodate both 'John' and 'the equator' which by his criterion are of different types; that it is therefore actually a different framework in the two applications. Russell could then make two points: first, if ' - - runs through Brazil' is taken as a framework that fits 'the equator,' then it isn't the same framework (though it uses the same, ambiguous, words) that fits 'John.' Secondly, the framework ' - - thinks about bananas' fits 'John,' but admittedly not 'the equator.' He could conclude, therefore, that even with the revised criterion these terms ('John,' 'equator') are of different types, for each has a framework the other does not fit. This is consistent but desperate. It appeals to the notion that running through Brazil is a different sort of affair for John and for the equator, and it tries to make this less obscure by saying that 'runs through Brazil' is systematically ambiguous depending on what type of thing is doing the running. (This is the sub-

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terranean way in which Russell's own rather vague doctrine of ambiguity is connected with his doctrine of ontological types. Though he did not explicitly connect them this way, the fact that they are so connected is what lies behind his unrevised general criterion of type.) But there has to be a more specific reason for the suggested manreuvre than merely a resistence to revision; for the revision comes couched in terms of the original criterion type and framework are indefinable - and it is resistance to it that alone introduces obscurity. We might try this: Whatever runs through Brazil as John runs can also trot or stroll through - and the equator cannot; therefore 'runs through Brazil' is ambiguous. But to make this work, we need more theory, not less. Russell did not go into it. He would have had to bring in such facts as that the Amazon, though it can run through Brazil, cannot stroll or trot through; and this brings in still more ambiguity on his criterion, though not on the revised criterion. On the other hand, both criteria are admittedly circular. The revision only makes this more explicit, and makes explicit too that the old vicious-circle principle is impotent to help on this. Typing of terms depends, for both, upon frameworks, but what does the identification of factual frameworks depend upon? Upon the type of terms? Beyond that, we are cast upon ad hoc devices, our presumption, for example, that we know roughly the shape of a fact as compared with the shape of nonsense. Nevertheless, some further things are evident too. Clearly, ' - - runs through Brazil' is not by any means as obviously a proxy for two different frameworks (depending on which term is fitted in) as '--- thinks of bananas' would obviously be two frameworks (one a metaphor) if we really tried to fit in 'the equator' as well as 'John.' So the doctrine of systematic ambiguity, if applied, would inevitably assault our intuitions. This shows perhaps ('perhaps' - just because these are isolated examples) that our intuitions and "the metaphysics of the stone age to which common sense is due" are after all more attuned to the revised criterion than to Russell's original. Furthermore, a new problem looms on the horizon for Russell's unrevised criterion. Suppose A and B are of different types. Suppose also the law of transitivity for types: If X is of the same type as K, and K of the same type as Y, then X is of the same type as Y. Now consider the framework ' - - is of the same type

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as K.' Suppose A fits this framework, yielding a true or false proposition. It follows that B cannot fit that framework since it, by hypothesis, is of different type: on Russell's criterion terms of different types cannot fit the same frameworks. Yet since A and B are of different types it should be the case that both fit this framework, though (or rather, because) both could not yield a true proposition. We cannot escape a contradiction. If A fits the framework, B both does and does not; if B fits the framework, A both does and does not. If neither A nor B fits the framework, they are neither of the same nor of different type. A problem somewhat like this was raised for Russell by Professor Black in 1944 (Schilpp, pp. 235 ff.). The supposition of both philosophers was that the contradiction could be avoided by thinking not of entities but of words as being typed. The proposal remained brief and obscure enough to sound more like a hope than a theory. Another alternative, to reject the law of transitivity outright did not, apparently, even sound like a hope. But Russell might have said, appealing to his earlier viciouscircle principle, that the troublesome framework itself has to be rejected. Could he say on these new grounds that the theory of types could not be self-referential, could not be made to apply to statements of itself, as frameworks employing the concept 'same type' attempt? On that view no such typing frameworks are permitted. As the Wittgenstein of the Tractatus might have put it: the theory of types only shows the truths about sameness and difference of type but it cannot state them. (He did not, at any rate, take the theory of types as such at all seriously in the Tractatus.) It would be a way of sa:ying in Wittgenstein's language that transitivity tor types is not a justifying rule for the logic of type theory, in somewhat the way that he held (see Section 6, above) that modus ponens is not a justifying rule for the propositional calculus. What these higher level 'rules' of logic are sometimes presumed to justify can in fact merely be exhibited. It is perhaps a limitation Russell would not have welcomed, but might have accepted, either for his own or for Wittgenstein's reasons.

23 The foregoing discussion has essentially followed the path of Russell's thought, beyond the point where he stopped, to a

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crossroads; the alternatives he suppressed are in full view. The possibilities are two-fold. (1) To work forward through or around the revised definition and find a still more sophisticated way to compare and control threats of circularity (in formulating criteria), or systematic ambiguity (of frameworks), or contradictions in making type distinctions; or (2) in the face of the too general character of Russell's types, the threat of circularity and contradiction, to withdraw all claims to a general philosophical definition of type based on the logical theory of types; all these devices then become unsystematic rules of thumb for collecting obvious category paradoxes. There are advocates for each of these paths of thought among philosophers influenced by Russell. He had wanted a general theory of types, good for logic and good for ontology, preferably applicable to one just because it was applicable to the other. To press forward would be to engage new problems in the theory of predication and to attend more precisely than he thought wise to the niceties of idiom; to go backward would be, contrary to his nature, to limit the horizon of what he knew to be one of his profoundest and richest sets of insights. Nevertheless, Russell's original goal, to develop a logical scaffolding which would handle the class, semantic, and category paradoxes as systematic variations in one unified programme, has remained attractive in many quarters and, though still unrealized, it continues to inspire new ideas. Three logical questions have been pivotal in this chapter, each surrounded by concentric and intersecting circles of other questions, and they have identified the three distinguishable layers of Russell's logic: What is the meaning of 'implies'? What is the meaning of 'all'? What is the meaning of 'class'? The questions were located in the context of Russell's own philosophical progress. And this enables us to identify the prevailing theme. For at the beginning of this chapter it was metaphysical ideas that were setting up the stage for logic; at the end logic is, as it were, upstaging the metaphysics - a metaphor for Russell's own development. Specifically, that line of advance in Russell's early work shows up in the systematic liberation of logic from its vassalage to metaphysics, giving to logical theory an independence that quickly turned to domination. The triumph of this trend

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was announced, if not achieved, in his logical atomism: "I hold that logic is what is fundamental in philosophy, and that schools should be characterized rather by their logic than by their metaphysics" (L&K, p. 323). It was within the poles of this development, wherein logic first received and then gave direction to metaphysics, that Russell's philosophy of mathematics took shape.

CHAPTER 4 -

THE PHILOSOPHY OF MATHEMATICS

The philosophy of arithmetic was wrongly conceived by every writer before Frege. Russell, My Philosophical Development (1959) I set out with a more or less religious belief in a Platonic eternal world, in which mathematics shone with a beauty like that of the last Cantos of the Paradiso. Russell, "Reflections on My Eightieth Birthday" in Portraits from Memory (1956)

CHAPTER 4

THE PHILOSOPHY OF MATHEMATICS

Russell's logical investigations in the foundations of mathematics produced a fund of novel ideas, techniques, and possibilities. His powers of conception and exposition have made the subject an engaging area of study for generations of students. His preoccupation with the connections between logic and metaphysics (in which he was joined by Wittgenstein), and between logic and mathematics (in which he was joined by Whitehead) made the philosophy of logic into a central topic of philosophy itself. In these areas Russell was at once technical innovator, international scholar, expositor, popularizer - and philosopher: a combination sufficiently rare so that even his brilliant predecessors, for example Frege, Peirce, and Leibniz, could not have had so pervasive an influence, however they deserved it. Indeed, perhaps no logician besides Aristotle has. Nevertheless, when at the turn of the century Frege, Russell, Whithead, and others were puzzling out the relations of logic to mathematics, they were labouring in a venerable tradition. For many philosophers, ancient and modern, it had been a dogma, a speculation, or at least a pious hope that logic and mathematics might have a deeper than surface unity, a unity that might possibly involve great technical insight and manipulation to expose - and perhaps great value to contemplate. Unlike the others who also pursued this vision as a technical idea, Russell incorporated a larger theme, one that Descartes, for example, would have thoroughly approved. In a sentence, the idea is this: just as logic had been based upon metaphysics, mathematics was to be based upon logic. Russell added a component Descartes would not have approved, though Plato might have: a deliberately created atmosphere of romance and high adventure, sometimes, in semi-popular writings enshrining mathematical thought in a halo of mystical piety. It would be in the spirit of his researches to expound his technical themes, which 180

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will occupy most of this chapter, after we have located it in this uniquely Russellian setting. Section A therefore considers aspects of his pre-Principia frame of mind, and the work of Peano as one of his starting points. Section B (Logicism) expounds part of the thesis that mathematics and logic are one, with some brief definitions as illustrations. Section C (Counting and Reasoning) considers applications of this thesis in two different areas. Section D (Critical Perspectives) puts some of Russell's results in the perspective of his related work and that of others.

A.

THE

POE TRY AND

E SSE N CEO F

MATHE MATICS

1 For many philosophers the very existence of mathematical knowledge - so certain, so immutable and eternal- is a problem. This is especially so for empiricists such as Hume and Mill who, ideally, would look to experience as the source of everything. But experience does not show how these qualities could arise. So what actually is mathematics? What is the ultimate nature of its subject matter, if it has one? Is it invented? discovered? 'or are these both misleading metaphors? What kind of reasoning is mathematical reasoning? Is it about objects? Does it depend upon the senses, or is it a matter of pure reason? or of intuition? Does truth in mathematics possess special, perhaps ineffable qualities, spiritual power, and a value more than practical? "I had thought of mathematics with reverence," said Russell. More blandly: what exactly are numbers? Do they exist? What is it that makes counting possible? Such questions, because they seem so perplexing and futile, are less a call for answers than a way to focus agitations. They have annoyed and stimulated thinkers since records were kept, turning mathematicians into philosophers, and stretching the imaginations of students of those other disciplines that depend upon mathematical truth, but not upon understanding it. The questions take on a still deeper and more persistent attraction for just two sorts of minds: those disposed to see mathematics as mysterious or sublime (for example, Plato and Pascal), and those whose philosophical investigations have yielded tools for formulating exacting answers (for example, Leibniz and Peirce). Now, for a time Russell met both conditions.

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The questions were of deepest interest to him precisely because he was both a visionary and a philosophical technician. "For the health of the moral life, for enobling the tone of an age or a nation, the austerer virtues have a strange power, exceeding the power of those not informed and purified by thought. Of these austerer virtues the love of truth is the chief, and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith." This is from Mysticism and Logic (p. 73). Russell gave this title in 1918 to a collection of essays written during the preceding decades, a title which bespeaks the poles of his feeling and thought. The book as a whole, one of Russell's most widely circulated, is less a unity than a dialogue where logic, like mathematics, epitomizes impersonally the claims of reason, and mysticism the co-equal claims of the spiritual life. Though mysticism suggests to the popular mind an assumed direct avenue to non-cognitive truth of sufficient cogency and value to render normal insight second rate, it was for Russell first of all a personal mood with which he coloured his celebration of the particular quality of mathematical truth. In "The Study of Mathematics" (1902) he dilates with a nice blend of the frivolous and the solemn. "Mathematics, rightly viewed, possesses not only truth but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of paintings or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exultation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry" (M&L, p. 60). (In Russell's perception of the idealized mathematician there is something like Heidegger's idealized view of the poet. Though both of their outlooks are held within a firm rationalistic framework, yet both hint at modes of access to the ultimate truth of things which are more direct, powerful, sublime, even if ineffable, and closer to the final spirit of reality than the rest of their philosophies can articulate. The comparison cannot of course be pressed very far.) "Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity, embodying in splendid edifices

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the passionate aspiration after the perfect from which all great work springs" (pp. 60-1). Such soaring romanticism, we may be inclined to say, is itself less philosophy than poetry. Though it was first expressed in 1902, we find Russell in a backward glance forty years later saying: "the eternal Platonic world gave me something nonhuman to admire. I thought of mathematics with reverence .... I have always ardently desired to find some justification for the emotions inspired by certain things that seemed to stand outside human life and to deserve feelings of awe. I am thinking in part of very obvious things . . . the edifice of impersonal truth, especially truth which, like that of mathematics, does not merely describe the world that happens to exist" (Schilpp, p. 19). Russell has done better than most in the attempt to chart the ineffable Pythagorean impulses which have moved some of the world's greatest thinkers. Late in life he remarked: "My philosophical development, since the early years of the present century, may be broadly described as a gradual retreat from Pythagoras. The Pythagoreans had a peculiar form of mysticism which was bound up with mathematics. This form of mysticism greatly affected Plato and had, I think, more influence upon him than is generally acknowledged. I had, for a time, a very similar outlook . . . " (MPD, p. 208). He reports that his earliest interest in mathematics as a tool to manipulate the world had given way to conception of it as "an abstract edifice subsisting in a Platonic heaven and only reaching the world of sense in an impure and degraded form. My general outlook, in the early years of this century, was profoundly ascetic. I disliked the real world and sought refuge in a timeless world, without change or decay or the will-o' -the-wisp of progress . . . this outlook was very serious and sincere" (pp. 209-10). So intense an esthetic and moral posture toward "the world of pure reason" - mathematics, logic, relations, universals is doubtless one which it would never occur to most of us to adopt, except for brief moments of deliberate indulgence; and it has consequently been too easy to underestimate its effects upon Russell's work. In My Philosophical Development (1959) Russell himself, with the deflationary wisdom and severity of age eighty-seven, quoted from those early writings, including some passages reproduced above, and remarked: "All this, though I still remember the

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pleasure of believing it, has come to seem to me largely nonsense, partly for technical reasons and partly from a change in my general outlook upon the world. Mathematics has ceased to seem to me non-human in its subject-matter. I have come to believe, though very reluctantly, that it consists of tautologies. . . . I cannot any longer find any mystical satisfaction in the contemplation of mathematical truth" (pp. 211-12). As to the merits of mathematical truth, mystical satisfaction is one thing, and largely emotional; philosophical suggestiveness is another, largely conceptual. But the two may be combined. Both had their hand in Russell's development, though his explicit awareness and memory of the former is keener than the latter. If as Russell says, Plato's mathematical studies influenced his general philosophy more than he knew, the same is true of Russell himself. That, in fact, is one thing this book is about. 2 The truth is that mathematics can be so warmly imagined because it can be so coldly calculated. In Principia Mathematica (1910-13) the noble haze of mathematical truth is dispelled in a frigid blast of symbols. There, in three huge volumes, we get more than two thousand pages of technical definitions, explanations, and deductions - a massive work of relentless complexity and elegance. Russell and Whitehead wrote the work together, though Whitehead, unlike Russell, was also teaching for a living, and thus the greater burden inevitably fell on Russell. "Whitehead invented most of the notation, except insofar as it was taken over from Peano," says Russell (MPD, p. 74). It is widely supposed that most of the ideas were Russell's and this is doubtless correct, though Russell has made generous gestures toward denying it. Evidently much of the important and novel work on infinite numbers in Volume II comes from Whitehead since he had published early versions of this under his own name; for similar reasons we know that the theories of descriptions and logical types were Russell's. At any rate he has said: "Broadly speaking, Whitehead left the philosophical problems to me" (p. 74). They divided the symbolic labour, exchanged drafts and each rewrote his own original for the finished form. The final manuscript for the printer was almost entirely in Russell's longhand. Originally, the work was to be merely the second volume of Russell's Principles of Mathematics; with Whitehead's help it

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was to take a few years at most. It took ten years; and both "turned aside from mathematical logic with a kind of nausea." Professor Quine has called Principia "one of the great intellectual monuments of all time." A monument it certainly is. But books that mark epochs may cease to be documents as they become monuments. Russell wrote in 1959, "I used to know of only six people who had read the later parts of the book. Three of these were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated" (MPD, p. 86). The work begins with an eighty page introduction by Russell. Two hundred and fifty pages are devoted to expounding the philosophy of logic, including the lines described in the previous chapter. From their own premises Russell and Whitehead deduced by the hundreds theorems also attainable in the logic of Boole, Schroder, Peirce, McColl and others, consolidating work done on many fronts. Cardinal arithmetic begins only after more than three hundred pages; the cardinal number one is defined twenty pages later. Definitions of numbers, including ordinals, rationals, reals, ratios, etc. are produced, finally quantity and measurement are discussed. A fourth volume, never published, was to be devoted to geometry. Summarizing Principia would be a little like summarizing the dictionary; but we can consider a few of the more philosophical themes. Russell and his followers have often described Principia's programme as "the reduction of mathematics to logic." The word "reduction" has been provocative, if for no other reason than this: it suggests that mathematics is not just what it appears to be, that it is really at bottom something else, that investigators before this century did not know what mathematics actually is, did not know perhaps just what numbers are. "The philosophy of arithmetic was wrongly conceived by every writer before Frege," Russell said. But he was rather less preoccupied with the blunt question 'What is a number?' than, for example, Frege was. Frege had said (Grundlagen) that no one had ever given even a plausible answer to the question, and that he, Frege, had finally given the true answer. Russell said, that they (he, Frege, Whitehead) were at least giving an account of number that served all the purposes served by the prior unanalysed concept. This is not merely the difference between an Englishman and a German;

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partly it reflects Russell's somewhat modified conception of what they had done after they had done it. The first point is that Russell later developed a new and more generally applicable notion to describe what had been done: he thought of his definitions as examples of what he began calling 'logical constructions' - replacements of antecedently certain but unclear entities with logically refined substitutes. This will be discussed later. Secondly, it is worth remembering that most disputes about 'reduction,' 'construction,' and 'foundations' are more separable than Russell supposed from the enterprise of establishing a conceptual and technical link between arithmetic and logic. Let us now work our way into some of the thoughts that generated Principia. 3 In July of 1900 Russell attended the International Congress of Philosophy in Paris, precipitating what he called "an intellectual honeymoon such as I have never experienced before or since" (MPD, p. 73). While there he met the Italian mathematician G. Peano, and was so impressed with his rigour and profundity that he asked Peano for copies of his published works, quickly mastered them and returned home to write and rewrite more than three hundred pages of The Principles of Mathematics before the year ended. He had virtually finished the work but not published it when he discovered Frege's work. The impact of these thinkers on Russell's mind during the composition of his own book, and at a time of his greatest creativity, created something of a revolution for him; it is not surprising that it also introduced a multitude of minor incoherencies in his own book (see last note of Chapter 1, above). Peano's system of arithmetic and Russell's response to it provide a convenient place to enter the discussion. Fortunately, the relevant part of Peano's work is not difficult to describe. Russell has himself given a particularly elegant exposition and criticism of it in his 1919 Introduction to Mathematical Philosophy. The three primitive terms of the Peano system, undefined, are, 'zero,' 'number,' and 'successor.' The five postulates of the system are these: (1) Zero is a number (2) The successor of any number is a number (3) No two numbers have the same successor

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(4) Zero is not the successor of any number (5) If P is a property such that (a) zero has P and (b) whenever a number n has P, then the successor of n also has P, then every number has P. The last postulate, having a hypothetical form, gives the principle of mathematical induction. Together, the postulates show the interrelations among the three primitive terms, but they do not, of course, define these terms. Nevertheless, it is evident that Peano supposes '0' or 'zero' to mean just what it does in common parlance, and also by 'number' he means the familiar natural number forming the series, 0, 1, 2, 3, 4, etc. These suppositions, in the absence of further definitions, are extremely crucial, and Russell was to fasten on just this fact. Here is a very rapid survey of what can be done with these elements of the Peano system. Given the primitive terms and postulates, we can define the individual natural numbers quite simply. 1 is defined as the successor of 0, 2 is the successor of 1, 3 of 2, etc. It can be seen that by postulate (2) this process can be carried on indefinitely, and the other postulates ensure that two numbers will never be defined in the same way. With the natural numbers in hand one can proceed to define arithmetical operations. The successor relation makes it possible to conceive addition as merely the repeated addition of 1, and this in turn may be used to define multiplication, which is considered (when nand k are the multiplicands) as the sum of k terms each of which is equal to n. The distributive, associative, and commutative laws can then be proved. Also substraction and division can be defined on the basis of addition and multiplication. This requires the creation of negative and rational numbers, but this too can be done by defining negative and positive integers as ordered couples of natural numbers, and defining rational numbers as classes of ordered couples of integers. Further expansions are possible and are required: real and complex numbers can be defined without requiring anything new; further arithmetical operations for them can also be defined, the concepts of function, limit, integral, etc., can be spelled out, together with their theorems. And so on. All the concepts and truths of elementary number theory can be elicited by logic alone from the three primitives and five postulates. Though Russell and Whitehead introduced a very large number of refinements in conception and

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technical exposition, most of this kind of work was familiar before Principia pulled it all together. Russell's most important contribution in the area was his definition of irrational numbers as classes of natural numbers. Two distinguishable programmes, both incorporated in Russell's work, have just been alluded to. One programme is the presumed reduction of all traditional pure mathematics - essentially arithmetic, analytical geometry, calculus - to the theory of natural numbers. The crucial breakthrough in this programme was made possible in the latter half of the nineteenth century by Dedekind, whose delineation of irrational numbers also provided the key to the ancient worry uncovered by Pythagoras about incommensurable lines. This programme - whether it is thought of as a 'reduction to' or a 'generation from' - may be called the arithmetization of mathematics. The second programme, which is specifically Peano's, is the reduction (generation) of the theory of natural numbers to (from) the three primitives and five postulates - the axiomatization of arithmetic. This is where Russell's work fits in. From the point of view of Russell's interest in what is now called the foundations of mathematics, Peano's nice achievement in systematization has fatal shortcomings. Fatal not in the sense that it cannot generate some mathematical truths which it aspires to reach, but in this sense: because it .starts with undefined terms it is open to any interpretation of these terms which is consistent with the rest of the system. And there are many such interpretations, radically different from one another. For Russell this raises the question whether Peano's axiomatization of arithmetic gives us access to the specific nature of arithmetic at all. Peano, remember, has assumed that 0 was just what we normally take it to be. Part of Russell's criticism may be indicated by noticing the consequences of arbitrarily specifying that the symbol '0' in Peano's system denotes what is normally denoted by '100.'* Postulate (4) tells us that 0 (that is, 100) is not the successor of any number, as indeed in the new system it is

* Perhaps a defender of Peano will say that the suggestion is unintelligible since it makes nonsense of the question whether the familiar numbers less than 100 are supposed to exist or not. And this is right; but on the other hand it is intelligible because we can work consistently in Peano's system with our arbitrary assumption. So there is a problem, which Russell does not heed, in formulating his criticism in a neutral way.

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not. In fact everything goes through as before. Nothing is inconsistent; and that is the important point. In this exotic system, we get theorems, for example, stating '103 + 103 = 106,' and we can prove that every positive real number has 102 real square roots, and so on. And it seems that Peano has 105 postulates. This suggests that Peano's system does not yield our ordinary notion of number but rather something more abstract of which our daily number concepts may be illustrations. (It is only a suggestion because it is not clear just what has been described above. Was it more than playing with words? Was it really playing with Peano's numbers?) To Russell, what proved that Peano has not pinned down the standard concepts of arithmetic is this: Any series whatever in which there is a first term, a successor for each term, no last term, and no repetitions, and such that any term can be reached from the first term in finite steps is what Russell called a 'progression'; and Peano's postulates can be consistently applied to any progression. (A progression, more briefly, is anything with a first term and a successor operator that yields something new on every application.) A progression might begin with anything, but the numbers, surely, must begin with zero. Moreover, there are an infinite number of progressions and therefore, in Peano's system, an infinite number of candidates for being, let us say, the positive integer 6, or any other whole natural number we choose. But for Russell there are not an infinite number of such 6's; there is only one of them. So Peano has not found this one. Russell says : "Progressions are of great importance in the principles of mathematics. As we have just seen, every progression verifies Peano's five axioms. It can be proved, conversely, that every series which verifies Peano's five axioms is a progression. Hence these five axioms may be used to define the class of progressions: 'progressions' are 'those series which verify these five axioms.' . . . The progression need not be composed of numbers: it may be composed of points in space, or moments of time, or any other terms of which there is an infinite supply. Each different progression will give rise to a different interpretation of all the propositions of traditional pure mathematics; all these possible interpretations will be equally true. In Peano's system there is nothing

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to enable us to distinguish between these different interpretations of his primitive ideas" (Intro. to Mathematical Philosophy, pp. 8-9, hereafter IMP). There is an important point which Russell, for understandable reasons, does not make, but which is this: Whether this criticism matters for the philosophy of mathematics depends upon your point of view - which is not to say that it is arbitrary, but is to say that it does not matter for any part of the practice of mathematics. Russell speculates that we might think of Peano's primitive terms as 'variables' concerning which the postulates make 'hypotheses,' and all derived theorems will then deal not only with the natural numbers but with a set of terms with certain interesting formal properties, among them the formal properties of numbers. But Russell says that this too fails to give an adequate basis for arithmetic. "In the first place, it does not enable us to know whether there are any sets of terms verifying Peano's axioms; it does not even give the faintest suggestion of any way of discovering whether there are such sets. In the second place, as already observed, we want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties" (p. 10). There are two important points made in this quotation and Russell does not explain either of them. One is about the existence of sets, and the other about counting. The significance of the remark about counting will become evident after we have indicated how Russell's theory would explain counting, and its place in the philosophy of mathematics. Though Russell does not here say what is involved in "discovering whether there are such sets," his thought is probably on these lines. He is imagining a theory with universally quantified statements about all sets, or, allowing for the theory of types, all sets of a certain type, and unless we have explicitly provided for the existence of appropriate sets, we shall not know what such statements are saying, or if they have a chance of being either true or false. For example, the axiom of infinity mentioned in the previous chapter was an embarrassment not only because it was needed and could not be proved, but also because it was needed at all: for why should a philosophy of mathematics depend on the 'size' of the universe?

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Indeed, Russell's general assumption that every meaningful axiom in a consistent set theory is either true or false has turned out to be very doubtful. (See Section D below on these points.) Thus Russell's remarks above both turn out to reflect, on a deeper level, his own philosophical position, rather more than he lets on. The criticism of Peano has been cited at some length, first, because it is essential for identifying Russell's position - and precisely the source of still-continuing controversy - that mathematical entities must and can be more precisely determined than mere progressions allow. Secondly, Russell's criticism of Peano has certain analogies with some scruples we shall want later to express about his own achievement. But we must be careful not to exaggerate: for it is not the case that Russell in any of his books dwells, as Frege certainly did (and as my exposition onesidedly does), on the Teutonic task of 'finding out what the numbers essentially are.' (In a sense he just wants to build up to numbers from base metals, for he already knows what the numbers are and where they are: they are the objective and perfect realities heralded by true mathematical propositions; and they dwell in the land of Being.) Rather his response to Peano's programme makes sense only in the context of his having his own answers, and believing answers are necessary, to questions unasked by Peano. 4 The Frege-Russell-Whitehead thesis, essentially that mathematics is logic, can evidently incorporate a good bit of Peano's work, but only by getting beneath it; for example, by defining the primitive terms. In several places Russell discusses various obstacles prior to Peano to a correct philosophical understanding of the question, what is a number? In Principles he argues that the chief of these was the long association of number with quantity. Indeed the whole of Part III of this work is given over to a powerful and detailed critique of this association, and the result is thoroughly negative: quantity "is not definable in terms of logical constants, and is not properly a notion belonging to pure mathematics at all" (p. 158). Russell points out for example that this illegitimate marriage gave birth to problems, insoluble for centuries, concerning measurement (e.g. are all spatial magnitudes susceptible to numerical measure ?). Against this long

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tradition the theory of irrationals was a decisive blow, and the almost concomitant development of branches of mathematics, such as Projective Geometry (to which Russell had devoted an earlier book), which deal even less with quantity than with numbers, dissolved the marriage of numbers and quantity for good and all. Later, in Introduction to Mathematical Philosophy (1919) he puts the historical blame for retarding influences on mathematical philosophy on another point: "Many philosophers, when attempting to define number, are really setting to work to define plurality, which is quite a different thing. Number is what is characteristic of numbers, as man is what is characteristic of men. A plurality is not an instance of number, but of some particular number. A trio of men for example, is an instance of the number 3, and the number 3 is an instance of number; but the trio is not an instance of number. This point may seem elementary and scarcely worth mentioning; yet it has proved too subtle for the philosophers, with few exceptions" (p. 11). This criticism is hard to make out. Certainly one can perceive the truth of Russell's positive remarks; but does he see his predecessors as defining something insufficiently abstract: instances of particular numbers, perhaps the outcome of exercises in counting, but not number itself? We know what Russell has up his sleeve: a definition of particular numbers such that it gives the general pattern for defining number itself. Later still, in My Philosophical Development (1959), he was inclined to dismiss pre-Frege philosophers of mathematics wholesale: "the mistake that all of them made was a very natural one. They thought of numl::lers as resulting from counting, and got into hopeless puzzles because things that are counted as one can equally be counted as many" (p. 68). This is too glib, but it is important because the place Russell's theory assigns to counting is one vital element in any final assessment of it. We have now accumulated quite a bit of negative material. In summary, Russell's view is that at the outset at least numbers are not to be philosophically discovered by attending to: progressions (they are too indefinite), quantity (this is too partial to reach the essence), plurality (this is insufficiently abstract), counting (this presupposes numbers). So where do we turn? How ought we tl) track numbers down for philosophical scrutiny? The answer is that we turn to things like classes, collections, sets, groups, or

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clusters, piles, packs, heaps - these are the places where numbers have their lair. Numbers, in short, are properties of classes. B.

LOGICISM

5 We have now to explicate the positive part of Russell's doctrine, known as logicism - essentially that numbers are properties of classes, and that mathematical propositions are expressible as logical propositions. I shall begin by saying something about classes as these relate to mathematics, and this will supplement what was said in the previous chapter about the paradoxes of class theory in logic. The idea that numbers are essentially properties of classes is not difficult to understand, even if it is a shade deviant. (For Frege numbers are essentially properties of the concepts of these classes - no less deviant.) There is the class of Snow White's dwarfs, they have the property of being seven; and the days of the week have the same property. The class composed of the months of the year has the property twelve, and so on. This is not quite the way we might have thought of it antecedently, but of course we hadn't thought of it antecedently at all. Both numbers and classes are a bit recondite. If we are told that the grapes in the cluster are seventeen (cf. 'the grapes in the cluster are green') we know that that cannot be true of each of them (as each is green), so we translate: as a class, they have the property seventeen. 'There are seventeen grapes' is the way we would have put it before philosophy intruded. So numbers are somewhat oddly but not implausibly conceived as properties of classes. To insinuate these vague qualms while introducing the idea of numbers as properties of classes is deliberately to anticipate. There is a long philosophical association of properties with predicates (cf. 'seventeen' and 'green'); but we are not about to explicate these number-properties of classes in terms of predicates. They are going to be explicated by means of quantifiers, which we know from the previous chapter to be a piece not of predicate machinery but of subject machinery. Classes, which we just now associated with mathematics, have also long been associated with logic, though in a rather primitive way. The obvious example is just that propositions ('Socrates is mortal,' 'men are mortal') can be interpreted in terms of class G

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membership (Socrates is a member of the class men), or class inclusion (the class mortals includes the class men). Since George Boole in the mid-nineteenth century, the possibility of a logic of classes has been explored and developed with great detail and finesse, and Russell has sometimes cited the Boolean class algebra as the real turning point in logical theory. But the programme of ax iomatizing class theory, including especially specifications for class identity, existence, and so on, was a further development, and it is from this - a more or less 'Euclidean' developmentthat Russell takes his departure. The idea now is that we are to think of classes as comprising a potential subject matter antecedently given, to be studied by means of a formal system. (The presentation of class theory in terms of axioms, rules, and theorems, which Russell builds upon, had introduced at the end of the nineteenth century a profoundly new conception of logic, and one that has largely determined its course in the twentieth century. A certain dialectic has emerged from this - to be discussed briefly at the end of this chapter - between what was called 'axiomatic methods' and 'natural deduction' methods in the exposition of logical theory.) The intuitive and historical association of classes with mathematics on the one hand and with logic on the other is a crucial coincidence. That connection is the central point in Russell's programme to build a highway from logic to mathematics. It may be fortunate that classes are such esoteric items that we know almost nothing about them apart from the formal system; we have almost no check on our investigation of them other than disciplined common sense and formal consistency. But this only reinforces the suggestion that classes, pure and abstract entities, least encumbered with our intuitions, may with appropriate formalizing open the door to the mysteries on the border between logic and mathematics. The point is that to some extent we can make them behave as we demand; and in view of the Russell paradox discussed in the previous chapter we need circumspection in our demands. A formal system - terms, rules, axioms, theorems - is interpreted by assigning meanings to its key terms so that its theorems will deliver the truths that are wanted. What Russell needs is a system of sufficient simplicity and power (free of contradiction) so that it can be interpreted in terms of mathematical entities and

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yet whose principles and primitives are among the recognized arsenal of logic. On this view such a system will have to incorporate the theory of types, and, as we saw earlier, other special axioms. There are crucial questions in the wings here, which Russell and Whitehead were to leave to their successors. How could we know whether a formal system can be interpreted in such a way that no mathematical truths can escape its interpretation? (Completeness) How can we guarantee beforehand that we will not turn up a contradiction at some distant point as we work out the interpretation? (Consistency) Are their automatic strategies for determining, of any formula fashioned in the terms of the system, whether it or its negation is a theorem? (Decision Procedure) In these and adjacent regions there have turned out to be unimaginably rich veins of research, virgin territory ploughed up almost inadvertently by Russell and Whitehead. Since this largely post-dates Principia we can put it to one side, along with other refinements of the idea of formal systems as such. What we want to see is simply how the materials of class theory are used to define the natural numbers. Peano's system enables us to organize, and present in a deductive way, the truths about the natural numbers. But for Russell it gives too general an answer to the question, What entities are the numbers? And it does not even attempt to answer the question, What is a number? "One of the chief triumphs of modern mathematics consists in having discovered what mathematics really is," says Russell (M&L, p. 75). Given Peano, the question what mathematics really is might be compressed into the question, What is a number?, an ancient question, correctly answered only recently, says Russell. "The answer was given by Frege in 1884, in his Grundlagen der Arithmetik. Although this book is quite short, not difficult, and of the very highest importance, it attracted almost no attention, and the definition of number which it contains remained practically unknown until it was rediscovered by the present author in 1901" (IMP, p. 11). 6 Russell, Frege, Whitehead and hosts of other philosophers who would disagree on many other points in the philosophy of mathematics are all prepared to agree that, niceties aside, a number may be viewed as a property of a class. What property? The number property of course, but what is that? Consider a

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way of viewing certain collections or classes of terms, in particular, those classes which have a certain given number of terms. Thus we have the collection of couples, the collection of trios, of quadruples, and so on. But a number is plainly not identical with just any given class of terms having that number: the number nine is not the same thing as the class of planets. But is the number nine, that is, the property of 'being nine,' identical with the collection of aU the classes which, like the class of planets, have just nine members? As propounded by Russell, the logicist's striking answer to this question is Yes. Number properties are classes of classes. We can explain Russell's doctrine - in a way more simplified than he has, but adequately for present purposes - by considering in turn a series of interconnected ideas. They are these: (a) (b) (c) (d)

Number of a class The same number A class has two members Two

(a) Number of a class. Numbers, we are agreeing, may be considered as characteristics of classes of objects. Thus a number, says Russell, is anything which is the number of some class. This seems intelligible enough as a first step. It may also seem circular - unless we can explain 'number of a class' in some new way. And we can. Each class of objects (the Apostles, the things on this desk, the days of the week, the planets) taken as one whole will inevitably itself belong to a larger collection composed of all the classes which have that same number of members. Now collections of classes, each class in any given collection having the same number of members, are, we will say, collections of similar classes, classes similar to each other. Thus, as was said, we have the collection of all couples, the collection of all trios, of all quadruples, and so on; the couples are similar to each other, the trios are similar to each other, etc. (The words 'class,' 'set,' and 'collection' are used by Russell interchangeably; but it will usually be clearer for us to continue to speak of the classes of classes as collections of classes.) Now, says Russell, in effect, this collection of classes is itself in each case the number of a given class. Russell is aware that this seems paradoxical at first. "We naturally think that the class of couples (for example) is something different from

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the number 2. But there is no doubt about the class of couples: it is indubitable and not difficult to define! whereas the number 2, in any other sense, is a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples, which we are sure of, than to hunt for a problematical number 2 which must always remain elusive. Accordingly we set up the following definition: The number of a class is the class of all these classes that are similar to it" (IMP, p. 18). The question now is: Which classes are to belong to any given collection of classes? They are the ones having the same number of members, that is, the similar classes. But what does this involve? We cannot blithely say that the same number is what is determined by counting, for some classes may have infinitely many members; and we cannot presuppose the notion of number, for this is what is being explained. So how are we to explain this idea, 'having the same number,' which is essential for identifying the classes that belong to any given collection? (b) The same number. This can be explained without presupposing anything illegitimate. It involves giving a precise meaning to 'there are just as many A's as B's even though we have no idea how many. As Russell points out, it is, from a logical standpoint, vastly simpler to find out whether two collections have the same number of terms than it is to define what that number is. He illustrates: "If there were no polygamy or polyandry anywhere in the world, it is clear that the number of husbands living at any moment would be exactly the same as the number of wives. We do not need a census to assure us of this, nor do we need to know what is the actual number of husbands and of wives. We know the number must be the same in both collections, because each husband has one wife and each wife has one husband. The relation of husband and wife is what is called 'one-one'" (IMP, p. 15). Perhaps it should be pointed out, for the benefit of Russell's example, that it is not necessary that each husband have his own wife; what is essential for one-one relations is just that he have some one wife or other. As Carnap has said in not quite the same connection: "In logic there are no morals." Now, two classes are like-numbered, that is, in the technical sense, are similar, when there is a one-one relation which

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correlates each of the members of one class with each of the members of the other class. (This relation of similarity is reflexive, transitive, and symmetrical.) The notion of similarity is then what is used, and it is the only thing that is used, in determining which classes belong to the collection of classes which makes up the number of a class. We have thus avoided the circularity of presupposing number in fashioning its definition, for the notion of 'same number' as here defined involves only the relation of one-one correspondence between members, or similarity of classes. (It is the relation of similarity which is also used in Principia to exhibit a sharp cleavage between finite and infinite classes. Infinite classes are similar to proper subsets of themselves: the members of the infinite class 1, 2, 3, 4, . . . have a one-one relation to the members of the infinite class 2, 4, 6, 8, ... Finite classes are not similar to their subsets.) (c) A class has two members. According to Russell, then, the class of all couples will be the number two, the class of trios will be the number three, and so on. Therefore, for a class to have two members is first of all for that class to belong to the class of all two-membered classes.* But this notion represents progress .* It is easy to get sidetracked here. It might, for example, be asked: According to the Russell account being given, what belongs to what? Does the number belong to the class? Or does the class belong to the number (the class of classes)? We started out by saying, what seems natural, that a number belongs to a class. And we seem to have ended by saying that a class belongs to a class of classes, which is a number. So a class belongs to a number. It seems like chasing one's own tail. Two points: first, we said that both numbers and classes are recondite; so abstract surprises are in order. Second, and more fundamental: we must not rest any of our theory on the unanalysed notion of 'belonging.' And this is no loss; we can say that a number is a property of a class and that a class is a member of a class of classes. So then a number is a property of a class of classes, and when we are shown that it just is the class of classes, we may say with Russell that properties are 'reduced' to classes. And this is not an awkward by-product, but the aim in view. That is then the answer as to why classes and numbers seem to 'belong' to one another in the puzzling circular way. Grossly stated: they are one another. This is what explains the problem we raised earlier: numbers are properties but they are now about to be explained not by means of predicates, as might have been expected, but by means of quantifiers. This will shortly be exhibited. It is the reduction of properties first to classes that renders this possible - a very nice and profound trick, if not an uncontroversial one. (See also next footnote). Vie must not, however, pretend that there are no other problems in interchanging number and class talk. The interpretation of 'There are two numbers between two and five' reminds one that though the last 'two' is a class of couples the first 'two' refers to a class of classes of classes.

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only if we can pick out the initial couple. The one-one relation is used to identify which other classes belong to the class of couples once we are given a couple. We must therefore secure our initial couple, which will, by similarity, give us the rest of them, and therewith the number two. Let us state exactly, with a brief and forgivable lapse into symbols, what it is for a class, call it C, to be a couple, that is to have two members. In Russell's prose it comes to tllls: there is an x which is a member of C, and a y which is a member of C, x and yare not identical, and any z is a member of C if and only if it is identical with x or identical with y. In Russell's poetry it comes to this :

(3x) (3y) [(x e C) . (ye C) . (x =1= y) • (z) [(z e C) = (z = x) v(z = y)]]

A little reflection quickly shows that the pattern of this symbolic representation can easily be adapted to specifying exactly a class of any given number of members. All we need is enough quantified terms to do the trick. For example, that a class C' has one member (is a unit) comes to this: There is an x which is a member of C and anything, say y, is a member of C' if and only if it is identical with x. And that a class C" has no members (is a null class) comes to: There is no x which is a member of C". Put into symbols, these latter statements become respectively:

(3x) [(x e C,)· (y) [(y e C') - (y = x)]] - (3x) (x e C") (d) Two. We have now explained what it is for a class to have two members, that is, to be a couple. The first abstract definition given above, when interpreted, asSIgning particular things to the x and to the y thereby picks out a particular couple. It does not matter which couple. Now the x and the y which comprise it may be put in one-one correspondence with other things, any other things, a process repeatable indefinitely, each yielding another couple, the total succession of such 'acts' yielding the collection of all couples. This collection is the number 2; or more modestly, it performs every clear task ever asked of the number 2. Accordingly, our symbolic definition of a couple, C, given above, is used in our definition of two. Given that definition of C, we can say two = df the class of all classes similar to C.

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7 Let us now recapitulate all the steps we have just traversed in the preceding section. First, we explained the notion of 'a number' as being a property or characteristic of a class, the number-characteristic as it were. This in turn was explained by the company it keeps: having the number characteristic is belonging to the larger total collection of all those who have it. These other member classes are then identified as just those whose terms can be put in one-one correspondence with the terms of the given class. Any given number, now unmasked as a class of classes, can be picked out by defining which condition its member classes must meet. Thus we defined a couple, a unit, and a null class, giving the general pattern for defining any such class we please; and lastly we defined in prose the number two.* There remain items in the penultimate stage which require explanation. There are, for example, the poetic notions of identity, membership, material implication, equivalence, and quantification, symbolized by =, e, >, =, (x). But these notions - and this is one of the principal points of the story - belong to general logic. We began with these materials of logic, proceeded through a hierarchy of interconnected definitions, and ended with the materials of mathematics. The branches of the tree are mathematics, the trunk is the theory of classes, the roots are the logical principles of quantification and the propositional calculus. If we like the figure we may say that the roots of the tree grow, according to Russell, in metaphysical soil, and the leaves of the tree are applied mathematics. At any rate, we can say, by drawing on Peano and others, that mathematics does not have any of its own primitive terms, or any of its own primitive postulates; for all pure mathematics, says Russell, "insofar as it is deducible from the theory of natural numbers, is only a prolongation of logic" (IMP, p. 25).

* In the discussion of Russell's metaphysics in Chapter 2 we took note of his thesis that for two or more terms 'having a common property' was to be analysed in terms of 'sameness of relation' to another term. This is part of what he called the principle of abstraction, a very far-reaching idea, both a metaphysical and a logical principle. Here it is in action on behalf of logicism. The Apostles and the months of the year, each of them a class, and each collectively a term, have the common property of being 12, which is to be understood as sameness of relation to some further term. Which relation, and which further term? Answer: membership in the class of 12-membered classes.

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8 The idea that numbers are classes of classes can be represented in another way. Imagine that the boxes below are indefinitely large, shown by their open-endedness. The first box is empty, the second contains all the single items, the next one all the couples, the next all trios, etc. Think only of the contents, not of the boxes themselves. (The admonition to ignore the boxes themselves comports picturesquely with another technical development in Russell's theory: "the symbols for classes ... are, in our system, incomplete symbols: their uses are defined but they themselves are not assumed to exist" (PM, p. 71). Classes are what classes contain.) Imagine that the row of box-contents continues infinitely.

1

1

1

1 1

1 1 1

1 1 1 1

1 1 1 1 1

V

X

X

+ ++

o o o o

X

V

V

o o o

X

X

X

X

s s s s s o o o o o

o

+s + s

V

V

V

V

V

s s s

s s s s

o o

X X X

++++

+ s

V

V

+V +V + ++ V V V xxxxx

If we consider the contents of each of these boxes as representing a class, that is, a class of similarity classes, then each may be seen as a number: respectively, 0, 1, 2, 3,4, 5, ... To note that a class has, say, three members is to note which box it belongs to, viz. the one which has only classes similar to it. And to note which one this is is to note that it is the box which has all and only the classes meeting this general condition: there is an x, a y, and a z which are members, none of which is identical with any other, and that any w is a member if and only if it is identical with either x, y, or z. Numbers are classes, that is, classes composed of classes; the number 3 is the class of classes containing all the members just described in purely logical terms.

C.

COUNTING AND REASONING

9 Numbers, we were all brought up to believe, are sometimes useful for counting: for example, fingers. They can also be involved in stretches of reasoning, such as concluding that two

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round trips which take a fortnight each way will consume eight weeks. Russell's philosophy of mathematics can be made to explain these and similarly obscure matters. First as to counting. Russell argues that the logical analysis of the notion shows that it is a complex affair, and that since it presupposes numbers it cannot, as some philosophers have thought, be used to generate numbers (Principles, p. 133). Counting involves, for example, taking things in order and avoiding repetitions, both of which may be simple as processes but still logically complex. Indeed, the logical complexities of a practically simple process threaten to multiply out of hand, for when infinite terms are involved matters may be infinitely harder. This makes it important that we have a conception of number whose byproduct renders counting (not of course easy, but) intelligible. Russell nowhere develops these matters at any length, but it seems to be his idea that the process of counting presupposes a rather more complex analysis, which itself presupposes numbers. We can speculate about this in the following way, using his idea of numbers. Consider some lines, / / / /, the counting of which, let us suppose, I wish to understand in terms of the Russell theory. This means not merely that I must identify the right box to which they belong (which would be merely to depict their number), but I must actually have a systematic way of specifying their number. This is to be understood in terms of assigning subsets of the lines to the occupied boxes in an orderly nonrepetitive way. This in turn is explained by the method of finding similarity classes in terms of one-one correspondence. 'rhus I am imagined to find the box for /, then to find the box for that plus the successor line, viz. / /, then for those plus their successor, viz. / / /, and lastly for / / / /, thus exhausting the lines. I now note which class of classes the lines joined on their last assignment. This wilI be the total number of lines counted. How do we know what the number is? Not certainly by looking into the last box used in order to count the members of its various classes! No, we already know, because of the preceding exposition, that the contents of the box just is the number. The box contains everything the preceding box contained, and just enough more to make it the successor number. It is just because of this that we can say that the number of numbers from 1 to 4

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is 4. In Principles Russell appeals (p. 133), somewhat unclearly, to a contrast between the 'analysis' of counting and the 'process' : it is the analysis we are now considering, and it is brought in to give an indication of the kind of reasons that may lie behind certain important platitudes. For example, that it isn't counting if we take some things twice, or omit some, or start with zero, or don't use each of the numbers up as we go, and that the number of (finite) numbers used up in counting from 1 to n is n. The process as such, Russell implies, need not appeal to the analysis, or even to the central notion in the analysis, viz. similarity classes. For the process of counting, says Russell, we simply "correlate a class of objects with a class of numbers." Doubtless what he has in mind is that the process correlates numbers with objects by correlating numerals; and vice versa. We can imagine the numerals - names of the classes of classes - perched on the boxes; and to report the result of the process of counting the lines we merely read off the name of the last used box. By assigning the lines to the boxes in the orderly way which was suggested above, we also set up a series of one-one correspondences between the lines and the names of the natural numbers (numerals). To put it now plainly: the 'process' of counting works with numerals and the analysis shows how it works by its series of similarity classes among objects and numbers (i.e. classes of classes). For the process of counting we seem to need the numerals: if we try, for example, to get by simply with similarity classes for the counted items, we would have to count the members of that class to get the number of the counted items. Russell's analysis of number shows that they are not numerals; his analysis of counting shows that numerals are not numbers, nor are they similarity classes of the classes which are numbers. We need not worry that all this makes counting seem more obscure than it is; we should not expect anything else from a logical analysis. After all, we know that children who can reason moderately well would be baffled by the structure of their own syllogisms. However, this has been said on behalf of the Russell of Principles. It is doubtful that the post-Principia Russell would accept all of it. There is something curiously cavalier in what is said about counting in Introduction to Mathematical Philosophy (1919). He says that the notion of similarity is "logically presupposed in the operation of counting," and logically "simpler

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though less familiar." But is counting logically complex only in its appeal to similarity? "In counting, it is necessary to take the objects counted in a certain order, as first, second, third, etc., but order is not of the essence of number: it is an irrelevant addition, an unnecessary complication from the logical point of view" (p. 17). It seems odd to say this in the context of counting. The only reason he gives is that the notion of similarity does not demand an order, nor finite classes, which is true enough. It may also be true, in a more obscure sense) that "order is not of the essence of number," but since it is of the essence of counting, this threatens to make a mystery out of how counting is connected with numbers conceived as classes. And in that case it is somewhat curious to find Russell (in his 1937 Introduction to Principles) criticizing other definitions of numbers for having failed to make intelligible their "connection with the actual world of countable objects." This last point will return.

10 Counting presupposes numbers, Russell said, giving only the briefest of explanations, and apparently changing his mind on just how. Do children learn to count their fingers? Or do they learn to count - on their fingers? Both. But these are different. It may be a psychological question which they learn first, but it is a logical question which is basic. The former counting may be called 'transitive' counting, counting things, which is what I have just discussed; the latter may be called 'intransitive' counting. Russell does not go into any of the questions that come up here. Certainly intransitive counting is less picturesque, and less lucid. Does that suggest that intransitive counting is derivative from transitive counting? Could it be supposed that intransitive counting consists merely in reciting the numerals? Then what makes it counting, rather than, say, poetry? Only its connection with transitive counting perhaps. Fortunately for us, answers to these interesting questions do not seem exclusively dependent on a particular philosophy of mathematics. Yet Russell probably presumed answers to them. He seems to have assumed, in his early poetic or theological moments, that 'intransitive' counting is really transitive counting - of numbers themselves, timelessly composed in rows in Plato's heaven. There is something rather profane, then, in saying that they are really after all only classes. However, classes too were objective, pre-

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sumably metaphysical entities; so secularism did not really set in until he found a technical way to dispense with classes. Later, when Wittgenstein persuaded him - or almost did - that mathematical propositions were not about a subject matter (i.e. object matter) at all, that they were tautologies, he had new reasons for being cautious about the analysis of counting. His caution took the form of silence. In explaining counting we confined ourselves to finite numbers. For actual counting it makes no difference whether there should turn out to be an infinity of numbers or just finitely more than anybody should ever need. For the analysis of counting, that is, for this part of the philosophy of mathematics, it does make a difference. So we have to identify the natural numbers, an infinity of them. Russell's way of doing this is not unique to him, but to indicate it will enable us to take a valuable detour through the connection between counting and mathematical induction. When Russell said that philosophers of the past had wrongly "thought of numbers as derived from counting," he apparently referred to what we just called intransitive counting; though still vague, this claim is clear enough for us to see that he is not accepting it. He takes numbers as antecedently given, to be merely used in counting; hence, counting cannot be explained except in terms of the explanation of numbers, not the reverse. Now the way he conceives counting - where he places it on the philosophical map, as it were - is analogous to the way some earlier philosophers had placed mathematical induction: it was for them as if numbers were somehow given (it was unclear just how, but perhaps they derived from counting) and then it appeared to be a remarkable and curious further fact that mathematical induction holds, that whatever is true of n and of n + 1 is true of any natural number; just as for Russell it is a further curious fact (which must and can be explained) that counting 'works.' The point then is this: what Russell effectively does is to reverse things, to advantage all around: for natural numbers, it is their utility in counting which is, as it were, by-product of the way they are theoretically conceived, and their role in mathematical induction is, by contrast, of the essence. "Mathematical induction is a definition." This outcome is what might have been expected from the analysis of counting given above. There, by identifying subsets of lines by reference to a given set 'plus' the successor,

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implicit appeal was made to part of the idea (shared by Russell with Frege and others) that the class of natural numbers is the class of which zero is a member and of which all successors of members are members. Russell says: "The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid .... We now know that . . . mathematical induction is a definition, not a principle. There are some numbers to which it can be applied, and there are others ... to which it cannot be applied. We define the 'natural numbers' as those to which proofs by mathematical induction can be applied, i.e. as those that possess all inductive properties" (IMP, p. 27). 11 There are other ways than counting to make simplicity profound. Let us consider the logistic use of mathematics for a bit of non-mathematical subject matter, an ordinary garden variety application of pure mathematics. I know, that (when I am in the garden) it there is a bird in my hand and two birds in the bush and no other birds in the garden then there are three birds in the garden. Any child knows this. But a philosopher can know something more; he can know how to make it lucid by making it seem obscure. Call the argument K. Now we know how to form in purely logical terms this just-mentioned class of the birds in the bush: it is by means of the class C depicted a few pages back. And we know how to form the just-mentioned class of birds in the hand: it is by means of the class C' also depicted above. Similarly we know how to form the mentioned threemembered class of all the birds in the garden. Consequently, we have the materials for translating the 'mathematical' argument K, bit by bit, until it is an inference of logic alone. We shall not even require the pure mathematical principle 'one plus two equals three.' We can do the mathematics without numbers. They will be theoretically eliminated from the picture while we perform the operation in terms of unalloyed logic, involving ideas from three levels: set theory, quantification, and truth functions. This logistic trick may be (imperfectly) exhibited and described as follows. ('H' - 'bird in the hand'; 'B' - 'bird in the bush'; 'G' - 'bird in the garden.') :

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1. There is just one bird in the hand a (3x) [Hx' (y) (Hy > (x = y))] 2. There are just two other birds in the bush b (3z) (3w) [[(Bz' Bw)' (z -=1= w)] • (u) [Bu > (u = z) v(u = w)]] 3. There are just these and no other birds in the garden c (s) [(Bs v Hs) - Gs' [(s = x) v (s =z) v (s = w)]] 4. Therefore: There are just three birds in the garden d (3x) (3z) (3w) [[(Gx' Gz' Gw)' (x -=1= z)· (x -=1= w) . (z -=1= w)] • (t) [Gt > (t = x) v (t = z) v (t = w)]] Here we have expressed all the required propositions in purely logical terms.* But we have not thereby deduced 4 and d from the other three. To do so formally is a fairly routine, though lengthy, operation in some one of the several standard formal techniques in which logic books now abound. It consists in showing that the joint assertion of (the corrected versions - see footnote) a, b, c with the denial of d is inconsistent. A purely mechanical affair, it depends upon explicitly formulated but wholly logical rules. (Essentially, what is to be proved is given in I below.) In sum, what seemed like a mere application of pure mathematics, that is of elementary number theory, becomes expressible in terms of logic. The work of numbers is being done by their logical surrogates. In just this way are logic and the more elementary parts of mathematics spliced together. In the previous chapter we took note of the fact - and the problems surrounding the fact - that modus ponens cannot be a premise for the propositional calculus, though deductions are inevitably made in accordance with it. In a similar way, the familiar logic that derives from Aristotle has principles of pure logic which we apply to the subject matter, but they are not premises for our deductions. (For example: 'If all A is Band all B is C then all A is C' is a principle involved in this argument - 'If all cats like milk, and everything that likes milk dies, then

* It must be assumed that 1, 2, and 3 constitute one single proposition as they did in argument J( above, ~ince 3 refers back to the birds mentioned in 1 and 2 (,other' means 'other than x, z, w'). Accordingly, the variables 'x,' 'z,' ow,' in c must be thought of as being reached by their quantifiers in a and b. Furthermore, and for similar reasons 'other' in 2 is not represented in b. To represent this accurately would be so cumbersome as to obscure the intuitive point being made.

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THE DEVELOPMENT OF BERTRAND RUSSELL'S PHILOSOPHY

all cats die.') In a somewhat similar way 'one plus two equals three' is not a premise for our 'deduction' above, though the deduction may be said to have been made in accordance with it. (Roughly: truths in applied mathematics are derived from other truths in applied mathematics in accordance with truths of pure mathematics, so that nevertheless applied truths are possible only because pure ones are.) Now it was a piece of applied mathematics that we re-expressed above; it turned out to be a piece of applied logic, quantification theory. The question may therefore come up whether the pure mathematical statement ('1 +2 = 3') kept in the background, has also its analogue in 'pure' logic, also kept in the background, as a principle in accordance with which the inference proceeded. The answer is that there is such a principle. In fact it can be extracted from the argument and expressed in several ways. Putting the refinements aside, we can grasp the matter intuitively by giving several increasingly abstract expressions to the principle in accordance with which our original argument K proceeds, and by which d is deduced from a, b, c. The following sequence of expressions, each of which is valid only if each of the others is, illustrates how we build up to pure mathematics:

1. II. III.

If only x H's, and only z and w B, and none of x, z, w are identical, and whatever H's or B's also G's, and nothing else does; then just x, z, and w G. For any H, B, and G if just one thing H's, and just two things B, nothing does both, and whatever H's or B's also G's and nothing else does; then just three things G. For any classes H, B, and G, if H has just one member, and B has just two members, and they have no common members, and G is a conjunction of Hand B; then G has just three members.

Now III is just a fancy way of getting us an equivalence for 'one plus two equals three,' in the jargon of set theory; I says it in the language of quantification theory and II says it in a combined way. Though details are suppressed, the general picture is basically accurate from Russell's point of view: it shows that a series of logical abstractions, involving truth functions, quantifiers, and sets (the three levels of Russell's logic considered in the previous chapter) can be discerned between the original argument

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K and the pure mathematics concerned. Doubtless, this was the kind of thing Russell had in mind when he wrote: "The logical definition of numbers makes their connection with the actual world of countable objects intelligible ... " Principles, Intro. 2nd ed., p. vi). And this is also what he had in mind when he wrote that "a careful analysis of mathematical reasoning shows . . . that types of relations are the true subject-matter discussed" (Principles, p. 23). I, II, and III are bits of mathematical reasoning, and they exhibit only types of relations. He means that the relations involved are essentially logical ones, and to see what he means by that is to see, as in the examples above, that propositional theory, quantification theory, set theory, elementary number theory, merge smoothly into each other, parts of one whole.

D.

CRITICAL

PERSPECTIVES

12 In his 1937 "Introduction to the Second Edition" of Principles, Russell said, "The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify. This thesis was, at first, unpopular, because logic is traditionally associated with philosophy and Aristotle, so that mathematicians felt it to be none of their business, and those who considered themselves logicians resented being asked to master a new and rather difficult mathematical technique. But such feelings would have had no lasting influence if they had been unable to find support in more serious reasons for doubt. These reasons are, broadly speaking, of two opposite kinds: first, that there are certain unsolved difficulties in mathematical logic, which make it appear less certain than mathematics is believed to be; and secondly that, if the logical basis of mathematics is accepted, it justifies, or tends to justify, much work, such as that of Georg Cantor, which is viewed with suspicion by many mathematicians on account of the unsolved paradoxes which it shares with logic. These two opposite lines of criticism are represented by the formalists, led by Hilbert, and the intuitionists, led by Brouwer" (p. v).

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Hilbert and the formalists, Brouwer and the intuitionists, Russell and the logicists - this is the way the scene shaped up in the early decades of the century. To enter this complicated set of controversies would be too vast and remote an adventure; moreover, the lines between the schools seem less sharp and helpful today than they once did. Staying closer to home, we can indicate something of the kinds of critical questions whiCh appear within and on the periphery of Russell's programme. The task will be only to raise a few interesting qualms, and so to keep intuitively alive the kinds of questions Russell raised. There is an alternative conception of numbers (one among several) to be mentioned that has certain affinities with Russell's, certain technical interests and liabilities. This is the view of von Neumann according to which a number is not a class of classes but rather the class of all preceding numbers. The boxes from an earlier page will illustrate: we are to think of each box, whose contents is a number, not as containing vast quantities of similar classes but as containing merely the set of boxes to its left. Thus, for example, 1 is the class whose sole member is the number 0; 2 is the class of 0 and 1; 3, the class of 2, 1 and O. The boxes have to be stacked inside each other. We imagine an infinite stack of Chinese boxes, on the outside of each is a name1 a numeral, telling us that inside there are just that many boxes, each box a number. The last box, named '0', is empty. This theory might be said to generate the numbers out of zero: we get 1 from 0, 2 from 1 and 0, and so on. The theory makes counting a curious business. Do I, for example, count the fingers on my left hand by associating one each with one each of these: 0, 1,2, 3, 4? Then how do I know they are five? Perhaps it is by counting up all the numbers I had to use up for the first task, starting this time with I! Here we would not1 as with Russell, indicate the numbers by saying that the number of numbers from 1 to n is n, but by saying that the number of numbers less than n is n. In any case, Russell would have to say that this theory does violence to his conviction that the numbers are antecedently given, that they are used in, but not derived from, counting.

13 For von Neumann1 numbers are classes of numbers or possibly classes of numerals; for Russell numbers are classes of classes. It was to be expected therefore that some philosophers

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would venture the view that numbers are numbers. Perhaps this is what some 'intuitionists' have had in mind. They have held that we cannot be said to understand the Russell definitions without a prior understanding of the universal notion number. We do, it seems, have some such prior understanding: we take ourselves to know from the beginning such things as that a number, for example, has neither weight nor colour and it may possibly be supposed that civilized men know well enough what numbers are, always have and always will; know beforehand therefore that they are not classes. There are many deep problems here which we cannot go into, but one allied problem that sometimes springs from the intuitionist camp can be discussed. There are various forms of the accusation that Russell's definitions are circular: for example, that the notion of number has to be appealed to to define it; that Russell's notions of similarity and one-one relations appeal to the number 1, and so on. Recall that similarity was defined for classes in terms of establishing one-one correspondences between members of the classes; and similarity so understood was essential for the definition of numbers as classes of similar classes. Does this involve an illegitimate use of the number one, its being used in its own definition? Critics often speculate about trickery here, suspecting that they see the rabbit being put into the hat. Russell clearly foresaw the arched eyebrows. He wrote: "It might be thought that a one-one relation could not be defined except by reference to the number 1. But this is not the case. A relation is one-one when if x and x' have the relation in question to y, then x and x' are identical: while if x has the relation in question to y and y' then y and y' are identical. Thus it is possible, without the notion of unity, to define what is meant by a one-one relation" (Principles, p. 113). Now this appears to be perfectly cogent and unexceptionable. So why is there a stubborn persistence of controversy on the point? This is worth explaining. One reason it persists is that critics who find the notion of 'one' presupposed in the notion of 'similarity' are thinking in terms of some vaguely understood and antecedent conception of 'one,' and also some vaguely understood notion of 'presupposed.' But since Russell's procedure is designed to elicit the exact conception of 'one,' which exact conception is not presupposed in any evident sense, and also to

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illustrate 'presupposition' by specific examples (e.g. the analysis of counting presupposes the concept of similarity) it is obvious why he finds this objection yirtually unintelligible. It may perhaps be replied to this that he is avoiding the circular reasoning only by circumventing it, and thus begs the question in a wider sense: the presupposition of 'one' is escaped only by substituting a 'different' and narrower conception of it. But it is not easy to give an exact meaning to this criticism, for it raises general and involved questions about the meaning and construction of any formal definitions. In some sense we certainly have to know of anything what it is before we can define it; so in some sense we do have to know what a number is. But in that sense we can define nothing without circularity. A second reason, more precise, for controversy here is that Russell in effect admits that 'number' presupposes 'same number' (which is. what looks suspicious to critics) but he would deny that this requires analysis in terms of 'same' and 'number;' it is analysed rather in terms of 'one-one relations' as explained above. To protest the legitimacy of this line of analysis is once more to get into the general controversy invoked above. A third and still connected reason for controversy lies in Russell's assumption that any two classes either are or are not similar, that their terms either can or cannot be put into one-one correspondence. This seems true enough, but it is not at all clear what would be involved in proving it. Maybe it presupposes something about one. But what? Anyway, intuitionist philosophies of mathematics would reject Russell's assumption for classes concerning which it is not possible to determine if they are similar or not. Two kind of problems beset formal systems: one is contradiction, the other is circularity. Contradiction is almost always clear and firm and fatal; it calls for rebuilding. Circularity is almost never like that; the idea is itself incurably vague, and so it sponsors both evasiveness and, sometimes, clarification of exposition.

14

In a vein somewhat different from the foregoing are a number of questions about the enterprise of Principia that have caused Russell a certain amount of logical annoyance. There is the presence of what, from an ideal point of view, he might have called extraneous elements in the system. I shall merely cite two

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of these to indicate the flavour of the topic. First, there are many points where a crucial proof in Principia would fail if a universe of infinite objects were not assumed. (Even so basic a notion as the definition of 'successor,' in requiring that no two numbers have the same successor, must assume an infinite sequence.) Again the need for infinite classes, or at least the need for infinite possible classes, is evident at many points. Russell thus felt obliged to postulate what he called 'the axiom of infinity,' providing himself in effect with infinitely many objects. The axiom "assures us (whether truly or falsely) that there are classes having n numbers, and thus enables us to assert that n is not equal to n + I." Without the axiom we are left "with the possibility that nand n + 1 might both be the null-class" (IMP, p. 132). The awkwardness, not to say embarrassment, of this procedure is evident even to the non-specialist, and especially for Russell who had said, in another context: "There are advantages in the method of postulation, but they are the same as the advantages of theft over honest toil." Another case of impurity, topic of much discussion, is the status of those axioms which assert the existence of classes. It has seemed to most philosophers, including Russell, that logic itself ought to be entirely independent of any existence claims (also independent of any claim about the existence of an infinity of things). The most critical of the existence assumptions required, beyond the axiom of infinity, was what was called the axiom of choice, one form of which states that given any collection of nonoverlapping classes there exists a further class composed of just one member from each of the given classes. It seems like a reasonable enough axiom, and on the surface much less adventurous than the axiom of infinity. But it would be controversial to call it true "in virtue of its form." The recent reference to impurity is more than metaphor. The idea was to base mathematics upon logic, but the burden of this vast mathematical superstructure has placed logic under enormous pressure, and it was expanded somewhat in consequence. What it is for a class to exist is a problem that got less clear as work went on. Another problem that bothered Russell more, however, was that of finding a single characterization of logic that will demarcate the required materials: the axioms, indefinables and definitions of the propositional, predicate, and

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class calculi; the axioms of infinity, choice, and reducibility; and the vicious-circle principle of the theory of types. What is it that is common and distinctive among all these materials? Is it just that they are used in the derivation of mathematical ideas, and that books about them are classified by librarians under 'mathematical logic'? But there ought to be, if Russell's goals are to be reached, some more fundamental identification of the logical subject matter. Yet no such identification has been uncontroversially made. Russell has an excellent, but inconclusive, discussion of these matters in the last chapter of his Introduction to Mathematical Philosophy (1919), and again in the Introduction to the second edition of Principles (1937). In these and other places he says that the characteristic of a logical proposition is that it is true in virtue of its form. But not all the axioms required for a logical deduction of mathematics appear to have this feature; and he knows that the characteristic itself wants explaining. Wittgenstein in the Tractatus explained it in terms of 'tautology,' but while Russell does not reject this, he also hesitates to accept it as final. The ambivalence is because tautologies are designed to show what cannot be stated - and what is this but logical form once more, truth under all empirical conditions whatever? Is truth despite any empirical condition really truth in virtue of form? Is form then non-empirical? 'P or not-P' is a tautology. But is the proposition reporting this a tautology too? As for other devices for identifying the logical character of logical propositions, we have seen long since that the notion of 'necessity' has no central place in his logic at all, except in the attenuated form of universality: the calculus of Principia is satisfied to have its rules be truth-preserving, and does not require that they be necessarily so; hence there is no point in saying that logic is the domain of necessity. The notion of 'analytic' does not help him either, first because he never did believe that logical truths were analytic in any important sense (this is to be discussed shortly) and, secondly, other philosophers had collapsed what was acceptable in this idea into 'tautology.' The fact is that Russell thinks of logical form in terms of its being an idealization of the most general possible features of the world, not only of propositions but of anything whatever. And this is why his speculations, in the "Introduction" to the second edition of Principles, hover

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round the idea that 'truth in virtue of form' is itself an indefinable 'logical constant,' a rockbottom notion. But, he reflects, indefinables have themselves given away in the past, and of the remaining ones, 'all' and 'neither/nor,' "neither looks very substantia!''' So he is tempted too to return to his first view, which was just to list the principles and primitives and let that be the characterization of logic. Two firm ideas emerge from all this. The first is that Russell was always strongly inclined to think of logic primarily as being a priori; and always aspired to some analysis of this venerable and vague idea, whether in terms of 'indefinables,' or 'logical constants,' or 'axioms systems,' or 'tautology,' or 'analytic,' or 'logical form,' or some combination of them; and always remained unsatisfied, and hopeful, about his own best suggestions. The second is that, despite all, he remained firm in his conviction that mathematics is reducible to logic even while remaining unsatisfied with any available characterization of logic. There is nothing illogical in this position, since it is abstractly possible to know that X and Yare identical without knowing precisely what either is; but it is less than what he had hoped for. However, on this topic there still today exists less than he had hoped for.

15 It has, since Principia, become customary to observe a difference in the presentation of formal logic, between 'Natural Deduction' methods and 'Axiomatic' methods -roughly, between putting the weight on rules (if they are conceived in the appropriate way, all the weight can be put on the rules, and logic becomes its rules), versus putting as much as possible into the axioms, and as little as' possible into the rules. One idea behind this distinction is the supposition that natural deduction methods maintain a more intimate contact with the actual structure of reasoning than do the artificial creations of formal axiom systems. The resulting divergence in logical conception and exposition is beyond our story, except for a few points. In Principia the rules are minimized at the expense of the axioms: there is essentially just one rule for the propositional calculus, modus ponens, and one further somewhat obscured rule for the predicate calculus. This emphasis, in the light of the subsequent history of logic, has led to the reading of Principia as simply an early axiom system, and the resulting idea has been that Na,tural Pecluction methods?

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in diverging from this, have diverged from the spirit of Russell's inquiry. There is a vital sense in which this image of things, though widely disseminated among logicians, is profoundly misleading for understanding the nature of Russell's intentions. Russell, as we saw in the last chapter, was trying to present the objective complement of valid reasoning, the science of thought. And this aspiration is entirely in harmony not with the idea of Axiomatic methods which grew up around Principia, but with the Natural Deduction methods. The upshot is that, from the point of view of the present day, Principia is looking both ways. The idea which it embodied gives support to natural deduction, but the practice which it stimulated was in the line of axiomatics. Properly pursued, this contrast opens up a quite different historical understanding of Principia than that anticipated by Russell. Consider, for example, its class calculus, or set theory. It is only mildly inaccurate to say that Whitehead and Russell were in search of the one true and correct system of set theory, which would be the set theory for mathematics; but later logicians see that system as just one among many, each of which has certain different and interesting formal properties, and all of which can be interpreted in ways that accommodate the truths of mathematics. Set theories differ principally in terms of the axioms with which they set out, but also in terms of what is meant by the existence claims made by the axioms; and they diverge in terms of how classes are to be formed, and this in turn is dictated partly by considerations of elegance and mainly by the need to avoid Russell's paradox of non-self-membered classes (or any other contradictions). Now to think of a set theory as being constituted by a group of axioms arbitrarily chosen within these parameters was not quite what Russell had in mind. And the recent discovery that an axiom about the existence of sets of certain size (Cantor's continuum hypothesis, for example) such that either it or its negation can be added to a consistent set theory without disaster - this goes against the grain of everything that was expected. There is no telling what Russell, that is, Russell the logician of fifty years ago, would say about this. For it presupposes a placing of Principia's set theory within a certain context which he did not imagine for it. He makes it clear in the 1937 Introduction to Principles that his understanding of the

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relation of set theory to mathematics requires that as a matter of objective fact axioms be either true or false. And whatever this does mean it was meant to mean more than just that they are consistent. The situation is somewhat comparable to that which appeared at the end of the nineteenth century with the development of non-Euclidean geometry. Russell's response, in 1903, to that recent development is relevant here. He then endeavoured to find a more general characterization of geometry such that it would be included in the principles of mathematics, and thus of logic, as he understood these, without entering the question of the truth of Euclid's axioms. Very briefly, this consisted in developing the appropriate logic for saying that the axioms implied the theorems; and though this involved him too in giving definitions of 'point,' 'straight line,' and so on, it did not involve any claim about the truth of the axioms. It is a controversial question, not now relevant, whether Russell does make a successful case for including geometry in the principles of mathematics: a fourth volume of Principia to be written by Whitehead and devoted to this topic was never published. Be that as it may, Russell's account of the situation in geometry at the turn of the century, besides being interesting in itself, can be applied to his own work of ensuing years. In the following passage, I take the liberty to interpolate the alternative terms that would make it relevant to this discussion of Principia. Russell writes in Principles: "It was then proved, with all the cogency of mathematical demonstration, that premises other than Euclid's [Russell's] could give results empirically [logically] indistinguishable, within the limits of observation [number theory], from those of the orthodox [PM] system. . . . But the investigation produced a new spirit among Geometers [Logicians]. Having found that the denial of Euclid's axiom of parallels [Russell's axioms in favour of Zermelo's, or Godel's continuum hypothesis] led to a different system, which was self-consistent, and possibly true of the actual world, mathematicians [logicians] became interested in the development of the consequences flowing from other sets of axioms more or less resembling Euclid's [Russell's]. Hence arose a large number of Geometries [logical systems], inconsistent, as a rule, with each other, but each internally self-consistent. The resemblance to Euclid [Russell] required in a suggested set of

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axioms has gradually grown less, and possible deductive systems have been more and more investigated on their own account. In this way, Geometry [Set Theory] has become (what is was formerly mistakenly called) a branch of pure mathematics, that is to say, a subject in which the assertions are that such and such consequences follow from such and such premises, not that entities such as the premises describe actually exist" (Principles, p.373). That this passage could have been put to this use to describe his own work is not something that would have occurred to Russell in 1903 when he wrote it, nor something that would have satisfied him in 1937 when he reprinted it.

16 If Russell's followers had come to understand Principia, technically, somewhat differently from what Russell expected, it is also true that he had himself come to understand it philosophically differently. And this, too, cast its own curious shadow over the entire enterprise of tracing mathematics back to logic. Russell wrote as recently as 1959 that "the primary aim of Principia M athematica was to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms. This was, of course, an antithesis to the doctrines of Kant, and initially [sic] I thought of the work as a parenthesis in the refutation of 'yonder sophistical Philistine,' as Georg Cantor described him, adding for the sake of further definiteness, 'who knew so little mathematics'" (MPD, pp 74-5). The particular doctrine of Kant's against which Russell here implies Principia was directed is important for understanding Russell's work. Briefly, it is the claim that all true mathematical judgments are synthetic and a priori. To understand Russell's relation to this still controversial doctrine it will be necessary to linger briefly over Kant's distinction between analytic and synthetic judgments. Kant speaks of judgments, taken as acts of the mind, rather than of more independent entities such as propositions or statements. Moreover, he thought of all judgments as consisting of a subject and predicate, a notion which is seriously defective it we apply it to propositions, for, as Russell often pointed out, it entirely neglects relational propositions or judgments. The failure to treat relational propositions as funda-

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mental, a failure implied by Kant's doctrine of judgments, is then the first point of Russell's objection. But it is not to this that Russell chiefly refers above. Put simply, the question whether a judgment is analytic is the question whether the predicate concept is contained either explicitly or implicitly in the subject concept. For example, on these grounds respectively both 'black cows are black' and 'black cows are bovine,' are analytic, whereas 'black cows are good milkers' is synthetic, and 'black cows are stupider than brown ones' is, according to Kant, synthetic (and according to Russell, relational, but not analytic). The important point is that the truth of an analytic judgment depends solely upon the meanings or analyses of its subject and predicate terms: nothing but logic and the dictionary are needed to show that they are true. Kant held that the truths of mathematics, like certain presumed truths of metaphysics (examples: 'events have causes,' 'attributes belong to substances'), were both synthetic and a priori: synthetic because the predicate in such judgments was not contained- even implicitly in the subject; a priori, because learning the truth of such propositions involves no appeal to empirical experience. This doctrine of Kant respecting the synthetic a priori character of mathematical truth, impossible as it is to make entirely clear in brief compass, has stimulated a vast amount of important philosophizing, and it may be said that (excepting a few empiricists such as Mill) it was the received opinion throughout the nineteenth century. It was explicitly supported by Russell in two of his first books: An Essay on the Foundations of Geometry (1897) and A Critical Exposition of the Philosophy of Leibniz (1900). Now Russell tells us in the statements quoted above that his own view in Principia, that mathematical entities could be defined in terms of logical ideas, and that mathematical truths could be deduced from and with "purely logical premises" was in opposition to the Kantian doctrine. But in fact the development of Russell's opinions is by no means as straightforward as that. This is important for various reasons, among them the special historic impact that Principia had in many circles, most notably among the logical positivists. After the first volume was published and while the remaining volumes were at the printer Russell wrote in T he Problems of Philosophy (1912) that it was Kant who "perceived that we have

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a priori knowledge which is not purely 'analytic' " and who saw that "all the propositions of arithmetic and geometry, are 'synthetic.''' "He pointed out, quite truly, that 7 and 5 have to be put together to give 12: the idea of 12 is not contained in them, nor even in the idea of adding them together. Thus he was led to the conclusion that all pure mathematics, though a priori, is synthetic" (pp. 127, 130). Russell enters no demurrer at all against this view even though (on the interpretation of Principia that later became current) he had just co-authored a three-volume refutation of it. What had happened? He has never paused to explain it, but it seems that the situation is approximately this: Russell believed before, and long after the writing of Principia, that (many, perhaps most of) the principles of logic, which provided the basis for all mathematics, were themselves synthetic, including what he took to be the most fundamental principle of all, the first primitive proposition of Principia, viz. "Anything implied by a true premiss is true"; and including, we must suppose, the logical translations we gave of the argument we called 'K' in Section 10 above. In short, what Russell believed in 1912 after Principia was precisely what he had written in 1903: "Kant never doubted for a moment that the propositions of logic are analytic, whereas he rightly perceived that those of mathematics are synthetic. It has since appeared that logic is just as synthetic as all other kinds of truth" (Principles, p. 457, my italics). Now let us pose a rather woolly question, designed only to emphasize Russell's problem. Suppose that mathematics can be extracted 'analytically' from logic, because it is implicitly contained within it in the way explained in this chapter. Suppose further that certain basic principles of logic are, as Russell believed, synethetic. Is mathematics itself then shown to be analytic, or synthetic? Does not the presumed synthetic character of the logical premises 'ride on through' to the mathematical results? To be more specific: In our example in Section 11 above, if I was pure logic and was synthetic, then was not III, which was set theory and equivalent to it, also synthetic? And was not 'one plus two equals three,' which was equivalent to both, also synthetic? Evidently Russell supposed so in 1912. But the question leaves him in an awkward position apropos the refutation of Kant. Mathematics from the point of view of the fact that its ideas are embedded in logic, is analytic;

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but from the point of view of its own propositions themselves, it is synthetic - a perplexing result. Russell, however, was in this way freed for a time to be both Kantian and anti-Kantian, but only at the price of rendering the whole problem ambiguous.* The above is a drastically curtailed account of a crucial and complex situation; the problem today has little of this nice and curious simplicity. The inconvenient truth is that Russell himself was not much agitated by a topic which Principia did most to stimulate and which has been noisily debated in the journals, viz. the distinction between 'analytic' and 'synthetic' : its criteria, importance, relation to a priori knowledge, and to mathematics and logic. Frege, whose philosophy of mathematics coincided essentially with Russell's, did not have Russell's problem, because he defined analytic truths (as Russell certainly did not) as those which followed from explicitly formulated laws of logic plus definitions fashioned in accordance with them. Frege was in this respect determinedly anti-Kantian from the outset even in his definitions of analytic truth. It first became possible for Russell to think of Principia as being radically anti-Kantian only years after its publication: under the influence of Wittgenstein in the late teens, he modified his conception of logic, suppressing instead of resolving questions about the analytic or synthetic nature of the principles of logic and, as it were, merely declined to reject the idea that the essence of logic was to be sought in the notion of tautology, which he despaired of defining. And, as we saw, he never did arrive at a firm conception of the boundaries of logic, other than the firm but vague conviction that it was somehow a priori. By no stretch of terminology can it be claimed that all the required principles of Principia are or could be made tautologies, though this was an interpretation, or an ideal, widely made by the first generation of Russell's followers, who read him through Frege's and Wittgenstein's eyes. Russell's latter day report on the anti-Kantian "primary aim" of Principia is therefore misleading and seriously anachronistic. One moral to be drawn here is that we must distinguish philosophical aims and claims made on behalf of Principia from the technical accomplishment that it was, and from the influence that it had. The logical positivists, for example, and possibly the later Russell as well, saw it as a successful assault upon the very citadel of rationalism - neglecting to note that it was written by

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committed rationalists. Mathematical truth had always been a paradigm of the a priori; but if mathematical truth is only logical truth, and if logical truth is based merely upon meaning and ultimately perhaps on rules and conventions (a further nonRussellian thesis) then the rationalist is deprived of his most prominent reason for thinking the mind is in contact with the special realm of non-empirical truth. For positivists at least, Principia was indeed a gallant 'reduction,' and they expected, even if Russell did not, that all that is a priori in metaphysics would be similarly demoted to the status of empty tautological meanings. For these students of Principia (better: for those prophets who pointed to this work for authority) the traditional 'grandeur' of both mathematics and metaphysics lay, not in objects for the mind's special insight, but in the mind's special blindness to the nature of the language. Russell himself was ambivalent on all this. Understandably so. Mathematics had first deeply captured his imagination, as it had Descartes', as a paradigm of knowledge. But upon analysis the very feature which had made it so impressive - its coerciveness, inexorability - seemed to be either inexplicable (when traced to the mysteries of logic) or not a very clear case of knowledge: thus the poignancy of his saying that he "suffered when Wittgenstein led me to regard it as nothing but tautologies." Not only was the notion of mathematics as especially excellent knowledge severely qualified, the mystical side in Russell's nature suffered a set-back as well- a theme with which this chapter began and which will recur in the last chapter of this book. We know that, more than metaphysics or logic or even life itself, it was mathematical truth that had fed the real flame in his soul. The truths of mathematics had been arrayed in timeless purity, perfect objects of the metaphysicians' contemplation, a refuge in Being for exiles from mere existence: "The true spirit of delight, the exultation, the sense of being more than man, . . . is to be found in mathematics as surely as in poetry" (M&L, p. 60). And so on. The promise of mathematics, sublime and remote, had been to emancipate Reason from the clutter of particulars, to uphold images of truth and eternity and incorruptible beauty, and to vouchsafe an occasional platonic hint that the hunger for perfection has somewhere an answering echo. . . . Well, these things are the faded embellishments on an enduring legacy. For

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there remains the fact that Principia M athematica, together with the work it incorporated and inspired, has been itself, for more than half a century, a fixed reference point for an entire dimension of intellectual life. However banal are the numerical entities of which it treats, one still rightly flutters its pages with a certain awe.

* In

about 1900 Russell wrote and left unpublished a short paper (now at the Russell Archives) on "Necessity and Possibility" in which he tentatively accepted the idea - rejected in published works - that the analytic might be definable in terms of deducibility from the (synthetic) laws of logic. He would not have thought that this made mathematical propositions analytic, and in Leibniz (1900) he states explicitly that they are synthetic. A still earlier and longer manuscript, more or less completed and abandoned in the summer of 1898, "The Nature of Mathematical Reasoning" (also at the Russell Archives) manages to circumvent most questions about analyticity and necessity in mathematics. Russell's early published views on necessity, especially in Leibniz, and Moore's views on the same topic are discussed in my "Analyticity and Necessity in Moore's Early Work" Journal of the History of Philosophy, October 1969.

CHAPTER 5

ATOMISM: THEORIES OF LANGUAGE

In the first flush of my belief in separate atoms, I thought that every word that can be used significantly must signify something, and I took this to mean that it must signify some THING. But the words that most interest logicians are difficult from this point of view. They are such words as 'if' and 'or' and 'not.' I tried to believe that in some logicians' limbo there are things that these words mean, and that perhaps virtuous logicians may meet them hereafter in a more logical cosmos. I felt fairly satisfied about 'or' and 'if' and 'not,' but I hesitated about such words as 'nevertheless.' My queer zoo 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 non-existence. There are still a number of philosophers who believe this sort of thing, and it is their beliefs which have become the philosophical basis of Existentialism. But, for my part, I came to think that many words and phrases have no significance in isolation, but only contribute to the significance of whole sentences. I have therefore ceased to hope to meet 'if' and 'or' and 'not' in heaven. I was able, in fact, by the roundabout road of a complicated technique, to return to views much nearer to those of common sense than my previous speculations. Russell, "Beliefs: Discarded and Retained" in Portraits from Memory (1956)

CHAPTER 5

ATOMISM: THEORIES OF LANGUAGE

The task of the present chapter is to examine the way in which the theory of descriptions and a small cluster of associated ideas about logic, language, and meaning supported the main developments of Russell's thought for more than two decades. The subsequent chapter will consider the epistemological side of the same themes. The implied distinction between linguistic and epistemological issues is an artificial division, but a possible and convenient one; and for both the theory of descriptions is basic. In this chapter, Section A explains the reasons Russell saw as creating a need for a new theory of denoting, later called the theory of descriptions, later still called the theory of incomplete symbols. Section B (Reference, Metaphysics, and Existence) pursues consequences of the theory into questions of reference and proper names. Section C (Acquaintance and Construction) indicates the ways in which he began to make epistemology out of his theory as well. Russell's atomism is more a series of projects, a collection of exciting possibilities, than a finished doctrine. He is drawing on Whitehead, Wittgenstein, Moore, and his own swiftly dancing logical imagination to map out work for himself - and, as it happened, a generation of followers. He recommends more doctrines than he defends, and he starts far more philosophical hares than he has time or inclination to run down. Accordingly, the maxim I laid down at the beginning of this book - to endeavour to keep faith first of all with the imaginative thrust of his mind - is especially pertinent here.

A.

THE

THE 0 R Y

AND ITS

0 F

DES C RIP T ION S

CONSEQUENCES

1 This theory, which dates from Russell's "On Denoting" in 1905, is probably his single most praised and influential H

2~

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philosophical idea. F. P. Ramsey is much quoted as having called it a "paradigm of philosophy," to which compliment the ageing G. E. Moore, after he had devoted his most prolix, most painstaking, and most boring essay to the theory, gave enthusiastic assent, a fact which obviously gratified and relieved Russell (see Schilpp, pp. 175ff.). Much of Wittgenstein's work in both acceptance and reaction is, as he frankly acknowledged, inspired by the theory of descriptions. Professor John Wisdom has said: "Although the theory of descriptions will not do everything, it opened a new era in metaphysics." Many have said that it marks the beginning of analytical philosophy. Russell himself, in his A History of Western Philosophy, modestly said that the theory "clears up two millennia of muddle-headedness about 'existence' "

(p.860).

An innocent reader confronted with this advertisement may well expect that the theory in question is recondite and impossibly profound and obscure, as so often happens with what philosophers praise. Like monads, for instance, or prime matter, or Absolute Substance. He may feel let down if he is told, as he now is, that in essence the theory is the recommendation to reformulate propositions such as 'the present king of France is bald' into their 'true logical form,' which is this: 'there is one and only one thing which is king of France and that thing is bald.' What, he may wonder, is all the applause about? Surely philosophers have been streamlining prose in the interest of logical form since Socrates forced the young scholars of Athens to state more clearly exactly what they meant. The theory of descriptions is essentially a technical device for exposing the logical character of certain basic expressions, the rationale for which merges with a larger network of controversial theories. In fact, as Russell worked it out, the theory impinges on semantics, logic, metaphysics, and theory of knowledge. As stated in the previous paragraph it is at bottom an exceedingly simple idea. But it embodies a thorough-going suspicion of ordinary language as potentially misleading, and a very firm reliance upon the clarifying power of logical uniformity. Let us reflect on this situation in Russell's thought before looking at the theory in detail. For, as elsewhere in this book, we want to see things as they presented themselves to Russell, rather than as they might appear in a textbook.

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Russell's suspicion of ordinary language may be illustrated by the central claim of his new theory, namely, that propositions containing descriptive phrases are not of the subject-predicate form as they appear to be. He did not begin philosophizing with this guilty-until-proved-innocent attitude toward language. Before the publication of "On Denoting" he often candidly deferred to the ordinary tutelage of ordinary language. "The study of grammar, in my opinion, is capable of throwing far more light on philosophical questions than is commonly supposed by philosophers . . . . On the whole, grammar seems to me to bring us much nearer to a correct logic than the current opinions of philosophers ... " (Principles, p. 42). This early attitude, whose morning innocence Russell was later to deplore, led him to his first theory of denoting (recall for example the a/some distinction) discussed in Chapter 3. Deplore it he did, though the rebuke, which came fifteen years later, was addressed not so much to his own former opinions as to philosophers generally: "Misled by grammar, the great majority of those logicians . . . have regarded grammatical form as a surer guide in analysis than, in fact, it is" (IMP, p. 168). Principia Mathematica and three other books intervened between these two statements, but most importantly it was "On Denoting" and the theory of descriptions that intervened. Within six more years (1925) Russell had settled into the more moderate opinion which happens also to summarize a cardinal tenet of much subsequent philosophy, and not only his own: "the influence of language on philosophy has, I believe, been profound and almost unrecognized. If we are not to be misled by this influence, it is necessary to become conscious of it ... " ("Logical Atomism," L&K, p. 330). To become conscious of an influence in order not to be misled - it is just this lesson which more than anything else is the major message of English-speaking philosophers of this century. It was Wittgenstein's judgment, first when he was a student and cohort of Russell's, and more emphatically when he was later a philosophical opponent of Russell's, that most, perhaps all, philosophical problems consist of little more than the results of being misled by language. To expose the linguistic facts which mislead us, and the intellectual cravings which abet us, is, on this view, the chief task of philosophy. Russell, despite the rhetoric about the muddles of millennia, has never really shared this somewhat

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nihilistic attitude about the nature of philosophical problems; but the theory of descriptions, the views which prompted it, and the commentaries which it inspired, are historically the principal source of this attitude. The theory of descriptions may be presented as a way to escape certain misleading influences. (See quotation at the head of this chapter.) Very well: Russell had said in Principles, "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. For what does not exist must be something, or it would be meaningless to deny its existence" (pp. 449-50). Here we have explicitly the view that a proposition whose subject term fails to refer to any entity at all is meaningless; so all meaningful ones do so refer. By virtue of his new theory, he was to reject this view entirely; but why should he have blamed language for being misleading in this respect? The answer is rather abstract. In the first place, it is plausible to suppose that the primary business of language is to refer to a reality beyond itself, and, secondly, language is the primary custodian and vehicle of meaning. Surely these factors are connected. A simple and plausible way in which they might be connected is just this: having meaning and denoting (referring to) an object are merely opposite sides of one semantic coin; words have meaning because they denote, and vice versa. This conception is simple and compelling, and for realists, from Plato to the present day, it has always been decisive. It was for some years decisive for Russell; and by 1905 it had already supported myriad arguments in several books. Verbs and prepositions denoted relations, names and noun phrases denoted persons or things, general words denoted classes, etc.; what is called logical constants (e.g. 'not,' 'or,' 'implies'), the so-called 'syncategorematica' of medieval philosophy, were the exceptions to the rule, they and they alone derived their meaning from their context instead of from denoted entities. (Russell never did quite believe what he attributes to himself in the quotation at the head of this chapter.) Russell did not make the mistake of assuming that the meaning of the word is the corresponding object - though this has long been a favourite way of misinterpreting him - but he did assume that, except for logical words, having a meaning and having a corresponding

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denoted object were interdependent factors, that each implied the other. The notion that words and unitary phrases have always the two interconnected aspects, meaning and denotation, each dependent upon the other, carries with it another assumption very natural to common sense and common grammar. In the proposition expressed by 'A bird is in the garden' we are concerned with three denotations, bird, garden and the relation in; and just as each word or phrase may be said to have its own denotation, so each may also be said to have also its own meaning, more or less independently of the meaning of the rest of the sentence; this is what dictionaries are all about. That meaning and denotation are always both involved is one assumption; that they are interdependent is another assumption; and that each meaning, like the denotation, is a separate affair detachable from the other parts of the sentence-meaning is a third assumption. Out of such presumptions does the philosophy of language arise. Enough of this. The last two paragraphs contain a number of spongy ideas, all plausible and vague. They had to be, for they are misleading, just as Russell said. Where the semantic picture which they suggest -let us call it 'meaning realism' - dominates one's philosophical imagination, and also leads to bizarre results, a clear and innovative way of breaking the picture will seem like a major philosophical breakthrough. One bizarre result that meaning realism leads to may be illustrated this way. 'The present king of France is bald.' 'The present king of France does not exist.' What do these propositions refer to? Not an existent, so they must refer to a subsistent being. "For what does not exist must be something, or it would be meaningless to deny its existence," Russell had said. This sort of doctrine, he said later, lacked "that feeling for reality that ought to be preserved in even the most abstract discussions" (IMP, p. 169). Let us now turn to therapy. 2 Russell has stressed, somewhat more than his followers, the 'principle' of this theory of descriptions, viz. "that denoting phrases never have any meaning in themselves, but that every proposition in whose verbal expression they occur has a meaning" ("On Denoting," L&K, p. 43). Reflecting on this principle, which

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breaks with meaning realism, enables us to reconstruct the nerve of his argument in a way that brings out an important feature, namely, that the rejection of a certain ontology is a presupposition and not a consequence of his theory. Here are the essential moves. Step 1: Consider the sentence: 'the present king of France is bald.' If its subject phrase 'the present king of France,' is meaningful, it has a denotation; but if it has a denotation, there is a present king of France. Yet in fact there is no present king of France (either in existence or subsistence), hence that phrase does not denote such a king, and therefore it is not meaningful. Step 2: But this result goes counter to our clear intuition that the entire sentence, at least, is meaningful, or significant. This intuition can be salvaged by recognizing that the denoting phrase 'the present king of France' is only misleading shorthand, inducing a mistaken subject-predicate analysis of the original sentence. If its correct logical form can be discerned, the entire sentence can be meaningful but not 'the present king of France' in isolation. Step 3: So, the theory says, the original sentence is actually of a disguised and complicated existential form, equivalent to a conjunction of the following three sentences; and since the first of these conjuncts expresses a proposition which is false, the original is false, though certainly not meaningless. (Alternative readings for (a) are: "Something is presently king of France'; 'For some x, x is presently king of France'; 'There is an x such that x is presently king of France.') (a) There is at least one present king of France: (3x) [Fx (b) At most one person is presently king of France: (y) (Fy > x = y) (c) That person is bald: Gx]. Collecting the pieces we say that 'The present king of France is bald' is now to be symbolized as (3x) [Fx· (y) (Fy > x = y). Gx] All of (a), (b), and (c) except the last two words of (c) are in effect what 'the present king of France' has now expanded into, and it is easy to see that this entire cluster of words, taken by itself, unlike 'the present king of France,' does not even appear to form a meaningful unit, any more than, say, 'king of ... and that' does. It struck Russell very forcibly therefore - as against the vague prejudices of meaning realism - that denoting phrases were to be thought of as inseparable parts of their contexts. Their

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meaning now had nothing to do with whether they successfully referred to something or not, but rather with whether they were organic parts of meaningful wholes. Hence the 'principle' of the theory cited above. In calling the original proposition false we were merely assuming that in fact there is no present king of France, and so it may be asked: even if no such king exists, does he not perhaps yet have being? Russell had supposed just this in Principles a few years earlier. But now he rejects it out of hand. The theory of descriptions provides no direct attack against a being-king; but it does pull the fangs of one argument for it, namely the argument that since the sentence was meaningful there had to be a denotation for the phrase 'the present king of France.' The presumed meaning of denoting phrases, Russell believed, had to be accounted for on some other principle than the meaning realist principle that they had a denotation. This other principle is that "a denoting phrase is essentially part of a sentence, and does not, like most single words, have any significance on its own account" (p. 51). In fact, if it has no meaning or significance in isolation, then the question of its denotation in isolation does not even arise. The phrase, indeed, has disappeared, upon logical analysis, into quantified variables and predicates. Thus Russell spoke of his theory as giving a "reduction [more accurately: expansion] . . . to forms in which no such phrases occur" (p. 45). Fragment (a) above is therefore not seen, as 'the present king of France' might be, as implying, via a vague semantic assumption, that there is somewhere, somehow, a present king of France; rather it expresses it plainly, and being false, makes the original proposition which embodied this part false, though meaningful. There is a more direct way to see the importance of Russell's new 'principle,' and this point is independent of the particular manner in which he arrived at it. We noted that the latter part of the original denoting phrase (viz. 'present king of France'), symbolized 'F,' is now treated as a predicate; it is as much a predicate in the analysed form as 'is bald.' The unity of the original sentence is recaptured in the symbolic form by the presence of the variable 'x' in each fragment, which serves as the link between the original predicate and the new-created predicate. Now this transformation of original grammatical subjects (e.g. 'the king of France,' 'centaurs') into logical predicates (e.g.

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'X is king of France,' 'x is a centaur'), with the variables doing the subjects' job, is a vital part of the theory. For example, it provides an antiseptic way to deny that something exists without any suspicion that the denial appeals to subsistence. Subsistence and existence can now, if we like, be denied at one stroke. Instead of saying 'centaurs don't exist,' 'the present king of France does not exist,' with the hazy worry that they thereby subsist, we can say this: 'is a centaur,' and 'is the king of France' apply to nothing, to nothing at all; more exactly, that 'x is a centaur' or 'x is presently king of France' has no true instance. Hence, to say 'centaurs don't exist' in this logic is to say effectively that they don't subsist either, viz. 'ro-' (3x) (ex).' Thus the principle of Russell's theory has applications beyond the realm of definite descriptions, and this extension of the theory will have to be taken up later.

3 We have now reconstructed those lines of thought, semantic in character, that motivated the formulation of the theory. Other lines of thought, more purely logical in character must be considered. So far we see that propositions are rewritten to expose misleading grammatical form, existential implications are exposed, subjects become predicates, denoting phrases get the same kind of analysis whether they start out as subjects or predicates, and meaning for denoting phrases is imbedded in its context. Here, semantic considerations shade into logical ones. Russell often says that the theory of descriptions shows that names are logically distinct from descriptions. Let us consider two examples of what this means. (a) Consider the ambiguity in such a proposition as 'it is false that the man next door is Dutch.' This could convey the idea that there is no man next door, or that there is but he is not Dutch. No such ambiguity appears if for the definite description 'the man next door' we had the name 'Smith.' The trick is therefore to give a symbolic representation of the alternative ways that the symbols for 'the man next door' relate to those for the rest of the proposition. Russell speaks of the primary occurrence (when the idea is that there is no man next door) and the secondary occurrence (when the idea is that he is not Dutch) of the descriptive phrase, and he represents the difference symbolically in terms of what, in Chapter 3, we called the scope of the expressions. So a description requires, and can get, a

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different analysis than a name even when both refer to the same thing. (b) Another thing the theory does with the logical distinction between names and descriptions is to give an explanation, as Russell put it, of "the usefulness of identity." "George IV wished to know whether Scott was the author of Waverley; and in fact Scott was the author of Waverley. Hence we may substitute Scott for the author of (Waverley,' and thereby prove that George IV wished to know whether Scott was Scott. Yet an interest in the law of identity can hardly be attributed to the first gentleman of Europe" ("On Denoting," WK, pp. 47-8). Russell's theory explains the puzzle in terms that can be symbolized along the lines given earlier. It is not that George IV wished to know if Scott was Scott, but rather it is this: one and only one person authored Waverley and George IV wished to know if Scott was this person; the point being that 'the author of Waverley' does not, like 'Scott,' get its meaning by naming, but in this indirect way. A name and a description that identify the same entity cannot, therefore, in general be substituted for each other, and Russell's theory makes this logically transparent. Of course it had been known for centuries that something like this had to be true. The question was, exactly what is true about identity statements - in particular, those identity statements where at least one of the terms is referred to by a 'the'-phrase, that has application elsewhere? To this Russell has given a systematic answer. Though Russell introduced the theory in "On Denoting" by emphasizing the way that the meaning of a denoting phrase was an organic part of its context, and later dramatized it in terms of the contrast of names and descriptions, he could use it as a logician to facilitate logical deductions by means of the quantifiers. Logic needs the notion of uniqueness; we need to be able to represent formally the idea of 'just one x' or 'the only x,' in a systematic way. It is this idea, says Russell, that is (usually, not always) behind phrases like 'the mayor of Chicago,' where 'the' implies 'the one and only.' Now the theory of descriptions defines the idea of uniqueness in terms of two others, quantification and identity. 'The mayor of Chicago' becomes 'something is mayor of Chicago and anything that is mayor of Chicago is identical to that thing.' So now there is a formal technique for diagnosing entire propositions containing definite descriptions. For example: (1) 'The

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clerk who assists the mayor of Chicago assists only short men' and (2) 'The mayor of Chicago is short.' By the rules of quantification theory we can do what Aristotelian logic, in bondage to subject-predicate logic, could not do, viz. produce a formal translation of these propositions and a formal deduction of (2) from (1 ). It is apparent now that the theory of descriptions, as a logical device, depends upon the background theory of quantification. But we saw in Chapter 3 that Russell had a very incomplete conception of quantification in Principles, published just two years before "On Denoting." In that silent interval he had simply given up parts of his old theory: for example, the three quantifiers we elicited (p. 146ff.) reduce to two, and the idea that the quantifiers 'denote' curious classes of entities is now wholly swept aside. He exhibits this, not in these words but by explaining the connection between what he called definite descriptions (denoting phrases) and indefinite descriptions (denoting phrases). It is necessary to explain his explanation and this will take us directly back to the text of "On Denoting." 'The'-phrases indicate definite descriptions, so called for obvious reasons. In a sense, however, indefinite descriptions are logically more important, because the more fundamental; they have been appealed to above. Indefinite denoting phrases may be indicated by such words as 'whatever,' 'whoever,' 'somebody,' 'anybody,' 'a man,' and the like. In any case a "phrase is denoting solely in virtue of its form" (L&K, p. 41), said Russell. The important point about this form is just that these are all reducible to variations on 'some' and 'every.' And this indicates what, from the point of view of the development of Russell's logical ideas, is a most important part of the theory of descriptions. The essay "On Denoting" and its theory of descriptions is the place where he first formulated for himself (and apparently still independently of Frege, who had done it before him) that part of quantification theory in what has ever since been its essential form. Gone are the various kinds of denoting phrases of Principles, gone the subtle distinctions between 'some x' and 'an x,' which gave to these expressions different meanings and different denotations; this and much more disappears because the entire attempt to apply meaning realism to denoting phrases (recall "the very paradoxical objects" associated with 'any x,' 'an x,' 'every x,' 'all x,' as discussed in Chapter 3

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above) is discarded. In the following central passage from "On Denoting," into which I have bracketed comments, it is evident that Russell's theory of indefinite descriptions is, from a logical point of view, simply the applying of quantification to erstwhile denoting phrases. This involves, to use the current jargon, that their meaning is now thought of syntactically and not semantically, and that in turn means that the quantified variable and not the denoting phrase is the primary vehicle of denoting. "My theory, briefly, is as follows. I take the notion of the variable as fundamental; I use 'C(x)' to mean a ... [propositional function] in which x is a constituent, where x, the variable, is essentially and wholly undetermined. Then we can consider the two notions 'C(x) is always true' and 'O(x) is sometimes true.' [This is his version of the now more common 'for all x, Cx' and 'for some x, Cx.'] Then everything and nothing and something (which are the most primitive of denoting phrases) are to be interpreted as follows .... [The textbook rendering is then given.] Here the notion 'Cx is always true' [universal quantification] is taken as ultimate and indefinable, and the others are defined by means of it [departure from Principles]. Everything, nothing, and something are not assumed to have any meaning in isolation [and a fortiori no denotation in isolation, as they had], but a meaning is assigned to every proposition in which they occur. This is the principle of the theory" (p. 42). To apply this theory of indefinite descriptions to the case of definite descriptions is to define the definiteness or uniqueness conveyed by 'the x' by means of two other sets of ideas: that of 'some x,' the existential quantifier; and the idea of 'not more than one x,' where the latter is accomplished by means of the universal quantifier and identity. All this, extremely ingenious in its conception, has now become routine and almost a commonplace of logic books; but this is the passage in which it was first borne in on the attention of philosophers. Thus is illustrated what was implied in Chapter 3: though Frege and Peirce had first discovered quantification theory, it was Russell who really placed it on the map. It was for Russell a special kind of liberation from the cruel battles with propositions, constituents, and unities that we

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discussed in earlier chapters. For example, the old puzzle as to whether 'I met a mermaid' contains as constituents besides me and met one mermaid, all mermaids, or a denoting concept for mermaid - the question now fades. If it is constituents that we want, they are: 'is a mermaid and is met by me' and 'sometimes true'; that is, they are predicates, variables, and quantifiers. But these are things we talk about when we are talking not of the old idea of constituents but of the new idea of logical form. It is a mark of the depth of Russell's philosophy that a successful answer to one question simply raised new ones. B.

REFERENCE,

METAPHYSICS,

AND

EXISTENCE

4 By tracing the origin of our theory from Russell's own point of view we have managed to place it at a crucial intersection in his theoretical outlook. Among the main converging lines of thought were: descriptive phrases have no meaning in isolation; descriptions are logically distinct from names; the disavowal of subsistent beings; quantification theory moves to the foreground; the use of descriptions for logical diagnosis and deduction. It will therefore not be surprising to discover that the theory sent reverberations through the whole of Russell's philosophy. We recall that his early realism, the blend of logic and ontology expounded in Chapter 2 above, had been a rich and unstable mixture. The theory of descriptions and the theory of types crystallized his thought into what he called 'the philosophy of logical atomism.' We can trace the path of these developments if we bear in mind that Russell's thought moves forward on several fronts at once. So, a set of fairly large topics is now before us, all springing from the theory of descriptions. An implied theory of reference, a theory about 'existence' and 'being,' a doctrine of proper names, a vision of ontological simples and of an idealized language: these matters, in that order, will occupy us in the remainder of this chapter. Just as intimately associated with the theory of descriptions is the doctrine of knowledge by acquaintance versus knowledge by description; and the doctrine of logical constructions. These will be adumbrated at the end of this chapter and considered in the next chapter. The number and variety of these topics emphasizes how rich were the philosophical ideas that Russell extracted from one logical

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man