Vagueness [1 ed.] 0754620808, 9780754620808

Table of contents :
Contents
Acknowledgements
Series Preface
Introduction
Part I: Nihilism
1 Peter Unger (1979) ‘There Are No Ordinary Things’
2 Samuel C. Wheeler (1979) ‘On That Which Is Not’
3 David H. Sanford (1979) ‘Nostalgia for the Ordinary: Comments on Papers by Unger and Wheeler’
4 Bertil Rolf (1984) ‘Sorites’
5 Roy A. Sorensen (1985) ‘An Argument for the Vagueness of “Vague”’
Part II: Observational Predicates
6 Crispin Wright (1975) ‘On the Coherence of Vague Predicates’
7 C.L. Hardin (1988) ‘Phenomenal Colors and Sorites’
8 Christopher Peacocke (1981) ‘Are Vague Predicates Incoherent?’
Part III: Degrees of Truth
9 R.M. Sainsbury (1986) ‘Degrees of Belief and Degrees of Truth’
10 Dorothy Edgington (1992) ‘Validity, Uncertainty and Vagueness’
Part IV: Epistemicism
11 Richmond Campbell (1974) ‘The Sorites Paradox’
12 Timothy Williamson (1996) ‘What Makes it a Heap?’
13 W.D. Hart (1992) ‘Hat-Tricks and Heaps’
Part V: Higher-Order Vagueness
14 Mark Sainsbury (1991) ‘Is There Higher-Order Vagueness?’
15 Crispin Wright (1992) ‘Is Higher Order Vagueness Coherent?’
16 Dorothy Edgington (1993) ‘Wright and Sainsbury on Higher-Order Vagueness’
17 Richard G. Heck Jr (1993) ‘A Note on the Logic of (Higher-Order) Vagueness’
18 Timothy Williamson (1999) ‘On the Structure of Higher-Order Vagueness’
19 Dominic Hyde (1994) ‘Why Higher-Order Vagueness is a Pseudo-Problem’
20 Michael Tye (1994) ‘Why the Vague Need Not be Higher-Order Vague’
Part VI: Contextualism
21 Hans Kamp (1981) ‘The Paradox of the Heap’
22 Jamie Tappenden (1993) ‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’
23 Diana Raffman (1994) ‘Vagueness Without Paradox’
Part VII: Intuitionism
24 Hilary Putnam (1983) ‘Vagueness and Alternative Logic’
25 Stephen Read and Crispin Wright (1985) ‘Hairier than Putnam Thought’
26 Hilary Putnam (1985) ‘A Quick Read is a Wrong Wright’
27 Timothy Williamson (1996) ‘Putnam on the Sorites Paradox’
Name Index

Citation preview

Vagueness

The International Research Library of Philosophy Series Editor: John Skorupski Metaphysics and Epistemology

Identity Harold Noonan Personal Identity Harold Noonan Scepticism Michael Williams Infinity A. W Moore Theories of Truth Paul Horwich The Existence of God Richard M. Gale, Alexander Pruss

Knowledge and Justification, Vols I & II Ernest Sosa

Spacetime Jeremy Butterfield Mark Hogarth, '

Gordon Belot

A Priori Knowledge Albert Casullo Vagueness Delia Graff and Timothy Williamson Supervenience Jaegwon Kim

The Philosophy o f Mathematics and Science

The Ontology of Science John Worrall Mathematical Objects and Mathematical Knowledge Michael D. Resnik Theory, Evidence and Explanation Peter Lipton

The Philosophy o f Logic, Language and Mind Events Robert Casati, Achille C. Varzi Consciousness Frank Jackson Reason, Emotion and Will R. Jay Wallace

Understanding and Sense, Vols I and II Christopher Peacocke The Limits of Logic Stewart Shapiro

The Philosophy o f Value

Consequentialism Philip Pettit Punishment Antony D uff Meta-ethics Michael Smith Human Sexuality Igor Primoratz The Notion of Equality Mane Hajdin

The Ethics of the Environment Andrew Brennan Medical Ethics RS. Downie Feminist Ethics Moira J. Gatens Public Reason Fred D ’Agostino, Gerald F. Gaus

Vagueness

Edited by

Delia Graff Cornell University and

Timothy Williamson University o f Oxford

First published 2002 by Dartmouth Publishing Company and Ashgate Publishing Published 2017 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN 711 Third Avenue, New York, NY 1001 7, USA

Routledge is an imprint of the Taylor & Francis Group, an informa business Copyright© Delia Graff and Timothy Williamson 2002. For copyright of individual articles please refer to the Acknowledgements. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing in Publication Data Vagueness. - (The international research library of philosophy) I. Vagueness (Philosophy) I. Graff, Delia II. Williamson, Timothy, I 955160 Library of Congress Cataloging-in-Publication Data Vagueness/ edited by Delia Graff and Timothy Williamson. p. cm. - (The International research library of philosophy) Includes bibliographical references. ISBN 0-7546-2080-8 (hardcover) I. Vagueness (Philosophy). I. Graff. Delia. II. Williamson, Timothy. III. Series.

8I05.V33 V33 2000 160-dc21

ISBN 13: 978-0-7546-2080-8 (hbk)

00-029308

Contents Acknowledgements Series Preface Introduction

PART I

NIHILISM

1 Peter Unger (1979), There Are No Ordinary Things’, Synthese, 41, pp. 117-54. 2 Samuel C. Wheeler (1979), ‘On That Which Is Not’, Synthese, 41, pp. 155-73. 3 David H. Sanford (1979), ‘Nostalgia for the Ordinary: Comments on Papers by Unger and Wheeler’, Synthese, 41, pp. 175-84. 4 Bertil Rolf (1984), ‘Sorites’, Synthese, 58, pp. 219-50. 5 Roy A. Sorensen (1985), ‘An Argument for the Vagueness of “Vague” ’, Analysis, 45, pp. 134-37.

PART II

61 71 103

109 151 173

DEGREES OF TRUTH

9 R.M. Sainsbury (1986), ‘Degrees of Belief and Degrees of Truth’, Philosophical Papers, 15, pp. 97-106. 10 Dorothy Edgington (1992), ‘Validity, Uncertainty and Vagueness’, Analysis, 52, pp. 193-204.

PART IV

3 41

OBSERVATIONAL PREDICATES

6 Crispin Wright (1975), ‘On the Coherence of Vague Predicates’, Synthese, 30, pp. 325-65. 7 C.L. Hardin (1988), ‘Phenomenal Colors and Sorites’, Nous, 22, pp. 213-34. 8 Christopher Peacocke (1981), ‘Are Vague Predicates Incoherent?’, Synthese, 46, pp. 121-41.

PART III

vii ix xi

197 207

EPISTEMICISM

11 Richmond Campbell (1974), ‘The Sorites Paradox’, Philosophical Studies, 26, pp. 175-91. 12 Timothy Williamson (1996), ‘What Makes it a Heap?’, Erkenntnis, 44, pp. 327-39. 13 W.D. Hart (1992), ‘Hat-Tricks and Heaps’, Philosophical Studies, 33, pp. 1-24.

221 239 253

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VI

PART V

HIGHER-ORDER VAGUENESS

14 Mark Sainsbury (1991), ‘Is There Higher-Order Vagueness?’, The Philosophical Quarterly, 41, pp. 167-82. 15 Crispin Wright (1992), ‘Is Higher Order Vagueness Coherent?’, Analysis, 52, pp. 129-39. 16 Dorothy Edgington (1993), ‘Wright and Sainsbury on Higher-Order Vagueness’, Analysis, 53, pp. 193-200. 17 Richard G. Heck Jr (1993), ‘A Note on the Logic of (Higher-Order) Vagueness’, Analysis, 53, pp. 201-8. 18 Timothy Williamson (1999), ‘On the Structure of Higher-Order Vagueness’, Mind, 108, pp. 127-43. 19 Dominic Hyde (1994), ‘Why Higher-Order Vagueness is a Pseudo-Problem’, Mind, 103, pp. 35-41. 20 Michael Tye (1994), ‘Why the Vague Need Not be Higher-Order Vague’, Mind, 103, pp. 43-45.

PART VI

279 295 307 315 323 341 349

CONTEXTUALISM

Hans Kamp (1981), ‘The Paradox of the Heap’, in U. Mönnich (ed.), Aspects o f Philosophical Logic, Dordrecht: D. Reidel, pp. 225-77. 22 Jamie Tappenden (1993), ‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’, Journal o f Philosophy, 90, pp. 551-77. 23 Diana Raffman (1994), ‘Vagueness Without Paradox’, Philosophical Review, 103, pp. 41-74. 21

PART VII

355 409 437

INTUITIONISM

Hilary Putnam (1983), ‘Vagueness and Alternative Logic’, Erkenntnis, 19, pp. 297-314. 25 Stephen Read and Crispin Wright (1985), ‘Hairier than Putnam Thought’, Analysis, 45, pp. 56-58. 26 Hilary Putnam (1985), ‘A Quick Read is a Wrong Wright’, Analysis, 45, p. 203. 27 Timothy Williamson (1996), ‘Putnam on the Sorites Paradox’, Philosophical Papers, 25, pp. 47-56.

497

Name Index

507

24

473 491 495

Acknowledgements The editors and publishers wish to thank the following for permission to use copyright material. Blackwell Publishers Limited for the essays: Mark Sainsbury (1991), ‘Is There Higher-Order Vagueness?’, The Philosophical Quarterly, 41, pp. 167-82; Roy A. Sorensen (1985), ‘An Argument for the Vagueness of “Vague” ’, Analysis, 45, pp. 134-37. Copyright © 1985 Roy A. Sorensen; Dorothy Edgington (1992), ‘Validity, Uncertainty and Vagueness’, Analysis, 52, pp. 193-204; Crispin Wright (1992), ‘Is Higher Order Vagueness Coherent?’, Analysis, 52, pp. 129-39; Dorothy Edgington (1993), ‘Wright and Sainsbury On Higher-Order Vagueness’, Analysis, 53, pp. 193-200; Richard G. Heck Jr (1993), ‘A Note on the Logic of (HigherOrder) Vagueness’, Analysis, 53, pp. 201-8. Copyright © 1993 Richard G. Heck Jr; Stephen Read and Crispin Wright (1985), ‘Hairier than Putnam Thought’, Analysis, 45, pp. 56-58. Copyright © 1985 Stephen Read and Crispin Wright; Hilary Putnam (1985), ‘A Quick Read is a Wrong Wright’, Analysis, 45, p. 203. Copyright © 1985 Hilary Putnam; C.L. Hardin (1988), ‘Phenomenal Colors and Sorites’, Nous, 22, pp. 213-34. Copyright © 1988 Noûs Publications. International Journal of Philosophical Studies for the essay: W.D. Hart (1992), ‘Hat-Tricks and Heaps’, Philosophical Studies, 33, pp. 1-24. Copyright © 1992 by Philosophical Studies. The Journal of Philosophy for the essay: Jamie Tappenden (1993), ‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’, Journal o f Philosophy, 90, pp. 551-77. Copyright © 1993 The Journal of Philosophy, Inc. Kluwer Academic Publishers for the essays: Peter Unger (1979), ‘There Are No Ordinary Things’, Synthese, 41, pp. 117-54. Copyright © 1989 D. Reidel Publishing Co.; Samuel C. Wheeler (1979), ‘On That Which Is Not’, Synthese, 41, pp. 155-73. Copyright © 1979 D. Reidel Publishing Co.; David H. Sanford (1979), ‘Nostalgia for the Ordinary: Comments on Papers by Unger and Wheeler’, Synthese, 41, pp. 175-84. Copyright © 1979 D. Reidel Publishing Co.; Bertil Rolf (1984), ‘Sorites’, Synthese, 58, pp. 219-50. Copyright © 1984 D. Reidel Publishing Co.; Crispin Wright (1975), ‘On the Coherence of Vague Predicates’, Synthese, 30, pp. 325-65. Copyright © 1975 D. Reidel Publishing Co.; Christopher Peacocke (1981), ‘Are Vague Predicates Incoherent?’, Synthese, 46, pp. 121-41. Copyright © 1981 D. Reidel Publishing Co.; Richmond Campbell (1974), ‘The Sorites Paradox’, Philosophical Studies, 26, pp. 175-91. Copyright © 1974 D. Reidel Publishing Co.; Timothy Williamson (1996), ‘What Makes it a Heap?’, Erkenntnis, 44, pp. 327-39. Copyright © 1996 Kluwer Academic Publishers; Hans Kamp (1981), ‘The Paradox of the Heap’, in U. Mönnich (ed.), Aspects o f Philosophical Logic, Dordrecht: D. Reidel, pp. 225-77. Copyright © 1981 D. Reidel Publishing Co.; Hilary Putnam (1983), ‘Vagueness and Alternative Logic’, Erkenntnis, 19, pp. 297-314. Copyright © 1983 D. Reidel Publishing Co.

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The Philosophical Review for the essay: Diana Raffman (1994), ‘Vagueness Without Paradox’, Philosophical Review, 103, pp. 41-74. Copyright © 1994 Cornell University. Reprinted by permission of the publisher and author. Oxford University Press for the essays: Timothy Williamson (1999), ‘On the Structure of HigherOrder Vagueness’, Mind, 108, pp. 127-43. Copyright © 1999 Oxford University Press; Dominic Hyde (1994), ‘Why Higher-Order Vagueness is a Pseudo-Problem’, Mind, 103, pp. 35-41. Copyright © 1994 Oxford University Press; Michael Tye (1994), ‘Why the Vague Need Not be Higher-Order Vague’, Mind, 103, pp. 43-45. Copyright © 1994 Oxford University Press. Philosophical Papers for the essays: Timothy Williamson (1996), ‘Putnam on the Sorites Paradox’, Philosophical Papers, 25, pp. 47-56; R.M. Sainsbury (1986), ‘Degrees of Belief and Degrees of Truth’, Philosophical Papers, 15, pp. 97-106. Every effort has been made to trace all the copyright holders, but if any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangement at the first opportunity.

Series Preface The International Research Library of Philosophy collects in book form a wide range of important and influential essays in philosophy, drawn predominantly from English-language journals. Each volume in the Library deals with a field of inquiry which has received significant attention in philosophy in the last 25 years, and is edited by a philosopher noted in that field. No particular philosophical method or approach is favoured or excluded. The Library will constitute a representative sampling of the best work in contemporary English-language philosophy, providing researchers and scholars throughout the world with comprehensive coverage of currently important topics and approaches. The Library is divided into four series of volumes which reflect the broad divisions of contemporary philosophical inquiry: •

Metaphysics and Epistemology

•

The Philosophy of Mathematics and Science

•

The Philosophy of Logic, Language and Mind

•

The Philosophy of Value

I am most grateful to all the volume editors, who have unstintingly contributed scarce time and effort to this project. The authority and usefulness of the series rests firmly on their hard work and scholarly judgement. I must also express my thanks to John Irwin of the Ashgate Publishing Company, from whom the idea of the Library originally came, and who brought it to fruition; and also to his colleagues in the Editorial Office, whose care and attention to detail are so vital in ensuring that the Library provides a handsome and reliable aid to philosophical inquirers. JOHN SKORUPSKI General Editor University o f St. Andrews Scotland

Introduction If you’ve read the first 500 pages of this book, you’ve read most of it (we assume that ‘most’ requires more than ‘more than half’). The set of natural numbers n such that the first n pages are most of this book is non-empty. Therefore, by the least number principle, it has a least member k. What is k? We do not know. We have no idea how to find out. The obstacle is something about the term ‘most’. It is recognizably the same feature as the feature of ‘heap’ that prevents us from finding an answer to the question ‘How many grains make a heap?’ and the feature of many other expressions that prevents us from finding answers to similar questions involving them. Call this feature, whatever its underlying nature, vagueness. If for each number n we could answer the question ‘Are the first n pages most of this book?’ in a way that survived further reflection, we could identify k. Since we cannot do that, for some number n we cannot answer the question ‘Are the first n pages most of this book?’ in a way that survives further reflection. Such a number is a borderline case. An expression or concept is vague if and only if it has actual or potential borderline cases. By definition of k, the first k pages are most of this book and the first k- 1 pages are not most of this book. Thus, by reading just one page, you could pass from not having read most of this book to having read most of this book (an extension of the argument shows that you could do so by reading just one word or one letter). Many object to consequences like that, arguing that our casual use of ‘most’ does not permit such fine discriminations, and therefore that no such number k exists. That implies rejection of the least number principle. But that is not enough. Their reasoning supports: (1)

For every natural number n, the first n+ 1 pages are most of this book only if the first n pages are most of this book.

Moreover: (2)

The first 500 pages are most of this book.

From (1) and (2), 500 steps of elementary reasoning yield: (3)

The first 0 pages are most of this book.

Proposition (3) is certainly false. Yet if we deny (1), classical reasoning takes us back to the existence of k. This is a typical sorites paradox. Some take sorites paradoxes to show that classical reasoning is invalid for vague languages. The standard truth-table semantics for negation, conjunction, disjunction and other logical constants presupposes that truth and falsity are mutually exclusive and jointly exhaustive properties of sentences (relative to a context of utterance). If that presupposition fails in borderline cases, any original presumption in favour of classical logic disappears. If vagueness

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does demand a revision of classical reasoning, it will be a global revision, for virtually all our words and concepts are vague. We would face the major task of discovering what alternative forms of reasoning would be valid for vague languages. Vagueness is a hard test for any theory of logic, meaning or truth.

Nihilism Nihilists take sorites paradoxes at face value, as showing that if x and y are linked by a sorites series for a predicate F, then there is no real difference between x and y with respect to the correct application of F\ a vague distinction is no distinction at all. Several essays in Part I explore this view. In There Are No Ordinary Things’ (Chapter 1) Peter Unger supports his title by sorites reasoning. He argues that there are no stones, for a stone would consist of finitely many atoms, and removing one atom from a stone while minimizing collateral damage would not mean that a stone was no longer present. Unger says that it would take a miracle o f conceptual comprehension for the removal of one atom without further repercussions to make such a difference in the applicability of our concept stone; he deems it implausible that our concepts should discriminate so finely. The argument generalizes in various ways to most ordinary kinds of thing.1Samuel C. Wheeler III argues similarly in ‘On That Which Is Not’ (Chapter 2). He suggests that philosophers have been inhibited from drawing eliminativist conclusions about common-sense kinds by a conception of reference as a matter of fit, on which the methodology of interpretation guarantees that most of what we say is true because an assignment of reference without that upshot would count as too uncharitable to be correct. This conception is incoherent, according to Wheeler, for Saul Kripke and Hilary Putnam have shown it inconsistent with many common-sense judgements about reference, which by its own lights it cannot reject wholesale. He presents an alternative conception of reference as a causal relation to entities individuated by natural laws, unlike common-sense kinds on Wheeler’s view. David Sanford replies to Unger and Wheeler in ‘Nostalgia for the Ordinary’ (Chapter 3). He takes Unger’s reasoning to assume the Principle of Valence, that every statement has a truth value (Sanford allows any number of truth-values), and mentions forms of supervaluationism on which the Principle is held to fail. As Bertil Rolf points out in ‘Sorites’ (Chapter 4) it is not obvious that Unger’s argument does assume the Principle of Valence, for Unger does not formulate his argument in meta-linguistic terms; his premises and conclusion are about stones, not about statements about stones. Sanford might reply that Unger implicitly relies on meta-linguistic assumptions in dismissing the possibility that his premises fail to be true. But when Unger does address semantic issues, his discussion suggests that he does not need the Principle of Valence, for he implies that any difference at all in the semantic properties of ‘A stone is present’ with respect to two situations differing only by an atom would be a miracle of conceptual comprehension. Even the difference between having a truth-value and having none would be such a miracle. If for some sequence of situations s0, s 1, . . . , sn ‘A stone is present’ has exactly the same semantic properties with respect to si-1.and si (for each i from 1 to n), then it has exactly the same semantic properties with respect to s0 and sn, which is what Unger wants.

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Sanford raises another problem. For various ordinary predicates F, Unger and Wheeler argue for \/x~Fx (‘Nothing is a stone’, ‘Nothing is a tall person’) on the grounds that any line supposed to demarcate the extension of F would be arbitrary. But within the framework of classical logic that both Unger and Wheeler accept, demarcating the extension of F is equivalent to demarcating the extension of ~F; to exactly the extent to which we cannot do the former, we cannot do the latter. Thus if the original argument was good, so should be the result of replacing F in it by ~F. But the conclusion of the new argument is \/x~~Fx ( ‘Nothing is not a stone’, ‘Nothing is not a tall person’), which is incompatible with the original conclusion except in the empty domain. As not even Unger and Wheeler want to claim that there is nothing whatsoever, the original argument was too sweeping. But if Unger and Wheeler conceded that \/x~Fx is not always true when any line supposed to demarcate the extension of F would be arbitrary, why should we accept it in their favoured cases? Would Unger and Wheeler do better to draw the meta-linguistic conclusion that F suffers reference failure? For that is compatible with, and even supports (by compositionality), the further conclusion that ~F suffers reference failure. But ascent to the meta-language raises a further problem for Unger and Wheeler. If they are right, an extension of their sorites reasoning will show that there are no words and no predicates, for evolutionary considerations point to vagueness in ‘There was a linguistic token at least n seconds ago’. A similar argument would show that thinking never occurs. Nothing is left to suffer reference failure. Unger and Wheeler write as though fundamental theories and concepts of physics were immune from their attack. But if they are right about vagueness, no such theories or concepts have ever been entertained. A corresponding point can be argued without ascent to the meta­ language. Just as it is vague in which counterfactual situations there would be stones, so it is vague in which counterfactual situations there would be electrons - where is the line between situations in which electrons behave differently because there are different particles for them to interact with and situations in which the differences are too great for there to be electrons at all? Unger and Wheeler’s arguments, if compelling, seem to constitute not just a reductio ad absurdum of the common-sense view of the world but a reductio of any view of the world at all. Are their arguments compelling? Unger does not really explain why conceptual comprehension in his sense would be hard to achieve. It is certainly hard to anticipate each problem case in advance, but he does not show that sharply bounded concepts would involve such anticipations, rather than having their boundaries set partly by default principles inherent in the nature of reference. Wheeler’s nomological constraints on reference seem implausibly strong; although the term ‘metre’ does not refer to a nomologically special length, since we could just as easily have used it to refer to a slightly different length, that does not seem to be a good reason for saying that ‘metre’ suffers reference failure. In Chapter 4 Rolf argues for the related conclusion that a natural language is analytically inconsistent, in the sense that jointly inconsistent sentences are assertible by virtue of their meanings; sorites paradoxes provide examples. He concludes that the concept of truth does not apply to the part of the language containing such paradoxes. Does analytical inconsistency really have that consequence? If a subtle inconsistency is discovered in the rules of football, the language of football is presumably analytically inconsistent, but that scarcely shows that no one has ever said something true in the language of football - for example, in situations unrelated to the inconsistency. And is a sentence assertible by virtue of its meaning if it does not express something whose truth speakers are in a position to know? Roy Sorensen, in ‘An

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Argument for the Vagueness of “Vague” ’ (Chapter 5) proposes that such questions should make nihilists more than a little uncomfortable. Arguing that ‘vague’ is a vague predicate by showing that it is sorites-susceptible, he constructs a sorites series of disjunctively defined predicates, the first one of which clearly is vague, the last one of which clearly is not, though we cannot locate a boundary between the vague and the non-vague. He charges that because ‘vague’ is vague, any claim to the effect that sentences containing vague predicates are defective will be subject to its own criticism.

Observational Predicates One way of upholding some form of nihilism while avoiding Sorensen’s result would be to maintain that not every vague predicate is incoherent, (in particular, not ‘vague’), but only a certain subset of all the vague predicates - the observational ones. A predicate is observational just in case its applicability to a thing depends only on how that thing appears. Observational predicates like ‘red’ (or perhaps just ‘looks red’) seem to pose a special problem for any theory of vagueness. In a sorites series for a non-observational vague predicate like ‘tali’, it must be that between successive members in the series there is some difference in the quality relevant for applicability of the predicate, namely height. But it is commonly thought that one can construct a sorites series for observational predicates that lacks this feature, because it is commonly thought that appears the same as is not a transitive relation. If appears the same as is non-transitive, then there can be a series of colour patches ranging from red to blue, while each patch looks the same as its successor in the series. (The idea that there can be such a series is typically offered as grounds for thinking that observational indiscriminability is non-transitive.) But if the applicability of ‘red’ depends only on the look of a thing, and looking the same implies having the same look, then there can be no change in its applicability between any two patches in such a series. Wright takes up this problem in ‘On the Coherence of Vague Predicates’ (Chapter 6). There he argues for the falsity of what he calls the ‘governing view of language’, which is a conjunction of two theses: first, language use is essentially rule-governed; second, we can discover the rules of use of our own language by an appeal to our intuitions about its proper use combined with a knowledge of our limitations of perception and memory. Most of the essay is devoted to arguing that if the second thesis were correct, we would have to concede that vague colourpredicates like ‘red’ are incoherent. Our intuition tells us that ‘red’ is observational - that is, its applicability to a thing can be determined by looking. Yet, if ‘red’ is observational, its applicability must ‘tolerate’ any change that is indiscernible to us so that, by consideration of our perceptual limitations - in particular, the putative non-transitivity of observational indiscriminability - we must conclude that the applicability of ‘red’ tolerates discernible changes as well, even large ones, since if observational indiscriminability is non-transitive, presumably any significant change in look can be arrived at by a sequence of indiscernible changes. The tolerance of ‘red’, combined with the putative non-transitivity of observational indiscriminability means that ‘red’ must apply to things with any look whatsoever. But our intuition about the proper use of ‘red’ tells us that this is not the case. If this is right, then ‘red’ is incoherent in the sense that the rules imply that there are objects to which it both does and does not apply. Ultimately, Wright does not accept that observational predicates are incoherent;

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he argues instead that, if their use is rule-governed, then the means of discovering those rules are not those afforded us by the ‘governing view’. In ‘Phenomenal Colors and Sorites’ (Chapter 7) C.L. Hardin disputes the claim that observational indiscrim inability is non-transitive. He proposes that observational indiscriminability is properly understood as a statistical notion. In order to determine whether two patches are observationally discriminable for a subject, one cannot simply display the patches once and take the subject’s response on that occasion as decisive, since ‘Actual attempts to match and discriminate closely similar material color samples are . . . typical cases of decision making under conditions of uncertainty’ (p. 153). The pair of patches must be displayed in sufficiently many trials, and statistical methods must be employed in order to determine whether the patches are observationally discriminable for the subject. Hardin claims that any distinctly coloured patches will prove observationally discriminable for any subject, although the greater the similarity of the patches, the more trials will be required to reveal that they are observationally discriminable. So conceived, observational indiscriminability is transitive. The upshot is that the tolerance of observational predicates will not have the problematic consequence that they apply to objects with any look whatsoever. Even if Hardin is right, however, that statistical indiscriminability is transitive, it seems we can make perfect sense of a non-statistical notion of observational indiscriminability, naturally expressed by relations such as ‘jc and y look the same to me now’. Even if colour predicates like ‘red’ are not tolerant with respect to this relation - two things might look the same to me now even if they are quite different in colour - a predicate such as ‘looks red to me now’ seemingly must be. If the non-statistical notion of observational indiscriminability is non-transitive, then Wright’s argument for incoherence may be run with the new predicate. To block such an argument, we would need to show either that predicates such as ‘looks red to me now’ are not tolerant with respect to the non-statistical indiscriminability relation, that the non-statistical indiscriminability relation is not itself coherent, or that it is after all transitive, despite Wright’s argument to the contrary. In Chapter 8, ‘Are Vague Predicates Incoherent?’, Christopher Peacocke challenges Wright’s conditional argument for the incoherence of observational predicates on two fronts. On one front he argues that the reasoning which supports the incoherence of observational predicates can be run even if one does not accept what Wright calls the ‘governing view’ of language. If the argument for incoherence is to be blocked, a mere rejection of the ‘governing view’ will not do the job. On another front, Peacocke rejects Wright’s claim that the incoherence of observational predicates cannot be avoided by the adoption of a special logic. Wright had argued that there is an important difference between vague predicates and ones whose sense has not been fully specified. The imprecise boundaries of vague predicates cannot be understood as being just like the border between two states that have agreed that there is a border between them but have not agreed on its precise location. In accepting the existence of a border, the two states deny that whenever you are on one side of the border a sufficiently small step will not bring you to the other side. But to the extent that a vague predicate is tolerant, the imprecision of its boundary is precisely not like that. Wright here alludes to supervaluation semantics, according to which a sentence is true just in case it is true on all ways of making its expressions precise, and hence according to which it may be true that there is a boundary line between the two states, even though it is not true of any given line that it is the boundary.2He equally rejects the

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idea that the incoherence of observational predicates can be avoided by a rejection of bivalence in favour of a range of degrees of truth. The idea would be that on a colour continuum, for example, the applicability of ‘red’ would gradually decrease as one moved along the spectrum. The predicate would still be somewhat tolerant, but not completely so. No small move along the spectrum could greatly decrease the truth of ‘This is red’, but a sequence of such moves could still cause it to decrease from complete truth to complete falsity. But even if the degree of truth of ‘This is red’ can gradually decrease, Wright claims that the truth of ‘Application of “red” to this is on balance justified’ cannot gradually decrease. Yet the predicate in this sentence seems no less observational, and hence tolerant, than ‘red’ itself. Peacocke does not address this specific argument of Wright’s. He proposes nonetheless that we should adopt a logic for vague predicates based on degrees of truth, and be prepared as a result to give up the idea that ‘red’ (and presumably also ‘looks red’) is observational, at least in the sense in which Wright understood this notion.

Degrees of Truth According to degree-theorists, gradual change is a semantic phenomenon. They hold that just as there is a gradual change in redness across a spectrum, there is also a gradual change in the truth-value of the sentence ‘The spectrum is red here’ as we point at different parts of it. Degreetheorists typically identify the set of truth-values with the set of real numbers between 0 (complete falsity) and 1 (complete truth), although they often acknowledge that this involves something of an idealization. Accepting a range of truth-values allows one to block sorites reasoning, although exactly what a degree-theorist deems wrong with sorites reasoning depends on how degrees of truth are incorporated in the semantics of the connectives and quantifiers as well as in the notion of validity. Consider a particular instance of the inductive premise of a sorites paradox: ‘If n is F, then az+1 is F \ A degree-theorist holds that the degree of truth of the consequent of this conditional may be slightly lower than that of the antecedent. On one view about the semantics for the conditional, modus ponens preserves complete truth (truth of degree 1), but the drop in degree from antecedent to consequent is enough to render the conditional less than completely true. On another view, the conditional may be completely true, despite the small drop in degree from antecedent to consequent, but then modus ponens will not always preserve complete truth. R.M. Sainsbury in ‘Degrees of Belief and Degrees of Truth’ (Chapter 9) is not concerned to develop a logic of degrees, but rather to defend the acceptance of degrees of truth by developing an understanding of them as the semantic correlates of a certain kind of partial belief. I may feel somewhat confident that John is a spy, but even more confident that Sue is one, even though I lack full certainty in either case. On the assumption that ‘spy’ expresses a sharp concept, though, my lack of full certainty indicates that I don’t know all the facts. Sainsbury argues, however, that our confidence in the truth of statements containing vague predicates may also come in degrees, but that what is distinctive about the degree of confidence we have in the truth of these statements is that it need not change as we become better informed. When it comes to borderline cases of vague predicates, Sainsbury holds that even an omniscient being may have degrees of belief that are merely partial, and he calls such degrees of belief non-epistemic. But if non-epistemic partial beliefs are to support the existence of degrees of

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truth, it must be shown that they come in a range of degrees between 0 and 1 - that if two objects are borderline cases of ‘red’, for example, a fully informed person can believe to differing degrees the claims that each is red. Moreover, it must be shown that partial degrees of belief do not simply reduce to full beliefs about the degree to which an object possesses a property, since even a proponent of bivalence can accept that there are varying degrees of, for example, redness. Sainsbury argues for these conclusions by appeal to claims about the relation between partial beliefs and behaviour. In ‘Validity, Uncertainty and Vagueness’ (Chapter 10) Dorothy Edgington suggests an analogy between probabilities and degrees of truth. The degree-theoretic semantics she proposes is perhaps the most natural way of formally developing Sainsbury’s understanding of degrees of truth. Defining the uncertainty of A as one minus its probability, she points out that, in a classically valid argument, the uncertainty of the conclusion is at most the sum of the uncertainties of the premises. That is a useful guide in applying arguments from uncertain premises. If degrees of truth behave like probabilities, and the degree of falsity of A is one minus its degree of truth, then, in a classically valid argument, the degree of falsity of the conclusion is at most the sum of the degrees of falsity of the premises. That would be a useful guide in applying arguments from imperfectly true premises. On this approach, a valid sorites argument runs from ‘a0 is red’ and a thousand conditional premises of the form ‘If at is red, aM is red’ to ‘a 1000 is red' but the conclusion may be perfectly false while all the premises are almost perfectly true, because their slight degrees of falsity accumulate. Edgington is guarded about what degrees of truth are. Following Hans Kamp (1975), she raises the possibility of a supervaluationist treatment. A supervaluationist might even make degrees of truth a special case of probabilities, rather than merely analogous to them, by defining the degree of truth of A as the probability of its truth on a randomly chosen sharpening of the language (for a specified probability distribution). The probabilistic model of degrees of truth implies that the degree of truth of a conjunction or disjunction is not a function of the degrees of truth of its conjuncts or disjuncts, just as the probability of a conjunction or disjunction is not a function of the probabilities of its conjuncts or disjuncts. Edgington gives several examples in which this approach seems to accord better with intuition than does any approach (such as fuzzy logic) that treats the degree of truth of a conjunction or disjunction as a function of the degrees of truth of its conjuncts or disjuncts.

Epistemicism Epistemicists hold that when a is borderline for F, Fa is true or false, but one cannot know which. Epistemicists about vagueness need not reject the possibility of gradual change, but hold that gradual changes of one kind (say in degree of redness) may obscure sudden changes of another kind (in the applicability of ‘red’). They can thereby retain classical logic and standard principles about truth and falsity. Nevertheless, some philosophers find epistemicism highly counterintuitive. Richmond Campbell’s ‘The Sorites Paradox’ (Chapter 11) is one of the earlier epistemic accounts of vagueness in the modern period. According to Campbell, what prevents us from knowing the truth of the matter in a borderline case is semantic uncertainty. He says little about what semantic uncertainty is, except that it originates in the meaning of the term, not in speakers’

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linguistic incompetence or ignorance of underlying facts. For Campbell, semantic uncertainty about a borderline case of ‘short man’ ‘is not due to any lack of knowledge’. That is puzzling. For if a proposition is true and we do not know it, we lack knowledge; on Campbell’s view, if Harry is borderline, either it is true that Harry is a short man and we do not know that Harry is a short man or it is true that Harry is not a short man and we do not know that he is not a short man. Campbell introduces an operator D to mean something like ‘it is semantically certain that’, and uses it to explain away the apparent implausibility of some consequences of his view by the falsity of similar claims involving D with which they are easily confused. It is neither semantically certain that Harry is a bald man nor semantically certain that Harry is not a bald man. Campbell tries to avoid higher-order vagueness when D is iterated by treating it as a technical device which we can sharpen without altering the semantics of the object-language to which it is added. He seems to suggest that such sharpening would be a matter of deciding how much agreement in use is required for semantic certainty. It can hardly be that simple, for even competent speakers may be subject to systematic illusions or oversights, and agree or fail to agree on inadequate grounds. A more recent defence of an epistemic view is Timothy Williamson’s ‘What Makes it a Heap?’ (Chapter 12). Both epistemicists and their opponents allow that vague predicates supervene on relevantly precise predicates in the sense that, if two possible cases are the same with respect to the latter, they are alike with respect to the former too. According to epistemicists, we often cannot know which conjunction of vague predicates follows from a given conjunction of precise predicates, whereas their opponents expect such supervenience connections to be cognitively accessible. Williamson regards that expectation as unwarranted. He explores limits on our ability to identify a property given by an apparently vague predicate with a property given by a relevantly precise scientific predicate.3 Bill Hart’s ‘Hat-Tricks and Heaps’ (Chapter 13) challenges the epistemicist assumption that we cannot discover the cut-off point for a vague concept. He suggests that heaps form something like a natural kind. By detailed analysis of their statics and dynamics, he argues that the least number of grains to make a heap, when suitably arranged, is four. On Hart’s analysis, one grain can rest stably on three others, and that configuration has the essential properties of a heap. No such configuration is possible with three or fewer grains. Even if one is not completely convinced by Hart’s argument, or doubts that it will generalize to other vague concepts (such as enormous heap), it may well lead one to suspect that one’s grasp of a vague concept does not show that no cut-off point for it can be discovered. Epistemicists may have to concede that the impossibility of knowing in a borderline case is at least sometimes a remediable practical inability.

Higher-Order Vagueness We introduced the notion of vagueness by description: vagueness is that feature of an expression, whatever its underlying nature, that prevents us from discovering the boundaries of its application. Some find it helpful to approach this feature by means of a metaphor: the boundaries of application of vague predicates are blurred rather than sharp. This metaphor is sometimes wielded against epistemic theories of vagueness. One hears it said that epistemicists counterintuitively posit sharp boundaries for vague predicates; or worse, that epistemicists, in

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positing sharp boundaries, reject the existence of vagueness altogether. Even the first charge is unfair, however. Epistemicists have as much claim to the metaphor as anyone else, and will naturally want to cash it out in epistemic terms. Epistemicists may say that the boundaries they posit for vague predicates are blurred in that we can narrow down the region of their location but cannot pinpoint it exactly. One could, of course, just stipulate that a predicate has sharp boundaries just in case, in any sorites series, there will be an object of which the predicate is true adjacent to an object of which the predicate is false. Epistemicists accept that in that sense vague predicates have sharp boundaries, while degree-theorists, and those who accept truth-value gaps, such as supervaluationists, do not. They say that in an appropriately constructed sorites series for vague predicates, between those objects of which the predicate is (completely) true and those of which it is (completely) false, there will be a range of objects that fall into neither category. The problem is that, although these theorists thereby eliminate sharp boundaries for vague predicates, they do not seem to replace them with blurred ones. On a basic supervaluationist approach, for example, there will be a sharp boundary between the objects of which a predicate such as ‘tall’ will be true and those of which it is not true (although not all these objects will be ones of which ‘tali’ is false). But we are no more able to find such a boundary than we are able to find a boundary dividing the tall from the not tall. This leads to the problem of higher-order vagueness. To date, there is little consensus about what exactly higher-order vagueness is, whether there is higher-order vagueness, whether it is a problem and, if so, in what way. To discuss the issues, we need to introduce a definitely operator, which we will here symbolize as ‘D ’. Officially, ‘D ’ gets prefixed to a sentence to form a sentence. Informally, we express definiteness adjectivally or adverbially. Taking the notion of a borderline case as primitive, and letting ‘B ’ be a sentential operator representing it, we may define ‘Dp’ as ‘p & ~Bp' Thus Walt is definitely tall just in case Walt is tall and not a borderline case. On the assumption that ‘Bp’ is equivalent to ‘B~p’, (one is a borderline case of ‘tall’ just in case one is a borderline case of ‘not tali’), we can prove using classical logic that ‘Bp’ is equivalent to ‘~Dp & ~D ~p’. A predicate F is firstorder vague just in case it could have borderline cases with its given meaning, so just in case ‘Ek(~DFx & ~D~Fx)' could be true. What is higher-order vagueness? One thought is that just as first-order vagueness in F prevents us from finding a first-order boundary between the Fs and the not Fs, second-order vagueness is whatever prevents us from finding a second-order boundary between the borderline cases of F and those things that are either definitely F or definitely not. Higher-order vagueness would be whatever prevents us from finding higher-order boundaries. In ‘Is There Higher-Order Vagueness?’ (Chapter 14), Mark Sainsbury suggests that it will be impossible to make good on this first thought. Since it seems that we are prevented from discovering boundaries of any kind for vague predicates, the thought would go, it must be that they are rcth-order vague for every n. But if we use iterations of ‘D ’ to capture higher-order vagueness, it still remains, says Sainsbury, that ‘three sets single themselves out for attention’ (p. 281) - those that are definitely" F for any n, those that are definitely" ~F for any n, and the rest. ( ‘Definitely"’ abbreviates n iterations of ‘definitely’.) We are unable, however, to discover the boundaries between such sets. One could take Sainsbury’s remarks to suggest that no theory of vagueness, even one that accommodates higher-order vagueness, is complete without an explicitly epistemological component. Sainsbury does not draw this conclusion, however.

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Instead, he thinks that we must give up any conception of vagueness for which the problem of higher-order vagueness arises at all. Part of Sainsbury’s case for his claim rests on his rejection of another thought about higher-order vagueness, which goes as follows. Just as we can define first-order vagueness in F in terms of the truth of ‘Ek(~DFx; & ~D~Fx)’, we could define higher-order vagueness in F in terms of the truth of other sentences containing F and ‘D ’. Sainsbury argues that there is no obvious way to proceed with this second thought. Crispin Wright, in ‘Is Higher-Order Vagueness Coherent?’ (Chapter 15), argues that, on the most natural way of proceeding (as well as on other ways of proceeding provided by Sainsbury), higher-order vagueness proves paradoxical in that it permits us to generate sorites paradoxes. The argument begins by noting that we cannot accept the following rendering of the claim that F has no sharp boundaries: Vagueness Vagueness

( 1)

We suppose that the variables range over the objects in an appropriately constructed sorites series for F, and that x' is the successor of * in such a series. From (1) and the assumption that the last member of the series is not F, we can infer that the first member is not F. The reasoning is blocked by replacing (1) with: Vagueness Vagueness Vagueness Vagueness

(2)

Similarly, we cannot render the claim that F has no sharp second-order boundary as: Vagueness Vagueness Vagueness Vagueness

(3)

for, like (1), (3) is subject to sorites reasoning. But analogously to our replacement of (1) with (2), we may replace (3) with: Vagueness Vagueness Vagueness Vagueness

(4)

Generalizing Wright’s proposal, we may render the claim that F does not have sharp (n+ l)thorder boundaries (is (n+l)th-order vague) as: Vagueness Vagueness Vagueness Vagueness

(5)

Wright claims, however, that given a certain plausible principle about the logic of the definitely operator, we can deduce a sorites premise for the predicate ‘Definitely not definitely F ’ from the assumption that F is definitely second-order vague - that is, from the definiteness of (4). The principle in question, dubbed ‘(DEF)’, is formulated as a rule in a system of natural deduction. It says that if a formula P follows from some premises, each of which begins with the definitely operator, then DP follows from those same premises. Schematically: Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness

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Dorothy Edgington in ‘Wright and Sainsbury on Higher-Order Vagueness’ (Chapter 16) and Richard Heck in ‘A Note on the Logic of (Higher-Order) Vagueness’ (Chapter 17) each show that it is possible to block Wright’s reasoning without rejecting (DEF) wholesale. Edgington grants that (DEF) is a truth-preserving rule of inference. She does not regard this as sufficient, however, to render (DEF) valid. (DEF) licenses an inference from ‘Walt is definitely tall’ to ‘Walt is definitely definitely tali’. But if Walt is a borderline case of ‘definitely tall’ (which he just may be, if ‘tall’ is second-order vague), then, given the rejection of bivalence, the premise will be neither true nor false, while the conclusion will be false. But, on her view, no valid rule can permit the conclusion of a one-premise argument to have a ‘lower’ value than its premise. As she points out, we had better accept such a conception of validity if reductio ad absurdum, which Wright’s proof employs, is permitted without restriction. Heck, who accepts that validity is just preservation of truth, proposes instead that we accept (DEF) as valid, but that we block Wright’s reasoning by restricting the use of certain rules of inference: we are to prohibit the discharge of a premise A, by either reductio or conditional proof, if A occurs as a premise of a line obtained by (DEF). Drawing on the work of Kit Fine, he presents a semantics for the definitely operator on which although any sentence P entails DP, the conditional P —> DP may be untrue. The semantics in question treats ‘D ’ as a modal operator, and yields a comparatively weak modal logic. It is well and good to have a semantics for the definitely operator; if we are to understand and assess higher-order vagueness, we need at least that. But that is not enough. A generalized proposal such as Wright’s, which amounts to the claim that a predicate F is (rc+l)th-order vague just in case D" F has borderline cases, will not do. A predicate such as ‘definitely both tall and not definitely tali’ has no borderline cases - it is contradictory, given that ‘D ’ distributes over conjunction and that DP —> P is valid. But still, ‘tall but not definitely tall’ is second-order vague if any predicate is. In ‘On the Structure of Higher-Order Vagueness’ (Chapter 18), Timothy Williamson aims to remedy the deficiency. On Williamson’s approach, higher-order vagueness in a predicate is to be understood as first-order vagueness in one of a certain class of higherorder predicates. Defining vagueness for sentences, we call A first-order vague just in case ‘~DA & ~D~A’ could be true. (We could define vagueness for a predicate F by letting A be Fx.) We call A (ft+l)th-order vague, just in case there is a first-order vague sentence B with the following property: B is built up from just A, ‘D ’ and truth-functors, and every occurrence of A in B is in the scope of exactly n occurrences of ‘D’. (This presentation is an alternative equivalent to that presented in ‘On the Structure of Higher-Order Vagueness’.) One upshot of Williamson’s discussion is that, while having a precise characterization of higher-order vagueness increases our understanding of it, it also forces us to realize that we do not yet understand it as well as we may have thought we did. So why exactly is higher-order vagueness thought of as a problem? A number of reasons have already emerged in our discussion. If higher-order vagueness is not thought of as an explicitly epistemological phenomenon, then it is far from clear that it will have the epistemic implications expected of it. Namely, it is far from clear that it will explain our inability to discover boundaries for vague expressions. Wright thought higher-order vagueness to be intrinsically paradoxical. Heck’s discussion makes clear that no simple approach to the semantics of the definitely operator used to express higher-order vagueness will do, since iterations of it are not otiose. Williamson’s discussion renders doubtful the idea that we have many clear intuitions about the nature of higher-order vagueness at all.

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Dominic Hyde, in ‘Why Higher-Order Vagueness is a Pseudo-Problem’ (Chapter 19) considers whether higher-order vagueness is problematic for yet another reason. Many philosophers take the defining feature of vagueness in a predicate to be that it has borderline cases. But if this is right, then it seems that a predicate could count as vague even if it were perfectly knowable which thing in a sorites series was the first borderline case of the predicate and which was the last. But vague-predicates are not like that. For those unwilling to accept epistemicism, it might seem that vagueness just is higher-order vagueness. If so, then the characterization of vagueness as simply having borderline cases is off the mark. Hyde argues that once we realize that having borderline cases is itself a vague notion, this problem disappears. Michael Tye, in ‘Why the Vague Need Not be Higher-Order Vague’ (Chapter 20), replies that Hyde cannot be so sure that the vagueness of ‘borderline case’ is as pervasive as he needs it to be.

Contextualism On most theories of vagueness, the universal premise of a sorites paradox is deemed defective. Let us call these universal premises ‘tolerance principles’. We would like to know not just that there is something defective about tolerance principles, but also why we were inclined to accept them in the first place. What is the cause of our mistake? Three selections appeal to the contextdependence of vague predicates in order to provide an answer to this question. Whether or not a man is tall depends on his height. So whether or not a man is in the extension of ‘tall’ depends on his height. Whether or not a man is tall depends just on his height in the sense that if two men are the same height, one is tall if the other is. Whether or not a man is in the extension of ‘tali’, however, does not depend just on his height. It also depends on the context in which the predicate is used. If we are talking about basketball players, we may truly utter ‘Houston is not tali’, while if we are talking about men in general, we may truly utter ‘Houston is tali’, even if all heights remain stable. These facts combine to show that ‘tall’ is context-dependent. What features of a situation are relevant for contextual interpretation is by no means a settled matter. In ‘The Paradox of the Heap’ (Chapter 21), Hans Kamp makes essential use of the notion of the background of a context of utterance - ‘the information that is taken for granted by the participants in the discourse of which the utterance is part’ (p. 372). A background is a set of sentences. When a conversation proceeds normally, the background grows (and hence the context changes) as information provided by one speaker becomes accepted by the other. To say that the background of a conversation is a feature of its context is to say that the truth of an utterance may depend on what has already been accepted. Kamp proposes that the extension of a vague predicate may be dependent on the background in the following way: we have some licence in our use of vague predicates - it may be permissible, but not mandatory, to accept as true the claim that a particular man is tall (say if his height is six feet) but, once this claim has been accepted as true - that is, incorporated into the background - then any man of sufficiently similar height will be in the extension of ‘tall’ in such a context. What we accept as true affects what we may subsequently coherently deny. Kamp develops a formal semantics for vague predicates based on this idea. On his semantics, a conditional is true in a context c just in case either its antecedent is false in c or acceptance of the antecedent (incorporation of it into the background) would yield a new context c in which the consequent would be true. Given this truth clause for the conditional, as well as the condition on the truth of atomic sentences

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previously mentioned, each instance of the universal premise of a sorites paradox comes out true. In this way, Kamp accounts for our attraction to tolerance principles. One radical feature of his proposal, however, is that although each instance of the universal premise of a sorites paradox comes out true, the universal premise itself comes out false, since acceptance of it would lead to what Kamp calls an ‘incoherent’ context, which is sufficient, given his truth clause for universal generalizations, to render the universal premise false. It is, of course, possible, though, to construct a sorites argument in which no universal generalization figures, by replacing the universal premise of a standard sorites argument with the appropriate instances of it. Such ‘no-universal’ versions of sorites arguments have only true premises on Kamp’s account. So it seems that since Kamp would regard it as true that a seven-foot tall man is tall, but false that a two-foot tall man is tall; he must give up the validity of modus ponens. This would leave him in the awkward position of having given a radical semantics in order to verify claims we are inclined to accept as true, at the expense of invalidating principles of reasoning we are even more inclined to accept as valid. Kamp is not willing to give up the validity of modus ponens, however. Instead, he offers a definition of validity that does not include preservation of truth as a necessary condition - a valid argument may have premises that are true in some context, even though the conclusion is false in that context. Validity, on his proposal, instead requires only that acceptance of the premises - incorporation of them into the background - would yield a context in which the conclusion is true, if that context is ‘coherent’. The lesson Kamp would have us learn is that, when reasoning with vague terms, one should not always accept the conclusion of a sound argument. One might, however, doubt that what we have here is a genuine conception of validity. Jamie Tappenden’s discussion in ‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’ (Chapter 22) is also guided by the view that our attraction to the offending sorites premises must be respectfully accounted for. According to Tappenden, vague predicates admit truth-value gaps, but in the normal course of events we may have reason to narrow the gap. Although * may be a borderline case for ‘tali’, on occasion we may admissibly stipulate that he is to count among the class of tall things. The gappiness of ‘tall’ gives us licence to count it as applying sometimes to more, sometimes to fewer things. But if jc and 3^differ by just a millimetre, we may not at the same time count x as tall and y as not tall. Since, as we sharpen the predicate, we alter the assignment of truth-values to simple sentences, we also alter the assignments to complex sentences. If on a single occasion we count x as tall and y as not tall, where x and y differ by a millimetre, then we falsify the sentence ‘If x is tall and y is just one millimetre shorter than x, then y is tali’. But it is a feature of the meaning of ‘tali’, claims Tappenden, that this sentence cannot admissibly be falsified. We may narrow the gap between the tall and the not-tall, but we may not close it all the way. If someone inadmissibly stipulates there to be a sharp boundary between the tall and the not-tall, we may, admissibly, induce them to retract their stipulation by uttering ‘But if two people differ in height by just a millimetre, then if one is tall the other is’. Although the uttered sentence is not true - indeed, cannot be true since it has absurd consequences - it is distinguished, so the story goes, because it cannot admissibly be falsified either. Because it is not true, it cannot be correctly asserted. But because it may not be falsified, it can correctly be uttered with the purpose of averting, objecting to, or getting someone to retract a stipulated sharp boundary. In Tappenden’s terminology, it is ‘articulable’. The ordinary man goes wrong in taking his good reasons for uttering this sentence as reasons for believing it to be true. An unacknowledged feature of Tappenden’s account is that the

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distinguished status accorded to the universal premise of a sorites argument is also enjoyed by its negation. We are not initially inclined to accept the truth of the negation of the universal premise of a sorites paradox, however. Something other than articulability is required to explain the difference in our attitudes towards the sorites premise and its negation. Suppose you run the following psychology experiment: you choose pairs of adjacent objects from an appropriately constructed sorites series for a vague predicate F; you present these pairs to your subjects and ask them to judge whether F applies to either of the members of the pair. Each subject is presented with only one pair. Each pair is shown to a number of subjects. It would not be surprising if different subjects made different judgements. But it would also not be surprising if every subject made a uniform judgement - made the same judgement about each member of the pair presented to him or her. (After all, if subjects were not so inclined, sorites arguments would not be paradoxical.) In ‘Vagueness Without Paradox’ (Chapter 23) Diana Raffman proposes that the context-dependence of vague predicates exactly mirrors the judgements of such subjects (on the condition that they are ‘competent’ ones). If a competent subject judges a given colour patch to be red on a given occasion, then, in the context of that occasion, the subject’s judgement is true. The conditions under which the subjects make their judgements are counted by Raffman as features of the context. The fact that we may make opposing judgements about adjacent members of a sorites series if they are not presented together, combined with the fact that we do not make opposing judgements about adjacent members when they are presented together, is argued to yield the following result: the universal premise of a sorites argument is not true; our tendency to think it true is explained by there being a similar but true sentence for which we may mistake it - for all x, if x is F then x is F, insofar as x and x are judged pairwise. This claim, if true, does not support sorites reasoning.

Intuitionism Hilary Putnam’s essay, ‘Vagueness and Alternative Logic’ (Chapter 24), treats vagueness as an objection to metaphysical realism. His metaphysical realist claims to have a precise concept of truth as a special relation of correspondence between sentences (as used on particular occasions) and obtaining states of affairs. Such a philosopher is hard pressed to explain how it can be vague, as it often is, whether a given sentence as used on a given occasion is true. Putnam advocates a suitably vague alternative concept of truth as ‘idealized justification’. He does not consider a modified metaphysical realism on which our concept of the correspondence relation that constitutes truth is itself vague. What states of affairs a sentence corresponds to depends on what it means, and the concept of meaning is vague. At the end of the essay, Putnam suggests that one might reason with vague predicates as with undecidable predicates in intuitionist logic. Where ‘B(a)’ abbreviates ‘A man with n hairs is bald’, the argument from Vn(B(n) —» B(n+1)) and B(0) to \fnB(n) is simply an application of the intuitionistically valid principle of mathematical induction. Since one must assert B(0) and deny \/nB(n), one must deny Vn(B(n) —> B(n+1)). Putnam’s idea is that one still need not assert 3n(B(n) & ~B(n+l)), because the latter is not an intuitionistic consequence of ~Vn (B(fn) —» B(n+1)). Thus one denies the major premise of the sorites paradox without asserting that the vague predicate has a sharp cut-off. In ‘Hairier than Putnam Thought’ (Chapter 25),

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Stephen Read and Crispin Wright point out that if one denies that the vague predicate has a sharp cut-off, by asserting ~3n(B(n) & ~B(n+l)), one generates new sorites paradoxes. For example, one can intuitionistically derive \/n(~~B(n) —> ~~B(n+l)); since one must assert —'B(O), one must assert \/n — B(n), which is absurd, since for sufficiently large n one must assert -B (n). In ‘A Quick Read is a Wrong Wright’ (Chapter 26), Putnam replies that his proposal was not to deny that the vague predicate has a sharp cut-off but to refuse to assert it; in fact, Read and Wright’s argument shows that one must assert its double negation. This is not the end of the matter, for Read and Wright are challenging Putnam to express the vagueness of B within an intuitionist framework. In asserting — Ek(B(n) & ~B(n+l)), Putnam does not assert anything that would not be assertible if B were precise, for he would then assert 3n(B(n) & ~B(w+l)), which entails its own double negation. In asserting ~~3n(B(n) & ~B(n+1)) without asserting 3n(B(n) & ~B(n+l)), one might in some sense show that B is vague, but one does not say that it is. A similar issue arises if one wishes to express the borderline status of a particular number n with respect to B; Putnam suggests asserting — B(n) without asserting B(n), but that does not enable one to say that n is borderline. Although one can explicitly refuse to assert 3n(B(n) & ~B(n+l)) or B(n), that implies nothing about the vagueness of B in the absence of a reason for one’s refusal; one’s reason for refusing to make an assertion sometimes consists in accidental ignorance. We want an explicit statement of vagueness, not just a conversational implicature. Even to say ‘B is vague’ in so many words is dangerous for, on Putnam’s approach, ‘B is vague’ and 3rc(B(fl) & ~B(n+1)) seem inconsistent; by standard intuitionist reasoning, ‘B is vague’ therefore entails ~3n(B(n) & ~B(n+l)), contrary to Putnam’s approach.4Similarly, ‘n is borderline for B’ and B(aî) seem inconsistent so, by the same reasoning, ‘n is borderline for B’ would entail ~B(n), again contrary to Putnam’s approach. Although Putnam might challenge the move from P,Q |- 1 (1 a contradiction) to P |- ~Q, the extra apparatus needed to explain the putative failure of the move might suffice to handle vagueness on its own, and make the retreat to intuitionist logic redundant. Read and Wright point out that, in the proof-theoretic semantics for intuitionist logic, a proof that p is unprovable constitutes a proof of ~p . If the vagueness of B proves that 3n(B(n) & ~B(w+l)) is unprovable, and the borderline status of n for B proves that B(n) is unprovable, that semantics gives us proofs of the negations of those formulas, again contrary to Putnam’s view. Putnam replies that he was proposing intuitionist logic not intuitionist semantics for vague language. He notes the difficulty of generalizing the proof-theoretic ideas in the intuitionist semantics for mathematical language to vague empirical language, and suggests that if p is empirical, the assertibility of the unassertibility of p does not entail the assertibility of ~p. In the absence of anything like intuitionist semantics, the soundness and completeness of intuitionist logic for a vague language would be something of a mystery. In the final essay, ‘Putnam on the Sorites Paradox’, Williamson argues that even if Putnam rejects intuitionist semantics, his commitment in ‘Vagueness and Alternative Logic’ to an epistemic conception of truth exposes him to a modified version of Read and Wright’s objection. More recently, Putnam has distanced himself from epistemic conceptions of truth, although he has not applied that distancing to the problem of vagueness. The abandonment of an epistemic conception of truth removes one barrier to an epistemic conception of vagueness.5

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Notes 1 Unger has subsequently moderated his view (Unger, 1990, pp. 192, 330-32). 2 Kit Fine’s ‘Vagueness, Truth and Logic’ (1975) contains the most comprehensive application of supervaluation semantics to the problem of vagueness. 3 There have been several recent exchanges on epistemicism. Sorensen (1995) and Williamson (1996) reply to Wright (1995). Gómez-Torrente (1997), Horgan (1997), Tye (1997) and Williamson (1997b) form a symposium. Williamson (1997a) replies to Horwich (1997) and Schiffer (1997). Burgess (1998) replies to Williamson’s ‘What Makes it a Heap?’ (Chapter 12, this volume). Williamson (1999a, 1999b and 2000) reply to Schiffer (1999), Mott (1998) and Andjelkovic (1999) respectively. McGee and McLaughlin (1997), Sainsbury (1997) and Bonini, Osherson, Viale and Williamson (1999) are also relevant. 4 See also Chambers (1998). For references to other work on intuitionist treatments of vagueness see Williamson, ‘Putnam on the Sorites Paradox’, Chapter 27 in this volume. 5 Recent monographs on vagueness include Bums (1991) and Williamson (1994); Pinkal (1995) and Soames (1999) also have much relevant material. Keefe and Smith (1996) is another anthology of essays on vagueness. Several recent issues of journals have been devoted to vagueness: The Southern Journal o f Philosophy, 23 (1995), supplement; The Monist, 81 (2) (1998); Philosophical Topics, 28 (2000); Acta Analytica, 14 (1999). We have avoided overlap with these collections since students of vagueness will probably want to possess them anyway. We have also excluded essays on vague objects and identity since the topic is well covered by another volume in this series, Noonan (1993). Keefe and Smith (1996) and Williamson (1994) contain extensive bibliographies on vagueness. The references below include many recent papers in English on vagueness.

References and Further Reading Andjelkovic, Miroslava (1999), ‘Williamson on Bivalence’, Acta Analytica, 14, pp. 27-33. Andjelkovic, Miroslava and Williamson, Timothy (2000), ‘Truth, Falsity and Borderline Cases’, Philosophical Topics, 28, pp. 211-44. Bonini, Nicolao, Osherson, Daniel, Viale, Riccardo and Williamson, Timothy (1999), ‘On the Psychology of Vague Predicates’, Mind and Language, 14, pp. 377-93. Burgess, John A. (1998), ‘In Defence of an Indeterminist Theory of Vagueness’, The Monist, 81, pp. 23352. Bums, Linda (1991), Vagueness: An Investigation into Natural Languages and the Sorites Paradox, Dordrecht: Kluwer. Chambers, Timothy (1998), ‘On Vagueness, Sorites, and Putnam’s “Intuitionistic Strategy” ’, The Monist, 81, pp. 343-48. Cleveland, Timothy (1997), ‘On the Very Idea of Degrees of Truth’, Australasian Journal o f Philosophy, 75, pp. 218-21. Cooper, Neil (1995), ‘Paradox Lost: Understanding Vague Predicates’, International Journal o f Philosophical Studies, 3, pp. 244-69. Copeland, B. Jack (1997), ‘Vague Identity and Fuzzy Logic’, The Journal o f Philosophy, 94, pp. 514-34. Dummett, Michael (1995), ‘Bivalence and Vagueness’, Theoria, 61, pp. 201-16. Edgington, Dorothy (2002), ‘Williamson on Vagueness, Identity and Leibniz’s Law’, in P. Giaretta, A. Bottani and M. Carrara (eds), Individuals, Essence and Identity: Themes o f Analytic Metaphysics, Dordrecht: Kluwer. Everett, Anthony (1996), ‘Qualia and Vagueness’, Synthese, 106, pp. 205-26. Field, Hartry (1998), ‘Some Thoughts on Radical Indeterminacy’, The Monist, 81, pp. 253-73. Fine, Kit (1975), ‘Vagueness, Truth and Logic’, Synthese, 30, pp. 265-300. Reprinted in Keefe and Smith (1996), Vagueness: A Reader. Fodor, Jerry and LePore, Ernest (1996), ‘What Cannot be Evaluated Cannot be Evaluated and it Cannot be Supervalued Either’, The Journal o f Philosophy, 93, pp. 516-35.

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Gómez-Torrente, Mario (1997), ‘Two Problems for an Epistemicist View of Vagueness’, in E. Villanueva (ed.), Philosophical Issues 8: Truth. Gómez-Torrente, Mario (2002), ‘Vagueness and Margin for Error Principles’, Philosophy and Phenomenological Research, 64, pp. 107-25. Graff, Delia (2000), ‘Shifting Sands: An Interest-Relative Theory of Vagueness’, Philosophical Topics, 28, pp. 45-81. Graff, Delia (2001), ‘Phenomenal Continua and the Sorites’, Mind, 110, pp. 905-35. Graff, Delia (2002), ‘An Anti-Epistemicist Consequence of Margin for Error Semantics for Knowledge’, Philosophy and Phenomenological Research, 64, pp. 127—42. Hawley, Katherine (1998), ‘Indeterminism and Indeterminacy’, Analysis, 58, pp. 101-106. Heck, Richard G. Jr (1998), ‘That There Might Be Vague Objects (So Far as Concerns Logic)’, The Monist, 81, pp. 274-96. Horgan, Terry (1997), ‘Deep Ignorance, Brute Supervenience and the Problem of the M any’, in E. Villanueva (ed.), Philosophical Issues 8: Truth. Horgan, Terry (1998), ‘The Transvaluationist Conception of Vagueness’, The Monist, 81, pp. 313-30. Horwich, Paul (1997), ‘The Nature of Vagueness’, Philosophy and Phenomenological Research, 57, pp. 929-35. Hyde, Dominic (1992), ‘Rehabilitating Russell’, Logique et Analyse, 35, pp. 139-73. Hyde, Dominic (1997), ‘From Heaps and Gaps to Heaps of Gluts’, Mind, 106, pp. 641-60. Hyde, Dominic (1998), ‘Vagueness, Ontology and Supervenience’, The Monist, 81, pp. 297-312. Kamp, Hans (1975), Two Theories about Adjectives’, in Edward L. Keenan (ed.), Formal Semantics o f Natural Language, Cambridge: Cambridge University Press. Kamp, Hans and Partee, Barbara (1995), ‘Prototype Theory and Compositionality’, Cognition, 57, pp. 129— 91. Keefe, Rosanna (1998), ‘Vagueness by Numbers’, Mind, 107, pp. 565-79. Keefe, Rosanna and Smith, Peter (1996), Vagueness: A Reader, Cambridge MA: MIT Press. Leeds, Stephen (1997), ‘Incommensurability and Vagueness’, Nous, 31, pp. 385-407. Levi, Don (1996), ‘The Unbearable Vagueness of Being’, The Southern Journal o f Philosophy, 34, pp. 47192. Lowe, E.J. (1995), ‘The Problem of the Many and the Vagueness of Constitution’, A nalysis, 55, pp. 179— 82. Lowe, E.J. (1997), ‘Reply to Noonan on Vague Identity’, Analysis, 57, pp. 88-91. Manor, Ruth (1997), ‘Only the Bald are Bald’, in Georg Meggle (ed.), Analyomen 2, Volume II: Philosophy o f Language, Metaphysics, Hawthorne: de Gruyter. McLaughlin, Brian (1997), ‘Supervenience, Vagueness and Determination’, in James Tomberlin (ed.), Philosophical Perspectives, 11: Mind, Causation and World, Boston: Blackwell. McGee, Vann and McLaughlin, Brian (1997), ‘Review of Vagueness’, Linguistics and Philosophy, 21, pp. 221-35. Morreau, Michael (1999), ‘Supervaluation Can Leave Truth-Value Gaps After AH’, The Journal o f Philosophy, 96, pp. 148-56. Morton, Adam (1997), ‘Hypercomparatives’, Synthese, 111, pp. 97-114. Mott, Peter (1998), ‘Margins for Error and the Sorites Paradox’, The Philosophical Quarterly, 48, pp. 494504. Müller, Vincent (1997), ‘Real Vagueness’, in Georg Meggle (ed.), Analyomen 2, Volume II: Philosophy o f Language, Metaphysics, Hawthorne: de Gruyter. Noonan, Harold (ed.) (1993), Identity, Aldershot: Dartmouth. Osherson, Daniel and Smith, Edward (1997), ‘On Typicality and Vagueness’, Cognition, 64, pp. 189— 206. Parikh, Rohit (1996), ‘Vague Predicates and Language Games’, Theoria (Spain), 11, pp. 97-107. Pinkal, Manfred (1995), Logic and Lexicon: The Semantics o f the Indefinite, trans. G. Simmons, Dordrecht: Kluwer. Priest, Graham (1998), ‘Fuzzy Identity and Local Validity’, The Monist, 81, pp. 331-42. Raffman, Diana (1996), ‘Vagueness and Context-Relativity’, Philosophical Studies, 81, pp. 175-92. Sainsbury, R.M. (1997), ‘Easy Possibilities’, Philosophy and Phenomenological Research, 57, pp. 907-19.

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Schiffer, Stephen (1997), ‘Williamson on our Ignorance in Borderline Cases’, Philosophy and Phenomenological Research, 57, pp. 937-43. Schiffer, Stephen (1998), Two Issues of Vagueness’, The Monist, 81, pp. 193-214. Schiffer, Stephen (1999), T he Epistemic Theory of Vagueness’, in J. Tomberlin (ed.), Philosophical Perspectives 13: Epistemology, Oxford and Boston MA: Blackwell. Simons, Peter (1997), ‘Vagueness, Many-Valued Logic and Probability’, in Wolfgang Lenzen (ed.), Das weite Spektrum der analytischen Philosophie, Hawthorne: de Gruyter. Soames, Scott (1999), Understanding Truth, New York and Oxford: Oxford University Press. Sorensen, Roy (1995), ‘Commentary: The Epistemic Conception of Vagueness’, The Southern Journal o f Philosophy, 33 (supplement), pp. 161-70. Sorensen, Roy (1997), ‘The Metaphysics of Precision and Scientific Language’, in James Tomberlin (ed.), Philosophical Perspectives, 11: Mind, Causation and World, Boston: Blackwell. Sorensen, Roy (1998a), ‘Sharp Boundaries for Blobs’, Philosophical Studies, 91, pp. 275-95. Sorensen, Roy (1998b), ‘Ambiguity, Discretion and the Sorites’, The Monist, 81, pp. 215-32. Tye, Michael (1996), ‘Fuzzy Realism and the Problem of the Many’, Philosophical Studies, 81, pp. 215— 25. Tye, Michael (1997), ‘On the Epistemic View of Vagueness’, in E. Villanueva (ed.), Philosophical Issues 8: Truth. Unger, Peter (1990), Identity, Consciousness and Value, New York and Oxford: Oxford University Press. Villanueva, Enrique (ed.) (1997), Philosophical Issues 8: Truth, Atascadero CA: Ridgeview. Williamson, Timothy (1994), Vagueness, London and New York: Routledge. Williamson, Timothy (1996), ‘Wright on the Epistemic Conception of Vagueness’, Analysis, 56, pp. 3945. Williamson, Timothy (1997a), ‘Reply to Commentators’, Philosophy and Phenomenological Research, 57, pp. 945-53. Williamson, Timothy (1997b), ‘Imagination, Stipulation and Vagueness’ and ‘Replies to Commentators’, in E. Villanueva (ed.), Philosophical Issues 8: Truth. Williamson, Timothy (1999a), ‘Schiffer on the Epistemic Theory of Vagueness’, in J. Tomberlin (ed.), Philosophical Perspectives 13: Epistemology, Oxford and Boston MA: Blackwell. Williamson, Timothy (1999b), ‘Andjelkovic on Bivalence: A Reply’, Acta Analytica, 14, pp. 35-8. Williamson, Timothy (2000), ‘Margins for Error: A Reply’, The Philosophical Quarterly, 50, pp. 76-81. Williamson, Timothy (2002a), ‘Epistemicist Models: Comments on Gómez-Torrente and G raff’, Philosophy and Phenomenological Research, 64, pp. 143-50. Williamson, Timothy (2002b), ‘Vagueness, Identity and Leibniz’s Law’, in P. Giaretta, A. Bottani and M. Carrara (eds), Individuals, Essence and Identity: Themes o f Analytic Metaphysics, Dordrecht: Kluwer. Williamson, Timothy (forthcoming/a), ‘Vagueness in Reality’, in M. Loux and D. Zimmerman (eds), The Oxford Handbook o f Metaphysics, Oxford: Oxford University Press. Williamson, Timothy (forthcoming/b), ‘Horgan on Vagueness’, Grazer Philosophische Studien. Wright, Crispin (1995), ‘The Epistemic Conception of Vagueness’, The Southern Journal o f Philosophy, 33 (supplement), pp. 133-59.

Part I Nihilism

[1] PETER UNGER

T H E R E A R E NO O R D I N A R Y T H I N G S

Human experience, it may be said, naturally leads us to have a certain view of reality, which I call the view o f common sense. This view is tempered by cultural advance, but in basic form it is similar for all cultures on this planet, even the most primitive and isolated. Accord­ ing to this prevalent view, there are various sorts of ordinary things in the world. Some of these are made by man, such as tables and chairs and spears, and in some ‘advanced’ cultures also swizzle sticks and sousaphones. Some are found in nature such as stones and rocks and twigs, and also tumbleweeds and fingernails. I believe that none of these things exist, and so that the view of common sense is badly in error. In this paper, I shall argue for this negative belief of mine. It shall not be my business here to offer arguments concerning the question of whether there are any people, or conscious beings. I contrast these putative entities with mere things, and trust that my usage of the latter term follows one common way of allowing for such a distinction. Further, among such things, I shall discuss only those which are not living or alive; perhaps I may call them ordinary inanimate objects. Nothing of basic importance depends upon any such a division; it serves only to restrict my topic conveniently. A second restriction I impose on myself is not to discuss certain more general concepts which are intended to delineate in a ‘thing-like way’ suitable portions or aspects of ‘the external world’, or of ‘physical reality’. Accordingly, while I shall argue that our concept of a stone, for example, is devoid of application, I shall not make any such claim for our concept of a physical object, or for any similarly general idea. So far as these present arguments go, then, there may well be various physical objects, indeed, even of a great variety of shapes and sizes. But whatever the shapes and sizes of any such objects, none will ever be a table, a stone, or any ordinary thing. At Synthese 41 (1979) 117-154. 0039-7857/79/0412-0117 $03.80. Copyright © 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and B oston, U.S.A.

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the same time, my arguments do not require the existence of any physical objects, but leave that question entirely open. The arguments I will offer for my negative beliefs are variations upon the sorites argument of Eubulides, that incomparable Greek genius who also disclosed the paradox of the liar, the problems of presupposition and those of intentionality.1 In its original form, the sorites argument appears to have concerned how many items, say beans, or grains of sand, or even some of each, will be sufficient to constitute a heap. None or one is insufficient. But, if there isn't any heap before us adding a single grain or bean, it seems, will not produce a heap. Hence, even with a million beans quite nicely arranged, there will be no heap of them. By generalization, this is a compelling argument that there are no heaps, and that our concept of a heap is relevantly incoherent. It is, we might say, a direct argument for this idea and, I believe, it is a sound one. Conversely, we may begin by supposing that there are heaps, and that a million beans typically arranged gives us an instance of that concept. But, then, removing a single peripheral bean gently from such a typical heap, it seems, will not leave us with no heap before us. Hence, we must conclude that even when we have but one bean left, or none at all, we still have a heap of beans. But this is absurd. Hence, we have reduced the original supposition of existence to an absurdity, and we may generalize accordingly. This, we may say, is an indirect argument that there are no heaps, and that our concept of them is not a coherent one. It is also, I believe, an adequate argument. Now, Eubulides’ seminal contribution has long labored under the misnomer of ‘the sorites paradox’. But, in any philosophically important sense, there is no paradox here. Rather, we are given two demonstrations of the non-existence of heaps, while no important logical problems come from accepting the conclusion. It is hoped that as this paper develops, we shall better appreciate our true inheritance from Eubulides. As a clarificatory note, let me point out that the sorites arguments just presented did not involve the notion of identity in any interesting way; we never said, or cared, which heap was present. Indeed, if that idea is involved at all, which I doubt, it is only in the manner in which any terribly general idea may be presupposed by, and so involved in,

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any argument at all, or virtually any. The arguments that I shall presently deploy similarly avoid iany interesting involvement with identity. By introducing the notion, I suggest, we may obtain further arguments to the same effect; so our avoidance of it will only make things harder for our Eubulidean efforts. But even with sparse materials, it will be seen, the existence of all ordinary things may be disproved. Indeed, this may be done several times over, in each of a variety of complementary ways. In each case I shall try to keep the reasoning quite simple and straightforward: for the fundamental issues, 1 believe, are themselves of such a nature. In the final section of this essay, I will discuss what I take to be the implications of these present rather restricted reasonings. 1. O R D I N A R Y T H I N G S A N D T H E S O R I T E S O F D E C O M P O S I T I O N

By ordinary things, as I have indicated, I mean such things as pieces of furniture, rocks and stones, planets and ordinary stars, and even lakes and mountains. This is not the only way that this expression may be used, but it surely represents no philosophic eccentricity on my part. For example, my use of the expression appears fairly close to that of W. V. Quine in his influential book, Word and Object.2 Despite certain engaging departures from accepted common sense, such as his views on “the indeterminacy of translation” , Quine’s book does, nevertheless, operate on a foundation of common sense assumptions. The first section of the work is called, aptly enough, 'Beginning With Ordinary Things’, and the book’s body begins with this sentence: “This familiar desk manifests its presence by resisting my pressures and by deflecting light to my eyes.” For Quine, then, a desk is a paradigm of an ordinary thing; his usage is much like mine. The difference between us, of course, is that while he thinks, along with almost everyone else, that there are such objects, I hold that there are no desks, nor any other ordinary things. It will not serve much of a point, I suppose, for me to list those philosophers whose usage is similar to my own. Nor shall I try to catalogue the various philosophies which rely on the supposition that there are ordinary things, however inexplicit they may be on the matter. For the nature

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of the issue is clear enough, and that it is of moment to various philosophers is also rather obvious. To jolt our minds away from common sense thinking, and toward the denial of desks and stones, a bit of ‘general science’ may be of more help than any celebrated philosophy. Even from the early grades, we are given some simple scientific learning which in broad outline, and with fatal incoherence, is this: our ordinary things, like stones, which most certainly exist, comprise or consist of many atoms, and even many more sub-atomic particles. The point here has little to do with any niceties of such a term as ‘consist’, but may be put this way: in any situation where there are no atoms, or no particles, there are in fact none of our ordinary things. This should move us to deny, with proper reasoning, the existence of all alleged ordinary things. The reasoning for this denial does not require atoms or particles. But for jolting the mind, I have found it helpful to cast it in such terms. I will do so here, choosing stones as my ordinary things and atoms as removable constituents. Accordingly, we may express these three propositions, which reasoning informs us form an inconsistent set: (1) There is at least one stone. (2) For anything there may be, if it is a stone, then it consists of many atoms but a finite number. (3) For anything there may be, if it is a stone (which consists of many atoms but a finite number), then the net removal of one atom, or only a few, in a way which is most innocuous and favorable, will not mean the difference as to whether there is a stone in the situation. The reasoning here is simple. Consider a stone, consisting of a certain finite number of atoms. If we or some physical process should remove one atom, without replacement, then there is left that number minus one, presumably constituting a stone still. Whether what is left is the same stone, as it presumably is, or whether it is another one makes no difference to our cautious reasoning here: thus do we make

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good our resolve to foreswear reliance on considerations of identity. (Indeed, we may be more cautious still, writing our third premise so that we require only that at least one stone is left.) Now, after another atom is removed, there is that original number minus two; so far, so good. But after that certain number has been removed, in similar stepwise fashion, there are no atoms at all in the situation, while we must still be supposing that there is a stone there. But as we have already agreed, in (2), if there is a stone present, then there must be some atoms. There is, then, a rather blatant inconsistency in our thought. However discomforting it may be, I suggest that any adequate res­ ponse to this contradiction must include a denial of the first pro­ position, that is, the denial of the existence of even a single stone. Whatever one then thinks of the other two propositions is a further, much more minor matter. Whether one eventually deems them straightforwardly true, vacuously true, without truth-value, or what­ ever, will not be of any surpassing importance. I call this argument, the sorites o f decomposition or, more fully, the sorites o f decomposition by minute removals. It is an indirect argument for the conclusion that there are no stones and, by general­ ization, no other ordinary things.3 I believe this indirect argument to be not only compelling but sound. Let us consider a number of points of commentary which may help us to assess this belief of mine. The first point we may consider is that, while this sorites of decomposition works well against the supposed existence of our ordinary things, it does not, in contrast, work as compellingly to deny the existence of physical objects. For no matter how small a thing is removed, if anything is left, which that first has been removed from, that remaining item, for all we can compellingly argue, may be a physical object. It takes some doing to argue to the contrary, and any such extra effort will only begin to approach, at best, our argument against any ordinary things. Now, none of this is to say that physical objects cannot be made to cease to exist, by cutting into them, so to say, or in any other way; nor is it to deny that. I wish only to notice the difference between how compellingly our sorites works to deny an

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ordinary thing, a stone, for example, and how much less powerfully it works against physical objects (if, indeed, it has any power at all with regard to the latter). Now, if one appreciates this contrast, it follows that he will be finding our sorites of decomposition quite compelling as deployed against our ordinary things: against stones and rocks, desks and tables, and planets and certain stars. While he may wish to alter the argument thus deployed, the alterations are, then, only a matter of details and niceties. While he may wish an explanation, in some depth, of why supposed ordinary things fall prey, that too presupposes agreement that our argument surely appears quite sound. The correct explanation of this appearance is, I suggest, the simplest one: the argument is as it appears to be, that is, it is a sound one. A contrast may also be drawn between ordinary things and certain particulars which are prominent in the physical sciences. A compelling argument of this sort may be given to deny stones, planets and at least certain stars, but not electrons, hydrogen atoms, and water molecules, or so it now appears. A second point worth noting is that the central idea of this argument does not depend on atoms, or on anything else so very minute. For example, we may remove ‘a speck of dust’s worth’ at a time to the detriment of any putative stone. For certain artifactual items, like a table, possible future cases might require decremental units which fall below the level of of unaided perception. For exam­ ple, someone might conceivably construct a table ‘smaller than a speck of dust’. With the proper equipment on his part, I am not sure that facts of material structure would prevent him from getting things into the intended shape, etc. And, perhaps the resulting item would be termed a table by common sense judgement. But the removal of tiny units would show this judgement also to be erroneous. Now, in imagination we may contemplate a table, or a stone, it seems, shrinking down to the size of an atom. But that is not real. When you remove more and more small units from an alleged table or stone, you don’t keep getting smaller and smaller tables or stones, or anything relevantly preservative.4 Further, our argument implies no particular, not to say particulate, theory of matter. For all we care,

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the only physical reality may be a single plenum, modifications of which are perhaps poorly labeled as atoms or as particles. As a third point, we may agree that it is somewhat arbitrary how many propositions we display in order to generate the contradiction. As we have it here, the third proposition is not one which entails that any stone consists of atoms, but it does entail that any stone which does consist of them may have an atom, or a few, removed without replacement. Alternatively, we might have conjoined our second proposition with our third, to pit against the assertion of existence a single more complex proposition. Alternatively yet again, we might unravel our third proposition into one which says that any stone consisting of atoms allows for such a net removal and another which says that whenever such a removal should happen to such a stone we are left with a stone.5 It is an interesting task, then, to spell out more explicitly the specific assumptions which underlie this argument, thus increasing the number of members of the inconsistent set we may exhibit. But we may confidently say even now, I suggest, that however finely we demarcate things, the relevant propositions are accepted by almost all educated human beings. The problem, then, is to respond to the inconsistency, and the most compelling solution, I suggest, is to deny the existence of ordinary things. As a fourth point, we should allow that our third premise, and even our second, has not been stated in a manner which is very clear or explicit. But the statement of a premise may be refined, while no substantial change in the argument will result from any relevant alteration. This is not to suggest that, in our present system of concepts, such a premise may ever be made adequately precise. It is only to notice that any attempt to move in that direction will mean no important problem for our argument. For example, one may squabble over the word ‘many’ or over the word ‘few’. I chose these vague words because they rather faithfully express, I supposed, the unreflective beliefs on these matters which most people have, so few of us being scientists. But we may of course replace them by more definite expressions which, upon reflection, must be admitted to yield acceptable propositions. Thus, we may say, by way of illustration, that any stone consists of at least one billion atoms, and that

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removing no more than ten thousand leaves a stone. Again, one may complain about our use of an expression so vague as ‘a way which is most innocuous and favorable’. But then we may be more explicit, always pretending, so to say, that there is real substance to the matter of whether there is a stone present or not, while always on the way toward showing how insubstantial is that very matter. In this direc­ tion, we might say, for example, that no atom is to be removed forcefully from a central position, but one may be taken gently from a peripheral location. We may say, again, that we are to remove in a manner most favorable to there being a stone left there after the removal, supposing such an actually absurd thing to have some bearing on reality. Further, we may point out that we are not supposing that in any given case one way is the most favorable or innocuous. At almost any choice-point, so to say, any atom or relevant group, of many millions, might be just as favorable to remove as any other. Certain conditions, which we may include under ‘way’, or which may be mentioned distinctly, may all be equally as favorable, and more so than many others; and so on. We might say, further, that there is nothing mysteriously ideal about these favorable ways, as my remarks about removing dust specks (supposing them to exist) make quite clear. For such ways and conditions, or ones near enough to them, appear to occur all the time on the face of the earth. Finally, to put a convincing cap on this whole matter, we note that all we need for our argument is this: for any putative stone, there is always at least one way of removing at least one minute item without going from (at least one or) a stone to none. With even one relevantly gradual path to follow, we can ‘peel our onion’ down to nothing. In fact, it seems, there are an ‘enormous’ number of these paths always open to us. Hence, we make our point with powerful overkill. As a fifth point, we may reply to the Mooreian gambit of clutching onto common sense at the expense of anything else, most especially any philosophical reasoning.6 According to this way of thinking it is always most appropriate to reply to philosophical challenges as follows. We are more certain that there are tables than of anything in the contrary philosophic reasoning. Hence, while we may never be

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able to tell what is wrong with the reasoning, at least one thing must be wrong with it. But while such a generalization may prove a useful guide in addressing many philosophical challenges, is it to have no exceptions at all? I think that an unquestioning affirmative answer here is not only likely to be untrue, or incorrect, but is extremely dogmatic. What of the present case, then, might not that be just such an exception? The merits of the case must be judged in terms of the particulars. That is, of course, we consider those points, some already made, which have much more to do with the issues here at hand. We have an inconsistency to which to respond. If we persist with our belief in ordinary things what rational responses are available? As a sixth point, we may note that any response other than our suggested one involves us in the acceptance of a miracle, in a fair employment of that term. The miracle expected will be of one of these following two kinds, though of course someone might expect both sorts of miracle. First, tables and stones might be preserved by natural breaks in the world order, so to say, by disjoint happenings whose occur­ rence prevents nature from being relevantly gradual. For example, after a few atoms were successively removed, or a few minute chips, it might be physically impossible to remove another. Or, for another example, after the sixth atom or chip was removed, the removal of the seventh might occasion a drastic result: the remainder might ‘go out of existence’*or turn into a frog, or whatever. Such happenings as this go against our daily experience, as with sanding a piece of wood or smoothing stones. They also conflict with our scientific perspective which, taking things down to deeper levels, fits nicely with this everyday experience. To expect tables and stones to be saved by such cooperative breaks in nature is, I say, to expect a miracle o f metaphysical illusion. Thinking nature relevantly gradual, this res­ ponse has little appeal for me. Now, given that the world is in fact relevantly gradual, and ap­ parently quite uncooperative, the only hope for ordinary things will lie with the human mind. We must suppose, contrary to what our intuitions seem to be telling us now, and contrary to what we believe to be the rather limited power of our everyday conceptions, that we

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are all the time employing ideas that have precise limits. We must suppose that with, say, a trillion trillion atoms there, in a certain case, there really is a stone, whether anyone can ever tell or not. But, with one or a few, say fifty, gingerly removed from the outside, the situation suddenly changes, even if no one can ever tell. And this means that with any one, or any fifty, of the atoms gone, there is no stone there. That’s the sensitivity of our word ‘stone’ for you! To believe in this is, I say, to believe in a miracle o f conceptual comprehension. Thinking of our everyday thought as relevantly im­ precise and unrefined, this alternative response also has little appeal for me. Accordingly, I must abandon my belief in stones. A seventh point will now be helpful to consider. This is the point that whatever holds true of such allegedly familiar things as stones and rocks, and desks and tables, insofar as it is relevant to our topic, also holds true of swizzle sticks, sousaphones, withered tumbleweeds and hundreds of other ordinary things. The ideas of these others are less familiar and frequent for us. The ideas of stones and tables, in comparison, are like old and trusted friends. To think these familiars inapplicable may thus, for most of us, occasion greater discomfort than thinking the same about, say, ‘swizzle sticks’. But the logical situation can scarcely sustain any such emotional difference. If tables will be left behind with the innocuous removal of a single atom, presumably any one of millions, then the same miracle must hold for swizzle sticks. The idea that nature may favor tables against swizzle sticks defies credibility, I hope even that of the most adamant defenders of common sense. But does anyone really think that by taking away peripheral atoms we shall ever encounter such a sensitive swizzle stick? Does anyone imagine that our concept of a swizzle stick discriminates at the required atomic level? Surely, this is quite absurd. But, then, it is just as absurd in the case of tables, and of stones. The last matters for us in this first section come from an idea of Donald Scheid, offered by him to me in conversation. He notes that, according to common sense, if we have an ordinary thing, there is some sufficiently large ‘part’ of it which would not leave such a thing

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were it taken away. For example, if one ate 97 percent of an apple, in the typical fashion, one would not still have an apple there. Following Scheid, we might say that that 97 percent was an ‘obliterating part’ of the apple. But, then, he notes, if something is an obliterating part, then one atom less should still give us an obliterating part. In stepwise fashion, we must conclude that a single atom is an obliterating part, for any ordinary thing. But doesn’t this contradict our third premise, which says that if you have a stone, then removing an atom leaves a stone still? I think not. Scheid’s argument does not give us the negation of our conditional but, rather, a conditional with the same antecedent and the negation of our previous consequent: if something is a stone, then the het removal from it of an atom will not leave a stone. So far from jeopardizing our third premise, then, this gives us an additional argument from it against ordinary things. For now we may conclude that if something is a stone, then the net removal from it of an atom both will and will not leave a stone there. And so, we may conclude, nothing is a stone. Two further points may be noticed with respect to Scheid’s reason­ ing. First, by extending it further, we may conclude that if something is a stone, then removing nothing from it at all will leave no stone. For if one atom is an obliterating part, and so is one atom less than that, the removal of nothing will mean the obliteration of such an object. Thus we may conclude, again, that there are no stones or other ordinary things. Though I have argued otherwise, Scheid’s reasoning might be supposed to refute our third premise. The supposition that it should do so lets us make our final point. For we have just seen that if this reasoning should refute our third premise, it will do so in a manner which is at least as effective against our first premise, that is, against ordinary things. To preserve rational belief in such objects, however, it can never be enough to undermine some contrary propositions in such a manner as that. Rather, we must refute such contrary state­ ments by reasoning which does not do as much, or worse, for our belief in ordinary things. And this, I suggest, can never be effectively accomplished.

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2. M A T H E M A T I C A L T H I N K I N G A N D T H E S O R I T E S O F S L I C I N G AND GRINDING

In recent years problems of vagueness have suffered the attempts of philosophers and logicians to afford them a solution in mathematical terms. Such an employment of technical devices is, I shall argue, almost certainly out of place in this connection, and it will not provide any rational counter to our sorites arguments. This is not to place any significant limitations upon mathematics. On the contrary, it is to recommend that its proper use here is, not to rescue hopeless concepts from demonstrations of their inadequacy, but to aid in the development of better, precise ideas with which those concepts may be replaced. To evaluate my recommendation, let us look at the situation. First, to judge by recent writings, the device more widely expected to help with vagueness, of those with a mathematical inspiration, is the assignment of new and exotic truth-values, most particularly of numerical ones. Along these lines, David Sanford, for example, has tried to develop a logic o f vagueness. As he himself says, he is rather representative: . . . I shall proceed from som e of the basic assum ptions shared by previous w orkers on the logic of v ag u en ess. . . The first assum ption I share is that an infinitely valued interpretation is appropriate in dealing with the application of sentential logic to vague sentences. T he values are the real num bers betw een 1 and 0 inclusive.7

The key idea in this sort of approach, for it to be at all plausible, is that the application of a vague term appears gradually to become more questionable as we move further from its accepted paradigms. The gradual change in truth-values is to mirror this appearance. But in reality this device, whatever else one may think of it, leaves wholly untouched the miracles which any failure of the sorites requires. The miracle of metaphysical illusion is no more to be expected now than before, when, presumably, truth and falsity were the only values in question. But, then, what of the miracle of conceptual comprehend sion; is it more to be expected now? There is some slight appearance to this effect. Now, with the removal of a peripheral atom, we do not go from truth to falsity, from one thing to its diametric opposite, so to

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say, but only from, say, unity to 0.999. So now, we go from one thing to another which is quite close by it. But as regards our required miracle, this appearance is quite deceptive, and accomplishes nothing. For any departure from unity, or indeed any change in truth-value, with the removal of a single atom, requires that our terms ‘stone’ and ‘swizzle stick’, be sensitive at least on the atomic level. That our expression ‘swizzle stick’ should be that sensitive quite defies credi­ bility. For it to be that discriminating is a miracle which surpasses my capacity for belief. The matter is not changed if we focus, not on statements, or on such things as might properly have a truth-value, but on sentences. Thus, in a relevant context, we may consider such a sentence as ‘There is a swizzle stick here’. Discounting the fatal vagueness inherent in the word ‘here’, and perhaps also that associated with the present tense use of the verb, we may ask when this sentence first ceases to express something which is true, whatever it then does instead. At the beginning, we are assuming, there is a swizzle stick before us; so at that point the sentence doesn’t fail to express a truth. As peripheral atoms come off, one by one, we are always asking whether it still does the same or whether now, for the first time, it does something else. For something else to be done, upon the removal of a peripheral atom, requires the same incredible sensitivity on the part of ‘swizzle stick’ as we previously noted. The distinction, then, between sentences and statements, which might prove helpful for other logical topics, does nothing to diminish the great problem here at hand. And in like manner, I suggest, further distinctions and devices of logic only get our required miracle further from our focus, and do not get its enormity to decrease by one jot. For me, then, no attempt to make it look like little is happening here can be of any use at all. Similar considerations, I believe, will suffice to refute any attempt at making our miracle look small, and so look to be expected, however complex that attempt might be. So far as my acquaintance goes, the following attempt is as complex as any available. It is suggested by an idea of Hartry Field, which is in turn based on an approach of David Lewis.8 Seeking no new logic, Lewis would assign

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a range of semantic values to sentences with vague terms. For Field these are “ successively higher degrees of truth” , to account for the fact that relevant sentences do “not jump suddenly from absolute falsehood to absolute truth” . But suppose two people differed as to where absolute truth is first to appear? There may be vagueness here too, Field suggests, in the term ‘true’, and various resolutions of it may yield various results for first appearance.9 Thus, it is suggested, there may be a great deal of play in the whole business. Letting qualms about these ‘resolutions’ have no say, we might amplify upon this idea. We might say, then, that different resolutions of ‘swizzle stick’ and ‘true’ together will yield different results, and this joint yield may let the look of suddenness fade further into the back­ ground. But what is accomplished by all of this? For our sorites to be thwarted, some resolution(s) of ‘true’ and ‘swizzle stick’ must apply up to a point, atomically counted, and then, quite suddenly, after a single peripheral atom is removed, presumably any one of millions, the whole business no longer applies! (Otherwise, the whole business will still apply with only one atom left, which is absurd.) But that, in a new guise, is our miracle of conceptual comprehension all over again. And, it is no more to be expected now than before, despite the more complex formulation in which it appears. For those who expect no such miracle at all, there is no response but to deny ordinary things. And this, I think, means that such a denial is the only rational response for anyone. In order to dissuade others from trying similar attempts to escape our arguments however, more may be required than to point up the reappearance of our miracle in various complex contexts. To alter the motivation behind such maneuvering, it may be best to take another, complementary tack. On this tack, we notice the ‘range of phenomena’ present in typical cases where philosophers have examined sorites arguments, cases where we have taller and taller ‘short’ men, or hotter and hotter ‘cool’ objects. And, we notice, also, that we have such an enticing range where we take atoms from a swizzle stick, one atom at a time, or only a few. We shall endeavor, then, to leave all such ranges well behind us. To do that, in fact, we need only return to the beginnings of all this» that is, to Eubulides and his heaps.

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As fairly as I can, I will try to approach the Eubulidean problem with the notion of degrees of truth, and the range of available values that it implies. Now, according to my own idiolect, or my linguistic intuitions, I can feel comfortable, whatever that may be worth, in saying with five beans suitably before me, ‘H ere’s a very small heap, and so, a heap’. Perhaps, then, I should assign the value of 1, or at least 0.95, to the proposition that Oscar is a heap, where ‘Oscar’ is supposed to name the hopeful heap of five now before me. (When 1 first wrote this, I was actually playing with grains of rice, not beans in fact, to try seriously to get a feeling for the matter.) Taking away a bean from Oscar, to produce Felix (who may or may not be the same entity as Oscar), I fell less comfortable in saying a heap is before me. But it isn’t all that unsettling. What am I to do; assign a value of 0.9 to the proposition that Felix is a heap? Taking away one again, now to yield Leo, 1 am at a loss. My sensitive intuitions seem to desert me; I know not which way to turn. Perhaps a value of 0.5 is now in order, or is that a bit too high, or too low, and just a fake at compromise? With two beans, and only Alex before me, I feel like 0 is the value for me. But can such a sudden and great drop stand scrutiny? Perhaps we’d better go back and re-evaluate, or better yet, give up this game. A familiarity with mathematical thinking may engender a second way of attempting to counter our attack on common sense. It may be urged that our lengthy arguments, with a step for each atom or chip, may require us to employ mathematical induction, and that this may be unwarranted in ordinary contexts.10 Now, I suggest that an objec­ tion focusing on mathematical induction is, indeed, only a special case of a more general worry about our sorites of decomposition by minute removals. The more general worry, or objection, would be that our argument is too long: “ In pure mathematics, and even in certain forms of empirical and everyday reasoning, long arguments are all right. But in certain other contexts, and you have clearly hit on one of them, restrictions of length must be imposed.” This objection may be elaborated upon in a manner reminiscent of mathematical logic. It may be said that we have employed certain ‘rules of in­ ference’, or that we have presupposed them. These rules, while perhaps good for unlimited use in some contexts, must surely be

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restricted in others, notably in those where we would challenge common sense. A system of logistic might even be concocted to codify formally the restrictions thought necessary and proper, and so on, and so forth. I can think of no good reasoning in support of such an objection. But more importantly perhaps, it misses the fundamen­ tal point upon which our sorites argument revolves, and which appears already to have been anticipated by Eubulides, with his heap. For we have seen that short arguments cut heaps down to size soon enough. To bring the point home for our more ‘cohesive’ ordinary things, I shall provide a variant upon my original sorites of decomposition, which I call the sorites o f slicing and grinding. I will present this sorites in two versions, the first being more suitable for most artifactual objects, like tables, as well as for the more highly structured natural items, for example, twigs and pine cones. With regard to any table, then, no matter how large, I suggest that there will always be at least one way of partitioning by volume, say, into roughly equal fifths, or if one likes into eighths, so that the Eubulidean bafflement matches that lately encountered with the heap. Having made our partitioning imaginatively, we then envision a physical process occurring to the putative table as follows. First, one fifth of the table is sliced off and ground to a find dust, perhaps even rendered into separated atoms. The dust, if not atoms, is then scat­ tered to the winds, or sunk speck by speck into widely dispersed regions of the sea. Then a second fifth is sliced, ground and scattered, and so on. Now, when we are down to our last fifth, there is, quite clearly, no table present. But, then, when did we first have no table? I submit that there was none in the first place. The idea of this sorites is, then, quite simple, and very easy to comprehend. And while the argument only requires that there be at least one way of effecting a relevant partitioning, for at least one relevant fraction, say, fifths, there are in fact for any such fraction one chooses, sixths, sevenths and so on, an ‘enormous’ number of ways to divide bafflingly the putative table in question. Accordingly, with what might be described as powerful overkill, this sorites of slicing and grinding eliminates any objection to our chal-

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lenge as requiring reasonings that are ‘too lengthy for the context’. And, of course, mathematical induction plays no part here. Let us exhibit the argument, to see that it is, indeed, not a very lengthy one: if the original item is a table, then so is what remains after an appropriate fifth has been sliced off, ground fine and scat­ tered widely. If what remains then, about four fifths of the original, is a table, then so is what remains after another such fifth is sliced off, ground and scattered. If what remains then, about three fifths, is a table, then so is the two fifths or so that remains after the next fifth is thus treated. If that is a table, then we have one when we only have a fifth left of the original, say, a small part of the top, most of one leg and a bit of another. But, this last, we have agreed is no table. Hence, the original item is no table either, contrary to popular misconcep­ tions. This argument is quite fully laid out. And it employs no peculiar ‘rules of inference’ or ‘logical principles’. Moreover, it is rather shorter than many reasonings in everyday life, as well as in mathe­ matics, science and philosophy, which are widely accepted, and scarcely ever questioned. To deny this argument, it should be clear, is to become involved in absurdity. Some persons on whom the foregoing argument is tried might, nevertheless, attempt to resist it by thinking that an important line was crossed with the removal of a certain fifth, no matter how cleverly baffling the design of the partitioning. Such a one might, then, deny a particular intermediate premise on such putative grounds. For example, he might deny this premise: if there is a table with three fifths left, then there is still a table with two fifths left. To my mind, such a denial appears quite absurd. But even if it is granted some momentary plausibility, such a denial will be only a delaying device. For after we have removed easy fifths, we may imaginatively divide the problematic fifth itself into, say, fifths. And then we may slice and grind again. (If we have already destroyed that fifth with our original putative table, we may use a duplicate of it.) Now, if there is a table there with three fifths, i.e., with fifteen twenty-fifths, then there should be one with fourteen twenty-fifths; and so on down to ten twenty-fifths. This should decide matters well enough. If not, which I find hard to conceive, we may perform again and again the same

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UNGER

partitioning procedure until any resister will find absurd his own putative sensitivity. But with any rationality on the part of our resisting subjects, no great length will be necessary. Fifths should often suffice; the twenty-fifths should take care of any remaining laggards. An entirely similar argument will not work compellingly against stones, for hardly anyone, I suppose, would be convinced, without further reasonings, that a fifth of a stone was not a stone. We might get further if we asked him first to imagine something which was ‘just about the smallest’ item he would count as a stone, bearing in mind the realities of material structure. But even this might well fail to convince. If so, we may get a lot further by following this procedure: We cut that ‘approximately smallest’ stone in, say, tenths, more or less simultaneously grind nine and scatter their dust, or atoms. Is the remaining tenth a stone? We then repeat the procedure. After only ten steps, the remaining item, while presumably larger than the scattered particles, is only one ten-billionth of the original, which, we remember, the person originally thought of as just about as small as a stone could be. If at any place our subject thinks an important line has been crossed, we may then backtrack, if need be using a relevant duplicate. Starting with the larger item, on the ‘first side of that line’, we may then slice and grind in tenths, one at a time, in much the same manner as that lately employed with alleged tables. Thus, we may reduce to absurdity the existence of this approximately smallest stone. As it was alleged to be that small, our backtracking may well occur with the first cutting in tenths. To convince further, we may then start well on the other side of this ‘small stone’; we may start, for example, with a supposed stone ten times the size of it. This latter, for our subject, would clearly be a stone if anything ever is. And, beginning with it will not add many steps to our argument. Accordingly, whether the items we treat are more like alleged tables or more like supposed stones, no suspiciously long or complex reasoning is needed to deny compellingly assertions of existence. It may be useful to interject at this point a few remarks regarding the relation of our arguments to time. While we are casting them in a

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temporal form this is not crucial to our arguments. For one thing we may put our reasonings in a counter-factual form: if there is a table before us, there would still be one without a peripheral atom, and without one of the baffiingly imagined fifths. For another, we can consider differing items existing at one time, each in a different region of space. Each item might differ from the next to be considered by one peripheral atom, the last being only one atom itself. With such small differences ás that, our formulation may best be, not only spatial, but also counter-factual. More realistically, we may have five objects, say, corresponding to the steps of the argument lately envisioned for tables and, if need be, some extras for our baffling twenty-fifths. For stones, as noted, a few more items might be needed. A third mode of mathematical objection is one based on a putative distinction between fully extensional systems and those which are, in part at least, intensional. Our sorites, it might be thus objected, relies on at least one premise which is intensional, that is, which fails to be fully extensional. This occurs, perhaps, with the third premise, where we are talking about a series of decrements which may never have been effected. Which putative table, after all, has undergone decom­ position atom by atom, or even in a way much like that? Our premise only speaks, it may be said, of what would be the case if such decompositions were carried out, or were to obtain in nature, or something of that sort. Further, as understood extensionally, our premise may be true and yet generate no contradiction with our other exhibited propositions. For those minute, perhaps atomic, removals which actually do occur to stones may leave them all clearly stones, as it happens. And, it might be urged, it is only an extensional sentence which makes any proper sense, and which should be used in the formulation of any worthwhile sorites argument.11 Now, granting that there is a clear distinction of the sort intended, I can see no reason for denying importance to sentences which are intensional in the manner of our third premise. Indeed, without such sentences, we abandon any attempt to express causal relations, lawlike connections, and so on, amongst events and processes in nature. If this is the only

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way to escape our sorites, then we are on a path very nearly as radical as the one marked by a denial of ordinary things. It is doubtful, then, that many would wish to follow this reply. But even if one takes the heroic stance here, no true escape from the sorites attack will be effected. For supposing that they existed in the first place, actual processes of destruction and deterioration have befallen a great many ordinary things. And, in a great many o f these cases, it cannot truly be said that in any given second the situation changed from one where there was a stone, or table, or whatever, to one where there was not. If there was really a stone there at an earlier time, then so in the next second, and so on. But during seconds near the end, so to say, there clearly is no stone or table or whatever. Hence, in the actual situations in question there never were any stones, or tables, or whatever. But there is no relevant difference between these putative objects and any other ordinary things. If there ever were, are or will be any stones, or tables, or whatever, then some of these things were among them. Hence, there never were, are or will be any ordinary things at all. And, here we have reasoned only as regards actually occurring processes and, so, I suppose, in a wholly extensional manner. Concerning the relation of our sorites to mathematical thinking, perhaps I may sum up my position in this way: the sorites, now given in two main variations, points up a contradiction in our beliefs. The introduction of mathematical complexities may draw our attention to certain other propositions, perhaps interestingly related to the ones in question, which are, in contrast, mutually consistent. But nothing can make a contradiction itself disappear, and not just disappear from our view. Our belief in ordinary things, then, may be rescued by mathe­ matical thinking only insofar as the latter makes credible at least one of the two miracles we have already discussed several times over. This, of course, is not to be expected. Hence, in all likelihood, no mathematical system can make rational for us a belief in ordinary things. None of this, of course, is to disparage mathematical thinking. On the contrary, it is to suggest that the proper service for it lies, not in the rescue of incoherent common sense, but in the formulation of more adequate descriptions of reality.

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3. T H E S O R I T E S O F C U T T I N G A N D S E P A R A T I N G

There is another variation of our sorites which I should like to present and consider, the sorites o f cutting and separating. My introduction of it will be a bit roundabout; I hope instructively so. When focusing on our sorites of decomposition by minute remo­ vals, which proceeds atom by atom, or at least tiny chip by tiny chip, we may easily get the idea that fairly sizable physical objects are more stable, or better able to endure changes, than are ordinary things. As the atoms come off one by one, or a few at a time, we get to a situation where, even according to common sense judgements, there is no table, stone or sousaphone. It seems, however, that there is still a physical object before us, one consisting of many atoms, perhaps even many millions of them. Moreover, though I have foresworn serious inquiry here into matters of identity, we do have the thought that this remaining physical object might well be the same one as the bigger item with which we started, and which we wrongly called a table, a stone or a sousaphone. One who thinks along these lines, then, may well get the idea that ‘table’ names a state or phase which a given physical object may occupy, whether during a certain portion of its career or throughout its entire history. The term ‘table’ may thus be thought to bear much the same relation to ‘physical object’ as ‘infant’ or ‘philosopher’ may bear to ‘man’ or to ‘human being’. Finally, it may be thought that terms for ordinary things, like ‘table’, really do apply after all, and that our conclusion to the contrary was based on a confusion as to what sort of logical role the terms have in our conceptual scheme for things, on a category mistake if you please. Whatever we may think of the previous thoughts which may thus lead to it, however, this last idea is a non sequitur, and is in any case badly in error. For whatever the category or categories in which one may place a term, whether one identifies it as a term for an object, a process, a quality or whatever, if that term is incoherent, so it will remain. If ‘table’ is best thought of as a term for a state or a phase, it will be an incoherent term of that sort, and so it will apply to no real state or phase, just as it will apply to no real object, quality, or

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whatever. In any case, then, as our sorites arguments have already shown, terms for ordinary things will apply to no reality at all. Placing this most important point to the side now, it is still a mistake to suppose that any of these terms, ‘table’ for example, has much to do with putative phases, or with anything of the kind. Indeed, in many cases, quite the opposite impression attends these terms from that somewhat peculiarly generated by our sorites of decomposition by minute removals. Rather than looking to denote any transitory phase, terms for many sorts of ordinary things, ‘table’ for example, appear to transcend particular physical objects. Thus, the term ‘table’ purports to preserve identity for a given table, it appears, even as sizable physical objects, in which the table may be said to consist, move hither and yon, come into being and cease to be. For example, let us consider a putative table which is, we shall suppose, initially made of a single piece of metal, say, of iron. Let us cut off a substantial piece of the metal from the rest, say, a piece somewhere between two fifths and one-half of the whole, perhaps measured by volume, and let us send it miles away. Now, we may well think that we have two rather substantial physical objects, whereas before we had but one. But the same table may be thought to exist throughout, first consisting in the one physical object, and then in the two created by the cutting, which we may call its separated ‘parts’. If the parts are brought back together, and joined by solder, we may think of the result as the original table, now again consisting in one substantial material object. It is, then, not only very natural and ordinary, but in a wider respect sustained by common sense, to think of the table as existing in the interim, part here and part miles away, perhaps in California. And the same thought will occur, of course, for such putative tables as are thus cut apart but are never made whole again. According to our common view of the matter, then, it is not very easy to get rid of tables by such a procedure of cutting and separat­ ing. Such putative ordinary things, even if they don’t really exist at all, appear at least to be rather stable, and hardly like the potentially fleeting phases we extracted from our original sorites of decom­ position. This newly encountered appearance of stability, however, is

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also an illusion. To make this plain, we may construct an additional piece of reasoning, the sorites o f cutting and separating. This applies, first, to those ordinary things whose identity seems to transcend any particular material object, as recently indicated, like tables and sousaphones, and such rocks and stones as are ‘important or well known’, like Plymouth Rock and the Rosetta Stone. It applies equally, reflection will reveal, to those ordinary things whose identity appears more ephemeral, like a rock or stone of little importance or famili­ arity. Let us take these two cases in turn beginning with those things whose identity appears rather more stable. We begin by assuming that if we cut a table into two roughly equal parts, and do so most innocuously and favorably, then there is still a table left in the case, no matter how widely the parts are separated. Common sense has us make this assumption. Further, even where we always choose a ‘largest available part’, as we always shall, no single operation of cutting and separating will be enough to take us from a situation involving a table to one involving none. By a series of such operations, however, we shall eventually have upon us a situation where there are only resultant specks of dust, or even atoms, scat­ tered all over the solar system or even into regions far more remote. In such a situation as this last, however, we quite clearly have no table at all. To suppose that we still have one is to be committed to all sorts of absurdities, even according to the view of common sense. For one will, presumably, then suppose as well that every table that ever was still does exist and also, presumably, every mountain and every lake, every star and every planet. Hence, we have again uncovered a contradiction in our ordinary thinking. The only adequate response to it is, I suggest, to conclude that we have reduced to absurdity yet again the idea that there are, or ever were, any such things as tables.12 The intuitive correctness of this reasoning makes vividly clear the futility in the thought that certain ordinary things, because they are ‘functionally defined’, will withstand a sorites attack. But, of course, this Aristotelian approach gets ordinary thinking wrong at the start. A fake rake is not a rake (which is fake) but, supposing there to be any rakes at all, a broken rake is a rake which is broken. Further, a rake

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may be first made in a lucite cube, and in such a way that the rake will shatter if the cube does. Thus, it will never be much good for raking, and so on, and so forth.13 This sorites of cutting and separating adapts easily to application with things whose identity does not appear to transcend one sizable physical object, for example, to unfamiliar, unimportant stones and rocks. Now, I do not mean to suggest that there is a rigid distinction here between, say, such stones and typical tables. Rather, I wish only to remark a certain tendency in our thinking, and to show that it makes no difference to the matters under discussion. For the im­ portant point is that, according to our ordinary thinking, an operation of cutting and separating leaves us with at least one stone. No such operation performed upon any stone, including any stone which results from such an operation, is enough to mean the difference between a situation with at least one stone involved and a situation where there isn’t any stone at all. Accordingly, when choosing largest resultants, we shall still have at least one stone present, we must conclude, even where all we have is specks of dust, or even atoms, widely scattered throughout the solar system and even far beyond. But, on the contrary, in such a case as that, there will truly be no stone. Thus we disclose again a contradiction in our thought, the only rational response to which is, I suggest, to abandon our supposition of existence for stones. Now it may be that this sorites will not work compellingly on every putative ordinary thing. But it will, I suggest, work on such an ample sample of things that we may say that, if none of them exist, then no ordinary things do. Whatever escapes direct application of this argument, I suggest, will do so for a superficial reason, which does not ensure its existence. Historically, problems concerning ordinary things, as well as those regarding sizable material objects, have often been discussed in a context concerning perception. It is thus easy to suppose that our sorites by minute removals importantly concerns the perceptual recognition of a dwindling entity. To a certain extent, this thought is undermined by our sorites of slicing and grinding, discussed in the section just previous. Our sorites of cutting and separating, I suggest, even more powerfully shows the inadequacy of such an assessment.

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We have just mentioned one valuable feature of our newest sorites variation. Another, as we remarked before, is that it undermines the appearance of stability in the identity of certain sorts of ordinary things. Still another is the negative lesson it gives us about ‘functional definition’. But the main value in exhibiting a variety of sorites arguments goes beyond the particular lessons each serves most clearly to teach. For as the arguments accrue, the thought that our ordinary things are real entities becomes ever more rapidly a des­ perate one. The devices to be imagined, to save our swizzle sticks and sousaphones, look ever more ad hoc and irrational. At the same time, we can more fully appreciate, from the various aspects of our present conceptions which thus prove vulnerable, the genuine difficulties one must overcome if one is to obtain ideas which truly are coherent. Rather than continue to add to these lessons by constructing further destructive sorites, I shall now discuss a variant upon, or complementary argument to, any such decompositional piece of reasoning. This will allow us to discuss certain broad logical issues which will surely have occurred to some astute readers by now, and which we have so far ignored almost entirely. 4. A C C U M U L A T I O N A R G U M E N T S AND THE PLACE OF PARADOX

Near the beginning of our essay, we saw that the original sorites, concerning a heap, worked as a two-edged sword. On one edge, starting with what seemed to be a heap of things, we were forced to conclude that we still had a heap there even with nothing before us. On the other edge, starting with nothing before us, we were forced to conclude that even with a million things nicely arranged before us, there was no heap present. I think that the correct assessment of this is that the sword cuts common sense clear through both ways. And, I think that an appropriate way of putting the situation is to say ‘There are no heaps’. The same duality is present in the case of any of our ordinary things. I think that the correct response is the same, and that the matter may be summed up by saying “There are no ordinary things: no tables, no stones, no planets and no sousaphones” . But as

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there may be some objection to my thoughts, these matters merit some discussion. As we have labeled the one edge of our Eubulidean sword the sorites o f decomposition, so we may call the other the sorites of accumulation. Our original sorites of decomposition by minute removals proceeded by the stepwise removal of very small items, tiny chips or even atoms, from the putative ordinary thing in question. The sorites of accumulation which we shall here examine involves the reverse of that procedure: A series of very small items will be accumulated, in some putatively relevant manner, upon some small beginning item, or in some chosen region. It should be clear, however, that our other variations upon our sorites of decomposition also admit of reverse procedures. Accordingly, there are accumulation versions as well of our sorites of slicing and grinding, and of our sorites of cutting and separating. The points which we shall now discuss, then, while related directly only to the minute removal version of decom­ position, look to be quite general in their application to these matters. Our relevant sorites of accumulation will be a direct argument for the idea that ordinary things do not exist. We shall derive this result from acceptable beginnings. Each of our variations of decomposition was an indirect argument for that same idea; there we began by supposing existence and, then, with acceptable auxiliary premises, we reduced the supposition to absurdity. Before we proceed to the direct arguments of accumulation, I pause to note, in contrast, that there are accumulation arguments which are indirect.14 One such indirect argument, which proceeds by small stepwise increments, is as follows. Begin with an alleged table. Now, the addition of a single atom to what we already have, providing it occasions no substantial disruption, will never take us from a situa­ tion where a table is before us to one where there is none. But, by appropriate increments, we shall still have a table even when we have before us a spherical object many times larger than a house. Then we have no table; therefore, we didn’t have any at the start. This sorites of accumulation is of interest for us in that it complements and reinforces our previous reasonings. But as it is like them in being a reductio ad absurdum, it presents us with no new logical form.

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In turning to our direct sorites of accumulation, I shall again begin by focusing on stones as my example of putative ordinary things. And I shall again use the atom as my unit, now of increment rather than decrement. We may begin with an empty region, and say that there is no stone in it, or we may begin with a single atom somewhere, and say that it is something which is not a stone; the upshot will be the same. I will choose the latter beginning. Now, if we add a single atom to something which is not a stone, it seems that such a minute addition, however carefully and cleverly executed, will never in fact leave us with a stone. For a single atom, I suggest, will never mean the difference between there being no stone before us and, then, there being one there. (A process which supposedly produces a stone may be thought sometimes to involve no physical additions at all but, instead, only a rather substantial rearrangement of that matter which is already in the relevant region or situation. But, of course, that does nothing to refute what I am advancing.) Now, there is an asymmetry, which may be worth noting, between the proper way of formulating a premise of addition here and that of stating our third premise, of removal, with our sorites of decom­ position. Before we said that if we removed an atom in a way most favorable to there continuing to be a stone, there would still be a stone. And this allowed us to derive our absurd result. But if we add an atom in a way most favorable to there continuing to be only something which is not a stone, no absurdity will ever be felt by anyone. For we shall never thus construct anything which, even according to quick common sense judgements, will be even remotely like a stone. We might well thus construct, for example, what quick common sense would call a wooden table, or a planet, or perhaps even a duck. But we will come nowhere near to producing any stone in the process, for our premise would be so formulated as to ensure such a result. Accordingly, our new premise will say, rather, that if there is, in a certain situation, only something which is not a stone, or some things which are not stones, the addition of any single atom, no matter in what way, will not mean the difference. Presumably, the ways now most relevant will be those least favorable to there continuing to be no stone there. But this asymmetry scarcely affects

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our argument, for even according to common sense, our new premise is as hard to deny as was our older one. In any event, then, by repeated application of this new principle, we must conclude that there is no stone before us no matter how many atoms we add to our original one, and no matter how they are arranged. Even when we have before us something which ‘looks for all the world like a stone’, and which would prompt people to think that there is a stone there, we must conclude that there really is no stone. We have again, this time by accumulation, exposed a con­ tradiction in our ordinary beliefs. This raises again for us the question of how to respond to such a contradiction. I submit that the proper response, as before, is to deny the supposition that stones exist. Our accumulation argument applies most immediately to such putative stones, if any, which have, or will be formed, through such a gradual stepwise process. But it also applies to any other putative stones. For any putative stone may have a duplicate of itself con­ structed via the gradual sort of process here envisioned. The one object, surely, will really be a stone only if the other is; as the duplicate is not, neither is the original candidate. In other words, as I have submitted, there are no stones at all and, similarly, for any other of our ordinary things. In the parlance of logic, we may say, then, that our sorites of decomposition is an indirect argument for the idea that there are no ordinary things, while this correlative sorites of ac­ cumulation is a direct argument for that same conclusion. Some people, however, may object to this understanding of these related sorites arguments. They would try to deny us our negative conclusion about ordinary things, perhaps claiming that the true situation allows neither a positive nor a negative proposition to prevail but, instead, presents us with a paradox. One form their objection may take is by way of this following argument, where a new way is suggested for combining a sorites of accumulation with a sorites of decomposition. In this way, we may make it appear that every ordinary thing, at least, is a stone. For example, we may start with a feather and, by decomposition, work down to an atom, still having to hold that a feather is before us. We may, then, by ac­ cumulation, work up to something which looks to be a stone. This

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time, our accumulation premise will be that adding an atom to a feather leaves a feather there still, providing only that no substantial collapse or similar result is effected in the process. Confronting what we have produced, we say that it is a stone. Reasoning by our sorites we may say it is a feather. But this may be done with any feather. Accordingly, we must now hold, not that there are no stones, but that all feathers are stones. And, similarly, so are all planets, and swizzle sticks, and sousaphones. While this reasoning is engaging, I think that it is weak. One place it goes wrong, I suggest, is where we confront what is before us. Whatever the look of things, we need not say that there is a stone before us. Indeed, we already have compelling arguments, both direct and indirect, for the conclusion that no such statement will be true. Another weakness, of course, is in insisting that a feather is still there with only one atom present. While Eubulides’ contribution has often been labeled ‘the sorites paradox’, there is nothing here which is a paradox in any philosophically important sense. I remarked on this near the outset; I hope I have supported it by now. Accepting our negative conclusions here does not mean important logical trouble for us; we only think we have troubles while we refuse to admit their validity. At this point, and in the second place, it may be objected that we beg the question against common sense in denying the combined argument just presented and, in particular, in denying the ‘claim of observation’. But it is very unclear what this charge can mean. For common sense does not speak with one consistent voice on any of these matters. What we have done is to disclose contradictions in our beliefs. Beliefs about atoms, to be sure, do not have an ancient claim to being part of common sense. But they now seem to be part of it, at least in advanced cultures. And, further, our arguments may be conducted wholly in terms of removing, and adding, tiny chips or specks. The premises concerning these visible items would surely be part of common sense, albeit reflective or implicit common sense. This would be true even of rather ancient times, and even rather primitive cultures. The most rational way of responding to the con­ tradictions, I submit, is to deny application for the ordinary concepts: the concepts of stones, tables, feathers and sousaphones. These

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concepts do not apply. We cannot say, then, ‘All feathers are stones’, unless we mean it in some peculiar sense of the logicians. But a vivid way of putting our point is to say ‘There are no stones’. At this present juncture, and in the third place, it may be said that if the notion of a stone is indeed incoherent, then no genuine pro­ position is expressed by saying ‘There are no stones’. Accordingly, it may then be urged, such a form of words is at best misleading, and in any case quite inappropriate for expressing our philosophy. As regards the part about propositions, I have little to say in reply, as I have little idea as to what a proposition may be, and only speak in such terms as a means for convenient exposition. But, in any case, I do not think that the negative form of words is inappropriate for expressing our philosophy. The following example may be helpful on this point. Suppose that some children tell stories to each other regarding certain imaginary entities, which they term ‘nouls’. Now, their imaginary world, in which nouls are supposed to exist, may be so described by them that it could not possibly contain any nouls, given their own descriptions. For example, they may so use the term that on certain days nouls, if they existed, would be people, on other days mere objects, and so on. Their concept of a noul, we might say, is quite incoherent, if indeed they have so much as any genuine concept here at all. The question whether there is a genuine pro­ position to the effect that there are no nouls, I will not attempt to decide. But if someone, perhaps on hearing the children, asked me what nouls were, one thing I would say to him is simply this: ‘There are no nouls’. In this essay, I have been trying to offer arguments, both direct and indirect, for the denial of the existence of ordinary things, and for all that that entails. The sorts of objections to this attempt we have been recently considering have it that a negative remark is out of place. Now, I think that we have effectively countered them. But even if I am wrong in this thought, things will not be altered much for the philosophy I seek here to advance. For we may say ‘There are no stones’ at least as a sub-conclusion in a lengthier argument on these matters. And, then, we might advance as another sub-conclusion something like ‘There are ever so many stones; indeed, all ordinary

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things are stones’. And a more final conclusion may then be put by saying that none of the previous remarks, as well as ‘There is at least one stone’, make any clear sense at all. Now, I should think that anyone disposed to cling to ideas of ordinary things would find little comfort indeed in these most recent radical recommendations. But even if they are not literally what I have been arguing for, they are surely in the same spirit as my own philosophy.15 5. S O M E I M P L I C A T I O N S O F T H E S E R E A S O N I N G S

The arguments we have considered are effective, I believe, against our beliefs in ordinary things. Before we close our essay, however, it will be well to make a few remarks as to what our reasonings imply and what they do not. For it is easy to misinterpret our results so as to underestimate their importance. And, for that matter, it is also easy to misconstrue them in such a way as to exaggerate their implications. We shall try, then, to encourage a balanced and accurate appraisal. One way to misunderstand our arguments is to take them as concerning words but not things. For while our arguments do concern words they likewise concern things (which are not words). It is true that we have shown that, in a relevant manner, terms for ordinary things are incoherent. In that that is so, those terms cannot apply to anything real. And from that it follows that there are no such ordinary things as those words might purport to designate. Accordingly, our results concern words and things alike. They thus differ from points of grammatical distinction which concern only words, and thus our points are more comprehensive. But they are also thus more com­ prehensive than points pertaining only to things and not to words, if such points there be; for our points concern things and something else besides, namely, words. The realization that our reasonings concern words, then, can hardly detract from whatever substance and im­ portance they might have. A second misunderstanding is to suppose that our arguments concern kinds as opposed to things. For in that they concern kinds of things, our reasonings concern both things and kinds. We have argued that certain kinds are never instanced; there are, then, never any

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things of those kinds. Things of those kinds do not exist. Accordingly, if there are arguments which concern only things, and not kinds, they fail, in that respect at least, to be as comprehensive as these present reasonings, which concern both. The observation that our arguments concern kinds, then, can scarcely deny them even the least bit of significance.16 As we have just remarked, our arguments concern words and kinds, as well as things which are neither. Concerning words and kinds, now, we might say this. First, we might say that it is in connection with semantics that our reasonings have what are their most obvious implications and, second, that their most obvious semantic implications concern certain sortal nouns, namely, those which purport to denote ordinary things. Thus, it appears quite obvious to us now that there will be no application to things for such nouns as ‘stone’ and ‘rock’, ‘twig’ and ‘log’, ‘planet’ and ‘sun’, ‘moun­ tain’ and ‘lake’, ‘sweater’ and ‘cardigan’, ‘telescope’ and ‘microscope’, and so on, and so forth. Simple positive sentences containing these terms will never, given their current meanings, express anything true, correct, accurate, etc., or even anything which is anywhere close to being any of those things. Various other words and expressions will similarly fail to serve with distinction. Amongst these unfortunate devices are certain ones which are only marginally counted as words, namely, certain proper names. Accordingly, in the most relevant sense, we must count as vacuous such names as ‘Venus’ and ‘Everest’, in those uses of them which dot the philosophical lit­ erature.17 For we may be confident that, in relevant uses, if ‘Venus’ names anything, it will be a planet which is in fact thus named. Similarly, it seems quite nearly certain that, in relevant contexts, if ‘Everest’ names anything at all it will name a mountain. But, as we have argued several times over, there are no planets or mountains. Hence, with respect to their relevant uses, these names are empty. What amounts here to the same: Venus does not exist and neither does Everest. In a similar vein, it may be noted that, in very many occurrences, certain pronouns, such as ‘it’, ‘they’ and ‘them’, either serve to make reference to some existing ordinary thing or things, or else fail in contributing to any true or realistic comment. As we have

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been arguing, the first of these two disjuncts never holds and, so, in all these occurrences, these pronouns do thus fail. Now, these negative ideas, concerning names and pronouns, I take liberty in noting, can be confirmed more directly, and somewhat independently. We may do this by constructing sorites arguments directly upon the alleged named entities, and upon those which are apparently the reference of the pronouns. To be sure, these confirming arguments, in contrast to the reasonings herein advanced, do importantly involve the notion of identity. Thus, they fall outside the limits on our reasoning which we have here imposed on ourselves. I take the liberty of mentioning them despite this violation because, in the present context, their role is only an auxiliary one. What we have said about words pertains immediately to our thought, for much of our thought is in terms of such words. For example, when we are under the impression that we are thinking about an object in the world, I suggest that our impression is mis­ taken. If we suppose that we are thinking of Venus and, thus, are thinking of some existing thing, I suggest that we are similarly in error. At best, we are thinking of something, but only in much the way we do when thinking of a fictional entity. Alternatively, it might be that we are here not really thinking of or about any (finite) entity at all. I leave these matters for future discussion, noting that, whatever their detailed outcome, our thought must make much less contact with reality than we have commonly assumed. To turn a phrase along a familiar line, we may say that our arguments have been pertinent to descriptive semantics, though they concern other things besides. When done properly, we have argued, descriptive semantics shows the poverty of our language and our thought and, thus, it shows the need for the invention of new terms, that is, for good prescriptive semantics. It is quite unclear to me, however, how we should go about finding a suitable replacement, or replacements, for one of our ordinary terms, for ‘log’, to take a representative example. With respect to atomic removals, to cite one difficulty, at a given juncture in a given case, there are millions of removals which seem quite innocuous and favorable. The item result­ ing from one such does not seem any more ‘loggy’ than that resulting

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from any other of them. Which steps are to be ruled out; and why? I leave these matters to others for further discussion, noting that other sorites arguments must also be avoided. It may be much harder than one might first suppose, I would suggest, to achieve coherence while adequately serving anything much like our everyday concerns. However difficult it might be, prescriptive semantics takes us in those directions which are somewhat practical. It does not touch on those issues that are philosophically most profound. For these philosophical issues concern the general features of the view, or views, which our arguments allow us to find acceptable. In this connection, it is important for us to notice that the arguments we have here exhibited, while they show common sense to be badly in error, do not force upon us a world view which is far removed from common sense. Now it may well be that extensions of, or variations upon, these present arguments will indeed require departures which are remarkably radical. But it is important to notice that the present reasonings do not themselves require so much. For one simple example, there is nothing in these arguments to deny the idea, common enough, that there are physical objects with a diameter greater than four feet and less than five. Indeed, the exhibited sorites allow us still to maintain that there are physical objects of a variety of shapes and sizes, and with various particular spatial relations and velocities with respect to each other. It is simply that no such objects will be ordinary things; none are stones or planets or pieces of furniture. In reflection upon world views, it is easy to suppose that the challenge to ordinary things, to the existence of stones, for example, must come from some ‘mentalistic’ philosophy: from idealism, from phenomenalism, and so on. The present arguments show this sup* position to be false, however, for they do not require us to embrace any such radical metaphysics as that. It may also be supposed, conversely, that a mentalistic philosophy must abandon ordinary things.18 Bishop Berkeley, the chief exponent of a mentalistic view, was of an opposite opinion: he never meant to deny ordinary things but intended to reveal their true mentalistic nature. Let us now grant that Berkeley was right in thinking that his idealism did not require him to deny ordinary objects.

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But, especially with visible chips in mind, rationality would require even him to accept these present reasonings. Hence, while his mentalism seems not to have required it, Berkeley was wrong, nevertheless, to affirm the existence of ordinary things. Given the limits to which we have here confined them, our Eubulidean reasonings allow us to hold a world view much like that of common sense. First, in that we have foregone arguments with regard to living things, we may still believe in plants and animals, and the organs, tissues and cells presumed to pertain thereof. In that we may regard them as mere products of living things, twigs and logs and fingernails may be considered and denied by us, but perhaps that is a loss which may be accepted with equanimity. Most importantly, it may still be held that there are people, including ourselves, and, with that, also the mental or psychological items typically characteristic of persons, such as thoughts, feelings and experiences. At the same time, as we have already remarked, we may still hold that there are physical objects of various shapes and sizes, including many of such size and duration as to be suitable for comment, even if in fact such comment rarely has been made. We may see, then, that it is rash to suppose, as some may do, that these reasonings thrust upon us some esoteric ontology where only events, or processes, or facts, or what­ ever, are allowed to exist, and where no physical objects or people are allowed any genuine place in the world. We can allow, of course, that some such ontological doctrine may indeed be correct; but to show it is will require arguments which differ from those advanced in this essay. Indeed, so far are our present arguments from forcing us to acknowledge only events, or processes, or whatever, that they require us to deny many such items which are ordinarily ac­ knowledged, namely, all those which involve ordinary objects. Thus, for example, the eruption of a volcano is something which can never occur. For its occurrence logically requires the existence of a volcano and, as we have argued, there will never be any such thing. Likewise, the alleged fact that a certain cat is on a certain mat never can obtain. For that would require the existence of at least a mat, and we have seen that there never is any at all. With these brief remarks, we have outlined, well enough I suggest,

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the situation with which our exhibited reasonings present us. Our reasonings, I have submitted, are as unobjectionable as they are simple. And their implied situation, it now seems clear, makes no great demands upon either our credibility or our imagination: on the one hand, our reasonings demand of us only a ‘rather small’ departure from common sense; on the other hand, as this required departure has not yet been made, our existing thought is in quite a bad way. If we are rational, we shall recognize two main options. First, we can proceed to engage in prescriptive semantics, so that a detailed view, at least largely commonsensical, may once again appear at hand. Or else we can press on with Eubulidean reasonings, in various new directions, to see whether such simple, persistent thinking might require further, much more radical departures. For my own part, that second option is compelling. But of course I am a radical skeptic, and simple, persistent thinking is all the sense I seem ever to desire.19 New York University NOTES 1 See, for example, William and M artha Kneate, The Development o f Logic, Oxford, 1962, p. 114ff. Eubulides has no writings extant, it appears, and scholarship pertaining to him is som ew hat difficult, as is indicated by, e.g., Jon M oline's 'A ristotle, Eubulides and the Sorites’, M ind 78, N .S., No. 311, 1969, pp. 393-407. Given that these four attributions are deem ed highly likely, how ever, and that no direct writings remain, it seem s likely that Eubulides discovered other things as well. But even if these four were all he discovered, they com prise a staggeringly brilliant achievem ent. It is a secondary intention of this paper that we come to better appreciate the brilliance of the Megarian m aster, as well as his great profundity. 2 W.V. Quine, W ord and Object, New York, 1960, p. Iff. 3 Especially for generalizing, I have been careful not to say, in the third premise, that rem oving a peripheral atom will always leave a stone. Such a condition would be faulted, for exam ple, by a putative stone containing a time-bomb about to explode. Innocuously removing an atom w on’t leave a stone there, for it has no chance to do so. But that removal w on't mean the difference as to w hether there is a stone there. So, even with threatening time-bombs, our careful prem ise will let us conclude that there aren’t any stones and, similarly, any time-bombs either. Having this distinction before us, in w hat follows I w on’t bother to employ it explicitly. H ere, I am indebted to T errence Leichti. 4 On these points, I have profited from discussion with Raziel Abelson.

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5 I am indebted here to discussion with Michael Lockw ood. 6 G. E. M oore, ‘Four Form s of Scepticism ', in his P hilosophical Papers, L ondon and New York, 1959, p. 226. 7 David H. Sanford, ‘Borderline Logic’, A m erican Philosophical Quarterly 12, No. 1, 1975, p. 29. For his treatm ent of sorites argum ents see p. 38. For references to various other thinkers o f similar persuasion see his footnotes. * See the appendix to David K. Lewis, ‘G eneral Semantics* in Donald D avidson and Gilbert H arm an, eds., Sem antics o f N atural Language, D ordrecht, H olland, 1972, pp. 215-216. Also, see the appendix to H artry H. Field, ‘Quine and the C orrespondence Theory’, The Philosophical Review 83, No. 2, 1974. This attem pt, or som ething much like it, has been brought to my attention by David Lewis in conversation. 9 I argue that ‘true* has no such vagueness, and is instead a kind of ‘absolute term ’, in Chapter 7 of my Ignorance, O xford, 1975, pp. 272-319. But, with Field, I am supposing the opposite here; my purposes are illustrative. 10 There are two papers, both appearing in Synthese 30 (1975), which treat o f sorites arguments, and of mathem atical reasoning in connnection with them , in a m anner conformable with that recom m ended here: Michael D um m ett’s badly m isnam ed 'Wang’s P aradox’, pp. 301-24 and, though perhaps the connection with m athem atical reasoning is som ew hat less, Crispin W right’s ‘On the C oherence o f Vague P red ictates’, pp. 325-65. N either of these authors, how ever, adopts a view as radical as th at which I espouse. Further from mathem atical considerations but, by my lights, the m ost farreaching paper in that issue of Synthese, is Samuel C. W heeler’s ‘R eference and Vagueness’, pp. 367-79. All of these papers are to be recom m ended. n These m atters w ere brought to my attention by David Sanford. 12 Underlying this sorites argum ent is a comm on sense assum ption which is itself incoherent. This assum ption is that we may properly distinguish betw een (1) rem oving so little from a thing that w hat is thus rem oved does not count as part o f th at thing but is only, say, an isolated atom or speck of dust, and (2) cutting off enough from the thing so that w hat is thus cut off rem ains as part o f the thing, part of the thing now being here and part of it over there. The incoherence of such a putative distinction may be quickly shown by a sorites of accum ulation: first, if w hat is rem oved is an atom , then it is not part of the ordinary thing, say the table, from which it was taken; but, second, if, instead, w hat is rem oved is only one atom greater, the additional atom on the side of that which is rem oved cannot m ean the difference; so, w hat is rem oved is still not part of the thing. Accordingly, by stepwise reasoning, no m atter how great the thing removed it will still not be part of the ordinary thing. The apparent stability of such things as tables, then, as well as our new sorites, rests on a comm on sense assum ption which is actually quite absurd. We passed over this in order to present our new sorites. This does not, of course, reveal any defect on the part o f our sorites o f cutting and separating. On the contrary, it show s this reasoning to be an argum ent a fortiori. 13 I am glad to find som ew hat similar thoughts expressed by such an influential philospher as R oderick M. Chisholm in his ‘The Loose and P opular and Strict and Philosophical Senses o f Identity’, in N orm an S. C are and R obert H. G rim m , eds., Perception and Personal Identity, Cleveland, 1969, p. 97. 14 In the w idest respect, I consider all o f my argum ents to be indirect proofs of the incoherence in comm on sense thinking. This is because I m ust conduct my argum ents

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in an available natural language, the existence of which may be exposed as incoherent by suitable sorites reasoning. But the exposure of such faults takes us beyond the confines of this present paper. 15 V arious points discussed in this section emerged from discussions with Saul Kripke and w ith Ralph Silverman. 16 This paragraph emerged from discussion with Samuel W heeler. 17 I discount, for example, uses of ‘Venus* in connection with various women. Presum ably, some thoughtless parents may have attached this moniker to a baby daughter. 18 This is the view implicit in Dr Johnson’s futile attem pt to refute Berkeley by “ kicking a stone” . In Word and Object, W. V. Quine endorses Johnson in these m atters; p. 3ff. and p. 17ff. 19 This present essay is m eant to advance but a small part of a nihilistic viewpoint in m etaphysics, ontology and the philosophy of language. For a concise sketch of more of the w hole, see my paper, ' 1 Do N ot E xist’, forthcom ing in Graham M acdonald, ed., Epistem ology in Perspective, (London: The Macmillan Press), which volume is the festschrift for Professor Sir Alfred Jules Ayer. For a detailed analysis and discussion of nihilistic sorites argum ents, I refer the reader to my ‘Why There Are N o People* forthcom ing in M idwest Studies in Philosophy, Vol. IV: Studies in Metaphysics. Nihilism fits well, in a variety o f w ays, with skepticism in epistemology. For an extended developm ent of skepticism , see my book, Ignorance, O xford, 1975. For some relations betw een these tw o views, see my paper, ‘Skepticism and Nihilism’, forthcom­ ing in N ous.

[2] S A M U E L C. W H E E L E R

O N T H A T W H I C H IS N O T

This paper presents arguments that, very probably, none of the ordinary ‘middle-sized’ objects of the ‘given’ world exist. In parti­ cular, there are no persons, as ordinarily conceived, nor, perforce, any psychological states of them. Since this may conflict with what seems to be thought, the way will be prepared by a sketch of the main presuppositions of the theory of reference which lies behind the general argument that all such counter-common sense claims must be false. The point of the long digression is to begin to destroy the rational grounds for resistance to the sorites1 arguments which est­ ablish the main point. After presenting the sorites arguments, a brief criticism of ways of avoiding the conclusion is presented. I. T H E A R G U M E N T T H A T M O S T O F W H A T M O S T P E O P L E B E L I E V E ÏS T R U E

In this section I sketch two theories of reference, the second a modification of the first. I then show how the second theory of reference entails the conclusion that most ‘ordinary’ common-sense beliefs are true; that is, that common sense is for the most part correct and that what appears to conflict with it either doesn’t actually or is false. A . Theory o f R eference I: Frege-Russell R esem blance What, after all, is it for a term to apply? Our terms have as their extension whatever fits their sense. Reference is a function of sense. So, given that we are talking about anything when we are using a given term, what we are talking about when using that term is determined by the sense expressed by that term. The internal features of the concept determine its reference. If the term applies, then, by what it is to apply, the object we are talking about will have the important features that are built into the concept, i.e. will fit the sense of the term. (Imaginary quotation)

This is the basic form of what I call ‘the resem blance theory of reference’. Exactly how ‘features’ or ‘senses’ of concepts are conSynthese 41 (1979) 155-173. 0039-7857/79/0412-0155 $01.90. Copyright © 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

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ceived and which features of concepts are part of the sense and so have to belong to the objects the terms apply to vary from version to version of this theory. Theory I has the weakness that it does not provide an empirical way of determining what the sense of a term is. What sense a term-for-a-person has, that is, might well be a private matter, even when the sense itself is an objective entity. The ‘expression’ relation between a word or thought-component and a particular sense is left to some kind of intuitive insight - we know what we mean, when it is our own word. More importantly, for our purposes, the basic form of the resemblance theory of reference does not give any guarantee that we are talking about anything at all. What is to prevent the natures out there from diverging in essence from the senses our terms express to such an extent that nothing fits our terms? Since the sense expressed by a word is, as it were, not clearly connected to what is outside when the term is used, nothing in principle prevents massive failure of reference. In short, the basic form of the resem blance theory of reference does not provide a rejoinder to skepticism or to m etaphy­ sical revisionism. B. Theory la: Q uine-Davidson and British Resem blance Theories

This theory overcomes the above difficulties by getting something functionally analogous to sense into empirically available phenomena. I will briefly describe the Quine-Davidson version of this theory, and then argue that W ittgensteinian and ‘ordinary language’ philosophers presuppose virtually the same principles about how language con­ nects referentially to the world. Version a: Q uine-D avidson 2 If we are talking at all. what we are talking about is determined by what we say in which situations. Roughly, our occasion-sentences have a stimulus-meaning or an ‘outside correlate’ meaning. A language has been translated or radically interpreted as far as empirical data goes when the appropriate correlations between what is out there (for us) and the person’s responses have been established. The reference of a term in an occasion sentence is constrained by these outside correlates. The constraint may not be sufficient to determine reference, but at least what there is to go on in hypothesizing

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reference is given by the outside correlates, the empirical substitute for Fregean senses. All there is to the ‘sense’ of a term is manifested in these outside correlates, i.e. in a person’s or a culture’s dispositions to use that term. So, look for senses in the pattern of a person’s or a culture’s speech behavior. (Imaginary quotation)

On this theory, given that a sentence is true if and only if what it is best translated or interpreted as obtains, most of what most people are inclined to say is true. (This holds in general, for Davidson; for observation sentences, for Quine.) That is, translation and inter­ pretation must, by the nature of reference, be ‘charitable’. Given that thought is language-like insofar as the same requirem ents must be met for thought-tokens to refer, most of what most people in a culture think will be true as well.3 Version b: British

It is somewhat difficult to pin down a ‘British’ theory of reference in a form acceptable to its practitioners, since so many of the philoso­ phers I have in mind, such as the later W ittgenstein,4 Austin,5 and Ryle6 tend to eschew theories in favor of detailed descriptions of the ‘ordinary’ use of our terms. Since they eschew theory, they eschew any technical use of ‘refers’ or ‘applies’, so that their ‘theory of reference’ can be called such only via locutions of indirect discourse. Their implicit views of language and its relation to the world, however, lead them to the most uncompromising defense of com­ mon sense. According to their conclusions, virtually every central belief ‘built into’ language by way of the judgements we learn in learning a language is true. So it is important for my purposes to bring their theory of reference into some kind of relation with theory la in its Quine-Davidson form. What further premise would yield a valid argument from the premise that a certain philosophical theory violates the rules for the use of a given term or concept or family of concepts to the conclusion that the philosophical theory is mistaken? I think that only a version of theory la will make this argument valid and that theory la is behind almost all dissolvings of problems and ‘analyses of the grammar of’ in this tradition. To argue that theory la is behind every such analysis would require

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detailed analysis and argument for the case of each of the philoso­ phers in question. I think it is clear, though, that a theory of essen­ tially similar to Davidson’s in ‘On the Very Idea of a Conceptual Scheme’,7 is implicit in most such philosophers’ work. The British theory, though, contains some modifications, and is complicated by a different conception of how strongly empirical sense determines reference. On the generalized ‘British’ theory, ignoring individual differences, concepts have a sense which is identified with their use, a complicated socialized and contextualized version of ‘outside correlate meaning’. The use of a term is, roughly, given by the sort of situation in which the term is to be applied, according to the rules of the language. ‘Correctness of application’, which seems to be the general surrogate for truth, is determined by the rules of the language, which them ­ selves seem to be a function of what most people in the language community say in paradigm situations. An application of a term is correct (i.e. a sentence is true) if and only if what is in fact the situation is in the set of situations where the term ‘is to be used’. For reasons I do not fully understand, these philosophers differ from Quine and Davidson in not finding reasons to believe that there is ‘slack’ between use and reference. ‘Rules’, possibly by some subtle normative force, are sufficient alone to give determinate results as to what is being referred to. This might be regarded as a consequence of their strategy for avoiding the paradoxical results of an unrestricted application of the resemblance theory of reference. Suppose reference is strictly determined by sense, and sense is determined by what people say in what circumstances. Then, if it makes sense to apply this to an isolated individual, all, not most, of what this one-person culture says will be true. This is because there is always some extension, given by exactly the set of situations in which the person applies the term, which will, by the theory, be what the person means by the term. Since it is senseless that a person can be speaking truths when any alternative utterance he could make would also be a truth, it has to be denied that there can be one-person linguistic communities. Thus ‘private languages’ are declared to be impossible. That is, to apply the resemblance theory of reference to get the result that most of what most people say is true rather than that anything

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that anyone could say is true, the unit of rule-discovery or inter­ pretation is made the culture as a whole. A culture as a whole generates rules by some kind of majority practice, so that error by individuals is possible. Since there can be no private language, and so no reference which is not everyone’s reference, speculation about ontological relativity makes no sense. So ‘aquiescence in the back­ ground language’ is the only alternative that makes sense. On the ‘British’ theory, reading ‘use’ for ‘sense’, truth is a function of use, and so, reference is a function of use. Once again, the correlate of the sense of a term is brought out into the world, so that meaning or use determines correct application. Thus correct ap­ plication, over a culture as a whole, is guaranteed. On both versions of theory la, then, reference is still a function of sense. But sense is constrained, if not determined, by what is there in a situation in which the speaker is disposed to use a given term. Sense has its criterion, if not its being, in the outside world. Thus sense and reference are virtually correlative on theory la, at least in that each puts limits on the variation of the other. What distinguishes this theory from theory I is that what a person or culture means is determined by seeing what is true when an expression is used and making meaning correspond to what is the case. Then since reference is a function of sense, most of what a culture agrees on will turn out to be true by the very nature of what it is for a term with a given meaning to be true of an object. Now, given the above picture of language and thought and its relation to the world, there is a standard reply to philosophical doctrines which challenge widely held beliefs in great numbers: ‘You’re misusing language’. Alternatively, ‘You’re misinterpreting the truth-conditions of this predicate/construction for English speakers.’ If theory la is true, the revisionary metaphysician and the skeptical epistemologist must be misusing or mis-paraphrasing language because, by the nature of the case, most of our beliefs must be true. Thus argum ents which conclude that we are mostly wrong in a whole area of belief are, provably, invalid or unsound. How they go wrong according to this theory of reference may take subtle and skillful analysis; the conclusion that they are wrong is foregone. Such refutations include paradigm case arguments, arguments about how a

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concept is learned, arguments about when we say a person has a concept, analyses of suspect arguments in terms of ‘extending a concept beyond its range of use’, arguments against the very idea of a conceptual scheme, and many others. Attacks on common sense based on supposed inconsistencies be­ tween science and common sense are similarly treated by adherents of theory la. Since our ordinary beliefs can’t be radically wrong, we just have alternative descriptions, different families of predicates with their associated application-constraints, or different purposes for different equally correct predicate-systems. II. T H E D E A T H O F T H E R E S E M B L A N C E T H E O R Y O F MIND-WORLD RELATIONS

K ripke8 and Putnam 9 have shown that the above theory of how language links up to the world does not coincide with our ordinary use of terms such as ‘refer’, ‘about’, ‘nam es’, ‘discussed’, etc. Theory la, then, is self-contradictory. That is, the theory that says that our ‘use’ of a term determines its meaning is not the theory of meaning and reference that the use of our idioms of reference embodies. A ‘use’ analysis of our referential concepts shows that use is not meaning. So reference is not a function of sense, in general, accord­ ing to the theory that it is. I should explain how K ripke’s and Putnam ’s dem onstrations work. Consider the intuition we have about what we would say in situations in which it turned out that, for instance, we had accepted the sentence ‘Aristotle was a Megarian philosopher who proposed the paradox of the heap and invented other fallacies’. In the appropriate circum ­ stances, we would say we had a false belief about Aristotle. This is a manifestation of our use of ‘about’ or of the sense of ‘about’. Intuitions about what we think is the case in such situations are intuitions about what we ‘would say’ in such situations. By most theories of reference, such intuitions are the basic data for a theory of the sense of the term ‘about’. Analogous remarks apply to the other referential terms Kripke and Putnam discuss. By the resemblance theory of reference, the relation of aboutness and the other referen­ tial relations must be whatever sets of ordered pairs accord with these

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dispositions to apply referential terms. But any such relations contain ordered pairs of terms and entities such that the sense of the term doesn’t fit the entity as well as it fits some other entity. The relations the resem blance theory of reference assigns to referential notions contain pairs of concepts and entities the resemblance theory would not predict. Thus a ‘use’ analysis of reference shows that use does not determine reference. Some rem arks are in order about the scope of these results. Kripke, by examples such as the Aristotle example, has shown that the resemblance theory is false for proper names. Putnam, by his ‘Twin E arth’ exam ples10 and others, has shown that Theory I and la are false of natural kind term s, such as ‘w ater’, ‘cat’, etc. His arguments seem to apply to any property-words which pick out what we regard as real properties. W henever we have a case of a term such that we hold that its correct application is not a m atter of our decision but rather of how things are, we have a case in which our intuitions are in disagreement with the resem blance theory of reference. K ripke’s and Putnam ’s results may not directly apply to terms for which our intuitions are that whatever our society chooses to say is correct. Terms such as ‘is m arried’, ‘is a bachelor’, ‘was duly elected’ seem to designate properties for which the resemblance theory is correct, if they designate properties at all. In such cases, there is, intuitively, no possibly recalcitrant objective fact to pose a danger of making most of our applications of a term mistaken. This is because, prima facie, it is our agreement on what to say that ‘defines’ these terms. K ripke’s and Putnam ’s results fail to apply, if at all, then, only in cases of properties which are social artifacts, properties whose being is social. Even for properties which seem to be social artifacts, the resem ­ blance theory may not be correct. Putnam uses the example of ‘pediatrician’,11 which does not seem to name a natural kind, but rather a kind of socially defined occupation. ‘Pediatrician’, though, does seem to have come to rigidly designate a group of people. Thus it could turn out that pediatricians are not doctors, if pediatricians all turn out to be M artian spies and to have just made a pretense of rendering medical aid, while in fact shipping little children to Martian forced labor camps.

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From a realistic point of view, the resemblance theory claims that all property-terms have reference in the way that ‘bachelor’ and ‘duly elected’ appear to have reference, i.e. by socially deciding an exten­ sion. Only on an idealistic conception of the world, though, could it be claimed that all of our terms for kinds and properties are social artifacts. On a realistic view, there is a world out there which can deviate from our conceptions of it, so that the content of a conception doesn’t determine its object. Even if the resemblance theory is right for sentences using social artifact terms, though, we don’t get the result that truths about the world are guaranteed. Every artifact-term seems to require in its definition some reference to intuitively real kinds, such as ‘bachelor = df unmarried male person ’. Thus there is no guarantee that most particular universally agreed on applications of the term are true. ‘Person’ is a natural kind term, so that the theory embodied in that concept may not be true. So it may be guaranteed that most widely agreed-on sentences of the form, ‘If A is a male person, then he is a bachelor’ are true, given that ‘m arried’ is a socially defined term known not to apply to A. But it will not be guaranteed in any social way that most agreed-on sentences of the form, ‘A is a bachelor’ are true. If we are wrong about what it takes to be a person, it could turn out that most of the things we all agree in calling bachelors are not. If, for instance, a thing has to have a soul to be a person, even though our concept involves no such thing, none of our paradigm bachelors will be bachelors if none of them have souls. The hypothetical beliefs involving social artifact terms may be guaranteed to be mostly true by a ‘use’ analysis, but claims about how the world really is with respect to such properties get no such guarantee. Only on the view that all terms are social artifacts will any facts about the world follow from universal beliefs. K ripke’s and Putnam ’s results, then, show that in no case can there be an argument that reference is a function of sense. No replacem ent theory is established by K ripke’s and Putnam ’s results, however; only the negative result that the resemblance theory is wrong. It is true that the content of our referential concepts seems to embody a kind of causal theory of reference where the caused items are mentalistic and intentional items. Furthermore, various social phenomena seem,

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intuitively, to be built into real reference as conditions in this causal relation. But apart from a resem blance theory of reference, we have no compelling reason to conclude that therefore reference is some kind of causal relation. We certainly have little reason to give our theory of reference the complexities that accommodating our varied intuitions about what refers to what, when, would entail.12 An apparently unnoticed consequence of K ripke’s and Putnam ’s results is that the invulnerability of the purported truth of the ordinary views of men is destroyed. If a concept’s reference is not determined by making its content both determine reference and be determined by w hatever is out there when we use it, then a concept amounts to a theory which may be radically mistaken. ‘Criterial’ features of a concept may be mostly false of what the concept is true of. Conceptual analysis will be merely that, with no very clear consequences for what is the case. If the replacem ent theory for the resem blance theory is some causal account, by now familiar storifes are available in which there is radical misinformation ‘built into’ the concepts, intuitions, and beliefs of a society. Furtherm ore, what can be true of one concept can be true of most of our concepts. On the causal alternative again, where the referent of a general term is the kind of which the causal sources of our concept are m em bers, we could have a situation where all special analytic contents of concepts were empirically false. Apart from a resem ­ blance theory of reference, we can be talking about the real world and getting it all wrong. If the resem blance theory is wrong, there is nothing impossible about truly massive error. A more poignant possibility is that our terms may not refer to anything. On a causal theory, if there is in fact no kind out there to which all or most baptism cases belong, then we are talking about nothing with that concept. Similarly with singular terms for fictional entities. If what ‘reference’ refers to turns out not to be very often instantiated, most of our terms will fail of reference. W hether or not a causal theory correctly describes the essence of reference, if there is such a thing, the fact that the resemblance theory is wrong eliminates the major arguments that massive error is impossible. Massive error, of course, will need to be supported by arguments from a better-off standpoint than common sense. The fact

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that we seem to make massive errors will have to be given an account from that standpoint. I argue in the following section that this last possibility, that most of our terms for objects don’t refer, is in fact the case. I believe that our errors and the apparent paradoxes that arise from this ‘vanishing of objects’ can be explained in terms of a theory that recognizes at most the micro-particles of physics and certain complexes of them as genuine objects. This is not a skeptical claim. I am not saying that we are as likely as not to be wrong about the existence of and features of ordinary objects. I argue that we are very probably in fact mistaken and that there are no such things. III. S O R I T E S A R G U M E N T S

Sorites arguments are generally regarded as sophisms or puzzles. I think the reason they have been so regarded is that by the resem ­ blance theory of reference, their conclusions are dem onstrably false. The resemblance theory is not the primary motivation for rejecting sorites arguments, but it is the main reason. The main motivation, I believe, is irrational nostalgia. With the death of the resemblance theory of reference, I think it is clear that sorites arguments are sound, for the most part. I use sorites arguments to make intuitive what I think is plausible on other grounds. I think that to be objectively real requires having an essence. For an object to have an essence is for there to be objective necessities true of it, that is, natural laws. There appear to be very good laws about micro-particles while the laws about medium-sized objects are very poor, so full of ceteris paribus clauses as to be mere rules of thumb. Since there seems to be little hope of a reduction of medium-sized object kinds to complexes of micro-particle kinds, there can either be two unrelated systems of objective kinds or the objects with the worse laws must go. For reasons I have explained elsew here,13 I think the medium-sized objects must go. I begin with a pair of premises: (a) If a putative property is a real property, then it is a m atter of fact whether an object has that property or lacks it. (b) W hether a purported object exists or not is a m atter of fact. A purported object either exists or doesn’t exist.

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I take these to be basic ‘realistic' principles of ontology, which state what it is to be and to have a property. There are two general kinds of sorites arguments relevant to my purposes in this section which I will present in turn.

A. Property-type Sorites Arguments No person who is not tall can become tall by growing one micron. By premise (a) though, at every micron-point in the growth of a person he either has the property of being a tall person or lacks it. Unless a single micron can make the difference between having this property and lacking it, no person can become a tall person by continuous growth. Since we are very sure that any precise borderline between having this purported property and lacking it is absolutely arbitrary, it seems clear that there is no property of being a tall person. Since it is up to us, it is not a m atter of any fact about the world. Since there is no property, nothing has it. There are no tall persons. There are, of course, in the range of cases where the question seems to arise, an infinity of properties of the form ‘is n meters in height’ where n is a positive real number. W hat has been shown is that no set of such properties constitutes the property of being a tall person. So there are no tall persons. It doesn’t help to have three or more truth-values or to decide that ‘neither tall nor not tali’ is a middle category. The same fuzziness that obtains between ‘tru e’ and ‘false’ and between ‘tall’ and ‘not tall’ will occur between any two adjacent truth-values and between any two adjacent categories along the dimension. And this fuzziness shows that there is no property there, if we are right that no precise borderline is correct. (Sophisticated versions of alternative logics are briefly dealt with in the next section.) All that is possibly real in such cases would be a relation on a dimension. In the case of a predicate such as ‘bald’, such a relation is probably not even there. That there is such a relation would depend on some kind of ratio of hairs to surface normally hairy combined with considerations about distribution. Several such relations might preserve transitivity, conform to our intuitions roughly, and give different orderings of pairs of men. No relation would be selected by

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our intuitions as clearly the relation that ‘balder than’ denotes. So ‘balder than’ unlike ‘taller than’, may not even denote a relation. Property-type sorites arguments can be extended to substanceterms and count-nouns, as long as there seem to be ‘defining proper­ ties’ which are fuzzy and for which borderlines are intuitively arbi­ trary. To show that there are no rational agents, imagine an entity becoming gradually less rational, believing fewer and fewer truths, making sounds which are harder and harder to translate without attributing inexplicable error, and behaving in ways that become more and more difficult to rationalize. Analyses of agenthood in terms of success with intentional explanation or interpretation, such as Den­ nett’s 14 lend themselves to a sorites-type evaporation of ‘agent’ as a substance-determiner. In the case of paradigm property-continua such as ‘tall person’, I have argued elsew here15 that we are sure that no place in which a line is drawn is objectively right, and have explained this as confidence that no laws of nature apply above any cutoff point which do not apply below. This amounts to confidence that there is no real cutoff point in the nature of things. With persons, our confidence that it does not matter objectively what one says is less clear, since a lot hangs on whether an entity is called a person. On reflection though, the property-sorites argument should con­ vince one that the only objects that exist are ones with a precise essence. Only precise essence can constitute the being of a genuine logical subject or of real properties of logical subjects. And objects with precise essences seem to exclude persons, tables, chairs, etc. It seems very implausible that, at a certain point in the elimination of the essential ‘property’ of such objects some drastic change should take place which made one of those objects an objectively distinct entity, where what kind of thing it was changed. The problem with generalizing property-sorites arguments is that we have to construct dimensions for putative essential ‘properties’ and have to have some grounds for thinking that we have the right essential properties. Since the resem blance theory of reference is wrong, however, the features that we take to be essential may well not be. It could turn out, that is, that something very unimportant to our concept of person is in fact essential to the nature of persons.

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B. U nger's Sorites A rgum ents

A more easily generalizable sorites argument, due to Peter U nger,16 is the composition/decom position sorites. This consists of taking a putative object such as a table and extracting in the most favorable ‘table-preserving’ way one atom at a time. (In this kind of sorites, such extraction is assumed to be physically possible.) Surely one atom cannot make the difference between a table being there and there not being a table there. But equally clearly, there are no 0-atom tables. So there are no tables. The only principle this argument needs is premise (b) that every ‘object’ either exists or not, after each diminution. The argument has the form: If there are tables, then if losing an atom most favorably preserves tablehood, then there are 0-atom tables. But there are no 0-atom tables and removing a single atom would preserve tablehood. So there are no tables. Tables are not beings. Such arguments can be interpreted as showing that where there seems to be thought to be a table, there is at most a complex of atoms. A sequence of complexes of smaller and smaller size is analogous to the dimension on which ‘taller than’ is defined. In this case also, there is no subset of that sequence which is the object of common sense, though there may be a definable artificial complex object at each point up to the last in the diminution. That there is even a single relation ‘is more tabloid than’ along this sequence is questionable, since many ‘acceptable’ relations could be defined. In the case of persons, Unger’s argument starts with brains, since almost everyone would say that after a brain-transplant, he has a new body, not a new brain. So, keeping the brain in the right sort of nutrient bath, the extraction without replacem ent proceeds. Now, the nutrient bath itself would not be considered to be part of the person and neither would further life-support systems that might have to be attached to the person as the decomposition progressed. When we are eventually down to one atom, most people would agree that there is no person there. And most prople would agree that a single atom ’s addition cannot turn a non-person into a person. A composition version of the same argument comes up in disputes about abortion. If a fertilized egg is not a person but a thirty year old is, then during

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some one-second interval a person must have come into being. Since this seems physically unlikely, there are probably no persons. What it would be for an ordinary object, a person, or any object to exist, would be for some one-atom reduction to make a natural, objective difference. That is, for an object to exist is for there to be a genuine law of nature which applied at one point but which failed to apply at the next point. W hat this sorites establishes is that ordinary objects and persons as we conceive them do not exist. It is logically possible that there is a precise molecular complex entity of some sort out there which is what we are referring to when we seen to refer to something with the qualities of, for instance, a person. But there seems to be no reason to think so and fairly good scientific grounds to think not. In any case, the ordinary conceptual scheme certainly does not support the pos­ tulation of such an entity, since ordinary intuitions about when an object survives a diminution, namely that it always survives tiny diminutions, are what the sorites argument uses to evaporate putative objects. I believe that every such argument is valid, both of the property type and of the composition/decomposition type. I believe that for nearly every ordinary object, and for most ordinary properties, the premise that a single atom will not make the difference between a property being there or not is true. Thus I believe that very few ordinary objects are real. IV. W A Y S O U T

(1) Alternative logics seem to have been the most frequent res­ ponse by people who take sorites arguments at all seriously. The alternative logics I have seen depend either on treating membership in a set as probabilistic, on treating membership in a set and truth as partial,17 or on multiplying truth-values. Probabilistic membership in a set, literally interpreted by a realist, seems to require violation of principles such as that no two persons with all the same physical dimensions can be such that one is a tall person while the other is not. On the probabilistic set-membership theory, if taken literally, if two persons have exactly the same height

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and are both in the set of tall persons with a probability of 0.5, then one may be tall while the other is not. If physically indiscriminable individuals must belong to the same kind, objectively, then prob­ abilistic set-membership makes no literal sense. Some proponents18 of this theory do not take it literally, but rather as a remark about how many acceptable arbitrary borderlines an object is above or below. But this is to recognize that there is no property there. Partial set-membership is a nearly equivalent version of the above theory which is immune to the above objection. However, the notion of partial set-membership seems to apply more properly to Masonic orders than to real objects in the world. An object must have an essence, that is, objective features losing which amount to its extinc­ tion. Similarly, for a real property, a thing either has it or lacks it. The notion of being partly in a kind or partly having a certain property amounts to the admission that what we have is a dimension rather than an essence which can determine the being of an object. For a real essence or a real property, either a thing has it or lacks it. Real propositions, likewise, are either true or false. More truth-values do not make sense ontologically for the obvious reasons. The above rem arks obviously beg the question against these wellthought out and elaborate theories. I deal with these theories so briefly, though, because I think it is clear that they presuppose a resemblance theory of reference and assume that realism is false. Furthermore, the motivation for accepting these complex alternatives to our very simple ontological principles is the assurance the resem ­ blance theory gives that our patterns of verbal behavior must be made mostly true. If reference were a function of societal dispositions to apply a term, we would expect that reference would be probabilistic, since response-patterns are. Properties would be statistical social artifacts. But on a realistic view, such probability-distributions have nothing to do with what is out there, only with our descriptionbehavior. (This is not to say that a theory of our description behavior is trivial or useless. I think Zadeh, Fine and others are dealing with interesting problems, but not with the problem of this paper.) Our description-behavior, apart from the resemblance theory, can deviate massively from what is the case. Our behavior in response to stimuli may be probabilistic, but what properties a thing has and whether a

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thing of a given kind exists or not is not. When there are no rational grounds for thinking ordinary logic is mistaken, its conclusions should be accepted. With the death of the resemblance theory of reference, there are no rational grounds for changing logic; only nostalgic ones. (2) People might respond to the conclusion we have reached via the sorites arguments with the claim that ordinary objects are ontologically primary while the objects of physics are instrumental parasites. Thus the fact that the essence of ordinary objects make no sense in micro-particle terms is taken as a mark against the reality of micro­ particles, and not vice-versa. This would amount to a super-Aristotelianism, where the real essences were those of medium-sized objects, with atoms being fictions. The fundam ental nature of the world is just as we perceive it. However such a theory might go in detail, essentially the same sorites arguments go through with small chips of tables, for instance, rather than atoms. Such a theorist would be committed to the claim that, after some minute chip has been removed from a table, the table has ceased to exist, even though we admittedly are not aware of this and it is not clear from our concept that this is so. Such a reactionary realism must concede, though, that this ceasing to exist will be a theoretically isolated phenomenon, even in terms of a ‘medium-sized object’ system of basic scientific terms. Not only will the existence or non-existence of the table not be connected with any laws of particle physics, it will not even be connected with any laws about ordinary objects. This is because there are no precise laws which connect ordinary object terms that we have any evidence for. So that there are precise but unknown breakoff points for tables and trees seems to be absolutely unfounded either intuitively or scientifically. The thesis is not supported by intuitions because our intuitions hold that every point is arbitrary. It is not supported scientifically because of the very unlawlikeness of the generalizations that look to be forthcoming about medium-sized objects. That is, if there is a distinct vanishing point of the tablehood of an object in terms of chip removal, no necessary connection to dinners or to chairs seems to break down. If you could pull a chair up to the thing before, you still can. The discovery of objective breaking points seems to be impossible

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in the case of medium-sized objects for exactly this reason. Since the laws are either imprecise or nonexistent, the discovery of exactly when an object has vanished cannot be made by checking anything else. Only the activity of nous, an intuitive apprehension of essence, can do this, but nous seems to tell us in this type of case that there is no precise essence. There are probably more ways people might try to avoid the conclusion. W hat I have been urging is that, apart from a resem blance theory of reference, there is no reason to think that the conclusion is false and therefore to be avoided. Apart from a superficial appearance of paradox19 and a negative emotional response to the disappearance of loved ones, nothing blocks acceptance of the result. It is not self-contradictory in any important sense to write this. I regard it rather as a tractarian ladder. The predictive success of physics gives us every reason to believe that the fundam ental objects are micro-particles. A little reflection on developments in the theory of reference leads us to entertain the possibility that we make massive errors in our ordinary judgements about what exists. Our intuitions give us reason to think that intuitive persons and other middle-sized objects are not the sort of thing that can be vanished by extracting an atom or inducing one error. Thus, by the compulsion of the sorites argum ents, we conclude that persons and their ilk are not any real sort of thing at all. W here we seem to see the properties and natural kinds of the ordinary world there are only fluxing clumps of micro-particles and continuous dimensions. The conclusion entails that there are no persons and so no ‘theory of reference’ whose results were used in the argument. But this is harmless, since the argument is a reductio ad absurdum of the ‘given’ world. In the past, bad science could allow souls, spiritual substances, etc., to be as reasonable as anything else. The main point to make about such bizarre theories and other tempting ways out is that there is no longer any reason to think one of them is more likely than the conclusion reached in this paper. Logically, there is no reason to adopt a faith or to change logic. The ‘bold stroke’ of accepting the conclusion of sorites arguments is only emotionally bold. Philoso­ phically and rationally, it is the most plausible course. Rescues from these conclusions, prima facie, are irrational, amounting to either

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blind faith in gods or unitary spirits or blind faith that science will find water-molecule-like accounts of personhood. They are motivated by the unpleasantness, not the implausibility, of the conclusion.20 University o f C onnecticut

NOTES 1 I use the term ‘sorites’ in this paper to refer to chain-arguments with paradoxical results, not just to any chain-argument. 2 Quine, W.V., Word and Object, Ch. 2, (M.I.T. Press: Cambridge, Mass., 1960); and Davidson, Donald, ‘On the Very Idea of a Conceptual Schem e’, Presidential address in Proceedings o f the American Philosophical Association, Vol. 47, 1973-74, pp. 5-20, and ‘Truth and Meaning’, Synthese 17, No. 3, 1967. 3 This argument is made most explicitly in Richard R orty’s ‘The World Well L ost’, Journal o f Philosophy 69, No. 19, 1972, pp. 649-665. 4 Wittgenstein, Ludwig, Philosophical Investigations, (The Macmillan Company: New York, 1953). 5 Austin, John, for instance ‘A Plea for Excuses’, in V.C. Chappell, ed., Ordinary Language (Prentice Hall, Inc.: Englewood Cliffs, New Jersey, 1964), pp. 41-64. 6 Ryle, Gilbert, for instance ‘The Theory of Meaning’ in V.C. Chappell, ed., Ordinary Language. The Chappell book, Richard R orty’s The Linguistic Turn, (University of Chicago Press: Chicago, 1967), and Charles Caton’s Philosophy and Ordinary L a n ­ guage, (University of Illinois Press: Urbana, 1963) give an adequate picture of the British theory of reference. 7 Davidson, Donald, ‘On the Very Idea of a Conceptual Schem e’, loc. cit. 8 Kripke, Saul, ‘Naming and N ecessity’, in D. Davidson and G. Harman, eds., Semantics o f Natural Languages, (D. Reidel: Dordrecht, Holland, 1972), pp. 253-355. 9 Putnam, Hilary, ‘The Meaning of Meaning’, in Minnesota Studies in the Philosophy o f Science, Vol. 7, (University of Minnesota Press, 1975), pp. 131-193. 10 Putnam, Hilary, op. cit., p. 139 ff. 11 Putnam, Hilary, op. cit., p. 163. 12 This account of what Kripke and Putnam have shown is given in Sam W heeler’s ‘Reference and Vagueness’, Synthese 30 (1975), 367-379. 13 Wheeler, Sam, ‘Reference and Vagueness’, loc. cit., p. 375. 14 Dennett, Daniel, Content and Consciousness (Humanities Press: New York, 1969). 15 Wheeler, Sam, ‘Reference and Vagueness’, loc. cit. 16 Unger, Peter, ‘I Do Not Exist’, manuscript. See also ‘There Are No Ordinary Things’ Synthese, this issue. The arguments I use in applying this sorites argum ent-type are condensations of his arguments. 17 An example of this alternative is L.A. Zadeh’s ‘Fuzzy Logic and Approximate Reasoning’, Memorandum No. ERL-M479, Nov. 12, 1974. (Electronics Research Laboratory, College of Engineering, University of California, Berkeley). 18 For example, Kit Fine in ‘Vagueness, Truth and Logic’, Synthese 30 (1975), 265-300. 19 It might seem paradoxical to reach the conclusion that there are no persons, since

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there has apparently arisen the illusion that there are persons, and it would seem that there must be real persons to have that illusion. I think the best that might be made of persons is that there may be a dimension of instants of regions of the Platonic flux on which the relation ‘is more personal than’ or ‘is more intentionally active than’ can be defined. The full account of exactly what status ordinary objects and persons might have when they are not entities is a large topic for another paper. Roughly, if we can imagine mistakenly that tables and computers are logical subjects, we can equally imagine that we are logical subjects. The theory of persons this paper accom modates resembles H um e’s bundle-theory. The constituents of the bundles are instantaneous states of complexes of micro-particles, though, rather than sense impressions. Also, no clear ontological line divides w hat’s in the bundle from w hat’s out of it. Thus illusory illusions can arise in ‘personal’ regions without there being any entities which are persons. 20 I would like to thank the philosophy departm ent at the University of Connecticut, and especially John Troyer and Jerry Fodor, for helpful discussion. The paper would not exist even on the ordinary view without the encouragem ent of Peter Unger and the use of some of his argum ents in the composition/decomposition sorites.

[3] D A V I D H. S A N F O R D

N O S T A L G I A FOR T H E ORDINARY: C O M M E N T S O N P A P E R S BY U N G E R A N D W H E E L E R

The furniture of the earth includes no furniture, according to Peter Unger and Samuel C. W heeler, III. There are no tables, chairs, or footstools. There is no planet Earth, for that m atter. N either are there sticks or stones, logs or boulders. Unger restricts his denials of existence in the paper under discussion to ordinary inanimate objects, but both he and W heeler are clearly inclined also to deny the existence of all plants and all animals including persons. Although both W heeler and Unger use sorites argum ents in their attem pts to prove that there are no ordinary things, there is a great difference betw een their arguments. This difference can be expressed by invoking the simple m etaphysical contrast betw een A ppearance and Reality. W heeler and Unger both think that the apparent exis­ tence of ordinary things is merely apparent. Unger argues that Ap­ pearance convicts itself of incoherence. His beliefs about reality are mainly negative: tables do not really exist, stones do not really exist, and so forth. On the question how Reality should be positively described, Unger can remain agnostic. W heeler apparently agrees with Unger that ordinary beliefs about Appearance are internally inconsistent. Unlike Unger, however, he has strong views about the positive nature of the Real, on what is required for genuine causation, genuine properties, genuine kinds, and genuine reference to things in the world. W heeler’s main reason for rejecting A ppearance as a sham and illusion is that he supposes there to be a conflict betw een the Real and the Apparent. I. U N G E R O N A P P E A R A N C E

Let us say that x and y are R -related if and only if either Rxy or Ryx \ and let us say that x and y are /^-linked if and only if x belongs to every set which contains y and everything R -related to any member Synthese 41 (1979) 175-184. 0039-7857/79/0412-0175 $01.00. Copyright © 1979 by D. Reidel Publishing Co., Dordrecht, H olland, and B oston, U .S.A.

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of the set. (If x is related to y by the ancestral of Ä, jc and y are Æ-linked; but x and y may be i?-linked without being related by the ancestral. Any two cousins are linked by the relation ‘parent of’.) A general form of the sorites argument may be sketched as follows: For any jc and only i f Fy.

y, if

x and y are J?-related, then Fx

if

and

Therefore, by the transitivity of ‘if and only if’. For any x and y, if x and y are R -linked, then Fx if and only if Fy. When R and F are appropriately specified, the premiss appears to be true because its denial requires, in U nger’s term s, either a miracle of m etaphysical illusion or a miracle of conceptual comprehension. Suppose that ‘F x ’ is interpreted to mean ‘x is a stone’ and ‘Rxy ’ is interpreted to mean ‘y results from the net removal from x of one atom, or only a few , in a way which is most innocuous and favorable for the continued existence of a stone’. To expect an explosion or disappearance or m etam orphosis or some other ‘break in the world order’ to result from such a slight removal of atoms from a stone is to expect a miracle of metaphysical illusion. To expect that the object resulting from such a removal from a stone will no longer be a stone, even though the resulting object is virtually indistinguishable from the original object, is to expect a miracle of conceptual comprehension. If we do not expect such miracles, we must accept the prem iss; and if we accept the prem iss, we must accept the conclusion: anything linked by the specified relation to a stone is itself a stone. Let us grant, for the purpose of Unger’s argument, that nothing which is really a stone could cease to be a stone merely by becoming linked by the specified relation to something which is not a stone. Any physical objects consisting of a finite number of atoms could be linked to something which is not a stone because it does not consist of enough atoms to be a stone. Therefore, no physical object consisting of a finite number of atom s is a stone. For any predicate ‘F ’, where F ’s are ordinary things, we can find some relation R which makes the premiss of the sorites argument appear true and which allows the possibility that anything we would

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call an F could be Æ-linked to something which is not F. For any predicate ‘F ’, then, where F ’s are ordinary things, we can conclude that nothing we would call an F is actually an F. There are no ordinary things. I suspect that many philosophers would gladly em brace a myriad of miracles of conceptual com prehension rather than accept U nger’s contentions that there are no stones, tables, or swizzle sticks. O thers, such as myself, would be reluctant to accept such m iracles but even more reluctant to deny the existence of tables and stones. I believe that we need accept no such miracles to resist U nger’s argum ents. Unger gives the following example of what it would be to believe in this miracle: We must suppose that with, say, a trillion trillion atom s there, in a certain case, there really is a stone, w hether anyone can ever tell or not. But, with one or a few, say fifty, gingerly rem oved from the outside, the situation suddenly changes, even if no one can ever tell. And this means that with any one, or any fifty, of the atom s gone, there is no stone there. T h at’s the sensitivity of our word ‘stone’ for you! To believe in this is, I say, to believe in a miracle o f conceptual comprehension. (Unger, this issue, p. 126.)

It is not only a sudden change from truth to falsity which Unger would regard as miraculous. The introduction of truth values inter­ mediate betw een truth and falsity is of no avail. If there are finitely many values, the change from truth to the next closest value with the removal of a single atom is still a miracle. If there are infinitely many values, densely ordered so that betw een any two values there is always a third, then there is never a next closest value, but this does not really diminish the need for an incredible miracle if we are to avoid U nger’s conclusions.1 For any departure from unity, or indeed any change in truth-value, with the rem oval of a single atom , requires that our term s ‘stone’ and ‘swizzle stick’, be sensitive at least on the atomic level. That our expression ‘swizzle stick’ should be that sensitive quite defies credibility. For it to be that discrim inating is a miracle which surpasses my capacity for belief. (Unger, p. 129.)

A stronger claim and weaker claim can be distinguished here. The stronger claim concerns any change in truth-value. It would be a miracle if ‘It is a swizzle stick’ said of an object x should have a

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value n and said of object y should have a value other than n when y is produced by removing a single atom from x. The w eaker claim concerns only any departure from unity. If the numerical truth-value 1 is identified with truth, 0 with falsity, and the real num bers between 1 and 0 with less than complete degrees of truth or falsity, the weaker claim is that it would be a miracle if ‘It is a swizzle stick’ said of an object jc should have value 1 and said of object y should have a value less than 1 when y is produced by removing a single atom from jc. 2 I accept the w eaker claim. I also believe that there are swizzle sticks, that ‘It is a swizzle stick’ said of some objects has value 1, and that continued rem oval of atoms from a swizzle stick would eventually destroy it, so that ‘It is still a swizzle stick’ would have value 0. Are these beliefs inconsistent? I think not, for I do not accept the principle that every statem ent has a truth-value. (In Sanford, 1976,1 called this the Principle of Valence.) The question ‘Is the truth-value of this particular statem ent (sentence token, assertion, or w hatever is sup­ posed to be the proper bearers of truth-value) 1 or something other than 1?’ does not always have an answer. There is no saying just when an object ceases fully to be a swizzle stick (when ‘It is still a swizzle stick’ ceases to have value 1) as it loses atom s one-by-one, and there is no saying just when the object is not a swizzle stick at all (when ‘It is still a swizzle stick’ comes to have value 0), although eventually it does cease to be a swizzle stick. If the Principle of Valence is rejected, it should be rejected at all levels. Not only may statem ents predicating ‘is still a swizzle stick’ of some object lack a truth value, for example, but so also may state­ ments predicating ‘has a truth value’ of statem ents predicating ‘is still a swizzle stick’ of some object lack a truth value, and so also may statem ents predicating ‘has a truth value’ of statem ents predicating ‘has a truth value’ of statem ents predicating ‘is still a swizzle stick’ of some object lack a truth value, and so on. There is no miracle of conceptual com prehension in the object-language or in the m eta­ language or in the meta-meta-language. There is no miracle of comceptual com prehension anywhere. The wholesale rejection of the Principle of Valence is independent of the num ber of values one admits into one’s semantics. Unger is

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right to claim that the introduction of intermediate values by itself does not diminish the appearance that some miracle of conceptual comprehension is required. A sudden drop from 1 to something less than 1 is as mysterious as a sudden drop from 1 to 0. If the Principle of Valence is rejected, one need not accept any kind of sudden drop of truth-value. The contrast I drew in Sanford 1976 between the semantics of super-valuations and many-value semantics now seems to me to be spurious. There is a real contrast between many-value semantics and two-value semantics. The method of super-valuations counts as true any sentence which is true on all admissible specifications. When the method of super-valuations is combined with a classical two-value semantics, only classical two-value specifications are admissible. When the method of super-valuations is combined with a many-value semantics, many-value specifications are admissible. There is another real contrast between different many-value semantics, since it makes a difference which one is used. (In Sanford, 1975, I developed one which counts all the classical tautologies as true.) In Sanford, 1976, I asked the following question, which was in­ tended to embarrass those who advocate the super-valuation ap­ proach to a logic of vagueness: G rant then that a certain statem ent is true if its predicates are made com pletely precise in any appropriate way. Why should the statem ent thereby be regarded as true if its predicates are not made precise in any of these appropriate w ays? (p. 206)

The question should also embarrass me, although perhaps not quite as much. It can be said in favor of a many-value semantics that any admissible many-value specification more accurately represents the actual ordinary use of the statement in question than any admissible two-value specification. A many-value semantics also permits a more accurate semantics of quantification (Sanford, 1976, pp. 207-209) and a truth-functional semantics of a determinacy operator (Sanford, 1975, pp. 34-39). There is still no commitment to assign some one value to any meaningful statement. One may instead assign a range of values. Any value within the range is an admissible value for the statement in question. ‘Less than 1’ and ‘greater than 0’ are the two

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widest ranges which exclude something. ‘Not less than 1’ and ‘Not greater than 0’ are the two narrowest ranges which can sometimes be assigned with complete confidence. When intermediate ranges are assigned, there may be no way to decide just how narrow a range one can safely assign. Unger’s experiment with a very small heap nicely illustrates a conscientious and sensible assignment of ranges of truthvalues. As fairly as I can, I will try to approach the Eubulidean problem with the notion of degrees of truth, and the range of available values that it implies. Now, according to my own idiolect, or my linguistic intuitions, I can feel com fortable, w hatever that may be w orth, in saying with five beans suitably before me, ‘H ere’s a very small heap, and so, a heap’. Perhaps, then, I should assign the value of 1, or at least 0.95, to the proposition that O scar is a heap, w here ‘O scar’ is supposed to name the hopeful heap of five now before me. (W hen I first wrote this, I was actually playing with grains of rice, not beans in fact, to try seriously to get a feeling for the m atter.) Taking away a bean from O scar, to produce Felix (who may or may not be the same entity as Oscar), I felt less com fortable in saying a heap is before me. But it isn’t all that unsettling. W hat am I to do; assign a value of 0.9 to the proposition that Felix is a heap? Taking away one again, now to yield Leo, I am at a loss. My sensitive intuitions seem to desert me; I know not which way to turn. Perhaps a value of 0.5 is now in order, or is that a bit too high, or too low, and just a fake at compromise? With tw o beans, and only Alex before me, I feel like 0 is the value for me. But can such a sudden and great drop stand scrutiny? Perhaps w e’d b etter go back and re-evaluate, or better yet, give up this game. (Unger, p. 131.)

Unger’s intuitions are as sensitive as I should hope to find. There seems to him to be an insoluble problem only because he clings to the Principle of Valence. So long as there is some truth-value assignment one is confident is incorrect, one can assign a range that rules out something. ‘Not less than 0.95’, ‘Not less» than 0.8’, ‘Not less than 0.25 and not more than 0.75’, and ‘0’ appear to be ranges Unger could conservatively assign to the statements that Oscar, Felix, Leo, and Alex are, respectively, little heaps. Why ‘Not less than 0.25 and no more than 0.75’ for the statement that Leo is a heap rather than a slightly wider, slightly narrower, or somewhat overlapping range assignment? The question is unanswerable. So what? The existence of ordinary things such as sticks and stones, swizzle sticks and tables is safe from all of Unger’s sorites arguments. No miracles of concep­ tual comprehension need be invoked if the Principle of Valence is rejected.

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II. W H E E L E R O N R E A L I T Y : A B R I E F A N T H O L O G Y . . . the real reason there aren’t any laws about tall men is that ‘tall m an’ do esn ’t designate a kind. For there to be a m atter of fact about w hether a predicate correctly applies to a new case, there must be an objective kind in the nature of things to which the predicate refers. The projectibility of a predicate, its law -boundness, is reflection of ontological fact: Laws are to be understood in term s of natural necessities and divisions in the nature of things, even though, in the order of knowledge, the nature of things is approached via the discovery of laws. Lawlikeness is our window on reality, not w hat ‘real’ actually is or means. (W heeler, 1975, p. 373.) To be real is to be able to enter into causal relations, and only law-bound sets can enter causal relations. The causal theory of reference is thus com m itted to rejecting the ontological liberalism which congeals the pattern of responses of any possible culture into a perfectly acceptable candidate for reality. (W heeler, 1975, pp. 375-376.) (a) If a putative property is a real property, then it is a m atter of fact w hether an object has that property or lacks it. (b) W hether a purported object exists or not is a m atter of fact. A purported object either exists or doesn’t exist. (W heeler, this issue, p. 164.) No person who is not tall can becom e’tall by growing one micron. By prem ise (a) though, at every micron-point in the growth of a person he either has the property of being a tall person or lacks it. U nless a single micron can make the difference betw een having this property and lacking it, no person can becom e a tall person by continuous growth. Since we are very sure that any precise borderline betw een having this purported property and lacking it is absolutely arbitrary, it seem s clear that there is no property of being a tall person. Since it is up to us, it is not a m atter of any fact about the world. Since there is no property, nothing has it. T here are no tall persons. (W heeler, this issue, p. 165.)

III. A P P E A R A N C E A N D R E A L I T Y

Wheeler’s views on the Nature of Reality have not all been published. He realizes, for example, that he is committed to there being a real distinction between having a property and lacking a property. Vague predicates come in pairs. If ‘is a tall person’ is vague, then so is its complement ‘is not a tall person’. If the vagueness of ‘is a tall person’ shows that there is no property of being a tall person, then the vagueness of ‘is not a tall person’ shows that there is no property of not being a tall person. Since there is no property, nothing has it. There are no things which are not tall persons.

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Something has gone wrong. It must be that ‘There are some tall persons’ says that something has a purported property, while ‘There are some things which are not tall persons’ says only that something lacks a purported property. There must be a real distinction between positive and negative terms, although Wheeler has not told us how to draw it. (If he wants to use the argument of Sanford, 1967, he is welcome to it.) If ‘is not a tall person’ is only a negative term, it is still a term. A sentence of the form ‘jc is not a tall person’ can be true even if there is no property of not being a tall person. It is not generally true, then, that the truth of a sentence of the form ‘jc is F ’ requires that there be a real property of being F Why should we suppose, then, that the truth of a sentence of the form ‘jc is a tall person’ requires that there be a real property of being a tall person? I have no argument here against Wheeler’s criteria for the reality of properties. But I do not see why, if these criteria are accepted, we should persist in supposing that an ordinary predicate truly applies to something only if the predicate refers to a real property. I have no argument here against Wheeler’s criteria for the reality of kinds. But I do not see why, if these criteria are accepted, we should persist in thinking that an ordinary singular causal statement such as ‘The stone broke the window’ can be true only if stones and windows both comprise real kinds. There is a problem, the problem of showing how stones and windows and the relations between them are phenomena bene fundata if neither stones nor windows are kinds. For a promising start at a solution to this problem, see Kim, 1978. According to Wheeler, “the resemblance theory [of reference] is not the primary motivation for rejecting sorites arguments, but it is the main reason. The main m otivation. . . is irrational nostalgia” (Wheeler, this issue, p. 164). This cannot be so. The vast majority of persons have no theory of reference at all, although they are confident in their beliefs about the existence of ordinary things. Their main reason for believing in the existence of ordinary things thus cannot be that they cling to an inadequate theory of reference. Neither can their beliefs in the existence of ordinary things justly be called irrational since they have no reason for doubting and abundant

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reasons for not doubting the existence of ordinary things. The charge of irrationality is best avoided in disputes of this sort since it is too easily repaid with interest.3 Duke University NOTES 1 Unger says that I correctly represent myself as being“ rather representative” of those who assign new and exotic truth-values in an attem pt to develop a logic of vagueness. He quotes the part of the following sentence which is not now printed in italics: “ I shall proceed from some of the basic assum ptions shared by previous w orkers on the logic of vagueness, give m y reasons fo r rejecting other such assum ptions, and then develop a new logic o f vagueness" (Sanford, 1975, p. 29, quoted in part by Unger, this issue, p. 128). This sentence is itself vague, in the pejorative sense of ‘vague’. Does it mean shared by some or all previous w orkers on the logic of vagueness? The shared assum ptions I go on to mention are shared by the previous w orkers I mention, L.A. Zadeh, K enton F. Machina, J.A. Goguen, and George Lakoff. They are not shared by some previous workers such as K. Fine and J.A.W. Kamp whom I did not mention and whose work on the logic of vagueness had not then been published. Although I retain a paternal fondness for the sem antics sketched in Sanford, 1975, I think it is more accurately called quite eccentric rather than rather representative. It differs in some non-trivial way from every other attem pt to develop a logic of vagueness. 2 In Sanford, 1976, I say that if there is no discontinuous change in truth value of a statem ent whose value falls from 1 to 0, then “ there is a last time when such a statem ent has value 1, a first time when it has value 0, and for each of infinitely many values betw een 1 and 0, both a first and a last time, perhaps identical, when the statem ent has this interm ediate value” (p. 198). I am grateful to John C orcoran for pointing out that this claim is quite mistaken. I should have said that there is either a last time when the statem ent has value 1 or a first time when it has a value less than 1, either a first time when it has value 0 or a last time when it has a value greater than 0, and for each of the infinitely many values between 1 and 0, either a first time when it has this value or a last time when it has a greater value, and either a last time when it has this value or first time when it has a lesser value. The correction of this mistake does not affect the point I wished to make, that it is absurd to suppose that there is always one correct answ er to a question of the form ‘W hat is the value of statem ent S at time tT 3 I am grateful to both Professor W heeler and P rofessor U nger for the help they have offered me, through correspondence and conversations, in understanding their views. N either has approved my summaries of their views in this paper. BIBLIOGRAPHY Kim, J., 1978, ‘Supervenience and Nomological Incom m ensurables’, American Philosophical Quarterly 15, 149-156. Sanford, D.H., 1967, ‘Negative T erm s’, A nalysis 27, 201-205. Sanford, D.H., 1975, ‘Borderline Logic’, American Philosophical Quarterly 12, 29-39.

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Sanford, D.H., 1976, ‘Competing Semantics of Vagueness: Many Values V ersus Super T ruth’, Synthese 33, 195-219. Unger, P., 1979, ‘There Are No Ordinary Things’, Synthese 41, this issue, 117-154. W heeler, S.C., III, 1975, ‘Reference and V agueness’, Synthese 30, 367-379. W heeler, S.C., Ill, 1979, ‘On That WTiich is N ot’, Synthese 41, this issue, 155-173.

[4] BERTIL ROLF

S O R IT E S

0. T H E SORITES PARADOXES

Paradoxes called “ sorites” (“ heap”) or ''falakros” (“ bald m an”) can be traced back to Eubulides of the Megarian school who was a contem ­ porary of Plato. For a long time, they were considered semantic curios only. But lately, they have been made the topic of many investigations. In this paper, I will first present some facts about them, then, in Sections 1-3, I will discuss some proposed solutions to them and in Section 4 present my own solution. A short conclusion states similarities and differences am ong the various solutions. We now proceed to some forceful examples of sorites: (SI) T h ere are mammalian ancestors. T here have not been infinitely many mammalian ancestors. T herefore, some mammalian ancestor has no ancestor who was mammalian. (Sanford 1975 p. 521) (C l) Suppose th at a movie cam era is focused on a tadpole confined in a small bowl of water. T h e cam era runs continuously for three weeks, and at the end of that time there is a frog in the b o w l . . . (the conclusion of the argument) means that there is some picture in the series S such that it is a picture of a tadpole, while the very next picture, taken one tw enty-fourth of a second later, is not a picture of a tadpole. (Cargile 1969 p. 193) (U l) . . . we may begin with a single atom somewhere, and say that it is something which is not a stone . . . Now, if we add a single atom to something which is not a stone, it seems that such a m inute addition, however carefully and cleverly executed, will never in fact leave us with a stone. For a single atom , I suggest, will never mean the difference between there being no stone before us and, then, there being one there. (U nger 1979 p. 143)

T hese exam ples indicate that sorites arguments may be generated by any actual or conceivable ‘almost-continuous’ process. T he main sem antic principle underlying sorites is called “ tolerance” by W right (1975)1. R oughly, the predicate “stone'’ is tolerant if for any objects x and y = (x plus one atom), “ stone” is as correctly applicable to x as to y. Suppose that the C IA plucks the hairs off poor T om ’s head, one by one. A t first, T om was not bald but when the CIA had pulled out his last hair, he definitely was bald. A situation such as the one described generates a good many sorites arguments. First, there are argum ents Synthese 58 (1984) 2 1 9 -2 5 0 . 0 039-7857/84/0582-0219 $03.20 © 1984 by D . R eidel Publishing Company

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employing m athematical induction: Tom was bald when he had 0 hairs; for each num ber, n, if Tom was bald when he had n hairs, he was bald when he had n + 1 hairs; therefore he was bald when he had 100.000 hairs. But m athematical induction is not necessary for generating sorites. We may, instead, formulate an argum ent containing 100.000 conditionals and 100.000 applications of Modus Ponens. Nor is it necessary to use “if-then” and Modus Ponens. T heir places can be taken by “ either-or” and the Disjunctive Syllogism.2 In this paper, I will refer to the following formulation of a sorites paradox: 0.1.1 0.1.2

Tom was bald when he had 0 hairs on his head. It is not the case that Tom was bald when he had 100.000 hairs on his head.

0.1.3

H ence, there is a number, n, such that Tom was bald when he had n hairs on his head but it is not the case that Tom was bald when he had n + 1 hairs on his head.

T he repugnancy of this argum ent lies in the fact that we seem to have as good reasons as we ever could have for counting the premisses true and the conclusion false or not true - just which num ber could the conclusion be talking about? Furthermore, the argum ent has a strong flavour of validity. Let “ P ” abbreviate the predicate “Tom was bald when he had — hairs on his head” . Certainly, if P is true of 0 (0.1.1) but not true of 100.000, then there must be a first number, n, such that P is true of n but not of its successor (0.1.3)? Let me form ulate the assumptions about the argum ent 0.1 which all seem true but which nevertheless contradict each other: (1) The argum ent 0.1 is valid; (2) Its conclusion is false and (3) Its premisses are true. Examples of proposed solutions denying (l)-(3) will be considered in Sections 1-3 respectively. In Section 4, it will be proposed that some of these alternatives are ‘meaningless’. 1. F U Z Z Y L O G I C

Fuzzy logic3 claims to solve the problem of the sorites paradoxes by denying that classical logic is valid when vague words occur in the argum ents. Let me sketch how this theory attem pts to solve the problem. Fuzzy logic assumes the existence of infinitely many truth-values

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which are generally represented by means of the closed interval from 0 to 1 on the real axis. The value 0 represents full falsity and 1 represents full truth. Before we plucked the hairs off poor T o m ’s head, the sentence “Tom is bald” would have the truth-value 0 and when we have plucked his last hair, it would have truth-value 1. We assume that the truth-values of the sentence would decrease approxim ately as shown below. shown below. shown below.

1)]

shown shown below. below. shown shownbelow. below. P(n 1+)] T he premiss “Tom was bald when he had 0 hairs on his head” would on this theory be fully true - i.e., have truth-value 1. T he sentence “Tom was bald when he had 100.000 hairs on his head” would have truth-value 0. Fuzzy logic generally assumes that the truth-value of the negation of A equals (1 - the truth-value of A). So the truth-value of the second premiss - 0.1.2 - would also be 1. But what about the truth-value of the conclusion? Let us assume that there is no pair (n, n + l) such that the truth-value of “Tom was bald” leaps m ore than 0,001 when we plucked off his n + l ’st hair. We assume - as is often done in fuzzy logic - that the truth-conditions of disjunction or existential quantification is given by the m ax-operator and the truthconditions of conjunction is given by the m in-operator. Then the truth-value of the conclusion 0.1.3 is given by: 1)] 1)]

1.1

Max

Min [the truth-value of P(n ), 1 - the truth-value

ssns=l ()().()()()

of P(n + 1)] (where P is the predicate “Tom was bald when he had — hairs on his head”).

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The value of 1.1 can never exceed 0,5. To put it in a m ore u n d er­ standable language, the conclusion 0.1.3 would be nearer falsity than truth. If truth is identified with, say, truth-values exceeding 0,85, it is clear that we have an argum ent which seems to be valid but which does not transmit truth from premisses to conclusion. Thus, the argum ent is not really valid. However, the theory of truth proposed by fuzzy logic is seriously deficient. The main problem it faces concerns whether truth assertions of the metalanguage will be vague. For instance, if “ tru e” is used in its ordinary sense, then “ A is true” is vague if A is. Will fuzzy logic adm it the vagueness of such sentences as “ A is true to a degree of 0 ,984” ? There are two possible replies. First, fuzzy logic might say that all such assertions of truth to a degree are fully precise. It seems Goguen takes this line: (G l) O ur models are typical purely exact constructions, and we use ordinary exact logic and set theory freely in their developm ent. This am ounts to assuming we can have at least certain kinds of exact knowledge of inexact concepts. (When we say som ething, others may know exactly what we say, but not know exactly what we mean.) It is hard to see how we can study our subject at all rigorously without such assumptions. (G oguen 1968 p. 327)

But saying that those assertions are precise does not m ake them so. As far as I know, fuzzy logic has never seriously dealt with this problem . One way to solve it would be to present a system of fully true biconditionals of the form: 1.2

“It is raining” is true to a degree of 0,984 if, and only i f , . . .

where “ . . . ” is to be a perfectly precise sentence. For obvious reasons, the difficulties for such an approach are considerable. But unless the truth-conditions of truth ascriptions are precisely fixed, the truth ascriptions are vague. Second, fuzzy logic might admit the vagueness of (some) sentences ascribing truth to a degree. But then the law of excluded middle would not hold in the m etalanguage, according to the principles of fuzzy logic. For say that “ It is raining” is close to being true to a degree of 0,984 and say that the following truth ascription is nearly true; m ore specifically, true to a degree of 0,89: 1.3

“ It is raining” is true to a degree of 0,984.

But if we assume negation and disjunction to be as before, the negation

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of 1.3 will be true to a degree of 0,11 and the sentence consisting of 1.3, disjunction and the negation of 1.3 would be true to a degree of 0,89; i.e., nearly true. However, these assumptions violate the principles of valuational systems by which fuzzy logic is justified and presented. For say that the valuational function v assigns 0,99 to “ It is raining” . Is it also the case that v assigns 0,984 to “ It is raining?” Of course not - v would not be a function if it did. But under the assum ptions above, it would be nearly true - true to a degree of 0,89 - that v assigns 0,984 to the sentence to which it assigns 0,99. T he whole idea of sentences having one truthvalue and not another vanishes, once vagueness occurs among the truth ascriptions; at least if we assume that the principles of fuzzy logic hold for vague languages. It is, I think, an open question w hether fuzzy logic' can be given any plausible form without using valuational systems - all reasons for fuzzy logic seem to rely on intuitive considerations about the assignm ent of truth-values. In fact, there may even be logical obstacles to defining fuzzy logic by axiomatic systems - cf. M organ and Pelletier (1977) and references therein. A nother fundam ental objection to fuzzy logic as a theory of vague­ ness is that it does not give a correct account of vagueness. T h e best reason for m ulti-valued logics is that they m ake it possible to represent discerned differences with different truth-values. This has been nicely stated by Sanford: (S2) Suppose that dying is at least som etimes a process involving a continuum of m om entary states no one of which is the last state at w hich the person is alive or the first state at which the person is no longer alive. If the transition from being alive to not being alive is continuous, it seems appropriate to have a continuum of values to assign to statem ents which assert that som eone is alive. T his sort of consideration can be generalized. W e should like to satisfy the following requirem ent: every discernible difference of sem antic status betw een statem ents about borderline cases should be reflected by a difference in assignable truth-values. (Sanford 1976 pp. 199-200)

This property of fuzzy logic lends some support to the view that it is better adapted to account for gradual changes than classical logic. Fuzzy logic does indeed m ake m ore subtle semantic distinctions than the standard m etatheory of classical logic. But the problem for the latter, illustrated by vagueness and sorites is not that it makes too coarse semantic distinctions. R ather, the problem is that it makes a distinction at all! T he difficulty for classical m etatheory is that it postulates a semantic difference where no relevant difference can be perceived.

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Fuzzy logic cannot possibly evade this difficulty by postulating, in principle, infinitely many im perceptible semantic differences. D. H. Sanford has, in various publications, proposed a sophisticated version of fuzzy logic. Let us consider whether his theory provides a better solution for sorites. He begins his (1976) with a distinction between the Law of Excluded Middle - that every disjunction of the form “ A or n o t-A ” is true - and the Principle of Bivalence which says that every statem ent is either true or false. He generalizes the latter to the Principle of V alence which says that every statem ent has a truth-value. Sanford wishes to deny both the Principle of Bivalence and that of V alence. His argum ent against them is based on an example of a patient who is alive but gravely ill on Tuesday and who is definitely dead on Friday. W e may think of the tim e-stretch in between as divided into arbitrarily short intervals. In Sanford’s view, Bivalence implies that, at each interval, “T he patient is alive at this interval” is either true or false. But then death m ust have been ‘instantaneous’ (ibid. p. 197). He does not accept this consequence. Bivalence implies the existence of a sharp cut-off point which there is not. M ore generally, a principle of n-valence implies the existence of n-1 sharp cut-off points, which there are not. T herefore, he rejects the Principle of V alence (ibid. Section 3). For the reasons given in (S2) above, he accepts having many semantic values. In his (1976), he leaves the details of his semantics unspecified. From what he says on pp. 202-203, it is clear that Sanford accepts the Law of Excluded Middle. Can Sanford’s theory be used to solve the sorites? Let us consider his attem pt to deal with U nger’s sorites: (53) I also believe that there are swizzle sticks, that “ It is a swizzle stick” said of some objects has value 1, and that continued rem oval of atom s from a swizzle stick would eventually destroy it, so that “ It is still a swizzle stick” would have value 0. Are these beliefs inconsistent? I think not, for I do not accept the principle that every statem ent has a truth-value. (In Sanford, (1976), I called this the Principle of Valence.) T he question “ Is the truth-value of this particular statem ent (sentence token, assertion, or w hatever is supposed to be the proper bearers of truth-value) 1 or som ething other than 1?” does not always have an answer. (Sanford 1979 p. 178) (54) If the Principle of V alence is rejected , it should be rejected at all levels. Not only may statem ents predicating “ is still a swizzle stick” of some object lack a truth value, for example, but so also may statem ents predicating “ has a truth value” of statements predicating “ is still a swizzle stick” of some object lack a truth v a lu e -----If the Principle of V alence is rejected, one need n o t accep t any kind of sudden drop of truth value. (Sanford 1979 pp. 178-179)

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(S5) There seems to him (i.e. to Unger; B.R.) to be an insoluble problem only because he clings to the Principle of Valence. So long as there is some truth-value assignment one is confident is incorrect, one can assign a range that rules out something . . . . The existence of ordinary things such as sticks and stones, swizzle sticks and tables is safe from all of U nger’s sorites arguments. No miracles of conceptual comprehension need be invoked if the Principle of Valence is rejected. (Sanford 1979 p. 180)

Sanford here considers the possibility of sorites in the metalanguage, in the m etam etalanguage and so on. He wishes to solve sorites at all levels in the same way - by a denial of Valence. This, he thinks, escapes all of U n g er’s sorites (S5). It may be interesting to find out whether it is so. How ever, it is even m ore interesting to find out whether a denial of V alence solves all sorites. I will argue it does not. First, what does the principle of Valence say? It says that every statem ent has a truth-value. But what is a truth-value? Is 0 one? Is 342 one? Is the Moon one? T he notion “ truth-value” is relational and should read “ truth-value in the valuational system V” . An entity might be a truth-value in one system but not in another. Unless Sanford’s discussion about V alence and truth-values is to be nonsense, we must assume there is a fixed valuational system. But is there a unique, correct system for English? T he answer is no, even if we leave all Quinean worries aside. T alk about truth-values is only an algebraically con­ venient way of replacing predicates like “— is true” with talk of solid objects as in “T he truth-value o f __is T ” . The use of such predicates is prim ary to the choice of valuational system. Now suppose we have the three predicates: “__is tru e” , “__ is false” and “— is neither true nor false” . T o these, a good many valuational systems correspond. For instance, one may contain, T , F and have a partially defined valuation function which is undefined when A is neither true nor false. Or one may have a system containing T , F and the Moon, assigning the latter when A is neither true nor false. All that matters is that the use of the corresponding predicates comes out true. Thus, there does not seem to be m uch philosophical substance to squeeze out of the concept of V alence, or of having a truth-value. (Sanford is by no means the only one to reify truth-values. Most discussions about distances between truth-values m ake such assumptions.) Talk about Bivalence does make good sense, however. For here, the adjectives “ true” and “false” are fixed. I therefore proceed to discuss Sanford’s rejection of Bivalence. I take it that Sanford, at all levels, accepts classical logic and hence the Law of Excluded Middle. W hat he rejects is the metalogical

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Principle of Bivalence (cf. (S3)). The difficulty for Sanford is that sorites occur in the object language while Bivalence belongs to the m etalan­ guage. But how could a paradox in the object language be eliminated by a change of only metalinguistic principles? (The same difficulty touches supervaluations, as we shall see in Section 2.) T he problem is that the speakers of the object language get themselves into a paradox by principles in their language. Although the speakers of the m etalan­ guage may change their descriptions of these principles, the principles rem ain what they were and the paradox remains where it was - in the object language. Sanford thinks that Unger gets himself and us into sorites because he clings to the erroneous Principle of V alence (cf. (S5)). But U nger need have no metatheory and probably has none, sceptic as he says himself to be. If Unger formulates a sorites in the object language, using only such logical principles of the object language as Sanford accepts, it will not do to accuse U nger of an error as to metalinguistic principles. And if Unger formulates a sorites in the m etalanguage, using such principles as Sanford accepts, it will not do to assert that U nger is mistaken about principles of the m etam etalan­ guage. A nd so on. And there are a good many bad sorites, form ulated in the object language. It would not seem that these can be solved by calling into question metalinguistic principles only. A nother problem for Sanford is that the acceptance of classical logical principles would make the sorites 0.1 logically valid. I take it that he accepts that its premisses are true. But then its conclusion will be true as well. But was not multivalued logic invented just to avoid this? I therefore do not think that Sanford’s theory solves the sorites. Before I leave the investigation of his theory, I wish to com m ent on his argum ent against Bivalence, mentioned above. Sanford observes that the choice of semantic principles depends on non-semantic principles; here the concept of change. This notion is defined by Russell as follows: (R l) C hange is the difference, in respect of truth or falsehood, between a proposition concerning an entity and a time T and a proposition concerning the same entity and another time T ', provided that the two propositions differ only by the fact that T occurs in the one where T ' occurs in the other. (Russell 1903 §442)

Russell’s definition can probably be given a formulation which, apart from some concept of time, only employs logical, or metalogical, concepts. T herefore, we ought to expect that considerations about

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change are relevant for logic and metalogic. I am inclined to agree with Sanford that Bivalence, together with obvious assumptions about time, implies that all changes are ‘in­ stantaneous’. For suppose that Fa is true at ti but not at t2, later than t, . Suppose that the time stretch betw een /, and t2 can be divided into very short intervals. By sorites - argum ents it seems to follow that there are two adjacent intervals, i, and ti+i such that Fa is true at r, but not at 1)] 1)] Thus, during the interval com posed of t and ti+u the entity denoted by a would seem to have changed with respect to the property denoted by F. But a, F , ti and t2 were taken arbitrarily. Thus, it seems that all change takes place during very short intervals and is ‘instantaneous’. Clearly, this is an intuitively unacceptable consequence. Sanford thinks he can get out of this quandary by giving up a metalogical principle - Bivalence. However, this will not do. F or it seems obvious that the general concept of change, as well as particular assertions of change, ought to be expressible in the object language. In a first order object language where a given change is describable, it seems we could use sorites to show it to be instantaneous. It is not only classical metalogic which conflicts with our opinions about change - it is classical logic itself. (The same point can be m ade against super­ valuations.) Finally, I rem ark on some other attem pts to solve sorites by giving up some principles of classical logic. First, sorites are not avoided only by a rejection of the Law of Excluded Middle. For, as we have seen (Section 0), there are versions of sorites which use “ if-then” and M odus Ponens. Second, it will not do to weaken the principle of m athem atical induction, as proposed by Weiss (1976). For there are versions of sorites which do not use that principle at all. Third, some have thought that sorites can be avoided by saying (a) that logic holds only for ‘state­ m ents’ and (b) that if a predicate is predicated of one of its borderline cases, the result is neither true nor false and hence not a ‘statem ent’. This would eliminate some versions of sorites, namely those which contain a chain of premisses asserting of each num ber that if T om was bald when he had that num ber of hairs on his head, he was bald when he had one extra hair on his head. But consider the paradox 0.1. In it, there occurs no sentence predicating a vague predicate of one of its b o rd er­ line cases. Therefore, this attem pted solution will not elim inate 0.1. Fourth, there is a solution proposed by Black in his (1963), the exact content of which is unclear to me. It has been aptly criticized by Campbell (1974), and I therefore pass it by.

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2.

SUPERVALUATIONS

The theory of supervaluations has been proposed by several different authors in order to solve problems about vagueness, for instance by Fine (1975), Kamp (1975) and Przelecki (1969), (1976), and (1977). Of these theories, Fine’s is the one most relevant for sorites, and I therefore limit my discussion to his theory. Fine accepts classical logic but not classical m etalogic which contains Bivalence. He states three reasons for having classical logic for a vague language (ibid. p. 287). The first is based on intuitive considerations concerning precisifications, the second on its account of ‘penumbral connections’ and the third on its simplicity. I agree with Fine’s third reason but consider it insufficient to establish classical logic. I therefore proceed to consider, first, pre­ cisifications and, second, penumbral connections. Fine presents his theory in a model theoretic fram ew ork unnecessary for a grasp of its intuitive content which I here transform into a simpler dress. I use “ T ” for “ is true” , “ F ” for “ is false” and “ Pr” for “ is precise” . T he relational sentence “ A=s B ” expresses that B is at least as precise as A, i.e., that B is a precisification of A. I assume “ *£” to be reflexive and transitive. My connectives obey the rules of classical logic. First, we assume that no sentence or proposition is both true and false:4 2.1

- ( T ( A ) & F(A )).

We assume that we move within a fragm ent of English where a sentence is precise, if, and only if, it is either true or false: 2.2

Pr(A) = T(A) v F(A ).

Now, it seems obvious that a sentence which is at least as precise as A must be true (false) if A is. Surely, if you say something true (false) and are requested to m ake your sentence m ore precise, it would be absurd to present a sentence which is not true but either vague or false (not false but either vague or true). We thus accept: 2.3

A T (ß).

(cf. Fine’s discussion of ‘stability’ p. 268). There is another axiom with “ F ” taking the place of “ T ” in 2.3. There are still two assumptions, both of which are m ore controver­ sial. The first says that for any sentence, there exists a precise sentence

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which is at least as precise as the former: 2.4 2.5

least as least Drecise as Drecise as A: as A:

(cf. Fine’s ‘com pleteability’ p. 272). T he other asserts that if an atomic sentence, A, is not true, then there exists a false sentence which is at least as Drecise as A: 2.5

least as Drecise least as as Drecise A: as A:

(cf. Fine’s ‘resolution’ p. 279). T h ere is another axiom arising from 2.5 by an interchange of “ T ” and “ F ” . From these assumptions follows the supervaluation principle saying that a sentence is true if, and only if, each of its ‘perfect’ precisifications is true:

2.6

least as Drecise asleast A: as Drecise as A:

(In order for the proof to go through, A must here be restricted to atomic sentences. Probably, this restriction can be dropped.) T here is an analogous proof of a principle with “ F ” in place of “ T ” . This elegant theory has interesting consequences. If we define A to be vague if, and only if, A is not (perfectly) precise: 2.7

least least as Drecise as Drecise as A:as A:

we can prove that A is vague if, and only if, there are (at least) two precisifications of A, one of which is true and one of which is false:

2.8

least as Drecise as least A: as Dreciseleast as A: as Drecise as A:

This is a very appealing result. A proposition like “Tom is bald” may be vague and neither true nor false because we could, in principle, m ake it precise in two different ways, one of which would m ake it true, the other would m ake it false. W e could, so to speak, draw an imagined borderline to distinguish its truth from its falsity in various ways, some of which would m ake it true, others false. The right hand side of 2.8 is used by Halldén (1961) in his definition of “ heterogeneous unclarity” . Fine notes that his theory could be used to account for am biguity, too (ibid. p. 282). W e thus have a theory with a very broad field of application. Those who wish to distinguish between vagueness and am biguity would presumably want to restrict the vari­ ables of the theory to unam biguous entities.5 L et me now point out th at the theory hitherto contains nothing from which it follows that sentences of the form “ A or n o t:A ” are true. For,

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first, there is the limitation on the axiom 2.5. Second, nothing has so far been said to determ ine what is a precisification of what. Fine proposes the following principles of precisification: (Fl) . . . an expression is made more precise through making its simple terms more precise. This assumption is correct for the language L. For the logical constants are transparent, as it were, to vagueness; any precisification of a constituent shines through into the compound. Indeed, the converse of the assumption also holds; an expression is made more precise o n ly through making its simple terms more precise. For the logical constants are already perfectly precise. Since the logical constants are also the gram­ matical particles, all vagueness can be blamed onto constituents as opposed to con­ structions. (ibid . pp. 274-275)

From this theory of precisification, Fine extracts a premiss to the effect that each precisification of the sentence “ A or no t:A ” would be logically equivalent to a sentence of the form “ A' or n o t:A '” where A' is a precisification of A. Now the Law of Excluded Middle can be ‘proved’. The sentence “ A or not: A ” is true if, and only if, all of its perfect precisifications are true (2.6 above). E ach perfect precisification of it has the form “ A' or n o t:A '” where A is a perfect precisification of A. Therefore “ A ' or n o t:A '” is true. T herefore, any sentence of the form “ A or n o t:A ” is true. H ow ever, this proof is not demonstrative. It relies on the following assumptions: (1) T he theory of supervaluations - 2.6, (2) Fine’s theory of precisification in (F l) and (3) that large portions of classical logic hold for reasoning about precisifications. As for (1), 2.6 is a con­ sequence either from 2.1-2.5 or from 2.1, 2.2, 2.7, and 2.8. These are all intuitive beliefs about precisifications. However, it is possible that such intuitive considerations are inconsistent. Hence, what follows from them need not be true. As we shall see in Section 4, there are strong reasons to expect that intuitive considerations involving vagueness are inconsistent. As for (2), Fine’s theory expressed in (F l) holds for the logical constants. But it is quite another m atter if it holds for words like “ o r” , “ and” , “ there is” , etc. Consider the sentence form “ If not:(A and jB), then either not: A or n o t:B ” . I think it is essentially indeterm inate w hether that sentence would be counted as true for all choices of A and B. Similarly for “ If every A is a B, then some A is a B ” . This indicates that some of the English words “ and” , “or” , “ not” , and “ if then” might be vague. If so, some sentences containing them can be made more precise by replacing them with logical constants, if these are available.6 B ut in arguing for the Law of Excluded Middle, one cannot assume that

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the English words “ not” and “o r” have the content of the logical constants of, say, classical predicate logic. For exactly this is one of the issues at stake. T he assumption (3) ought to be highly controversial. For, as we have seen in our discussion of fuzzy logic, vagueness invades the m etalanguage. I conclude that Fine’s first argum ent for the Law of Excluded Middle is not very strong. I now turn to the argum ent from ‘penum bral connections’. Consider the sentence “ H erbert is bald and H erbert is not bald” . Surely we would say that this sentence is false even if it were found out that H erbert is a borderline case for “ bald” . Thus, some logical, or logic-like, relations hold between sentences which are vague and thus neither true nor false. Fine asserts that all sentences of the form “ A & - A ” are false, even if A is vague (ibid. p. 270). Furtherm ore, in his view, the sentence “The blob is red or the blob is pink” would be true, if said of a blob which is a borderline case both for “ red” and “ pink” . How ever, this argum ent is not strong. I agree with Fine that there are som e ‘penumbral connections’. But it is unclear how far they support classical logic. Fine’s example of the blob certainly does not support the Law of Excluded Middle at all, for it is not an instance of it. W ould “ T he blob is red or it is not red” be intuitively true? I doubt this. And even if it were, one example does not prove a general law, especially not if the law is so burdened with counterexamples as the Law of Excluded Middle. Certainly, when a man loses the hairs on his head, one by one, there will be some stage in this process at which it is very counter­ intuitive to claim that the following is true: “ Either he is bald or he is not bald” . Fine himself does ‘not wish to deny that LEM (i.e., the Law of Excluded Middle; B.R.) is counter-intuitive’ (ibid. p. 286). He admits one could ‘attem pt to construct a logic that was m ore faithful to unreform ed intuition’ (ibid.), but he thinks such an attem pt would, not yield as simple a logic as the classical one. I agree that Fine’s elegant theory is best seen as a proposal for the reform of, and not as a description of, ordinary English. But proposals for reform are m ore likely to take effect if based on knowledge about the state of the sinner. Until I say otherwise, I therefore wish to consider supervaluations as a theory describing facts about English. C onsider the conclusion of our sorites 0.1: “There is a num ber, n, such that Tom was bald when he had n hairs but not when he had n + 1 hairs” . According to supervaluations, this sentence would be true. It

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would be true on the ground that, for each perfect precisification of the predicate P = “Tom was bald when he had — hairs on his head” , there would be a number, n, such that the precisification of P would be true of n but false of its successor. Each precisification would be true of some n and false of its successor. But it is not the case that there would be some number n, such that each precisification would be true of n but false of n + 1. Different precisifications would draw the borderline at different numbers, as it were. In this way, Bivalence can be denied while classical logic is kept intact in the object language. This is not really a satisfactory description of the logic involved in sorites. First, there is the problem that the solution dem ands ascent to a metalanguage, where the distinction between Excluded M iddle and Bivalence is expressed and where it is explained why the conclusion of our sorites 0.1 is true. But sorites arise in the object language. I do not see why the metalinguistic discussion involved in the theory of super­ valuations would be relevant for sorites, stated in the object language. T he argum ent from change which I turned against Sanford in Section 1 applies here as well. The problem about accepting the conclusion of the sorites 0.1 is simply that it runs counter to our conviction that T om did not turn bald by losing one hair. But this conviction can be expressed in the object language. So why should the elaborated m etalinguistic theory be relevant here? Second, does supervaluation give a true description of the truth-conditions of the English phrase “ there is” ? I think not. We have some grasp of what would make sentences of the form “There is a P ” true. Now, if supervaluations were right about the truth-conditions for “ there is” , we would find it completely acceptable that there was an n ’th hair, the loss of which turned Tom bald or, m ore generally, that all changes are instantaneous. T he very fact that we find conclusions of sorites with the form of 0.1 unacceptable shows that “ there is” does not have the truth-conditions which supervaluations says it has. Observe that even if there were some examples of sentences of the form “T here is a P" which could be true without P being true of any object, this would not touch my second argum ent. All this would amount to prove is that the truth-conditions of a sentence of the form “ There is a P ” depend on which predicate takes the place of P . If such were the case, it would be a disaster to the methods of formal semantics. There would, so to speak, not be any contributions to truth-conditions which depend only on “ there is” . (Exactly the same points can be m ade

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for other words like “o r” , “ not” , “ and” etc.) I conclude that supervaluations is not a true descriptive theory of the logic underlying the sorites paradoxes in English. E very theory which accepts the conclusion of the sorites 0.1 will get itself into problem s concerning change. This holds, for instance, of O degard (1965) and Campbell (1974). I therefore proceed to consider another type of theory.

3 . W H E E L E R ’S A N A L Y S I S O F A N D S O L U T I O N T O S O R I T E S

U nger (1979) and W heeler (1975) and (1979) present radical solutions to the sorites. Predicates like “ table” , “ chair” , “ bald m an” which give rise to sorites are, in their view, true of nothing. From this they conclude that there are no tables, chairs or bald m en. In the case of our sorites paradox 0.1, such a solution would am ount to a denial that Tom was ever bald. In this section, I will first present and analyse W heeler’s analysis of one (or: the?) source of sorites, then his claim that ‘none of the ordinary “ middle-sized” objects of the “ given” world exist’ (1979 p. 155), and finally his reasons for the denial of m iddle-sized objects. Roughly, W heeler’s view is that the paradoxial flavour of the sorites derives from a certain kind of theories of reference which he defines as follows: (W l) I call any theory of reference which claims that the reference of a c oncept o r term is determ ined by ‘internal’ features of the concept, the language-user, o r a com m unity of language-users, a resemblance theory. A ccording to such a theory, w hat a concept or term means is a function of features of the speaker himself o r th e concept itself. Nothing beyond, e.g., patterns of response of the organism to stim ulation, o r social interactions between a language-user and his fellows, need be consulted in determ ining what, if anything, a given term refers to. Features of the concept guarantee certain features of its referent. (W heeler 1975 p. 367)

In his (1979), Section 1, W heeler claims that the following philosophers hold resem blance theories of reference: Frege, Russell, Quine, D avid­ son, the later W ittgenstein, Austin and Ryle. H e contrasts such theories of reference with another kind which he defines below: (W2) I call a causal theory of reference any theory which construes the reference of a term as som ething determ ined not by internal features of the concept, prim arily nor by patterns of individual or group behavior, but rath er by external, historical facts about what in the external world is the source of the occurrence of that term . T h a t is, reference is a real physical interaction between a term -for-a-person and some entity in the world.

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W hether that interaction has taken place is not in general determinable by scrutiny of either of the participants in isolation. (W heeler 1975 p. 368)

W heeler considers that K ripke, Putnam and Donnellan hold causal theories of reference. He elaborates on some of K ripke’s and Putnam ’s well known argum ents which he takes to show the falsity of resem ­ blance theories of reference. W heeler thinks that sorites have a bearing on these two kinds of theories of reference. Roughly, the causal theory is compatible with them and perm its a solution to them while the resem blance theory is incom patible with sorites. I now modify what I take to be W heeler’s argum ent from sorites against resem blance theories of reference. Let us consider the soritesparadox 0.1. First, we observe that its two premisses and the negation of its conclusion are a logically inconsistent triad. Second, most people would doubtlessly affirm that Tom was bald when he had 0 hairs on his head, that he was not bald when he had 100.000 hairs on his head and that the loss of any one hair did not change Tom from not being bald to being bald. Thus, most people would affirm the two premisses and affirm the negation of the conclusion. Thus, most people would affirm an inconsistent triad. Third, it is W heeler’s opinion that resemblance theories of reference entail that what most people would affirm is true. Therefore, resem blance theories would entail that there is an in­ consistent triad of sentences which is true. (Cf. W heeler 1975 pp. 368-369 and 1979 Section I.) A lthough I wish to be uncom m itted to W heeler’s assertions about the theories of reference, I heartily agree with the rest of this argum ent. W hat most people would hold to be true cannot reasonably be said to be true. I will develop this contention in a somewhat different direction in the next section.7 I now proceed to W heeler’s solution to the sorites and his reasons for it. W heeler wishes to solve the sorites by denying that vague predicates are true of anything: (W3) . . . the property-sorites argum ent should convince one that the only objects that exist are ones w ith a precise essence. Only precise essences can constitute the being of a genuine logical subject o r of real properties of logical subjects. A nd objects with precise essences seem to exclude persons, tables, chairs, etc. It seems very implausible that, at a certain point in the elim ination of the essential ‘property’ of such objects some drastic change should take place which m ade one of those objects an objectively distinct entity, w here what kind of thing it was changed. (W heeler 1979 p. 166)

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(W4) I begin with a pair of premises: (a) If a putative property is a real property, then it is a matter of fact whether an object has that property or lacks it. (b) Whether a purported object exists or not is a matter of fact. A purported object either exists or doesn’t ex ist___Since we are very sure that any precise borderline between having this purported property (of being a tall person; B.R.) and lacking it is absolutely arbitrary, it seems clear that there is no property of being a tall person. Since it is up to us, it is not a matter of any fact about the world. Since there is no property nothing has it. There are no tall persons, ( ib id . pp. 164-165)

I propose to reconstruct W heeler’s argum ent of (W4) as follows: We let “ P ” be a variable over predicates, i.e., linguistic entities, and “/ ” a variable over real properties. 3.1.1

D es(P , f) → Vx(f x v —fx) (If a predicate designates a real property, any object either has that property or lacks it.)

3.1.2

Blc(x, P ) → - E f (D es(P, f) & (fx v -fx)) (Ifx 3.1.2 isa borderline case for P, P designates no real property which x either has or lacks.)

3.1.3

T ru e (P , jc)→ Ef (D es(P, f ) & fx) (If P is true of x, P designates a real property which x has.)

From these assum ptions, it follows that if a predicate is vague - i.e., has some borderline case - it designates no real property and can therefore be true of nothing. T his reconstruction is the m ost forceful one I can extract from W heeler’s (W4). Let me com m ent on its premisses. First, 3.1.1 attempts to reconstruct W heeler’s (a). It is a version of the Law of Excluded M iddle and is superfluous in a logical system which contains the Law of Excluded M iddle for properties. W heeler, as far as I can see, gives no argum ent for it. H ow ever, it has been denied that all properties, or all property-like entities satisfy it. For instance, the resemblance-classes of Stephan K ö rn e r’s (1966) do not satisfy it. Som ething akin to 3.1.2 is assumed in the last paragraph of (W4), w here the non-existence of a certain property is proved from the vagueness of ‘it’. If P is a vague predicate, it cannot, so to speak, designate a ‘precise’ property. I have found no intelligible argum ent for 3.1.2 in W heeler’s writings. How ever, it is highly controversial. Such theories on vagueness as Russell’s (1923), H alldén’s (1949) and super­ valuations deny it. A ccording to this kind of theories, a vague predicate

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may, for instance, designate (any of) a collection of perfectly specific or precise properties. For instance, “ bald” may designate (any of) the specific properties of having 346 hairs, of having 345 hairs, of having 344 hairs etc. In order for “ bald” to be true of x, it would be necessary and sufficient that x has one of these properties. In order for “ bald” to be false of x, one of the properties of having 789 hairs, of having 790 hairs etc., would have to be instantiated by x. In short, this kind of theories assumes a one-m any relation between a vague predicate and the properties it designates. Such theories would affirm the antecedent of 3.1.2 and deny its consequent, for some values of “ jc ” . T h e premiss 3.1.3 is needed for the reconstruction. I believe it is denied by Sanford (1979 p. 182). O ne might say that vague properties are like secondary qualities - they do not inhere in things. If g is such a property, “gx” m ight be false or meaningless, when “ jc ” stands for a real entity. It may be left open whether the Law of Excluded M iddle holds for such properties. A nother of W heeler’s controversial assumptions about truth should be observed. T h e conclusion of his argum ent would be that vague predicates like “ tall person” are true of nothing. This is a semantic assertion, involving truth. He then infers that there are no tall persons. T h e latter assertion does not involve truth. But the fact is that a good m any authors on vagueness deny that the inference from —T(A ) to —A is valid. In short, the argum ent 3.1 is an abundance of controversial semantic assum ptions. It therefore lends very little force to the claim that there are no tables, chairs, tall persons etc. It remains to be seen w hether there is a better reconstruction of W heeler’s argum ent than 3.1. Sometimes W heeler speaks as if it followed (cf. “ compulsion” , in his 1979 p. 171) that there are no tables, chairs, or bald men. However, it cannot be so. T here are alternative solutions to the sorites than those which involve the denial of middle-sized things. Actually, W heeler does argue against some alternative solutions. Above, W heeler gives a sem antic argum ent for his solution to the sorites. Below, we shall see a nother argum ent which has to do with scientific laws. He indicates that the two argum ents are related, but I do not understand how. (W5) I think th at to be objectively real requires having an essence. F or an object to have an essence is for there to be objective necessities true of it, that is, natural laws. T h ere appear to be very good laws about micro-particles while the laws about medium­ sized objects are very poor, so full of ceteris paribus clauses as to be mere rules of thumb.

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Since there seems to be little hope of a reduction of medium-sized o bject kinds to complexes of m icro-particle kinds, there can either be two unrelated systems of objective kinds or the objects with the worst laws must g o . . . . I think the medium-sized objects must go. (W heeler 1979 p. 164)

W heeler indicates that this argum ent against middle-sized objects is an adaption of an argum ent from Davidson’s ‘M ental E vents’. I here give a rough outline of W heeler’s argum ent from (W5) and from his (1975) pp. 375-376: (1) Everything real belongs to some system in which natural laws hold. (2) There are no strict laws for middle-sized objects. (3) But there are strict laws for m icro-objects. (4) T herefore, laws for middle-sized objects cannot be reduced to laws for m icro­ objects. (5) If there were middle-sized objects, they would be causally related to m icro-objects. (6) Causal relations involve strict laws. (D avidson’s lemma). (7) Thus, if there were middle-sized objects, there would be strict laws about them. (8) But there are no strict laws for them and so there are no middle-sized objects. T he validity of this argum ent essentially depends on the notion of law, which W heeler leaves obscure. Would he recognize statistical laws? A ccording to some theorists, the ultimate laws of m icro-objects would only be statistical. Some would even want to deny that W heeler’s ontological principle (a) of (W4) holds true of m icro-objects. I also fail to see that laws of middle-sized objects are more burdened by ceteris paribus clauses than are laws of m icro-objects (cf. (W5)). A nother problem for this argum ent arises from the fact that obser­ vation sentences typically contain vague predicates. Consider, for instance, “ This solution turned red ” or “T he needle of this am m eter points at a m ark representing 1,12” . All positive ascriptions of obser­ vation predicates would, on W heeler’s view, be false. H e realizes this.8 But how can W heeler then distinguish between true and false can­ didates for the laws of our universe without relying on true observation sentences? He never turns to this question. W e have now inspected W heeler’s two argum ents for his solution to sorites. T he argum ents are interesting because of the issues they raise, but they lend no support to his solution. I find W heeler’s position on middle-sized objects completely unacceptable. The very capability of getting oriented in the world by means of sentences speaking of such objects presupposes a crucial distinction between ‘correct’ and ‘incor­ rec t’ which W heeler’s theory obliterates. For instance, if W heeler asks where his pen is, he may be correctly informed that it is on his desk or

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incorrectly informed that it is on the floor. But in W heeler’s view, the sentence “The pen is on the desk” would be just as false as “T he pen is on the floor” . Clearly, this is a serious incompleteness of his theory. Finally, the reader who has observed W heeler’s craze for an ontology containing only the m icro-particles of physics and their properties will feel surprised that W heeler writes a philosophical treatise, containing vague terms. According to his own theory, his vague predicates are true of nothing! Marvellous! First H um e’s Enquiry issued an im perative which would commit itself to the flames, then W ittgenstein, in speaking nonsense, informed us that this was what he was doing and now W heeler, by means of vague predicates, true of nothing, lets us know that his theory is true of nothing! These peaks of sophism leave my mind boggling. 4.

DOES T H E C O N C EPT OF T RU TH APPLY TO ENGLISH?

In the preceeding sections, we have discussed some examples of various kinds of solutions to the sorites 0.1. When faced with that sorites, it seemed we had but three alternatives - to say that one of its premisses was not true, that its conclusion was true or that the argum ent fails to transport truth from premisses to conclusion; i.e., the argum ent would be (semantically) invalid. But we now see that these three alternatives have a common presupposition - that the concept of truth applies to English, or, m ore precisely, that it applies to a part of English containing sorites paradoxes. It is the aim of this section to define the concept of an ‘analytically inconsistent’ language (or theory; I see no need to distinguish these notions here), to show that a com m on notion of truth does not apply to analytically inconsistent languages and, finally, to give some argum ents which make it plausible that English is analytically inconsistent. Thus, it is plausible that the concept of truth does not apply to English - or to that part of English which contains sorites paradoxes. I believe that theses, similar to mine, have been asserted by Dum m ett (1975) and W right (1975). I first turn to the concept of analytical inconsistency. Like so m any other logical concepts, it can be fairly well understood for a formal language while its application to a natural language such as English is much more precarious. At first, we restrict attention to formal lan­ guages, i.e., languages for which there is a publicly accessible m ethod for determining of any object whether it is a sentence, for any sentence

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whether it is a m em ber of a collection of ‘axioms’ or basic assertions and for any collection of sentences whether they constitute a proof. I do not assume that the sentences of formal languages are ‘meaningless m arks’; all that is im portant for my discussion is that there are no hidden assumptions. A formal language is said to be ‘inconsistent’ if a contradiction is provable in it; i.e., a sentence which conjoins a sentence A with the negation of A, e.g., “ A and n o t:A ” . G iven a formal language L, we assume that there is a ‘definitional part’ of L, D(L). D (L ) contains logical axioms and rules of inference of L, explicit and implicit definitions and m eaning postulates. Given L, it is a m atter of epistemology, not of pure logic, to delimit D(L). It should be noted that the assum ption of a set D(L) am ounts to a belief that it is meaningful to speak of analytical sentences of L. W hile I do admit that the notion of analytic sentence of English is a vague notion in the sense that there exist borderline cases for it, I do not share Q uine’s suspicions that the notion of analyticity is meaningless. Finally, a language L is said to be ‘analytically inconsistent’ if it is inconsistent and if some contradiction has a proof using only elem ents of D (L ). In an analytically inconsistent language, there is a contradiction provable only from the elem ents of D (L ); a contradiction ‘follows from ’ the nature of certain m eanings, so to speak. It was a basic insight of the H ilbert school that a system of m eanings, e.g., in the form of implicit definitions, are not sufficient for truth. F or those definitions m ay give rise to inconsistency. This insight is obscured by talk of sentences which are ‘true only by virtue of their m eanings’. If we assume that contradic­ tions are false and that they can follow from a system of m eaning, it follows that a system of m eanings cannot in itself guarantee truth. At least, it needs to be supported by a consistency proof. I do not mind speaking about sentences which are assemble only by virtue of their m eaning, however, and we may call them ‘analytic’ or ‘analytically assertible’. W hat I reject is the notion of analytically true as defined by ‘true by virtue of m eaning’. Let me exemplify the concept of an analytically inconsistent lan­ guage. Suppose R to be the language (theory) of real numbers and that its axioms and rules of inference all belong to D(R). W e then extend R to R + by adding the following definition:

4.1

4.14.14.1 4.14.1 4.1 4.1 4.14.1

4.1

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We assume D ( R +) to be had from D(R) by adding 4.1. We can now derive a contradiction from D ( R +) if we take c = d = 1 and first take a = 1; b = 2 and then a = 2; b = 4, observing that h = |. T he reason why analytically inconsistent languages are im portant comes forth here one cannot eliminate the contradiction in R + by denying the foul definition 4.1. For “ ?” has no other m eaning than that bestowed upon it by 4.1. To deny 4.1 would destroy the m eaning of “ ?” without giving it any new meaning. We shall later see that an analytically inconsistent language cannot be faithfully translated into a consistent language. But first, the notion of faithful translation needs to be defined. A ‘faithful translation’ of the language L x into L2 is a function, Tr, which maps each sentence of L, on one sentence of L2 in the following way. Tr maps the negation of A on the negation of Tr(A); Tr maps the conjunction of A with B on the conjunction of Tr(A) with Tr(B) and similarly for the other logical words. Furtherm ore, if A is a sentence in D (L ,), Tr(A) is in D (L 2), otherwise not. Similarly for rules in D ( L X). Thus, faithful translation preserves logical structure and translates analytic sentences of Lt with analytic sentences of L 2. Furtherm ore, when a sentence follows from another by logical rules of inference of L u its translation does so in L 2 . I believe these are minimal requirem ents on translation. Perhaps one also would want translation to m ap elem ents of L, on elem ents of L2 which have the same m eaning, but such refinement is not necessary for a grasp of my argum ent about the interrelations between analytically inconsistent languages and faithful translation. Suppose that L, is analytically inconsistent and that it can be faithfully translated into L2. T hen, there is a proof in L | of a sentence “ A and n o t:A ” and this proof uses only elem ents of D (L i). But then this proof can be faithfully translated into a proof in L2 which uses only elem ents of D(L2). Thus, La will be analytically inconsistent. Now suppose that L, is inconsistent but not analytically inconsistent. Then, each provable contradiction will have to be proved using elements outside D (L |). This means that there m ight be faithful translations of L into another language which does not assert the translations of those elements and which m ight be consistent. It does not follow that those languages into which L\ faithfully can be translated are inconsistent. It can now be seen that the traditional concept of truth does not apply to an analytically inconsistent language, say L ,. For this concept of truth is only captured in a language in which one can prove all

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sentences of the form: 4.2

X is true if, and only if p

where “ X ” holds the place for a name of a sentence and “ p” holds the place for the faithful translation of that sentence. But as L, is analytic­ ally inconsistent, there is a sentence “ A and n ot:A ” which can be proved by using only elem ents from D (L X). Then, in the metalanguage to L |, M LU the following will be provable: 4.3

“ A and not: A ” is true if, and only if (Tr(A) and not: Tr(A)).

4.4

Tr(A) and not: Tr(A).

Thus, MLi would assert that “ A and n ot:A ” is true, which is absurd. Let me exemplify in which way this result is relevant. Frege’s system call it “ F ” - is known to be inconsistent, mainly due to assumptions in connection with his axiom V. Suppose we count all other axioms to D (F ). If axiom V also belongs to D (F ), F is analytically inconsistent and there is no consistent m etalanguage into which F can be translated. In this case, it would be wrong to say that axiom V is false or that its negation is true. O ne could not deny that axiom without destroying the m eanings of som e term s it contains. There could be no metalanguage, expressing the m eaning of axiom V which declares it to be false, unless that m etalanguage is inconsistent in which case its assertions are uninteresting. B ut if axiom V does not belong to D(F) - if it, so to speak, is not a consequence of m eanings - one can reject the axiom while preserving its m eaning. In this case, there may be a metalanguage to F , expressing the same m eanings as F and which asserts the falsity of axiom V. In such a case, one could say that the axiom is false and that Frege was w rong about, say, objects; using “ object” in the same sense as the object language of F . Frege’s own reaction when faced with Russell’s paradox indicates that he did not himself think axiom V an unavoidable consequence of the meanings of certain terms. Consider now the sorites argum ent 0.1. I assert that the following theses are tenable: 4.5.1

T h e prem isses of the sorites 0.1 are assertible by virtue of their m eanings; they would belong to D(English).

4.5.2

T his also holds for the negation of the conclusion of 0.1.

94

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In English, the conclusion follows from the premisses and is assertible if the premisses are.

If these three theses hold, English would be analytically inconsistent in a way analogous to the way in which a formal language can be analytically inconsistent. But the analogy between English and formal languages is precarious mainly in two respects: the notion ‘assertible by virtue of m eaning’ and the notion ‘follow from in English’. H ow ever, it seems reasonable to assume that there is such a thing as ‘being assertible by virtue of meaning’. For instance, ordinary language philosophers often investigate ‘what we would be entitled to say’ if such and such were the case; e.g., we would be entitled to say that Jack has prom ised to m arry Jill if he has uttered the words “I promise to m arry Jill” in normal conditions. Now, this entitlement cannot be a carte blanche to utter the sentence “Jack promised to marry Jill” in any circum stances, for other obligations may override this permission - e.g., the obligation to stick to the subject m atter in a certain kind of conversation. R ather, the entitlement permits us to accept such and such a sentence, given that this or that is the case or that this or that has com e to our knowledge. The entitlement under consideration concerns nothing beyond facts or knowledge, and in this it resembles the notion of assertibility of a formal system. But, in daily life, we are entitled to say things if the circumstances are so and so or are known to be so and so. Form al languages disregard such relativization to nonlinguistic circum stances. But there are words such that if all circumstances are agreed on and two persons still apply the word differently, then this would indicate that the two persons connect different meanings to the word. For instance, if someone were to deny that Nixon is a man, either he does not know enough about Nixon or he means something else by “ m an” than the rest of us. The former alternative can be eliminated and the latter remains. In order to account for this possibility, the notion of D(English) would need to be relativized to situations in which sen­ tences are assertible by virtue of their meanings; i.e., situations in which a sentence cannot be rejected or doubted without destruction of its m eaning. It is fairly clear that the premisses of our sorites 0.1 would be assertible by virtue of their meaning, given that a situation of the kind indicated were the case. When Tom had 0 hairs on his head, we would be entitled to say that he was bald. Anyone who doubts or denies his baldness in a situation like that without being misinformed about T o m ’s

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growth of hair would rightly be taken to use “ bald” in another sense than we others do. Similarly for the use of “ not bald” of Tom when he had 100.000 hairs. So we turn to the negation of the conclusion of the sorites 0.1. Given that Tom loses his hair in the way described, we would be entitled to deny that there was a num ber of hairs, n, such that Tom turned bald on losing his n ’th hair; i.e., we would be entitled to deny that there is a num ber of hairs, n, such that he was bald when he had n hairs but not bald when he had n + 1 hairs. T he conclusion of 0.1 asserts a kind of change which clearly has not taken place; nam ely that Tom changed from not being bald to being bald in losing one hair. It is, I think, part of the meaning of “ change” that T o m ’s gradual transition into baldness does, at the loss of no single hair, count as a change from not being bald to being bald. A person who asserts “ Now he w ent bald” at T om ’s loss of his n’th hair certainly uses language in a peculiar sense. I therefore conclude that there is a valid analogy betw een properties of English and properties of formal languages which m akes it reasonable to claim the truth of 4.5.1 and 4.5.2. We now turn to the problem whether the sorites argum ent 0.1 is ‘valid in English’. When I claim that this argum ent is valid in English, I do not mean to imply that a m ajority of English speakers would affirm its conclusion if they have affirmed its premisses. For the argum ent might be valid but the speakers confused - the validity of argum ents does not consist in a disposition to verbal behaviour. F urtherm ore, I do not mean just that the conclusion of 0.1 is ‘deducible’ from its premisses. It seems to me that ‘deducible’ must be relativized to some systematization or axiomatization. In my view, the conclusion of 0.1 would be deducible in any correct axiom atization of English, for “ correct” here means that the axiomatization m akes the conclusion of any valid argument deducible from its premisses. T he notion ‘valid argum ent of English’ must be used in the definition of ‘adequate axiomatization of English’ rather than conversely. (This point is missed by those authors who try to block the deduction of the conclusion of a sorites from its premisses. They can do so only with respect to a given systematization or axiomatization of English. E ven if the restric­ tions they propose work and can be justified for one axiom atization, they might fail for others. For instance, one has tried to block the deduction of sorites by demanding that all sentences in a legitim ate deduction must be either true or false. This restriction might w ork in Hilbert style axiomatizations, but it is much too strong for natural

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deduction systematizations. It would, for instance, rule out such reduc­ tio proofs where the non-existence of a is proved by showing that both Fa and not: Fa imply a contradiction. (Cf. C an to r’s theorem ).) C on­ sequently, we must not argue for the validity of 0.1 by proving it to be valid in some systematization. T he validity of 0.1 follows from rather m odest assumptions about logical form. First, 0.1 has a logical form which we may represent as below: 4.6.1

F(0).

4.6.2

not: F ( 100.000).

4.6.3

T here is a num ber, n, such that F (n ) and n o t :F(n + 1 ).

Second, there are argum ents which are valid by virtue of this logical form, e.g., the argum ents we obtain by taking F(x) as: 4.7

4.7

4.7 4.7 4.74.74.74.7

Third, if an argum ent is valid by virtue of a certain logical form, all argum ents having that logical form are valid. Hence, 0.1 is valid. My argum ent here is a specimen of a com m on type: one accepts that two given argum ents do not differ in any logically relevant respect; they have ‘the same logical form ’. From this one concludes that if one argum ent is valid, the other is. T he validity of the argum ent of the structure 4.6 with F taken as 4.7 is inescapable and so 0.1 would be valid as well. My argum ent here presupposes that vagueness alone is no ‘logically relevant respect’ in which two argum ents can differ. Now, it is common to distinguish between extensional and intensional vagueness. A predicate which actually has borderline cases is extensionally vague. A predicate is intentsionally vague if it is logically possible that it should have borderline cases. If substitution of an intensionally vague predi­ cate for a precise predicate could change the logical form(s) of an argum ent, almost all applications of logic to non-logical and nonm athematical subject m atters would go by board due to the intensional vagueness of almost all predicates outside these disciplines. This would be an unacceptable consequence. But if, instead, substitution of an extensionally vague predicate for a precise one could change the logical form(s) of an argum ent, logical validity would depend on what the

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world is like in m uch the same way as it would if only argum ents with true premisses could be logically valid. An argum ent containing the premiss 4.8

T om is bald

would be valid if, say, Tom is bald, valid if Tom is not bald and otherwise invalid. A theory with such consequences does not seem quite correct. So I think we m ay assume that vagueness - both extensional and intensional - is irrelevant to logical validity in English. (This is not to deny that logical form and logical validity might depend on contingent m atters of fact - cf. the sentence: 4.9

W hat B. R olf says on February 19th, 1982 is false.

Perhaps the logical validity of argum ents containing this sentence depends on who said it and when. B ut 4.9 seems very different from 4.8, e.g., with respect to the peculiar possibility of self-reference.) It has been seen that English probably is analytically inconsistent. This hypothesis has interesting consequences. First, it explains why sorites argum ents seem valid, that their premisses seem true and that their conclusions seem false. For a sorites like 0.1 is valid in English and its premisses are assertible and so is the negation of its conclusion. Furtherm ore, the hypothesis explains why these paradoxes are so hard to ‘solve’. For the principles of English m ake us steer into inconsistency and this cannot be avoided except by rejecting parts of the conceptual system of English. T h e sorites cannot, so to speak, be solved within English - one has to abandon parts of English in order to rem ove the inconsistencies. I now turn to som e objections to the hypothesis that English is inconsistent. O ne objection would say that, in an inconsistent language, a contradiction would be assertible. But there is a clear sense in which a contradiction like “ It is raining and it is not raining” is not assertible in English. This objection can, how ever, be avoided by a distinction betw een explicit and implicit assertibility. An implicitly assertible sentence ‘follows from ’ explicitly assertible sentences. In English, no contradiction would be explicitly assertible. But this does not imply that English contains no implicitly assertible contradictions. A nother objection derives from T arski’s remarks: (T l) At first blush it would seem that this language (i.e., everyday language; B.R.) satisfies both assumptions (I) and (II), and that therefore it must be inconsistent. But

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actually the case is not so simple. Our everyday language is certainly not one with an exactly specified structure. We do not know precisely which expressions are sentences, and we know even to a smaller degree which sentences are to be taken as assertible. Thus the problem of consistency has no exact meaning with respect to this language. We may at best only risk the guess that a language whose structure has been exactly specified and which resembles our everyday language as closely as possible would be inconsistent. (Tarski 1944 p. 21)

From a premiss that the structure of everyday language is not specified and that we do not know which objects are sentences and which sentences are assertible, Tarski leaps to a conclusion that there is no ‘exact m eaning’ connected with the claim that English is inconsistent. B ut this conclusion does not follow. For it is enough that we know of one object that it is an assertible sentence of the form “ A and not: A ” in order for us to know that English is inconsistent. We can know this without having specified all of English and all of its assertible sentences. Furtherm ore, w hether English is inconsistent ought to depend on what can truly be specified and what can be known rath er than on what has been specified and on what is known; it is not the case that our specifications create the logical properties of English. But I think that T arski’s guess points in a sound direction - the inconsistency of English probably follows also from the assumptions about truth present in English. T he thesis that English is inconsistent might be unfashionable during the present idolatry of com m on sense and it will be a disgrace to those philistines who think that the others always know best. But the thesis is not new and it ought not be neglected. In his criticism of the ontological proof of G od, Leibniz expresses the view that the use of logical principles on certain concepts m ight lead to contradictory conclusions. A nd K ant thought that, by its very nature, the hum an mind is prone to fall into contradictions and needs to be protected by critical philosophy. T he theory th at our com m on sense concepts lead to contradictory conclusions was a m ajor elem ent in the philosophy of Adolf Phalén. He saw the derivation of contradictory conclusions from these concepts as a m ajor driving force in the developm ent of philosophy and he tried to support this view by a careful study of the history of philosophy. A m ajor task for philosophy would be, according to him, to cure common sense thinking from its incoherence. It m ight seem that if ordinary language were inconsistent, the m an in the street would fall into contradiction m ore often than he does. But I doubt that it would be so.

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For, first, m ost people do not reason very much and they do not put their conceptual systems to severe tests. On the other hand, philoso­ phers do, and it is not surprising that they should manage to uncover the inconsistencies. Second, there is no reason to believe that, in arguing, m en blindly follow the rules of their language. Even though our language seems to be inconsistent, we might be equipped so as to avoid falling into inconsistency. For instance, C antor’s and Frege’s concep­ tual fram eworks were inconsistent and this was undiscovered for quite a long tim e in spite of the fact that these frameworks were severely tested. Both logicians were capable of rational manoeuvring within fram e­ works which, due to their inconsistency, were incapable of distinguish­ ing rational from irrational. T he inconsistency of a language or of a conceptual fram ew ork does not disable its users for rational discourse and thinking.

5.

CONCLUSION

I have here attem pted to give a description of the logical situation confronting us in sorites. I have not proposed any medicine against the inconsistency of English. T he inconsistency is not very devastating for our actual linguistic practice but, at most, for our theoretical descrip­ tions of that practice. T he solution to sorites we have scrutinized in Sections 1-3 can all be seen as attem pts to form ulate consistent m etalanguages in which classical logic holds. Fuzzy logic defines a logic for the object language which is w eaker than the one it uses in the m etalanguage. T he logic it proposes for the o bject language is also weaker than the one actually there. Fuzzy logic thus appears as a call for a reform by which not even the reformists want to live. As a theory of vagueness or as a proposal for the reform of a language containing vagueness, fuzzy logic is unacceptable. This detracts nothing from the value it might have in the many other fields where it is applied, however. Fine sees his own theory of supervaluations as, in part, a reform of ordinary language. I think it is best seen as an attem pt to change the m eaning which “ there is” and other logical words have in English. In particular, it seems that supervaluations tries to sever the connection betw een the m eaning of the logical words and the m eaning of “ change” . In affirming the conclusion of our sorites 0.1, super-

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valuations affirms a sentence which is normally taken to say that Tom changed from not being bald to being bald in losing one hair. Furtherm ore, it would propose to eliminate tolerant predicates from the m etalanguage. It is debatable whether this is possible.16 W heeler’s theory can be seen as a proposal for a metatheory which m arks all positive ascriptions of vague predicates as false, while a good many sentences about swirling molecules and continuous dimensions would come out true. T he main problem here is that we would not know which m etasystem to choose unless we could rely on vague observation sentences. A final word on self-applicability. The analytic sentences of the language I have used are, in some sense, inconsistent. I do not commit myself to all of them, but only to those I have actually produced. My thesis of the inconsistency of English does not entail its own in­ consistency.

NOTES

* Previous versions of this paper have been much improved by criticism from Professors Sören Halldén, Bengt Hansson and the anonymous referee of Synthese. The paper draws on ch. VI of my Ph.D. thesis. 1 Wright’s theses on tolerance are discussed in the chapter ‘On Vagueness’, Section 11, of my (1981). 2 The Disjunctive Syllogism permits us to infer B from “A or B” and not:A. 3 The ideas of fuzzy logic seem to stem from Black (1937). It has been proposed by Zadeh, for instance in his (1975), by Goguen (1968), by Machina (1976), and by King (1979). 4 The variables “A”, “B”, and “ C” are to range over sentences or propositions for which the axioms hold. 5 I argue for such a distinction in ‘On Vagueness’, Section 2. 6 In my (1980), which is reprinted in my (1981), I show that the logical constants are perfectly precise. In ‘On Vagueness’, Section 10, this matter is further discussed. 7 Wheeler (1975 p. 368) expresses awareness of alternatives like the one I formulate in Section 4. 8 In his (1973), which Professor Wheeler has kindly made available to me, he expresses knowledge of this.

BIBLIOGRAPHY

Black, M.: 1937, ‘Vagueness: An Exercise in Logical Analysis', Philosophy of Science 4.

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Reprinted in M. Black, Language and Philosophy, Cornell University Press, New York, 1949. Black, M.: 1963, ‘Reasoning with Loose Concepts’, Dialogue. Campbell, R.: 1974, ‘The Sorites Paradox’, Philosophical Studies 26. Cargile, J.: 1969, ‘The Sorites Paradox’, British Journal for the Philosophy of Science 20. Davidson, D.: 1970, ‘Mental Events’ in L. Foster and J. Swanson (eds.) Experience and Theory, University of Massachusetts Press. Dummett, M.: 1975, ‘Wang’s Paradox’, Synthese 30. Fine, K.: 1975, ‘Vagueness, Truth and Logic’, Synthese 30. Goguen, J. A.: 1968-1969, The Logic of Inexact Concepts’, Synthese 19. Halldén, S.: 1949, The Logic of Nonsense, Uppsala Universitets Ârsskrift. Halldén, S.: 1961, Universum, döden och den logiska analysen, Gleerup, Lund. Kamp, J. A. W.: 1975, ‘Two Theories about Adjectives’, in E.L. Keenan (ed.) Formal Semantics of Natural Language, Cambridge University Press, Cambridge. King, J. L.: 1979, ‘Bivalence and the Sorites Paradox’, American Philosophical Quarterly 16. Körner, S.: 1966, Experience and Theory, Routledge and Kegan Paul, London. Machina, K. F.: 1976, ‘Truth, Belief and Vagueness’, Journal of Philosophical Logic 5. Morgan, C. G. and F. J. Pelletier: 1977, ‘Some Notes Concerning Fuzzy Logics’, Linguistics and Philosophy 1. Odegard, D.: 1965, ‘Excluding the Middle from Loose Concepts’, Theoria 31. Phalén, A.: 1931, ‘Our Common Notions and their Dialectic Movements in the History of Philosophy’, Proceedings of the 1th International Congress of Philosophy, Oxford. Przelecki, M.: 1969, The Logic of Empirical Theories, Routledge and Kegan Paul, London. Przelecki, M.: 1976, ‘Fuzziness as Multiplicity’, Erkenntnis 10. Przelecki, M.: 1977, ‘The Concept of Truth in Empirical Languages’, Grazer Philoso­ phische Studien 3. Rolf, B.: 1980, ‘A Theory of Vagueness’, Journal of Philosophical Logic 9. Rolf, B.: 1981, Topics on Vagueness, Fil.Dr. Thesis, Lund. Russell, B.: 1923, ‘Vagueness’, Australasian Journal of Philosophy and Psychology 1. Russell, B.: 1903, The Principles of Mathematics. Sanford, D. H.: 1975, ‘Infinity and Vagueness’, The Philosophical Review 84. Sanford, D. H.: 1976, ‘Competing Semantics of Vagueness: Many Values versus Super-Truth’, Synthese 33. Sanford, D. H.: 1979, ‘Nostalgia for the Ordinary: Comments on Papers by Unger and Wheeler’, Synthese 41. Tarski, A.: 1944, ‘The Semantic Conception of Truth’, Philosophy and Phenomenological Research 4. Reprinted in L. Linsky (ed.) Semantics and the Philosophy of Language, University of Illionois Press, Champaign, 1952. Page references to the latter. Unger, P.: 1979, ‘There are no Ordinary Things’, Synthese 41. Wheeler, S. C.: 1973, ‘A Solution to Wang’s Paradox’, mimeo, University of Connecticut. Wheeler, S. C.: 1975, ‘Reference and Vagueness’, Synthese 30. Wheeler, S. C.: 1979, ‘On That Which is Not’, Synthese 41. Weiss, S. E.: 1976, ‘The Sorites Fallacy: What Difference does a Peanut Make?’, Synthese 33.

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Wright, C.: 1975, ‘On the Coherence of Vague Predicates’, Synthese 30. Zadeh, L. A.: 1975, ‘Fuzzy Logic and Approximate Reasoning’, Synthese 30. Filosofiska Institutionen Lunds Universitet Kungshuset, Lundagârd 5-22350 Lund Sweden

[5] AN ARGUMENT FOR THE VAGUENESS OF ‘VAGUE’

B y R o y A. S o r e n s e n

ALTHOUGH some commentators on vagueness have claimed ^tV vague’ is vague, none has presented a clear, intuitive argument for the thesis. I think such an argument can be provided with the help of a certain sequence of disjunctive predicates. After describing these predicates and presenting the argument, I will show how the vagueness of ‘vague’ creates difficulties for those who believe vague predicates are incoherent and those who wish to deal with the problems of vagueness by a demarcation between vague and nonvague predicates. Here is one version of the sorites paradox: (1) (2) So (3)

0 is a small number. If n is a small number, then n 4- 1 is a small number. One billion is a small number.

As lar as 1 know, all commentators agree that the defectiveness of this mathematical induction is somehow due to the vagueness of ‘small’. Although there is considerable disagreement over the nature of the defectiveness and the exact nature of vagueness, there is general agreement that predicates which possess borderline cases

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are vague predicates. Commentators provide conflicting definitions of ‘borderline case’. Some define it behaviourally, some epistemically, still others define borderline cases in terms of truth value gaps. My use of the notion of borderline case is intended to be compatible with all of the proposed definitions. Now consider the sequence of numerical predicates ‘1-small’, ‘2-small’, ‘3-small’ etc. The n-th predicate on the list is defined as applying to only those integers that are either small or less than n. These predicates can be used to construct a sorites paradox for the predicate ‘vague’. (1) Í2) So (3)

‘1-smair is vague. If ‘n-smalP is vague, then ‘n ‘One-billion-smair is vague.

4-

1-small’ is vague.

The predicate ‘1-small’ is just as vague as ‘small’ because both predicates clearly apply to 0 and apply precisely in the same way to all the other integers. The same holds for ‘2-small’ and ‘3-smair, but eventually we reach predicates where the ‘less than n ’ clause eliminates some borderline cases. Once we reach predicates in which all borderline cases are eliminated, we can tell that ‘vague’ no longer applies. For instance, it is a clear cut issue whether the predicate ‘one-billion-small’ applies to a given integer or not: since no integer greater than a billion is small, this predicate is as definite as the predicate ‘less than one billion’. But it is unclear as to where along the sequence the predicates with borderline cases end and the ones without borderline cases begin. In short, ‘vague’ is vague. The vagueness of ‘vague’ suggests that accounts of vagueness which describe vague predicates as incoherent are necessarily false. Such an account can be true only if the terms it uses to describe vague predicates are coherent. Since they use ‘vague’ in their thesis that all vague predicates are incoherent, their account can be true only if their account uses an incoherent term. Bertil Rolf, W. V. Quine, and Michael Dummett are recent examples of philosophers who are sympathetic to the incoherence thesis.1 The most straightforward acceptance of the incoherence thesis appears in the work of Peter Unger and Samuel Wheeler.2 Unger and Wheeler believe that the sorites paradox shows that vague predicates are incoherent. They point out that the predicates 1 See Rolf’s chapter on the sorites in his Topics in Vagueness for his rejection of the consistency of vague predicates. Quine’s ‘acceptance of the paradox and the consequent inconsistency of the pristine use of ostensively acquired terms’ is indicated in the sixth and seventh paragraphs of ‘What Price Bivalence?’ Journal o f Philosophy vol. LXVIII no. 2, February 1981, pp. 90-5. Dummett’s conclusion that a wide class of observational predicates are incoherent appears on p. 319 of ‘Wang’s Paradox’, Synthese vol. 30 no. 3/4, April/May 1975, pp. 301-24. 2 Unger and Wheeler have presented their views in a number of articles. Two representa­ tive articles are Unger’s ‘There Are No Ordinary Things’ and Wheeler’s ‘On That Which Is N ot’, both appearing in Synthese vol. 41 no. 2, June 1979.

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we have for ordinary things are vague predicates. Since incoherent predicates lack extensions, they draw the sceptical conclusion that ordinary things (chairs, flowers, people etc.) do not exist. If Unger and Wheeler were to concede that ‘vague’ is vague, they would be forced to concede that ‘vague’ lacks an extension. But then they would have to conclude that nothing is vague. This conflicts with their assertion that there are vague predicates, viz. our predicates for ordinary things. In other words, the following set of proposi­ tions are jointly inconsistent. (i) There are vague predicates. (ii) Vague predicates are incoherent.

(iii) Incoherent predicates lack extensions. (iv) ‘Vague’ is a vague predicate. Indeed, since (iv) implies (i), the subset {(ii), (iii), (iv)} is also a set of jointly inconsistent propositions. Other commentators have complained that proponents of the thesis ‘All vague predicates are incoherent’ have used vague predi­ cates in their arguments supporting the incoherence thesis. But it might be replied that the use of those vague predicates was an inessential use. However, proponents of the incoherency thesis must make essential use of a vague predicate since ‘vague’ or an equivalent term must appear in the statement of their thesis. For suppose they restate their thesis with another predicate such as ‘nonprecise’ or ‘borderline-case-inducing predicate’. The substitution will have to be synonymous with ‘vague’ in order to express the same thesis. Since all synonyms of vague predicates are themselves vague, the thesis will remain a self-refuting thesis. The vagueness of ‘vague’ also poses a problem for those who hope to avoid the difficulties of vagueness through a demarcation between vague and nonvague predicates. All complements of vague predicates are themselves vague. Hence, ‘nonvague’ is vague. There­ fore there is no way to determine the exact extension of ‘nonvague predicate’. Consider the Frege/Russell principle that logic only applies to nonvague predicates. Does logic apply to this principle? Since it contains a vague predicate, one is tempted to deduce from the principle that logic does not apply to the principle. But the deduction is possible only if logic applies to the principle. This self-referential difficulty is a symptom of the fact the principle radically restricts the scope of logic. Moreover, acceptance of the principle would leave the scope of logic indeterminate. For the vagueness of ‘vague’ ensures the impossibility of exactly determin­ ing which predicates are suited for logical evaluation. Matters are not helped by further restricting logic to clearly nonvague predicates because ‘clearly nonvague’ is vague. For in addition to borderline cases, vague predicates have borderline borderline cases and thus borderline clear cases. Thus the vagueness of ‘vague’ presents a dilemma for those who wish to restrict logic to nonvague terms.

105

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The statement of the restriction must contain a vague term. If logic applies to the statement, the statement is incorrect. If logic docs not apply to the statement, then the ‘restriction’ is without force; for it has no implications as to what is ruled in or ruled out. Since a restriction must rule something out, the ‘restriction’ would not be a genuine restriction. In general, it appears that the vagueness of ‘vague’ ensures that the worrisome features of vague predicates will also be features of our attempts to describe those predicates. Thus dissatisfaction with vagueness seems to commit us to dissatisfaction with out attempts to express that dissatisfaction.3 U niversity o f Delaware, N ew a rk , Delaware 1 9 7 1 6 , U.S.A.

©

R oy A. Sorensen

1985

3 In writing this paper, I have benefited from the encouragement and remarks of Peter Unger as well as from several important corrections and helpful suggestions from an anonymous referee.

Part II Observational Predicates

[6] CRISPIN W RIG HT

O N T H E C O H E R E N C E O F V A G U E P R E D IC A T E S *

I. I N T R O D U C T I O N

Frege came to believe that a language containing vague predicates was essentially defective - that it was philosophically intolerable that predicates should occur for which it was not always determinate whether or not they could truly be ascribed to an object. Expressions of this conception are scattered throughout his writings. When, more seldom, he argues for it, it is on the ground that logical transformations may fail when applied to sentences containing expressions whose range of application has been only partially defined. It is not just a matter of the Law of Excluded Middle. Let F be a predicate defined only among, and universal­ ly applying to, individuals which are G. Then anything G is F. But the contrapositive fails : we cannot say that anything not-F is not-G, since the concept of having or lacking F has been fixed only for things which are G.1 Frege does not seem to have spared a thought for the idea that vague terms might require a special logic. The vagueness of ordinary language is seen rather as a flaw both needful and capable of remedy. This con­ ception was endorsed by Russell2 in his widely-despised intioduction to the Tractatus. Ordinary language is always more or less vague; a logically perfect language, however, is not vague at all, and ordinary language is deemed to approach fulfilment of its function - that of having meaning only in proportion as it approaches logical perfection, i.e. in inverse proportion to its vagueness. Of course, we have long since abandoned the Frege-Russell view of the matter. We no longer see the vagueness of ordinary language as a defect. But we retain a second-order wraith of the Frege-Russell view in the notion that, even if the senses of many expressions in natural language are not exact, there is a precise semantical description for a given natural language, i.e., a theoretical model of the information assimilated in learning a first language or, equivalently, of the conceptual Synthese 30 (1975) 325-365. All Rights Reserved Copyright © 1975 by D. Reidel Publishing Company, Dordrecht‘Holland

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equipment in whose possession mastery of the language may be held to consist. Even if ‘bald’, say, is imprecise, this does not require any inexactitude in an account of its sense. We suppose our use of language to be fundamentally regular; we picture the learning of language as the acquisition of grasp of a set of rules for the combination and application of expressions. The task of a philosophical theory of meaning is naturally interpreted as that of giving a systematic account of the contribution that constituents of a complex expression make to its content; we are concerned, that is to say, when we attempt such a theory, with the nature of the path from familiarity with the senses of the sub-sentential components of a new sentence to recognition of the sense of the whole. Such a theory will normally only concern itself with describing the contribution of a constituent qua expression of a certain logical type; it is in this connection that problems to do, e.g., with the nature of the distinction between proper names and other singular terms, or that between singular terms generally and pre­ dicative expressions, or with the question whether the notion of reference may properly be extended to apply to predicative expressions, derive their interest. The completion of such a theory would thus only be a preliminary to a full semantic description of a natural language; for we think not just of the type of contribution but of the specific contribution which a constituent makes as determined by semantic rules. It is no obstacle to such a conception that we cannot in general in­ formatively state such rules, i.e., provide a statement which could be used to explain the sense of an expression to someone previously un­ familiar with it. Consider, for example, the following schematic rule for a one-place predicate, F :

‘F may truly be applied to an individual, ay if and only if a satisfies the condition of being 0’. How should we specify 0 if F is ‘red’? Plainly the only such condition which actually captures our understanding of ‘red’, rather than, in a merely extensional way, its conditions of application, is ... the condition of being red. So we may not in general expect instances of the schematic rule to be of any explanatory use if they are stated in a given language for a predicate of the same language; as a corollary, it will not in general be possible to appeal to such a rule to settle a dispute about the ap-

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plicability of an expression. It remains open to us, nevertheless, to regard such a rule as exactly encapsulating (part of) what is understood by someone who understands, e.g. ‘red’, for it states conditions recognition of whose actualisation is sufficient to justify him in describing an object as ‘red’; it is merely that such a capacity of recognition cannot be be­ stowed by stating the rule. Our picture, then, is that to use language correctly is essentially nothing other than to use it in conformity with a set of instructions, of semantic rules. Of course our handling of language is in general quite automatic, but so is a chess player’s recognition of the moves allowable for a piece in a certain position; it remains true that an account of his knowledge is to be given by reference to the rules of chess. The question now arrises, what means are legitimate in the attempt to discover features of the substantial rules for expressions in our language, the rules which determine specifically the senses of such expressions? The view of the matter with which we are centrally concerned in this paper is that we may legitimately approach our use of language from within, that is, reflectively as self-conscious masters of it, rather than externally, equipped only with behavioural notions. We may appeal to our con­ ception of what justifies the application of a particular expression; we may appeal to our conception of what we should count as an adequate explanation of the sense of a particular expression; to the limitations imposed by our senses and memories on the kind of instruction which we can actually implement; and to the kind of consequence which we attach to the application of a given predicate, to what we conceive as the point of the classification which the predicate effects. The notion that forms the primary concern of this paper - henceforward referred to as the governing view - is that we can derive from such considerations a re­ flective awareness of how expressions in our language are understood, and so of the nature of the rules which determine their correct use. The governing view, then, is a conjunction of two theses: that our use of language is properly seen, like a game, as an activity in which the allowability of a move is determined by rule, and that properties of the rules may be discovered by means of the sorts of consideration just described. The governing view does not involve a psychologistic con­ ception of understanding, according to which understanding would be regarded as an essentially mental state of which the correct employment

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of an expression was a mere behavioural symptom; but it provides means in excess of the behaviourist minimum, = the description of when an expression is actually used, for an attempt to investigate the nature of the semantic rules operative with respect to a given class of expressions. It may be that, as so far characterised, the view will seem platitudinous; the purpose of this paper is to question its coherence. A striking feature of the Philosophical Investigations is the hostility displayed by the author of the notion of a language-game to the idea that it is explanatory of our use of language to appeal to the concept of rulefollowing. It is not, of course, that there cannot be a resoluble dispute about the correct use of a particular expression. Rather Wittgenstein seems to argue for an indeterminacy in the identity of the rules which someone supposedly follows. Here it is irrelevant whether we can supply, e.g., what seems an informative completion of the above schematic rule for a particular predicate, F. The question is, what content is there to the claim that such a rule faithfully incorporates someone’s understanding of FI For he and we may sincerely agree on a particular formulation and then, sincerely and irresolubly, apply F in mutually inconsistent ways. It is useless to protest that all that follows is that he uses F in accordance with a different rule. We agreed on how the rule was to be characterised; now, it seems, we reserve the right to offer some other characterisation of what governs his use of the expression. (Though there is no necessity that any such characterisation should occur to us.) From his point of view, however, the initial characterisation remains perfectly adequate; it is of our use of F that a re-characterisation of the determining rule is required. But, now, what objectivity is there in the idea of the correct characterisation of his or our respective rules? And, if there is none, how can we penetrate, as it were, to the real nature of the rule which we follow, if we wish to explain our use of FI Thus the term­ inology of semantic rules, adequate perhaps for certain purposes as a picture, fails to express a domain of objective fact from which our use of language may be seen to flow. This line of argument poses a difficulty for the first thesis of the governing view. It is mentioned as a familiar type of criticism of the notion of a semantic rule, and by way of contrast with the character of the difficulty for the governing view to be posed in this paper. Here it is contended rather that the second thesis, concerning the means whereby

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features may be discovered of the semantic rules which we actually follow, constrains us to recognise semantic incoherence in our under­ standing of a whole class of predicates - elements whose full exploitation would force the application of these expressions to situations where we should otherwise regard them as not applying. The second thesis requires the recognition of rules which, when considered in conjunction with certain general features of the situations among which their associated expressions are to be applied, issue in contradictory instructions. Never­ theless we succeed in using these expressions informatively, and to use language informatively depends on using it, in large measure, consistently. It follows that our use of these expressions cannot correctly be pictured purely as the implementation of the rules which the second thesis yields for them - these rules cannot be obeyed by consistent behaviour. The governing view is therefore incoherent; for if its second thesis is true, the semantic rules which are operative with respect to certain predicates are capable by consistent beings only of selective implementation and thus, contrary to the first thesis, are not constitutive of what we count as the correct use of these expressions. Predicates of the relevant kind are all examples of a certain sort of vagueness : not exactly borderline-case vagueness, if that is understood, as Frege sometimes describes it, as the existence of situations to which it is indeterminate whether or not a predicate applies, but something which, under the guise of a favourite metaphor, he constantly runs together with the possession of borderline-cases, viz., the idea of lacking ‘sharp boundaries’, of dividing logical space as a blurred shadow divides the background on which it is reflected. The conflation is plausible be­ cause the image equally exemplifies the idea of the borderline-case, a region falling neither in light nor in shadow. But there seems no reason why having borderline-cases should imply blurred boundaries. Borrowing another of Frege’s analogies, we may assimilate a predicate to a,function taking objects as arguments and yielding the True or the False as values; in these terms a predicate with borderline-cases is simply a partial such function, and that is consistent with the obtaining of a perfectly sharp distinction between cases for which it is defined and cases for which it is not. Borderline-case vagueness simpliciter presents no difficulty for the governing view; it is merely that we are presented with situations to which no response is determined by the semantic rules of our language

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as the correct one. On the other hand, if the second thesis of the governing view is correct, then predicates with ‘blurred boundaries’ are, in typical cases, to be regarded as semantically incoherent. This incoherence resides in their vagueness as such. It is plausible to suppose that the vagueness of many expressions is not, as Frege and Russell thought, merely a reflection of our intellectual laziness. Rather, the utility and point of the classifications expressed by many vague pre­ dicates would be frustrated if we supplied them with sharp boundaries. (If it is an empirical truth that stress diseases are more widespread in highly concentrated populations, it is doubtful whether it would survive an exact numerical definition of ‘highly concentrated’.) The sorts of consideration admitted by the second thesis will transpire, in the suc­ ceeding sections of the paper, to yield support for the idea that such predicates are essentially vague. The thesis equips us to argue that lack of sharp boundaries is not just a surface phenomenon reflecting a hiatus in some underlying set of semantic rules. Lack of sharp boundaries is not the reflection of an omission; it is a product of the kind of task to which an expression is put, the kind of consequences which we attach to its application or, more deeply, the continuity of a world which we wish to describe in purely observational terms. Lack of sharp boundaries is semantically a deep phenomenon. It is not generally a matter simply of lacking an instruction where to draw the line; rather the instructions we already have determine that the line is not to be drawn. This conclusion might seem a welcome contribution to our understanding of the nature of vagueness, were it not that it is supplied in the form not merely that no sharp distinction may be drawn between cases where such a predicate applies and cases where it does not, but that no such distinction may be drawn between cases where it is definitely correct to apply the predicate and cases of any other sort. Thus it is that an adherent of the governing view simply has no coherent approach to the Frege-Russell view of vagueness. His second thesis requires him to reject the suggestion that vagueness is a superficial, eliminable aspect of natural language with no real impact upon its in­ formative use. But it does so by means of considerations which require him to regard many vague predicates as semantically incoherent, so that, unless the Frege-Russell view is right, he cannot maintain his first thesis with respect to such expressions. Only if their vagueness is an incidental

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feature can he maintain that the essential semantics of such expressions conform to his first thesis. The programme for the remaining part of the paper is as follows. In the next section, three examples will be presented of predicates to which the second thesis attributes semantic incoherence; specifically, it sustains for each of them the reasoning of the Sorites paradox. The character of the incoherence will be generalised, and arguments afforded by the second thesis for such an account of the semantics of these ex­ pressions will be developed. In Section III a fourth and deeper rooted example will be presented. In Section IV we shall consider an obvious strategy for solving paradoxes in the Sorites group which, if allowed, would undercut the considerations of Sections II and III. This strategy will be rejected. In Section V it will be argued that one class of predicate to which the second thesis attributes semantic incoherence is, in a certain sense, ineliminable. It will be shown that a seemingly promising adapta­ tion of certain of Goodman’s 3 ideas fails to provide an adequate re­ fashioning of the semantics of these predicates; and that a simpler suggestion, while indeed liberating them from semantic incoherence, does so at the cost of generating other predicates with the same feature. In the final section our conclusions will be drawn together. If the governing view is unacceptable, that is something which it is as well to know. The interest of the paper, however - if any - derives equally from that of the issues which we shall have occasion to discuss passim : the nature of vagueness and the correct logic for vague expressions; ostensive definition and observational language; the empirical sources of the concept of continuity, and the notion of order within phenomenal continua like the spectrum of colours. ii

Example 1 Our first example is the Megarian paradox itself.4 If we begin with a pile of salt large enough to be fairly described as a heap, the subtraction of a single grain of salt cannot make a relevant difference; if n + \ grains of salt may constitute a heap, so may n such grains. Of course, the plausi­ bility of this supposition derives from the species of sense with which ‘heap’ has been endowed; we have not fixed exact boundaries for the concept of a heap, either in terms of the precise number of grains con-

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tained or, indeed, in terms of any other precise measure. To allow that at some stage the subtraction of a single grain might transform a heap into a non-heap would be exactly to anticipate the determination of such a specific boundary. However, in the present semantic situation of ‘heap’ it would be merely an error to insist that, for some particular value of n, n + 1 grains of salt would amount to a heap while n grains would not; for that is simply not the sense of ‘heap’. If it were agreed in some particular case that n + 1 grains did amount to a heap, no-one could produce a telling reason for withholding the predicate from the same pile minus one grain; except perhaps avoidance of the incoherence implicit in the situation. ‘Heap’, then, would appear to be semantically incoherent, for its sense is such that it essentially lacks exact extensional boundaries; and its lack of such boundaries demands particularly that a transition from n + 1 grains to n grains can never be recognised to transform a case where ‘heap’ applies to a case where it does not. So we are seemingly equipped to force the application of ‘heap’ through successively smaller aggrega­ tions of salt-grains, terminating in items which amount not to heaps but merely, say, to pinches. Here we gravitate towards the idea that lack of exact boundaries is, as such, an essentially incoherent semantic feature. This idea, however, will need qualification.

Example 2 Predicates of degree of human maturity - ‘infant’, ‘child’, ‘adolescent’, ‘adult’, display the same peculiarity. They are mutually inconsistent yet lack sharp boundaries with respect to their neighbours in the scale of human maturation. More exactly, if we take some sufficiently small interval of time and suppose that someone matures in a typical fashion, then at no stage will he effect within such an interval of time a transition from one stage of maturity to the next. To illustrate the point with an example from Esenin-Volpin:5 take as the relevant interval the span of time from one heartbeat to its succes­ sor; then the concept of childhood - the sense of ‘child’ - is such that one does not, within a single hearbeat, pass from childhood to adolescence. To be sure, one is not a child forever; but at least childhood does not evaporate between one pulse and the next. If one’s wth heartbeat takes place in childhood, then so does the n+ 1th6.

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‘Infant’, ‘child’, ‘adolescent’, ‘adult’, are thus all semantically in­ coherent expressions; for the sense of each of these predicates is such that, in a typical process of growing-up, their correct application will always survive the transition from one heartbeat to its successor or to its pre­ decessor. So again, by appropriately many steps of modus ponens, we may force the application of each of these predicates to cases we should otherwise regard as falling within the domain of a competitor.

Example 3 Consider a series of homogeneously coloured patches, ranging from a first, red patch to a final, orange one, such that each patch is just dis­ crim inate in colour from those immediately adjacent to it, and is more similar to its immediate neighbours than to any other patches in the series. Marginal, uni-directional changes of shade are thus involved in every transition from a patch to its successor. Now, it is notable that the sense of colour predicates is such that their application always survives a very small change in shade. Given that one is content to call something ‘red’, one will still be so content if its colour changes by some just discriminable amount. There is a notion of a degree of change in respect of colour too small to amount to a change of colour. Only if a substantial difference intervenes between two patches shall we consider ourselves justified in ascribing to them incompatible colour predicates. Obviously this is an incoherent notion. In particular in view of the proximity in shade of neighbouring patches in our series, it provides an easy proof that all the patches are red, (or orange, or doubtfully either). Moreover any two colours can be linked by such a series of samples; so any colour predicate can likewise be exported into the domain of ap­ plication of one of its rivals. Colour predicates as a class are semantically incoherent. In these examples we encounter the feature of a certain tolerance in the concepts respectively involved, a notion of a degree of change too small to make any difference, as it were. There are degrees of change in point of size, maturity and colour which are insufficient to alter the justice with which some specific predicate of size, maturity or colour is applied. This is quite palpably an incoherent feature since, granted that any case to which such a predicate applies may be linked by a series of ‘sufficiently

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small’ changes with a case where it does not, it is inconsistent with the exclusivity of the predicate. More exactly, let 0 be a concept related to a predicate, F, in the follow­ ing way: that any case to which F applies may be transformed into a case where it does not apply simply by sufficient change in respect of 0; colour, for example, is such a concept for ‘red’, size for ‘heap’, degree of maturity for ‘child’, number of hairs for ‘bald’, degree of complexity for ‘memorable’ as applied to patterns, and so on. Then F is tolerant with respect to 0 if there is also some positive degree of change in respect of 0 insufficient ever to affect the justice with which F is applied to a particular case. These wholly intuitive ideas are sufficiently clear for present purposes. At this point it has to be conceded that the manner in which these examples have been presented has been wholly tendentious. It is not to be doubted that the predicates in question do lack sharp boundaries; and the antiquity of the paradox bears witness to how easy it is to interpret this as involving the possession by these predicates of a principle of reapplication through marginal change. But is this a correct interpretation? If ‘heap’, for example, lacks sharp boundaries, then certainly we are not equipped to single out any particular transition from n to n —l grains of salt as being the decisive step in changing a heap into a non-heap; no one such step is decisive. But that is not to say that such a step always preserves application of the predicate. Should we not instead picture the situation as comparable to that in which neighbouring states fail to agree upon a common frontier? Their failure to reach agreement does not vindicate the notion that e.g. a single pace in the direction of the other country always keeps one in the original country. For they have at least agreed that there is to be a border, that some such step is to be decisive; what they have not agreed is where. The analogue of this conception for the predicates which interest us is exactly that their vagueness is purely a reflection of our intellectual laziness. We have, as it were, decided that a disjunction is to be true - at some stage, n grains will be a heap whereas n —1 grains will not - without following up with a decision about which disjunct is true. The notion that these predicates are tolerant confuses a lack of instruction to count it the case that a proposition is false with the presence of an instruction to count it true. This conflation would be permissible only if our set of semantic rules were in a certain sense complete, that is, if it contained instructions

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for every conceivable situation. But for there to be vague expressions in our language is precisely for this not to be so. If we avail ourselves of the types of consideration afforded by the governing view, we shall reject this suggestion as a deep misapprehension of the nature of this species of vagueness. The lack of sharp boundaries possessed by these examples is correctly interpreted as tolerance, provided that we may so discover elements of their senses. The point is not that there is any decisive formal obstacle to providing general stipulations by means of which we might identify in such examples a last point at which the predicate in question could definitely correctly be applied. We could set upper and lower limits for ‘heap’ in terms of number of grains.7 We could do the same for ‘child’, etc., by setting precise age limits to the successive phases of life - perhaps in terms of numbers of heartbeats. For colour predicates it would prima facie be less easy to provide such a refinement, since the notion of being of the same shade encounters difficulties which those of being of the same age, or containing the same number of grains, do not. But suppose it is possible. Then what in the semantics of these examples is already inconsistent with our so refining their senses? ‘Heap’ is essentially a coarse predicate, whose application is a matter of rough and ready judgement. We should have no use for a precisely demarcated analogue in the contexts in which the word is typically used. It would for example be absurd to force the question of the execution of the command, ‘Pour out a heap of sand here’, to turn on a count of the grains. Our conception of the conditions which justify calling something a heap of sand is such that the justice of the description will be unaffected by any change which cannot be detected by casual observation. A different argument is available for colour predicates. We learn our basic8 colour vocabulary ostensively, that is, by exposure to samples illustrative of its application. Evidently it is a precondition of our capacity to do so that we can reasonably accurately remember how things look. We can imagine people who can recognise which simultaneously pre­ sented objects match in colour, and so are able to use a colour-chart, but who cannot in general remember shades of colour sufficiently well to be able to employ without a chart a colour vocabulary anything like as rich as ours. Such people, for example, having been shown something yellow, might later be quite unable to judge whether what they were earlier

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shown would match the orange sample now before them. Thus, for such a community, an ostensive definition of ‘yellow’ would not be feasible, and, in order to make the distinctions of our basic colour vocabulary, they would require to employ charts. Of course, we also use charts for some purposes; but never to make distinctions of the magnitude of those, say, between the colours of the rainbow. Any object to which one of these predicates definitely correctly applies may be recognised as such just on the basis of our ostensive training. But if this is so, it has to be a feature of the senses thereby bestowed upon these predicates that changes too slight for us to re­ member - that is, a change such that exposure to an object both before the change is undergone and afterwards leaves one uncertain whether the object has changed, because one cannot remember sufficiently ac­ curately how it was before - never transform a case to which such a predicate applies into one where such is not definitely correctly the right description. The character of the basic colour training which we receive, and which we hand on to our children, presupposes the total memorability of the distinctions expressed by our basic colour predicates; only if single, unmemorable changes of shade never affect the justice of a par­ ticular, basic colour description, can the senses of these predicates be explained entirely by methods reliant upon our capacity to remember how things look. With respect to ‘child’, etc., the governing view allows us a third type of consideration. Broadly, we do not make distinctions in phases of maturity in the fashion of a naturalist, just to record a variety with which we are confronted; rather, these classifications are of substantial social importance in terms of what we may appropriately expect from, and of, persons who exemplify them. Thus infants have rights but not duties, whereas of a child outside infancy we demand at least a rudi­ mentary moral sense; we explain the anti-social behaviour of some adolescents in terms of their being adolescents; and we make moral and other demands of character on adults which we would not impose on the immature. It is plausible that the predicates of Example 2 could not endure this treatment, were they not tolerant with respect to marginal changes in respect of degree of maturity. It would be irrational and unfair to base substantial distinctions of right and duty on marginal - or even non-

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existent9 - such differences; if we are forced to do so, for example with electoral qualifications, it is with a sense of artificiality and absurdity. There is another point: without tolerance these predicates could no longer sustain the explanatory role which they now have for us. Only if a substantial change is involved in the transition from childhood to adolescence can we appeal to this transition to explain substantial alterations in patterns of behaviour; if some adolescents differ barely, if at all, from some children, so that no significant change need be under­ gone in making the transition, to have done so can explain nothing. That predicates of degree of maturity possess tolerance is a direct con­ sequence of their social role; very small differences cannot be permitted to generate doubt about their application without correspondingly coming to be associated with a burden of moral and explanatory distinctions which they are too slight to convey. Our embarrassment about where to ‘draw the line’ with these examples is thus a reflection not of any hiatus in our semantic ‘programme’ but of the tolerance of the predicates in question. If casual observation alone is to determine whether some predicate applies, then items not distin­ guished by casual observation must receive the same verdict.10 So single changes too slight to be detected by casual observation cannot be permitted to generate doubt about the application of such a predicate. Likewise, if the conditions under which a predicate applies are to be generally memorable, it cannot be unseated by single changes too slight to be remembered. Finally, very slight changes cannot be permitted to generate doubt about the application of predicates of maturity without con­ travening their moral and explanatory role. The utility of ‘heap’, the memorability of the conditions under which something is red, the point of ‘child’ thus appear to impose upon the semantics of these predicates tolerance with respect to marginal change in their various relevant respects. N ot that, on the governing view, those considerations provide a whole account of their vagueness. For example, the considerations applied to ‘heap’ presumably apply to ‘red’ and, in some measure, to ‘child’ also. However, it is clear that to allow just the foregoing sketchy considerations is to concede both that the vagueness of our examples is a phenomenon of semantic depth - that is, it is sacrificed at much more than the cost of the intellectual labour of the stipulation - and that it is a structurally incoherent feature. Two things

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follow. First, there is no special logic for predicates of this sort, crystallising what is distinctive in their semantics in contrast with those of exact predicates; for what so distinguishes them is their inconsistency. A ‘logic’ of this species of vagueness is chimerical. Second, the manner in which we typically use these expressions needs some other model than the simple following of rules, if these are to incorporate all the features of their senses which we should wish to recognise. It is perhaps more nearly comparable to the behaviour of a public-spirited citizen in relation to fiscal law; for, as is familiar, the overall effectiveness of a system of taxat­ ion may well depend both upon the presence of loopholes and on people’s forbearance to exploit them. in Anyone who holds the second thesis of the governing view should re­ cognise a distinct and more profound source of tolerance in adjectives of colour than the inability of our memories to match the sharpness of discrimination possessed by our senses. Colour predicates, it is plausible to suppose, are in the following sense purely observational: if one can tell at all what colour something is, one can tell just by looking at it. The look of an object decides its colour, as the feel of an object decides its texture or the sound of a note its pitch. The information of one or more senses is decisive of the applicability of an observational concept; so a distinction exemplified in a pair of sensorily-equivalent items cannot be expressed by means solely of predicates of observation, for any ob­ servational expression applying to either item must apply to both. What is about to be illustrated with respect to colour-predicates is, under appropriate assumptions, a feature of any expression whose sense is observational in the fashion just sketched. Since colour predicates are observational, any pair of objects in­ distinguishable in point of colour must satisfy the condition that any basic colour predicate applicable to either is applicable to both. It is, however, familiar that we may construct a series of suitable, homo­ geneously coloured patches, in such a way as to give the impression of a smooth transition from red to orange, where each patch is indiscriminable in colour from those immediately next to it; it is the non-transitivity of indiscriminability which generates this possibility. So, since precise matching is to be sufficient for sameness of colour, we can force the

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application of ‘red’ to all the patches in the series, some of which are not red but orange. That is: since ‘red’ is observational, its sense must be such that from the premises, that x is red and that x looks just like y , it follows that y is red, no matter what objects x and y may be. This rule enables us to conclude that each successive patch in our series is red, given only the true premise that the first patch is red. The purpose of Example 3 was to illustrate, from the standpoint of the second thesis, a tolerance of predicates of colour with respect to marginal changes of shade. If we retain such an account for this new example - Example 4 - we shall be forced to regard identity of shade as a non-observational notion. We shall be admitting that changes in shade take place between adjacent patches where none seem to have taken place, where the most minute mutual comparison reveals no difference. But it is clear that we shall be driven to some such admission even if, rather than construe it as non-observational, we abandon the notion of identity of shade altogether. For some sort of non-observational changes are taking place in this example, however we choose to describe them. Example 4, if allowed, reveals colour predicates as tolerant with respect to changes which cannot be directly discerned in objects which undergo them; an object may suffer such a change without it being possible to discover that it has done so save by comparing it with some­ thing else. This feature differentiates this example from the others. Moreover we do not have ready to hand a concept in terms of which we can describe what these changes essentially are. N ot that we could not offer an account in terms e.g. of the physics of light; but it is contingent that the changes in question are associated with any particular physical changes which we might independently discover. On the other hand it is seemingly not a full account of the matter just to say that a patch now matches something which it did not match before, while not itself seeming to have changed in the meantime; this is unsatisfactory as a full account because we are inclined to say that, when objects come into a relation which they did not share before, one or both of them must have changed in independently specifiable respects; whereas, in this case, neither patch may seem to have changed. We are lacking, for example, a notion stand­ ing to the concept of shade as that of real position stands to phenomenal position. Had we such a concept, we could account for the changes in question in real terms.

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This lack would be a prima facie obstacle to stipulating away examples of this fourth type; for the stipulations envisaged in the other three cases each made use of a concept - number of grains, number of heartbeats, phenomenal shade - in terms of which we could describe the small changes involved. Later we shall consider how this obstacle might be overcome. For the present it is more significant that the difficulty arises at all. Had we a notion of shade comparable to that of real position, we might provide such a stipulation; but had we such a notion of shade, we should already have surrendered its observationality, and to provide the stipula­ tion would be to surrender that of colour as well. To stipulate away the tolerance of a predicate is to provide a general explanation of where, in a series of the relevant sort, it may be applied correctly for the last time. But if we did so with respect to ‘red’, ‘orange’ and the type of series illustrated in Example 4, we should have to forego our entire present conception, viz. the look of a thing, of what justifies the application of these predicates. To say that a predicate is observational requires that the conditions under which it may be applied to an object must be de­ terminable simply by observation of it; whereas we are contemplating a situation where ‘red’ may definitely correctly be applied to only one of a pair of cases which are observationally exactly similar - whose looks match exactly. Clearly, then, any observational predicate must display tolerance in a series in which it is initially but not everywhere exemplified and whose every member cannot be observationally distinguished (save numerically) from those immediately adjacent to it. Naturally, no series can satisfy these conditions unless the relevant relation of indistinguishability behaves non-transitively. These considerations are broadly analogous to what was said of the Heap : if we so fix the sense of a predicate that whether it applies has to do with nothing other than how an object seems when casually observed, then changes other than can be determined by casual observation of it cannot transform a case to which the predicate applies into one where it does not, or to which its application is somehow doubtful. The point remains good if we omit the word, ‘casual’. But Example 4 is seemingly deeper-reaching, at any rate for someone who, in the spirit of Frege, required that language should be purified of vague expressions. The cost of eliminating predicates of casual observation is convenience; but the cost of eliminating the use of expressions tolerant in the manner of

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Example 4 would be the abandonment of predicates of strictly observa­ tional sense. Might there not then be a higher price to pay, namely, the jeopardising of contact between language and empirical reality? We shall not pursue this thought immediately. First we need a deeper perspective for Example 4. We require an explanation of the observationality of certain predicates - or, what in this context comes to the same thing, some reinforcement of the supposition that their semantics are pure­ ly observational - and we require to know under what circumstances we may expect our sensory discriminations to be non-transitive. The inter­ section of these explanations will yield a general indication of the range of the type of case which Example 4 illustrates. That we do intuitively regard the semantics of adjectives of colour as purely observational is beyond doubt; and simply illustrated by the fact that we should regard it as a criterion of lack of understanding of such an adjective if someone was doubtful whether both of a pair of objects which he could not tell apart should receive the same description in terms of it. We regard it as a criterion of understanding such a predicate that someone, presented under suitable conditions with an object to which it applies, can tell that it does so just on the basis of the object’s appearance. But can an explanation be provided of why there should be predicates with such a semantics? It might be supposed that any ostensibly definable predicate must be observational. If an expression can be ostensively defined, it must be possible to draw to someone’s attention those features in his ex­ perience which warrant its application; and if this is possible there can be no question of the expression applying to some but not others among situations which he cannot distinguish experientially. It would be a poor joke on the recipient of an ostensive definition if the defined expression applied selectively among situations indistinguishable from one which was originally displayed to him as a paradigm. Unless it is disallowed that aspects of the semantics of an expression can be discovered by appeal to such considerations, we are bound to regard this suggestion as basically correct. Accordingly, we can no longer combine the conventional idea of the place which ostensively defined predicates occupy in our ‘conceptual scheme’ - the base of the epistemic pyramid - with the view that strict semantic coherence is a necessary condition of intelligibility. Rather, we have to recognise that such

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predicates are endowed by their very mode of introduction with a kind of original sin - a species-liability to tolerance. We are not yet, however, in a position to draw this conclusion. Certain­ ly an ostensive definition must be regarded as issuing a license, so to speak, to apply an expression to any situation relevantly matching that which the definition uses; an ostensive definition must do this or it is of no use. But that is not to issue a license to apply the expression to a situation which does not match the original situation, but merely matches an intermediate situation, indistinguishable from both. If we think of an ostensive definition as a command: ‘Apply F to situations like this’, the command obviously does not apply to situations which are not ‘like this’ but merely like something like this. This is so whether the required resemblance is conceived as indistinguishability or as some less exact likeness. What, then, is the connection between an expression’s being observa­ tional - its applying to both, if to either, of any pair of observationally indistinguishable situations - and its being ostensively definable? It is as follows. The picture of acquiring concepts by experience of various cases where they do apply and various cases where they do not - a picture which surely has some part to play in a philosophically adequate concep­ tion of the learning of a first language - cannot be wholly adequate for concepts which differentiate among situations which look, feel, taste, sound and smell exactly alike. So if that picture is wholly adequate for any concepts, they must be concepts whose range of application does not include situations which experience cannot distinguish from situations which may not definitely correctly be regarded as falling within that range. To master the sense of a predicate is, at least, to learn to differentiate cases to which it is right to apply it from cases of any other sort. If such mastery can be bestowed ostensively, a comparison of two such cases must always reveal a difference which sense-experience can detect. The notion, then, that the whole range of application of a predicate can be made intelligible by ostensive means, presupposes that it is never the case that only one of a pair of objects, which the senses cannot tell apart, is characterised by it. It is tempting to suggest that the trouble ultimately resides in our whole conception of what an ostensive definition tries to achieve. The point of an ostensive definition of ‘red’ is to communicate the concept

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of a certain look, a look which is to be understood as invariably justifying calling something which has it, ‘red’. Looks, however, like sounds and smells are phenomenal, so that they are imbued with the same instability which was noted earlier in the idea of phenomenal position. If it is sufficient to share a look that things seem exactly similar, then the non­ transitivity of indiscriminability provides a way of proving that everything has the same look. If it is not so sufficient, then the nature of a particular look cannot be revealed by mere display of something which has it, for something could appear absolutely similar yet not have the look; the ‘look’ of something is no longer a purely phenomenal notion, so it cannot be communicated by ostensive procedures, for all that they can capitalise upon is how things appear. One cannot, for example, give an ostensive definition of a real length. Be that as it may, there is a clear, general connection between observationality and ostensive definition. If there is in the conditions of correct application of a predicate nothing which is incapable of ostensive com­ munication, then the predicate must apply to both, if to either, of any pair of indistinguishable objects. But it seems manifest that adjectives of colour, and many others, do precisely not involve any such further condition of correct application; on the contrary, ostensive training would appear fully determinant of their meaning - or, if it is not, it is the only training which we get. Let us move, then, to the second question: under what circumstances can we expect our sensory discriminations to be non-transitive? It is natural to view it as a consequence of the coarseness of our perceptions that series of colour samples can be constructed in which we can directly discern differences in hue only between non-adjacent members. For we do admit in many cases, e.g. the concept of spatial position, the idea of a change too small to be directly perceived, par excellence too small to be perceived without special apparatus. This admission entails that indiscriminability will behave non-transitively in suitable circumstances, since small, imperceptible differences may add up to a larger, noticeable one. But it might not seem that this explanation is the one which we seek; for it presupposes the admission of a species of objectivity which cannot be exemplified by observational predicates. The observationality of colour words requires that looking the same colour is sufficient for the same

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colour predicate to apply: whereas the proposed explanation, applied in the case of spatial position, essentially distinguishes between when things seem to be in the same position and sufficient conditions for their being truly described as being so. A change of position need not be noticeable ; a change of colour must be. Naturally we may be driven to abandon this very aspect of the semantics of colour predicates as a result of the tension between observationality and non-transitive indiscrimina­ bility; but until we do so, it cannot be an explanation of the non-transitive indiscriminability of suitable colour samples that differences of colour exist too fine for our gross perceptions. This however is still not quite the point. For plainly, as we noted, the non-transitivity of matching requires that not every feature of colour patches can be a directly observational one; colour patches evidently allow of changes, whether these changes are described as changes in colour or not, of a kind which cannot be directly discerned. So we have no alternative but to admit a gap among such items between seeming not to have changed in any respect, and actually not having done so. We have to admit the described objectivity, if not in the notion of change in colour, at least in the notion of change in some respect. What is really wrong with the explanation is not the presupposition of objectivity, but the circumstance that the sole ground for affirming that there is a distinc­ tion between seeming and being here is that indiscriminability is behaving non-transitively; whereas we are trying independently to circumscribe the circumstances under which we may expect such behaviour. So we require an alternative account. Suppose that we are to construct a series of colour patches, ranging from red through to orange, among which indiscriminability is to behave transitively. We are given a supply of appropriate patches from which to make selections, an initial red patch Cu and the instruction that each successive patch must either match its predecessor or be more like it than is any other patch not matching it which we later use. Under these con­ ditions it is plain that we cannot generate any change in colour by selecting successive matching patches; since indiscriminability is to be transitive, it will follow if each Ct in the first n selections matches its predecessor, that Cn matches Ct . The only way to generate a change in colour will be to select a non-matching patch. When 344 344 C„> is complete, how will it look in comparison with

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the series of Example 4? It is clear that we shall have lost what is distinctive of that series : the appearance of continuous change from red to orange. In the new series the shades are exemplified in discrete bands (con­ taining perhaps no more than one patch) and all the changes take place abruptly in a transition from a patch to its successor. So it appears that, were our judgements of indiscriminability to be universally transitive among samples of homogeneous colour, no field of colour patches could be ordered in the distinctive fashion now possible; that is, so as to give the impression of a perfectly smooth change of hue. If matching generally behaved transitively among shades, no series of colour patches could give the impression of continuous transformation of colour; contrapositively, then, for matching to function non-transitively among a finite set of colour patches, it is sufficient that they may be arranged so as to strike us as forming a phenomenal continuum. This reasoning may obviously be generalised. Any finite series of objects, none of which involves any apparent change in respect of 0, may give an overall impression of continuous change in respect of 0 only if indiscriminability functions non-transitively among its members. Evidently, though, not all processes of seemingly-continuous change come, as it were, ready made out of finitely many stages. Seeminglycontinuous processes in time do not generally do so ; and nor do certain purely spatial seemingly-continuous changes, e.g. the convergence of a pair of near-parallel lines. The question now arises whether there is not a more general connection between seemingly-continuous change and non-transitive indiscriminability than that illustrated in the somewhat artificial case of the colour-patches. Let us consider the case of processes of change in time. Let D be such a process, and let a stage of D be the state of D at a particular point in time, an instantaneous exposure, as it were, of the process at that point. D is to be non-recurrent in the following sense: if 0 is the respect in which changes in D take place, no distinct stages, x, y, z in D are to be such that z is in respect of 0 more like x than y is when x < y < z ; (‘< ’ = is earlier than). It is analytic that any process of change is non-recurrent up to some stage. We claim that, on a very natural presupposition, D will be seemingly continuous only if there is some finite selection of stages of D among which indiscriminability behaves non-transitively. Suppose on the contrary that while D itself is seemingly-continuous,

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there is no way of selecting stages from D , however close together, so that indiscriminability behaves non-transitively among them. Consider a maximal set, S , of stages of D yielded by the following rule of selection: x is a member of S if and only if x is discrim inate in respect of 0 from every y previously selected as a member. (Let the first selection be regarded as satisfying this condition vacuously.) Since S is maximal, every stage in D must be indiscriminate in respect of 0 from some member of S. Suppose now that S is finite and consider the series, , of all its members in order of temporal succession. Let Di9 Di+i> be a pair of stages adjacent in this series. Plainly any stage occurring in D later than Di$ but earlier than D /+1 must be indiscrim inate in respect of 0 from either Di or Di+l but not both; it cannot match both, since matching is hypothesised to be behaving transitively; it cannot match neither, or it must match some other stage in !,...,/)„>, so violating the hypothesis that D is non-recurrent. Clearly if such an in-between stage matches, say, Z>„ then all stages lying temporally between it and Dt in D must likewise match Di9 or D will not in this region be non-recurrent; mutatis mutandis if it matches Z>i+1. So the region of D between 344 and 344 Di+ must divide into two contiguous segments, every stage in one of which will match D while every stage in the other matches Di+1. Evidently, then, D cannot present an impression of continuity of change as it moves through this region, contrary to hypothesis. To obstruct this reasoning it will not be sufficient to hypothesise that S is infinite; we need specifically that is densely ordered by temporal succession, that between any pair of stages of D discriminable in respect of 0 lies a stage discriminable from them both and from any stage outside the region of D which they flank. We have to suppose that we have in this sense infinite powers of discrimination in D , that we can always directly discern some distinction more minute than any discerned so far. The ‘very natural presupposition’ earlier referred to is that this is not so. To summarise our conclusions here: if D is a non-recurrent temporal process of change such that indiscriminability behaves transitively among every selection of stages from it, and if we can directly discern only a finite variety of stages of D - at least in some of its regions - then D must contain seemingly absolutely abrupt changes. Hence if D is everywhere

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to give an impression of continuous change, indiscriminability cannot behave transitively among every selection of stages from it; specifically, if Dt and D i+1 are adjacent in 344344 Z)w>, derived as above, at least one stage occurring between Dt and Di+i in D must match both. These considerations are of course incomplete. We have not considered purely spatial processes of change, though one might venture to expect that exactly analogous reasoning would apply; and we have not considered whether there is not a sufficiency condition for seeming-continuity of change in terms of non-transitive indiscriminability. But enough has been done to give this phenomenon a certain dignity. It is not something con­ fined to the psychological laboratory, comparable say, to peoples inability to judge the relative lengths of: and - something of which someone might reasonably require experimental confirmation that he too was subject to it. Granting the presuppositions of the above reasoning, we have rather to regard the non-transitivity of the relation, ‘is not discriminably different from’, as a reflection of a pervasive structural feature of our sense experience - the continuity of change. The general lesson then of Example 4 is this. If we attempt to mark off regions of a seemingly-continuous process of change in terms of predicates which are purely observational; - predicates of which it is understood that ostensive definition gives their whole meaning - these expressions are bound to display tolerance in a suitable series of stages selected from the process. In his book on the Philosophy of Mathematics, Körner several times characterises perceptual concepts as essentially inexact. This section can be regarded as an attempt to bring this insight into sharp relief. Absolutely any ostensively defined predicate may come to display tolerance, for absolutely anything which it characterises might undergo seemingly continuous change to a point where it could be so characterised no longer. Example 4 is but a tremor, signalling a basic fault, as it were, buried deep in the relation between the nature of our experience and those parts of language by means of which we attempt to give the most direct, non-theoretical expression to it.

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This conclusion rests upon two premisses which might be held open to question: that it is right to regard the senses of colour predicates, etc., as purely observational; and that this is a very fundamental fact about their senses, whose sacrifice would be possible only at great cost. For the first no further argument will be provided in this paper. The considera­ tions adduced earlier in the section are surely decisive, provided it is allowed that they are relevant - provided the second thesis of the govern­ ing view has not been rejected. For the second, however, no argument has so far been presented; we merely voiced concern that ‘contact’ be­ tween language and the empirical world might be attenuated if the use of purely observational predicates was abandoned. No special considera­ tions have been advanced in relation to Example 4 to correspond to the points about convenience, memorability, and social role in relation to the earlier examples. The matter must await the next but one section. First we must consider a general objection to our treatment of all four examples. IV

To concede that the vagueness of our examples is correctly interpreted as tolerance is to concede that there can be no consistent, non-classical logic for such predicates. But it is natural to suggest that the arguments for this interpretation may have overlooked an essential feature of this sort of predicate: that they typically express distinctions of degree. There are degrees of redness, of childishness and, if a smaller heap is regarded as less of a heap, of heaphood also. When is the distinction between being F and not being F one of degree? Typically, when the comparatives, ‘is less/more F than’, are in use and when iteration of one of these relations may transform something F into something not F, or vice versa. Moreover the semantic relations between the comparatives and the simple descriptions, ‘is F ’ and ‘is not F ’, are such that if a is less/more F than b, then the degree of justice with which a can be described simply as F is correlatively smaller or larger than that with which b can be so described. That is, a twofold classification of possible states of affairs into those which would justify the judgement, ‘a is F ’, and those which would not, misses what is distinctive about a predicate whose application is a matter of degree. For that to be so is exactly for there to be degrees of such justice.

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It is thus plausible to suppose that a logic for distinctions of this sort cannot be based upon simple bivalence. With such predicates there are, as it were, degrees of truth whose collective structure is that of the set of degrees of being F. We shall not speculate what this structure might be, but it does not seem too fantastic to suppose that an indefinitely large or even densely ordered series might be involved, so issuing in infinitely many truth-values. In this sense it is arguable that our examples do require a special, i.e. non-classical logic. The view that classical logic is inadequate for distinc­ tions of degree is not contested in this paper. What is contested is the idea that the seeming tolerance of the examples is generated by over­ looking that the predicates in question express distinctions of degree. This impression views the paradoxical reasoning as essentially depending upon the constraints of bivalence - thus no attention is paid to the point that distinctions of degree are involved. Consider a pair of objects one of which, a, we are happy to describe as F while b is slightly less F than a. How is b to be described? If our admissible descriptions are restricted to ‘F ’ and ‘not-F’, if we have to say one or the other, then presumably we shall describe b as F. For if something is more like something F than something not-F, to describe it as F is the less misleading of the two alternatives. But the justification with which ‘F ’ is applied in successive such cases successively decreases. We have no principle of the form: if a is F and b differs sufficiently marginally from a, then b is F; with dis­ tinction of degree there are no ‘small changes insufficient to affect the justice with which a predicate applies’; they are, on the contrary, small changes in the degree of justice with which the predicate may be applied. Of course we do have the principle: if the judgement that a is F is justified to some large degree and b is marginally less F than a, then the descrip­ tion of b as F will be better justified than its description as not-F. But that is not a paradoxical principle. Anyone who thinks he at last feels the cool wind of sanity fanning his brow would do well to be clear why we do not still have this principle : if b is marginally less F than a, then if the less misleading description o f a is ‘F \ the less misleading description of b is ‘F \ Yet if this principle is false there must, in any Sorites type series, be a last case of which we are prepared to say that, if we had to describe it either as F o r as not-F, the better description would be *F\ Why, then, is it usually embarrassing to

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be asked to identify such a case without any sense of arbitrariness? Let us say that ‘a is F ’ has a positive value just in case ‘F ’ is a less misleading description of a than ‘not-F\ Then our embarrassment is exactly to identify a last object to which the application of F would receive a positive value. But then the suspicion arises that tolerance is with us still; only it is not now the truth of the application of F that would survive small changes but, so to speak, its positivity. Is this suspicion justified? One thing is clearly correct about the as­ sumption of bivalence : faced with a situation and a predicate, we have only two choices - to apply or to withhold. There is not a series of distinct linguistic acts in which we can reflect every degree of justification with which a predicate may be applied. The crucial notion to be mastered for practical purposes is thus that of a situation to which the application of F is on balance justified. Without mastery of this notion, no amount of information about the structure of variations in the degree with which F applies entails how the predicate is to be used. Now of this notion may it not still be a feature that it always survives sufficiently small changes? that if a and b are dissimilar only to some very small extent, then if describing a as F is on balance justified, so is thus describing 6? It is clear that all the considerations adduced in the previous sections now apply. The introduction of a complex structure of degrees of justi­ fication has got us no farther; for among these we have still to distinguish those with which for practical purposes the application of the predicate is to be associated ; otherwise we have not in repudiating bivalence done anything to replace the old connection between justified assertion and truth. The distinction in question need not of course be exact, (so one’s embarrassment at having to identify a last case to which the application of F is preferable to that of its negation is understandable). On the con­ trary, in the kind of case which we have been considering the concept will be tolerant. To rehearse the reasons: if we are to be able to remember how to apply F, then differences too slight to be remembered cannot transform a situation to which its application is on balance justified into one which is not so; if we are to be able to apply F just on the basis of casual ob­ servation, the same applies to differences too subtle to be detected by casual observation; if the distinction between cases to which the ap­ plication of F is on balance justified and others is to be made just on the

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basis of how things look, or sound, etc., then any pair of indistinguishable situations must receive the same verdict; finally if F is associated with moral or explanatory distinctions which we are unwilling to tie to very small changes, we shall likewise be unwilling to allow such changes to generate doubt about the status of a situation previously regarded as on balance justifying description as F. Of course a quite uncritical use is here being made of the notion of a situation to which the application of F is ‘on balance’ justified. But this is legitimate. As remarked, there must be some such notion if a many-valued logic for distinctions of degree is to have any practical linguistic application. v We turn to the question whether we could not, at not too heavy a cost, eliminate the tolerance of observational predicates. W hat is necessary, it seems, is rejection of ostensively defined predicates; hence the initial doubt whether such a purified language could engage with the observa­ tional world at all. But, on reflection, it is clear that the dislocation of language and the world of appearance does not have to be as radical as that. When three situations collectively provide a counterexample to the transitivity of indiscriminability, there is nothing occult, as it were, in the circumstance that they do so. It is an observationally detectable difference between indiscriminable situations that one is distinguishable from a third situation from which the other is not; the relation, ‘a matches b matches c does not match a \ is an observational relation, i.e. one whose application to a trio of objects can be determined just by looking at them, listening to them, etc. Observational concepts evidently require narrower criteria of reapplication than indistinguishability, if they are to be purified of tolerance. But we should not jump to the conclusion that to provide such criteria will require surrender of observationality altogether, for the phenomenon which is causing the trouble is itself observational. Indeed, the only kind of observationally detectable difference which there can be between indiscriminable items is that one should be distinguishable from some third item from which the other is not. So if the class of expressions in question is to remain in contact with observation, we have to look for some form of stipulation which exploits the non-transitivity of indis-

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tinguishability to provide a basis for describing indiscrim inate situations differently. No other explanation can correspond to a distinction which sense-experience can determine to obtain, a distinction which we can simply be shown. One implication of this suggestion is already apparent. All along our discussion has centred on the use of certain predicates. Now there is a tradition, dating at least from Frege, in accordance with which a predicate of individuals is essentially an open sentence of only one individual argument place. The status of a predicate in this sense is indifferent to the occurrence of individual constants and bound variables within the expression, and so to the nature and scope of the procedure required to determine whether the predicate applies. But there is also the narrower traditional conception in whose terms predicates essentially express properties, whose application to an individual is a question of scope no broader than the boundaries, so to speak, of the individual concerned; it is this which distinguishes properties from relations. Such was the con­ ception of quality to which Locke and Berkeley appealed. Of course it may be appropriate to decide whether a property applies by comparison with cases where it does; but that only serves to emphasise that its ap­ plication cannot always be a matter of comparison. What is the point of this distinction here? Simply that it is not coherent to demand more than indistinguishability as sufficient for re-application of an observational quality. The question whether such a quality is shared by a pair of indistinguishable items cannot essentially turn on what relations they bear to other things ; it has to be possible to decide whether an object has the quality by observation of it alone. So the envisaged kind of modification to the senses of observational predicates requires that we abandon their use as expressions of observational qualities. Indeed the interest of the proposal depends upon our abandoning the notion of an observational quality altogether; for if there are such qualities, there can be no objection to introducing predicates to express them. This is not to say, of course, that language must cease to contain expressions whose syntax is that of a simple predicate, i.e. expressions containing no singular term or quantifier and having but a single in­ dividual argument place. But if the conditions of application of such an expression can be determined by observation, they will not be determinable

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by observation of a single individual. In a certain sense the world of observation is to be a world of relations. The semantics of any observa­ tional predicate will be implicitly relational. The sense of this last claim becomes clearer if we now take note of a striking aspect of the philosophical psychology of non-transitive matching. Summarily, it does not seem to be possible to conduct experiments with non-transitively matching triads in memory. For suppose that a predicate, F, is defined ostensively by reference to some individual, a, which, it is noted at the time, perfectly matches another individual, c ; it is understood that F is not to be applied to individuals which match a unless they also match c. Later the trainee comes across b which, so far as he can deter­ mine, matches a perfectly; the question is, does b match c? It is evident that the issue is only resoluble by direct comparison, and especially that it cannot be settled by memory, however accurate. For the most perfect memory of c can give no further information than that it looked just like a; which, when non-transitive matching is a possibility, is simply insufficient to determine whether it would match b. This, it should be emphasised, in contrast with our conclusions concerning Example 3, is not a limitation imposed by the feebleness of our memories; it is a limitation of principle. So, if we are to be able to exercise expressions whose application to matching individuals depends upon their behaviour in relation to a third, possibly differentiating individual, it is clear that we have to be able to ensure the availability of the third individual. Expressions of this species will be practicably applicable only in relation to a system of paradigms. Thus we can see, even in advance of attempting a specific stipulation to remove the tolerance of ‘red’ as displayed in Example 4, that the kind of semantic construction it will have to be is going to tie the application of expressions of colour to the use of a colour-chart. Let us then consider, as a test case, how we might go about the con­ struction of such a chart. What we require of the chart is that it should enable us to identify a last red patch in any series of the type of Example 4. How is this to be achieved? In The Structure of Appearance11 Goodman notes the difficulty that the non-transitivity of matching provides for his concept of a quale : ... this is somewhat paradoxical; for since qualia are phenomenal individuals, we can hardly say that apparently identical qualia can be objectively distinct.

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But Goodman sees a solution: ... the fact that some matching qualia are distinct can be accounted for without going beyond appearance; we need only recognise that two qualia are identical if and only if they match all the same qualia.

(The strategy is not original to Goodman. It was also adopted by Russell in Inquiry into Meaning and Truth and indeed there described as ‘famil­ iar’.)12 Let us say that patches a and b are of the same Goodman-Shade just in case they satisfy such a condition; that is, any c matches a if and only if it matches b. Then it would appear that an ideal colour chart - a Goodman Chart - would contain samples of all Goodman-Shades, together with instructions about how each is to be described. Evidently, however, we must abandon this ideal straight away. We might chart a region of colour comprehensively in the sense of being confident that any sample of colour which we should intuitively regard as falling in that region would match something on the chart; but we could never have reason to be sure that every Goodman-Shade falling within that region was represented. For there is no way of foreclosing the possibility of finding a sample which is Goodman-distinguishable from every sample on the chart from which it is not simply distinguishable; that is, it matches something which such a chart sample does not, or vice versa. It is, indeed, a peculiarity of the notion of a Goodman-Shade that, on a certain natural presupposition, a comprehensive Goodman Chart could not be achieved in principle. Imagine a band of colour varying uniformly in the following sense: that there is a constant distance, d inches, say, such that we can always distinguish the colours on the band more than d" apart but can never distinguish those less than d n apart. It follows that the colours at any distinct points on the band are Goodmandistinct, since associated with any pair of distinct points is a point less than d" from one and more than d ” from the other. On such a band the claim of identity of Goodman-Shade at any distinct points is defeasible. There seems a priori no reason why there should not be a band of colour varying uniformly in this sense. But then there can no more be a specific, finite, comprehensive totality of samples of Goodman-Shades than there can be such a totality of points in the linear continuum. The assumption that colour may vary uniformly thus generates a somewhat

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startling consequence for the notion of a Goodman-Shade : single Goodman-Shades do not require that regions in which they are exempli­ fied should be extended. This is, perhaps, paradoxical only if, like Goodman, we suppose that these notions are still in any meaningful sense phenomenal. Besides, the kind of comprehensiveness which we were contemplating is not neces­ sary anyway. What is necessary is the construction upon the set of Goodman-Shades of an effective total order ; such an order should, moreover, coincide with that generated by the intuitive relation of likeness among colour samples in any case where the latter issues a clear verdict. Given a decidable such order, there is no obstruction to stipulating intelligibly a determinate last red shade; and then we may in any series like that of Example 4 identify a last sample preceding or coinciding with this shade. Let us, then, consider what may be achieved in the direction of ex­ plaining such an effective order. Plainly, we have to be able to arrange a series of samples in such a way that we can tell of any new sample whether it falls between any pair of samples already located; such recognition will of course be crucial in cases where one of the samples is the last red shade. Conversely, if we are given the two outermost flanking samples, it will be sufficient for the task to have an effective notion of betweenness. When should a shade, b, come between a and c in an intuitively correct arrangement of colours? The notion we really need is that of b coming nearer to a than c does; for b comes between a and c just in case b is nearer to a than c is and nearer to c than a is. Plainly a colour-patch, 6, should come nearer to a than c does if and only if it is more like a than c is. Let us consider then the construction of a Goodman Chart in terms of this basic principle; that sample b is to be placed nearer sample a than sample c is just in case it is more like a than c is. The principle has one very clear cut application: among a trio of non-transitively matching samples, the sample which matches the other two must be located between them. And surely there is this much to be said for it: if on a Heaven-sent, correctly constructed Goodman Chart of sufficient degree of refinement of distinction we find a sample b located between a and c9 that it is so should be explicable by (iterated) application of the principle; if a comes to the left of b, then there should be something to the left of a which a matches and b does not, and so on.

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It is thus plausible to claim that the principle at least gives necessary and sufficient conditions for correctness of mutual location among samples in an existing Goodman Chart. This is not, of course, to say that it can serve as a principle of construction in an effective sense. We are, moreover, making a very substantial assumption. Suppose that a matches b matches cf while a and c do not match; what, then, if some d matches a and c but not bl The basic principle will require both that b and d be placed between a and c and that a and c be placed between b and d, contrary to the supposition of total order. It is thus a necessary condition of the possibility of a total order among the relevant GoodmanShades in terms associating ‘is nearer to’ with ‘is more like to’ that this cyclic matching situation should not be a possibility. It will be assumed, since our main conclusions will be unaffected if it does not, that this necessary condition obtains. Suppose then that we have been given a red and an orange flanking sample, and have advanced, using the basic principle and the kind of ability on which Hume remarked, to a Goodman Chart in which every sample matches just those immediately adjacent to it and no others. Consider a section of the chart:

...a; b ; c; d; e; f\

g; h....

Our task is to locate an ;t which matches both d and e ; does it go between them? Evidently in two cases it does: the basic principle puts x between d and e if it matches both or neither of c and/ . But suppose one of the other two cases obtains, for example that x matches c but not / . Still, we may get a decision if x matches b, for then it must lie between c and d. But suppose x does not match b ; suppose in fact that x and d match and are distinguishable from all the same samples located so far. Then all the basic principle tells us is that x must lie somewhere in the region between c and e. We have no way of locating x in relation to d on present information; apparently we need more samples on the chart. A sample between b and c, or between e and / , matching x or d but not both would do the trick. Yet we have just noted that the question whether something goes between samples adjacent at this degree of construction is effectively decidable only in two cases of four; so we have no guarantee of being able to locate such additional samples; indeed the spectre is introduced of reciprocally

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undecidable cases, where the location of x is decidable only by deciding that of y and conversely. This spectre at least can be banished. The discovery of a y matching x but not d or vice versa would actually guarantee a solution to the question of their relative locations in the region between c and e.13 Still we cannot convincingly claim to have an effective principle of construction, since we have no effective means of finding such a y. There is a yet more radical deficiency. If x and d are as a matter of fact of the same Goodman-Shade, then not only have we no effective way of determining that they are; we have no way of recognising the fact at all. This is a fundamental point. Goodman, we noted earlier, congratulated himself on being able to explain a distinction between matching and identity among qualia without going beyond the realm of appearance. Certainly, the notion of Goodman-identity is explained by means of a phenomenal relation and a quantification over ostensibly phenomenal individuals. Yet, evidently, it is only in a formal sense that we have not gone beyond appearance in this account; for, as Russell noticed, the resulting relation is not one which may appear to obtain. The concept of a Goodman-Shade is not merely not an observational notion; it is a transcendental notion. Nothing counts as discovery of identity of Goodman-Shade. This casts serious doubt on the suitability of the notion to serve in the construction of a semantic rule. For, surely, sense can be given to an expression only by reference to conditions whose satisfaction we can determine at least in principle. After a stipulation of the kind considered, the ability to recognise whether something was red would sometimes require the ability to recognise identity of GoodmanShade, viz. when what we have is in fact a sample of the last red shade. But this is an ability which we do not have. There is no effectively decidable order among Goodman-Shades based on likeness. A fortiori there is no effective stipulation of the sort con­ sidered: a stipulation based on the notion of a Goodman-Shade and providing a sharp, decidable red/orange distinction in any series of the type of Example 4. The concept of the last red shade is underdetermined, for nothing amounts to recognition of which shade it is. Indeed Good­ man’s whole strategy for surmounting the difficulty of nontransitive matching amounts to nothing other than the introduction of a spuriously phenomenal identity to which nothing in our experience can correspond.

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Not that Goodman introduced the notion with a view to providing the kind of stipulation which we are looking for; but we must look elsewhere. Could we not instead simply devise ad hoc paradigms? Consider a colour-chart complete in the sense distinguished earlier; i.e., we are confident that anything we should wish to regard as falling in the red/ /orange region will match something on the chart. It is plausible to suppose that likeness provides an effective order among any finite set of colour samples, so suppose the chart samples to be so arranged. Then we can generate a decidable red/orange distinction as follows. Select some patch towards the middle; then any colour patch matching some­ thing on the chart either matches the selected patch or it does not; if it does, it is red; if it does not but matches a sample to the left of the selected patch, it is red; otherwise it is orange. Naturally we could not guarantee that duplicates of a chart would always deliver the same verdict. Charts could look absolutely similar, and even satisfy the condition that the «th sample on either matched and was distinguishable from exactly the same samples on the other chart as its own «th sample, yet deliver discrepant results. But they would not often do so. Besides, the situation is nothing new. Rulers, for example, sometimes give different results. A final criterion for one system is de­ posited in Paris; and we could do the same with a colour-chart. Nevertheless the generalisation of this proposal seems quite ludicrous in practical terms. We are confronted with the imaginary spectacle of a people quite lost without their individual wheelbarrow loads of charts, tape-recordings, smell- and taste-samples and assorted sample surfaces. But this does the proposal an injustice. There will be no need for all this portable semantic hard wear. To pursue the analogy with the use of rulers: it is true that, without a ruler, we can only guess at lengths; but after the introduction of an ad hoc paradigm for colours, the use of colour predicates will presumably be analogous not to that of expressions like, ‘two feet long’, but rather to that of expressions like, ‘less than two feet long’, i.e. expressions of a range of lengths. Of such expressions the criterion of application is still measurement; but unless the case is peripheral, we can tell without measuring what the outcome of measure­ ment would be. Training in the use of paradigms might be essential if one is to grasp the sense of such expressions; but, once grasped, most cases

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of practical application could be decided without the use of paradigms; for most practical purposes the wheelbarrow could be left behind. It appears, then, that were we to adopt such stipulations as a general strategy, it would not have to affect our use of observational language very much at all. At present we can tell of anything red that it is so just by looking at it. This would still usually be true after the proposed stipulation; and if the new distinction was suitably located, cases where it was not true could generally coincide with borderline-cases of the old red/orange distinction. The use of predicates so refined could thus greatly resemble their present use; the distinctions which they expressed would be empirically decidable; and there would be one crucial disanalogy they would be tolerance-free. It is, indeed, apparent that exactly parallel considerations may be brought to bear upon our earlier treatment of Examples 1 and 3. Even after a precise re-definition of ‘heap’, we would be able to learn to tell in most cases just by casual observation what verdict the new criterion would give if applied; it would seldom be necessary actually to count the grains. And the distinction between red and orange, supposing an exact distinction were drawn by means of a chart, would be unmemorable only within that small range of shades which could not by unaided memory be distinguished from the last red sample. It would thus appear that the cost of eliminating tolerance in cases of these two types need not after all be high, since we could expect to be able to tell in general just by looking at, etc., an item on which side of the dividing line it would fall. If there need after all be no substantial sacrifice in endowing formerly observational predicates with exact boundaries, what has become of the alleged profound tension between phenomenal continuity and language designed to express how things seem to us? The answer is that we have simply swept it under the carpet. The possibility of our dispensing with paradigms for most practical purposes depends upon our capacity e.g. to distinguish between cases where we could tell whether or not ‘red’ applied just by looking and cases where we could not, where we should have recourse to a chart. If we are able to make such a distinction, what objection can there be to introducing a predicate to express it? But then, it seems, the semantics of this predicate will have to be observational. For on what other basis should we decide whether something looks as though comparison with a chart would determine it to be red than how

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it looks? Of any pair of colour patches which look exactly alike, if either looks as though the chart would deliver the verdict, ‘red’, both must. So the new predicate, introduced to reflect our capacity to make this dis­ tinction, will be applicable to both members of any pair of matching colour-samples if to either. Of course, there is no reason to have any such predicate; but equally there is no reason not to. If we were sometimes able to tell without using a chart whether something is red, it w'ould surely be possible to make intelligible to us a predicate designed to apply in just such circumstances. A language all of whose observational concepts were based on paradigms would avoid containing tolerant predicates only by not containing means of expression of all the observational distinctions which we are in fact able to make. Naturally it would make no difference to this point if we insisted on actually using the paradigms in every case, and hang the in­ convenience, as it were. We could dispense with them almost always, so we should just be insisting on a charade. Would it not, moreover, be quite absurd to propose that the tolerance of such new predicates - ‘looks as though it would lie to the left of the last red shade’, ‘looks as though it contains fewer than ten thousand grains’, etc. - might in turn be stipulated away? Their meaning will not permit it; it cannot be allowed of things which look exactly alike that one may look as though it satisfies some condition which the other looks as though it does not - unless how a thing looks may not be determined by looking! But the earlier treatments of Examples 1 and 3 involved overestimation of our interest in preserving the tolerance of the predicates concerned only if we posses a coherent understanding of these new pre­ dicates; if Examples 1 and 3 do not, after all, pose a substantial problem for the governing view, it is because of our capacity to handle expressions falling within the scope of Example 4. VI. C O N C L U S I O N

Let us, then, review the character of the difficulty which Example 4 would appear to pose for the governing view. It is a fundamental fact about us that we are able to learn to classify items according to their appearance, that we are able consistently, as it seems to us, to apply or to withhold descriptions just on the basis of how

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things strike the senses. In a discrete phenomenal world there would be no special difficulty - no difficulty, that is, not inherent in the idea of a semantic rule as such - in viewing our use of such expressions as essential­ ly nothing but the following of rules of which it was a consequence that indiscriminable phenomena should receive the same description. But if mutually exclusive use is made of a pair of such predicates, and if cases to which one applies permit of continuous transformation into cases where the other applies, it cannot be correct to represent the use made of either predicate just as the doing of what is required by a set of rules with such a consequence. Yet we are constrained - if the relevance is allowed of considerations to do with what we should regard as adequate explana­ tion of such expressions, or with certain criteria which we should accept of misunderstanding such an expression - to attribute to the rules govern­ ing these predicates precisely such an implication; and all the phenomena which we confront in our world impress us as capable of continuous variation. In the Introduction, the difficulty was presented, starkly, as that of the inadequacy of any inconsistent set of rules to explain a consistent pattern of behaviour. This needs a little refinement. It is, to begin with, unclear how far our use of e.g. the vocabulary of colours is consistent. The descriptions given of awkward cases may vary from occasion to oc­ casion. Besides that, the notion of using a predicate consistently would appear to require some objective criteria for variation in relevant respects among items to be described in terms of it; but what is distinctive about observational predicates is exactly the lack of such criteria. So it would be unwise to lean too heavily, as though it were a matter of hard fact, upon the consistency of our employment of colour predicates. What, however, may be depended upon is that our use of these predicates is largely successful; the expectations which we form on the basis of others’ ascriptions of colour are not usually disappointed. Agreement is generally possible about how colours are to be described; and this, of course, is equivalent to saying that others seem to use colour predicates in a largely consistent way. It is of this fact which the governing view can provide no account. A semantic rule is supposed to contribute towards determining what is an acceptable use of its associated expression. The picture invoked by the first thesis of the governing view is that there is, for any particular

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expression in the language, a set of such rules completely determinant of when the expression is used correctly; such a set thus provides a model of the information of which a master of the use of the expression may be deemed to be in possession. Clearly, however, the feasibility of such a picture requires that the rules associated with an expression, about whose use we generally agree, be consistent. For if they issue conflicting verdicts upon the correctness of a particular application of the expression, it cannot be explained just by appeal to the rules why we agree that the application is e.g. correct. The problem presented for the first thesis by the occurrence of tolerant predicates, or of any other kind of semantically incoherent expression, is not that, in a clear-cut way, nothing can be done to implement an in­ consistent set of instructions. It is true that, strictly, anything that is done will conflict with a part of them. But we can imagine a game whose rules conflict but which is nevertheless regularly and enjoyably played to a conclusion by members of some community because, for perhaps quite fortuitous reasons, whenever an occasion arises to appeal to the rules, the players concur about which element in the rules is to be appealed to, so that an impasse never comes about. We need not enquire whether they have noticed the inconsistency in the rules. The point of the analogy is that in practice they always agree whether a move is admissible, as we general­ ly agree whether something is red. The analogue of the first thesis in rela­ tion to this example is the notion that the rules completely determine when a particular move is admissible. But while it may be true that the authority of the rules can be cited for any of the moves the community actually makes, it is plain that the rules alone do not provide a satisfactory account of the practice of the game. For someone could master the rules yet still not be able to join in the game, because he was unable to guess what sort of eclectic application of them an opponent was likely to make in relation to any given move. An outsider attempting to grasp our use of a tolerant predicate would presumably not encounter exactly this difficulty; it would be clear that we were not prepared to allow remote consequences of its tolerance, inferred by means of reasoning of the Sorites type. But the difficulty of principle for the first thesis is the same. The rules of the game do not provide an account of how the game is played, for it is possible that someone might grasp them yet be unable to participate. The semantic

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rules for an expression are supposed to provide an account of its correct use; they cannot do so if someone whose use of it differed radically from ours could still be thought of as in possession of exactly the same brief as he can be if it consists in an inconsistent set of instructions. The comparison of language with a game is in many ways an extremely natural one. What better explanation could there be of our ability to agree in our use of language than if, as in a game, we are playing by the same rules? So we are attracted towards the assimilation of our situation to that of people to whom the practice of a game has been handed down via many generations but of which the theory has been lost. Our task is to infer the theory. The burden of this paper, however, has been that this image is in conflict with the principles of investigation into the semantics of an expression which we find it natural to allow. These prin­ ciples yield an account of the senses of certain vague predicates which pushes our agreement in their application beyond explanation by appeal to what the rules for their use require. And it is doubtful whether an intuitively satisfactory conception can be achieved of what an investiga­ tion into the semantics of an expression might be which did not admit these priciples, - unless the notion is abandoned that such an investiga­ tion is something which only a master of the investigated language is optimally placed to carry out. Such a conclusion would seem to force on us a more purely behaviouristic concept of how a theory of language-use should be accomplished, and a corresponding shift in the concept of a semantic rule. There would no longer be any room for the idea that such rules might be discoverable by means of such a sort that our use of the associated expression(s) could prove to be in conflict with or otherwise inexplicable by reference to them; there would no longer be any room for the idea that by reflection on the kind of training which he has received in the use of some expression, the criteria which he would employ to judge that someone misunderstands it, his concept of the purpose or interest of the classification which it effects, or his awareness of his own intellectual and perceptual limitations, a speaker of a language has access to sound conclusions about his un­ derstanding of an expression which a mere observer of his use of it has not. This would be one response to the difficulties generated by the richer, more natural methodology of the governing view. It can be expected to

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encounter severe difficulties of its own: in particular, it is unclear that an adequate characterisation can be achieved from so restricted a standpoint of in what the vagueness of the sorts of example with which we have been concerned consists. But if the notion of a semantic rule is not to be aban­ doned altogether, some more restricted account of the epistemology of semantic rules is required than that afforded by the governing view.

All Souls College, Oxford N O TE S * This discussion was largely motivated by two unpublished papers: Aidan Sudbury’s ‘Are Imprecise Terms Essential?* and Michael Dummett’s ‘Wang’s Paradox’. The paper has benefitted by extensive discussion with both Sudbury and Dummett. 1 Cf. G rundgesetze , Vol. II, Section 65. 2 Elsewhere Russell takes the vagueness of ordinary language more seriously; e.g. in ‘Vagueness’, A u stral. Jour, o f P sych, a n d P hilosoph y 1 (1923), 84-92, the notion that vagueness is a flaw is tempered by pessimism about its remediability. 3 In the S tru ctu re o f Appearance. 4 Traditionally, attributed along with its variant, the ‘Bald Man’ to Eubulides. (See e.g. Diogenes Laertius, Lives, ii, 108.) 5 ‘On the Ultra-intuitionistic Foundations of Mathematics’, in In fin itistic M e th o d s , Pergamon Press, Oxford, 1961, pp. 201-223. 6 Esenin-Volpin’s interest in the example derives from the possibility, which it suggests, of satisfying orthodox arithmetical postulates for the successor function in an openended but finite domain. The existence of such domains depends upon the admissibility of the concepts of which they are the apparent extensions; that is, it depends upon the correctness of the view that we may have a coherent understanding of predicates whose semantics are strictly incoherent. For unless a predicate of the genre in question really does have an incoherent semantics, it will not yield a structure in which the relevant postulates are satisfied; but unless such predicates are acceptable, we shall not be much interested in the pure mathematics of their extensions. There will be, unfortunate­ ly, no further discussion in this paper of the significance of these examples for the concept of infinity. 7 We spare the reader the irrelevant complications involved in how such a stipulation would apply to substances of varying granular coarseness, viscosity, etc. 8 Intuitively, a colour predicate is basic in a given language if it does not express a shade of a more general colour for which means of expression exists in the same language. Obviously, as it stands, the notion is fragile. 9 As they would sometimes be, if the distinctions were made in terms of precise numbers of heartbeats. 10 There is no presupposition here that a definitely correct verdict can always be re­ ached; but if it cannot, that in turn must be the situation with respect to each item in question. 11 Chapter IX, Section 2. 12 Chapter VI.

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13 P ro o f: Let y match d but not jc , and e but not c. Then, letting *M’ = ‘matches’, 13 = ‘does not match’, *a b c' = *b is between a and c’ : (i) (ii)

y

M dt d

M c, y M c;

y M d, d M x , y

13 x ;

y d c;

.*. y d x ;

hence y is right of d. hence by (i), x is left of d .

The case where y matches c but not e is quite similar. Suppose on the other hand that y differentiates x and d the other way about, i.e., that y matches x but not d. Let y again match e but not c. (The converse case is again essentially similar.) Then : (i)

y M e, e M d t y IÏÏ d;

.. y e d ;

(ii)

y M x , x M d, y M d;

.. y x d;

hence y is right of d. hence by (i), x is right of d.

So if a differentiating y matches only one of c and e , the question of the location of in relation to d is soluble. What, though, if y matches both or neither of c and e l Somewhat surprisingly the basic principle, conceived as determining a total order, proves strong enough to rule out both cases. Again, let y match d but not x : Suppose y M c, y M e; then : x

(i) (ii)

M x, y c M x, y

y M e> e

M x;

y M c,

M x;

:.y e x y ex.

Hence both c and e lie between y and j c ; s o x cannot l i e between c and e , contrary to the hypothesis that, save for 13x matches all and only the patches matched by d y (whence cxe).

Suppose alternatively y t â c , y X f e; then: (i) (ii)

y M d, d M e, y M e; y M d, d M c, y 1ÏÏ c;

y de.

/. y d c .

Hence y is required, absurdly, to lie both right and left of d. Now let y match x but not d: Suppose y M c t y M e; then : (i) (ii)

y M c t c M d yy f i f d; y

M

e, e

M d,y M

d\

y cd. :.yed.

Hence, absurdly, y is required to lie both left of c and right of e. Suppose y 1ÏÏ c , y A f e; then (i) (ii)

y

M x,

x M c , y f á c\

y M x, x

M e, y

] & e;

yxc. :.yxe.

Here (i) and (ii) are conjointly satisfiable only if both y and x lie to the left of the region between c and e; but x must lie within that region since, prior to the discovery of y , it matched and was distinguishable from the same samples as d.

[7] Phenomenal Colors and Sorites C. L.

H a r d in

SYRACUSE UNIVERSITY

I.

M ost philosophers seem prepared to accept the principle that the relation of indiscriminability between phenomenal colors is n ontran­ sitive. This principle, hereafter known as Nontrans, m ay be stated as follows: There exist triples of phenomenal colors x, y, and z, such that x is indiscriminate from y and y is indiscriminable from z, but x is discriminable from z.

N ontrans has been used as a stick with which to beat sense-datum theorists as well as to cast doubt on the possibility of giving a ra ­ tional reconstruction of the semantics of everyday color term s. In both cases, what is called into question is the coherence of phenom enal color predicates, or, alternatively, the concept of a phenom enal color. For if we m ust distinguish the colors of x, y, and z by sight alone, and we require that the phenom enal color of an object be the color that the object seems to have, then if x is indistinguishable in color from y, x and y m ust be the same (shade of) phenom enal color. Similarly, if y is indistinguishable in color from z, y and z m ust be the same phenom enal color. By the tra n ­ sitivity of identity, we m ust conclude that x is the same phenom enal color as z, but since x and z are distinguishable in color, x cannot be the same phenom enal color as z. It is evident that the contradiction disappears if we abandon the conception of a phenom enal color, for it does not follow that ju st because x seems to be the same (shade of) color as y, and y seems to be the same color as z, that x m ust then seem to be the NOOS 22 (1988) 213-234 © 1988 by Noûs Publications

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same color as z. Although it is plain enough that “ is identical w ith” is transitive, nobody is tem pted to suppose that “ seems to be iden­ tical w ith” is transitive. So, it is argued, if we are to avoid con­ tradiction, we m ust conceive of color properties as non-phenom enal (Cf. A rm strong 1968, p. 218). A similar line of reasoning seems to cast doubt upon any systematic account of the semantics of everyday color term s, since in ordinary life it is the look of things that decides w hether or not a color predicate applies to them . U n d er such circum stances, if x and y look to be just the same color, any basic color predicate ap­ plicable to x is also applicable to y. But then if y looks ju st like z in color, any basic color predicate applicable to y is also applicable to z. Given a sufficient array of intermediate color samples, a sorites argum ent enables us to take a color predicate which is applicable to the first m em ber of the array and force it in turn on each suc­ ceeding m em ber of the array, arriving at the absurd conclusion that any color predicate applicable to one sample is applicable to any other sam ple.1 In what follows, I shall argue that if we are prepared to count statistical ensembles of observations as observational data— a quite comm on practice in science— N ontrans and the argum ents which depend upon it m ust be rejected. T he coherence of a concept of phenomenal color will be defended, though the m anner of the defense m ay give scant comfort to lovers of sense-data. A deeper understand­ ing of the reasons for rejecting N ontrans will in tu rn enable us to gain theoretical purchase on the ubiquitous but frequently overlooked fact th at phenom enal colors, and not ju st phenom enal color predicates, are often indeterm inate. Finally, an exam ination of the actual practice of assigning color predicates will show how that prac­ tice is based upon the structure of phenom enal color and suggest some of the factors which determ ine the foci and boundaries of col­ or term s. II.

W hen philosophers write about whether or not one hom ogeneously colored patch is discriminable in color from another, it is easy to get the impression that one could decide w hether or not one had a m atch rather easily, just by giving a good straight look. If a dif­ ference between the patches falls above the threshold of discriminability and the conditions of seeing are optim al, the straight look will reveal that difference, but if the difference falls below the threshold, neither that look nor any succeeding look will uncover the discrep­ ancy, and the samples will m atch perfecdy. T he tru th of N ontrans depends upon the existence of a fixed, sharp discrimination threshold.

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Actual attempts to match and discriminate closely similar material color samples are, by contrast, typical cases of decision making under conditions of uncertainty. If one compares the matches a subject makes or rejects with the spectral characteristics of the material samples to be matched, one discovers that some of the rejected matches are of samples which are physically extremely close, while some of the accepted matches are between samples which are physically further apart.2 In fact, when the same samples are presented on another occasion, several of the previously accepted matches will now be rejected, and several of the heretofore rejected matches will now be accepted. How, then, is one to decide what the limits of discrimination are? The appropriate procedure seems fairly clear, and analogous to that generally applied in observational sciences such as astronomy, which must extract meaningful obser­ vational information from a babel of perturbing influences: assume that the sources of error are random, use that assumption to factor out biases, and then select and apply a criterion of statistical significance. But if we follow this procedure, we get a statistical distribution of discriminations. What is to become of the ‘‘classical’9 conception of the fixed, sharp threshold? Answer: it must be aban­ doned, and has in fact been abandoned in psychophysics for many years.3 We can clarify what is involved in a statistically conceived discrimination threshold experiment by representing the problem as one of extracting a signal which is transmitted over a noisy chan­ nel, where the noise may be understood as the combined effect of those factors which interfere with the detection of the signal.4 We shall assume that these factors are randomly distributed and thus representable by a normal (Gaussian) distribution curve. Suppose, for instance, that someone is trying to make out what is being transmitted over a static-filled telephone line. When the level of the message is very close to the average static level, the message will be very difficult to detect, but as the message strength increases relative to the background, larger portions of it become intelligible. But even when its level is very close to the noise floor, some por­ tion of the message can still be recovered, since there will be moments when the noise will fluctuate below its average value, leaving a “ win­ dow’’ through which a bit of message can come. As the message is repeated, more of it will make its way through. Although the receiver may not ever be able to reconstruct the message with cer­ tainty, the proportion of it which she gets right will increase with each improvement in the ratio of signal to noise, provided that she is given a reasonable number of message repetitions. For a low signal to noise ratio, her success in discriminating message from background

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NOUS

will not manifest itself in a single trial, but will become apparent over several. To fix our ideas, let us suppose that the task is a simple one: the receiver is asked to judge whether, in a given interval of time, a sound of a particular audible frequency has been transmitted. D ur­ ing some intervals such a sound will be transmitted, but during other (randomly presented) intervals there will be only noise. She is to answer ‘yes’ if she thinks she hears the sound, ‘no’ if she thinks that she hears only noise. She is obliged to make one of these two responses for each interval. Her situation is such that she will always be uncertain about the correctness of her judgement on any par­ ticular occasion, although we might expect that she will feel more confidence in some cases than in others. There are four ways in which the presence or absence of the signal might match her response: (1) The signal is presented; she says ‘yes’—a “ h it” . (2) The signal is presented; she says ‘no’—a “ miss” . (3) The signal is not presented; she says ‘yes’—a “ false alarm ” . (4) The signal is not presented; she says ‘no’—a “ correct rejection” . Figure (1) represents the receiver’s decision situation. The abscissa represents the level of the subject’s sensory activation upon the presentation of the stimulus, while the ordinate indicates the pro­ bability that a given level of activation will be achieved. The lefthand bell-shaped curve indicates the activation distribution for the presentation of noise alone, while the right-hand curve shows the distribution which is obtained when a signal is added to the noise. The net effect of adding a signal is to move the whole distribution configuration to the right.

A

.~ :.ä o

'Correet rejeel,

.0

o

cl:

Noise Alone

Hits

Signal plus NOi,se Folse Alarms

Misses

A

c Figure Figure 11

B

Level of oel ivo I i on

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How is the receiver to decide whether the sound she hears d u r­ ing a particular interval is to be labeled ‘signal’ or ‘noise’? She could adopt a conservative strategy and not answer ‘yes’ until she feels comparatively confident that she is hearing the message tone. This strategy, represented by decision criterion line B in figure (1) will minimize the false alarms, but it will also maximize the misses. O r she could make every effort to tag each bit of signal she thinks she might be hearing, even if it means giving a large num ber of ‘yes’ responses to stretches of noise. Line A represents this decision criterion. O r she might adopt some intermediate decision strategy, such as that indicated by criterion line C, which assigns equal value to scoring a hit and to avoiding a miss. In the figure, the shaded areas represent the proportions of misses and false alarms which are to be expected if criterion C is used. The location of the criterion line depends upon the receiver’s inclinations, conscious or u n ­ conscious; nothing in the physical situation dictates it. It expresses the receiver’s bias. If the receiver’s hit and false alarm rates are established by her performance over a set of trials, it is possible to determine her sen­ sitivity, which is represented by d '; the distance in standard devia­ tions between the mean of the noise curve and the mean of the signal plus noise curve (it is assumed that both curves are normal and of equal variance). If the criterion and sensitivity are known, the hit and false alarm rates may be calculated. The three factors are thus to some extent like belief, desire and behavior, a trio familiar to philosophers, and, like belief and desire, the sensitivity and the criterion are not fully open to public view. However, an important disanalogy is that the hit and false alarm rates suffice to determine the sensitivity uniquely, without an independent specification of the criterion. Furthermore, there are differential empirical handles which we have on the sensitivity and the decision criterion, such as the susceptibility of the criterion, but not the sensitivity, to rewards and punishments. The important points to remember are that the criterion may be set independently of the sensitivity, and the sen­ sitivity has no fixed lower limit. In the ideal case, for a fixed criterion, any two repeatedly presented signals above the noise level may be distinguished from one another by the difference in their propor­ tions of hits to false alarms. All that is required is that the num ber of trials be large enough to distinguish those two proportions. (In actual cases, things are not so simple, since the receiver’s sensiti­ vity is apt to change with time.) With this very sketchy introduction to signal detection theory, we are in a position to apply the theory to the question at hand, which is to understand what is involved in what we commonly

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describe as the discrimination of the color (or, more particularly, some dimension of the color, such as the hue) of one sample from that of another. Here, the signal is the color (better: wavelength) difference between the samples; if there is no difference, any percep­ tual variation will be due to the noise alone, but if there is a dif­ ference, the activation pattern will shift to the right.5 It should be clear in general how the concepts of a decision criterion and the sensitivity of the receiver (here, of course, the observer) will carry over to this case. Over a series of trials, the presence of a signal—a wavelength difference—will manifest itself in a statistical difference in the ratio of hits to false alarms. The smaller the difference, the greater the number of trials required to uncover it. Let us now return to the situation which is commonly described as one in which sample x is indistinguishable in color from y, and y from z, but x is distinguishable in color from z. We have seen how, in such a circumstance, y could in fact be distinguished from both x and z, given a sufficiendy large num ber of yes-no matching trials. The data represent nothing beyond comparisons of the ap­ pearances of objects, but no given pair of comparisons is sufficient to decide the distinguishability or indistinguishability of x and y, or of y and z. Then why is a pair of comparisons, or at least a very small number of trials, often deemed sufficient to establish the distinguishability of x and z? Because when x and z seem definitely distinguishable from one another on one trial, the mean values of the signal plus noise curve (which represents color difference) and the noise curve will be about three standard deviations apart,6 so it is overwhelmingly probable that repeated trials would lead to the very same result. The usual specification of a just noticeable color difference under standard conditions is set at three standard devia­ tions in order to achieve an optimal balance between minimal obser­ vable difference and reliability in detecting the difference across trials for a given observer. One might of course acknowledge all of these things, and yet insist that the color difference between x and y is not observable in the usual sense, since no probabilistic property, extending as it must over several cases, can be determined to obtain just by look­ ing. There is of course something to be said for this use of ‘obser­ vable’. However, there are several reasons for construing ‘obser­ vable’ in a broader way. The first is that the observer is, in this case, improving her epistemic situation vis-á-vis her surroundings just by looking. The role of statistics is simply to get a measure of the improvement. Organisms constandy register and appraise small ‘‘subliminal” sensory changes. These are very im portant to everyday life, although they are usually ignored in epistemological

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discussions. Second, when x and y neither match nor seem distinct, that fact registers in the consciousness of the subject in the form of reduced confidence in her judgement, so it is not without phenomenal significance in the individual trial. Third, to the ex­ tent that the question is one of nomenclature, the practice of the sciences must be given some weight. In astronomy, for example, the probable error is given as a part of an observation report; if a report were given without a probable error it would be regarded as meaningless. Equally, a report with a smaller probable error than could be generated by a correct application of the particular obser­ vational technique in question would not be regarded as yielding useful information. In ordinary life we of course report our obser­ vations without stating probable errors, but we do so when preci­ sion doesn’t matter very much, and we cover our tracks with coarse­ grained predicates, within the boundaries of which our observation can be comfortably situated. But in borderline cases our everyday behavior changes in various ways, one of which is to take several looks at the problematic match. Fourth, if correcting and calculating are not to enter into the ascriptions of phenomenal color predicates, phenomenal color properties cannot be the objects of scientific study in any but the most casual sense. But in fact, the ways that the colors of things appear to people are of considerable interest in the workaday world, and substantial sums of money are spent in the precise study and control of those ways. There is such a thing as color science—and color technology too—and one of the central pro­ blems of this science is the nature of visual discrimination of one color sample from another and the circumstances under which it occurs. In any case, the statistical representation of discriminability is the optimally precise7 one to employ whenever the outcomes of dif­ ficult discriminations are to be described, since it tells us the most about the observer’s sensory information. Correspondingly, the op­ timally precise color predicates in such cases should be thought of as predicates of approximation which are standardly used in science to report an observation along with its probable error. O ne could, in principle at least, assign all color predicates accordingly. Such predicates of approximation would employ a standard system of perceptual color order such as the Munsell system based on just noticeable differences (statistically construed, of course) together with an indication of probable error.8 The predicates could, again in prin­ ciple, be made as fine-grained as one wished, although in practice, as we shall see, the differences between observers would make poindess the construction of predicates beyond a particular level of perceptual precision. In the sort of situation envisaged in Non-

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trans, x, y, and z would be assigned different predicates of approx­ imation, the error range of the predicate assigned to x overlapping that assigned to y, and the error range of the predicate assigned to y overlapping that assigned to z, with little overlap in the error of the predicates assigned to x and z. Since y is then assigned a different predicate from x and z, the sorites problem does not arise. One feature of our entire approach may trouble the reader. It might be granted that repeated trials could successfully distinguish a stimulus interposed between two stimuli a just noticeable difference apart. But surely there must exist two stimuli which are close enough together that no num ber of observations could succeed in distinguishing them. And if that is true, a sequence of such distinct but observationally indistinguishable pairs could be constructed be­ tween two observationally distinguishable stimuli, and the sorites argument could be resurrected. To this one may reply that there is no compelling reason to suppose that there is such an absolute discrimination limit in prin­ ciple,9 but every reason to regard many physically distinct stimuli as indistinguishable in practice. This is because one rapidly reaches the point in which the number of trials which is required to make a discrimination exceeds the ability of the observer and the ex­ perimenter to maintain constancy in the experimental situation. Boredom sets in, the organism’s sensitivity changes, the alignment of the apparatus shifts, etcetera, etcetera. Over the long run—the very long run—these perturbing factors distribute themselves in a statistically normal fashion. But in the shorter, humanly possible run, they bias the experimental results. The signal gets buried in the noise. In a similar way, it is practically impossible for a heap of gray sand consisting of black grains and white grains to be separated into a white pile and a black pile just by random stirring, or for someone to be asphyxiated by air molecules randomly segregating themselves into the opposite corner of the room. But this does not call into question the characterization of the distribution of sand particles or gas molecules as essentially statistical. So one must not confuse practical with theoretical perceptual indistinguishability. Now the assignment of predicates to color samples on the basis of perceptual distinguishability is very much a practical matter. One of our concerns here is whether we can construct a perceptually founded semantics of actual color language, and it is plain that the fine-grained color predicates which we have been discussing are constructable only in very carefully controlled circumstances and could have little bearing on the linguistic color practices of the “ real world” . But if the issue now turns on the conditions which govern the prac-

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tical application of color terminology, the sort of situation called for in Nontrans is equally suspect. For it doesn’t arise in everyday color-attributing practice either, and if it did, it would teach us lit­ tle or nothing about what underlies that practice. Let’s see why. When fine-grained predicates of color appearance are actually employed in color science or technology—they usually take the form of interpolations between Munsell color chips—their limits of ac­ curacy are tacitly understood; hue interpolations by practiced observers are accurate to about half of a Munsell step. This is not the limit of hue discriminability, however, but it is something like the limit of useful naming of phenomenal hues for the purposes of communicating between people, since the perceptual differences between individuals at a given time and between observations by the same individual at different times have at this point come to swamp the fine discriminations of which any one observer is capable on a single occasion. Ralph Evans remarks (Evans 1948, pp. 196-97), It is not realized ordinarily how great is the variation of observers in this respect. A rough estimate indicates that a perfect match by a perfect *‘average” observer would probably be unsatisfactory for something like 90 percent of all observers because variation between observers is very much greater than the smallest color differences which they can distinguish. Any observer whose variation from the standard was much greater than his ability to distinguish differences would be dissatisfied with the match.

Even if the assumption that there is a nontransitive “ observa­ tional” indistinguishability of colors were correct, and a sorites argu­ ment could be launched, it would have little bearing on a rationed reconstruction of the rules governing color predicates in a public language since such predicates are necessarily much coarser than the fine grain of just noticeably different colors perceivable by particular individuals. It should, then, be useful to inquire as to what phenomenal features do seem to underlie the ascriptions of colors to objects in ordinary language, and, in particular, to ask how and why boundaries between colors get drawn as they do. I shall ad­ dress these questions in the last section of the present essay. III.

O ur discussion has so far been conducted in terms which would not seem objectionable to a hardened behaviorist. The signal detec­ tion analysis disposes of the problem, and some might be content to leave well enough alone. But it raises a deeper question, the answer to which must, at the present state of knowledge, be quite speculative. It is this. W hat is the source of the statistical spread in the color

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discrimination experiments? There are three possibilities. It could be that the color qualities as experienced by the subject are sharpvalued and remain the same across repeated occurrences of the same stimulus pair. The spread is then a spread across time of the sub­ ject’s judgements rather than of the color qualities as she experiences them. O r it might be that the sensory presentations are sharp-valued and the judgements consistent, but that the sensory presentations differ slighdy in character from one trial to the next. In this case there is a spread of experienced qualities which the judgements only reflect. I want to explore a third possibility, that the spread indicates indeterminacy in the individual color experiences, and that sensory experiences are rarely, if ever, perfecdy sharp. A paradigmatic in­ stance of such sensory indeterminacy is to be found in the percep­ tions of shape and color at the periphery of one’s visual field. But if there is such phenomenal indeterminacy, how could it be modeled by determinate neural processes? We can in fact discover suggestive similarities between the task faced by a human subject in separating signal from noise, and the signal detection “ problem” which must be “ solved” by assemblies of sensory neurons. M any sensory neurons display incessant spontaneous activity, discharging randomly in the absence of stimulation. In the case of lower-level visual neurons, the effect of a stimulus may be either to increase or to inhibit neural firing with respect to that spontaneous base rate. The success of higher-level neurons in detecting and decoding a sensory message will depend upon their being able to uncover changes in the frequency of the outputs of those lowerlevel neurons. Since the basal firing of lower-level neurons is ir­ regular, higher-level neurons will have to sample the input from these neurons over a stretch of time if they are to establish that a particular criterion value has been met. The fundamental prin­ ciples involved here are illustrated in figure (2). Although the figure depicts the behavior of a peripheral neuron which originates on a skin receptor and terminates on a second-order neuron in the spinal chord, the neuron’s mode of function is closely similar to that of a typical post-receptoral visual cell.10 In figure 2, (a) shows a train of action potentials, i.e., voltage spikes, resulting from the application of a stimulus. In either the stimulated state or the spontaneous firing state the interspike inter­ val is irregular. The two are therefore only to be distinguished by a difference in the average number of impulses in a given time period, (b) plots the average number of impulses over several 100 milli­ second periods following stimuli of various strengths. Notice that

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the average num ber increases smoothly with increases in stimulus strength. If we turn our attention to the distribution of pulses d u r­ ing each 100 ms period, we see substantial variation. In (c) two histograms are displayed, showing the proportions of impulses dur-

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ing several 100 ms periods, ranging from 2 to 17 impulses per 100 ms. The lower histogram is a no-stimulus (spontaneous firing) record, with an average of 6 impulses, while the upper histogram, with an average of 12.2 impulses, represents the effects of a moderate stimulus. Both of these distributions have a standard deviation of ± 3 impulses. The reliability of detecting a stimulus depends on the signal/noise ratio, which is (12.2-6)/3 * 2.1 for a stimulus of the strength shown here. Because these distributions overlap, er­ rors are unavoidable when judging whether a stimulus was present or not from the number of impulses in a pulse train, and (d) shows that for a signal/noise ratio of 2.1 the error rate would be at least 15% on average. A stronger stimulus would separate the distribu­ tions shown in (c), and the error rate would fall as shown in (d). Just how closely related are an organism’s sensing a color and a neural assembly’s detecting a particular firing rate? There are a number of reasons for taking such visual experience to reflect the operational characteristics of individual neurons. First, any constraints on the operation of the early stages of an information processing system will be constraints on the capabilities of the system as a whole; information once lost can never be recovered. Second, irreducible visual noise has its subjective manifestation: it is the experience of gray which one has in the absence of all visual stimulation when afterimages and other controllable noisy ingredients have disappeared from the visual field. Known in the trade as “ brain gray” or “ eigengrau” , this phenomenal color is brighter than black; we only experience black by virtue of contrast with white. Notice, for ex­ ample, that parts of a television picture can look black, and darker than the television screen looks under normal room lighting when the set is turned off, yet the television picture is generated by adding rather than subtracting light. The usual physiological explanation of this is that phenomenal blackness is coded by an inhibition of the basal rate of firing of an achromatic visual channel. It is that base rate which is the neural counterpart of brain gray. Third, the spon­ taneous activity of the visual system shows up in the appearances of small flashes in the visual fields of subjects in detection threshold experiments when no stimuli are presented. These are interpreted by visual scientists as relatively large chance fluctuations away from the average which constitutes brain gray. (In the presence of a signal, this intrinsic noise is substantially, but not totally, inhibited.) Fourth, visual detection does not take place instantaneously, and, given moderate stimulus intensities, the color which a stimulus object is experienced as having will, within limits, vary with the length of time during which the stimulus is presented.11 In physics, one can-

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not, for very fundamental reasons, meaningfully attribute a definite frequency to a signal of very brief duration,12 and we can now see that for equally fundamental reasons one cannot meaningfully speak of an organism’s hearing a definite pitch or seeing a definite color during a very brief interval, an interval too brief for obtaining a proper sample. If the neural correlate of sensing a color is detecting a particular firing rate, the presence of noise in the system would cause there to be fluctuations around that firing rate accompanied by a cor­ responding indeterminacy in the sensed color. And if there is substan­ tial overlap between the noise curve and the signal plus noise curve (which represents the color difference information between the two stimuli) this should manifest itself phenomenally as an overlap in the indeterminacies of the two colors. In a high-precision yes-no matching situation this could show up as a match made with some subjective discomfort. Such discomfort might be experimentally displayed in, for example, the observer’s making lower estimates of subjective confidence in the match. And in fact, as the ex­ perimenter varies the test stimulus from a physical match with the comparison stimulus to a point at which the subject can reliably d istinguish the tw o—a classically specified ju st noticeable difference—the subject’s confidence in her ‘yes, there is a difference’ answers, which is very low in the immediate vicinity of high-reliability matches, gradually increases. It turns out that subjects’ relative level of confidence in their reports generally correlates very well with their discrimination rates, although they tend to underestimate how ac­ curately they perform (Vickers 1979, p. 185). So lower-level processes—improvements of the signal-to-noise ratio in the subject’s visual system—may leave their footprints in higher-level subjective phenomena—the subject’s feeling of confidence. The felt confidence would be, as it were, an inward sign of outward grace, and would serve to link more closely the felt character of the individual percep­ tion to the organism’s performance across a series of trials. If the reader finds this talk about perceived indeterminacy of colors to be either puzzling or gratuitous, he is invited to observe carefully the colors in the periphery of his visual field, and to try to match them with colors presented in the center of his visual field. There is a large experimental literature on this subject, and it proves to be the case that several centrally-presented samples which look quite distinct in color will match a particular peripherally-presented sample of comparable size equally well.13 This is sufficient to show the falsity of the widely held view that every discriminable colored patch is characterized by exactly one determinate color.14 Some col-

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ored patches will have to be assigned a range of colors in any event, and the force of the preceding considerations is that this will be true to a varying extent of every colored patch. We could model the difference between relatively determinate and relatively indeterminate phenomenal colors as the difference between a signal plus noise curve of relatively narrow dispersion and one with relatively wide dispersion, which has a considerable range of almost equally probable neural discharge rates. If some such picture represents what actually goes on in the nervous system, this has the potential for bearing significant philosophical fruit. For not only do we have vague color predicates, we have phenomenal colors which are indeterminate in the sense that patches of such colors can match equally well (or badly, since they won’t seem just the same) several patches of color which are phenomenally distinct from one another. For anyone who believes that there must be bearers of phenomenal colors, indeterminate colors are an em bar­ rassment, since they seem to require admitting objects with indeter­ minate properties into his ontology. But if to sense colors is simply to have appropriate neural processes, there is no need for concern, since the appropriate neural processes are statistical processes which supervene on determinate neural events. In a similar fashion, the thermodynamics of gases must in the last analysis employ either vague predicates or precise predicates of approximation which have as their intensions indeterminate properties. This is because the pressure, temperature and volume of a gas cannot be specified beyond a certain level of precision owing to its ultimately discrete constitu­ tion. But, as we all know, the thermodynamic predicates can be replaced by almost coextensive predicates which take as their inten­ sions statistical properties of the population of molecules which con­ stitute the gas, so we can accept vague or approximate ther­ modynamic predicates without countenancing aui irreducibly indeter­ minate ontology. IV.

Let us now return to the question of how the foci and boundaries of color terms are fixed in everyday language. It has been estimated that a trained normal observer can, under optimal conditions, discriminate something on the order of ten million surface colors, and there are about half a million colors which are considered to be commercially different (Judd 1963, p. 359). But the num ber of color names in use in English is much smaller; the Inter-Society Color Council—National Bureau of Standards Method of Designating Colors lists some 7,500 color names, and reduces these to 267

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equivalence classes (Kelly 1976). The names assigned to these classes are, in turn, compounded of just ten hue names, three achromatic names and a few modifiers such as ‘dark’, ‘light’, ‘very’, and ‘-ish’. In a famous cross-cultural study of basic color terms, Berlin and Kay (1969) found that eleven English color terms satisfied their criteria of basicness; all of the terms on their basic list are included in the Bureau of Standards basic list. A survey of seventeen best sellers (Evans 1948, p. 230) showed that of a total of 4,416 color term tokens, 4,081 are occurrences of just twelve terms, and half of the total number are tokens of ‘white’, ‘black’ and ‘gray’. There are several reasons for the enormous disparity between the number of possible color discriminations and the number of color terms in normal use. One of them, variability between observers, has already been mentioned. A second is the great difficulty people have in comparing colors under normal as opposed to standard condi­ tions of observation. Changing light, shadows, contrast, inhomogen­ eity of color across surfaces, and surface differences among materials such as texture and glossiness all make color appearance comparisons highly problematic. A third and obvious reason is that it is far more difficult to compare a perceived color with a mental standard than with another color seen at the same time. We can, then, confine our attention to the semantics of a relative­ ly small number of color terms, such as Berlin and K ay’s eleven. Eleven proved to be the maximum number of basic color terms in any one of the twenty natural languages studied by Berlin and Kay. Furthermore, the paradigmatic or “ focal” color chips for the basic color terms of a given language were very similar to those which were selected as focal by the native speakers of other languages with a comparable number of basic color terms. This is persuasive evidence that human beings do not carve up color space arbitrarily, but according to some sort of perceptual saliences common to the members of our species. The nature of these saliences and their biological basis have come to be fairly well understood. There is a substantial literature on this subject, including some remarks of mine (Hardin 1984 and 1985), so I shall summarize the necessary facts as briefly as possible. The visual system is configured into three types of antagonistic color channels: a red-green channel, a yellowblue channel, and a white-black channel. Color is signalled in any of them by a deviation in either a positive or negative direction from its basal firing rate. The color sensation associated with baserate firing in any of the channels is a dark gray, the “ brain gray” which has been previously mentioned. If either the red-green chan­ nel or the yellow-blue channel deviates from its base rate while the

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other does not, the perception of a unique hue—red, green, yellow, or blue—is the result. If both types of channels deviate for the same retinal region, the system codes a binary hue such as orange (reddish yellow) or purple (reddish blue).15 Now suppose we ask someone to rummage around a pile of Munsell color chips and pick out a red of medium lightness which is neither bluish nor yellowish, i.e., a unique red. Then we ask that person to locate a chip of higher lightness which seems to her to be just as reddish as it is yellowish. She proceeds to find a chip in accordance with each instruction. Next, we ask some other peron to find a chip which is the best example of red and a chip which is the best example of orange. After he has made his selections, we compare his two chips with the two picked out by the first per­ son. The empirical evidence suggests that it is very likely that her unique red will be the same as, or very close to, his “ best” red, and that her chip which is “just as reddish as it is yellowish” will correspond closely to his “ best” orange. An exercise of the same sort could be carried out for the other Berlin-Kay basic color categories, and it would be discovered that the other unique hues would match up well with “ best” yellow, green and blue, and that paradigmatic purple would prove to correspond to a balanced redblue. (We shall have to specify that the unique yellow be of high lightness, since low-lightness yellows look brown to most people.) We can thus suggest the principles which govern the location of the Berlin-Kay foci. As the phenomenal building blocks of all of the other hues, the unique hues have a psychological primacy which no other hues have; all of the spectral hues plus the purples can be adequately described using only the unique hue names, but the unique hues cannot be adequately characterized if we are restricted to the use of the names of other hues. (Red cannot, for example, be seriously described as a “ purplish orange” .) In the so-called “ color naming” experiments which support this claim ,16 subjects were not only able to describe all spectral hues using com­ binations of just four hue names, but proved to be remarkably skilled in estimating the proportions of each of the unique hues in a given binary hue. The binary hues such as orange which are perceptibly composed of pairs of approximately equal unique hue constituents seem to be almost as easy to pick out and remember as the unique hues themselves, although the absence of cyan or turquoise (bluegreens) from the basic list thereby becomes puzzling. (The absence of yellow-greens is less puzzling, since they are perceptually less salient.). But the boundaries of color categories seem to be another mat-

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ter. Here is what Berlin and Kay have to say about them (Berlin 1969, p. 13): Repeated mapping trials with the same informant and also across informants showed that category foci placements are highly reliable. It is rare that a category focus is displaced by more than two adja­ cent chips [The chips used by Berlin and Kay were 2.5 Munsell hue steps apart—CLH]. Category boundaries, however, are not reliable, even for repeated trials with the same informant. This is reflected in the ease with which informants designated foci, in con­ trast with their difficulty in placing boundaries. Subjects hesitated for long periods before performing the latter task, demanded clarifica­ tion of the instructions, and otherwise indicated that this task is more difficult than assigning foci. In fact, in marked contrast to the foci, category boundaries proved to be so unreliable, even for an individual informant, that they have been accorded a relatively minor place in the analysis. Consequendy, whenever we speak of color categories, we refer to thefoci of categories, rather than to their boundaries or total area, except when specifically stating otherwise. (Emphasis in text.) Let us address the problem of fixing color boundaries by con­ sidering a simplified case: the hue boundary between orange and red. We shall examine in imagination the phenomenal hue stretch between unique red and unique yellow. (One must always bear in mind that imagination is but an ersatz for experience, and that pro­ missory notes issued here must (and can) be backed by experiment.) T he stretch will be represented by estim ated percentages of phenomenal red and yellow in each of five hues in the range along with the hue names which are associated with them. 100% red (unique red) 75% red, 25% yellow yellowish red 50% red, 50% yellow yellow-red (orange) 25% red, 75% yellow reddish yellow 100% yellow (unique yellow) The three intermediate hues exhibit a perceptual tension between red and yellow. Yellowish red seems more red than yellow, while reddish yellow seems more yellow than red. The boundary of red in the broadest sense extends to the immediate neighborhood of unique yellow, and the breadth of that spread we acknowledge by our use of the modifier ‘reddish’. But in a somewhat narrower sense, the boundary between red and yellow falls at the point at which the perceptual “ pull” of yellow is equal to that of red. This point is, of course, orange. But once we introduce orange as a distinct hue category, its boundary with red is at issue, and the extension of ‘red’ must be contracted to make room for the oranges. The

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natural red-orange boundary would seem to fall at the 75% red, 25% yellow region which was well within the scope which we took ‘red* to have when we were concerned to compare red with yellow.17 The principle which appears to be operating is that the unique hues are at the core of (humanly) natural color categories, the ex­ tent of which will be governed by the phenomenal tensions between them and by the number of intermediate categories which come to be defined by a person’s culture and subculture as well as by the pragmatic requirements of the task at hand. And, as is usual in such cases, experts draw the boundaries more precisely and reliably than laymen. In the instance of colors, the common system of classification is simplified and made more precise and regular for expert use up to the point of precision of discrimination at which it begins to lose its value (because, for instance, individual differences have come to play too large a role). These ideas are exemplified in the hierarchy of color terms developed in the Universal Color Language of the Bureau of Standards and the Inter-Society Color Council (Kelly 1976). The language is divided into six levels. Level 1, the least precise, consists of 13 terms, the 11 of Berlin and Kay and ‘violet* and ‘olive’. Level 2 consists of the terms of Level 1 plus sixteen combination terms such as ‘purplish pink’. Level 3 comprises the 267 color terms of the ISCC-NBS Method of Designating Colors to which we have previously referred. Level 4 is based on color order systems such as the Munsell system and uses their designations. It comprises about 5,000 terms. Level 5 has about 100,000 designations, which represent visuallybased extrapolations and interpolations from the standard samples of the Munsell or other systems and are expressed in their nota­ tion. Level 6 abandons visual criteria for defining the stimulus and instead relies upon instrumental specification.18 The question, ‘What are the boundaries of red?’ has, in and of itself, no well-defined answer. It is first necessary to specify, ex­ plicitly or tacidy, a context and a level of precision and to realize the margin of error or indeterminacy which that context and level carry with them. Perhaps some philosophers have so much trouble with the concept of a color boundary because they too easily abstract from considerations of level and context or because they are hyp­ notized by the dogma that color predicates at the same level of specificity and, indeed, colors themselves, exclude one another. For neighboring unique hues and the predicates which go with them, the latter simply isn’t so, as that yellow-red which we call ‘orange’ adequately testifies. Hue exclusions may be handled properly by observing two principles governing binaries: (1) no binary may have

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complementary hues as its constituents; (2) the sum of the percen­ tages of the constituents of a binary must be 100%. For its part, orange can and does have red as a phenomenal constituent, so worry­ ing about border line cases because orange excludes red is disturbing one’s peace quite gratuitously. For each real-life color boundary question, there is a level at which it may be asked and answered in a reasonably objective way, even if the answer is that one is free to legislate as one likes. To settle the question of whether a casually-viewed piece of material is orange or red, one could look at it again and consult one’s more color-aware mate, treating the question as one to be answered on Level 2, perhaps. If one were stymied and sought further assistance, the sample could be compared under ISCC—NBS standard condi­ tions to a set of Munsell papers, making as close a match as possi­ ble and then referring to the Method of Designating Colors for a deter­ mination. This would be to ask a Level 3 question. But suppose that one had been in a fussy frame of mind, noticed that the match was inexact, went to the Really Big Book of Munsell papers, and found that the closest match was to 7R 5/8, which is directly on the boundary between “ 15—moderate red” and “ 37—moderate reddish orange” . What to do? The answer is plain: call it red or reddish orange—as you like. At this stage, one could try visual in­ terpolation, to see if the sample is really a Munsell 6.5R or perhaps a 7.5R, and decide the issue that way. But this would be to misuse the nomenclature of Level 3 by invoking Level 4 standards of ac­ curacy with respect to which Level 3 concepts are undefined. There is, to be sure, still a fact of the m atter as to what color the object appears to be, and of how that color is to be specified, but there is no fact of the matter as to whether it is red rather than reddish orange. To recognize that there is, in such a situation, no fact of the matter is neither to abandon realism nor hope of formulating seman­ tic rules, but simply to realize that the nature of our purposes and the capabilities of our natures impose limits on the precision of our utterances.19 R

eferences

American Society for Testing and Materials, “ Standard Method of Specifying Color by the* Munsell System,” Designation D 1535-68, 1968. Accompanies the Munsell Book of Color. (Baltimore: The Munsell Color Company, 1970). Armstrong, D. M ., A Materialist Theory of the Mind. (London: Roudedge and Kegan Paul, 1968). Barlow, H. B., “ General Principles; the Senses Conceived as Physical Instruments, ” in Barlow, H. B. and Mollon, J. D. (eds.), The Senses (Cambridge: Cambridge Univer­ sity Press, 1982). Berlin, B. and Kay, P., Basic Color Terms (Berkeley: University of California Press, 1969). Dummett, M ., “ Wang’s Paradox,” Synthese 30 (1975): 301-24.

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Evans, R ., An Introduction to Color (New York: John Wiley and Sons, 1948). Fuld, K., Werner, J. S., and Wooten, B. R ., “ The Possible Elemental Nature of Brown,” Vision Research 23, n.6 (1983): 631-37. Gordon, J., and Abramov, I. “ Color Vision in the Peripheral Retina. II. Hue and Satura* tion.” Journal of the Optical Sociäy of America 67 (1977): 202-207. Hardin, C. L., “A New Look at Color,” American Philosophical Quarterly 21, n.2 (1984): 125-33. Hardin, C. L., “ Are ‘Scientific’ Objects Coloured?” Mind XCIII, n. 372 (1984): 491-500. Hardin, C. L., “ The Resembkuices of Colors,” Philosophical Studies 48 (1985): 35-47. Hood, D. C. and Finkelstein, M. A., “ Detection and Appearance of Small Brief Lights,” in Mollon, J. D. and Sharpe, L. T. (eds.), Colour Vision (London and New York: Academic Press, 1943). Judd, D. B. and Wyszecki, G., Color in Business, Science and Industry. 2nd edition (New York and London: John Wiley and Sons, 1963). Kelly, K. L. and Judd, D. B., Color: Universal Language and Dictionary of Names, National Bureau of Standards Special Publication 440 (Washington: U.S. Government Print­ ing Office, 1976). Linsky, B., “Phenomenal Qualities and the Identity of Indistinguishables,” Synthese 59 (1984): 363-80. Moreland, J. D. and Cruz, A., “ Colour Perception with the Peripheral Retina,” Optica Acta 6 (1958): 117-51. Nachmias, J ., “ Signal Detection Theory and its Application to Problems in V ision,” in Jameson, D. and Hurvich, L. (eds.), Visual Psychophysics, vol VII/4 of Handbook of Sensory Physiology 1. (Berlin, Heidelberg and New York: Springer-Verlag, 1972). Parikh, R. 1983. “ The Problem of Vague Predicates, ” in Cohen, R. S., and Wartofsky, M. W. (eds.), Language, Logic and Method (Dodrecht: Reidel, 1983): 241-161. Peacocke, C., “ Are Vague Predicates Incoherent?” Synthese 46 (1981): 121-41. Schrödinger, E., “ Thresholds of Color Differences,” translated in MacAdam, D. L. (ed.), Sources of Color Science (Cambridge, Mass.: MIT Press, 1970). The article was originally published in 1926. Swets, J. A. (ed.), Signal Detection and Recognition by Human Observers (New York, London and Sydney: John Wiley and Sons, 1964). Travis, C. 1985. “ Vagueness, Observation and Sorites,” Mind XCIV n. 375 (1985): 345-366. Vickers, D., Decision Processes in Visual Perception (New York, San Francisco and London: Academic Press, 1979). Werner, J. S. and Wooten, B. R., “ Opponent Chromatic Mechanisms: Relations to Photopigments and Hue Naming,” Journal of the Optical Society of America 69, n. 3 (1979): 422-34. Wright, C., “ On the Coherence of Vague Predicates,” Synthese 30 (1975): 325-65. Wyszecki, G. and Stiles, W. S., Color Science (New York, London and Sydney: John Wiley and Sons, 1967). N

otes

lA cottage industry seems to have grown up around this argument. See Dummett (1975), Wright (1975), Peacocke (1979), Linsky (1984), and Travis (1985). The discussion I like best is Parikh (1983). 2A word to those with more technical background in visual science: The kind of situa­ tion I have in mind here is not metameric matching in general, but the circumstance in which the subject has to decide, given a series of randomly-presented pairings of color samples (light spots, say) in which there are small differences between the dominant wavelengths of some of the samples, whether or not a particular pair match in color. It is the typical “ difference threshold” experiment. What I am concerned to do in what follows is to apply to such experiments the kind of signal detection analysis which is commonly applied to the determination of “ absolute thresholds.” 3For instance, one finds Erwin Schrödinger explicidy but matter-of-facdy employing a statistical measure of color discrimination thresholds in 1926. See Schrödinger (1926; 1970). ♦Signal detection theory, on which the following discussion is based, was first ad­ vanced as an analysis of communication problems in electrical engineering in the early 1950s

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and was applied to psychophysics a few years later. An early collection of papers edited by one of the pioneers in the psychophysical applications is Swets (1964). For a more recent discussion of theoretical issues, see Vickers (1979). sTo talk about “ color differences” between the samples can be profoundly misleading, for it suggests the mistaken ontological view that colors are properties of physical objects. The appropriate stimulus difference which is to be detected by the subject by virtue of changes in her color experiences is a difference in the dominant wavelength of the reflected light. See Hardin (1984) for an argument against the identification of colors with wavelengths or with any other set of physical properties. T h e three standard deviation figure is based on the discussion of D. F. Mac Adam’s work in Wyszecki (1967), p. 529: “ MacAdam’s extensive auxiliary experiments on justnoticeable color differences indicated that for the same observer these are proportional to the corresponding standard deviations of color matching, the just-noticeable difference being about three times as large as the corresponding standard deviation.” In signal detection terms, when d' is about 3, there will be a criterion such that for every 75 hits there will be just one false alarm. 7An “ optimally precise” representation I take to be one which is the maximally precise meaningful representation under a given set of conditions of observation. •The Munsell system is described in the Munsell Book of Color and exemplified in a set of colored papers (chips) made to exacting specifications by the Munsell Color Company, Baltimore, Maryland, The Munsell Hue circuit is divided into 100 steps which are percep­ tually evenly spaced. For the precision with which the chips may be visually compared, see American Society for Testing and Materials (1968). This accompanies the Munsell Book of Color. Here, for the technically initiated, is an example of a Munsell-based predicate of ap­ proximation: ‘is of Munsell Hue 5 ± .5 Red’. •Of course when information is once lost, as wavelength information for an individual photon is lost when it is absorbed by a visual photopigment, no amount of subsequent pro­ cessing can recover it. The chromatic information available to higher-order visual cells con­ sists of the ratios of receptor excitations. The argument being made here is that in principle any two distinct ratios of receptor excitations are distinguishable given a sufficiently large number of trials. Some may disagree with this claim. There have been several hybrid signal detectionthreshold theories propounded, and there is a continuing controversy over whether absolute thresholds are limited by intrinsic noise. These are largely disputes about detection rather than difference thresholds. Furthermore, two points seem not to be in dispute: sensory systems arc characterized by intrinsic noise, and thresholds are statistical constructs. These points arc really all that are required to support the principal burden of the argument of the present paper. For a review of some of the controversies about the application of signal detection theory to visual problems, see Nachmias (1972). 10Figure (2) and the description of it which follows are adapted from Fig. 1.1 of Barlow (1982). “ A small, brief (10 minutes of visual diameter, 40 milliseconds duration) spot will look different from a larger, longer lasting (49' diameter, 500 msec, duration) spot of the same wavelength composition. Cf. Hood (1983). 12 Fourier analysis tells us that the shorter the time of emission of a signal, the greater the range of frequencies it must contain. 13In a private communication, visual scientist Davida Teller disputes this way of put­ ting it. In her view, what happens is that small peripherally-presented chromatic stimuli look more desaturated than foveally-presented ones (red, it is generally agreed, is an excep­ tion), so that they can presumably be matched one-to-one to desaturated foveal stimuli. There are several comments to be made in reply. First, one can to some extent judge for oneself. The present writer finds the chromatic deliverances of his peripheral vision to be unequivocally “ generic” . Second, it is plain that peripheral visual shape does have this generic quality, and the philosophical freight could be delivered by this fact alone. Third, the number of distinguishable hue steps is reduced as a sample becomes progressively less saturated, so it would be true that in foveal (central) vision, a given desaturated sample could equally well (or badly) match in hue several close, but distinct saturated samples. So even if Teller is

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right, the problem of generic hues could be rooted in desaturation. But finally, at least some of those who have done the experimental work on peripheral hue-naming seem to think that there is a factor besides desaturation in peripheral color vision. Cf. Gordon (1977): “ At these eccentricities-(50, 40, and 72 degrees), color-matching functions are grossly dif­ ferent from foveal ones, and lights, especially from the middle of the spectrum, appear desaturated and of uncertain hue.” (p. 202) “ Subjects often commented that, in the periphery, hues were often somewhat uncertain; the exception was red: whenever any red appeared in a stimulus, subjects had much more confidence in their judgements. . . . in the periphery, the small (1.5 degree) target was generally achromatic and of uncertain hue” (p. 204). 14See, for example, “ observational shade principle (iii)” in Peacocke (1981), p. 131. l4Notice that we have used nine color names: ‘red’, ‘green’, ‘yellow’, ‘blue*, ‘orange*, ‘purple’, ‘black’, ‘white’, and ‘gray’. O f the eleven Berlin and Kay basic terms, only two have been omitted: ‘pink* and ‘brown’. Pink is easily described as a light, bluish red of low saturation. Brown poses a more difficult and controversial problem. Cf. Fuld (1983). Fortunately, its resolution will not be required for our present purposes. 1#There have been several such experiments. I consulted Werner (1979). ,7As a rough check on this conjecture, one may find the spectrum locus of the 75% red, 25% yellow phenomenal estimate in a color naming experiment (I used one by Werner 1979) and locate a Munsell hue chip which has a dominant wavelength close to that spectral locus. The association of a dominant wavelength with a Munsell chip may be made by means of the Munsell-C.I.E. conversion charts in American Society for Testing and Materials (1968). In chis case, the locus is about 620 nanometers and the roughly corresponding Munsell hue for a chip of high Chroma and middle Value is 7.5 Red. If we then consult the National Bureau of Standards color designation system, we find that such a chip is described as “ moderate reddish orange” and is immediately adjacent to the region specified as “ moderate red” . As a further check, it is interesting to look up the red and orange category boundaries which Berlin and Kay mark down for American English. Their informants included this same Munsell chip within the boundaries of both ‘red* and ‘orange’. Cf. Berlin 1969, p. 119. ,aThe instrumental specification, such as the 1932 C.I.E. standard, will be useful for predicting color matching, but not color appearance. Level 6 is thus somewhat differendy conceived than the other levels. 19I would like to thank Jonathan Bennett, Tom McKay, Peter Van Inwagen, Davida Teller, Austen Clark and George Graham for their useful comments. This paper was written during a leave of absence made possible by the National Endowment for the Humanities and Syracuse University.

[8] CHRISTOPHER PEACOCKE

A R E V A G U E P R E D IC A T E S IN C O H E R E N T ? *

Does the Sorites paradox show a wide class of observational expres­ sions to be incoherent? Michael Dummett has argued that it does.1 To accept Dummett’s argument is not to be forced to abandon observational predicates altogether. His argument does not apply to all observational predicates: one can retain ‘is discriminably different from’ and such comparatives as ‘is yellower than’ and ‘is balder than’. But clearly we could not express everything we originally wanted to express. A description of an object using vocabulary to which Dummett’s argument does not apply will not always settle the question of whether it is green, or whether it is bald if it is a person. My aim in this paper is to offer a diagnosis which does not blame the Sorites paradox on the incoherence of certain vague predicates, and which allows it to be literally true that an object is green. First I will consider some other reactions to Dummett’s argument. They are reactions with which it is hard to rest content. In consider­ ing why this is so we will discover properties which must be posses­ sed by any more satisfying reaction. i

Crispin Wright has suggested that one possible reaction to Dummett’s argument is to say that the paradox establishes the following con­ ditional conclusion: if we regard understanding an expression as grasping certain kinds of rules which are to govern its use, then these vague observational predicates are incoherent. This theorist Wright envisages contraposes and concludes that such a conception of understanding is mistaken. The use of such predicates cannot be completely determined by a set of incoherent rules; for, the theorist will say, “our use of these predicates is largely successful', the expectations which we form on the basis of others’ ascriptions of * I am greatly indebted to Crispin Wright for comments on an earlier draft of this material. Synthese 46 (1981) 121-141. 0039-7857/81/0461-0121 $02.10. Copyright © 1981 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

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colour are not normally disappointed. Agreement is generally possible about how colours are to be described. . . ” .2 So he says that “the methodological approach to these [vague observational] expressions, at any rate, must be more purely behaviouristic and anti-reflective, if a general theory of meaning is to be possible at all” (ibid., p. 247). This suggestion is not sufficient to defuse the Sorites paradox. For consider this predicate C of objects: C(x) iff x is such that the community will agree in calling it ‘red’. Now suppose too that a difference d in the wavelength (w/1) of light is not visually discrimin­ able by any member of the community; and that light of w/1 k is definitely red. Then we can still construct this paradox: If a reflects light of w/1 k, then C(a). If an object differs in the w/1 of light which it reflects by just d from something that is C, it too is C. All visible objects (reflecting pure light) are C. Here an absurd conclusion has been drawn in terms of the vocabulary we use for describing the linguistic practices of the community. The presence or absence of behaviourism is not obviously the problem: this paradox must be resolved even if the predicate C has a behavioural definition. Thus the paradox seems to arise even if we do not suppose that the use of these expressions is governed by rules.3 It might be objected that since ‘C ’ is not an observational predicate, the second premise of this new paradox is not true; and so it might be concluded that the reaction Wright considers is not vulnerable to this difficulty. Now it is certainly true that ‘C ’ is not literally an obser­ vational predicate. But it is related to an observational predicate, viz., ‘red’, in such a way that the reasons given for saying that the second premise of a Sorites argument must be true are applicable to the second premise of this new argument too. The reason given for accepting the second premise of a standard Sorites argument using the predicate ‘red’ is that it would be inconsistent with the observationality of ‘red’ to suppose that a difference d of wavelength which is not visually discriminable could make the difference between a situation in which the predicate applies and one in which it does not. But it would be equally inconsistent with the observationality of ‘red’ to suppose that a difference d of wavelength, something ex hypothesi not visually detectable by any member of that community, can make

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the difference between an object being such that the community will agree in calling it ‘red’ and an object without that property. One proposal for avoiding this metalinguistic paradox might be based upon a comparison between a person or a community using an observational predicate with a measuring instrument the readings of which are displayed in digital form. A given small change in the magnitude presented to the instrument may or may not produce an alteration in its reading, depending upon the internal state of the instrument: there will be an alteration if the instrument is internally sufficiently near a threshold of sensitivity. For such an instrument, an analogue of the major premise of the metalinguistic paradox would be false: whether the instrument produces a different response to a given change in what is presented to it must depend on its internal states. Can we not give a similar description of the linguistic practices of a person or community using a vague predicate? There is, however, a disanalogy between the cases which prevents the instrument from being used as a model to avoid the metalinguistic paradox. An observational predicate is one whose application to an object can be determined from the kind of experience produced by that object in standard conditions: in particular, if two objects produce experiences which are not in quality discriminably different from one another, it cannot be that an observational predicate definitely applies to one of the objects and does not definitely apply to the other. Now any model for an application of vague observational predicates must provide analogues of three things involved in such application: there must be states which are the analogues of having experiences, there must be something analogous to the continuity of experience as reflected in the nontransitivity of nondiscriminable difference, and there must be some analogue of the application of an observational predicate upon a particular occasion. It seems impossible to provide for all three of these in the model of the digital instrument. Suppose we took the instrument’s internal states as analogous to the having of experiences and the display of a digital reading as analogous to the application of an observational predicate. What could be the analogue of continuity of experience? Are we to say that the analogues of experiences which are not discriminably different in quality are in­ ternal states of the instrument produced by physical magnitudes differing by less than a specified amount d? This would destroy the analogy. There are internal states either side of the threshold cor-

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responding to a particular reading which are produced by physical magnitudes differing by less than d. So under this analogy there would be no difficulty in the idea of an observational predicate (correspond­ ing to one reading) definitely applying to one but not to the other of two objects which produce experiences which are not discriminably different in quality. But in fact we can make no sense of this idea. It may seem tendentious to use the concept of experience in this argument against one attempt to avoid the metalinguistic paradox. Was not Wright explicitly concerned with a more ‘behaviouristic’ characterization of the use of language? But ‘behaviouristic’ here did not mean: behaviourally specifiable. It meant: not based on the suggestion that a speaker’s application of predicates is governed by rules he uses to guide him in the use of language. Suppose, perhaps per impossible, there were some way of blocking the paradox involving ‘C ’ which could not equally be applied directly or indirectly to the original object language paradox involving ‘red’. Should we then be satisfied with saying: “The community will agree in calling certain things ‘red’ and not others. We can describe this state of affairs without paradox, but we cannot say under what conditions they will agree in calling something red.” ? No: to stop here would be to fail to say what information is conveyed by an utterance containing ‘red’. A theory of meaning which did stop here would not be something which, if known, would put someone in a position to understand the language of the community. Mark Platts has also reacted to Dummett’s argument.4 Platt says that “We grasp the use of a vague predicate at least m part through a group of paradigm exemplars of them.”5 If a patch of colour is not discriminable from a red patch, it is itself red. But, Platts holds, one cannot construct a Sorites paradox with the predicate ‘is a paradigm red patch’: he writes that “indiscriminability from a paradigm of red does not of necessity mean that we have another paradigm of red.”6 This suggestion seems to ignore the role of observationality in the claims of Dummett and Wright. Dummett and Wright made a case that an observational predicate must be applicable to both or neither of a pair of objects which are not discriminably different. Having in no way undermined or qualified the principle, Platts’ view is open to a simple dilemma. Either ‘is a paradigm of red’ is an observational predicate, or it is not. If it is, then the Sorites paradox can be restated with respect to it. If it is not, the account he gives of what it is to

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understand ‘red’ makes ‘red’ not an observational predicate, which seems wrong. Moreover, while the principle relating observationality and indiscriminability stands in unqualified form, the original paradox using ‘red’ itself has not been defused.7

il

It is sometimes suggested that the Sorites paradox can be neutralized by making proper use of the point that vague predicates are predi­ cates of degree: it can be that one thing is red to a greater degree than another. Wright has argued that such considerations cannot block the paradoxes.8 Let us initially explain the notion of degree thus: two objects are red (say) to the same degree iff any object not discrimin­ ably different from one of them in respect of colour is not discrimin­ ably different in respect of colour from the other. There are many ambiguities and indeterminanacies in this definition, but let us ignore them just at present, since the main point we want to make holds under all ways of resolving them. The important point is that the degree to which an observational predicate applies to an object is not itself an observational matter, in the following sense: two objects can be not discriminably different from each other, and yet the degree to which a given observational predicate applies to the two objects may be different. There is a familiar argument for that conclusion. Suppose for reductio that it were not so; that is, suppose that any observational predicate applies in exactly the same degree to any pair of objects which are not discriminably different. Now consider a triad of red objects a, b, c where a is not discriminably different (“d.d.” ) from b b is not d.d. from c a is d.d. from c; in particular a is redder than c. Then the degree to which ‘red’ applies to a must be the same as the degree to which it applies to b, under the hypothesis of the reductio ; since the degree to which ‘red’ applies to b is similarly the same as the degree to which it applies to c, it follows that the degree to which ‘red’ applies to a is the same as the degree to which it applies to c. But that is not consistent with the fact that a is redder than c. Hence

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we must conclude that a predicate can apply in different degrees to objects not d.d. from one another. The notion of degree we are using here is close to Goodman’s concept of identity for qualia. Suppose we are prepared to quantify over colours as universais and to treat matching as a relation between such universais. Then we can say that object x and object y are red to the same degree iff any colour matching the colour of either one of them matches the colour of the other. Similarly, the degree to which x is red is greater than the degree to which y is red iff some colour matching the colour of x is redder than the colour of y. If we want to be free of the quantification over colour universais, or want at least to define the matching relation between them in terms of relations between particulars, there are severe difficulties if we do not want to take identity of shade as a primitive notion; but since these difficulties equally affect sharp observational notions, and so cannot be the source of the Sorites paradox, I will relegate them to a footnote.9 In any case, we should note that although difference of degree to which an observational predicate applies is not always an observational notion, the notion of degree has been explained in terms of obser­ vational notions such as ‘matching’ (nondiscriminable difference), plus logical notions. In this sense the notion of degree does not go beyond distinctions manifested in the abilities exercised by the speakers of the language; this is in contrast with those who are prepared to employ, for instance, a sharp notion of a family of admissible valuations in giving a semantics for vague expressions. If we relativise the major premise of the traditional Sorites argument by using this notion of degree, do we still obtain the paradox? When we relativise we obtained this: If an object is red and a second object is not d.d. from the first, then the second is red to a degree which one cannot determine just by looking at those two objects to be different from the degree to which the first is red. Our point was that nevertheless these degrees may be different. So we cannot relativize the premises of the Sorites argument and show that an orange is red to the same degree as a British pillar box. What of the original major premise in the colour version of the paradox? Is it still true that anything not d.d. from a red object is itself red? We now have to hand the materials for arguing that it can

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be simultaneously true that a is not d.d. from b while the conditional if a is red, b is red has a consequent with a lower degree of truth than its antecedent. I claim that the Sorites paradox shows that there cannot be a con­ ditional possessing both of these properties: (1) (2)

modus ponens inferences for this conditional are valid without restriction for some observational predicate F, and any objects x and y (named by a and b respectively), if x is not discrimin­ ably different from y, then the conditional with antecedent ‘a is F ' and consequent ‘ft is F ' is true.

One can interpret the conditional in such a way that the major or conditional premises of the Sorites argument are true, by (say) counting a conditional as true if its antecedent and consequent do not discriminably differ in their degree of truth; but then of course modus ponens will not be unrestrictedly valid. Alternatively one can retain modus ponens and require (consequentially) that such a conditional is true only if its antecedent and consequent have exactly the same degree of truth: and then the major premise of a Sorites paradox is false. But one must stick to one of these two courses consistently. So my suggestion is that the paradox results from the use of a con­ ditional taken to satisfy incompatible conditions, rather than from any incoherence in vague predicates. (In fact (2) here is stronger than is necessary for the properties to be incompatible: it suffices to use as antecedent the condition that x and y do not differ sufficiently for F to be applicable to one and not to the other of the two.) The question now remains of how on this proposal we can give some positive characterization of observationality. It was, after all, such characterizations which led Dummett and Wright to find the major premise of the paradox so compelling. But there are ways of stating the connection between nondiscriminable difference and observationality which do not lead to paradox. In particular, we can say that if F is an observational property, then if two objects x and y are not discriminably different (in respect of a given kind of pro­ perty), then it is not the case that x is definitely F and y is not definitely F ; in symbols, ififaa isifred, a isb red, is red b is red

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This last conditional has a sharp antecedent and a consequent whose truth can be a matter of degree: it resembles “ If the ball is in this urn, it is red” . If F is an observational property, then whenever the antecedent 7xy’ is true, the consequent will be true enough for the whole conditional to be true (all this relative to a given assignment to the variables). But to obtain a Sorites paradox from this conditional, we would need something equivalent to the principle Vagueness Vagueness

128 128 128 128

This principle should be rejected if Ixy is to be sufficient for ~(DFx & ~DFy): for if y is F to a lesser degree than x, then the conclusion will have a lesser degree of truth than the premise DFx. Again, we should not be surprised to obtain paradoxes from this principle. There is an apparent problem for this general diagnosis over the metalinguistic paradox. The predicate C is not a predicate of degree. For any given object, either all members of the community will agree in calling it ‘red’, or not all members will call it ‘red’. Here there is no room for talk of vagueness or matters of degree. C is a sharp predicate, and it does not make sense to say that it is true in a higher degree of some objects to which it applies than it is of others to which it applies. Does not the metalinguistic paradox show then that my diagnosis does not cover all examples of Sorites-like paradoxes? I reply that it covers the metalinguistic paradox indirectly. Some­ times we have to state a theory about the extension of a sharp predicate by using a vague predicate. In particular is this true of the sharp predicate C and the vague predicate ‘red’. We have to say that: Any object which is red the community will agree in calling ‘red’; any object which is not red the community will not agree in calling ‘red’. I suggest that the only reason that we feel tempted to accept the major premise of the metalinguistic paradox is that we employ reasoning using conditions with properties (1) and (2) and which contain ‘red’, and then go on to apply these two general principles to draw conclusions which contain the predicate C.

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Without using the conditional which has to have incompatible pro­ perties, we have no reasons for believing the major premise of the metalinguistic paradox. Insofar as inductive evidence might support that major premise, it must, if enlarged by further investigation, eventually refute the premise. Since C is sharp we have no option but to interpret the conditional in the major premise as a classical material conditional (rather than anything whose semantics mentions degrees), and then some instance of this premise must be false if the principles displayed earlier in this paragraph are true. Since C itself is not an observational predicate, there is no pressure against this conclusion: all we used earlier in raising the metalinguistic paradox as a problem for one position is that it shared a property with the obser­ vational predicate ‘red’, viz. that if x is definitely red and y is not discriminably different from x, then it is not the case that y is not red and not the case that y is not C. Let us return to the object language paradox. To say that (1) and (2) are incompatible properties is not to imply that there is no inter­ pretation of the conditional on which modus ponens is valid without restriction and on which ‘if a red b is red’ is sometimes true where a and b name observationally indistinguishable objects. We do not need to dispute the validity of the inference Vagueness

not the case that y is not C. Vagueness Vagueness Vagueness

notnot thethe case case thatthat y isy not is not C. C. Vagueness VaguenessVagueness Vagueness Vagueness

To account for the validity of these inferences, we need only to construe the conditional rif A then B 1 as true when either A is definitely false or B is definitely true, when A and B have definite truth values. For such a partially defined conditional, modus ponens will always preserve definite truth; this conditional does not have property (2). Indeed, for such a conditional the truth value of the whole depends just on the truth values of its constituents, and not upon their degrees of truth. Using a distinction of Dummett’s we may

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say that the content sense and the ingredient sense of A and B with respect to this conditional are identical: .. we must distinguish between knowing the meaning of a statement in the sense of grasping the content of an assertion of it, and in the sense of knowing the contribution it makes to determining the content of a complex state­ ment in which it is a constituent: let us refer to the former as simply knowing the content of the statement, and to the latter as knowing its ingredient sense”.12 In the case of a conditional the truth value of which has to be specified as a function of the degree of truth of its constituents, ingredient sense and content sense would come apart. We might attempt to extend the interpretation of the conditional for which the inferences displayed in the last paragraph are valid so that its truth value is determinate in other cases too. But there is not a new source of paradox here. If the extended specification has property (2), as it would if we said that rif A, then ß -1 is to be definitely true when A and B differ in degree of truth indiscriminably, then modus ponens will no longer be valid in general. It would be implausible to claim that the “ if” of English determinately either abandons property (1) or abandons property (2). In practice we are prepared to reject false conclusions arrived at by Sorites-like reasoning, while not being prepared specifically to blame either the form of the inference or some conditional instance of the major premise. As a consequence, no conditional with a determinate formal semantics and possessing either property (1) or property (2) can claim to express precisely the meaning of the English conditional used in the presence of vague predicates. The vague expressions we have considered so far here have been observational predicates. But an expression can be vague and feature in Sorites-like reasoning essentially while being neither a predicate nor in any natural sense observational. The quantifier ‘many’ is an example. We can often reach false conclusions from the true pre­ mises of the form: Many F ’s are G For any object x which is F and G, if many F ’s are G then many F ’s are G and distinct from x. For example, suppose the complete list of members of some society is a, . . . , z, and that many members of the society, c , . . . , z, say, voted for the resolution. Repeated application of the two displayed premises

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can lead us from these true suppositions to the false conclusion that many members of the society voted for the resolution and are distinct from each of c , . . . , z. The diagnosis of the paradox as resulting from the incompatible conditions placed on the conditional still applies, since whether many F ’s are G is a matter of degree. It is most important that to argue for the coherence of some observational predicates where nondiscriminable difference is non­ transitive is not thereby to argue for the coherence of a notion of an observational shade. Indeed, Dummett does show that the notion of an observational shade is incoherent. Let us consider just the case of colour. The concept of an observational shade is intended to conform to these three principles: (i) (ii) (iii)

if x and y are discriminably different, they are not of the same observational shade if x and y are not discriminably different, they are of the same observational shade. an object has at most one observational shade (at a given point on its surface, in the case of colour).

It is obvious that these principles lead to contradiction in the case of a triad a, b, c where a is not discriminably different from b and b is not discriminably different from c. a and c (at given points) have different observational shades (by (i)). But what shade can b have? It has to be both the same as that of a and the same as that of c (by (ii)); yet the shades of a and c are distinct. The argument to this contradiction does not employ any condition­ als whose consequents have a lower degree of truth than their antecedents. There is no obstacle to accepting this proof of in­ coherence while retaining our resolution of the Sorites paradox. (Ordinary colour predicates do not, of course, conform to a principle analogous to (i).) With this distinction between observational shades and vague predicates in mind, let us consider Dummett’s example of the slowly moving pointer: I look at something which is moving, but moving too slowly for me to be able to see that it is moving. After one second, it still looks to me as though it is in the same position; similarly after three seconds. After four seconds, however, I can recognize that it has moved from where it was at the start, i.e. four seconds ago. At this time, however, it does not look to me as though it is in a different position from that it was in one, or even three, seconds before. Do 1 not contradict myself in the very attempt to

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express how it looks to me? Suppose I give the name 'position X ’ to the position in which I first see it, and make an announcement every second. Then at the end of the first second, I must say, ‘It still looks to me to be in position X’. And I must say the same at the end of the second and the third second. What am I to say at the end of the fourth second? It does not seem that I can say anything other than, ‘It no longer looks to me to be in position X '; for position X was defined to be the position it was in when I first started looking at it, and, by hypothesis, at the end of four seconds it no longer looks to me to be in the same position as when I started looking. But, then, it seems that, from the fact that after three seconds I said, ‘It still looks to me to be in a different position from that it was in after three seconds’, that I am committed to the proposition, ‘After four seconds it looks to me to be in a different position from that it was in after three seconds'. But this is precisely what I want to deny.1’

What is going on here? Dummett introduces a notion of position which conforms to principles (i)-(iii); and then for that notion derives the contradiction as reached in the previous paragraph. He then remarks One may be inclined to dismiss Frege’s idea [that the use of vague expressions is fundamentally incoherent] if one does not reflect on examples such as these.

But this example points up only the incoherence of observational shades, and not vague expressions more generally. The argument Dummett gives in the case of observable position could not be reproduced for such predicates as “in the left of one’s visual field” without using Sorites-type reasoning.14 iii

Does it make sense to suppose that the world itself is vague? If so, does it undermine our resolution of the Sorites paradox? It is natural to construe the suggestion that the world itself is vague as the suggestion that the world has to be described by (inter alia) vague expressions, where this need is not in some way a result of limitations on our capacities.15 An uninteresting way of interpreting this suggestion is as the denial that vague expressions can have sharp translations. This is uninteresting because it is reasonable to require that translations preserve vagueness: so this interpretation of the suggestion hides substantive philosophical issues. One formulation which does not make the issue vanish is this. Suppose we have a language L containing vague expressions. Then the suggestion that the world itself is not vague is the suggestion that there will be some conceivable language L 1 which contains no vague expressions and which has the following property: it is a priori that if two situations agree in all respects describable using the

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language L \ then they agree in all respects describable using the language L. This is a form of supervenience. I shall say that the vagueness of a vague expression E is superficial if for any language L whose sole vague expression is E, there is some language L 1containing only sharp expressions, and such that the descriptions of L supervene on those of L 1in the sense just explained. It would not be disputed that the vagueness of some expressions is superficial. The quantifier ‘many’ is an example. The truth values of sentences containing ‘many’ will supervene on those sentences not containing ‘many’ but containing cardinality quantifiers: there cannot, for example, be two situations with respect to one of which some sentence of the form ‘Many F ’s are G ’ is true and with respect to the other is false, if the situations have the same number of F ’s being G. On this construal, then, the thesis that the world itself is vague would be the thesis that not all (possible) vague expressions have merely superficial vagueness. I mention this interpretation to distinguish it, in thought at least, from another sense in which it might be said that the world itself is vague. This is the vagueness that is denied by the principle that for any simple property the presence of which in an object is a matter of degree, the relation “x is more (or -er) than y " is a total ordering: thus the degree to which one object is must be either greater, less or the same as the degree to which some other object is . There cannot be incomparable degrees. The restriction to simple properties would be difficult to make precise, but is clearly necessary for the principle to be plausible: no one would expect in advance the orderings ‘x is a better novel than y' or 'x is a better city to live in that y’ to be total, and this seems to be a result of the fact that so many different comparisons would have to be made before one could make reasonable judgements of such orderings between particular objects. There is no reason in advance why some combinations of these component comparisons should not result in there being pairs of cities which are incomparable in respect of which is better to live in, while one also has no reason to say that they are equally good. On the other hand, it may be said that it is one interpretation of the claim that the world itself is not vague that this situation cannot arise for simple properties. The claim is certainly plausible for a wide range of simple properties: we find it hard to make sense of the possibility that of two rods neither is physically longer than the other and yet they are not of equal length.

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Are the degrees to which something may have the colour red totally ordered? If every coloured object had (at each point and at each time) a determinate Goodmanian shade, then, provided we hold such fac­ tors as brightness and saturation constant, there would be a total ordering of degrees of redness. But does every object have a deter­ minate Goodmanian shade? There seems to be a radical indeterminacy in applying Goodmanian shades to actual objects or experiences, of a kind which suggests that the circularity we discussed some pages back is indicative of an important point, rather than being a reflection of limited ingenuity in formulation. Suppose, to give the Goodmanian shades as favourable a chance as possible, we do not question that there are objects of any arb­ itrary Goodmanian shade (though this may already commit us to the existence of infinitely many things). Consider two objects a and b which are simultaneously perceived by the same person, and which are not juxtaposed and which are not discriminably different from one another. Suppose too that if they are moved into juxtaposition, there is a colour boundary between th e m -th e y are then discriminably different. (Alternatively for the conclusion of the present argument it suffices that some third object which matches one but not the other in colour is moved in between the two given objects and is in jux­ taposition with both of them.) One might take the presence of this visible boundary as conclusive evidence that a and b had different Goodmanian shades all along, since things have identical Goodmanian shades only if they match in colour exactly the same things. But one would be wrong to do so. There is nothing in the situation as described to rule out the supposition that the Goodmanian shades of a and b were the same until they were moved into juxtaposition, and just before this moment the shade of one of them (which one?) altered indiscernibly so that they do not match when juxtaposed. There is a serious question whether there is anything for the difference between these two hypotheses to consist in. It is important that the problem here is a constitutive one, one of meaning, and not one of verification. Dummett has remarked “ Although, as is wellknown, some philosophers have gone down this path, it will seem quite unreasonable to deny that someone who was capable of telling, by looking or feeling, whether or not a stick is straight knew what it was for a stick to be straight, on the ground that he would not thereby show that he knew what it was for a stick which no one had seen or

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touched to be straight” .16 Straightness is a primary quality; if some object alters in respect of this property while it is unobserved, there is no difficulty in saying what the alteration consists i n - i t must involve some redistribution of matter in space. But constancy of secondary qualities must, in given observational conditions, depend on constancy of experience. An indiscriminable alteration in the shade of some object produces ex hypothesi no change in the experiences produced by the object. There seems to be nothing for such alterations to consist in. The impossibility this produces of assigning determinate, totally ordered Goodmanian shades to objects has nothing to do with the inexpressiveness or inadequacy of our language: the point remains however much we refine the language, for the point concerns the nature of experience itself.17 The conclusion to be drawn from these considerations is this, if we wish to continue to hold that all objects which are red are red to some degree or other, then we must confine ourselves to making those statements of degree which are true under every assignment of Goodmanian shades to objects consistent with the requirement that if one object is perceptibly redder than another, then it is red to a greater degree. But this has the consequence that in our example of a and b in the previous paragraph, neither is red to a greater degree than the other, nor are they equally red. For some assignments of Good­ manian shades (consistent with the requirement just mentioned) make a red to a greater degree than b, and some make b red to a greater degree than a. Thus the resulting final degrees of redness are not totally ordered.18 If this is correct, then our resolution of the Sorites paradox can stand: it is just that we must not naively take the degree to which objects may have some property as always totally ordered. One can accept this resolution while holding that on one interpretation of the phrase, it is indeed true that the world itself is vague. New College, Oxford.

APPENDIX: FORMAL SEMANTICS

If ‘red’ were an incoherent predicate, we could hardly use it in any theory of ours, including a theory of truth for a language. Even if a

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theory containing it did not also have the resources for demonstrating the incoherence, there would be an acceptable extension of the theory that would have them. But if we accept that the previous con­ siderations suffice to block the Sorites paradox, then there is no obvious objection to using ‘red’ in the metalanguage of a truth theory for a language containing ‘red’. (It is not as if there were much else available which we could use in stating the semantic contribution of ‘red’ to the sentences in which it occurs, other than ‘red’ itself.) Consider a fragment of a language with several proper names, the predicate ‘red’, and sentential negation. We want the degree to which the predicate “ is true” applies to the sentence ‘Uranus is red’ to match precisely the degree to which Uranus is red. In the strictest form of representation, we would use here a variable-binding operator G y which applies to a pair of predicates, which yields something which takes two terms to form a sentence: CHRISTOPHER PEACOCKE

This formula would be true iff and tjf are true to exactly the same degree of the objects denoted by t and t'. (All this would need relativisation to a sequence in a first-order language.) Thus we want it to come out that CHRISTOPHER PEACOCK CE HRISTOPHER PEACOCKE

We noted earlier that there is a qualitative aspect to the notion of degree on which vague predicates are predicates of degree. G is to be understood in such a way that the truth of this last formula requires the qualitative aspects of the degrees of the two predicates to coincide: if Uranus is not red enough to be said to be red, then rred (Uranus)1 is not true enough to be said to be true (and so forth). In fact in order to employ an operator of a syntactic category with which we are more familiar, I will write the last displayed formula with a special biconditional ‘«V thus: CHRISTOPHER PE CA HCRO I SCTKOEP H E R P E A C O C K E

rA ** B 1 is true iff A and B are true to exactly the same degree. (‘«V thus has property (I) and lacks property (2) we discussed earlier.) Since we will here be concerned only with examples in which vague­ ness in sentences is produced by vague predicates, the substitutivity of identity will hold unrestricted in such contexts; indeed it is obvious

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given that we can regard the last displayed formula as an abbreviation of the one before it. W is degree-functional: the degree to which rA ** B 1 is true is determined by the degree of truth of A and B. For closed sentences A and B. rA ** B 1 will be definitely true or definitely not true. But of course that will not be the case with other degreefunctional connectives, for instance negation. This degree-functional negation we will write ‘1’. We shall not use any operators in the theory of truth which are not degree-functional. Degree-functionality is the analogue for a simple vague language of truth-functionality for simple classical languages. We can then offer the obvious truth-theoretic axioms: 137 137 137

interestinterest here.interest here.interest here. here. interest here.interest interest here. here. interest interest here. here. interest here.

We can derive the T-sentence for ‘red(Uranus)’ from A1 and A3, using the noted transparency of *«->’. By the degree functionality of ‘Y we can have inferential principles allowing the substitution of ‘«-»’-equivalents within contexts governed by ‘Y. Hence from the already proved T-sentence for red (Uranus) and A2 we can derive that for r~lred(Uranus)n. Plainly there is nothing of much technical interest here. What of the model theory? One’s conception of the appropriate notion of a model for a vague language is naturally crucial for one’s attitude to the validity of the paradoxical arguments and the claim that I have made about conditionals in vague languages. The question about models could be side-stepped entirely if we adopted a replacement definition of validity. If we say that a schema is valid iff no uniform substitution of nonlogical expressions for its schematic letters yields a sentence which is not true, we have a definition of validity which is as intelligible for vague as for sharp languages. Nevertheless, it is obvious that the old objection to this definition in the case of sharp languages applies too in the case of vague languages: if the back­ ground nonlogical vocabulary is rather weak in its expressive power, some invalid schemata may be counted as valid. This objection is ordinarily taken to motivate the familiar settheoretic definition of validity. But it does not of course tell against a modal definition of validity which counts a schema as valid iff it is not possible that there be nonlogical vocabulary in some extension of the

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language which makes the schema not actually true. In fact I have considerable sympathy with the view that not only is this modal definition to be preferred to the set-theoretic one, but that the set-theoretic notions themselves should be explained in modal terms. Nevertheless I shall not discuss validity in vague languages in terms of that modal definition, for what I have to say about such validity should be acceptable to anyone who is able to understand the settheoretic definition, by whatever route. I shall develop an analogue of the set theoretic conception for vague languages. A sentence of a sharp language is valid iff it is an instance of a schema that is true under all suitable assignments of sets to its schematic letters. What should stand in the same relation to vague languages as sets thus stand to sharp languages? To play this role I shall introduce the notion of a sea of objects. Seas of objects stand to vague predicates as sets stand to classical predicates. There is a binary predicate ‘f l n £ ’ true of pairs of objects and seas. This predicate is itself a predicate of degree, and the identity condition of seas is given in terms of it. Where ‘a ’, ‘b ’ . . . range over seas, we can say that 138

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Thus seas are ‘extensional’: if the same objects are in a and b to the same degrees, a is identical with b. Note that this is a sharp identity condition: once the degrees are fixed, either it definitely holds or definitely does not hold. Seas are not ‘vague objects’ if that phrase is taken to imply that under a given distribution of degrees it can be indeterminate whether the relation of identity holds between a pair of seas. We can also introduce a sea abstraction operator x(___x . . . ). An object is in the sea x($(x)) to just the degree to which it is 0 : 138

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From SI and S2 it follows that 138

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This should suffice to convey the fundamental idea of a theory of seas. An axiomatic theory of these entities could be developed, with various existence and comprehension axioms. Provided we operate with degree-functional notions of negation, alternation and con-

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junction, we can make sense of sea-theoretic operations of com­ plementation, union and intersection respectively. Employing the notion of a sea, we can then say: a valid schema of a vague language is one which comes out true under all suitable assignments of seas to its schematic letters. A schema is a logical consequence of a set of schemata if all assignments of seas making all elements of the set true also make true the given schema. Note that in these definitions we exploit the qualitative aspect of the notion of the degree to which an object is in a sea. This model theory takes vagueness very seriously. Not only do the assigned entities, the seas, have their identity conditions given in terms of a vague predicate; we also of necessity use connectives appropriate to a vague language, the degree functional correctives, in giving the model-theoretic account of truth under an assignment. We say for instance that for an atomic sentence Ft, and a sequence s of objects from model M : true (rF / \ M,s) ** s(t) In M (F ). Here M (F ) is M ’s assignment to F and s(i) is s ’s assignment to the term t : the important point is that '*+' is our degree-functional biconditional. (Truth in a model would of course be truth in that model relative to all sequences of objects from the domain of that model.) A similar point holds for the other degree-functional connectives. From this point on it is a mechanical matter to construct the model theory, by mimicking the classical forms with the alterations we have indicated. There is though one final caveat: if one is considering a language with more than one vague predicate, one must not assign seas of a uniform kind to different predicates unless one is prepared to accept the consequence that it makes sense to compare the degrees to which those different predicates apply to an object.

NOTES

1 See Dummett (1975). I presuppose familiarity with his argument. J Wright (1976), p. 245. It should be emphasized that this is not the only reaction consistent with Wright’s arguments. Another option is to hold that we need to change our views about the ways in which governing rules can be identified; yet a third is to suggest that it is not impossible to make coherent use of semantically incoherent expressions. i This metalinguistic paradox raises a question for Dummett too: would he wish to say that the predicate C is incoherent? 4 Platts (1979), chapter IX

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5 Ibid, p. 230. ‘ Ibid., p. 230. 7 Platts does say that “indiscriminability from an instance of red justifies us, other things being equal, in saying that an item is red" (Platts, 1979, p. 231) and says that other things are not equal when I have been persuaded by a Sorites argument (or chain, of corresponding instances) into calling a manifestly orange object lred \ But the supporter of the incoherence thesis will just say that this means that observational judgements take precedence over conclusions reached by valid means from true premises: that is, it is already revisionary, and is consistent with the incoherence thesis. 8 Wright (1976), section IV. 9 The problem is that we cannot say: particulars x and y are red to the same degree iff ~ 0 3 z(z matches x & ~ z matches y). For we would need to add that in this possible circumstance the degrees of redness of x and y do not differ from their actual values: and thus the condition becomes circular. A similar problem arises with counterfactuals: for a parallel reason we cannot explain the matching of two objects as their being indiscriminable were they to be juxtaposed. We return to some of these issues in a later section. 10 We can note, too, that this relativized major premise just displayed bears on Wright’s arguments (1976) that the utility and point of vague predicates would disap­ pear if we made them precise. His arguments show that precise predicates could not be applied on the basis of casual observation. I agree. But I disagree with his view that his arguments provide reasons for believing the unrelativized major premise of the Sorites paradox. Since two objects a and b can be observationally indistinguishable and yet be red to different degrees, we cannot conclude from Wright’s arguments that in such a case *b is red’ must be just as true as is ‘a is red*. All we can conclude is that any difference in degree of truth of these two sentences must correspond to a difference in degrees of redness not detectable by causal observation. But there are such differences. 11 Wright holds that if we introduce degrees of application, the paradox can still be stated for the predicate “it is on balance justified to predicate ‘red’ of138 (1976, p. 239). If this is not a predicate of degree, I would treat it as I have treated C; if it is a predicate of degree, I would treat it as I shall go on to treat 'red' 12 Dummett (1973), pp. 446-447. 13 Dummett (1975), p. 316. 14 It is not everywhere clear how to distribute temporal indices in Dummett’s argument, and some may wonder if there is a resulting fallacy. This is not so - and in any case the point can be made in a spatial example with a disc:

138138 138 The colours of a, b and c are as described in the text earlier. The solid straight line is the only visible boundary in the disc. (One can also reproduce the argument for a patch slowly changing in colour.) 15 Dummett (1979), p. 9 16 Dummett (1976), p. 97. 17 One could also reach the conclusion of this paragraph by considering the hypothesis that the shades of all objects are vibrating simultaneously between two indiscriminably different shades.

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18 I have taken it not to be satisfactory to avoid the indeterminacies by appealing to physical magnitudes which are the ground in the objects of their secondary qualities. Such an appeal would not be avoiding indeterminacy by appeal to the nature of the experiences themselves.

BIBLIOGRAPHY

Dummett, Michael: 1973, F re g e : P h ilo s o p h y o f L a n g u a g e, Duckworth, London. Dummett, Michael: 1975, ‘Wang’s paradox’, S y n th e s e 30, 301-24. Page references are to this printing: the paper is reprinted in his T ru th a n d O th e r E n ig m a s , Duckworth, London, 1978. Dummett, Michael: 1976, ‘What is a theory of meaning? (Il)’ G. Evans and J. McDowell, eds. T ru th a n d M e a n in g , Clarendon Press, Oxford. Dummett, Michael: 1979, ‘Common sense and physics’, in G. McDonald, ed.. P e r c e p ­ tio n a n d id e n tity , Macmillan, London. Platts, Mark: 1979, W a y s o f M e a n in g , Routledge, London. Wright, Crispin: 1976, ‘Language-mastery and the Sorites paradox’, in G. Evans and J. McDowell, eds., T ru th a n d M e a n in g , Clarendon Press, Oxford.

Part III Degrees of Truth

[9] Degrees of Belief and Degrees of Truth

R.M. Sainsbury University of L ondon, King’s C o lleg e

Ramsey said that a statement of the form ‘p is 1/3 true’ would be ‘sheer nonsense’.1 The aim of this paper is to dispute that view, and thereby to establish the coherence of the notion of degrees o f truth. What would motivate such an attempt? There are ordinary idioms which seem explicitly to invoke the notion. But appearances are, in every case I know, superficial.2 We do indeed say of statements that they are ‘very nearly true’ (and there is a range of similar idioms). However, there is no need to see this as attributing a degree of truth close to the maximum. It could as well be seen as saying that the statement requires only minor modifications to be (completely) true, which is consistent with its being (completely) false as it stands. The statement approaches the truth not by being partly true, but by diverging only a little in content from a truth. We must distinguish between nearly stating a truth and stating something which is nearly true. It is not obvious from the idioms in question that we see ourselves as doing the latter rather than the former. My motivation comes from reflection on vagueness. The central thought is that a sentence containing a vague expression may fail to be completely true and fail to be completely false, yet not be without truth value. Such a sentence represents the world as being a certain way, so there must be a question of whether it represents it correctly or incorrectly. But there may be no definite answer: it is a partially correct representation. (And this does not mean that parts of the representation are wholly correct.) This is the idea I shall try to justify. 1. Degrees of Truth in Semantic Theories Vagueness breeds paradox. A simple example, which has given its name to the family of paradoxes of vagueness, is the paradox of the heap. Take away one grain of sand from a heap of sand, and you must still be left with a heap.

P h ilo so p h ic a l P apers

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So, however many grains you take away, even if you take away all of them, what remains is a heap of grains. The premise seems hard to dispute, the reasoning is quite standard,3and the conclusion is absurd. One constraint on a semantic theory for a language with vague expressionsis that it block the paradoxes. It has to entail that the premises are, despite appearances, not true, or that the reasoning is, despite appearances, defective. One kind of theory which delivers this entailment exploits what it calls degrees of truth. So here we have an initial motivation for finding philosophical room for degrees of truth. This initial motivation will not, by itself, take us far. There are other kinds of semantic theories which deliver the required paradox-blocking entailment,5 so anxiety about the paradoxes in itself should not be allowed to bully us into a degree of truth theory. In any case, having the paradox-blocking entailment is far from sufficient for the adequacy of a semantic theory. In addition, the theory must, of course, match the sentences of the language with their meanings. Moreover, the semantic concepts the theory uses must be justifiable. To justify exploiting the notion of degrees of truth in a semantic theory, one must elaborate some special connection between a degree of truth and some aspect of speakers’ use. 2. Degrees of Truth and Comparisons One natural place to look for this connection, at least in the case of vague predicates, is the language users’ comparative judgments. Many, though not all, vague predicates are linked a priori to comparatives. For example, one who has mastered the sense of ‘adult’ is in a position to know that if a is an adult and b is older than a, b is an adult.6 One who has mastered the sense of ‘red’ is in a position to know that if a is redder than b but is not red, nor is b. And so on. Sometimes the a priori link between the vague predicate and the comparative will be semantically visible (e.g. ‘redder than’ and ‘red’); sometimes it will not (e.g. ‘heap’ and ‘more numerous than’). The degree theorist’s heart will warm to comparisons because they induce an ordering, and so does the degree theory. Comparative judgments will order visible objects by how red they are. Degree theoretic semantics will order sentences ascribing ‘red’ to these objects by how true they are. Assuming that truth degrees are, or are represented by, numbers, say the reals in the interval 0 to 1, the ordering of the degrees will be formally different from the ordering induced by the comparatives. But perhaps the degree theorist can somehow throw away the unwanted strictness in his ordering, and the unwanted precision in his degrees. That problem is not my concern here. The following philosophical issue remains: all that the comparisons

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uncontroversially entitle the theorist to is degrees of the property the predicate expresses: degrees of redness or degrees of adulthood. To make good degree theoretic semantics, its central concept, degrees of truth, has to be justified. Accepting that the data demand a recognition of degrees of redness does not, in and of itself, require accepting that they demand recognition of degrees of truth. There is no obvious path from the fact that things can vary in how red they are to the claim that sentences can vary in how true they are. On the degree theorist’s behalf, I propose that one way of justifying the concept of degrees of truth takes as its starting point degrees of belief. 3. Degrees of Belief Let’s suppose that the degree theorist uses 1 to represent complete or maximal truth, O to represent complete or maximal falsity, and the numbers in between to represent the intermediate cases characteristic of vagueness. Thus ‘a is bald’will be accorded degree 1 if a’s scalp is hairless, degree O if a has a full head of hair, and some number in between if a is neither definitely bald nor definitely not bald. The question is: why should we think of these numbers as representing degrees of truth, rather than degrees of baldness? The basic idea behind the answer I propose is simple: borderline cases for a vague predicate will give rise, in fully informed and rational beings, to degrees of belief. These resemble epistemic degrees of belief in that they are related to strength of tendency to act. But they are unlike epistemic degrees in that they represent no ignorance. Rather, they reflect the fact that there is no definite right or wrong: it would be as wrong to believe with total confidence that a borderline case is a positive case as to believe, with total confidence, the negation of this. The next stage in the argument involves showing that these kinds of degrees of belief force upon us the notion of degrees of truth. Suppose you know that all the intelligent members of a certain group are spies. It is important to you to have as many collaborators as possible, drawn from the group. Intelligence is not, in itself, a quality that matters to you. But it is important that you should not select a spy as a collaborator. Suppose a is a borderline case of intelligence. If we could measure the strength of your disposition to engage a as a collaborator, we would have a measure of your degree of belief in the proposition that a is intelligent. It might be that you will attempt to engage a as a collaborator if and only if the benefits you think you will reap if he is not a spy are three times greater than the disbenefits you think will accrue if he is a spy. If you have nothing else to go on than your total conviction that all the intelligent ones are spies, together with your knowledge of a ’s intelligence-related qualities, then we have reason to assign 0.75 as a measure of the degree of your belief in ‘a is

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intelligent’. This is simply an application of an idea which is standard in discussions of epistemic degrees of belief, starting with Ramsey’s classic discussion mentioned in note 1. Indeed, my thought is this: a vague belief will affect behaviour just as an uncertain one would. Falling short of certainty and falling short of precision both diminish the power of a belief to be realized in behaviour. In the example, we would also have reason to assign 0.75 as a measure of your degree of belief in ‘a is a spy’. But this is an epistemic degree, measuring incompleteness in your information. Supposing that ‘spy’is a sharp predicate, a is definitely a spy or definitely not a spy and the trouble is that you aren’t completely sure which. But, by hypothesis, a is not definitely intelligent nor definitely not intelligent. Let’s suppose also that you have full knowledge of a’s personal qualities: there is nothing about his intelligence you do not know. Yet, despite your total confidence in the proposition that all the intelligent ones are spies, you cannot with total confidence activate with respect to a, or refrain from activating, the generalization that all the intelligent ones are spies. In contrast to the epistemic degrees to which ignorance gives rise, let us call the degrees to which vagueness gives rise non-epistemic. The two kinds of degrees could be present together, with respect to a single belief. Suppose some problem with your eyesight made you only 80% confident in the deliverances of visual perception. Suppose that some object is, so far as you can tell, about as good a case of red as of not-red. Then, behaviourally, your belief that it is red would be as if it had degree 0.8 X 0.5, i.e. 0.4. The different kinds of degrees of belief are similar in one respect: they represent, as Ramsey put it, ‘the extent to which we are prepared to act on (a belief)’ (p.31). But in a vital respect my thesis requires that they be different: there is no sound inference from epistemic degrees of belief to degrees of truth, whereas I believe that there is a sound inference from non-epistemic degrees of belief to degrees of truth. 4. The need for degrees of belief

Truth is what belief aims at. What should one aim at in belief, when vagueness is present? I suggest that the answer is: degrees of truth. Establishing this answer involves two stages. The first, to be tackled in this section, involves showing that degrees of belief are what we must have, if we are both to use vague concepts and optimize the quality of our beliefs: I call this the thesis of the ineliminability of non-epistemic degrees of belief. The second stage of the argument takes this as a premise, and purports to go on and establish the coherence of the notion of degrees of truth.

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W hat would an omniscient being believe in the case of the story told earlier? He would believe that a is a spy, if a is a spy, and that a is not a spy, if a is not a spy; and a is either a spy or not. An omniscient being will have degree of belief 1 in one of these propositions, degree of belief 0 in the other. Here, only epistemic degrees are relevant, and omniscience will have no use for epistemic degrees other than 1 or 0. But what would he believe about a ’s intelligence? The hypothesis is that a is a borderline case, neither definitely intelligent nor definitely not intelligent. If omniscience believes with either degree 1 or degree 0 that a is intelligent, something is not quite right. Som ething in a correct belief needs to differentiate a both from people who are definitely intelligent, and from people who are definitely not intelligent. Omniscience m ust either have no belief on the subject of a ’s intelligence, or else go in for a degree of belief other than 0 or 1 — though not, of course, an epistemic degree. The degree m ust reflect how intelligent a is. It m ust locate him correctly between what is required in order to be definitely intelligent and what is required in order to be definitely not intelligent. I can envisage only two routes which m ight seem to avoid this claim. Either it will be said th at omniscience need have no belief at all concerning a ’s intelligence; or that he can have a degree of belief 0 or 1 concerning it. One way to take thefirst route is to say that an omniscient being need make no use of concepts like ‘intelligent’or ‘bald’, for he will instead be able to form beliefs that relate directly to the underlying sharp facts on which vague properties supervene. (In the case of baldness, they are something to do with the num ber and length of the hairs; it is less clear what they are in the case of intelligence.) One response to this objection is to say th at this would im pugn the being’s omniscience, for there will then be some fact, viz. a vague fact concerning a ’s intelligence, which the being would not know. I will not m ake this response, for it involves the obscure and controversial thesis that there are vague facts ‘in the w orld’. My preferred response involves clarifying my overall strategy. The aim was to ask what one should believe in certain circumstances, given that one possesses a coherent but vague concept, and given that, so far as the application conditions of this concept go, one has full knowledge (so there is no question of epistemic degrees of belief). If what the objector envisages is that an omniscient being would simply eschew all vagueness, then it is beside the point. But the objection can be understood differently. It may be taken to say: the underlying facts are sharp. If you knew them , there would be no room for any kind of degree of belief. So if beliefs about intelligence come in degrees, these are really epistemic degrees, reflecting your ignorance of the

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basic sharp facts. This version of the objection simply overlooks the fact that everything relevant to the application conditions o f ‘intelligent’ is, by hypothesis, known. Even if we knew the underlying sharp facts (supposing there always to be such), the question of applying ‘intelligent’ remains open, for the supposed supervenience relation is vague. If the reply is that we should simply forget about ‘intelligent’ and think only in terms of sharp properties, then the objection reverts to its first form, already dismissed as beside the point. Let me give the objection a final reformulation. It is not that what we should do is abandon vague expressions like ‘intelligent’ altogether; nor is it that ignorance is the source of hesitation about whether they should be applied to borderline cases. Rather, we should quite firmly hold that neither ‘a is intelligent’ nor ‘a is not intelligent’ is true; and that is the end of the m atter. This, presumably, is the position that would naturally go with a supervaluational approach to the semantics of vagueness (see n.5 above). This position seems to me untenable for the following kind of reason. Suppose that b is also borderline for ‘intelligent’. Nonetheless, it can be quite definitely true that a is more intelligent than b. This can properly have a decisive effect on action. For example, if you have to choose one of them to collaborate, and you know that all the intelligent ones are spies, you must, if rational, prefer b. I cannot see how the rational strength of this preference can flow from anything other than contrasting degrees to which it is rational to believe ‘a is intelligent’ and ‘b is intelligent’. The obvious way to take the second route is this: we can, and in practice do, introduce qualifiers into the content of beliefs which enable us to hold them to degree 1 orO. Intermediate degrees are not required. For example, the subject may believe, in the case discussed, that a is a fairly good case o/intelligence, or that a is a person who comes close to being intelligent. Such qualified beliefs are rationally believable to degree 1 or 0. There is an analogy with epistemic degrees. A belief, p, of intermediate degree, say 0.5, may be replaced by a qualified belief, of degree 1, which has the form: it is 50:50 whether or not p. But, as the analogy shows, the fact that a qualification can remove the need for intermediate degrees of belief, by replacing candidates for intermediate degrees of belief by beliefs in which counterparts of the interm ediate degrees figure, as qualifications, in the content, should not be taken as a proof of the needlessness of degrees of belief. Just the reverse: that you hold that p is 50% probable proves that, if you are consistent, you believe that p to degree 0.5. We need to take the unqualified beliefs as basic for the following reasons. First, the explanation of the impact upon behaviour of the qualified beliefs m ust, so far as I can see, run through an account of the subject’s response to

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the contained unqualified belief. Thus explaining what in behaviour is involved in believing that it is 50:50 whether or not p must involve some Ramsey-like relationship between the subject and p. This is what is fundamental. Secondly, as Ramsey showed for epistemic degrees, there can be variations in degree of unqualified beliefs which are too fine to be reflected in our current everyday ways of introducing qualifiers. A person’s condition with respect to ‘the next throw of this pair of dice will contain a 6 upperm ost’ may be different from his condition with respect to ‘the next throw of this pair of dice will sum to 3 ’. He may be prepared to take bets at different odds. It does not follow that he has the conceptual apparatus to think qualified thoughts which precisely reflect these conditions - and he will not have the apparatus, if the best he can manage are qualifying idioms like ‘There’s a slight chance, a good chance, a fair chance’. The reasons apply as much to non-epistemic as to epistemic degrees. A final clarification: the thesis is not that whenever we have borderline cases, any belief ought to have an intermediate degree, only that this is sometimes so. It is quite consistent with what I am claiming that rationality requires belief of degree 1 in ‘if a is intelligent then a is intelligent’, even if a is a borderline case of intelligence. To sum up: if vague expressions are coherent, there must be a question about how sentences containing them should figure in one’s beliefs. There are two ways to include borderline cases. You can use qualifiers (‘f airly intelligent’, etc), and then vagueness will not lead you to beliefs of degrees other than 1 and 0. Or you can go directly for intermediate degrees. I argued that the form er reduced to the latter, so there is no cognitive reflection of the phenom ena of vagueness without degrees of belief. Given this as a premise, the next stage is to show th at it leads to a coherent concept of a degree of truth. 5. From degrees of belief to degrees of truth I envisage at least the following three obstacles in the way of the move from degrees of belief to degrees of truth: ( 1) Degrees of belief involving a vague property, f, are a response to degrees of f-ness but do not introduce degrees of truth. (2) Epistemic degrees of belief do not license degrees of truth. So why should non-epistemic ones? (3) The phenom enon th at seems to call for degrees of tru th can be interpreted as a m atter of degrees of closeness to truth, in the m anner envisaged in P art 1. Two initial points. One, relevant to (l), is this: of two definite cases of f, one can be much f-er than the other. Sometimes, then, big differences of f-ness are

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not appropriately accompanied by any difference in degree of belief in a proposition of the form a is/ . So it cannot be said that degrees of belief are simply a response to degrees of f-ness. A second initial point is this: where there is numerical measurement, there is comparability. Given that degrees of truth are grounded in com parison, failures of comparability would tell against them. People have claimed to find heterogeneous comparisons, like ‘Henry is balder than the meeting is long’, unintelligible. The claim seems to me surprising. But, even if it is accepted, it will not upset the notion of an underlying comm on measure. F or there will still be a fact of the m atter of how likely ‘Henry is bald’ is to be realized in action, as compared with ‘the meeting is long’. The fundam ental fact is not propensity to assent to judgments which are explicitly comparative in form. It is the essence of Ram sey’s account of degrees of belief that there is in principle evidence for them which is less directly accessible to the subject. Turning now to (2), epistemic degrees of belief reflect some lack in us (lack of inform ation). They need give rise to no tem ptation to suppose that they answer to some feature of the world independent of ourselves.7 By contrast, the necessity for vagueness-induced non-epistemic degrees of belief, if we are to have as accurate and complete a picture of the world as possible, shows that these degrees do answer to some feature of the world. Omniscience, if it allows vague concepts at all, as we m ortals do, could do no better than have an intermediate degree of belief about a borderline case. In the case of an epistemic degree, the content of the belief may (if there is no vagueness) reflect the world completely correctly, or completely incorrectly. This is not so in the case of a non-epistemic degree. If non-epistemic degrees are ineliminable, as has been argued in Part 4, we need a notion of a degree of fit between statem ent and world, a notion that makes room for the partially correct representation that is achieved by a belief about a borderline case. The goal of a rational person is to hold such a belief with a confidence matching the degree to which it represents the world correctly: in short, a confidence answering to its degree of truth. In sharp cases this goal am ounts to: being certain of what is true (to degree 1) and rejecting with certainty what is false (i.e. true to degree 0). The argument is still incomplete. One who admits, as I think has been established, the need for the notion of partial correctness in a representation, may still object that this is to be understood in terms of how much modification the representation requires in order to become (completely) true, rather than in terms of degrees of truth. I offer two responses to this objection. The first is defensive: I renew the building of a bridge between the ineliminability of non-epistemic degrees of belief and degrees of truth. The

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second attacks the suggestion in the objection, that non-epistem ic degrees could be understood in terms of how much m odification is required to achieve total truth. T ruth is used in assessing a person’s cognitive condition. The simple model is this: if the attitude is belief and its content is that p then the subject pictures the world correctly if, and only if, p is true. The general form ula is: the nature of the attitude (belief, disbelief, mere entertaining) plus an assessment of its content with respect to truth, yields an assessment of the subject’s cognitive condition. Notice that the assessment has no epistemic dimension: a subject may picture the world correctly entirely by accident. The very different considerations required for an epistemic assessment are brought out by the fact th at a subject who is certain that a given horse will win the D erby is in error even if it wins. Turning our backs firmly on the epistemic, how does the form ula apply when the content of the belief attributes a vague property to a borderline case? By the premise to this section, there is an n such that, if the attitude is belief to degree n , the subject pictures the world correctly. W hat assessment of the content with respect to truth should combine with this input (belief to degree n) to yield this output (correct picturing)? The only possible answer seems to be: tru th to degree n. Once the general applicability of the form ula is accepted, there is no room for an interpretation of the phenom enon in term s of how much the content of the belief must be modified in order to state a truth. This interpretation can be independently attacked. If one can do no better than have a degree of belief reflecting the extent to which the borderline case has the property in question, why does one’s belief-content need m odification at all? O ne’s success as a w orld-picturer is already complete, and cannot be improved. 6. Conclusion The upshot is that we can make sense of the notion of degrees of truth. To show this is not to offer a theory of degrees, but only to show that such a theory is possible. The theory itself would have to speak to m any deep questions ranging from what the appropriate logic for vagueness is, to whether there are vague facts. But these are not topics for this paper8.

NOTES

1. F. Ramsey, “Truth and Probability”(1926), in D.H. Mellor’s collection of Ramsey’s papers, Foundations , London, 1978, p.58-100, at p.89. For a recent attack on the idea

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of degrees of truth, see e.g. D. Miller, “Impartial Truth”, in Skala, Termini and Trillas, eds. Aspects o f Vagueness, Reidel, 1985. 2. Cf. S. Haack, “Is Truth Flat or Bumpy?”in D.H. Mellor, ed., Essays in Memory o f F. P. Ramsey , Cambridge, 1980. Some writers, however, appear to attach some weight to these considerations of idiom: cf. L.A. Zadeh, “Fuzzy Logic and Approximate Reasoning”, Synthese 30, 1975, p.402-428; G. Lakoff, “Hedges: A Study in Meaning Criteria and the Logic of Fuzzy Concepts”, Journal o f Philosophical Logic 2, 1973, p.458-508. 3. There are various ways in which it could be spelt out in detail. For example: (1) A 1000-grained collection is a heap (2) If a 1000-grained collection is a heap, so is a 999-grained collection etc.... (1002) If a 1-grained collection is a heap, so is a 0-grained collection. 1001 applications of modus ponens , starting with the categorical ( 1) applied to (2), and using the result as input to the next application, yield the absurdity that a 0-grained collection is a heap. The truth of the conditional premises is supposed to be guaranteed by the generalization that taking away a grain cannot turn a heap into a non-heap. 4. I take as my paradigm the theory propounded by J. A. Goguen in “The Logic of Inexact Concepts”, Synthese 19, 1969, p.325-373. 5. For example, the supervaluation kind. See, e.g., K. Fine, “Vagueness, Truth and Logic”, Synthese 30,1915, p. 265-300; J. A. W. Kamp, “Two Theories about Adjectives”, in E.L. Keenan, ed., Formal Semantics o f Natural Language, CUP 1975. 6. This involves an oversimplification. Being an adult is a more complex property than a merely chronological one. Or consider being tall, a property which is relative to some envisaged group (to be tall for a Swede involves being taller than does being tall for an Eskimo): shifting the relativization could lead to a case in which a is taller than b, b is tall (for an F) and a is not tall (for a G). That there is a connection between mastery of some vague predicates and such comparative judgments remains untouched by these considerations, but the connection requires some ceteris paribus qualification. 7. Consider, for example, a degree of belief in whether a horse will win a race. However, under certain further conditions, in some ways parallel to those I discuss, ineliminable degrees of belief can arguably be used to construct an objective feature of the world: probability. The present project, however, is independent of that one. 8. I make a start on the question of the appropriate logic for vagueness in Sc 2 of “Evidence for Meaning”, Mind and Language 1, 1986.

[10] OCTOBER 1992

ANALYSIS 52.4

VALIDITY, UNCERTAINTY AND VAGUENESS By D o r o t h y

E d g in g t o n

I TA THAT’S THE USE of knowing that an argument is valid when VV I think, but am not sure, that the premisses are true? In such a situation — a common enough one — what am I entitled to think about the conclusion of the argument? It won’t do to reply: if each premiss is very likely to be true, and the argument is valid, the conclusion is very likely to be true. This is most vividly shown by the Lottery Paradox: many premisses, each very probable — Ticket 1 won’t win’, Ticket 2 won’t win’, and so on. Conclusion: none of the tickets will win. This extreme example illustrates a general point: even with only two premisses, the conclusion can inherit some uncertainty from each, and be less certain than either. The correct answer — if we concede that degrees of confidence should, ideally, behave like probabilities — is this: call the uncertainty of a proposition one minus its probability; then we can establish that the uncertainty of a conclusion cannot exceed the sum of the uncertainties of the premissesfrom which it is validly derived. First, if you recognize the validity of a one-premiss argument, you cannot consistently think the conclusion is less probable (more uncertain) than the premiss.1 Second, a many-premiss argu­ ment can be reduced to a single-premiss argument by conjoining the premisses. Third (as I show below) the uncertainty of A&B cannot exceed the sum of the uncertainties of A and of B. Fourth, conjunction and addition being associative, we can generalize, obtaining the Uncertainty Principle: u(A) may be low yet u(A) u(C) may high. be low yet u(C) high.

when A x, ..., An entail Cr A model of the third part of the proof: take a matchbox of length 1, and two matches of lengths 0.9 and 0.8, to be put length­ wise in the box. What is the minimum overlap of the two matches? Well, placed end to end with no overlap, their length totals 1.7. So, to get them in a box of length 1, they must overlap by a minimum of 0.7. (The maximum overlap is 0.8, the length of the shorter 1Recognizing that A entails C, you have p(A & ~C ) = 0; so p(A) = p(A Sc C). p(C) = p(A 8c C) + p (~ A &: C) £ p(A). 2Note, the converse of this result is unproblem atic. If the argum ent from A to C is invalid it is possible that (A 8c ~C), and it is consistent with probabilistic struc­ ture that p(A 8c ~C ) be arbitrarily high. So both p(A) and p( ~C ) may be high, i.e. u(A) may be low yet u(C) high.

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194 ANALYSIS match.) Now let the lengths of the matches represent the prob­ abilities of two propositions, A and B, the lengths of the box occupied by just A, just B, both, and neither represent respectively the probabilities of A& —B, —A&B, A&B and —A& —B. (These four probabilities must sum to 1: that is the substantive assump­ tion of probabilistic structure.) p(A) = 0.9 [u(A) = 0.1]; p(B) = 0.8 [u(B) = 0.2]; p(A &B)^ 0.7, or u(A & B)^0.3 = u(A) + u(B).3 194

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Suppose you think A and B are each 99% probable. So A&B must be at least 98% probable. Suppose A&B entails C: it’s impossible that (A&B)&~C. So, given the values you assign to your premisses, C must also be at least 98% probable. The answer we considered first and rejected is a first approximation to the truth: a valid argument with a few highly probable premisses does guarantee a high probability for the conclusion — though not quite as high as each premiss. It is only if there are lots of premisses, or a few which are only fairly probable, that the ‘conjunctivitis’ becomes serious. A result like this is needed: arguing from contingent premisses, absolute certainty is not only rare, it is hard to distinguish from the practically-but-not-quite certain. A logic which validated confidence in the conclusion in the former case but had nothing to say about the latter would be very limited in its use. In what follows, I consider further applications of the Uncertainty Prin­ ciple — to conditionals, and then to vagueness.

n Puzzlement about conditionals is easily generated by this dilemma: either the falsity of its antecedent is sufficient for the truth of a conditional, or it is not. Suppose it is not. Then it would be wrong to infer from the mere information that at least one of two proposi­ tions A and B is true, that if A is not true, B is true. For if the 3 The proper proof: p(A & —B) + p(A & B) + p( —A & B) = p(A) + p(B) - p(A & B) £ 1. Add 1 to each side of the inequality, subtract (p(A) + p(B)) from each side, and we have: 1 —p(A& B),p(A) (so FAB doesn’t entail A, i.e. is compatible with ~A); the latter because we can have p( ~A) > p(B | A) [= p(FAB)] (so ~A doesn’t entail FAB, i.e. ~(FAB) is compatible with ~A). We now know enough about the range of possibilities generated by A, B and FAB for our purposes. Divide the range of possibilities into ‘the A-part’ — the possibilities in which A is true — and ‘the ~Apart’. Now consider all consistent distributions of probabilities over these possibilities which give a non-zero probability to A. p(B IA)[= p(A & B)/p(A)] depends only on how probabilities are distributed in the A-part But p(FAB) depends also on how probabilities are distributed in the ~A-part — between (~A&(FAB)) and (~A& ~(FAB)). But there are many probability distributions which agree in all prob­ abilities in the A-part, hence agree on p(B|A), yet disagree in the ~A-part, hence disagree on p(FAB). So there are probability dis­ tributions in which p(B | A) * p(FAB). FAB is fabulous! The message, really, comes down to this. If the Hypothesis is correct, conditional judgements are irreducibly hypothetical They are judgements about what is the case under a supposition. And the proof shows that such judgements are not equivalent to judge­ ments about what is the case full stop. ‘But isn’t validity the necessary preservation of truth? How can we do logic with conditionals if we give up truth?’ This conception of validity may be too narrow, independently of conditionals. It’s beyond doubt that there are valid arguments involving moral judgements, for example, but it’s controversial whether these judgements have truth values. Legal experts spend their lives deriving consequences of laws, yet it’s not obvious that laws have truth values. A way forward with conditionals is to use the Uncertainty Principle of Section I: call the uncertainty of ‘If A, B’, one minus p(B|A) [which is p(~B|A) = p(A& ~B)/p(A)]. An argu­ ment is valid iff it’s impossible for the uncertainty of the conclu­ sion to exceed the sum of the uncertainties of the premisses.6 We already know this is correct for arguments consisting entirely of propositions. What emerges? We’ll look at a logic with conditional sentences, but without conditional sentences as parts of longer sentences. (Embedded conditionals are still a headache — or else, they’re 5 This was first proved by David Lewis in [14], See also Carlstrom and Hill [2]. 6This, again, was Em est Adams’ idea. See [1] ch. 2.

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197 eliminable by paraphrase; that’s a disputed issue.7) Writing ‘A — B’ for the ‘real’ conditional, and ‘A 3 B’ for the material conditional, we can show that always (with p(A) # 0), u (A 3 B )iu (A -B ): the former is just p(A & ~B); the latter is p(A& ~B)/p(A). So p(A 3 B) is always ^p(A—*-B). Now, classically valid arguments with no con­ ditionals in the conclusion remain valid: if the conclusion follows from premisses including the weaker ‘A=>B\ it will still follow when this is replaced by the stronger conditional. But not all classically valid arguments with conditional conclusions remain valid: the premisses may entail the weaker A OB, but not the stronger A —*-B. The rule of inference which fails is Conditional Proof. In the following list, an inference on the left is valid, its partner on the right, derivable by a step of Conditional Proof, is not. VALIDITY, UNCERTAINTY AND VAGUENESS

Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness

Vagueness Vagueness Vagueness VaguenessVagueness Vagueness VaguenessVagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness

Counterexamples to the forms on the right are now well known, being common to other treatments of conditionals.8 On this approach, they are counterexamples because they violate the Uncertainty Principle: the premisses can have a high probability or conditional probability, the conclusion a low conditional prob­ ability. Examples: (1) I’ll go swimming tomorrow; so, if I have a heart attack tonight, I’ll go swimming tomorrow. (2) It will either rain or snow in London in July; so, if it doesn’t rain it will snow. (3) I won’t be hit by a bomb and injured today; so, if I’m hit by a bomb I won’t be injured. (4) If you strike the match, it will light. If you dip it in water and strike it, you strike it; so, if you dip it in water and strike it, it will light. (5) If Bush wins, he won’t win by much; so, if he does win by much, he won’t win. How can we explain why these forms have seemed valid, and have seemed to serve us well, for so long? Part of the answer is that we can do for the others what we did earlier for (2) — state extra conditions under which the conclusion does follow — and these will be conditions which are normally met when the premisses are useful to us. Another part is this: the counter­ examples depend crucially on at least one of the premisses being,

7For valiant efforts to cure the headache, see McGee [17] and Jeffrey [11]; for criticism, see Edgington [4], pp. 199-205; for the paraphrasing strategy, see Jackson [9], Appendix, and Gibbard [7], pp. 236-8. 8See Stalnaker [22], Gibbard [7].

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however slightly, less than certain. The right-hand list satisfies what we might call the Certainty Principle — for premisses of probability 1 (and non-zero antecedents), the conclusions must have probability 1. Where uncertain premisses are not an issue — in mathematics, say — these forms won’t let us down. (The great founding fathers of modem logic were interested primarily in mathematical reasoning. They weren’t interested in the question: What follows when the premisses are slightly uncertain? On the contrary, their aim was to banish uncertainty from mathematics. For their purposes, the simplicity and clarity of their conditional were advantages which greatly outweighed any slight artificiality in odd contexts. It is when we focus on uncertainty that the distor­ tion becomes intolerable.) Also, when we consult our intuitions about the validity of a form of inference, we tend to picture ourselves as certain of the premisses, and consider what attitude to the conclusion is justified. Then the counterexamples don’t show up. When an argument consists entirely of propositions, the Uncertainty Principle holds whenever the Certainty Principle holds. With conditional judge­ ments, though, there is a sharp discontinuity between what follows from premisses of probability 1, and premisses of probability arbitrarily close to 1. How to explain this weird discontinuity? An analogy will help: the sharp discontinuity between the logical powers of all’ and almost all’. If all As are B and all Bs are C, then all As are C. But we can have: all As are B, almost all Bs are C, yet all As are ~C! Suppose there are 10 As, all of which are B and —C, and 1000 Bs, 990 of which are C. That is the analogue of (4) on the right. We could go through the whole list, showing the validity of the analogues on both sides for all’, but only on the left for almost all’ (Exercise!) — permitting, as the Uncertainty Principle does, a slight weakening of the ‘almost’ for a conclusion derived from more than one premiss. Now, if we carve up our range of possibilities (or our matchbox) into small enough equaliy-probable bits for the problem at hand, we can interpret p(B|A) = l ’ as all the A-parts are B-parts’, p(B|A) is high’ as almost all the A-parts are B-parts’, etc. Then the analogy becomes a model of the sharp discontinuity. We could, of course, stipulate that an argument is valid if it satisfies the Certainty Principle. Then we could reinstate the. prin­ ciples on the right. But then there would be no answer to the question with which we began: nearly certain premisses would guarantee nothing about the conclusion. Given the rarity of absolute certainty about contingent matters, and conditionals in particular, and the unclarity of the line to be drawn between it and its near neighbours, knowing that an argument is valid would be of little use. Given our epistemic powers, adhering to the Uncertainty Principle yields the more robust classification of argu­ ments. The issue is just one of usefulness.

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m The problem made vivid by the paradox of the heap (bald man, patch of red) is an everyday problem — somewhat as the problem made vivid by the Lottery Paradox is an everyday problem. A solution to the Sorites should help us to understand the general structure of reasoning with vague terms. There’s the ‘manana paradox’: the unwelcome task which needs to be done, but it’s always a matter of indifference whether it’s done today or tomorrow; the dieter’s paradox: I don’t care at all what difference to my weight one chocolate will make! The phenomenon is exploited by auctioneers, taking bids in intervals whose difference is trifling; and by seducers, of an old-fashioned kind. Where to draw the line, when we’ve got to draw it somewhere, and any particular place is arbitrary? If the patient’s blood pressure is too high and/or his cholesterol level is too high, a drug should be administered. Equivalently, the machine is in a dangerous state if certain variables like pressure and temperature are too high, and should be switched off. How high is too high? We could state a precise value, but it would usually be somewhat arbitrary: a system does not normally switch from safe to dangerous as a result of a minute change of the relevant variables; and the ‘degree of dangerousness’ of one variable may depend on the values of others. One can be criticized for being too cautious, and for being too rash, but it’s often not easy to see how good judgements are to be made. The Sorites paradox involves a long series of, say, patches of colour each insignificantly different from its neighbour — indeed, you can’t see any difference by just looking at two neighbouring patches; so, it would seem, neighbouring patches are the same colour — if one is red, so is its neighbour. But at one end, the patches are red, at the other end they are not. Yet an apparently valid argument, starting with the true premiss that a, is red, and with many conditionals of the form ‘If an is red an+j is red’, yields the false conclusion that aXi^, say, is red. The two best-known solutions are presented as rivals/ The first, the supervaluations approach, says that the argument is valid but has one false premiss, although it is indeterminate which premiss is false. It says this because on any permissible sharpening of the concept ‘red’, one premiss would be false, but there is no premiss which would be false on every permissible sharpening. And the argument is, of course, valid on all permissible sharpenings.10 The second approach takes seriously the fact that, among things which are neither definitely red nor definitely not-red, some are redder than others, and employs talk of the degree to which some­ thing is red, or the ‘degree of truth’ of ‘x is red’. As we progress 9 For example, by Mark Sainsbury in [20], ch. 2. 10See Fine [5].

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200 ANALYSIS down a portion of the series, this degree of truth slowly declines. The conditionals ‘If an is red, an+x is red’ have consequents slightly less true than their antecedents, and are themselves slightly less than fully true. It is then said that the argument is invalid: modus ponens does not hold for premisses which are less than fully true, for the conclusion may be less true than either premiss. Here I object. Why not (following the lessons of the Uncertainty Principle) say that the argument is valid, and it’s just what we would expect of a valid argument with more than one less-thanfully-true premiss, that the conclusion may import some falsity from each, and hence be less true than either individually? To say that the argument is invalid obscures an important distinction: between the case where the fall in value of the conclusion is con­ strained by the values of the premisses, and the case where it is not — the ‘genuinely invalid’ argument which might have two almost-true premisses and a fully false conclusion. How do degrees of truth work? The widely-held view is that we generalize from a two-valued truth-table, retaining the idea that the value of a complex proposition is a function of the values of its parts.11 The value of ~A is one minus the value of A. The favourite proposal for conjunction and disjunction is: (a) v(A & B) = Min[v(A), v(B)] v(A v B) = Max[v(A), v(B)]. An alternative proposal is: (b) v(A & B) = v(A).v(B) v(A v B) = v(A) + v(B) - v(A).v(B). The degree of falsity of a conditional is held to be the amount by which the antecedent exceeds the consequent, so v(A —►B) = 1 if v(B) ^ v(A), 1 - (v(A) - v(B)) otherwise. Apart from the negation rule, I think these proposals are wrong. First, we can rule out the alternative proposal (b) for conjunction and disjunction by letting A and B be synonymous: let A be ‘x is small’ and B be ‘x is little’. Then A&B, AvB, A and B should have the same value. But if v(A) = v(B) = 1/2, v(A&B) = 1/4, v(AvB) = 3/4, on this proposal. Second, although proposal (a) works in the above case, it is wrong, I think, when A and B are independent. Let the objects x, y, z be balls of various colours and sizes. Suppose: v(x is red) = 1, v(x is small) = 1/2 v(y is red) = 1/2, v(y is small) = 1/2 v(z is red) = 1/2, v(z is small) = 0. “ See Peacocke [19], Forbes [5], pp. 170-6, and much work under the heading ‘fuzzy logic’.

Vagueness

201 Intuitively, ‘x is red and small’ has a higher degree of truth that 'y is red and small*; and ‘y is red or small’ has a higher degree of truth than 'z is red or small’; yet all of these are 1/2 true, on proposal (a). In trying to fulfil the command ‘Bring me a ball which is red and small’, would I not do better to bring x (than which nothing redder can be conceived) than y ? And for the command ‘Bring me a ball which is either red or small’ would not y be a better choice than z? Third, the conditional: take two balls very close in colour, x and y ; let v(x is red) -0 .5 and v(y is red) = 0.49. v(If x is red, y is red) = 0.99. So far so good. Now let y also be small to degree 0.49. It is less compelling that v(If x is red, y is small) = 0.99. One might think: supposing that x is red does strongly incline me to classify y as red, but has no effect on my inclination to classify y as small: v(If x is red, y is small) = v(;y is small) = 0.49. Fourthly, return to the two balls similar in colour, red to degree 0.5 and 0.49 respectively. Why is it plausible to give a high value (0.99) to ‘If x is red, y is red? Because they are so close in colour that it seems far-fetched to maintain that one is red and the other is not. We should, therefore, give a very low value to ‘x is red and y is not red’; but on proposal (a) for conjunctions, its value is 0.5, on proposal (b) its value —0.25. A conclusion validly derived from one premiss should not have a lower value than the premiss. The inference from ‘If A, B’ to ‘~(A& ~B )’ must be invalid. Does it not seem strange to give a middling value to ‘A & HB* yet to consider it almost completely true that if A, B? We can do better with degrees of truth by giving them a probabilistic structure. p(A&B) and p(AvB) are not determined by (but are constrained by) p(A) and p(B); p(A&B) = p(A).p(B| A). The maximum value of p(A&B) is the smaller of p(A) and p(B); its minimum value is 0 if p(A) + p(B)^l, p(A) + p (B )-l otherwise. p(AvB) = p(A) + p(B)-p(A&B). Its minimum value is the greater of p(A) and p(B); its maximum value is 1 if p(A) + p(B)^ 1, p(A) + p(B) otherwise. I do not say that the indeterminacy of vague concepts is an epistemic matter.12 There exist different applications of prob­ abilistic structure. Objective chances apply if and when the future is physically undetermined by the past. Relative frequencies also satisfy the principles of probability. I propose that it is also the right structure for theorizing about the indeterminacy of applica­ tion of vague concepts. The idea behind supervaluations provides one model for think­ ing of degrees of truth in this way: there are many permissible sharpenings of a vague concept. We could interpret the degree of truth of ‘x is red’ as the proportion of permissible sharpenings in VALIDITY, UNCERTAINTY AND VAGUENESS

12Unlike Timothy Williamson in [24]. 13See Mellor [18], Lewis [16], Von Mises [23].

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which this sentence is true.14 A weighted proportion of permissible sharpenings might be better — a ‘bell-shaped curve* over permis­ sible sharpenings. On some such model, the relation between supervaluations — talk of truth on all, some or no sharpenings — and degrees of truth is structurally analogous to the relation between modal logic — truth in all, some or no possible situations — and probability as a weighted proportion of possible situations in which a proposition is true. Ignoring the sophistication of weights, however, the simple proportion of sharpenings gives us a model for the probabilistic structure of degrees of truth enabling us to see how the difficulties with the logical constants are overcome. Explain v(A —* B) [or v(B|A)] as the value to be given to B if we decide that A is true. Define: v(A & B) = v(A). v(B IA) v(A vB) = v(A) + v(B) - v(A & B). First, if A and B are synonymous, v(A—►B) = l, so v(A) = v(B) = v(A & B) = v(A vB). Second, the independent vagueness of ‘red’ and ‘small’: deciding that x is red makes no difference to the proportion of sharpenings in which x is small — v(x is small if x is red) = v(x is small). So we have (a) v(x is red) = 1, v(x is small) = 1/2; v(x is red & small) = 1/2. (b) v(y is red) = 1/2, v(y is small) = 1/2; v(y is red & small) = 1/4; v(y is red or small) = 3/4. (c) v(z is red) = 1/2, v(z is small) - 0; v(z is red or small) = 1/2. Third, take two balls x and y, similar in colour, red to degree 0.5 and 0.49 respectively. If it is supposed fixed that x is to count as red, the proportion of remaining sharpenings in which y is red is high. These are not independent: v(x is red→ y is red) is high. But if y is also small to degree 0.49, fixing x as red leaves the propor­ tion of sharpenings in which y is small where it was: v(x is red —►y is small) = 0.49. Fourth, in the above example, v(x is red & y is not red) = v(x is red).v(y is not red if xis red). The second term being close to 0, v(x is red & y is not red) is close to 0; more directly — on a very small proportion of sharpenings does (x is red & y is not red) come out true. In the Sorites, the premisses other than the first may take the form ‘~ (a n is red & ~ ( an+ is red))'; or they may take the form ‘an is red-*an+1 is red*. Either way, each premiss has a high degree of truth, and the argument is valid, but the large number of premisses, each with a small degree of falsity, enable the conclu­ sion to be completely false. 14This is discussed in Kamp [12].

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On this view, the Sorites Paradox and the Lottery Paradox are structurally the same phenomenon. Each is a pathological case of a phenomenon we need to understand: reasoning with uncertainty, and reasoning with vague concepts. I have given a mere sketch of the structure I believe the latter should follow. Of course, the numbers are to be taken with a pinch of salt, for both degrees of truth and degrees of certainty. That is to say that degrees of truth and certainty are themselves vague. But the numbers afford us the luxury of addition, multiplication, etc., in exhibiting a model of the structure of such reasoning. Compare this: suppose our conceptual resources allowed us to classify the lengths of desks as merely ‘long' and ‘short’; we refine them, by adding the category middling in length’ — and further, by locutions such as ‘three feet nine inches long’. Our use of this last is still imprecise — the problem of vagueness reiterates — but it does not follow that no progress has been made. (There may be no such thing as the exact length of a desk, across time, or even at a time.) Where it matters, we can apply probabilistic structures to probabilistic structures. Similarly, vagueness and uncertainty can interact. Imagine evidence mount­ ing by small intervals that, say, parents are mistreating a child. There is the question of degree of certainty, and the question of at what degree of certainty action should be taken. I can pursue such matters no further, but see no difficulties in principle within the framework I have proposed. I was told that in Japan washing machines are programmed with fuzzy logic. More serious applications are at least gleams in the eye of the AI fan. Whether or not they are more than that, we should try to get the basics right.15 Birkbeck College, Malet Street, London WC1E 1HX R eferences

[1] Ernest Adams, The L ogic o f C onditionals: a n A pplication o f P robability to D edu ctive L ogic (Dordrecht: Reidel, 1975). [2] I. Carstrom and C. Hill, Review of Adams [1], P hilosophy o f Science 45 (1978) 155-58. [3] Dorothy Edgington, ‘Do Conditionals Have T ruth Conditions?’, C rítica 18 (1986) 3-39, reprinted in [10]. [4] Dorothy Edgington, T h e Mystery of the Missing M atter of Fact’, Proceedings o f the A ristotelian Society Supplem entary Volume 65 (1991) 185-209. [5] Kit Fine, ‘Vagueness, T ruth and Logic’, Synthese 30 (1975) 265-300. [6] Graeme Forbes, The M etaphysics o f M o d a lity (Oxford: Oxford University Press, 1985). 151 am grateful to Peter Smith, Jerem y Butterfield, W. D. Hart, and to audiences at the British-Polish Logic Colloquium, Oxford 1991, at Sheffield University, Simon Fraser University and the University of New Mexico, for comments on earlier versions of this paper.

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204 [7] [8] [9] [10] [11] [12] [13] [14] [15]

ANALYSIS Allan Gibbard, 'Two Recent Theories of Conditionals’, in [8], 211-47. W. H arper, R. Stalnaker and G. Pearce (eds.), Ifs (Dordrecht: Reidel, 1981). Frank Jackson, C onditionals (Oxford: Blackwell, 1987). Frank Jackson (ed.), C on dition als (Oxford: Oxford University Press, 1991). Richard Jeffrey, ‘Matter-of-Fact Conditionals’, Proceedings o f the A ristotelian Society Supplem entary Volume 65 (1991) 161-83. J. A. W. Kamp, T w o Theories About Adjectiuves’ in [13], 123-55. E. L. Keenan (ed.), Form al Semantics o f N a tu ra l L angu age (Cambridge: Cam­ bridge University Press, 1975). David Lewis, ‘Probabilities of Conditionals and Conditional Probabilities’, P hilosophical R eview 85 (1976) 297-315, reprinted in [8], [10] and [15]. David Lewis, P hilosophical Papers , Vol. II (New York: Oxford University Press, 1 9 8 6 >-

[16] David Lewis, ‘A Subjectivist’s Guide to Objective Chance’, in [8], reprinted in [15] 83-132. [17] Vann McGee, ‘Conditional Probabilities and Com pounds of Conditionals’, P hilosophical R eview 98 (1989) 485-542. [18] D. H. Mellor, The M a tte r o f Chance (Cambridge: Cambridge University Press, 1971). [19] Christopher Peacocke, ‘Are Vague Predicates Coherent?’, Synthese 46 (1981) 121-41. [20] Mark Sainsbury, Paradoxes (Cambridge: Cambridge University Press, 1988). [21] Robert Stalnaker, ‘Probability and Conditionals’, P hilosophy o f Science 37 (1970) 64-80, reprinted in [8]. [22] Robert Stalnaker, ‘A Theory of Conditionals’, Studies in L ogical Theory: Am erican P hilosophical Q uarterly , M onograph Series No. 2 (Blackwell, 1968), reprinted in [8] and [10]. [23] Richard Von Mises, P robability , Statistics a n d Truth (London: George Allen and Unwin, 1957). [24] Timothy Williamson, ‘Vagueness and Ignorance’, Proceedings o f the A ristotelian Society, Supplem entary Volume 66 (1992) 145-62.

Part IV Epistemicism

[11] R IC H M O N D C A M P B E L L

T H E S O R IT E S P A R A D O X

(Received 5 September, 1973)

In the sorites paradox application of principles of classical logic to propo­ sitions containing a vague term leads to absurd conclusions. Philosophers are divided on the significance of this ancient paradox. Some, like the logician Goguen, see it as a successful challenge to certain principles of classical logic; for others, like Max Black in ‘Reasoning with Loose Concepts’, the challenge is interesting but unsuccessful; for still others, notably Quine, the idea that reasoning with vague terms could create a problem for, say, the classical true-false dichotomy is too obviously mis­ taken to be interesting .1 What follows falls into the middle category, though I begin with a critique of Black’s solution. I shall argue that the absurd conclusions of the sorites paradox can be avoided without resort to a system of many-valued logic or some other non-classical system. I. B L A C K ’S S O L U T IO N

Partially symbolizing Black’s version 2 of the sorites, we have the premises : ( 1) (2 )

A man of height 4 feet is a short man; (h) (if a man of height h feet is a short man, then a man of height (A +1/120) feet is a short man.

Using only universal instantiation and modus ponens, we eventually reach the absurd conclusion that a man of height (((4 + 1 / 1 2 0 ) + 1 / 1 2 0 ) ... +1/120) feet, i.e., a man of height 7 feet, is a short man. W hat has gone wrong? Confident that premise (1) is true and the deduction valid, we might deny the truth of premise (2). Its negation, on Black’s interpreta­ tion, can be put: (N2)

(3 It) (a man of height h feet is a short man and a man of height (h -f 1 / 1 2 0 ) feet is not a short man).

P h ilo so p h ic a l S tu d ie s 26 (1974) 175-191. A ll R ig h ts R e se r v e d C o p y rig h t © 1974 b y D . R e id e l P u blish in g C o m p a n y , D o rd re c h t-H o lla n d

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But (N2) appears plainly false, since it implies a fairly precise cutoff point (within 1 / 1 2 0 of a foot) between the heights of short men and those of men who are not short. The term ‘short m an’ is not so precise. If, on the other hand, we deny the truth of both (2) and (N2), then it seems we have to give up the Principle of Excluded Middle (PEM). Black does, in fact, reject both (2) and (N2). “ The view I have been sketching,” he says, “ requires us to rcject both the inductive premise [(2 )] and its negation [(N2)] as illegitimate” (p. 12). At the same time, he wants to retain PEM and permit its application in reasoning with loose concepts. He believes this is possible provided we understand inter alia “ that the use of customary logical principles in application to proposi­ tions involving loose concepts presupposes an ad hoc demarcation, some­ where in the region of cases that are not indisputably clear” (p. 1 2 ). There are a number of questions to be raised here. First, consider the true proposition ‘Every man shorter than some short man is short’. Surely PEM applies here - that is: either every man shorter than some short man is short or not every man shorter than some short man is short. Yet, notwithstanding the looseness of the concept of short man, we can recognize the truth of this disjunction without making any arbitrary divi­ sion among borderline cases. But then how does the application of this logical principle presuppose an ad hoc demarcation? Does Black want to say that only some applications to loose propositions have this pre­ supposition? Second, it might be objected that once a demarcation among borderline cases is made the reformed conception of short is not loose in the relevant sense. Given the new use of ‘short’, the proposition expressed by the sentence that originally expressed premise (2 ) is false and its negation is true. PEM would clearly apply to this new proposition, though in this case apparently not to a proposition involving a loose concept. Hence, the application of PEM where there is an ad hoc division among border­ line cases does not explain how PEM applies in reasoning with loose concepts. It may be, however, that ‘short’ can be used loosely or im­ precisely even when its use implies an ad hoc division among borderline cases. That is: contrary to the objection, the implied demarcation among borderline cases may not entail a new, precise conception of short. But how this is possible remains to be seen. Finally, there is the problem that if PEM is to apply to premise (2) in its

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177

original sense, Black needs to explain how he can consistently ‘reject’ both (2) and (N2). Presumably, rejection of both entails the denial that either (2) or (N2) is true-which is tantam ount to declaring that PEM does not apply after all. II. D E P T H OF T H E P A R A D O X

An important feature of the sorites that Black ignores is that there are good reasons for thinking (2 ) not true other than its leading to the absurdity that a seven foot man is short. For if (2) were true, then given the truth of (1), there would be no borderline cases of short men. (Any putative instance of a borderline case can be proved from these premises to be a case of a short man.) Notice, moreover, that if a four foot man is short and a seven foot man not short and there were no borderline be­ tween them, there would have to be a cutoff point at which the addition of a small fraction of an inch makes a man lose his shortness. Yet if the latter were so, (2) would be false. Together with certain truisms, (2) thus entails not only that a seven foot man is short, but also that there are no borderline cases of short men and indeed that (2 ) itself is false. To make matters still worse, since (2), together with (1), rules out borderline cases, it rules out the only consideration that would warrant its truth. For (2), being a universal proposition, appears true precisely because there seems to be no counterexample, i.e., no height which marks a cutoff point (within 1 / 1 2 0 of a foot) between short men and men not short. But, as before, if a four foot man is short and a seven foot man not short and there were no borderline cases between, then there would be such a point. Premise (2), then, surely cannot be true. But recall that the negation of (2 ) entails a sharp boundary between the short and not short and hence appears to eliminate borderline cases (except within 1 / 1 2 0 of a foot3) just as much as (2). Implicit in the sorites, therefore, there appears to be the following dilemma: if PEM applies to (2), either (2) or its negation is true; if either is true, there are no borderline cases; hence, either PEM does not apply to (2 ) or the concept short applied to men does not admit of border­ line cases. It might be doubted that the negation of (2) does exclude borderline cases. For although (N2) entails a sharp boundary between the short and not short, one might doubt that (N2) and the negation of (2) are equiva-

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lent. After all, (2 ) contains an if-then construction and this has not been given a precise interpretation. To allay any doubt, let us take the if-then to be material implication so that (2 ) could be rendered: 178

R IC H M O N D C A M PB E L L

(‘5(A)’ stands for the propositional function ‘a man of height h feet is a short man’.) So interpreted, (2), in conjunction with (1), still excludes borderline cases. On the other hand, the negation of (2) becomes by the rules of Quantifier Exchange, Definition of Material Implication, and Double Negation: 178

R IC HRMIC OH NM D OCNADM C PB AEMLPB L ELL

Given these rules, then, the dilemma of choosing between PEM and the existence of borderline cases arises as before. The trouble is that from the standpoint of classical logic, these rules are as unexceptional as PEM. I I I . E X C L U D E D M ID D L E VS B IV A L E N C E

But does the dilemma really force a choice between PEM and the exis­ tence of borderline cases, granted that the rules just mentioned are sound rules of inference? For if, contrary to the implications of (2) and its negation, such cases do exist what follows is that neither (2 ) nor its nega­ tion is true. That is: it follows that (2) is neither true nor false (since to say that the negation of (2) is true is to say that (2) is false). But it may be objected that this consequence is not incompatible with PEM on the ground that PEM does not imply that every proposition is either true or false. The Principle of Bivalence (PBV) does imply this; hence, the di­ lemma seems to force a choice between the existence of borderline cases and PBV. But if PBV cannot be derived from PEM, we can retain PEM in the face of borderline cases, dispensing with PBV instead . 4 PBV could be symbolized: Vagueness

where ‘T* is an operator meaning ‘it is true that’ and ‘i 7’ is defined. Vagueness Vagueness

Vagueness T H E S O R IT E S P A R A D O X

225

179

The proposition that PEM is true could then be rendered : Vagueness Vagueness

Obviously, if the T-operator can be distributed into a disjunction, then PBV can be derived from PEM. Also, if Tarski’s equivalence is accepted, Vagueness

again PBV can be derived from PEM. Hence, to retain PEM but not PBV has at least two counterintuitive consequences. First, the assertion of premise ( 2 ) will not be equivalent to the assertion that (2) is true. Second, the disjunction of (2 ) and its negation can be true, even though neither disjunct is true. These consequences may not be regarded as fatal to the attempt to separate PEM and PBV . 5 But a method of resolving the origi­ nal dilemma that avoids these consequences would be preferable to one that doesn’t, other things being equal. I shall try to develop such a solu­ tion in the following. I wish to retain both PEM and PBV, while at the same time acknowl­ edging the existence of borderline cases. To resolve the original dilemma, I shall have to show, therefore, that either premise (2) or its negation does not exclude borderline cases. In view of the sorites argument, it is clear that ( 2 ) does exclude them (given the principles of classical logic), since any putative borderline case of a short man can be proved by this argu­ ment to be a case of a short man. I shall argue, however, that despite appearances, the negation of (2) - or its equivalent (N2) - is not incompat­ ible with the existence of borderline cases. The original dilemma, then, would not force a choice between PEM or PBV and the existence of borderline cases. If (N2) is compatible with such cases, there will be no objection to denying the truth of premise ( 2 ) for the reasons detailed in the last section. The sorites paradox will be thus dissolved in a straightforward way: the sorties argument will be unsound, containing a false premise. To show that (N2) is compatible with borderline cases, I shall first sketch an interpretation of some truisms about borderline cases of short men. Then I shall construct a proposition which is apt to be confused with (N2) and which does exclude these cases. The negation of this false proposition will be a true proposition apt to be confused with (2). The recognition of

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these distinctions should remove the temptation to think that (2 ) is true because its negation is incompatible with borderline cases. IV. B O R D E R L IN E CASES

To begin with, a borderline case of a short man is one where (a) it is uncertain whether or not it is a case of a short man, (b) the uncertainty is not due to lack of information about tlic man’s height or about the dis­ tribution of heights among men in general, and (c) the uncertainty is not due to linguistic incompetence, e.g. failure to understand the meaning of the expression ‘short m an’ or how the term is used in English. This I take to be a truism - as it stands. Nothing as yet follows about the implication of (N2). In (a) I am not using ‘uncertain’ in any special or technical sense. I refer simply to the uncertainty that most of us justifiably feel when faced with the problem of deciding whether a borderline case falls within or without the category on whose ‘borderline’ it lies. We say that we do not know where to place the case in question, that it is impossible to tell, that it is uncertain how the case is to be classified. There is nothing extraordinary about this use o f ‘uncertain’. But although the expressions ‘uncertain’ and ‘uncertainty’ are not given any technical meaning in (a), (b), and (c), the kind of uncertainty specified by the conjunction of these conditions is of a special kind. For convenience I shall speak of this uncertainty as semantic uncertainty, since the uncertainty is apparently due to the fact that the meaning o f ‘short man’ is vague or inexact. I prefer this label to ‘epistemic uncertainty’, since the uncertainty defined by (a), (b), and (c) is not due to any lack of knowledge. No amount of knowledge of empirical facts about men or of the meaning of ‘short man’ will remove the uncertainty. One might ask, of course, whether this uncertainty, so defined, ever obtains in reality. But I take it for granted that it does and that (a), (b), and (c) are mere truisms about borderline cases of short men. I interpret this semantic uncertainty as applying to the classification of something under a description, e.g., the classification of a particular indi­ vidual as a short man. Put another way, semantic uncertainty applies to the predication of a property (in the broadest sense) to a particular thing. The uncertainty is a function of the uncertainty that most competent speakers would feel about making the predication even when they are cognizant of

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all the relevant evidence. The uncertainty is thus relative to the language used to make the predication. It is, nevertheless, an objective m atter whether the uncertainty obtains. It is possible for someone to be mistaken about whether a particular predication is semantically uncertain, or not to be mistaken for the wrong reasons. Moreover, it is possible to have strong or weak evidence that a predication is uncertain. Semantic uncertainty may be viewed under this interpretation as an operator applying to predications to form new predications . 6 We might symbolize it by ' B' If ‘h’ refers to a particular height and ‘Sh ’ means that a man of height A feet is a short man, then the proposition that a man of height A feet is a borderline case of a short man would be symbolized: 227

227 227

If semantic certainty, symbolized by the operator ‘C \ is defined as ‘~ B \ then (3) will be equivalent to : 227

2272 27227

Now it shoud be clear from the above interpretation of a borderline case that a borderline case of a short man is likewise a borderline case of a man who is not short. That is: a predication is semantically uncertain if, and only if, its negation is semantically uncertain. For it will be uncertain whether or not someone is a short man just in case it is uncertain whether or not the individual is not a short man. The following equivalences thus have the status of logical truths: 227

Vagueness Vagueness

227

Vagueness Vagueness

From ( 6 ) it is clear that semantic certainty does not imply truth. I f p is a proposition predicating a property of an individual, the following are n o t tr u e in g e n e r a l: 227 227

Vagueness Vagueness Vagueness

For if they were true, then a proposition predicating a property (or its negation) to an individual would have to be both true and false in order for the predication to be semantically certain. We would be forced to con-

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elude that all predications are semantically uncertain - every predication involves a borderline case. From (5), however, it does not follow that semantic uncertainty does not imply lack of truth. That is : (5) does not lead to a rejection of the converses of (7) or ( 8 ) : 182 182

Vagueness Vagueness Vagueness Vagueness Vagueness Vagueness

Nevertheless, I shall interpret semantic uncertainty in such a way that ( 9 ) and (10) are not true in general. For if it is uncertain whether or not a proposition (or its negation) is true, that should not imply (even materially) that the proposition is not true. The fact that the uncertainty is semantic entails that the uncertainty cannot be eliminated by additional knowledge. But this should not make the proposition untrue any more than the lack of uncertainty should make a proposition true. After all, what is seman­ tically uncertain is the truth of the proposition. To assume that semantic uncertainty is incompatible with truth is to assume that the truth of the proposition is not really in doubt when the proposition is uncertain in this way . 7 A consequence of this interpretation of semantic uncertainty is that the existence of borderline cases, e.g., of a short man, is not logically in­ compatible with either PBV or PEM. The following are each possible : 182 182

R IC H M O N D C A M PB E L L R IC H M O N D C A M PB E L L

Since, indeed, I maintain PBV, I imply for any height h the following are each impossible : 182 182

R IC H RM ICO HN MDO NCD A MCPB AM E PB L LE L L R IC H M O N DR IC CH A M PB O NEDL LC A M PB E L L

These are impossible, since PBV is incompatible with the second and third conjuncts in each case. Now if every height is such that either a man of that height is short or a man of that height is not short and if every height less than a height of a short man is also a height of a short man, then between the heights of short men and those of men not short, there has to be a cutoff point separating the heights of short men from those of men not short. The

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existence of this cutoffpoint is implied by (N2) and is not incompatible with the existence of borderline cases throughout the range of heights in which the cutoffpoint lies. Notice that (N2) implies only that there is such a point (somewhere) - there is no implication that anyone knows or could know where the point is located. The location of the point within the range of borderline cases is as semantically uncertain as the predication of short­ ness to any height within that range. At this point it will surely be objected that if the existence of a cutoff point is compatible with the existence of borderline cases, then the so-called semantic uncertainty that attaches to the latter is really an epistemic uncertainty of a very peculiar sort. For there will be a ‘right’ and a ‘wrong’ answer to the question of whether a borderline case is truly a short man, despite the uncertainty that most competent speakers feel. But if no acquisition of knowledge, linguistic or otherwise, can possibly alleviate this uncertainty, what sense can there be to talking about right and wrong answers? In what sense can there exist the one, true point of demarcation which cannot possibly be discovered even in principle? My reply is twofold. First, there seems to be no inherent contradiction or incoherency in the supposition that a proposition is true even though it is ‘in principle’ impossible to discover whether it is true. The proposition that if I had flipped a coin just now (although I did not), it would have landed heads might be true, even though it would be, I should think, ‘in principle’ impossible to verify its truth. The example may not be satisfy­ ing; yet I know of no convincing a priori argument to show that such an example must contain a contradiction or some other kind of incoherency. (I assume that the objection cannot draw support from some version of the notorious verifiability theory of meaning.) Second, pragmatic considerations weigh in favour of countenancing an undiscoverable point of demarcation within a range of borderline cases. Taking this course, we need not tamper with classical logic, at least not for the sake of accommodating vague predicates, yet we need not look upon borderline cases as anything but what they are: they are no less problematic in virtue of there being an undiscoverable demarcation point and any neat division of them into the short and not short will be perforce ad hoc. On this interpretation of borderline cases, vague language turns out to be no less vague for conforming to the requirements of classical logic.

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This conclusion is not very different from Black’s. Let us reconsider two of the questions raised earlier regarding Black’s proposed solution to the sorties paradox. Applying PEM, we concluded that either every m an shorter than some short man is short or not every man shorter than some short man is short, and we noted that recognizing the truth of this dis­ junction does not require making an ad hoc division among borderline cases. Yet Black says that such a division is presupposed by an application of PEM to a proposition involving loose concepts, in a sense, whal Black says is true: any use of loose or inexact terms, not to say the application of PEM to a propositioji involving them, docs imply that there exists such a division. But recognizing that there exists such a division does not require making some particular division among borderline cases. Any division we made would be ad hoc - but the existence of such a division is not an ad hoc assumption. This is very close to what Black says: “ I must understand that the use of the customary logical principles presupposes an ad hoc demarcation, somewhere in the region of cases that are not in­ disputably clear; and, finally, I must understand that there are no rules for drawing such lines” (p. 1 1 ). The second question was whether the ad hoc demarcation presupposed robbed the concept in question of its looseness. Even if PEM would apply to a proposition involving concepts rendered exact by arbitrary stipulation, it was pointed out that this would not show that PEM applies to reason­ ing with loose concepts. We can now see that a concept is not rendered exact just because its use in a proposition entails the existence of an exact demarcation among borderline cases. The concept remains loose or inexact since the location of the demarcation is uncertain. Although this uncertainty cannot be removed without altering the concept, the applica­ tion of PEM does not require any such removal. V. D IS S O L V IN G T H E P A R A D O X

On the present interpretation, then, the negation of the inductive premise of the sorites argument, 184

V. D IS S O L V IN G T H E P A R A D O X

turns out to be true, making the inductive premise, 184

of thesoritesargument,

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false. This is how the paradox is dissolved. But there arises an important objection. This objection, I believe, lies at the heart of the sorites paradox since it reveals the primary philosophical motivation for rejecting (N2) and thus rejecting PBV (or else accepting (2) with all its unfortunate consequences). The objection is simply put: no m atter what height A you choose, it is never the case that A satisfies the open sentence 185

T H E S O R IT E S P A R A D O X

For if any height did satisfy (15), then all actual heights S (4 + 2/120)’, etc. To shortcut these irrelevant complications, I permit the notation ‘5 (A +1/120)’ for ‘5* (A)' below. 3 Generally I shall omit this qualification below, since an arbitrarily small quantity may be used in place of 1/120 of a foot. Still the need for some qualification is theoretically important. For how can the negation of the inductive premises be generalized to ex­ clude borderline cases absolutely? Black’s generalization may be symbolized thus:

190

Here the trouble Here isthethat trouble its own is that negation its own negation

This generalization entails an absolutely precise cutoff point. The trouble is that the corresponding inductive premise, which is the negation of (N2a), does n o t generate the s o r ite s argument:

190

Here the trouble Here the is trouble that its own is that negation its own negation

In fact, from (2a) and *S(4)’ one cannot even infer *S (4 +1/120)’. There is however, another way to generalize the inductive premise which will generate the s o r ite s argument:

190

Here the Here trouble theistrouble that itsisown thatnegation its own negation

Here the trouble is that its own negation

190 190

Here the trouble Here the is that trouble its own is that negation its own negation

does not exclude borderline cases absolutely, even when coupled with the truisms that any man shorter than a short man is a short man and any man taller than a man who is not a short man is not a short man. There remain possible borderline cases within 1/120 of a foot. Interestingly enough, there appears to be no generalized version of the negation of Black’s inductive premise such that it excludes borderline cases absolutely and its negation generates the s o r ite s paradox. 4 Black, for his part, rules out this option when he says : “ The rule of excluded middle is intended to apply to statements having one or other of two determinate truth values...” (P. 10). 5 None of the usual difficulties with Tarski’s equivalence concern the kind of proposi­ tion here in question. Still, it could be regarded as question begging to rely s o le ly on the consequences mentioned in objecting to the retention of PEM without PBV. Storrs

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M cCall, in ‘A Non-Classical Theory o f T ruth, with an A pplication to Intuitionism ’, A m erica n P h ilosoph ical Q u a rte rly 7 (1970), 83-88, chooses to abandon both T arsk i’s

equivalence and the distribution o f truth into a disjunction in o rd er to retain PEM w ithout PBV. Also, Bas van Fraassen argues in F o rm a l S e m a n tic s a n d L o g ic , the M acm illan Company, N ew Y ork, 1971, pp. 163ff., th at we can retain som ething like T arski’s equivalence for a nonbivalent language by granting th at ‘7>’ is true if and only if *p' is true. But I do not take the consequences m entioned, ju st in themselves, as a conclusive objection to the separation o f PEM and PBV. 6 The introduction of operators at this stage is intended only as a convenient shorthand. There is no attem pt to locate the resulting form ulae in som e system o f logic. 7 R. R. Verma considers this interpretation o f sem antic uncertainty or indeterm inacy in ‘Vagueness and the Principle o f Excluded M iddle’, M in d 79 (1970), 67-77. 8 A height h would be in the range o f first order borderline cases provided th a t it satisfies the open sentence ‘ ~ D S (h) 8l ~ D ~ S ( h ) \ T here w ould be two ranges o f second order borderline cases corresponding to the open sentences ‘ ~ D D S (h) & D ~ D SLogic’. (hy and t ~ D ~ D ~ S ( h ) & ~ D D ~ S (h )\ T he first range lies between ‘B orderline the heights h satisfying ‘JDDS (hy and those satisfying *D ~ D S (/*)’, while the second range lies between the heights h satisfying lD ~ D ~ S ( h ) \ an d those satisfying *DD ~ S (h)\ There would be four open sentences defining third o rder borderline cases: 4~ D D D S (hi) & ~ D ~ D D S (h )\ * ‘B ~ orderline D ~ D ~ DLogic’. S ( h‘B )& ~ D D ~ Logic’. D S (hy orderline D D ~ D Logic’. ~ S (h) D ‘B ~ orderline D ~ D ~ SLogic’. (h )\ an d ‘B orderline D ~ DLogic’. D ~‘BSorderline (h) DDD ~ ‘B orderline ‘B orderline Logic’. Logic’. ~ S (h )\ Theoretically, this process o f defining higher o rder borderlines can be con­ tinued indefinitely. T he procedure is explained by D avid Sanford, its au th o r, in ‘B orderline Logic’. 9 F o r exam ple: G oguen, M achina, and Sanford, op. c it. 10 In G ogucn’s paper, for example, given any height h the conditional ‘If a m an o f height h is short, then a man o f height h plus a small increm ent is sh o rt’ is given a truth value equal to the quotient o f the truth value o f the consequent clause divided by the truth value of the antecedent clause. It is shown th at even if the tru th value o f the conditional claim is very nearly, but not quite, equal to 1, repeated use o f m odu s p o n e n s , as occurs in the s o r ite s argum ent, results in the falsity o f the claim th at a seven foot m an is short. But this ingenious approach provides no explanation why the tru th value o f the conditional should drop from 1 to som ething less a t som e critical point in the range o f increasing heights. ( I f the truth value is 1 throughout, then the s o r ite s paradox reappears. ) It may be suggested that the critical point can be chosen arbi­ trarily (within a certain range), but then I fail to see how G oguen’s approach is preferable to retaining classical logic, dissolving the s o r ite s by affirming the negation o f B lack’s inductive premise, and explaining the existence o f borderline cases by reference to degrees o f sem antic uncertainty (rather than degrees o f truth).

[12] TIMOTHY WILLIAMSON

WHAT MAKES IT A HEAP?

ABSTRACT. On the epistemic view of vagueness, a vague expression has sharp boundaries whose location speakers o f the language cannot recognize. The paper argues that one of the deepest sources of resistance to the epistemic view is the idea that all truths are cognitively accessible from truths in a language for natural science, conceived as precise, in a sense explained. The implications of the epistemic view for issues about the relations between vague predicates and scientific predicates are investigated.

1. On the epistemic view of vagueness, a vague expression has sharp bound­ aries whose location speakers of the language cannot recognize. This is not to deny that vagueness exists; it is to assert that its underlying nature is epistemic. The view may go back to the great Stoic logician Chrysippus.1 It permits the application of classical logic and bivalent truth-conditional semantics in their full strength to vague languages. Nevertheless, many philosophers are willing to forego these advantages and reject the epis­ temic view, because they believe that it outrages compelling intuitions. This paper does not attempt to rehearse a full-scale defence of the view, still less to survey all the alternatives. That has been done elsewhere; it involves an investigation of the general cognitive principles which explain our ignorance in borderline cases.2 A more limited project will be under­ taken here. It will be argued that one of the deepest sources of resistance to the epistemic view is the idea that all truths are cognitively accessible in a sense explained below from truths in a language for natural science, conceived as precise. Once the idea is stated clearly, its questionable nature becomes apparent. Vagueness has too often been studied as though from the standpoint of a privileged perfectly precise language. Such an external standpoint on vagueness is arguably impossible, not even fully imagin­ able. One has to study the phenomenon from within. Doing that is just what many formal theories of vagueness attempt to avoid. Section 2 makes the connection between resistance to the epistemic view and the language of natural science. Section 3 investigates the impli­ cations of the epistemic view for issues about the relations between vague predicates and scientific predicates. Erkenntnis 44: 327-339, 1996. © 1996 Kluwer Academic Publishers. Printed in the Netherlands.

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2.

The epistemic view of vagueness strikes many philosophers as simply outrageous. The elaborate arguments against it which they sometimes go on to construct seem to be rationalizations of a more basic feeling. To say this is not to deny that the arguments require a serious answer, but the present concern is the feeling itself. It is often expressed like this: The epistemic view permits this situation: h is a heap, although h is a borderline case of a heap. But what makes h a heap? The suggestion is that h is a heap only if something makes it a heap, and that the epistemic theorist can oifer nothing to make it a heap. Alternatively, the feeling may be expressed metalinguistically, like this: The epistemic view permits this situation: the word ‘heap’ applies to h, although h is a borderline case of the applica­ tion of ‘heap’. But what makes ‘heap’ apply to hi The suggestion is that ‘heap’ applies to h only if something makes it apply, and that the epistemic theorist can offer nothing to make it apply. The two objections are not mere rephrasings of each other. A descrip­ tion of the use of the word ‘heap’ would be directly relevant to the latter objection but not to the former, for whether h is a heap depends on its physical properties, not on the use of a word. Nevertheless, the two objec­ tions spring from the same source, and raise similar issues. For simplicity, the focus will be on the first objection. After all, if it is unobjectionable for h to be a heap, then it is equally unobjectionable for the word ‘heap’ to apply to /i, given that the word ‘heap’ applies to all heaps. Suppose that the result of subtracting one grain from h is a non-heap h~, which is also a borderline case. It may further be asked what the relevant difference is between h and h~, which makes the difference between a heap and a non-heap. Presumably, if X makes h a heap, and Y makes hr a non-heap, then the relevant difference is the difference between X and Y. Thus the epistemic view can satisfactorily answer the question ‘What makes the relevant difference between h and h~V if and only if it can satisfactorily answer the questions ‘What makes h a heap?’ and ‘What makes hr a non-heap?’. Moreover, the question ‘What makes h~ a non­ heap?’ raises the same theoretical issues as does the question ‘What makes h a heap?’. It is therefore sufficient to focus on the latter question as an objection to the epistemic view. The most austere reply to the objector is that what makes h a heap is simply the fact that h is a heap. In answer to the semantic objection,

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the corresponding reply is that what makes the word ‘heap’ apply to h is simply the fact that ‘heap’ does apply to h. An equally austere but slightly more structured reply is also possible: what makes ‘heap’ apply to h is the combination of the semantic fact that the condition for ‘heap’ to apply to something is that it be a heap with the non-semantic fact that h is a heap. These replies are likely to leave the objector baffled yet quite unsatisfied. They have somehow met the letter of the objection but missed its spirit. What, then, is the spirit of the objection? The objector is likely to complain ‘It cannot be just a brute fact that h is a heap’; what does this mean and why should it be true? Questions of the form ‘What makes it the case that P T or ‘In virtue of what is it the case that PT or ‘In what does its being the case that P consist?’ should never be allowed to go unchallenged. They demand the replacement of ‘P ’ by some other statement ‘Q ’ that is somehow prior to ‘P ’, without explaining what kind of priority is in question or why we should expect to be able to find anything prior to ‘P ’.3 The best way to elicit the objector’s tacit constraints on an acceptable answer to the rhetorical question is to ask it in a non-borderline case. Let /i* be clearly a heap. What makes h* a heap? The austere reply is again that it is simply the fact that h* is a heap. The objector should find this reply unsatisfying, too, for if it is legitimate here, then it is equally legitimate in the borderline case. Of course, the objector is likely to accept that h* is a heap (since h* is clearly a heap), and unlikely to accept that h is a heap (since h is a borderline case). However, to conclude from this that the austere reply is question-begging in the borderline case would be to mistake the dialectical position. The objector is trying to articulate a devastating objection to the epistemic view, to render the view untenable by asking a question that proponents of the view feel the need but lack the ability to answer on their own terms. They willingly grant that some object h is a heap but not clearly a heap (of course, when we see h we cannot recognize it as such an object). Since the challenge is to reply on that hypothesis, they are entitled to assume that h is a heap. The objector seeks more than the austere account of what makes something a heap, whether it is a clear or borderline case of a heap. The difference is that the objector assumes that a richer account is possible only in the clear case. What kind of rich account does the objector envisage for the clear case? What makes h* a heap? The objector may want to say something like this: what makes h* a heap is the arrangement of its constituent grains. Having exactly the microphysical properties that h* actually has is a sufficient (although not necessaiy) condition for being a heap; h* is a heap in virtue of those more fundamental facts. More generally, the truths of everyday life supervene

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on those of natural science; situations cannot differ in the former without differing in the latter too.4 In Sellars’ terms, the manifest image supervenes on the scientific image. The problem for the objector is that the premise of the objection is now quite consistent with the epistemic view of vagueness. For the epistemic theorist can grant that situations cannot differ in everyday ways without differing in scientific ways, too; in the borderline case, what makes h a heap is the arrangement of its constituent grains. Thus having exactly the microphysical properties that h actually has is a sufficient (although not necessary) condition for being a heap; h is a heap in virtue of those more fundamental facts. The epistemic theorist is not forced to treat it as a brute fact that h is a heap. Similarly, if the objector were to use everyday terms and say that what makes h* a heap is the fact that h* has enough grains, then the epistemic theorist would reply that what makes h a heap is the fact that h has enough grains. Trying a slightly different line, the objector might say that in the clear case what makes h* a heap is speakers’ agreement in applying the word ‘heap’ to it. After all, there is no agreement in applying the word to h, for it is a borderline case. This line runs an obvious risk of confusing use and mention, h* could easily have been a heap even if speakers had not agreed in applying the word ‘heap’ to it, for they could easily not have used that word in that sense, while h* retained the same microphysical properties. To avoid this mistake, the objector might specify that what makes x a heap in a possible situation s is not agreement in s in applying the word ‘heap’ to things with the microphysical properties of x in s but agreement in the actual situation in applying the word to things with the microphysical properties of x in s.5 As it stands, the new formulation is still too crude, for agreement in applying a word to a thing does not generally guarantee the correctness of that application. Whole societies can sometimes be mistaken. It takes only the mildest dose of realism to recognize that there can be agreement without truth, and truth without agreement.6 That h* is a heap does not follow from the assumption that speakers in the actual situation agree in applying the word ‘heap’ to h*, unless one adds the dubious further premise that this is not the kind of matter agreement about which can rest on a mistake. Indeed, it would be uncharitable to interpret the objector as attempting to reduce the semantics of a word to its use, non-semantically described, for the reducibility claim is too contentious to ground an effective objection to the epistemic view. What the objector may more reasonably assume is that semantic truths supervene on non-semantic truths about use: situations cannot differ in the former without differing in the latter too (given a

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fixed contribution from the environment). Supervenience does not entail reducibility. At least, there is no entailment when the reducing formula is required to be of finite length; the objector implicitly required a finite answer by asking the epistemic theorist to say what makes h a heap. Given such a supervenience thesis, the total use of ‘heap’ with respect to h* in the actual situation is sufficient for h* to be a heap.7 For the sake of argument, set aside the suspicion that the new view irrelevantly introduces linguistic facts into an account of the wholly nonlinguistic fact that h* is a heap. The more pressing problem is that the premise of the objection is now quite consistent with the epistemic view of vagueness. For the epistemic theorist may grant that situations cannot differ semantically without differing in non-semantic facts about use, too (given a fixed contribution from the environment); in the borderline case, the total use of ‘heap’ with respect to h in the actual situation is sufficient for h to be a heap. Thus the dialectical position does not greatly depend on whether the objector appeals to the microphysical properties of h* or to the use of the word ‘heap’ in saying what makes h* a heap. The epistemic theorist can accept the relevant supervenience thesis and apply it to the borderline case. In both cases, the crucial feature of the underlying truths is that they are conceived as precise (whether perfectly precise or just precise in relevant respects is an important question that will be left open here).8 Once a precise sufficient condition has been given for something to be a heap, the issues raised by the further question ‘What makes it meet that condition?’ are not directly relevant to the problem of vagueness. Thus the pertinent supervenience thesis is that vague truths supervene on precise truths: situations cannot differ vaguely without differing precisely, too. Call this claim simply ‘supervenience’. Although the epistemic view does not entail supervenience, the two are quite consistent. It will be simplest to assume that the epistemic theorist accepts supervenience. Supervenience attributes a coarse-grained metaphysical completeness to precision. It implies that any vague truth is a metaphysically necessary consequence of some set of precise truths. The epistemic theorist and the objector agree on the metaphysical completeness of precision. What they disagree on is its fine-grained cognitive completeness. In the clear case, the conclusion that h* is a heap is cognitively accessible from some precise truth about /i*, in the sense that, given knowledge of the latter, we are in a position to know that h* is a heap. Although the precise truth does not itself contain the concept of a heap, we can recognize it as entailing that h* is a heap by using our grasp of that concept. In the borderline case, by contrast, the conclusion that ft is a heap is not cognitively accessible

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from any precise truth about h. Given knowledge of such a truth, we are not in a position to know that h is a heap. The objector regards a vague statement as true only if cognitively accessible from a precise truth; the vague statement passes this test only in a clear case. The epistemic theorist regards vague statements as cognitively autonomous, and as such capable of truth in both cases. Belief in the cognitive completeness of precision is outraged by the epistemic view of vagueness. It is worth noting what the cognitive completeness of precision does not imply. First, it does not imply that all precise truths are knowable, or that all vague ones are. It says that if we knew all precise truths then we should be in a position to know all vague ones, too, but it offers no guarantee of the antecedent, and as a result none of the consequent either. For all it says, there may be precise unknowable truths of physics. Second, the cognitive completeness of precision does not imply that all vague statements are analytically equivalent to precise ones (could they be vague if they were?). It says that every vague truth is a corollary of a precise one, as ‘Most As are Bs’ is a corollary of ‘97% of all As are Bs’. It does not suggest that the vague truth entails the precise truth, as it clearly does not in that example. ‘Most As are Bs’ need not be analytically equivalent to any precise statement. Only in a weak sense is it claimed that whatever can be said can be said clearly. Third, it is not even implied that vague knowledge is always derivable from precise knowledge. If a vague truth is known, then it is implied that there is a precise truth knowledge of which would have put one in a position to know the vague truth, but it is not implied that knowledge of that precise truth is possible. In spite of what it does not imply, the claim that precision is cognitively complete should be suspect. For it does imply that all truths can be told in language of a privileged kind: precise language. The vague truths can be read off the precise ones. In this sense, the claim embodies a sort of reductionism. Such reductionism undoubtedly has its attractions. But consider some of the vague words that we use to talk about what matters to us: ‘cause’; ‘reason’; ‘say’; ‘know’; ‘desire’; ’pain’; ‘person’; ‘society’; ‘environment’. Any truth formulated in such terms can be read off precise truths, if precision is cognitively complete. The objector may concede that some concepts permit us to formulate cognitively inaccessible truths, but must deny that vague concepts are an autonomous source of cognitive inaccessibility. Any inaccessible vague truth is a mere corollary of an inaccessible precise truth. This is clearly a contentious view. One may suspect it of being at bottom a form of scientism, on which all questions can be replaced without significant loss by questions of natural science. Be

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that as it may, the epistemic theorist can deny the cognitive completeness of precision without absurdity. It would be hard to show that the claim of cognitive completeness lies behind all objections to the epistemic view of vagueness. But if you are inclined to object to the view by asking ‘What makes h a heap?’ or ‘What makes “heap” apply to ft?’ in a borderline case, then you should ask yourself what you would accept as an answer to such a question in a clear case, and why you think that it must have such an answer there.

3. On the epistemic view of vagueness, vague predicates stand for properties that each thing has or lacks. The cognitive incompleteness of precision raises questions about these properties. What is their metaphysical status? Can they be identified with robust natural scientific properties? If so, what stops us from locating their cut-off points? If not, are they somehow minddependent? These questions will be addressed on the basis of the epistemic view. As a convenient over-simplification, the language of natural science will be assumed to be unified and precise. This false assumption is harmless here, for it in no way helps the epistemic theorist. The same property can be presented in different ways. The property of being part of Hesperus is the property of being part of Phosphorus. For definiteness, it will be assumed that the property of being F is the property of being G if and only if it is necessary that everything is F if and only if it is G.9 The necessity here is to be understood as metaphysical necessity: what could not have been otherwise. Thus to ask whether the property of being a heap is a natural scientific property is to ask whether ‘It is necessary that everything is a heap if and only if it is G ’ is true on some substitution for ‘G ’ in the language of natural science. Some vague predicates do stand for natural scientific properties. Some molecules contain many atoms; others do not. If the molecule x contains at least as many atoms in a possible situation 5 as the molecule y does in a possible situation t> and y contains many atoms in £, then x contains many atoms in s.10 Let n be the least number i such that for some molecule y and situation t , y contains exactly i atoms in t and y contains many atoms in t. The definition of ‘n ’ assumes the least number principle, that every non-empty set of natural numbers has a least member, but that principle is unproblematic on the epistemic view of vagueness, whose consequences are being explored here. It follows that for every molecule x and situation s, x contains many atoms in s if and only if x contains at least n atoms in s. Thus the property of containing many atoms is the

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property of containing at least n atoms. Since ‘contains at least n atoms’ is a predicate of natural science (where ‘n ’ goes proxy for a numeral for n, not the definite description used to define ‘n ’), the vague predicate ‘contains many atoms’ stands for a property of natural science. We may know exactly how many atoms a molecule contains, yet be unable to find out whether it contains many atoms, because unable to find out which natural number n is. One might say therefore that we are unable to find out which property the property of containing many atoms is. However, one should not put much explanatory weight on that description, for the ‘know which’ locution is notoriously slippery. It works best in the context of a system of canonical terms for the items in question, one for each item. The numerals are canonical names for the natural numbers, for example. In contrast, we lack a system of canonical predicates standing for all properties. The distinction between predicates that acquaint us with the properties for which they stand and predicates that do not is unclear. An adequately explicit account will avoid the ‘know which’ construction. What needs to be said is that the predicate ‘contains at least n atoms’ is more firmly embedded in our physical theory than the predicate ‘contains many atoms’; the former has richer inferential connections with other expressions, making it easier for us to know its extension. We cannot measure the application of vague concepts because we cannot calibrate our measuring instruments to them; we lack the initial connections between our theory of the instruments and the concepts whose application is to be measured.11 We do know some property identities expressed by means of an every­ day and a scientific predicate; for example, we know that the property of being some gold is the property of being some of the element with atomic number 79. It might therefore be asked why no such knowledge is avail­ able for vague predicates. To answer the question, one must first identify the conditions for such knowledge to be possible, and then explain how they can fail to be met. The following account is certainly too simple, but indicates at least the general lines of the epistemic theorist’s answer. Let ‘jF’ be an apparently vague pre-scientific predicate that we use in talking about some field of interest. We then develop a scientific taxonomy of the properties relevant to that field, which we think of as natural prop­ erties. Each property is expressed by a canonical predicate in the language of the taxonomy; these predicates form a set I \ We develop workable tests for the application of predicates in I\ Suppose, indeed, that each predicate ‘G* in T is perfectly precise, so that it is necessary both that everything that is G is clearly G and that everything that is not G is clearly not G (the use

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of ‘G ’ outside quotation marks should be read as subject to the condition that ‘G ’ means G). Call a predicate ‘G ’ F-eligible just in case it is necessary both that everything that is clearly F is G and that everything that is clearly not F is not G. The two occurrences of ‘clearly’ here refer to what is clear prior to any theoretical identification of the property of being F. F-eligibility is a necessary but not sufficient condition for necessary coextensiveness with ‘F ’.12 It is necessary, for being clearly the case entails being the case: necessarily, everything that is clearly F is F and everything that is clearly not F is not F. Moreover, if a predicate ‘G ’ in T is not F-eligible then a clear counter-example to the supposed coextensiveness of ‘F ’ and ‘G ’ is possible, for it is possible that something is either both clearly F and not G (and therefore clearly not G, because ‘G ’ is precise) or both clearly not F and G (and therefore clearly G, because ‘G ’ is precise). Thus i f ‘G ’ in T is not F-eligible, vagueness does not prevent us from recognizing that the property of being F is not the property of being G. In the other direction, F-eligibility is not sufficient for necessary coextensiveness with ‘F \ Prior to any theoretical identification of the property for which it stands, ‘F ’ appears vague: it is possible for something to be neither clearly F nor clearly not F; the mere F-eligibility of ‘G ’ does not require such a thing to be F if and only if it is G. Now suppose that the predicate ‘G ’ in T is F-eligible, and that no other predicate in T is. By the remarks just made, ‘G ’ is the only predicate in F that stands a chance of being necessarily coextensive with ‘F \ Moreover, with luck, most of the things that we have agreed in applying the predicate ‘F ’ to will be G, because most of them will be clearly F , and most of the things that we have agreed in applying the predicate ‘not F ’ to will be not G, because most of them will be clearly not F . Thus, if our scientific taxonomy of the field is a good one, the only natural property underlying our use of the predicate ‘F ’ as a serious candidate for its reference is that marked out by the predicate ‘G \ If ‘F ’ functions like a natural kind term in the way described by Putnam and Kripke, then it applies to just what has that property with respect to any possible world. There are no special obstacles to its so functioning. We may assume that, either by our intention or (more likely) by default, it does so. On this basis, we are defeasibly warranted in asserting that the property of being F is the property of being G. By relying on that assertion, we can clear up the apparent vagueness in ‘F \ The property of being F could be theoretically identified because there was exactly one F-eligible predicate in T, the set of scientific predicates for the relevant properties. But nothing in our pre-theoretic use of ‘F ’

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gave us an a priori guarantee that F would contain exactly one F-eligible predicate. If it had contained none, or more than one, no such theoretical identification would have been possible. We should therefore have been unable to clear up the apparent vagueness in CF ’; it would have been deep vagueness. How could T fail to contain exactly one F-eligible predicate? If it contains more than one, they must be mutually compatible. For all in­ eligible predicates apply to whatever is clearly F , and it may be assumed that something is clearly F. If the predicates in F partition the field at each level of generality into mutually exclusive and jointly exhaustive sets, then two predicates in T are compatible only if one is at a higher level of generality than the other; it will not be easy for predicates at different levels of generality both to be F-eligible. However, there is no general reason to expect the natural properties to be mutually incompatible; in an example below, they are not. In the other case, no predicate in F is F-eligible. Thus for any predicate ‘G ’ in T, if ‘G ’ is general enough necessarily to apply to everything that is clearly F , then it is so general that it can apply to something that is clearly not F. This outcome is quite feasible, indeed common. Every natural property diverges clearly from the property of being F at some point. Even if ‘F ’ is necessarily coextensive with a scientific predicate, there need not be exactly one F-eligible predicate in T. Suppose, for example, that ‘F ’, the vague predicate ‘contains many atoms’, is necessarily coex­ tensive with the F-eligible scientific predicate ‘Gn’, ‘contains at least n atoms’ (assume for the sake of the example that if any F-eligible predicate is in T, ‘Gn’ is). Since ‘F ’ is vague, for some number m it is unclear whether molecules that contain exactly m atoms contain many atoms. It follows that the predicates ‘Gm’ (‘contains at least m atoms’) and ‘Gm+i ’ (‘contains at leastm -f 1 atoms’) are both F-eligible. For let a molecule x contain exactly i atoms; if x is clearly F , then i > m, so x is both Gm and Gm+ j ; if x is clearly not F , then i < m, so x is neither Gm nor Gm+]. If ‘Gn ’ is in T, then presumably ‘Gm’ and ‘Gm+i ’ are too, and F contains more than one F-eligible predicate. If ‘Gn ’ is not in F, then presumably ‘Gm’ and ‘Gm+i ’ are not either, and F contains no F-eligible predicate. Either way, F does not contain exactly one F-eligible predicate. Thus the standard way of identifying a property as presented by an everyday pred­ icate with a property as presented by a scientific predicate is blocked. We are in no position to discover that the property of containing many atoms is the property of containing at least n atoms. Many vague predicates may fail to be necessarily coextensive with any (finite) scientific predicate at all, even unknowably. Call the properties that

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such predicates stand for unscientific properties. The property of being mountainous may be an example.13 Perhaps creatures whose sensibilities differed from ours would never talk or think about the unscientific proper­ ties that we talk and think about. Should we then say that those properties depend on our sensibilities? In the counterfactual sense, they need not do so. If something is mountainous, it would still have been mountainous even if our sensibilities had been different, provided that the difference in our sensibilities made no measurable difference to the lie of the land. So much is obvious on the basis of a reflective understanding of the word ‘mountainous’. That we might have used the word differently is a quite separate matter. If unscientific properties do depend on our sensibilities in some deeper sense, it is quite unclear what it is. One may suspect the idea that unscientific properties must be mind-dependent of a kind of scientism. Although such a suspicion is no substitute for an argument, it will make one hesitate to join the search for an etiolated sense in which unscientific properties are mind-dependent. Equally, one will doubt that the properties, unscientific or not, for which vague predicates stand are mind-dependent in any interesting sense. The present sketchy remarks are a challenge to explain such a sense. The epistemic theorist takes ordinary vague terms at face value, insisting that we can understand them theoretically only by engaging in the practice of using them. From this perspective, borderline cases confront us with unknowable mind-independent truths. There is a popular inteipretation or misinterpretation of the later Wittgenstein that may be summed up in the slogans ‘Practices must be viewed from within’ and ‘Nothing is hidden’. The phenomenon of vagueness suggests that those two slogans cannot both be right.14

NOTES 1 On the Stoic treatment of Sorites paradoxes see Barnes 1982, Bumyeat 1982, Williamson 1994:12-27. 2 See Williamson 1992 and 1994. Earlier defences o f the epistemic view include Cargile 1969, 1979, Campbell 1974, Scheffler 1979, Sorensen 1988: 217-252. 3 Quotation marks will be used as corner quotes where appropriate. 4 The most controversial aspects o f the supervenience thesis in the text do not directly impinge on this paper: for example, the supervenience of the mental on the physical. The argument is also not very sensitive to the technical details of the supervenience claim, with respect to which there are many options. For more on the supervenience claim see Williamson 1992: 152-153, 1994: 202-204. 5 A similar move is often made in discussions o f secondary qualities; the idea goes back to Davies and Humberstone 1980. 6 The point is forcefully argued in the context o f vagueness in Putnam 1983: 281-283.

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7 For more on the supervenience of semantic truths see Williamson 1992: 154-155, 1994: 205-207, Putnam 1994: 290. 8 That scientific predicates are not perfectly precise is argued in Williamson 1994:169-170; see also Putnam 1983: 277. 9 Sober 1982 argues against the view of property identity in the text. If he is right, causal factors may play a more central role. The arguments in the text could be adapted accord­ ingly. 10 The possible objection that the standard for what counts as many in a situation s should depend on what is normal in s probably confuses the context of evaluation with the context of utterance. Even if the principle in the text does over-simplify, it does so harmlessly. 11 See Williamson 1994: 216-247 for an explanation on the basis of independently con­ firmed epistemic principles o f our inability to judge the application of vague predicates directly in borderline cases. 12 ‘Necessary coextensiveness’ should be taken in the sense of the account o f property identity, i.e. the predicates ‘F ’ and ‘ (3h) (h is a hat-trick & p scores h in m)) This consequence says that there is what we might call a threshold factor reaching which suffices uniformly for any single player in any single match to have brought into being (or less grandly, to have scored) a hat-trick. The word “sorites” is derived from a Greek word for heaps, and the heap is a well known sorites. Suppose that sand is the stuff in the offing. Sand comes in grains as well as heaps. Following Wang,1 we may put the heap as an inductive argument. The basis step is that (3) Zero grains of sand do not make a heap of sand, and this premiss seems obviously true. The next claim is that a single grain of sand is never enough by itself to make the difference between what is not a heap of sand and what is. This claim yields the induction step that (4) For all n, if n grains of sand do not make a heap, then neither do n + 1 grains. (The corresponding induction step for goals and hat-tricks would fail at 2.) It follows by mathematical induction from (3) and (4) that (5) For all n, n grains of sand do not make a heap. The heap is seen as a paradox because while this seems a valid argument from true premisses, there are nevertheless heaps of sand made of large (but finite) numbers of grains of sand. On the other hand, (5) seems to be jiist plain true. Consider an infinite flat plane whose points have rectangular Cartesian coordinates and where the unit distance is one light-year. The lattice points on the plane are those both of whose coordinates are integers. Suppose that at every lattice point there is a single grain of sand, but that elsewhere the plane is void. There are infinitely many grains of sand on the plane. But it seems clear that the array of sand on the plane is too flat and sparse to make a heap there. If so, not even infinitely many grains of sand make a heap, and for each natural number n, there is a finite region of the plane with n grains that shows that n grains do not make a heap. (5) states a consequence of this last observation. The obvious moral of this story is that a heap of sand is not constituted merely by a number of grains alone; their configuration matters as well as their number. A number of grains of sand will make a heap only if they are heaped up. Consider an area in which

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three but not four grains will fit stably each touching the other two, and suppose a fourth will rest stably supported by the three below. This conical or pyramidal configuration seems to one person to be a heap of sand, albeit a small one. But number seems to matter as well as shape. For a single conical or pyramidal grain of sand does not seem to be a heap either of sand or of grains of sand. Neither does a single such grain seem to be a heap of molecules. It seems that items in a heap should be bound loosely, as by friction and gravity, rather than tightly, as chemically in a crystal. But even if a heap should be only loosely bound, it also seems that it should be stable (under gravity): one grain balanced on another seems to make a stack as contrasted with a heap; and even one grain resting on two side by side seems potentially too unstable under gravity (and the breeze) to be a proper heap. Since four is the least number of grains that will form a loosely bound but stable cone or pyramid, four seems to be the threshold of heaphood for sand.2 That, at any rate, is how it seems to one speaker. But no single speaker has infallible access to common usage of a word like “heap”. Evidence that common usage of a word is as it seems to a single speaker should, it seems, include his articulation of that usage seeming right to most other speakers. But such seeming right can also seem undecidably poised between, on the one hand, recognizing a prior but implicit meaning and, on the other, going along with a decision. Now let us consider for a moment a very special sort of heap; this sort is special not because it is somehow ideal or perfect, but because some aspects of its structure are easier to discern and describe. The simplest regular polyhedron is a pyramid with four faces each of which is an equilateral triangle congruent with the other faces. Imagine such a solid set on one of its faces and filled with sand. Or instead, imagine a triangular pyramid of grains: one grain at the top rests on a layer of three, which rest on a layer of six, and so on. In this structure, there is a single grain at the pinnacle, and the (n + l)th layer is got by putting a row of n +1 grains at one edge of a copy of the nth layer. So the number of grains in the nth layer3 is a acopy copyofofthe thenth nth a

acopy acopy copy copy a

Thus the number of grains in the whole pyramid of n layers is nth layer. So the number of grains in the anth nthlayer. layer.So Sothe thenumber numberofof

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4

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Let us now use this calculation to introduce some terms of art. Call a positive integer a pyramid number if and only if it is STUDIES STUDIES

4 for some positive integer n. (At present, this still includes the degenerate case of unity.) For any positive integer k, let us say that the largest pyramid number in k, £(k), is the largest pyramid number less than or equal to k, that the first pyramid number after k, f(k), is the least pyramid number greater than k; and that the closest pyramid number to k, c(k), is £(k) if k - €(k) is less than or equal to f(k) — k, but f(k) otherwise. These terms of art are as sharply defined as any in number theory. One reason why people make heaps is to store things or stuff quickly and compactly but accessibly. Man-made heaps tend to be thrown together pretty much slap-dash, and an air of the slap-dash may be part of how we think of heaps. As the number of layers grows, our pyramids may seem rather more regular than slap-dash; it seems very unlikely that a pyramid number of grains just thrown together on a patch should fall into a pyramid of sand there. Nevertheless, we can imagine that they should so land, which ought to convince us that they could so land, and thus that despite the unlikelihood of arising without deliberate engineering, our regular pyramids (of at least four grains) are heaps, like regular heaps of tins in a supermarket display. Gravity and opposition to it by friction among the bits seem to be important in heaping up things or stuff. Grains of sand close enough in empty space but not yet touching would under their mutual gravitation tend to form a ball, which seems not to be a heap even if their number is not so great that heat and pressure would fuse them chemically or otherwise into a single lump but would leave them only loosely packed. People make heaps on smallish simply connected regions of the earth’s surface. (Roughly speaking, a region is connected if it is not made up of two or more regions separated by territory not part of the region, and it is simply connected if as

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well the region omits no patches inside it.) According to Newton the vectors of the earth’s gravitational force through such a region converge in a point inside the earth, so those vectors are not mutually parallel. But if the region of the earth’s surface is a tiny fraction of the whole surface, then the distance between the region and the point of convergence is so large that to a very good approximation we can think of that region as flat and of those vectors as mutually parallel, and in this way we can pretty much ignore the inaccuracy in speaking of such a region as perpendicular to, or athwart, the earth’s gravitational field.4 This field is the familiar one, which is not so weak as to let grains float away into a ball, nor so strong as to squash them into a thin veneer of quarks and leptons. Let us now turn from gravity and the region to friction and the grains. We are thinking of the grains as pretty much symmetrical and globular, and any two of them as pretty much of a muchness (so if we need more than we are given to get up to a pyramid number of grains, we can get some more pretty much like those given). We seem to understand friction less well than Newton understood gravity. To some extent, friction seems to be the secret hideout of quantities required to balance the books of conservation, but anyone who has slid too fast in bare hands down a rope has felt the effects of outlaw friction. Feynmann says somewhere that it is only impurities on the surfaces of bricks of supposedly pure gold that keeps a stack of such bricks from merging into a single column of gold, for how, he asks, does gold know where one brick ends and another starts? Grains of sand in a heap should not fuse into a single lump or honeycombed tracery of quartz by sheer contiguity alone, so the sand should not be too pure. But neither should the sand be so impure that grains are only sparsely dotted about here and there in dirt like sparse raisins in a parsimonious pudding, for then barring an unlikely separation in being tossed it would not make a heap properly of sand. The grains should be somewhat rough. For example, heaping up things or stuff quickly and cheaply mostly rules out surrounding the region with a wall, and without such a wall around its base, a heap of greased ball bearings on a steel plate seems very unlikely. Perhaps such a pile held up by a retaining wall would be a heap, but friction among the grains seems a more natural counter to gravity than such a retaining wall. To be held up by friction, the grains should be a bit rough. (We shall concentrate on hard grains like those of sand, not squashy ones like grapes.) But they should also be smooth enough to slide among each other to settle compactly and stably under gravity. We expect a person heaping them up to be unlikely to find them sticking together in a fat jumble, or a weird tangle of curlicues, on a tiny base. Perhaps

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inside a heap vertical stacks of grains, in which each above the ground rests on just one grain below it, might occur held up by the surrounding heap as a retaining wall. But if the grains are globular and smooth enough, grains at the surface of the heap seem likely to slide down to rest on more than one grain. Heaps slope; that way, higher grains sit more stably, under gravity and forces like the breeze, on the grains below them. At the very least, in the absence of external forces like the wind, a heap should not shift all by itself. In the small here, stability is stable equilibrium; in a pyramid of one grain on three, a light breeze might raise the top grain by a hair, but if the pyramid is stable, that grain will fall back into its original position when the breeze has passed. But on a larger scale, dunes shift in the desert wind without falling back into their earlier configuration when the wind dies, yet though one can picture a tall stack of sand one grain across on a rock in the desert, such a column seems less stable than a dune. Perhaps the dune seems stabler because moderate winds produce moderate shifts in its configuration, while the column seems less stable because a light breeze produces a big shift, collapse. It should be clear that these physical considerations concerning gravity, friction and stability are vaguer than our earlier number theoretic terms of art. We have seen that no number of grains suffices all by itself to make a heap of sand. Configuration matters too. Maybe one might wonder whether that configuration is just a matter of purely geometrical three dimensional shape, the shape being a hump like a truncated cone tapering smoothly away from one flat surface toward a point at the apex, or like the volume enclosed by the revolution of a bell-shaped statistical curve about its vertical axis of symmetry. But, while it is very hard to be sure, the physical considerations alluded to above, which were roughly the statics of heaps, might intimate that the formation of heaps under forces like gravity and friction may matter as well as the form eventually assumed. Heaps are physical objects, and we should expect them to have a physics, even if on reflection it seems to be a rather complex physics. In that way, though it is not supposed to be obvious, we might think of a heap less as a pure geometrical solid than as the configuration somewhat rough, but also somewhat smooth, things or stuff tend naturally to take in space on or over a simply connected region athwart a gravitational field. Do heaps have dynamics as well as statics? Suppose many grains are dropped successively toward a point A on a plane. The grains may bounce, but any bouncing is damped, successive bounces are lower and thinner across, and bouncing grains eventually come to rest on the plane or on grains earlier landed there. It seems sensible to start by supposing that the

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probability that a grain comes to rest directly over a spot in the plane will fall off somehow as the distance from the spot to A increases. One also expects that the number of grains that come to rest over such a spot will also fall off somehow as the distance from the spot to A increases. (Perhaps one has both expectations because one expects the ratio of the number ending up over a spot to the total number dropped to vary with the probability of a grain ending up over a spot.) Suppose this falling off is linear, so that, for example, when about a foot of grains rest over A, about six inches of grains rest over the circle in the plane with centre A and radius six inches, and almost none wind up a foot or more from A. Then a roughly conical heap of grains forms on the circle in the plane with centre A and radius one foot. This cone peaks over A. But why should any grains wind up over A, let alone more than over spots farther out? Must some, and thus many, fail to bounce at all? But we should not build any unexplained asymmetries in ab initio. Whether, how and how much hard grains bounce on the ground depends on how hard and resilient the ground is. If it readily absorbs momentum and retains it as dissipated heat, the earliest grains landing around A should stay thereabouts. Then later grains should start bouncing on the hard grains now accumulated about A. It seems least unsymmetrical to suppose most of these to fan out within a certain distance in all directions across the ground from A. If they mostly bounce all the way out to that distance, we get a wall of a sort considered below. If they spread evenly through this distance, a disc harder and more resilient than the ground takes shape around A. Then it is as if the grains had fallen on harder and more resilient ground to begin with, so it seems reasonable to expect them to bounce. So suppose that all, or nearly all, the grains bounce (at least early on), but that while a grain is as likely to bounce away from A in any one direction as any other, the lengths of first bounces cluster statistically around an average, that subsequent bounces also cluster each around an average length, and that the numbers of bounces also cluster about an average. Then (at least earlier on) the distances from A at which grains come to rest will cluster around an average, so given our symmetry of the directions of their travel from A, a circular wall of grains will form cresting over the average distance from A that grains travel. More grains will land against the inside of this wall than the outside, so the wall will thicken inward more quickly than outward. As this wall rises and thickens inward, it will start to reflect grains back into the hollow it encloses. For sometimes as the wall bulges inward gravity will overcome friction and grains will cascade into the hollow, and sometimes grains will strike the wall thickening inward hard enough to bounce back into the hollow

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(in which case the momentum such grains impart to the wall will mostly dissipate in jiggling among the grains there). So one expects something like a phase during which the hollow fills more quickly than the wall rises. In this phase the configuration should tend toward a hump or flattened mound; the bell-shaped profile is beginning to emerge. It seems reasonable to suppose that as the number of grains at or over a spot to absorb momentum from a grain striking there increases, the damping of bouncing there will also increase. In that way wall building and hollow filling could continue on the shrinking flattened top of the mound, so it would become more conical. As well, if grains are still dropping from high enough above the top of the hump over A, early bounces could still be forceful enough for the middle of the hump to rise more slowly than the mid-sides, while around the edges grains will lie enough more shallowly to damp bounces less there, so the edges will also rise more slowly than the mid-sides. This trend toward a bell-shaped profile should be offset by occasional landslides to be expected as gravity overcomes friction here and there around the bulging mid­ sides. Such avalanches would tend to spread and point the growing heap, so there should also be a trend toward a conical shape whose perhaps irregular base records recent avalanches. We have sketched one rough qualitative dynamics of how a heap might form and it does not seem to describe too badly how a heap of poured grains looks to form. The longer one dwells on our picture, the more mechanisms occur to one as candidates for inclusion in our dynamics, and as the number of mechanisms increases, the more one wants (or feels obliged) to be able to calculate to check one’s qualitative speculations, though one guesses that such calculations might be very difficult to frame realistically and to carrry out. Nevertheless, perhaps one should become convinced that heaps have dynamics as well as statics. Heaps have a nature we might be able to explain; heaps constitute a natural kind. In addition to gravity and friction,' the region, the number of grains and the stability of their configuration seem to be important elements in the nature of this kind. The basic linkage of these elements seems to be that if enough grains rest stably on or over a suitable region, they will make a heap there. How precise can we make this model? Can we in particular be precise about the number of grains in a heap in anything like the way we can be precise that three goals (by a single player in a single match) make a hat-trick? Can our model tell us anything about the paradox of the heap? The salience of the number of grains in a heap is as old as the paradox of the heap, but the region on which a heap sits is hidden under the heap. Not every grain in a heap on the ground rests on

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the ground. Some of its grains may rest both on the ground and on other grains; the whole bottom tier may be thus jointly placed. But it seems to be part of the very idea of a heap that there be in it some grains that rest not on the ground but on other grains.5 So the seat of a heap, the outline of its base, is not in contact with every grain in the heap. Let us say that a grain in a configuration on the ground is over all and only the points in the ground in which perpendiculars to the ground from points in the grain intersect the ground, and that the configuration is over all and only the points in the ground over which a grain in the configuration lies. The region including exactly these points is like the shadow cast at high noon, but its choice is more gravitationally motivated. If there are never holes through a grain, it will always be over a simply connected region of points.6 Any one heap we assume to be over a connected region of points. If a heap could be empty in the middle, this region would not always be simply connected, but we shall concentrate on heaps over simply connected regions. To a certain extent, this is a matter of the compaction and stability of the grains in the heap, for were a heap very loosely packed, and thus easy to disturb significantly, it might have “air shafts” straight down from above to the ground.7 It is a consequence of our assumptions that a heap sits on all patches within the region on which it sits, but on no larger territory including that region, and this seems reasonable. Comparing the size and shape of the region below a configuration of grains with the number of grains in the configuration can give us some idea about whether that configuration is a heap. Imagine, for example, a stack of grains one grain across but ten feet tall. Here a large number of grains stand over a tiny (one grain across) patch, and they do not make a heap on it because, it seems, in even a light breeze they would collapse (perhaps into a heap on a larger patch), and thus are too unstable to make a heap. But one cannot infer the configuration just by comparing the size and shape of the region below it with the number of grains in it. For example, a high arch one grain across that touches ground at two patches could contain exactly as many grains as a lower wall one grain thick that marks out the same long thin strip of ground as the arch. While the second configuration may be more stable than the first as measured by sizes and placements of forces required to knock each all down, neither seems stable enough to be a heap, though each might collapse into a long, thin heap. One natural way to try to grasp the structure of a configuration of grains on a patch starts from the idea of a path through the configuration. Just for a few sentences let a thing be a grain or the ground. A path through a configuration of things between one thing

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and another is a finite sequence of things beginning with the one, ending with the other, and in which each thing (except the last) touches its immediate successor in the sequence. The length of a path between two things is the number of things excluding the first in the path. (A configuration of grains is unified if and only if for any two grains in it there is path between them of grains in the configuration and excluding the ground; heaps are, we assume, unified.) The remove of a grain in a configuration is the least number among lengths of paths between it and the ground. We are concentrating on configurations of finitely many grains, in which case the set of removes of grains in the configuration is a finite initial segment of the positive integers, that is, there is a positive integer k such that for any positive integer p, p is less than or equal to k if and only if some grain lies in the configuration at remove p from the ground. This k is unique to the configuration, and is called its height. A tier in the configuration is the totality of grains in it at the same remove from the ground. (A configuration is stratified if and only if for each tier in it and any two grains in that tier there is a path between the two grains composed just of grains in that tier. A more loosely packed heap need not be stratified.) For each p less than or equal to the height of the configuration, let n(p) be the number of grains at remove p; n(p) is the number of grains in the pth tier from the ground.8 Heaps slope. For grains all pretty much of the same size, sloping requires that for each p less than the height of a heap, n(p +1) < n(p). So at most n(p) = n(p + l) + 1. If that could hold for all positive integers less than the height n of the heap, then the number of tiers and the number of grains in the first tier would both be n, and the number of grains in the whole would be |n(n +1), But no such configuration of height greater than one and of globular grains would be stable enough to be a heap. What seems lacking is what we might call local stable equilibrium, that is, that each grain off the ground sit in a well walled around by grains below.9 The patterning of wells seems tricky. In a triangular pyramid of height three, the six grains in the bottom layer make four wells only three of which are filled by grains in the next layer up. Again, seven grains can be arranged in a hexagon of six around a seventh in the middle. While in general at most three grains will sit in a tight, level and still triangle on the seven, four grains will sit on them in a wobbly, uneven and less stable square. Were the second structure of eleven grains imbedded in a larger configuration, then in higher and higher tiers of the configuration above the imbedded structure, remove and altitude could be expected to separate and stability to diminish. How such instability starts at an imbedded structure and amplifies from it up the configuration toward collapse (or is somehow

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compensated toward stability) depends not just on the number of wells in a tier, but also on their separations, array and how precariously they might be filled. This in turn seems to depend more exactly on, at least, the sizes and shapes of the grains. But we wish not to be more specific than to suppose them globular and much of a muchness in volume. That vagueness or generality seems to seal us off from the fine structure of heaps. So paths seem to be routes that will take us only so far into the nature of heaps. But our earlier dynamics left us with a somewhat less focused or larger scale picture of not so much the structure as the construction of one reasonably representative sort of heap. This picture includes a picture of the seats of such heaps. From farther away they have largely circular seats. Let us now try to qualify this basic thought toward greater plausibility and generality. During some phases of growth, outgoing landslides are to be expected. Then the seat of such a heap would have a circular core from which pseudopods radiate.10 We should include pyramids too. Their seats are equilateral triangles, squares, and perhaps other regular polygons. Here too we should allow for slides, especially for taller pyramids. Since tradition landed us among heaps of sand, it would be a shame not to include dunes too. The wind is a main agent in the dynamics of dunes.11 Consider a sandy territory like a beach or desert across which blow winds strong enough to lift and carry grains of sand but not strong enough to carry them all out of the territory. The grains first lifted should be those high enough to be caught by the wind; let us not worry about mechanisms like earthquakes or burrowing animals to raise these grains. The nucleus of a dune might be a bush or an outcrop of rock; grains carried by the winds collide with this nucleus and fall to the ground around it. But the dynamics shaping dunes concerns not so much their original accretion as their transformation and migration. Broadly speaking, there are two kinds of dunes. Transverse dunes extend farther across than with the wind. These occur where the wind blows from a nearly constant direction. Their slope is gentler on the windward side. Grains higher on this side are more exposed to the wind and winds are swifter at greater elevations, so more grains tend to be lifted by the wind higher on the windward side, which is thus flatter. Toward the more sheltered lee side, these grains tend more just to drop. Here the slope is more like that of a heap in a builder’s yard, and steeper. Barkhans, one of the commonest sort of transverse dunes, have the symmetrical crescent-shaped seats the dynamics leads one to expect. Longitudinal dunes are ridges of about the same slope on either side. They are elongated in the direction of the prevailing winds. Grains along the upper edge of the profile that a dune presents to the wind would seem very exposed to

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the wind. Where the wind blows from a nearly constant direction, this edge is very thin and the proportion of grains in a dune exposed along it to the wind is small, so the action of the wind in shifting them should be only a small part of how it shapes the dune. But where the prevailing winds swing ceaselessly back and forth across a given quadrant, this exposed edge swings back and forth along the two sides athwart the winds. Then a larger proportion of grains along these sides should be lifted by the wind, many of them to drop in the more sheltered lee of the dune, so the dune lengthens with the wind as it thins athwart it. Longitudinal dunes can thus be expected to have oval seats (perhaps with pseudopods for slides). We might try to think of the two windward-pointing horns of a barkhan as two “wind-slides” advanced by the migration of the windward centre of the dune with the wind. Then perhaps we could picture its seat too as roughly oval with pseudopods. All these seats seem to shape up as convex figures perhaps with pseudopods; maybe, even just ovals perhaps with pseudopods might accommodate all these kinds of heaps fairly well.12 Admittedly, for all one knows, clever workmen might make heaps with remarkably concave seats. But a heap should be heaped up, and one might expect slides on a higher heap with a concave seat to tend to fill its concavities, since the slope should be steeper by concavities in the seat. (Besides, such perversity in piles seems pretty peripheral.) Note too that it is sufficient conditions for the existence of a hat-trick given in (1) and (2) above. We are after sufficient conditions for the existence of a heap to compare with those for a hat-trick. So if most heaps have convex, or even oval, seats (perhaps with pseudopods), then committing ourselves to regions of those shapes should be good enough in both generality (of heaps) and specificity (of content) for us. Besides, dynamics seems to turn those shapes out naturally. Having settled on a shape for seats, we next want to connect such regions systematically with a number of grains. Area in grains might be one such connection. Let us take ‘the’ area in grains of a region to be ‘the’ number of grains just enough to cover it all. No grain is allowed to stick much out over the edge13 of the region. Each grain should sit flush beside its neighbours (but not be pushed so hard against them that any are crushed or broken). Each grain should rest on the region, but none may rest at all on any of its neighbours. We assume the grains uniform enough that varying which grains are laid out on the region, or how grains are arranged on it, hardly varies the numbers of grains just needed to cover it. So these numbers peak sharply in an average in which we may take, say, the greatest natural number. For a region r we take this number, call it s(r), as the size or area in grains of r. This is clearly rough. For

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example, any circle a good bit smaller in square inches than the typical cross section of a grain will have the same size in grains, namely zero, as its left half. But then neither will a whole heap of grains sit on such a small region. Observe too that the size of a union of disjoint neighbouring regions can exceed the sum of the sizes of the regions unioned, so size is not what analysts call a measure. Next we want systematically to connect the number of grains in a whole configuration athwart a gravitational field with the number of grains in the configuration’s bottom tier. This we can do most precisely for pyramids, so to illustrate let us return briefly to the numerology of triangular pyramids. Given a number k (of, say, grains in a configuration) we can effectively calculate c(k), the pyramid number closest to k, by generating pyramid numbers in order of magnitude until we trap k between two successive pyramid numbers; c(k) tells us how many grains there are in a triangular pyramid approximating the configuration. Having calculated c(k), we can effectively calculate the number n such that c(k) = £n(n + l)(n + 2); in fact, our way of calculating c(k) leaves us with this n in hand. From this n we can effectively calculate |n(n + 1), the number of grains in the bottom layer of a pyramid of c(k) grains. So there is as recursive (and in fact primitive recursive) function a such that for any k, a(k) is the number of grains in the bottom layer of a pyramid of c(k) grains. One can think of a(k) as something like the area in grains of the base of a pyramid of c(k) grains, and one can compare this area with the size of one of our regions. We said before that the main elements of the nature of heaps of hard, globular grains seem to be the forces of gravity and friction predominant in configuring the grains, the region on which they are configured, the number of grains in the configuration, and the stability of the configuration. Time probably matters too, and not just because a heap tossed momentarily into the air by an earthquake probably ceases to be a heap for at least a moment. Suppose superglue is poured on a heap and penetrates thoroughly among the grains. When the glue sets, the configuration of the grains may well be unchanged. But that configuration has probably then ceased to be loose enough to be a heap; it is now probably rigid enough to be a single lump. Besides, time is the dimension of dynamics, so since dynamics has been prominent in our thinking about configurations, we should allow for time. Let us begin by rethinking some of our earlier ideas about paths. For any region r, any grain g, and any instant t, a path from r to g at t is a finite sequence of grains ending with g, whose first member touches r at t, and each of whose members with a successor touches its immediate successor at t. For

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any region r and any instant t, the grains on r at t are all and only the grains to which there is a path from r at t.14 The grains on r at t are unified if and only if for any two grains on r at t there is a finite sequence of grains on r at t beginning with one of the two, ending with the other, and such that any grain in the sequence with a successor touches its immediate successor at t. Let us write UGF(r,t) to mean that the grains on r at t are unified (predominantly) by the action on them of gravity and friction (alone).15 Let g be the function whose value for any region r and instant t is the number of grains on r at t. Let us write P(r,t) to mean that r is the locus of points of intersection with the ground of projections along gravitational vectors from points in grains on r at t. Let us write C(r) to mean that r is an oval or a regular polygon perhaps with pseudopods; the letter “C” is a mnemonic for central convexity. Let us write S(r,t) to mean that the grains on r at t are stable. Stability has to do with more than just the instantaneous configuration of grains on r at t. It also has to do with what happens to grains on r during intervals of time around the instant t; stability is a dynamical property. During longish intervals around t, the grains then on r should hardly move at all under the influence of gravity and friction alone. Call the forces acting on the grains other than gravity and friction (and perhaps a bit of dampness among the grains) external forces. Thinking of dunes, we might take the wind to be a central example of an external force. A moderate wind is one of up to about Beaufort force four, the number Admiral Beaufort assigned to a moderate breeze. In light air of force one, the grains may rise with the air, but they should mostly fall back pretty much as they were when calm returns; the grains on r should be in pretty much stable equilibrium under very small external forces applied to them during intervals of a few minutes or even hours or days around t. Under stronger and stronger winds, the outside grains on r may be more and more shifted by the wind (as in the dynamics of dunes) during shorter and shorter intervals around t. Tornadoes may blow up the configuration of grains on r very quickly. But moderate winds during

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moderate intervals around t should modify which grains are on r, and the chains along which they are unified by gravity and friction, only moderately.16 One expects a heap of the grains on a region to be at least as stable as any unified configuration of those grains that lies on all or part of that region but that is not a heap. Dead calm seems very rare, at least in the temperate zones, so terrestrial heaps out of doors are almost always subject to external forces. We can use our arithmetical apparatus to build up a crowding condition. For suppose that a(g(r,t)) = s(r). Then generally speaking, if the grains on r at t were arranged in a triangular pyramid whose number of grains best approximates g(r,t), then the number of grains in the base of this pyramid would already suffice to cover r tightly. So however the grains on r at t are actually configured, many of them are not in contact with r itself, and so rest on other grains. If winds of no more than moderate force are blowing across r at t but nevertheless the grains on r at t are stable, then it seems very likely indeed that the grains on r at t will lie in a heap; grains collapse from weird and wonderful configurations into plain old heaps because that is the most stable configuration for them under moderate external forces. Let us write W(r,t) to mean that moderate external forces are acting across r at (or a bit before) t; a central example would be winds of Beaufort forces from one to four. Then our basic idea is that, as a matter of fact, when a(g(r,t)) = s(r), so r is crowded with grains at t, and W(r,t), so it is a bit windy across r at t but not too windy, and S(r,t), so that the crowd of grains on r at t is stable in these winds, and UGF(r,t), so the grains have not, for example, been fused into a rigid lump with superglue but are loosely unified by gravity and friction, and C(r), so r has the sort of shape we expect of the seat of a heap, and P(r,t), so r is really the region over which all the grains on r rest at t, then there is a heap of grains that sits pretty well on r at t. This basic idea needs some refinement and comment. It is not obvious that even all our conditions together require that there be any grains at all on r at t.17 Moreover, nothing we have required forbids that g(r,t) = 1 (so a(g(r,t)) = 1) where s(r) = 1. In that case, all our other conditions can be met too, and yet one grain does not by itself make a heap. We said before that four is the least number of globular grains that will make a heap, the triangular pyramid consisting of one grain sitting in the well walled by three grains on the ground below. With that in mind, let us add to our conditions that g(r,t) 4.

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Our crowding condition that a(g(r,t)) = s(r) is fairly demanding. There are two worries here. First, pyramids are pretty steep, so in a pyramid more grains rest on a smaller seat; natural heaps squat on broader seats.18 Such slouching may be important for stability. To allow for slouching, we might remove the conditions that P(r,t) from the antecedent.19 At the same time, we might change the consequent to say that there is a region r1 such that C^r1), P(r\t) and that includes r, and a heap of grains on r1 at at t. But r1 should not be so much bigger than r that the grains on r1 can spread out in a sheet one grain thick. With the smallest pyramid in mind, we might require that s(V) < 4/gs(r).20 Since r is then a large part of r1, we might require that they be roughly concentric. Second, our crowding condition applies only to regions whose sizes happen to be numbers of grains in the base of a triangular pyramid. That might seem an unnatural restriction. So we might demand by way of crowding that a(g(r,t)) ^ s(r) to take in some regions whose sizes do not fall in exactly with the numerology of bases of pyramids, but without reducing crowding.21 Drawing the pieces together, our idea has become that Drawing the pieces Drawing together, the pieces our idea together, has become our ideathat has become that Drawing the Drawing pieces the together, pieces our together, idea has ourbecome idea has that become that Drawing the Drawing piecesthe together, pieces our together, idea has our become idea hasthat become that that that that The nature of heaps seems to be deep enough to make it hard to be sure about (6). It might, for example, be more informative to add that the grains on r1 at t are the grains on r at t,22 and that the heap on r1 consists of these grains. But perhaps one can drum up enough confidence that something roughly like (6) is roughly right for it to seem worthwhile to compare (6) with (1), the dictionary’s condition for the existence of a hat-trick, and (5), the conclusion of the inductive argument in the paradox of the heap. One putative difference between (6) and (1) should be dismissed right away. Those who think that (1) is analytic seem hardly likely, even if they believe (6), to think that (6) is analytic. Since almost none of those confident in their ‘intuitions’ about meanings make much serious effort to describe how to justify claims about the meanings of “hat-trick” or “heap”, debating the analyticity of (1) or (6) would be like arguing about how tall elves would have been: fun perhaps, but frivolous. It is also irrelevant. Even if (1) were analytic, what matters is its truth. Similarly, it is not to the point whether (6) is analytic, necessary or known a priori. Approximating to simple truth is hard enough.23

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It may next be urged that “heap” is vague where “hat-trick” is not. It is true that there is a number, three, such that it takes that number of goals by a single player in a single match for him to have scored a hat-trick in the match. But does a player who scores just two goals in a match score two thirds of a hat-trick? If Frankie Bunn scored two hat-tricks against Scarborough on 25 October 1989, were they respectively his first, third and fifth goals, and his second, fourth and sixth? Here it seems natural to decide that in the string of goals scored by a single player in a single match, his first three goals would be his first hat-trick, and his next three goals would be his second hat-trick; fractional hat-tricks seem a bit of a joke to one who has never negotiated professional players’ salaries. Both points in this last sentence express decisions not forced by the pure numerology of hat-tricks or the dictionary definition. There is a precise ‘aspect’ to the relation between hat-tricks and goals that nevertheless need not all by itself make that relation an entirely precise reduction. Part of our effort has been directed toward finding or deciding what things are to heaps as goals are to hat-tricks.24 Grains (of sand) are, of course, prominent on our list, but along the way we have also taken note of regions, times, crowding, numbers of grains, sizes and shapes of regions, pyramids, the forces of gravity and friction, unification mostly by them, external forces like the wind, stability, and being directly over a region. Very little that we have said under most of these headings deserves to be called precise, let alone a reduction. Stability, for example, is central for us, and yet we have left that pretty vague. If we have increased precision in anyone’s thinking about heaps, then it has been our intention that this have come about not through arbitrary, wilful or capricious decisions, but rather that if decisions got made, they were natural. Is there a lesson here about vagueness? It is hard to be sure, but if there is, it could be that the conventional wisdom, which might go, “The fault, dear Brutus, is not in our world, but in ourselves, that we are vague,” presumes a dubious contrast between us and the world, a contrast blurred by natural decisions about matters of fact. At any rate, (6) is shot through with vagueness and not intended to reduce heaps to anything else. The paradox of the heap is a more definite test for our account of heaps than the nature of vagueness. In (6) replace the numeral “4” by the variable “n”. The existential quantification of the resulting predicate with respect to this variable follows from (6) as (2) follows from (1), and is like (2) in its logical anatomy.25 Moreover, if we have been on something like a right track, the extension of this predicate is at least a species of the kind of number n such that n grains make a heap. So (6) would commit us to the conclusion that there is a

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number n such that n grains make a heap. This conclusion looks to contradict (5) even though we said that (5) is just plain true. But equally plainly there is something like an equivocation here. We agreed to (5) on the grounds that no number of grains alone makes a heap whatever their array. But (6) is mostly about array. Very roughly, an idea behind (6) is that a heap on a region is a stable, crowded configuration on it, so since four is the least number of globular grains that have a stable, crowded configuration (the smallest non-degenerate pyramid) on a region (an equilateral triangle of size three), four is the least number of grains that make a heap.26 So our diagnosis is that the fallacy in the paradox of the heap is, roughly, equivocation. Allowing any old array, no number of grains makes a heap. But there is in the nature of heaps a least number n such that the crowded, stable configuration of n grains is a heap on the ground. Of course, no one who ever thought about the paradox of the heap for more than a nanosecond would have allowed just any old array. But, partly for that reason, it is somewhat surprising how often people here speak vaguely of “being in the right sort of arrangement” without going into the matter more deeply. Our claim is that when one goes deeper, one finds a sort of array and a least number n of grains such that it is natural to decide that n grains in an array of that sort make a heap. This is only roughly equivocation because the ‘sense’ in which (5) is false is as much discovered in nature (including ours) as already latent in conventions of usage alone.27 But of the traditional forms of fallacy, equivocation seems best to fit our diagnosis of the paradox of the heap. The only sorites to which anything we have said is intended to apply directly is the paradox of the heap. But one seems to be able to generate sorites almost at will. For example, an appointment to an academic department that takes one day from start to finish is quick; no single day is ever enough by itself to make the difference between a quick and a slow appointment; so however many days an appointment takes, it is quick. It can feel as if one is following a largely inarticulate procedure in generating such paradoxes. This somewhat odd feeling may lead one to conjecture that sorites are all of a piece, and that consequently there should be a uniform procedure for solving them. Our examination of heaps might suggest a rule of thumb like, “Probe the nature of the subject matter of the sorites for a number that it is natural to decide is the relevant threshold.” Such a vague or general recipe is guaranteed always to work only if there always is such a threshold. That much may follow by modus tollens from the falsity of the conclusion of the inductive argument in a Wang-style presentation of a sorites. But whether we can recognize such thresholds may depend on how deeply we have probed

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the nature of the subject matter of a sorites. How it might be probed, and whether we are up to probing it deeply enough, might well vary considerably from one sorites to another. At this level of lesser generality, perhaps the lesson of our examination of heaps is that not much by way of a uniform procedure for solving sorites is to be expected. There is no guarantee that we will ever understand ourselves well enough to solve the paradox of the quick appointment, but then neither has knowing how to get into trouble always been guaranteed to be enough for knowing how to get out again.28 One might want to say, “Maybe there is a least number of grains that make a heap, and it is four, and so much for the paradox of the heap. But perhaps that just shows that the heap was an atypical bad example of the sorites. There is still the paradox of the bald man. Indeed, we can generate sorites pretty much at will. What about the general run of sorites?” The heap might be atypical, but it is certainly an old sorites. It does back to Eubulides of Megara in the 4th century BC to whom discovery of the sorites in general, and the heap and the bald man in particular, are generally credited. Our word “sorites” comes from the Greek word “acopóç” for heap. So to have solved the heap would be to have solved an old and prominent sorites. Admittedly, it is just one paradox. But how natural a kind are the sorites? Maybe there is no general method, like a chapter in a theory of vagueness, that suffices on its own apart from the subject matter of this or that sorites to solve all sorites uniformly. For all one knows, there may be perhaps as yet unknown facts in the physiology and sociology of hair that suitably articulated and arranged will yield of some number a natural decision that it is the least number of hairs to make a head of hair. To have solved the heap should add vivacity to this epistemic possibility about the bald man. So no one should any longer presume that he can ju st say there is no least number of hairs such that a man with that many hairs is not bald, and expect to be believed automatically merely for having said so. Nobody really knows any such thing. Maybe for each sorites there are perhaps as yet unknown facts in its particular subject matter that suitably arranged would issue in a natural decision solving the sorites. That might be true even if there were no method, or even guarantee, of our ever recognizing such facts. In a typical sorites we have a mass noun S, like “sand” or “hair”. We also have a pair of count nouns T and U, like “grain” and “heap” for sand and “strand” and “head” for hair. (Actually “hair” occurs both as a mass noun and as a count noun in English, but what is going on is clearer if we say strands of hair for hairs.) The

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crux of the recipe for getting a sorites up and running is to choose T and U so as to be able to sell the claims that Zero (or one) Ts of S do (does) not make a U of S. Since a single T of S does not make the difference between what is not and what is a U of S, if n Ts of S do not make a U of S, neither do n + 1. Thence we conclude that no number of Ts of S make a U of S. This is paradoxical if there are Us of S made up of a number of Ts of S. If we grant that there are such Us of S and we hang on to our first premiss and classical logic, we are obliged to conclude that for some n, n Ts of S do not make a U of S, but n + 1 do. Admitting such threshold factors can feel embarrassing and quixotic if one has no hope of recognizing such a number for a particular S, T and U. But faint heart ne’er solved fair paradox. One could let one’s confidence in classical logic lend vivacity to the epistemic possibility of recognizing facts in the physiology and sociology of hair that make it natural to decide of some number that it is the least number of strands to make a head of hair. But maybe there is a real choice here. Suppose Eubulides had been confronted with three grains of sand in a triangle with a well in the middle in which sits a fourth, and told that this shows that four is the least number of grains to make a heap of sand. What if Eubulides just digs in his heels and says it is not big enough to be a heap? We can go into our routine about stable crowded configurations of grains and say that crowding already does sufficient justice to bigness. But if Eubulides mulishly digs in his heels, there is no guarantee we can force him to agree with us. Neither is it written that when we go into our routine about truth, falsity, negation and disjunction, we can force Michael Dummett to consent to the law of the excluded middle. So perhaps it is genuinely a decision that four is the least number of grains that, so configured, make a heap of sand. But even if so, neither does it seem an arbitrary or capricious decision to be made by Eubulides or any other single person willfully. Our routine about crowding and stability seems to make a certain sense of heaps and our thinking about them. Truth tables bring a certain order and perspicuity to some of our calculations with propositions and to our thinking about such reasoning. So even if it is genuinely a decision that four is the least number of grains to make a heap of sand, nevertheless it is a natural decision. Part of the relevant stretch of nature here might be what we call common or good sense, for the choice is not up to Eubulides alone or even lots of isolated individuals, but should emerge as consensual good sense; it should

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catch on for good. Any such appealingness might be diminished if we came up with an alternative routine about features other than crowding and stability that make it natural to decide of some number other than four that it is the least number of grains to make a heap. If each routine retained some of its appeal despite acknowledgment of the other, we might decide naturally that we had discovered an ambiguity in the nature of heaps, perhaps as we discovered a distinction between two kinds of jade. Still, so long as we have just the one routine about crowding and stability, four seems the natural choice. We are part of nature, and natural decision in the face of fact is one attempt to describe a way we conspire with nature. In the absence of a trustworthy, received, consensual, good sense articulation of human nature generally, maybe that description is about the best now to be expected.

University College London NOTES 1. See the footnote on page 324 of Michael Dummett, “Wang’s Paradox,” Synthèse , XXX (1975), 301-324. 2. But suppose one is piling cylindrical objects like logs lengthwise. Then three logs, one on a pair, seem more likely to make a loose but stable and tidy small pile than four. As noted below, such a lengthwise pile of cylinders seems importantly twodimensional. Here we concentrate on more three-dimensional heaps of more symmetrical globular objects like grains of sand. 3. This is also the number of logs in an n-layered pile of the sort considered in the last note. It is because the number of logs in the pile grows like the number of grains in a layer that the pile seems two-dimensional while the pyramid seems threedimensional. 4. There can be vagueness about where a heap stops and the ground below begins. A dune is a heap of sand, but the ground in a desert is mostly sand too; where does the dune stop and the ground start? Let us reduce this vagueness by supposing that the flat ground on which our heaps sit is rock, or some stuff like clay whose particles are very much smaller than grains of sand. 5. But there need not be more above than on the ground. Pyramids with two, three or four layers are smallish heaps, but it is not until pyramids with five layers that there are more grains off the ground than on. Or, if one insists that there be more grains off than on the ground in a heap, then at most pyramids with at least five layers will be heaps. On the other hand, triangular pyramids are steeper than square ones because the well or cup within four grains is broader and more accommodating than the well within three grains. So in a triangular pyramid the pressure on grains along an outside edge of the base increases more quickly with the number of layers than it does in a square pyramid. Presumably there comes a number of layers at which this pressure comes to exceed the friction and inertia holding such edge grains at the base in place against the ground, and some of them pop out, perhaps precipitating a cascade among some of those they helped to support. In this way a tall triangular pyramid may be less stable than an equally tall square pyramid.

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Suppose at least a degree of stability is required of a configuration of grains for it to be a heap. Then there may be a number of layers beyond which a triangular pyramid ceases to be a heap, and an analogous number of layers for square pyramids, and, if the increases in pressures with numbers of layers are different enough in the different shapes, the first number may be less than the second. Maybe a narrower band of triangular pyramids than of square ones includes heaps. 6. Here we are pretending that grains are globules of continuous stuff. It seems idle to worry about whether such globules are open or closed, though it may not be idle to ask such questions about electrons and quarks. 7. Air shafts are more likely in square pyramids than in triangular ones. 8. While we counted layers in a pyramid from the top down, we are counting tiers in a configuration from the bottom up, 9. Every pyramid is everywhere in local stable equilibrium, but we saw on note 5 above that tall pyramids may nonetheless be unstable on a larger scale. 10. Let us agree not to try to take into account sideways slides from the talus of older outgoing slides. But in a big heap of small grains, we might get finite approximations of fractal branching. 11. What follows in the text is drawn in part from page 605 of the third edition of The Columbia Desk Encyclopedia , William Bridgewater and Seymour Kurtz, eds., Columbia University Press, 1963. The smallest pyramid can roll. Suppose grain d sits in the well walled by grains a, b and c, and the wind is blowing toward a along its breadth that, projected, passes through the contact between b and c. The wind could push a and, both as well and via a, d so that d falls through the crevice between b and c while a rises to sit in the well now walled by b, c and d. If the wind then swings through 60 degrees, the process can repeat with b coming to sit in the well walled by c, d and a. So if the wind swings back and forth across an arc of about 60 degrees, the pyramid can roll along a zig-zag track, the zigs at 120 degrees to the zags, and each a bit shorter than the breadth of a grain. The trend of the path is, on a larger scale, in the direction from the midpoint of the wind’s arc, even though the pyramid is not rolled over by the wind when it blows from the midpoint of its arc. The speed at which the pyramid rolls depends not so much on the speed of the wind as on the speed at which the direction of the wind swings back and forth across its arc. 12. The seat of a pyramid will become more oval as the number of sides in the regular polygon at its base increases. 13. Ovals and regular polygons always have edges. 14. We could instead have spoken here of the set of grains on r. But abstract objects like sets seem largely irrelevant to our concerns, so it seems less unnatural to use plurals instead. However, the definite article with the plural should not cause one to suppose that there always are grains on any old region whatsoever. For more on plurals see George Boolos, “To be is to be a value of a variable (or to be some values of some variables)”, The Journal of Philosophy , LXXXI (1984), 43(M9. 15. Imagine a weak glue. Suppose that after setting, it is strong enough to hold a hump of grains together against the wall of a ship in outer space, but that even after setting it is weak enough that the stamping footsteps of passing crew members suffice to shock grains from the outside of the hump enough to float off into the void. If being unified by gravity and friction is, or were, a necessary condition for being a heap, then this hump is not, or would not be, a heap. Suppose this hump is nonetheless a heap. Then it might be a good idea to replace our reference to gravity and friction by name with a general reference to forces only strong enough to unify the grains reasonably well on a region through shorter periods against moderate external forces like winds up to Beaufort force four. But it is really the issue of stability that seems

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to be trying to surface here. Besides, the typical terrestrial heap is unified by gravity and friction (plus perhaps a bit of dampness). 16. A good measure of stability should include a measure of changes among paths and chains among grains on the region over the interval as well as reflecting which grains abide and which do not, and it should relate this measure to the magnitude and direction of the external forces. 17. Suppose g(r,t) = 0. Is it the case that P(r,t)? How does one project from nothing? 18. If the length of a side of an equilateral triangle is e, then the length of its altitude is e^/3/2', and the length of the radius of its inscribed circle is e/2^/3. So the cosine of the angle between two faces of a triangular pyramid is 1/3, and thus the angle between them is about 70J degrees. The sand from one four-year-old’s sandbox pours out in conical heaps whose sides slope at about 30 degrees. Let us make some more comparisons. If the height of a conical heap of this sand is h, the radius of its base is h tan 30, so the area of its base is nh2 tan2 30, and the volume of the cone is l/37ch3 tan2 30. This sand is fine. Grains large enough for middleaged eyes to keep track of line up side by side at about three to the sixteenth of an inch. Such grains figure at about 48 per inch, or 2,256 per square inch, or 108,288 per cubic inch. Our calculations being rough, let us approximate n by 3. Then since the tangent of 30 degrees is 1/√ 3 ,there are on the order of 36,000 grains in a heap one inch high. The pyramid number closest to 36,000 is 35,990, which is the number of such grains in a pyramid with 59 layers. The length of the edge of such a pyramid would be about 1.23 inches. If the edge of a pyramid is e, its height is e√ 2/3. Since the root of 2/3 is about .81, such a pyramid would be about .996 inches tall, which is surprisingly close to one inch. The difference emerges at the base. For while there are about 2,256 grains in the base of the conical heap, there would be only 1,770 grains in the bottom layer of the pyramid; the heap slops over the sides of the seat of the pyramid. It also looks as though the grains in the heap are a good deal less efficiently packed than they would be in the pyramid. 19. If UGF(r,t) and P(r,t), then for many r1 interior to r it may happen that UGR(r1t) but not P(r1t) since there can be grain chains arching from grains touching r1 over to grains resting on the ground in r but not r1. 20. 2,256 divided by 1,770 is about 1.27, so the factor accords with the figures in note 18. 21. Perhaps what one really wants is a less numerological and more empirical understanding of what crowding is. The number of grains in a pyramid is one sixth of, roughly, the cube of the length in grains of its edge, and its height is √2/3 of the length of its edge. So the number of grains in a pyramid varies directly with, roughly, the cube of its height; the constant of proportionality is the product of 1/6 and √3/2 Perhaps the number of grains in a natural heap also varies directly with, roughly, the cube of its height. The empirical constant of proportionality here might also be the product of information about crowding and slouching. If slouch is measured by the slope of the side, then perhaps crowding can be calculated. Unlike local stable equilibrium, crowding is an overall feature of heaps. 22. Suppose a large pyramid number of grains are erected on a triangle r of size s(r). Note 5 leads us to expect that for larger and larger numbers of grains, the pyramid is more and more likely to slump into a heap on a larger and more oval patch r1 roughly concentric with r (or its circumscribed circle). If sir1) varies directly with s(r), the constant of proportionality might contain information about the packing and stability of natural heaps. Cp the previous note. 23. If there are laws of nature, and if such laws are natural necessities, and if (6) holds by natural law, then perhaps (6) is a natural necessity. But there is no better explanation of what natural necessity has to do with necessity than of what canaries have to do with cans.

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24. What was it that seemed vague about heaps? Did it lie, more precisely, in (our conception of) the relation between heaps and grains? Is the locus of vagueness often relations between kinds or properties, or schemes of these? There is a known number of ounces in a pound, and of gills in a pint, but not, it seems, of dollops in either a pint or a pound, though one good cook is firm that there are ten smidgens to a dollop. Authors of cookbooks for the American market tend to write in terms of measures by volume where authors of cookery books for the British market tend to write more in terms of measures by weight. Highly skilled cooks seem hardly to use either in practice, but just to look and feel and taste and smell. 25. Clearly one should not generalize thus on “g(r,t)” in (6), for while (Vn) (n + 1 > n), it is false that (3k) (Vn) (k > n). 26. It does not matter for the truth of the claim in the text, but are there heaps of five through nine grains? Start from a pyramid of grain 1 sitting in the well walled by grains 2, 3 and 4. Grain 5 might fall to nestle beside 2 and 3 as a small pseudopod, or, blown in from one side, it might push in among 2, 3 and 4 to make the smallest non-degenerate square pyramid of 1 sitting in the deeper well walled by 2, 3, 4 and 5. Grain 6 might make a small pseudopod touching on 2 and 3 on this square pyramid, or nestle beside 2 and 5 to extend the pseudopod on the triangular pyramid, or sit in the well walled by 2, 3 and 5. Grain 7 could sit in the well walled by 2, 5 and 6 in the augmented triangular pyramid, or perch a bit precariously on 2, 3 and 6 (and probably lean a bit on 1) in the square pyramid with a pseudopod. Narrowing our attention just to the triangular pyramid, grain 8 could nestle beside 5 and 3, while grain 9 could then sit in the well walled by 3, 5 and 8 awaiting a capstone for the well then walled by 1, 7 and 9. Of these configurations, only those of eight or nine grains are crowded enough for (6) to certify them as heaps; (6) is only a sufficient condition, and need not be the whole story of heaps. 27. This contrast probably will not withstand critical scrutiny. 28. A remote ancestor of this essay appeared at a discussion group that met in Roger Scruton’s room at Birkbeck College a decade or more ago. At this remove, it is hard to be sure, but Martin Davies, Dorothy Edgington, Sam Guttenplan, Ian McFetridge, Colin McGinn and Hans Kamp probably also participated in that discussion. That ancestor more recently revived and transmogrified partly out of thinking about the dissertation of my former student, Dr George Galfalvi, and partly out of thinking about some excellent lectures that Timothy Williamson of University College Oxford gave at University College London in October of 1989. Williamson’s third lecture paid due attention to James Cargile’s “The Sorites Paradox”, The British Journal for the Philosophy of Science , XX (1969), and to chapter 6 of R.A. Sorensen’s Blindspots , Oxford University Press, 1988. I am also indebted to my friend and colleague Jonathan Wolff for helpful comments.

Part V Higher-Order Vagueness

[14] The Philosophical Q uarterly Vol. 41 Mo. 163 I S S N 0031-8094 $2.00

IS THERE HIGHER-ORDER VAGUENESS?

B y M ark S ainsbury

I. INTRODUCTION In the classical conception, it is no accident that a concept draws sharp boundaries. Concepts are used in classification, classification is the assignment of things to classes, and classes are sharp: for any class and any object, there is a definite fact of the m atter whether or not the latter belongs to the former. A concept must have sharp boundaries because there are no such things as unsharp boundaries: the extension of a concept is like a geometrical area and there is no such thing as an unsharp area. In applying a concept in a simple sentence, we select a single possible state of affairs and exclude all others, and this presupposes a sharp demarcation between states of affairs, and thus between things which fall under a concept and things which do not. T he classical conception finds vagueness a problem: at best an aberration, at worst (as Frege is reputed to have believed) an impossibility and therefore an illusion. Those who argue that vague predicates are incoherent are, conditionally, right: vague predicates are incoherent if the classical conception is correct. So we must either turn our backs upon vague predicates, or else upon the classical conception. My preference is for the latter. In this paper I show how the concessive classicist - one who would try to find a place for vagueness within the classical landscape - is forced to misunderstand vagueness; and I gesture towards a non-classical way of looking at concepts and classification, a way upon which vagueness is the norm, and sharpness an artefact. II. HIERARCHICAL VAGUENESS Suppose that one imbued with the classical conception looks with sympathy upon the project of encompassing vagueness within his approach: he is what I call the concessive classicist. Suppose further that, not unnaturally, he thinks the prime feature of a vague predicate is that its sense is such as to permit borderline cases. For such a concessive classicist, it is natural to describe a vague sense in terms of its effecting a tripartite partition. There are the positive cases, the negative cases, and the

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borderline cases. Since the standard semantic apparatus lies within classical set theory, these are thought of as three sets or extensions. This natural tendency emerges even in sophisticated formal treatm ents, of which the two best known are supervaluations and degrees of truth. We can divide the atomic sentences containing a vague predicate into three sets: those which are supervaluationally-true/true-to-degree- 1 ; those which are supervaluationally-false/true-to-degree-O; and the remainder. In forging a link between the formal semantics and intuitive notions, it will be hard for these theorists to prevent an association between these three sets and, respectively, the set of the positive, negative and borderline cases for the predicate in question. In this paper, I will not consider formal approaches to vagueness. Rather, I will focus on the notion of higher-order vagueness. I believe that this notion is essentially a manifestation of the grip which the classical conception has upon us, a grip which I am trying to loosen. Here is one way in which the notion of higher-order vagueness might arise. T he first thought is to represent the sense of a predicate like ‘green’ or ‘child’ by its effecting a division of categorially appropriate objects into three sets. T his is supposed to do justice to the actuality or possibility of borderline cases: surfaces intermediate between blue and green, people intermediate between childhood and adulthood. It thus does justice to ‘tolerance’ intuitions: a very small difference in shade cannot make the difference between something being green and being blue, so we need a class of borderlines; a very small difference in age cannot make the difference between childhood and adulthood, so we need a class of borderlines. On the classical conception I am investigating, this is an adequate representation of the lowest level of vagueness - vagueness,. However, with most or even all vague predicates, it soon appears that the idea that there is a sharp division between the positive cases and the borderline ones, and between the borderline cases and the negative ones, can no more be sustained than can the idea that there is a sharp division between positive and negative cases. We find a new type of borderline case: for example, those things which seem intermediate between being definite cases of children, and being borderline cases of children. We decline to accept that there can be any sharp boundary here. I f there were, it would remain true that there would be such a thing as the last heartbeat of my childhood, or at any rate the last heartbeat of my definite childhood, and that seems as crazy as the idea that the predicate ‘child’ divides the universe into a set and its complement within the universal set. Hence we allow for the more than theoretical possibility of a higher order of vagueness - call it vagueness, - which consists in a five-fold division: into the definite positive cases for the predicate, the definite borderlines and the

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definite negative cases, together with the cases which are borderline between being definite positive cases and definite borderlines, and those borderline between being definite borderlines and definite negative cases. A predicate whose sense was correctly described by its effecting this five­ fold division would count as vague2. T he generalization is straightforward: a vague„ predicate is one whose sense can be described by its drawing 2 ” boundaries, thus dividing the categorially appropriate objects into 2” + 1 sets. A vague predicate is vague„ for some n > 0. A higher-order vague predicate is vague„ for some n > 1. A radically higher-order vague predicate is vague„ for all n. In these terms, it is a theoretical possibility that there be predicates which are vague, without being higher-order vague. Arguably, some predicates meet this condition, but, intuitively, meeting it is inconsistent with being a paradigm of vagueness. W hatever one may think of the framework, the arguments against counting everyday predicates like ‘green’ and ‘child’ as merely vague, were good. So the substantive question, from within this conception, is: what is the order of vagueness of such predicates? Or are they radically higher-order vague? I shall argue that these questions cannot be adequately answered, and that this is reason enough to abandon the conception within which they arise. Suppose that ‘green’ and ‘child’ are radically higher-order vague. Such predicates are associated with a dimension of comparison (younger than, greener than) which will impose a structure upon the infinite set of sets associated with them. These sets will be ordered: the first set - call it the super-positive set - will be the set of unimpugnably definite green things (children), the last set - call it the super-negative set - will be the set of unimpugnably definite non-green things (non-children). But there would appear to be nothing in the use of the predicates corresponding to these super sets. There is no least red thing of which ‘red’ is unqualifiedly and unimpugnably true. Were there such an object, its identification would be an essential prerequisite for the full and perfect command of the language; yet evidently nothing plays this role. Moreover, we have in fact returned to a three-fold classification: there is the set of unimpugnably red things, the super-positive set; the set of unimpugnably non-red things, the super­ negative set, and the set of the remainder, the union of the remaining sets associated with the predicate. Finally, while there might theoretically be a finite encoding of the information represented in the infinite hierarchy, it seems extremely implausible that we have internalized it. Many of these points hold against the thought that such familiar predicates are higher-order vague. Increasing the num ber of the divisions does not stop it being the case that three sets single themselves out for attention, corresponding to the super-positive, super-negative and the

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union of the remainder, in a way that is not m irrored in the use of the predicates. These sets demand attention, because they cannot but be interpreted, within the framework, as corresponding to the things of which the predicate is true beyond the shadow of vagueness, the things of which it is likewise false, and the things touched by vagueness. But sharp boundaries between these categories is just what the hierarchy was originally introduced to mitigate. This conception of higher-order vagueness shows the inadequacy of one classical conception to give a proper account of vagueness. It would be a harder task to show that every working out of the classical conception m ust founder on the rock of higher-order vagueness. However, classical semantics typically treats a language as if it contained a set of true sentences, and this is just what a vague language cannot do. Sets are sharp. If there were a set of truths, one of them would truly apply ‘red’ to the least red thing to which the predicate can be truly applied, so there would be a least red red thing; this is what we have just seen to be impossible. In the next section I consider an approach to vagueness that is as neutral as possible on matters semantic and philosophical, and which promises to be as available to the classicist as to anyone else. T he official reason for discussing it is to help complete the case against the classicist: if this approach does not enable the classicist to find room for vagueness, I am nearer establishing my conclusion that no approach available to the classicist can do so. The unofficial reason is that it is an approach that has been adopted by Crispin W right, in an interesting recent discussion of higher-order vagueness . 1 III. THE CHARACTERISTIC SENTENCE Suppose we could find a sentence-schema, containing a schematic predicate position, such that the sentence resulting by replacing the schematic element by a predicate is true if that substitute is a vague predicate. Then we could characterize vagueness in a way apparently consistent with classicism, but without at least in these early stages bringing to bear the classical apparatus of sets in any harmful way. And perhaps then we could make sense of some notion of higher-order vagueness after all. T o connect this question with the unofficial reason for introducing it: W right argues that on this approach to characterizing vagueness, higher-order vagueness is immediately paradoxical; so the hope, were it raised, would be immediately dashed. 1 Crispin Wright, ‘Further Reflections on the Sorites Paradox’, Philosophical Topics , 15 (1987), p. 227-90.

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I am going to argue that the characteristic sentence approach is unsatisfactory and that, largely for this reason, no party to the issue should gain comfort or discomfort from W right’s claim that higher-order vagueness is paradoxical. W hat should the characteristic sentence schema be? Perhaps we should take tolerance intuitions as our starting point: for a vague predicate, but not for a sharp one, a small difference makes no difference to whether the predicate is true of an object or not. T o set the case up in more detail, let us restrict our target (if this is indeed a genuine restriction) to those vague predicates which are associated with a dimension of comparison, as ‘green* is associated with ‘greener than’, ‘child’ with ‘younger than’. It would be wrong to suppose that it is a simple m atter to state the details of this association, and how it impinges upon use; it would also be wrong to take as empirically serious some philosophers’ idealizations on this point (e.g., that chronological age is the only factor which determines the applicability of ‘child’). But with these caveats in mind, we can proceed with the attem pt to find a characteristic sentence which would select out those vague predicates associated in the following sort of way with a dimension of comparison. We will think of the dimension as made up of points ordered by the comparative relation, with satisfaction-entailing ones towards the left. T hus for each predicate O associated in the relevant way with a dimension of comparison there is a relation to the left of( abbreviated 4 > ’) meeting the following conditions: (i) if y (definitely) satisfies ‘O ’ and x > y then x (definitely) satisfies ‘O ’; (ii) if x (definitely) does not satisfy ‘O ’ and x > y then y (definitely) does not satisfy ‘O ’; (iii) if neither x > y nor y > x , then ‘O ’ has the same semantic relations to both x and y ; (iv) if neither x nor y (definitely) satisfies ‘O ’, and x > j/, then x is intuitively a better candidate than y for satisfying ‘O ’. Some associated dimensions of variation are clearly discrete (like numbers of grains for ‘heap of grains’); some are theoretically continuous (like height for ‘tall’); for some, like shade, there is neither obvious discreteness nor obvious continuity. However, one can take small intervals in a continuous dimension, so that it is intelligible to speak of adjacent positions; and I shall adopt the convention that an interval position x has the interval x adjacent to it on its right. For continuous dimensions,

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the prime is implicitly relative to some division of the dimension into intervals. Setting the tolerance intuitions within this framework, a natural first attempt at a suitable characteristic sentence would be: if x then Ox' (where ‘í>.r’ abbreviates ‘any object occupying x satisfies 0 ’). This indeed seems to be a way of expressing the thought that slight changes cannot make a difference to whether ‘í>’ applies or not. However, this sentence leads immediately, at least classically, to the paradox that every x is 3>. We should not accept a characterization of vagueness which has this consequence, if there are alternatives. In any subject matter involving truth-value gaps, or a third truth value, and so in particular in the case of vagueness, we m ust recognize the possibility of two operators each having a claim to the title of negation. One of these will turn only falsehoods into truths, the other will turn any non­ truth into a truth. Thus if we hold that a sentence failing to meet the standards of categorial appropriateness - say, ‘This piece of chalk is hungry’ - is truth-valueless, we must distinguish the claim that it is false from the claim that it is not true, for in claiming that the sentence is neither true nor false we want to reject: it is false that this piece of chalk is hungry But accept: it is not true that this piece of chalk is hungry. Let us assume that our object language contains an expression, ‘N eg’, for the kind of negation which turns only falsehoods into truths. If a is borderline for ‘$ ’, then it will satisfy neither ‘O ’ ‘nor ‘Neg ’. T he difference between ‘Neg’ and ‘not’ is like that between ‘false’ and ‘not true’, which, as we have seen, can be introduced independently of vagueness. T he envisaged doctrine about this piece of chalk can then receive an object language expression of the form: ‘N ot p and not Neg p \ In a reasonable three-valued logic, if a sentence S has a hideous consequence, we m ust reject it, that is, assert ‘not S'; but we do not necessarily have to deny it, that is, assert ‘Neg S \ W hat is correct in the tolerance intuitions is that no O-entailing location is adjacent to a Neg--entailing one. So an improvement on the previous paradoxical sentence is:

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if Ox, then not Neg Ox' where the variable ranges over locations on an associated dimension, and ‘Ox’ is read as before. However, this does not capture vagueness, precisely because a predicate with three sharp extensions will verify 2 the sentence without being vague. T hus suppose that ‘child*’ is true just of persons who have not reached their sixteenth birthday, false of persons who have reached their eighteenth birthday, and neither true nor false of all other persons. Intuitively, this is not a vague predicate, despite the existence of borderline cases. Yet it verifies the proposed characteristic sentence-schema. We are in a predicament akin to that faced in the previous section. T he characteristic sentence cannot distinguish between real-life vagueness and vagueness, as defined by the hierarchy of the previous section. We therefore are once again driven to find some way of handling the fact that a vague predicate does not divide its significance range into three sets. L et us follow W right, and suppose the object language to contain some expression for definiteness, to be written ‘D ef’. Then one approach would be to treat the previous characteristic sentence as good for defining vagueness,, and cast around for ways of using ‘D e f’ to characterize vagueness,, for n > 1. One intuitive idea is that you should not have, on adjacent positions, first a definite O and then a definite borderline. A borderline satisfies the condition: ‘not Ox & not Neg Ox’. T his suggests taking the characteristic sentence schema to be: ( 1 ) if D ef” Φx, then not D ef (not Ox' & not Neg Ox') where ‘Defn’ abbreviates n occurrences o f ‘D ef’. In the previous hierarchy, vagueness0 was precision. Here we find the same: an instance of (1) free of ‘D ef’ will require that there are no borderlines. However, (1) raises the question of how one should integrate the ‘D e f’specified phenomena with the phenomenon specified in terms of ‘N eg’ alone. One could avoid this question if one replaced (2) by (2) if D efn IOx, then not Neg D ef”_1Ox' though this would have the disadvantage that vagueness 0 is undefined. Both suggestions can be criticized in a num ber of ways. Given the envisaged operation o f ‘not’, it will yield a sentence with a truth value from 2 Or at least fail to falsify. Which alternative is appropriate depends upon how conditionals fare in an appropriate three-valued logic. This reflection already shows that the advertised neutrality of the approach is more apparent than real.

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any sentence, so the conjunction in ( 1 ) will always have a definite truth value, so it is unclear what role ‘D ef’ has. T urning to (2), if ‘D ef Otf’ always polarizes to true or false (and who at this stage is to say it does not?), then ‘Neg’ will be effectively equivalent in the context to ‘not’, and by double negation we will have the paradoxical. if DePO x then D ef”Ox'. It is worth noting, then, that the way forward for the characteristic sentence approach is not entirely clear. In particular, it cannot both eschew serious semantics, and also convince us that one rather than another characteristic sentence-schema is correct. We need pursue the m atter no further, for there are two further reasons for dissatisfaction. First, it supposes that any language containing a vague predicate also contains equivalents of ‘Neg’ and ‘D ef’, and these assumptions appear quite unjustified . 3 Secondly, the approach simply takes ‘D ef’ for granted, whereas vagueness and definiteness are too intimately connected (as intimately as contingency and necessity) for this to be allowed to persist. A correction would require the introduction of a semantic story, which would than replace the characteristic sentence-schema as the basis of the account. IV. IS HIGHER-ORDER VAGUENESS INTRINSICALLY PARADOXICAL? Crispin W right 4 has argued that it is. Adopting the characteristic sentenceschema approach, he represents O ’s having the lowest order of vagueness in terms of the truth of the sentence: if D ef Ox, then not D ef not Ox' where ‘D ef’ is an undefined sentence operator intended to express a concept of definiteness. T he next order of vagueness, according to W right, is defined by the truth of the sentence. if D ef D ef Ox, then not D ef not D ef Φx'. W right argues that from this sentence, together with a relatively uncontentious assumption about the logic of ‘D ef’, one can derive the paradoxical sentence: 3 Steven Williams suggested that this could be circumvented by defining vagueness for a language relative to an extension of it containing ‘D e f and ‘Neg’. But then we could not rely on the intuitions of the language-users to determine the truth-values of instances of the schema. 4 Wright, ‘Further Reflections on the Sorites Paradox’.

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if D ef not D ef Φx' then D ef not D ef Φx. T he sentence is paradoxical because it immediately entails that there are no definite cases o f ‘O ’. T h e derivation of the paradox depends essentially upon the presum ed validity of the inference rule:

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175175 175 75 175 75 175 1175 175 1175 provided every member of T is prefixed by ‘D ef’,

which, he says, leads at once to the theorem if D ef A, then D ef D ef A" and so to IT : D ef A iff D ef D ef A. It m ust at once be apparent that a system containing IT is unlikely to provide a happy home for a higher-order vague language. W right’s proof that it is actually paradoxical goes like this. Using the contrapositive of the characteristic sentence, viz: if D ef not D ef Ox', then not D ef D ef Ox we assume the antecedent and derive the consequent. IT then yields not D ef Φx and since all the assumptions are (or could unproblematically be) preceded by ‘D e f’, an application of D E F gets us from this to D ef not D ef Φx A step of conditional proof yields the paradoxical sentence. From the standpoint of this paper, there are three things to notice about W right’s proof. First, one cannot feel happy with the introduction of the undefined ‘D ef’, followed immediately by an assumption about its logic 3 In fact, it leads only to: Def Ah Def Def A, and who is to say whether the Deduction Theorem holds in a suitable logic for ‘D ef’?

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which leads to paradox. It would seem a clear possibility that there should be a conception of ‘D ef’ upon which it demands progressively higher standards. Such a conception would fail to validate IT or the definitization rule (D EF), and would need to be argued against if higher-order vagueness is to be shown paradoxical by the argument. Second, it remains to be shown that W right’s characteristic sentence is just the right one. I gave two alternatives in the previous section. One cannot, by a proof of the same general structure as W right’s, derive anything paradoxical from them. So their entitlement to represent vagueness needs to be underm ined before any conclusion antithetical to higher-order vagueness can be drawn from W right’s proof. Third, it remains to be shown that some characteristic-sentence approach to the question what is vagueness, or what is higher-order vagueness, is correct. V. DOES ‘RED’ MAKE A TRIPARTITE DIVISION? Because W right is, in my view, still to some extent in the grip of classical conceptions, there is for him no difference between the above question and the question whether ‘red’ is vague, but not vague 2 (in the sense of the characteristic-sentence approach). Rejecting the characteristic-sentence approach, I accord no interest to the second question. But I certainly accord interest to the first, and indeed it is a central thesis of this paper that it is to be answered negatively. W right answers it affirmatively. It is a familiar claim that red is a phenomenal quality, and that this is registered by an essential connection between looking red and being red. Setting aside various necessary qualifications: to be red is nothing but to look red or to be judged red. As W right puts it: For an object to be (definitely) red is for it to be the case that the opinion of each of a sufficient num ber of competent and attentive subjects who are appropriately situated to command a clear perception of the object, functioning normally, and free from interfering background beliefs - for instance some doubt about their situational competence - would be that it was red. . . . We have therefore to acknowledge, surprising as it may seem, that a Sorites series of indistinguishable colour patches can contain a last patch which is definitely red: it will be a patch about whose redness there is a consensus . . . and its immediate successor will be a patch about which the consensus breaks down .6 6 Wright, ‘Further Reflections on the Sorites Paradox’, pp. 244, 245.

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At first sight, this looks like a ‘proof’ of the conclusion that ‘red’ is not vague at all. T he original equation was between a phenomenal quality and consensus, not between a definite phenomenal quality and consensus, so the parenthetical ‘definitely’ in the first sentence appears out of place. But then what failure of consensus establishes is a sharp borderline to the things which are red, and that seems to amount to the view that ‘red’ is sharp all the way down. I think that W right does not read his remarks in this way because he takes the vagueness o f ‘red’ to consist in its having a three-fold extension. T here is a penumbra of borderline cases. W hat the quoted passage addresses is whether this penumbra is vaguely drawn, and the answer argued for is that it is not. I suggest that we must reject at least one premise of the argument, and also its conclusion. T he premise we m ust reject is that the correct use o f ‘red’ is determined by a sharp consensus rather than a vague one (‘almost all agree W right, perhaps, was moved by the thought that a vague consensus would underwrite vagueness2 (in the sense of the characteristic-sentence approach), which he believed he had shown to be paradoxical. The premise we must accept is that for every trial or group, there is a left­ most point (on the associated dimension) at which their consensus breaks down. We must be careful not to argue from this to the conclusion that there is a leftmost point for every group, a conclusion we would all agree with W right in believing to be empirically false. But even if there happened to be such a universal leftmost point, it is unrealistic to suppose that this could govern the use o f ‘red’. What you have to do, as a good user o f ‘red’, is to make your use align reasonably well with others’ use. There is no require­ ment upon you to align your usage with that of the maverick who first breaks ranks. Moreover, there could be no such requirement, or we could never know whether we are conforming properly or not. The sharp line between the last point with complete consensus and the first without would become of crucial importance for our use: we would need to know what it is, to avoid the irrationality of applying ‘red’ further to the right, or witholding it further to the left. Plainly, we could have no such knowledge. We can, however, reason­ ably be credited with a knowledge of approximately how others use ‘red’. It perhaps does W right no service to consider a putative 3V proposition; let us return to the V3 premise. W right explicitly discusses the difference: there is, a priori, no reason to suppose that ‘the last definitely red patch’ would turn out to have a stable reference; if it did not, that would disclose an element of context-relativity in the concept of red which we normally do not suspect.' ' Wright, ‘Further Reflections on the Sorites Paradox’, p. 245.

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I interpret this somewhat elliptical passage as follows: granted, there is a phenomenon that superficially looks like what people have wanted to label higher-order vagueness: there is no appearance of speakers using a vague predicate as if there were a definite sharp boundary between the positive cases and the borderline ones. But this appearance is an illusion: it is not a matter of vagueness or blurring of boundaries; it is a m atter of variability of context. It is, indeed, important to distinguish between these phenomena. T here are many ways in which a predicate, even a vague predicate, can be sensitive to context. A context in which Swedes are under discussion sets greater absolute height requirements upon ‘tali’ than a context in which Eskimos are under discussion; and the failure of a context to specify needed relativizations can lead to truth-value gaps, just as vagueness can. A major difference, however, is that vagueness can induce truth-value gaps even in ‘complete’ contexts and can lead to divergent but non-erroneous judge­ ments even in indistinguishable contexts. Adding contextual information will not eliminate the instability of the consensus from group to group. There will remain ‘complete’ contexts which elicit divergent judgements because from the subjects’ viewpoint it becomes arbitrary whether to respond yes or no, so that there is no point which strikes them as having any special claim to m erit a switch of response. W right attempted to reinterpret the evidence against the view that ‘red’ makes a tripartite division as a manifestation of context-dependence rather than a manifestation of vagueness; but this attem pt m ust be rejected . 8 VI. AN ALTERNATIVE MODEL. PARADIGM AND SIMILARITY One classical approach starts by seeing vagueness in terms of extra borderlines: two rather than one, at the first attem pt, and if that does not do justice to how a predicate works, four rather than two, then eight rather than four, and so on. But if an idea is bad, as is the idea of thinking of vagueness in terms of boundaries, you make it no better by iterating it. Even an infinite iteration of the bad falls short of the good. Another approach, which promises at least to make room for classical conceptions, starts by characterizing vagueness through a characteristic sentence. T his approach, too, has to manoeuvre to allow for the fact that a tripartite conception is inadequate. T he manoeuvres call for a concept of definiteness, but various troubles attend its introduction. Characterizing 8 I should stress that there is no evidence that Wright would regard his account o f ‘red’ as applicable to all vague predicates.

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vagueness in terms of uncritically assumed definiteness is not m ethodo­ logically optimal. Further, it becomes unclear what the right characteristic sentence is. Finally, we do not know the logic of definiteness. W right has shown that there are choices under these last two heads which lead to paradox, but one does as well to conclude that one has made the wrong choices as that there is something wrong with ‘higher-order vagueness’. And, finally, W right’s attempt to persuade us that ‘red’ effects a tripartite division is unsuccessful. So where should we go from here? I think that things went wrong at the outset, when we allowed, without criticism, that one could characterize a predicate’s vagueness in terms of its allowing for borderline cases. We saw as we went along that this could not be a sufficient condition, since there are or could be predicates (e.g., ‘child*’) which meet it by effecting a tripartite partition, but which we intuitively do not count as vague. (Let us not be difficult about terminology: call them vague if you please, but admit that they function very differently from paradigms of vagueness like ‘child’ and ‘green’.) T he right way to characterize the vagueness of a predicate is by the fact that it classifies without drawing boundaries: it is boundaryless. A boundaryless predicate allows for borderline cases, but this is not its defining feature. A boundaryless predicate draws no boundary between its positive and its negative cases, between its positive cases and its borderline cases, between its positive cases and those which are borderline cases of borderline cases. T he phenomena which, from a classical viewpoint, lead to notions o f‘higher-order vagueness’ are accounted for by boundarylessness. But these phenomena are not bolt-on options; they are integral to the very nature of vagueness. On my view, the phenomena which those of classical inclinations classify as ‘higher-order vagueness’ are real enough; it is just that there is no real hierarchy here. Nothing gives rise to substantive issues about the level of vagueness appropriate to our familiar examples of vagueness, and there is no multiplication of sets. How could a boundaryless concept be used in classification? Surely classifying must exclude as well as include, and how could this not involve a boundary? Among many urgent questions about the nature of boundary­ lessness, this is the most pressing, and the one I shall focus upon in the remainder of this paper . 9 T o convince you that boundaryless classification is possible, I would ask you to think of the colour spectrum. It contains bands but no boundaries. 9 Another urgent question is whether boundarylessness entails Sorites paradoxes. The materials for a negative reply at the level of formal semantics, within a framework quite congenial to the approach here, is given by Michael Tye in ‘Vague Objects’, M in d , XCIX (1990), pp. 535-57.

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The different colours stand out clearly, as distinct and exclusive, yet close inspection shows that there is no boundary between them. T h e spectrum provides a paradigm of classification, yet it is boundaryless. Boundaryless classification is most at home within a system of boundaryless contraries, like colour concepts and biological taxonomies. Red excludes yellow, though there is no boundary between them , no least red location on the spectrum, immediately followed by yellow. A positive account of how such systems work would naturally look to the notion of a paradigm, quite commonplace in psychology and psycholinguistics, and not unfamiliar to philosophers . 10 Because this kind of classification works by grouping things around paradigms in accordance with some similarity relation, it makes vagueness the norm; it is departures into sharpness which require special explanation. One would expect that paradigms for a concept would be associated with ‘anti-paradigm s’ - clear cases of non-application. But what would these be? Would a suitable anti-paradigm for ‘child ’ be a dog, a television or a F rench horn? Not any clear example of a non-child will do. T he notion of an anti­ paradigm applies only within a system of mutually exclusive concepts. In the case o f ‘child’, the anti-paradigm is obviously ‘adult’. And this is the structure we in fact find —not universally but very commonly —with vague predicates: ‘Iarge’/'smalP, ‘hot’/‘cold’, ‘quick’/‘sIow’, ‘true’/‘false’, etc., as well as the more complex structures of contraries provided by colour classifications and the taxonomy of species. Everyone should agree that grasping a concept involves knowing both what it includes and also what it excludes. It is natural for the classicist to give an account of this knowledge in terms of negation: to understand O one must know what it is to be O and what it is not to be í>. From this point of view, it is hard to see how a concept could be vague. At worst, it would have a restricted significance range. By contrast, the alternative account can see knowing what a concept excludes in a positive way, in terms of things subsumed under contraries at the same level. It has rightly been stressed that vagueness in a predicate serves various useful ends. T he vagueness o f ‘child’ enables us to use it in the expression of duties of care, whereas an analogous sharp predicate would draw lines in places which seem to have no moral significance. T he vagueness of ‘red’ enables us to use it, with at least reasonable accuracy, on the basis of observation, whereas an analogous sharp predicate would draw lines in 10At the time of writing this paper, I was unaware of the considerable body of work, by such psychologists as E. Rosch, referred to as ‘prototype theory’. However, prototypicality is orthogonal to vagueness. Atypical birds like penguins can be definite birds, and even sharp predicates like ‘even number’ are associated with a spectrum of typicality rankings. Many approaches other than prototype theory fall within the paradigm and similarity genre.

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places where we could not detect them. Rather similar considerations apply when we try to give an account of how concepts, especially those closely related to perception, arise from the rich flow of information supplied by an organism’s perceptual system. T he natural expectation is boundaryless concepts, governed by paradigms rather than extensions. T o support this point, let us speculate about the mechanisms of concept formation. T he speculations are within a framework which would be absurdly idealized for serious empirical purposes, but this does not prevent them from making a philosophical point. Suppose you are building a creature. So far it can perceive - at least it is sensitive to its environm ent but it is conceptless. How do you add concepts? You want the creature to classify, to distinguish its own kind from others, predators from prey, the nutritious from the harmful, and so on. How could you build in sharp, bounded concepts? And would it be desirable? Consider what would be required to attain a concept, red*, similar to our familiar concept red except sharp. Suppose that its perceptual system is essentially similar to the human visual system. T his means in particular that there is a probabilistic element in the nervous responses: some physically identical conditions of stimulation will sometimes produce one response in the optical system, sometimes another. I f red* is to be linked to observation in the way that red is, it would be necessary that the creature see a boundary in the spectrum where we see none, and this is inconsistent with the supposition that its visual system is hum an . 11 So red* will relate differently to observation. T he probabilistic nature of the visual system means that we would have to say that the creature has mastered a concept with respect to which there is a right response that it has no reliable means of attaining; and this is inconsistent with even a modest version of D um m ett’s ‘manifestation requirem ent’. So there appears to be no way of adding a sharp ra/-like concept red* to a creature with a human-like optical system. Even if one could do it, one might not want to, if only because no distinction of use to the creature will hang on whether something is red* or not. An almost red* fruit will be at least almost as ripe as a red one. By contrast, one can see, at least in the roughest outline, how one m ight add a boundaryless concept, by the mechanism of paradigm and similarity. Colour paradigms will be roughly central on perceived spectral bands. T he relevant dimension of similarity will be visually available. Remembering the paradigm for subsequent applications will not require precision, since 11 would It be good to be able to locate this claim in a system of description which allows for such points as that those trained in the identification of birds ‘see birds differently’, at some level of description, from the novice.

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no fixed degree of similarity to the paradigm is required for correct application. Philosophers like Putnam have used a similar idea in connection with natural kinds, without, as far as I know, saying that a concept thus governed is boundaryless. Perhaps, even, they not only failed to say it but failed to think it. I f so, this is because of a fortunate accident about at least biological natural kinds: their ‘spectra’ in fact have gaps, which give the illusion of boundaries. T he gaps are an accident, which genetic engineering may one day reveal as such. T here is no a priori reason why there should not be series of extremely similar species linking, say, strawberries and raspberries, where some intermediates would be borderline with respect to both concepts. Boundarylessness comes in different forms, and a proper exploration of the notion would be a considerable project. In this paper, I seek only to show the importance of the project. We must shift away from the classical perspective. We are carried away by images which make us find boundarylessness problematic. We think of a system of classification as like a grid, a net, a system of pigeon-holes, a way of drawing a line, dividing a field. In this way of thinking, the Fregean idea that a boundaryless concept is no concept at all seems irresistible. But we should shift images. Classification is better likened to providing magnetic poles around which some objects cluster more or less closely and from which others are more or less repelled; some fall between a num ber of poles, drawn by more than one but specially close to none. Elaborating the alternative picture is required not just for an understanding of the standard examples of vagueness, but also for understanding a wide range of concepts and predicates, like those used in biological taxonomy, not normally thought of as vague . 12

Kings College London

12 I have benefited from comments by Samuel Guttenplan, and by participants at seminars in Sheffield, at the 1990 conference on Language, Concepts and Communication at the University of Sussex, and at the Oxford Philosophical Society in Summer 1990.

[15] JULY 1992

ANALYSIS 52.3

IS HIGHER ORDER VAGUENESS COHERENT?

By C rispin

W r ig h t

I T is widely assum ed not m erely that the Sorites afflicts vague expressions only, but that it is a paradox of vagueness — that vagueness is what gives rise to it. Since alm ost all expressions in typical natural languages are vague, that belief brings one uncom ­ fortably close to the thought, advocated by such philosophers as Peter U nger , 1 that natural languages, and the conceptual systems which they embody, are typically incoherent. But I think the thought that vagueness, per se, generates Sorites-susceptibility is a m uddled thought .2 W hen spelled out it goes, presum ably, som ething like this. If F is vague, its very vagueness m ust entail th at in a series o f appropria­ tely gradually changing objects, F at one end b u t no t at the other, there will be no nth elem ent which is F while the n +1 st is not; for if there were, the cut-off betw een F an d not-F w ould be sharp, contrary to hypothesis. Accordingly, the vagueness o f F over such a series m ust always be reflected in a tru th o f the form:

I

(i) —ï (9x)(Fx & —•Fx'), (where x' is the im m ediate successor o f x.) That, o f course, is a classical equivalent o f the universally quanti­ fied conditional which is the m ajor prem iss in standard form ula­ tions o f the Sorites — a thought which p ro m p ted H ilary Putnam to suggest that a shift to intuitionist logic m ight be o f value in the treatm ent o f vagueness. So indeed it m ight, in o th e r connections. But it won’t help here; for intuitionistic logic will yield a paradox from (i) (as will any logic with the standard 3- and & -Introduction rules + reductio ad absurdum)? This form o f the paradox — the No 1See for instance U nger [4]. - And that the most convincing and troublesom e versions o f the Sorites draw on other features o f the (vague) expressions they concern. See sections I and VII o f my [5]. *Of course, what intuitionistic logic does make possible is a response to the paradox which treats it as a reductio o f the m ajor prem iss, hence as a p ro o f that -*-»(3x)(Fx 8c —«Fx'), w ithout sustaining the d o uble negation elim ination step which forces that conclusion into a statem ent the precision o f F. B ut the basic difficulty rem ains: w hether o r n o t the double negation is sustained, to trea t the paradox as a reductio is to deny a prem iss which seems to say m erely th a t F is vague. Any genuine solution to the paradox has therefore to explain how th at appearance is illusory — how the m ajor prem iss fails as a schem atic description o f vagueness. T here is no way around the obligation and no reason to think, once it is m et, that any fu rther purpose will be served by im posing intuitionistic restrictions in this context.

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Only appears* though. W hat, it seems to me, the No Sharp Boundaries paradox brings o u t is that, when dealing with vague expressions, it is essential to have the expressive resources afforded by an o p erato r expressing definiteness o r determinacy. Som eone m ight think th at the introduction o f such an operato r can serve no p o in t since there is no app aren t way whereby a state­ m ent could be tru e w ithout being definitely so. T hat is undeniable, bu t it is only to say that — in term s o f a distinction o f Michael D um m ett ([1 ], pp. 446-7) — the content senses o f ‘P* and ‘Definitely P’ coincide; w hereas the im portant thing, for o u r purposes, is that their ingredient senses — their contribution to contexts em bedding them — differ, the vital difference concerning the behaviour o f the two statem ent-form s w hen em bedded in negation. Equipped with an appropriate such operator, we can see that a p ro p er expression o f the vagueness o f F with respet to the relevant sort o f series o f objects is n o t provided by (i) bu t ra th e r requires a statem ent to the effect that no definitely F elem ent is imm ediately succeeded by one which is definitely n ot F; that is N othing untow ard follows from that.

And this principle generates no paradox. The worst we can get from it, with o r w ithout classical logic, is the m eans for proving, for successive x', that N othing untow ard follows from that.

N othing untow ard follows from that. II If, however, we take seriously the idea of higher order vagueness, then a case can be m ade that this m erely postpones the difficulty. For if the distinction betw een things which are F and borderline cases of F is itself vague, then assent to N othing N othing untow untow ard follows ard follows from that.from that.

would seem to be com pelled even if assent to (i) is not. So once again the m aterials for paradox seem to be at hand, each ingre­ dient move taking the form o f a transition from ~^Def{Fx') to ~^Def(Fx). But the following is the obvious reply. O f any pair of concepts, F and H, which share a blurred boundary, we shall want to affirm N othing N othing untow untow ardard follows follows from that. from that.

when x ranges over the elem ents of an appropriate series in which the b lurred boundary betw een F and H is crossed. The original

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problem occurred in (i) when, with —«F in place o f H, we over­ looked the need to prefix the predicates with a definiteness opera­ tor. And now we are guilty o f the same oversight again in (iv); it is merely that this tim e H has been replaced by —Def(F). As soon as the inclusion o f the definiteness o p erato r is insisted on, all that emerges is HIGHER ORDER HIGHER VAGUENESS ORDER VAGUENESS

which yields nothing m ore than the harm less HIGHER ORDER HIGHER VAGUENESS ORDER VAGUENESS

Evidently the trick will generalize; so we need never, it seems, be at a loss for a way o f form ulating F’s possession o f vagueness, of whatever order, in a way that avoids paradox.

m But this is too quick. It is possible to be confident that the sort o f form ulation illustrated by (vii) avoids paradox only because we have so far no semantics for the definiteness operator, and are treating it as logically inert. W ithout considering in detail what form a semantics for it m ight take, a crucial question is w hether it would be correct to require validation for this principle: HIGHER

VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS

provided {A1 . . . A j consists o f propositions all o f which are ‘defini tizecT. For, in the presence o f DEF, and assum ing th at the corrected for­ m ulation, (vi) above, o f what it is for the borderline betw een F and its first-order borderline cases to be itself blurred, is itself definitely correct, the harm less (vii) gives way to VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS

whose generalization will enabíe us to prove th at F has no definite instances if it has definite borderline cases o f the first o rd er .4 By

mm m m m m m m m m m

VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS VAGUENESS HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHER HIGHERHIGHER HIGHERHIGHER

m m mm m mmmm mmmm m mmm mm mm

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contrast, (iii) gives way, by parallel reasoning, only to the innocuous o f his discussion. o f his discussion. T he trouble is thus distinctively at higher-order. DEF says, in effect, that the tru th of each o f a set o f fully definitized propositions ensures that every consequence o f th at set is likewise definitely true. This may get some spurious plausibility from conflation with the distinct and indisputable principle that w hatever is a consequence of a set of propositions each o f which is definitely true is itself definitely true. But DEF is plausible in any case. In effect it comes to the claim that w hen a proposition o f the form: it is definitely the case that P, is true, it cannot be less than definitely true. If DEF is valid, and (vi) is a satisfactory charac­ terization, th en higher-order vagueness — always a difficult and vertiginous-seeing idea — would seemingly be an intrinsically paradox-generating phenom enon (ergo, presum ably, a delusory one). IV Interesting recent work of Mark Sainsbury’s raises a n um ber o f points bearing on this paradox ’ I shall com m ent on three aspects o f his discussion. First, Sainsbury objects that in (vi), which he takes as its classical equivalent o f hiso fdiscussion. his discussion. o f hiso fdiscussion. his discussion. I picked a needlessly vulnerable characteristic sentence for higher o rd er vagueness. The m otivation for (vi), recall, was that if x' is a borderline case o f F, it will at least be true that it is not definitely F; and that if it is a definite borderline case, then the same will be definitely true. Thus (vi) or, if you will, (vi)1 says that no (definitely) definite F thing is succeeded by a definite borderline case — that the distinction betw een the Fs and the definite borderline cases is not one with an abrupt threshold, not a sharp one. Isn’t that ju st what second-order vagueness ought to be? ’Mark Sainsbury [2]. Sainsbury’s paper is a reaction to my [5], in which the No Sharp Boundaries paradox for higher-order vagueness was first, to the best of my knowledge, presented. See also Sainsbury’s [3]. There is a slight infelicity here, in so far as Fx)’ is not actually definitive of X’s being a borderline case of F, but will also be true if x is a negative case. But nothing important hangs on this. The thought in the text is restored — if good at all - by restricting the range of 'x’ to positive and borderline cases of F. Alterna­ tively, the reader may prefer to treat '—'D efifx)' as characterizing the agglomerate of borderline and negative cases together: (vi) and (vi)1 then plausibly capture what it is for the distinction between F’s and this agglomerate to be vague — which is just what it is for F to be second-order vague.

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Sainsbury ([2], p. 176) believes that there are o th e r equally well m otivated candidates for the characteristic sentence, from which one cannot by a proof o f the same general structure as Wright’s, derive anything paradoxical. . . So their entitlement to represent vagueness needs to be undermined before any conclusion antithetical to vagueness can be drawn from Wright’s proof.

W hat candidates? Well, it is natural, w hen vagueness is at issue, to want to work with some notion, however intuitive, o f truth-value gaps, o r o f tru th values o th er than tru th and falsity. E ither will set u p the possibility o f a distinction within the n o tio n o f negation. O ne notion, the proper negation o f A, may be defined as tru e if A is false, and false if A is true. The broad negation o f A, by contrast, while false if A is true, will be true in any o th er case — tru e ju s t in case A is o th er than true. In these term s, it is n atu ral to charac­ terize a borderline case o f F as som ething such th at n e ith e r the claim th at it is F no r the p ro p er negation o f th at claim is true. W riting ‘Neg A' for the p ro p er negation o f A and ‘N ot A’ for the broad, x is thus a borderline case of F if N otFx & N otN egFx This gives us som ething else to play the role o f i—'Defi 2. This paper provides a rigorous definition in a framework analogous to possible worlds semantics; it is neutral between epistemic and supervaluationist accounts of vagueness. The definition is shown to have various desirable properties. But under natural assumptions it is also shown that 2 nd-order vagueness implies vagueness of all orders, and that a conjunction can have 2 nd-order vagueness even if its conjuncts do not. Relations between the definition and other proposals are explored; rea­ sons are given for preferring the present proposal.

/ People can be bald or not bald. That classification is vague. People can be definitely bald or definitely not bald, but they can also be neither definitely bald nor definitely not bald. That classification is vague too. People can be definitely definitely bald or definitely definitely not bald or definitely nei­ ther definitely bald nor definitely not bald, but they can also be neither def­ initely definitely bald nor definitely not definitely bald, or neither definitely definitely not bald nor definitely not definitely not bald, or neither definitely neither definitely bald nor definitely not bald nor definitely not neither def­ initely bald nor definitely not bald. That classification is vague too. People can be ... . Vagueness in the first classification is first-order vagueness in “bald”, vagueness in the second classification is second-order vagueness in “bald”, and so on. Higher-order vagueness is «th-order vagueness for some ri> 1. We have as much reason to acknowledge higher-order vague­ ness as we have to acknowledge first-order vagueness; the difficulty of applying the higher-order classifications to a sorites series is of just the same kind as the difficulty of applying the first-order classification, which was what led us to recognize the problem of vagueness in the first place. The terminology of “«th-order vagueness” presupposes that a hierarchy of orders of vagueness has been defined. The foregoing paragraph con­ tains no such definition. Instead, it left a string of dots with “and so on”. We naturally assume that we could supply an explicit definition if the need arose. One is given below, although it is not the only possible way of fill­ ing in the dots. This explicitness enables us to confront some troubling Mind, Vol. 108 . 429 . January 1999

© Oxford University Press 1999

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128 Timothy Williamson

questions about the structure of the hierarchy, to which intuition turns out to be a much less useful guide than it initially feels. The aim of this paper is to clarify these structural issues without addressing deep questions about the nature of vagueness. By far the best developed framework for the analysis of higher-order vagueness is to be found in classical propositional modal logic with the necessity operator read as “definitely”. To preserve the formal analogy with “necessarily”, “Definitely P ” must be understood as “It is definite that F \ not as “It is definite whether P ”; the latter is equivalent to “Either it is definite that P or it is definite that not F \ This formalism is consistent with both an epistemic and a supervaluationist view of vagueness. Both can make sense of the equation “necessity = truth in all possible worlds” by reinterpreting it. As a first approximation, for the supervaluationist, definiteness is truth under all sharpenings of the language consistent with what speakers have already fixed about its semantics (“admissible sharp­ enings”); for the epistemicist, definiteness is truth under all sharp interpre­ tations of the language indiscriminate from the right one. In both cases, we hold everything precise constant as we vary the interpretation. For example, if it is precise how many hairs Jack has on his head, we do not vary the number as we evaluate “Jack is definitely bald”. What we vary is, for example, how many hairs it takes not to be bald. Formally, higher-order vagueness corresponds to contingency in which worlds are possible. The relativized equation “necessity in world w = truth in all worlds possible in w” can be reinterpreted too. For the supervaluationist, definiteness under a sharpening s is truth under all sharpenings admitted (deemed admissible) by s ; a sharpening t may be admitted by a sharpening sxand not by a sharpening s2 (Fine 1975, p. 146; Williamson 1994, pp. 156-9). For the epistemicist, definiteness under an interpreta­ tion i is truth under all interpretations indiscriminable from z; an interpre­ tation j may be indiscriminable from an interpretation /, and not from an interpretation i2(Williamson 1994, pp. 230-44, 270-2). This paper is not concerned with the choice between epistemicism and supervaluationism; it uses their shared formal framework. As for other views of vagueness, particularly those that reject classical logic, it is for their advocates to attempt an analysis of higher-order vagueness with comparable rigour and detail. 1 Such analyses, if forthcoming, would most likely face problems analogous to those discussed here. The object language will consist of sentence letters /?, q, r , ..., the usual truth-functors (including the truth and falsity constants T and i.) and the “definitely” operator A. a , ß, y ,... are metalinguistic variables over all sen1 Supervaluationists may reject classical rules of natural deduction in which as­ sumptions are discharged (Fine 1975, p. 143; Williamson 1994, pp. 150-2). Such rules are not at issue here.

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tences of the object-Ianguage. For simplicity, predicate and quantifier structure is suppressed. Although we mayinterpret the sentence letters as containing such structure, we will investigate vagueness in sentences without attempting to apportion it amongst their subsentential constituents. A is used to represent vagueness within the object-Ianguage. Of course we can al~o represent vagueness metalinguistically, but in the case ofhigher-order vagueness that would involve a cumbersome hierarchy of metalanguages. 2 To say that a formula 0. is valid we write "t= 0.". We can think ofvalidity as a semantic guarantee of truth; this explication raises more philosophical questions than it answers, but will do for present purposes. Validity is a matter ofmeaning, notjust form, for the sentence letters are to be taken as interpreted, and t= can reftect differences in their meanings. For exampIe, if r is interpreted as precise and s as vague, we shall have t= Ar v A-'r but not t= A sv A-'s; thus validity need not be closed under uniform substitution. But these formal principles about validity will be used: PC If 0. is a truth-functional tautology then t=

0.

K t= A(a -7 ß) (Aa -7 Aß) T t= Aa -7 0. B t= 0. -7 A-,A-'a

MP In 0. -7 ß and t= 0. then t= ß RN If t= 0. then t= Aa. PC and MP reflect the episternicist and supervaluationist commitment to classical logic. K says that definiteness is closed under modus ponens and RN that it is semantically guaranteed by validity. Together they imply that definiteness is closed under semantic consequence in the sense that ift=(al & ... & an) -7 ß then t=(Aal & ... & Aan) -7 Aß. In the formal semantics, a model is an ordered tripie where W is a set (of"points"), Ra binary relation (of"accessibility") on Wand n~a mapping from formulas to subsets ofW; think of[aß as the set ofpoints at which 0. is true. Truth-functors behave as usuaI: [0. & ßß= [[aß n [ßll; ['aß =W-{[a~. Au is true where 0. is true at all accessible points: [Aaß = {we W: V'xeW(wRx:::::> xeUall}. We say that t= ajust in case [a~ =W (0. is true at every point). Given the relevant class C ofmodels, we characterize validity by specifying that t= 0. just in case M t= 0. for each M e C. 2 Sainsbury (1991, p. 174) objects to approaches based on use ofa "definitely" operator that not all vague languages contain such an operator (which is true) and that if we expand the language by adding one we cannot rely on native speakers' intuitions to determine the truth-values of sentences (which is also true). Certainly speakers of a vague language may lack the concept of a borderline case or refuse to acknowledge borderline cases as such (Williarnson 1997a, pp. 945-6). But this just shows that we must expand the perspective of unidealized native speakers of the language in order to address higher-order vagueness: for exarnple, by working in an expanded language.

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130 Timothy Williamson We can think of each model as corresponding to a fixed precise state of the world, including for instance the number of hairs on Jack’s head; what vary from point to point in the model are the vague facts, for instance whether Jack is bald. Alternatively, we could combine all these models into a single large model, of which the original models would constitute mutu­ ally disjoint sub-models, inaccessible from each other by means of R; the same formulas would come out valid in the singleton class of the new model as in the class of old models.3 The examples in this paper use a single model each, without such disjoint sub-models; this corresponds to the special case in which all the relevant precise facts are necessary ones. For example, it is presumably not contingent whether it is possible for there to be a bald man with n hairs on his head. The focus on single models is merely for simplicity; the phenomena to be discussed arise equally for classes of many models. To simplify the exposition in what follows, we will distinguish explicitly between models and classes of models only where it is useful to do so. We will therefore sometimes speak of validity with respect to a single model, rather than the singleton class of that model. On the formal semantics, PC, MP, K and RN hold automatically. T embodies the platitude that it is semantically guaranteed that whatever is definitely so is so. T follows from the assumption that the accessibility relation R is reflexive. For the supervaluationist, every sharpening admits itself, otherwise it would be self-defeating; for the epistemicist, every inter­ pretation is indiscriminable from itself. The principles PC, K, T, MP and RN define the analogue of the modal logic KT, also known as T, and as M. B is the least obvious of the principles listed above. It follows from the assumption that the accessibility relation R is symmetric. For the epistemi­ cist, R is something like indiscriminability, which is symmetric. For the supervaluationist, a sharpening s has R to a sharpening t just in case s admits t\ if sharpenings deem admissible just those sharpenings that differ from them by at most some fixed amount, then R will again be symmetric. Intuitively, the crucial feature of a sorites series is that successive members differ at most slightly, and differing at most slightly seems to be a reflexive, symmetric, non-transitive relation. Although these considerations in favour of B are by no means decisive, we will provisionally include it as contributing to a particularly simple conception of the semantics, and call it into question again if it proves to have dubious consequences. The addi­ tion o f B to KT gives the modal logic KTB, also known as B. 3 We should need the unified model if we wanted to study the interaction of “definitely” with “necessarily”, for however definitely Jack is bald, he is only con­ tingently bald. The evaluation of “Necessarily Jack is bald” at a point at which Jack has no hairs on his head requires the evaluation of “Jack is bald” at points at which Jack has many hairs on his head.

Vagueness On the Structure o f Higher-Order Vagueness 131

We define a formula a to be precise if and only if N A a v A~»a: it is semantically guaranteed that it is definite whether a holds, a is vague if and only if a is not precise. Correspondingly, a is precise or vague in a model M according to whether Ml= A a v A^a. Given schema T, we could equivalently have defined a as precise if and only if 1= (a -> Aa) & (“«a -» A^a). Given K and RN, that condition implies that NAm(a -> A"a) and h Am(~la -> A ^ a ) for all natural numbers m and n, where An is a sequence of m occurrences of A. Then a is precise in a model just in case whenever ÍBJ, i g [[a]) if and only if/ e fa]], for the validity of a -> Aa requires the “only i f ’ direction and the validity of ~la -» A^a requires the “if* direc­ tion. Thus an accessibility relation permits only precise interpretations of formulas just in case each point is accessible only from itself. If the lan­ guage is completely precise, A is a redundant operator because h a Aa for every sentence a; the corresponding modal logic is known as Triv. In the models discussed in most detail below, every point bears the ances­ tral of R to every point, so a is precise just in case either [[a] = {} or [[a]] = W. In these models, only vacuous distinctions are precise. If we make a precise tripartite distinction between points which bear the ancestral of R only to points at which a is true (so a is perfectly definitely true), points which bear the ancestral of R only to points at which a is false (so a is perfectly definitely false) and points which bear the ancestral of R both to points at which a is true and points at which a is false (so a is neither per­ fectly definitely true nor perfectly definitely false), the first two categories are empty whenever [aflis strictly between {} and W.4 More generally, a is precise in any reflexive generated model just in case [[a]]= {} or [[a]]=W, where a generated model is one with a point bearing the ancestral of R to every point. Formally we lose nothing by restricting our attention to generated models, for validity with respect to a class C of models is equivalent to validity with respect to the class of all generated sub-models of members of C (Hughes and Cresswell 1984, pp. 78-SO). Vagueness is closed under semantic equivalence, in the sense that if f= a Aa and h a -» “«Aa, so t= Aa -> AAa and NA^a -» A“»Aa, so N AAa v A“»Aa because N Aa v A~*a. In general, a complex sentence of our object-language is vague only if at least one of its sentence letters is vague. But if a has firstorder vagueness without second-order vagueness, then precision in Aa does not imply precision in a, as will be confirmed below. If we specify the class of appropriate models vaguely then “h a ” will itself be vague. Its vagueness could in principle be analysed in the same manner as that of a. We do not do that here because “N” is not a symbol of our object-language.

II We can think of higher-order vagueness as follows. We have a first-order classification of states of affairs according to whether a or "«a holds. Vagueness in this first-order classification is first-order vagueness in a. In discussing it, we make a second-order classification of states of affairs according to whether members of the first-order classification definitely hold, definitely fail to hold or are borderline cases. Vagueness in this second-order classification is second-order vagueness in a. More generally, we have an (rt-fl)th-order classification according to whether members of the wth-order classification definitely hold, definitely fail to hold or are borderline cases. Vagueness in the wth-order classification is wth-order vagueness in a. Let us be more precise. Define a classification as a set of sentences closed under all truth-functors. For any set X of sentences, let CX be the smallest classification including X: the closure of X under all truth-functors in the language. We define the nth-order classification for X, C„X, inductively: (C,X = CX; C ^ X = C {Aa: aeC„X}. For every member a of the classification C„X, we have a member Aa of the classification Cfl+,X; we also have a member A^a of C„+1X because CnX is closed under negation, and therefore a member “'Aa & ^A^a of C„+IX because C„+IX is closed under truth-functors. Thus C„+1X is the smallest classification that can classify each member of (C^X as definitely holding, definitely not holding or borderline. A classification is precise if and only if all its mem­ bers are precise, a is wth-order precise if and only if Cn{a} is precise; a is Hth-order vague if and only if a is not /2th-order precise. We can define «th-order vagueness in the language as a whole similarly, by letting X be the set of all formulas.

Vagueness On the Structure 0/Higher-Order Vagueness 133

Membership of 2). Can we construct intuitive examples of higher-order vague conjunc­ tions without higher-order vague conjuncts? The trick in the model is that all cases are borderline for p and q separately while p & q definitely fails in a vague range of cases. It is not easy to see how a simple sentence could acquire meaning without definite application anywhere. But we can assign p and q complex interpretations. Here is a fanciful example. Imag­ ine that we use the integers to form an extremely vague popularity scale. Very popular individuals have highly positive popularity ratings; very unpopular individuals have highly negative popularity ratings. The scale 10 It would be futile in this example to require [[a]] to contain all points between points it contains, unlike {0, 2} and {0, 3}, for such sets are not closed under the operators in the language. For instance, if flrj] = {0,1} then |(r v A t) -> Aa| = {0, 2} and [[Ar v A~7 *]J= {0, 3}.

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is so vague that an individual’s rating never clearly falls within a range of less than ten. Now read p and q thus:

p: John’s popularity rating is either even and highly positive or odd and not highly positive. q: John’s popularity rating is either even and highly negative or odd and not highly negative. Whether John’s popularity rating is highly positive or not, it is quite indef­ inite whether it is odd or even; thus every case is borderline for p . Simi­ larly, every case is borderline for q. As for p & qy it implies that John’s popularity rating is even only if it is both highly positive and highly neg­ ative, which it cannot be. Thus p & q is equivalent to “John’s popularity rating is odd and neither highly positive nor highly negative”. Although this cannot definitely hold, it can definitely fail to hold, when it is definite that John’s popularity rating is either highly positive or highly negative. Moreover, this clause is itself vague; if John is popular enough for it to be borderline definite that his popularity rating is highly positive, A^(p & q) will also be borderline. Thus p & q has higher-order vagueness even though its conjuncts do not. There is the possibility in principle of higher-order vagueness in a truthfiinctional compound without higher-order vagueness in its constituents. A more promising approach is to question that which makes the possibil­ ity look problematic: the structural analogy between higher-order vague­ ness and (first-order) vagueness. A conjunction is vague only if at least one of its conjuncts is vague, but it may be misleading to think of higherorder vagueness in a as a species of vagueness in a. Higher-order vague­ ness in a is first-order vagueness in certain sentences containing a (com­ pare Fine 1975, p. 141). The sentences whose first-order vagueness constitutes higher-order vagueness in a conjunction are not in general sen­ tences whose first-order vagueness constitutes higher-order vagueness in any of its conjuncts. For example, first-order vagueness in A-'(p & q) con­ stitutes second-order vagueness in p & q, but not in p or in q. We tend to imagine higher-order vagueness as a subtle kind of infection or impurity, which cannot be present in a compound sentence without being present in one of its parts, and surely is not present in & itself. The image is not even accurate for first-order vagueness, since a precise conjunction can have vague conjuncts. Even if we imagine the infection in one conjunct as somehow neutralizing the infection in the other, that does not prepare us for an infected conjunction without infected conjuncts. It is time to aban­ don the infection model of higher-order vagueness.11 11My chief debt is to Delia Graff; much of the material emerged in the course of extended correspondence and discussion with her. Audiences at Cornell Uni­ versity, New York University and a conference on vagueness at Bled, especially Kit Fine, David Sanford and Roy Sorensen, also helped.

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Appendix: Proofs Notation: Let [i,j] = {k: i ~ k ~j}, where the variables range over all integers.

(i) Let n ~ 1. We must show that in some model of KT p has vagueness of any order below n and none of order n or higher. Let W = [l, n], iRj just in case either li11 ~ I orj = I, and [[PD= [I, n-I]. We show by induction on k that Uall E {[I, n-k], [n-k+I, n], {}, W} whenever aECk{P}' for I ~ k ~ n. Basis: If aEC I{P}, then a is a truth-function of p, so Uaß E {[I, n-I], {n}, {}, W}. Induction step: Suppose thatk< n and [aD E {[l, n-k], [n-k+ I, n], {}, W} whenever aECk{P}. IfUaD= (1, n-k] then [.:lall= [1, n-k-l]; if UaD= [n-k+ I, n] then [[.:lall= {} (because UaD does not contain I); ifUall= {} then [.:lall = {}; if Uall = W then I[ .:lall = W; the only new set produced by Boolean operations over W on [l,n-k-l], {} and W is [n-k,n]. Thus [[all E{[l,n-k-I], [n-k,n], {}, W} whenever aECCk+I{p}. This completes the induction. In particular, when k = n, Uall E {{}, W} whenever a E {P} . Thus C k {P} is vague when k < n and C. {P} is precise, so ce k {P} is precise when k ~ n, as required. (ii) We must show that for n ~ 0 there is a KTB model in which Bp is vague just when i < n. We need only show that this holds for arbitrarily large n, for if m ~ n and in so me KTB model Bip is vague just when i < n then BiB'··p (= Bi+ •.•p) is vague just when i < m; thus B'··p behaves in the way required for the case of m, and so does p in a new model in which it is true at the same points as B'··p in the original model. Moreover, we need only show that B'p is precise and (for n > 0) B'·'p vague, for (since Bi+'p is precise if Bp is precise) that entails that Bop is vague just when i < n. Let h ~ o. Consider a model in which W = [0, 2'+'-1], iRj just in case li11 ~ 1 and [[PII= [2'-1, 2']. We will show that HB'pD= W where n = 2'-1; thus B"p is precise and (for n > 0) B'·'p vague, for if B'·'p were precise then UB'pD= {}. For X [PD = [0, 2"'-1] =W. The basis ofthe induction for (*) (m =0) is trivial. Induction step: Assume (*) for m, and suppose thati+2m +2-2 ~j ~ k-2"+2+ I, [j_2 m +'+ I,j+2 +'] X = [j-2"·'+1,j+2"·'J, as required. (iii) We show in KT and therefore in KTB that for n 5, 2 nth-order vagueness in U implies nth-order borderline-vagueness in u. Trivially, 1storder vagueness implies Ist-order borderline-vagueness. We show in KT that 2nd-order vagueness implies 2nd-order borderline-vagueness. Suppose that U is 2nd-order borderline-precise. Thus Bu is precise, so 1= t..Bu v t..-,Bu. Since 1= Bu ~ ...,t..u, 1= t..Bu ~ l1""l1u. By T, 1= l1...,Bu ~ ""Bu, so 1= t..""Bu ~ (l1u v l1""u). Also by T, 1= ...,u ~ ""l1u, so 1= l1""u ~ l1""t..u; moreover 1= (u & ""Bu) ~ l1u, so 1= (l1u & l1""Bu) ~ l1l1u. Thus 1= l1""Bu ~ (Mu v l1""l1u). Putting the pieces together, 1= l1l1u v l1""l1u. Thus the precision ofBu entails the precision of l1u. Equally, the precision ofB""u entails the precision of l1""u. Since 1= Bu ~ B""u, the precision ofBu also entails the precision of B""u, and therefore of l1'u. But, as noted in the text, if l1u and l1""u are precise then C 2 {u} is precise, so u is 2nd-order precise. By contraposition, if u is 2nd-order vague then u is 2nd-order borderline-vague. (iv) To show in KTB and therefore in KT that for n > 2 nth-order vagueness in u fails to imply nth-order borderline-vagueness in u, consider the KTB model in (ii) in whichp is 2nd-order borderline-vague but nth-order borderline-precise for n > 2. By (iii), p is 2nd-order vague in this model, and therefore nth-order vague by the proof (in KTB) in the text.

Department 0/Philosophy The University 0/Edinburgh David Hume Tower George Square Edinburgh EH8 9JX UK

TIMOTHY WILLIAMSON

REFERENCES Burgess, John A. 1990: "The Sorites Paradox and Higher-Order Vagueness". Synthese, 85, pp. 417-74. Dummett, Michael 1959: "Wittgenstein's Philosophy of Mathematics". The Philosophical Review, 68, pp. 324-48.

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Edgington, Dorothy 1993: “Wright and Sainsbury on Higher-Order Vagueness”. Analysis , 53, pp. 193—200. Fine, Kit. 1975: “Vagueness, Truth and Logic”, in Keefe and Smith 1997, pp. 119-50. Originally published in 1975 in Synthese, 30. Heck Jr., Richard G., 1993: “A Note on the Logic of (Higher-Order) Vagueness”. Analysis , 53, pp. 201—8. Hintikka, Jaakko 1962: Knowledge and B elief Ithaca, NY: Cornell Uni­ versity Press. Hughes, G.E., and Cresswell, M. J. 1984: A Companion to Modal Logic. London and New York: Methuen. Humberstone, Lloyd 1988: “Some Epistemic Capacities”. Dialéctica , 42, pp. 183-200. Keefe, Rosanna and Smith, Peter (eds.) 1997: Vagueness: A Reader. Cambridge, MA and London: MIT Press. Rolf, Bertil 1980: “A Theory of Vagueness”. Journal o f Philosophical Logic , 9, pp. 315—25. Sainsbury, R. M. 1990: “Concepts Without Boundaries”, in Keefe and Smith 1997, pp. 251-64. Originally published in 1990 by King’s Col­ lege London. ------ 1991: “Is there Higher-Order Vagueness?”. The Philosophical Quarterly, 41, pp. 167—82. Sanford, David 1975: “Borderline Logic”. American Philosophical Quar­ terly , 12, pp. 29-39. Villanueva, Enrique (ed.) 1997: Philosophical Issues, 8: Truth. Ridgeview: Atascadero, CA. Williams, Bernard 1978: Descartes : The Project o f Pure Enquiry. Lon­ don, Penguin. Williamson, Timothy 1994: Vagueness. London: Routledge. ------ 1995: “Is Knowing a State of Mind?”. Mind, 104, pp. 533-65. ------ 1997a: “Reply to Commentators”. Philosophy and Phenomenologi­ cal Research , 57, pp. 945—53. ------ 1997b: “Replies to Commentators”, in Villanueva 1997, pp. 255-65. ------ 1998: “Conditionalizing on Knowledge”. British Journal fo r the Philosophy o f Science , 49, pp. 89—121. Wright, Crispin 1987: “Further Reflections on the Sorites Paradox”, in Keefe and Smith 1997, pp. 204-50. Longer version originally pub­ lished in 1987 in Philosophical Topics, 15. Wright, Crispin. 1992: “Is Higher Order Vagueness Coherent?”. Analysis, 52, pp. 129-39.

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[19] Why Higher-Order Vagueness is a Pseudo-Problem DOMINIC HYDE

Difficulties in arriving at an adequate conception of vagueness have led many writers to describe a phenomenon that has come to be known as “higher-order vagueness”. Almost as many have found it to be a problem that needs to be addressed. In what follows I shall argue that, whilst we must acknowledge its presence, it is a pseudo-problem. The crucial point is the vagueness of “vague”, which shows the phenomenon to be unproblematic though real enough.

1. Higher-order vagueness

Vagueness is usually introduced as a phenomenon affecting predicates. On this account the signature of vagueness is our inability to draw a sharp line between those things in the predicate’s positive extension and those in its negative exten­ sion. This initial focus on predicate vagueness reflects one aspect of the paradig­ matic concept o f vagueness: vagueness as applied to predicates and characterised by the presence of “border cases”. The apparent lack of a sharp boundary to the predicate’s extension is typically explained by the presence of border (borderline, or penumbral) cases for the predicate in question: cases which jointly constitute the border region {borderline region or penumbra) for the vague predicate. Even given qualifications to the notion of a border case in order to exclude cases of inexactness, meaninglessness, ambiguity, context sensitivity, etc. the notion of a border case still appears too broad to properly characterise the phenomenon of vagueness. Consider an example of Mark Sainsbury’s: the predicate “child*”. It is to count as true of all those people who have not yet reached their sixteenth birthday, false of all those who have reached their eighteenth birthday, and neither true nor false of all other people. Now, for seventeen year olds it is neither determinately true that they are children* nor is it determinately false that they are children*: they do not determinately satisfy the predicate nor do they determinately satisfy its negation. So it would seem that a seventeen year old counts as a border case for the predicate “child*” thereby making it vague, even though, intuitively, the Mind, Vol. 103 . 409. January 1994

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predicate is perfectly precise.1To distinguish such cases from vagueness proper, something more needs to be said. Russell acknowledged an extra ingredient in vagueness: The fact is that all words are attributable without doubt over a certain area, but become questionable within a penumbra, outside of which they are again certainly not attributable. Someone might seek to obtain preci­ sion in the use of words by saying that no word is to be applied in the penumbra, but unfortunately the penumbra itself is not accurately [pre­ cisely] definable, and all the vaguenesses [sic] which apply to the pri­ mary use o f words apply also when we try to fix a limit to their indubitable applicability. (Russell 1923, p. 87; my italics)

Alston more recently echoes this in the Encyclopedia when, after having endorsed the border case conception, he continues: ... “middle-aged” is vague, for it is not clear whether a person aged 40 or a person aged 59 is middle-aged. Of course there are uncontroversial areas of application and nonapplication. At age 5 or 80 one is clearly not middle-aged, and at age 45 one clearly is. But on either side of the areas of clear application there are indefinitely bounded areas of uncertainty. (Alston 1967, p. 218; my italics) The now common response to the vagueness of the penumbra itself is simply to say that the penumbra has border cases. Thus, in an attempt to get at this extra ingredient by means of the notion of a “border case”, i.e., from within the para­ digmatic concept of vagueness, talk moves to a hierarchy of border cases and the paradigmatic concept is iterated. Since the mid-seventies this phenomenon of higher-order vagueness (HOV) has come to the fore in discussions of vagueness. Higher-order vagueness arises, as we have seen, because vague predicates typically fail to draw any apparent sharp boundaries within their range of significance. The paradigmatic concept we have been discussing initially attempts to accommodate the intuition that there is no apparent sharp boundary between the positive and negative extension of a predicate in terms of the presence of a penumbra or border region (or border cases). So, for example, with the predicate “red” the absence of any apparent sharp boundary between the red and the non-red is initially described by refer­ ence to border cases. The requirement that there be no apparent sharp boundary is thus satisfied by eliminating the sharp boundary and replacing it with a border region. Yet the region itself might paradoxically appear sharply bounded—unless it too has border cases. There is no more an apparent sharp boundary between the positive extension and the border region than there was between the positive extension and the neg1See Sainsbury ( 1991, p. 173). Wright ( 1987, p. 244f) has argued that the vague pred­ icate “red” d o e s effect a tripartite division of its range of significance; it has a penumbra of border cases but this penumbra is itself precise. Sainsbury convincingly rebuts Wright’s argument by denying Wright’s criterion for the correct use of “red”; cf Sainsbury (1991, p. 176). Even were Wright’s argument successful, there is no reason to think that it would hold in general. In fact, as I shall go on to show, such a claim must be false since there are higher orders of vagueness.

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ative extension. So, if the presence of a penumbral region is taken as definitive of vagueness then it is not itself characterised merely by the existence of border cases; there must be border cases of border cases. But why stop here? There appears to be no more reason to suppose that there is a sharp boundary between the determinately determinately red and the vaguely determinately red than there was to suppose a sharp boundary between the determinately red and the vaguely red. “At no point does it seem natural to call a halt to the increasing orders of vagueness” (Fine 1975, p. 297), so the iteration seems endless: border cases echo up through the hierarchy. The real lesson of higher-order vagueness is that vague predicates draw no apparent sharp boundaries, not merely that they apparently fail to draw a sharp boundary at the first level, or the first and second levels, or ... After having said all of the above one might think, like Sainsbury, that the iter­ ative conception of vagueness is both inescapable on the paradigmatic approach to vagueness and misguided, thus motivating a search for an entirely new approach which avoids the so-called “problem of higher-order vagueness” (Sainsbury 1991, p. 179). Alternatively, one might think that the iterative concep­ tion, with qualifications ad infinitum, constitutes an adequate reply. These two responses presuppose there to be a real problem to be addressed. I want to argue that the iterative conception captures a feature of vagueness that is real enough—the phenomenon of higher orders of vagueness—but that this phenomenon is ultimately an echo of a more basic feature of border cases. There is no real problem. Recourse to an infinite string of qualifications, like those above, betrays an ignorance of the ambiguity of “border case”.

2. The problem dissolved Before we are in a position to see the problem aright we need to acknowledge two important points made in the literature on vagueness. The first point is that “vague” is itself vague—it is an homological term. A convincing argument to this effect is provided by Sorenson (1985). Consider the sequence of predicates “1-small”, “2-small”, “3-small”, etc. defined on the natu­ ral numbers. The nth predicate on the list is defined in such a way as to apply to only those integers that are either small or less than n. Using these predicates as arguments we are able to construct a sorites paradox for the predicate “vague”: “1-small” is vague. If “n-small” is vague then “n+1 -small” is vague. “106-small” is vague. The predicate “1-small” is as vague as “small” since both predicates clearly apply to 0 and both apply in exactly the same way to all other integers. The same is the case with “2-small” and “3-small”. Each of these two predicates apply in exactly the same way to all other integers as “small” does with the

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exception that “2-small” has 0 and 1 as clear instances whilst “3-small” has 0, 1 and 2 as clear instances; since 0, 1 and 2 are all clearly small it follows that “2-small” and “3-small” are as vague as “small” itself. However, we eventually reach predicates where the “less than /z” clause has the effect of making some integers clear instances of the predicate “n-small” whereas they were border cases for the predicate “small”. Some border cases are eliminated. Still further down the series of predicates, at “#-small” say, we find that all border cases for “small” have been eliminated for the border region of “qsmall”; the predicate “g-small” has no border cases at all and is therefore pre­ cise. For example, it is clear cut whether or not to apply the predicate “106small” to any integer; if the integer is less that 106 then the predicate clearly applies and if the integer is 106 or greater then, since it is clearly neither small nor less than 106, the predicate clearly does not apply. Yet, as Sorensen points out, it is a vague matter where along the sequence the predicates with border cases end and the ones without border cases begin. Consequently “vague” is vague. Since “vague” is vague, it cannot be defined in purely precise terms. This fol­ lows from the second point requiring acknowledgment. According to Rolf’s general theory of what it means to describe phrases of any grammatical category as vague, the concept of vagueness is recursively extended to non-denoting categories where any notion of a border case seems inapplicable (i.e., from names, predicates and sentences to logical constants, modifiers, etc.).2 The recursive extension is based, correctly I think, upon “[t]he leading idea ... that a non-predicate is vague iff it sometimes ‘causes’ vagueness of a sentence of which it is a part—more precisely: if there is a vague sentence which, except possibly for the phrase under consideration, contains only precise phrases” (Rolf 1980, p. 320). More precisely and more generally: if all but one o f the constituent sub-phrases o f a complex phrase are precise then, if the com­ plex phrase is vague, so is that one remaining constituent sub-phrase.

As a consequence we can establish the following principle: The Inheritance Principle

If all the constituent phrases of a complex phrase are precise then the complex phrase is precise. (Rolf 1980, p. 321, Theorem 3.5) For example if a precise predicate is predicated of a precise subject term then the resulting sentence will be precise. As Rolf remarks elsewhere, the fact that precision is inherited has interesting consequences itself for definability (Rolf 1981, p. 92). If we have purely precise phrases to hand, we can define only precise phrases by means of them; if the 2 olfR(1980, §3). Prior to the extension, an important modification is made to the char­ acterisation of predicate-vagueness in an attempt to rule out deviant border-case situations that are attributable to vagueness surrounding what to count as the su b jec t of predication rather than any vagueness in the predicate. This issue is orthogonal to that of higher-order vagueness; the constrained notion of a border case is still subject to the same worries as before. To keep the main issues clearly in view I shall ignore this modification (important though it is).

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definiens is vague then there must be some vagueness in the definiendum. So “vague” itself must be characterised in partially vague terms. I take this to be something we can all agree on. So, thus enlightened, let us turn to the so-called “problem of higher-order vagueness”. Recall how the ascent up the hierarchy was motivated: requiring that vague predicates not only possess border cases but border cases of border cases and bor­ der cases of border border cases, etc. The worry was that the simple requirement that vague predicates have border cases could not be sufficient for vagueness since incomplete predicates seemed to have border cases (cases to which the predicate did not appear to determinately apply nor determinately not apply). “Child*” was cited as an example of a predicate which is precise though it has border cases in this limited sense. This suggested that the general idea of charac­ terising vague predicates in terms of border cases needed modification: the requirement that there be border border cases, etc. In other words, an adequate characterisation of the notion of predicate-vagueness via border cases was seen to be available only if it was explicitly specified that “border case” is vague, and “border border case” is vague, etc., resulting in the inadequacy of anything short of an infinite iteration within the characterisation. I think that there are higher orders of vagueness, but that this is already entailed by the paradigmatic conception and can be seen to follow when the notion of “border case” employed therein is properly understood. Enter the vagueness of “vague”. When the concept being analysed is vague­ ness we need some vagueness in the analysans to capture the vagueness in the analysandum. Hence, implicitly, when characterising predicate-vagueness in terms of border cases we must suppose the notion of a border case (and associated notions like penumbra and border region) to be itself vague.This is because there is no other candidate for vagueness in the analysans. Predicate-vagueness is char­ acterised by “there being border cases”, yet the existential quantifier is precise3 so the notion of a border case must be vague.4 What are we to say with regard to the predicate “child*”? Simply that it does not possess a border case, at least in the sense of “border cases” used in the char­ acterisation of vagueness—the vague sense of “border case”. “Child*” does have apparently indeterminate instances; however, the term “indeterminate” here is precise, whereas the apparent indeterminacy required for vagueness is itself vague. (It would be preferable, I think, if some other terms could be used to dis­ tinguish the precise and vague sense of “determinate”. Given that border case ter3 Rolf (1980, p. 322) offers the precision of the existential quantifier (standardly inter­ preted to mean “at least one”) as Theorem 3.7. 4 This puts paid to any attempt to avoid the supposed difficulties associated with HOV by either denying there to be any higher orders of vagueness (as Wright has tried to sug­ gest; cf. n. 1), or by claiming it to be vague whether there are higher orders of vagueness: a line of defence presumably advocated by Michael Tye who, seeing the admission of higher orders of ontological vagueness as problematic, claims it to be vague whether there are any such higher orders; cf. Tye ( 1990, p. 535). The phenomenon of higher-order vague­ ness is definitely real.

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minology is now quite firmly entrenched in the literature on vagueness, a new term could be used to play the role of “determinate” in the precise sense and then the term “border case” can be defined by means of “determinacy” in the vague sense, thus ensuring that when one speaks of border cases one is speaking vaguely.) In requiring that the indeterminacy characteristic of vagueness is vague one might be tempted to think that a worrisome circularity is lurking in the back­ ground. I think this worry is misplaced. Were we to characterise “vague” using the term “vague” then a vicious circularity would indeed arise, just as it would were we to characterise “meaningful” using the term “meaningful”. However we are not characterising “vague” using the term “vague”; rather we are characteris­ ing “vague” using vague terms and this is no more a problem than characterising “meaningful” in meaningful terms. In fact, far from being viciously circular, it is (as I have already pointed out) required of us that we characterise “vague” in vague terms for exactly the same reason that we are required to characterise “meaningful” in meaningful terms—both “meaningful” and “vague” are homological expressions. The apparent circularity is simply the misplaced recognition of this homological aspect of vagueness. It is misplaced recognition of the fact that any characterisation of what it is for a term to be vague will itself partially characterise the term “vague”, since “vague” is a vague term; the analysis catches itself within its scope. Summing up then: the problem surrounding higher orders of vagueness arises when one tries explicitly to state something about the nature of vagueness that manifests itself in the characterisation anyway—the phenomenon of higher-order vagueness. It is not necessary to explicitly state any extra conditions in one’s characterisation of vagueness to ensure compatibility with the phenomenon unless one thinks that the notion of a border case is precise-unless-explicitlyqualified—and this is simply false! Ignorance of the lurking ambiguity in the term “border case” (ultimately as a result of the ambiguity of “determinately”) creates unnecessary difficulties. There are border border cases for vague predi­ cates, but this need not be stated as part of the analysis of the concept of predicate-vagueness any more than, having said that “red” is a predicate, one must then go on to state that “‘red’ is a predicate” is a predicate and that ““ red’ is a predicate’ is a predicate” is a predicate, etc. One is simply repeating oneself and adding nothing new. The vagueness of “vague” can be seen to be built in right from the very start, when we characterised vagueness in terms of border cases, once we have conceded (as we must since “vague” is vague) that the sense of “border case” employed in the characterisation is the vague sense. Philosophy Program Research School o f Social Sciences Australian National University Canberra, 2601 Australia

DOMINIC HYDE

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347 41

REFERENCES Alston, W.P. 1967: “Vagueness”, in P. Edwards (ed.) The Encyclopedia o f Phi­ losophy. London: MacMillan, p. 218. Fine, K. 1975: “Vagueness, Truth and Logic”. Synthese, 30, pp. 265-300. Rolf, B. 1980: “A Theory of Vagueness”. Journal o f Philosophical Logic, 9, pp. 315-25. Rolf, B. 1981: Topics on Vagueness. Ph.D. Thesis, Lund University. Russell, B. 1923: “Vagueness”. Australaisian Journal o f Philosophy, 1, pp. 8492. Sainsbury, R.M. 1991: “Is There Higher-Order Vagueness?”. Philosophical Quarterly, 41, pp. 167-82. Sorensen, R. 1985: “An Argument for the Vagueness of ‘Vague’”. Analysis, 45, pp. 134-7. Tye, M. 1990: “Vague Objects”. Mind, 99, pp. 535-557. Wright, C 1987: “Further Reflections on the Sorites Paradox”. Philosophical Topics, 15, pp. 227-90.

[20] Why the Vague Need Not be Higher-Order Vague MICHAEL TYE

Is higher-order vagueness a real phenomenon? Dominic Hyde (1994) claims that it is, and that it is part and parcel of vagueness itself. According to Hyde, any gen­ uinely vague predicate must also be higher-order vague. His argument for this view is unsound, however. The purpose of this note is to expose the fallacy, and to make some related observations on the vague, the higher-order vague, and the vaguely vague.

/ Hyde tells us that vagueness, of the sort associated with predicates, is character­ ized by border cases (where a border case is one in which it is indeterminate whether the predicate applies). He then argues as follows: since “vague” is vague (by the argument of Roy Sorensen’s he cites on p. 37), we must suppose that the notion of a border case is itself vague. For “predicate-vagueness is characterised by ‘there being border cases’, yet the existential quantifier is precise”. So, “there is no other candidate for vagueness in the analysans” (p. 39). But if “border case” is vague, then it follows that there are border border cases and hence higher-order vagueness. So, as soon as we admit that a predicate is vague, we must grant that it also admits of higher-order vagueness. As is so often the case in discussions of vagueness, this argument moves too fast. The reasoning may be reconstructed as follows: (1) “‘F ’ is a vague predicate” may be analysed as “‘F ’ has border cases”. (2) “Is a vague predicate” is itself a vague predicate. From(l) and (2), (3) “Has border cases” is vague. From (3), (4) “Border case” is vague. So, (5) There are border cases of border cases. So, (6) For any vague predicate “F ”, “F ” has border border cases. The argument contains a non-sequitur. (6) does not follow from (5), unless (5) is read as saying that for any vague predicate, there are border cases of border cases Mind, Vol. 103 . 409. January 1994

© Oxford University Press 1994

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of that very predicate. The trouble is that if (5) is read in this way, then it does not follow from (4) and it is completely unsupported by anything Hyde says. All that follows from the fact that “border case” is vague is that some border border cases exist, just as all that follows from the fact that “red” is vague is that some border red things exist. So, all that Hyde is entitled to conclude from his premises is that some vague predicates have border border cases. It could be, for example, that the only vague predicates admitting border border cases are very peculiar and atypi­ cal. So, for all that Hyde has shown, it could be that no everyday vague predicates are also higher-order vague. What, then, has Hyde demonstrated? It seems to me that, on the assumption that Sorensen’s argument establishes premise (2), he has shown that there is higher-order vagueness somewhere. This result is not uninteresting. Unfortu­ nately, considerations similar to those that Hyde himself adduces give us reason to doubt that Sorensen’s argument is sound. So, the existence of higher-order vagueness remains very much in question. Let me explain.

II Suppose for the moment that Sorensen’s argument is sound and that (2) is true. Then given also (I)1, (7) “Is a vague predicate” has border cases may be inferred. So, by the usual understanding of the notion of a border case, (8) There is at least one predicate such that it is indeterminate whether “is a vague predicate” applies to it follows. Obviously, “is a vague predicate” applies to any given predicate if, and only if, the predicate is vague. So, from (8), we arrive at (9) There is at least one predicate such that it is indeterminate whether it is vague, and hence ( 10) There is at least one predicate such that it is indeterminate whether it has any borderline cases of application. I call such predicates “vaguely vague”2. So, once we admit that “vague” is vague, by Sorensen’s argument, we appear to need to countenance both the higher-order vague and the vaguely vague. However, once the vaguely vague is admitted, Sorensen’s argument becomes suspect. For vaguely vague predicates admit of 1 In place of (1), I actually prefer the weaker claim that being a vague predicate entails having border cases (see Tye 1990), but for the present purpose either claim will do. 2 So, vaguely vague predicates are such that it is indeterminate whether they have any border cases. By contrast, vague predicates have border cases, and higher-order vague predicates (if any there be) have border border cases. Formally, the difference is of the sort exhibited in the following sentences: where “V” abbreviates the operator “it is indetermi­ nate whether”, “V(3jc)Fjc”, as compared with “(3jc)VFx” and “(3*)VVFjc”.

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sorites sequences just like the one Sorensen generates for “vague” (see Tye 1990, pp. 551-552). So, an alternative explanation of the Sorensen sequence for “vague” is that “vague” is vaguely vague. So, if Sorensen’s argument is sound, then we may reasonably deny that his argument is sound. So, we may reasonably deny that Sorensen’s argument is sound. So, Hyde has not succeeded in demon­ strating that there is any higher-order vagueness. Of course, if Sorensen’s argument is not sound then the argument given above for the existence of vaguely vague predicates is unsound. But there are other rea­ sons for believing in the vaguely vague. In my view, the vaguely vague, unlike the higher-order vague, plays an important role in dissolving the sorites para­ doxes, as they apply to common or garden vague predicates. That is another story, however, and one that I cannot take up here (see Tye 1990, and forthcom­ ing). MICHAEL TYE

Department o f Philosophy Temple University Philadelphia, PA 19122 USA Department o f Philosophy King ’s College London Strand London WC2R 2LS UK

REFERENCES Hyde, D. 1994: “Why Higher-Order Vagueness is a Pseudo-Problem”. Mind , 103, pp. 35-41. Tye, M. 1990: “Vague Objects”. Mind , 99, pp. 535-557. Tye, M. 1994: “Sorites Paradoxes and the Semantics of Vagueness”, in Philo­ sophical Perspectives, 8, Tomberlin, J. (ed.). (Volume on Philosophy of Language and Logic).

Part VI Contextualism

[21] HANS KAMP

T H E P A R A D O X O F T H E H E A P*

1. (I) (1) (2)

One grain of sand cannot make a heap. If n grains of sand cannot make a heap, then n + 1 grains cannot make a heap either; and this is so irrespective of the choice of n.

So: (3) No matter what n, n grains of sand cannot make a heap. This argument is paradoxical. For from what appear to be true premises we derive an evidently false conclusion by means of what would seem to be a valid form of inference. The principle of inference employed, the so-called principle of mathemati­ cal induction, is an essential tool in mathematics. While important in just about any branch of mathematics it is strictly speaking a principle of number theory, as is clear from the following description of it: In order to prove that all positive integers possess a given property P it suffices to show that (II) (1') the number 1 has P, and (2') for any number «, if n has P then n + 1 has P too. Here is a proof - the standard proof! - that the principle of mathematical induction is valid: Assume that (1') P (l) and (2') (V«)(P(n)-> P(n + 1)) are both true. Now suppose that (Vn)P(n) is false. Then there is an n for which P fails. But then there must be a smallest such n,n0 say. Since P (l) and not P(nQ),nQí 1. So n0 = k + 1 for some positive integer k. Since n0 is the smallest n such that not P(n), we have P(k). But then by (2')P{k + 1). i.e. P(/i0), which contradicts the assumption that not P(n0). If this argument demonstrates to our satisfaction that the principle of mathematical induction embodies a sound method for showing univer­ sality of the numbertheoretic properties considered in mathematics, then why should it fail to establish its validity in connection with such a property of n as is expressed by the phrase ‘n grains cannot make a heap’? An answer which comes to mind immediately is that this last property 225 U. Mönnich (ed.), Aspects o f Philosophical Logic, 225-277 Copyright © 1981 by D. Reidel Publishing Company

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is vague whereas mathematical properties are sharp. Indeed, if P is a vague predicate then the justification of induction we just rehearsed seems to break down. For suppose we argue as above. Again we are led to consider the smallest number n0 such that ~iP(/?0). Again we can conclude that n0 = k 4- 1 for some k. But now, since P is vague, it is no longer obvious that we can infer from the fact that n0 is the smallest number for which P fails that P must be true of k. E.g. if by n0 we were to understand the smallest number for which P definitely fails, then there would be no reason to expect k to unambiguously possess P. For in the cases in which we are here interested the vagueness of P will show up in the form of a certain ‘no-manVland’ situated between the opposing territories of, respectively, these numbers which definitely have P on the one side and of those which definitely lack the property on the other. In fact it may not even be meaningful to speak of the ‘smallest n such that n definitely lacks P ’ in the first place. So, for all we know the principle of mathematical induction may be valid for sharply defined properties, and yet fail when applied to properties whose extensions are blurred. 2.

To see whether this is indeed so would involve us in a lengthy analysis. It seems however that we can spare ourselves that effort. For there is a quicker route towards the conclusion which that analysis would establish, that is, that the invalidity of (I) cannot be blamed on the principle of mathematical induction alone. Besides the defense of the principle which we rehearsed above there is another, almost equally familiar, justification for it, favoured in particular by those mathematicians who are of a constructivist persuasion. It goes as follows: From the premises (V) and (2') I can, for any number k whatever, construct a valid proof that P(k). For (a) I obviously have a ‘p roof for the case where k = 1 : all I have to do is write down premise (I'); (b) let k be any number and suppose I can construct a proof of P(k) from (T) and (2'); then I can also_construct a proof of P(k + 1). All I have to do is to take the proof of P(k ), then obtain from (2') the special instance H A NHS A K NA S MKPA M P

and finally infer P(k + 1) by means of Modus Ponens. kBut waif you will say, This is begging the question. You have given me a proof of P (l) and you have shown me how to convert a proof of

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P(k) into a proof of P(k + 1). But in order to infer from this that there is a proof of P (ii) for all 11, it is precisely the principle of induction one has to appeal to'. This objection is correct: as I have stated it, the justification is circular. And yet, when we apply this line of argument to the cases that involve vague predicates we find that something has been gained. For whether or not P itself is vague, the predicate Q(n) expressed by the phrase 'there is a proof of P(ii) from (1') and (2')' is not vague. Given accepted rules of inference the question whether a given sequence of lines constitutes a proof is a perfectly well-defined - and indeed a decidable - question. So the earIier justification of mathematical induction will warrant the inference from Q(I) and (V'n)(Q(Il) - Q(11

+ 1))

to (V'Il)Q(Il). Moreover, from (1'), (2') and the statement: 'for any 11 there is a valid proof of P(ii) from (1') and (2')' we can infer (3')

(V'Il)P(n).

To be more precise let Q be defined by: Qis true ofthe number 11 iffthe sentence P(ii) is provable from (1') and (2') by means of the inference principles of classicallogic. Then it follows from obvious properties of formal proofs that (4)

Q(l)

(5)

(V'I1)(Q(I1) - Q(1l

and

+ I)).

As Q is sharp the induction principle allows us to infer (6)

(V' I1)Q (11).

From (1'), (2') and (6) we can obtain the conclusion (3')

(V'I1)P(I1)

if we adopt in addition the principle (7)

(V'Il)Q (11) - «(1') & (2') - P(n)).

I claim that (7) is valid; or, rat her, that it must be accepted as valid by anyone who assumes that the rules of classicallogic are valid in the context in which they are here employed. For to accept these as valid is precisely to accept a derivation of peil) from (1') and (2') which only makes use ofthese rules as a guarantee that either one of (1') and (2') is false or else that 11 satisfies P.

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This shows that we cannot hope to explain the paradox by putting the blame exclusively on the principle of induction, while maintaining that the rules of the predicate calculus - specifically those to which the above argument appeals - apply to vague predicates no less than to predicates which are exact. 3. The existing literature contains two proposals for a resolution of the paradox which I shall briefly review before presenting the account which 1 have come to prefer. All three theories which we shall consider agree in denying that the second premise of the argum ent-i.e. (2), or ( 2 ') - is true. According to the theory we shall consider first the premise is not false but' only ‘nearly true’. Here ‘nearly true’ is to be understood in the terms offered by many-valued logic. In fact the entire account constitutes an appli­ cation of that branch of many-valued logic which has come to be known as ‘fuzzy logic’. It would carry us too far to give a detailed exposition of all those elements of fuzzy logic which are relevant to this application. But the central ideas are easily conveyed. Suppose P is a vague predicate, say ‘bald’, and that there are lots of individuals to whom we hesitate to attribute this predicate. Among those individuals there will nonetheless be some that are nearer to deserving the predicate than others. Suppose a is nearer to being bald than b. Then you might consider that in a sense the proposition P(ã) is nearer to unqualified truth than P(B). Indeed, if one assumes that between the completely true and the completely false there are many ‘intermediate truthvalues’, linearly ordered by their respective distances from perfect truth, then what I just said about the ‘truthvalue’ of P(ã) can be recast as the claim that P(ã ) will differ less from the value of a perfect truth than does the ‘truthvalue’ of P(E). It is common practice to give perfect truths the value 1 and perfect falsehoods the value 0. The intermediate truth values will then correspond to numbers in the open interval (0,1 ), closeness to 1 signifying closeness to perfect truth. In particular, for the a and b of our example P(û) would have a higher value than P(S). Even if we accept the contention that (1.2) is not perfectly, but only approximately true, it still isn’t obvious how from this near-truth and the unqualified truth (1.1) we can pass by means of the familiar inference rules to a totally false conclusion. According to fuzzy logic this is possible

359

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because the rule of M (odus) P (onens), which we need to use many times to arrive at a definitely false conclusion P(n\ no longer preserves truth the way it does in 2-valued logic. In 2-valued logic the truth value of the conclusion one draws by means of M P is never smaller than the minimum of the truth values of the premises from which it is drawn. In many-valued logic this is no longer so. If the values of cp and (p -> i// are both, say, 1 - e (for some small e) then the value of \j/ will be 1 - 2e. So the result of applying MP may have a value lower than that of either of the premises; and even where each individual application of MP leads only to a small decrease, a large number of applications may lead from the almost true to the com­ pletely false. Thus we may conclude that the ‘proofs’ of P{k) from (T) and (2') aren't correct proofs; indeed they get ‘worse’ as k increases: the larger /c, the less we can say, on the basis of the proof and the given truth values of the premises, about lower bounds for the truthvalue of the conclusion we have reached.1 How plausible is this account? That depends on the acceptability of what we have just said about the conditional: that where cp and (p -» if/ both have the value 1 - e, the value of can be as low as 1 -(e.2). In the most common many-valued treatment of the conditional this claim is a consequence of the stipulation that if the truthvalues of (p and ^ are respectively a and ß then the value of a ß is 1 iff a ^ ß, and l-( a -/? ) otherwise. In my opinion this characterization has little to recommend itself. This is not so much because of inferiority to some alternative specifi­ cation which could be stated within the same mould, but because the many-valued approach towards the analysis of the familiar logical operators is in my view misconceived from the very start. I argued this point at some length in an earlier paper, [7], and will not repeat the argu­ ment in the very same form here. Instead I shall first give a brief sketch of the semantics for vague predicates which [7] develops in detail. We need a statement of this type of semantics anyway for our discussion of the second version of the paradox. I shall return to the question of multi­ valued truthtables after that. 4. Let P be any vague predicate. The vagueness of P manifests itself in there being objects of which it cannot be decided whether or not they belong to the extension of P. When I use the phrase ‘cannot be decided’

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I do not just mean that we lack the practical means of finding the answer to the question ‘is P true of this object?’ I mean that there simply isnt any answer: The predicate P does not partition the domain of objects into two sets - the set of objects of which it is true and the set of those of which it is false - but into three sets, the set of objects of which P is true (we will refer to this set as the positive extension of P), the set of objects of which P is false (the negative extension of P), and the set of objects a for which the question fcis P true of a ? has no answer (the truthvalue gap of P). Suppose now we wish to study the logic of vague predicates within the context of a simple formalized language-say a language of first order logic whoseonly non-logicalconstantsareindividualconstantscp c2,c3, ... and the 1-place predicate P, which is to stand for some vague propertyand suppose that we investigate the logic of this language along model theoretic lines.2 In order that our investigation shall reveal the logical features which are connected with the vagueness of P this vagueness must be in some way manifest in the semantic representation P receives in the models for the language on which our analysis will be based. These models must therefore at the very least draw the tripartite division which I just mentioned. For the moment this is the only aspect of vagueness that I will explicitly represent. Precisely how we do this is not important; but the exposition of what follows requires that we settle for some particular way of doing it. This is the way I have chosen: Instead of specifying just the extension of P (as is done in the models familiar from the standard model theory for the first order predicate calculus) the model provides two sets, the positive extension of P and the negative extension of P. These extensions are always mutually exclusive; but they need not jointly exhaust the universe of discourse. What remains if we subtract from the universe of discourse both the positive and the negative extension of P is called the truth value gap of P. This implies that for certain sentences P(ct) the model may leave the truthvalue undecided; this indeterminateness will in turn affect the complex sentences which contain such sentences P(c,), or corresponding open formulae, as parts. Precisely what the truthvalues of these complex sentences are is no easy question; it is from an attempt to give a satis­ factory answer to this question that the theory I am about to explain was born.3 Vagueness is typically experienced as a flaw. Often when we come upon an object a and our criteria for the predicate P do not give us a clear verdict as to whether a satisfies P or not; and when, moreover, it appears

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that this is not because of a lack of empirical information about a, or some other incidental inaptitude on our part at applying the criteria, then we feel a certain pressure to modify the criteria so that they will decide the case which confronts us as well as similar cases we might come upon subsequently. Generally we shall wish to extend or sharpen the criteria, i.e. to modify them in such a way that the cases decided by the old criteria are still decided in the same way. In this way we make the associated predicate more precise, its truthvalue gap has been narrowed. Just as what is determined by the criteria before we have modified them is conveyed by a certain model M, so we may assume that what is deter­ mined by the criteria after modification is represented by some new model M'. It seems unreasonable to suppose that truthvalue gaps generally are, or even that they could be, completely eliminated through a single modi­ fication. However, it will simplify the exposition at this stage if we assume that such modifications are possible and restrict attention to them. Where moreover we consider only a single predicate P the effect of such a ‘definitive’ modification will be a model which is essentially like a two­ valued model for classical predicate logic. Suppose once again that we are confronted with the object a and that the criteria for P do not decide whether a satisfies P. What are we to say in such a situation about the truth value of the sentence Vagueness Vagueness

This is not quite such a simple question as it may seem. One answer that might come to mind is ‘Well, P{â) is not fully true; nor is it fully false, so ~~\P{ã) isn’t fully true either. A disjunction is true only if at least one of its disjuncts is true. But in the present case neither disjunct is true. So the disjunction cannot be true either. In fact, the value of the disjunction could not simultaneously exceed those of each of its disjuncts.’ This is essentially what the fuzzy logician would say. It expresses a view which is, so far at any rate, coherent and beyond direct refutation. It is not the only possible answer, however. To see this, observe first that in a case where the criteria do decide whether a satisfies P, but where we are not able to see this, the truth value of P(ã) v ~\P(â) must be 1. Now it is simply a fact about language use that the situation where there really is no answer almost always looks, from the user’s point of view, just like one where there is an answer which the user cannot find. One reason for this is implicit in an important observation of Putnam ’s. Putnam has pointed out that we rely in our use of many technical and semi-technical terms on

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the knowledge of those more expert than ourselves to tell us, whenever a real need arises, what precisely falls under those terms and what does not. Indeed such reliance on the expertise of others is a practical necessity which none of the speakers of a reasonably expressive language such as our own can escape ; and it would be setting standards squarely at variance with what we normally consider linguistic competence to insist that only the entirely self-reliant are genuinely competent speakers of their language. Putnam makes this observation in the context of an argument that our own application criteria - those which he refers to as our ‘stereotypes’do not in general suffice to determine the extensions of the correspon­ ding terms. This is so, according to the realistic doctrines advocated by him and by Kripke, for those terms which purport to refer to natural kinds. The extensions of such terms are, on this theory, determined by the essential characteristics which demarcate the kind that comprises those individuals with reference to which the term has come into its use, from other kinds. These characteristics, however, may be hidden from the view of most, or even all of us, in which case it may require significant scientific progress to discover to which individuals the term in fact applies. Even if we accept this form of realism (as in fact I think we should) there remains the possibility that some terms which we take to refer to natural kind terms have truthvalue gaps. This could arise either because the field of entities to which the term has been used to refer turns out to be more or less evenly distributed over several natural kinds, or because the demarcation criteria do not yield the neat partition into equivalence classes which the theory must presuppose (some such complication seems to arise for instance, in connection with the classification of certain types of finches). It must be concluded from the possibility of such cases that we can never be quite sure that the extensions of natural kind terms are fully determined. It is nevertheless reasonable, given the enormous success enjoyed by the scientific enterprise, that we assume natural kind terms to be welldefined irrespective of our own ability to decide what falls within their range. The situation is different with regard to terms which do not stand for natural kinds but for, say, artifacts, professions, character traits, activities or any other universais which have no significant place in the fabric of lawlike regularities. Such predicates cannot transcend, through the power of natural law, the limits of what usage can conventionally impose. Of any such predicate it would therefore be quite unwarranted to

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assume that it has no truth value gap at all. Nonetheless, when we are faced with a question concerning the extension of the predicate which we are unable to settle, we can rarely if ever be sure that someone more knowledgeable would not know the answer, and so we are disposed towards treating even such predicates as if they were sharp. Consequently we are prepared to acknowledge as true, or false, all those sentences which will come out true, or false, no matter where the dividing line between positive and negative extension will exactly fall. P(â) v “~|P(ã) and P(ã)& ~]P{â) are two such sentences; the former will be true no matter where the division runs, and the latter always false. Let us for subsequent use state this principle for the determination of the truth values of com­ pound sentences containing vague predicates explicitly. We shall refer to it as ‘SV’ (for ‘Super Valuation') (SV) Speakers tend to feel committed to the assumption that the con­ cepts they use are well defined; this implies in particular that they will accept as true any statement which can be shown to be true on the hypothe­ sis that the predicates it contains do not have truthvalue gaps; and simi­ larly that they accept a statement as false if on this hypothesis it can be shown to be false; they will accept these things irrespective of whether the relevant predicates really have gaps, or whether they only appear to have gaps to them. Formally the commitment postulated in SV comes to this: Besides the model M, which specifies the extensions of P determined by the criteria as they are already accepted there are other models M' which represent the various ways in which the truth value gap could be eliminated and in which therefore P behaves as a sharply defined predicate. A sentence will count as true if it is true in all these models M' and false if it is false in all of them.4 5. We saw that the many-valued account of the paradox depended on the possibility of distinguishing between different degrees to which objects may fall short of belonging to the positive extension of a predicate. The theory we are in the process of presenting now involves this idea too, albeit in a slightly different form. Let us consider another, but equally notorious, version of the paradox, according to which every human male is bald, a conclusion we are invited to draw from the premises that (1") any man with one hair on his head is bald : and (2") for any n if any man

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with n hairs on his head is bald then so is any man with n + 1 hairs on his head. Suppose you are presented with two heads, sparsely crowned with hair, and that you are unable to decide of either whether or not it is bald; even so you may be positive that the first head is balder than the second. So that, if somehow you were to reach the conclusion that the second head is bald, then you would take this to imply that the first must be classified as bald too. On the other hand you might be led to conclude that the first head is bald without feeling compelled thereby to regard the second head as bald. This means that a modification of the criteria for ‘bald* would simply not be acceptable to you if it rendered the second head bald but not the first. Such a modification would be incompatible with your present understanding of the concept. Your understanding of the predicate P thus imposes certain restrictions on the ways in which the modified criteria may partition the truth value gaps. In formal terms, it limits the set S of models A4' which represent the semantically admissible partitions. This set S therefore embodies by virtue of excluding certain models which are possible from a strictly formal point of view, important semantic information about the use of P. Let P(n) mean e.g. ‘h grains of sand cannot make a heap’. Let us assume the positive extension of P to consist of the numbers {1,... ,/i0} and the negative extension of the numbers {nl ,nl + 1,...}. Mathematically speaking, there are all sorts of ways in which the truth value gap {nQ+ 1,..., nj — 1} can be distributed over the positive and negative extension. But only some of these distributions are semantically admissible: only those, to be explicit, which put, for some number n2 in the gap, all num­ bers {n0 + 1,... ,n2} into the positive, and the remainder into the negative extension. S will thus contain only those models Af ' in which the positive extension of P is some initial segment {1,... ,n2} of the positive integers and the negative extension contains all the others. It is easily seen that if a satisfies the predicate P ‘to a higher degree’ than b then the subset Sa of 5, consisting of those models Mf in which a belongs to the positive extension of P, is a proper superset of the set Sb of models M' in which b belongs to the positive extension of P. So we might hope to find the ‘degree of satisfying P ’ in some fashion reflected by the ‘size’ of the corresponding subset of S. In [7] I tried to make this idea explicit by introducing a measure function p on the appropriate field of subsets of S. With the help of such a function p we can, if we wish, define the “intermediate truthvalue” of any sentence, simple or complex, as the p-measure of the set of all M' in S in which the sentence is true. It is well

Vagueness

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THE PARADOX OF THE HEAP

worth remarking that if we think of these p-measures as truthvalues the connections &, v, ---> are 110t truth-functionaL For example, suppose P(il) has the value 1/2. Then so does I P(ii). It follows that P(ii) v I P(il) has the value 1 while P (il) v P (i1) has the value 1/2; that P (il) & I P (il) has the valueOwhile P(il) & P(il)has the value 1/2;and that P(ii) ---> IP(il)has the value 1/2, while P(il) ---> P(il) has the value 1. These examples show that the truthvalues of the components of a cOlnpound sentence are not by themselves sufficient to determine the truth value of the compound. Structural relationships between the components are relevant too - and this seems intuitively correcL It is difficult to give a fully satisfactory motivation for the function p. In general there is very little that favours the choice of one such function over anolher 5 . Concerning the predicate Po' however, our intuitions abaut degrees of satisfaction are somewhat c1earer than they are in most cases. Indeed it seems quite natural to assume that the degree to which a number 11 satisfies Po is 1 when 11 is one of the numbers 1, ... ,11 0 ; that it is owhen /l is ~ /ll; and that for 11 = 11 0 + 1, ... ,111' it diminishes stepwise. For the purpose of discussion I shan also assume, although this does not seem to follow c1early from our intuitions, that these stepwise reductions are an of the same magnitude e. This implies that e

1

= --/lI -

Let us put in addition, for each

11

in

{11 0 ,"" 11 1 -

11 0

I},

Sn = dI the set of an those M' in S in which the positive extension of Po includes {1, ... ,11}.

If it is agreed that the truthvalue of Po (il) should coincide with the degree to which 11 satisfies Po; and if moreover the function p is to yield these truthvalues in the way indicated, then we must cIearly have, for each 11 in {11 0 ' ... ,ni -l} p(Sn) = 1 - (e.(11 -

11 0 )}

This implies that for each such

11

p(Sn - Sn+l) =e.

If we assume in addition that the domain of pis simply the field generated by these sets (Sn - Sn+ 1)' then p is completely determined by the stipulations we have made. Let us assume for the sake of argument that the semantics of Po is correctly captured by the model M, the set Sand the function p as we

366

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H A N S KAMP

have just characterized them. Consider any sentence H A NHSA N KSA M KPA M P

If k < n0 then P(k 4- 1) is true in all members of S ; so P(k) -►P(k + 1) is also true in all members of S and so has p-value 1. If k ^ nx then P(k) is false in all members of S and so P{k) -► P(k + 1) is true in all members of S, and so again has p-value 1. In the remaining, and most important, case, where n0 ^ k < nl9P(k) is true in the members of Sk while P(k 4- 1) is true in the members of Sk+ x. So the only Mr in which P(k) is true while P(k 4- 1) is false are the members of (Sk —Sk+1). In all other models P(k) -» P(k 4- 1) is true. So the p-value of the sentence is 1 —e. Now consider the sentence H A N S K A MHPA N S K A M P

Let M' be any member of S. Suppose the positive extension of P in M' is the set {1,... ,h 2}, for some n2 in {n0, ... ynl - 1}. Then P(n2) -+ P{n2 -h 1) is false in M'. So (Vn)(P(n) -> P(n -f 1)) is false in M'. As this holds for arbitrary M' in S the sentence is false in all members of S, and so has p-value 0. This solves the paradox of the heap: the second premise of the argument is perfectly false. No wonder then that we can derive from it the plainly false conclusion (V/i)P(j7). Moreover, the solution explains why we could have been misled into supposing that (V «)(P(n)-^P(n + l)) was true, or nearly true: although it is correct to say that for any k the sentence P(k) -+ P(k + 1) is either fully true or nearly true, it is something else, and indeed wrong, to say that the sentence (Vn)(P(n) -* P(n + 1)) is nearly true. This solution to the paradox, found independently by Fine [4] and by Richmond Thomason [unpublished] struck me when I first learned of it as quite satisfactory. In particular it appears superior to the many-valued account to which I must now devote a few more words. The many-valued theory would attribute the same intermediate values to the sentences P{k) -> P(k 4- 1). But it diverges from the above account in attributing to (V/i)(P(n) P(n 4- 1)) the minimum of the values of the individual sentences P(k) -►P(k 4- 1). Why does the many valued theory do this? Well, in giving a semantic account of the quantifiers, many­ valued logic finds itself in the same predicament that arises from the attempt to find many-valued truthtables for the connectives. Indeed if one wants to preserve the equivalence between, say, universal quantifica­ tion and conjunction in the case of finite domains, and has already

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accepted the definition of & according to which the value of ((φ& ) is always the minimum of the values of φ and of t/^, then there is nothing for it but to accept the definition of the previous paragraph. Once we regard the second premise of the paradox as approximately true it becomes, as we have already seen, inevitable to block the inference by faulting the inference rules which are used in the proofs of the sentences P(k). The Fine- Thomason account shows, however that once we admit that the conditionals P(k)→ P(k + 1) are not absolutely true, there is no evident need to put the blame for the paradox on classical logic and to replace that logic by one which rests on very problematic foundations. 6.

But does it really solve the paradox? Gradually I have become persuaded that it does not. This change of heart has undoubtedly been caused at least in part by the scepticism with which my initially enthusiastic exposi­ tions of the supervaluation account often met. 1 do not know whether the analysis which I want to discuss in the remaining sections of this paper will satisfy the sceptics I have encountered. But at any rate it does dispose of what is perhaps the most telling criticism of the supervaluation account. The point is made by Sanford, who notes in [15] that the very same truth definition which confers upon (Vn)(P{n) -►P(n + 1)) the value 0 gives the value 1 to its contradictory (3n)(P(n) & ┐ P(n + 1)). But if this latter formula is indeed to be seen as the symbolization of the state­ ment that there is a number n such that n grains of sand can make a heap whereas n + 1 grains cannot, then this is a dubious result at best. For what number n could possibly satisfy this condition? The defense of the supervaluation theorist must be that it is immaterial that we cannot supply such a number; that there must be such a number, although we could never say which one it is, in as much as any resolution of the truthvalue gap of P will yield one. But this is quite unconvincing. The difficulty is not just that we cannot point to some particular number n for which the condition is satisfied. It is rather that for any n we might care to consider, the idea of its satisfying the conditions seems contradictory or incoherent. (In [3] Dummett makes an important point about what he calls ‘observational predicates’.6 By an observational predicate we understand a predicate P with the following property, to which I shall refer as E q u i­ valence of) O(bservationally) I indistinguishable entities): (EOI) Suppose the objects a and b are observationally indistinguishable

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H A N S KAMP

in the respects relevant to P ; then either a and b both satisfy P or else neither of them does. Dummett concluded that many observational predicates must be (i) vague; and (ii) incoherent. I will not retrace their argument in its most general context but consider the properties of observational predicates given some further assumptions, which are pertinent to the paradox which occupies us here. Consider an observational predicate P such that: (i) the object a definitely belongs to the extension of P; (ii) the object b definitely does not belong to the extension of P; (iii) There is a finite sequence of objects a = a x, then is true in a context c iff, provided the evaluation of q> in c is positive the evaluation of ^ in the context modified by this evaluation is positive too.

(13) is applicable, I maintain, not only to conditionals whose consequents contain context-dependent elements of the sort we just considered, but whenever the consequent involves context-sensitive material. It applies, in particular, to those conditionals whose consequent contains contextsensitive predicates such as P; in that case the modification effected by the evaluation of the antecedent should be understood in the sense in which we talked about modification in the preceding section, vz. through acceptance of the evaluated sentence as true. Thus for instance if k and k + 1 count as not discriminable in any way that is relevant to P, the conditional P(k) -+ P(k + 1) will be true in any normal context c. For acceptance of P(k) creates, as we argued already, a context in which, k + 1 being not relevantly distinguishable from the number k which has already been committed to the positive extension of P, P(k + 1) must be true as well. Note by the way that (13) reduces to the familiar truth condition for the conditional in all those cases where the evaluation of (p has no effect on that of i.e. where it makes no difference whether we evaluate ÿ in the modified context or in c itself. There is not a very great deal that I can say in support of the principle that allows (V/z)(P(n)-→P(n + 1)) to be false even though all conditionals P(k) -►P{k + ï) are true. The principle embodies the idea that a sentence such as (2') must be reckoned false whenever its acceptance would lead to inconsistency. Although I personally find this idea intuitively plausible I have no real argument to show that the principle is correct; and I am aware that this, more than any other, part of my proposal is likely to meet with scepticism. However, leaving the intrinsic merits of the principle momentarily aside, we can note at least that it will be operative only where acceptance may affect subsequent evaluation. To be explicit, suppose we stipulate that a sentence of the form (Vx)

and

[j/ to have a different truth value from that of i/* in any context c such that B(c') |-

Vagueness

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HANS KAMP

(i)

v0 =

(ii)

F;(P)

(iii)

F~(P) =

(iv)

Co = {CI(S') :D M S;;; S' S;;; So}, where CI(S') is the c10sure of S' under I- c; Coh o = {S' S;;; Co :('Vn)((P(n)ES' -+ n < n\)& ( -, P(ii)E S' -+ n2 < 11 )) } ;

(v) (vi)

{ 1,2,3, ... } ;

= {I, ... ,no}; {n 3 ,

Inc o = {S'

S;;;

... };

Co :(3n,n') (n' - n ~ r &P(ii}ES' &

-, P (ii)E S') }; (vii)

for all CEC o Bo(c)

= c;

(viii)

for all CEC o and r

S;;;

(ix)

for all positive integers n, n' n - on' iff

So mo(c' f) = CI(c ur);

In - 11' I ~ r.

Let Co be the set CI (DMo)(co is what one might ca)) a 'minimal' context of Mo' the context in which the minimum of commitments have been made as regards the extensions of P.) For this context we have (a)

(b)

+ I)]~I< = I for each positive integer k; [('Vn)(P(n) -+ P(n + l))]~~.co = 0 (as CI (co u {('V n)(P(n) -+ P(n + I}}} )

[P(k) -+ P(k

tl'JO,C{)

belongs to Inc o );

(c)

(d)

[P(k)]~~.co = I iff k ~ no + r; [I P(k) ]~~.co = 1 iff k ~ n 3 - r.

That (a) - (d) are true is some indication that our semantics captures at least part of the intuitions which we are attempting to formaJize.

15. As usual we can define validity by quantifying over those parameters on which the truth-value of a sentence depends - here, models and contexts. Before putting down adefinition along these lines we should remind ourselves, however, that the not ions 'true' and 'false', as we have specified them, can be expected to behave reasonably only with respect to contexts which are coherent. It is advisable therefore to take only such contexts into account. Thus we co me to:

Vagueness THE PARADOX OF THE HEAP

391

261

DEFINITION 4. A sentence ep of L o is a logical consequence of a set of sentences r, relative to f-, in symbols r F= ( f- )ep, iff for every contextsensitive model M relative to f-, and every cECoh M if for all l/I E r [l/I]f-M.< = 1 then [ ep ]f-M.< = 1. Within the present framework this is however not the only possible way of characterizing validity. It might be argued that the statement that 'cp can be validly inferred from r' is not really captured by Definition 4 but should be interpreted, rat her, as the claim that ep is true in any context in wh ich r /ws already beeil accepted. When formally implementing this interpretation we should again make sure to involve coherent contexts only. Thus we are Ied to the following definition, of a relation to which I shall refer as 'ep can validly inferred from r' and which I shall symbolize as F : DEFINITION 5. A sentence ep of L o call be validly inferred !rom a set of sentences r relative to f-, in symbols r F (f- ) ep, iff for every contextsensitive model M relative to f- and every CEC M if m(c,r)ECoh then = I 12 [ rn]f'f" M,m«.n . Having defined these relations we are bound to ask the familiar completeness questions: Are they axiomatizable; and what would an axiomatization look like? As the matter stands, however, these questions are indefinite. Before we can make any attempt at answering them we must first fix the extension of the relation f- ; the problem is: which extension should we choose? In a way this is a kind of completeness question too. For ideally fshould be chosen in such a way that it coincides with the semantic consequence relation which it generates. Let us suppose that it is F, rat her than F=, which captures the intuitive concept of valid inference. Then f-ought to be such that for all r, ep r f-- ep if and only if r F (f-- ) ep. We can recast this requirement as folIows. Both f-- and 1= denote subsets of the set ',ß(So)@ So·1= , moreover, is determined by f--. i.e. we can think ofF as a function which maps certain subsets of '.l3(So) @ So to such subsets. Our problem is to find a fixed point of this function. Let us note the part icular ways in which f- influences the extension of F. In the first place the extension of f-- may affect the size of the class of context-sensitive models. Inspection of the definition of that class shows that where f-- 1 and f-- 2 are two relations such that f-- 1 ~ f-- 2 the class of context-sensitive models

392

Vagueness

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HANS KAMP

relative to ~ 2 is included in the class of context-sensitive models relative to ~ I' In the second place ~ enters into the truth definition via the clauses which concern atOlnic formulae. Here it is clear that if ~ 1 ~ ~ 2 and M is any context-sensitive model relative to ~ 2 then for any sentence cp and CE Coh u if [ cp ]~I,c = {~ then [cp ]~/r = {~. It is easy to infer from these two observations that if ~ I ~ ~ 2 then F (~I) S; F (~2)' Thus F is a monotone function from some power set 'l3 (X) into 'l3 (X) and so has a fixed point. But what might these fixed points be like? This is an issue on which I have little to say at present. The problem appears to be technically non-trivial and to have many ramifications. I expect that there may be a great many fixed points and that the logics represented by most of them will be quite unreasonable. One way to secure fixed points which satisfy certain predetermined constraints is the following. Suppose C is a condition such that (i) whenever f- satisfies it, then so does F (~); (ii) whenever E is a set of relations satisfying ethen nE satisfies C ;(iii) whenever E is a ~ - chain of relations

U

satisfying ethen E satisfies C. Then we can first form the intersection ~ 0 of all relations which satisfy C. By (ii) this relation satisfies C. According to (i) F (~o) will satisfy C too and so ~ 0 S; F (~o). From this it follows that the union of the relations

F

(~ 0)' F

(F (~ 0))' F (F ( r= (~ 0)))' ...

is a fixed point. By assumption (iii) this union satisfies C. Among the conditions for which (i), (ii) and (iii) hold there are in particular those of the form R S; ~, where R is some fixed relation. Such conditions are of particular interest where R is what we might call an 'inference rule', i.e. the set of all pairs< r, cp> wh ich correspond to some inference schema. Rather than make the general notion of 'inference rule' I have in mind precise, let me just present the particular instances which I wish to consider here. The inference schema which we know as Modus Ponens can be given in the form; schema;

262

262

To this schema corresponds the inference rule R I consisting of all pairs < r,

where for some t/J both t/J and t/J ~ cp belong to 1. Similarly to the schema;

262

schema; schema;

262

393

Vagueness

263

THE PARADOX OF THE HEAP

corresponds the relation R 2 whose members are the pairs< r, cp such that for so me t/I, I(ep --+ t/I) is in r. In this fashion we can also form relations R 3 and R 4 corresponding to the schemata

>

ICP

I(t/I--+ ep)

ICP 4

Let R o =

UR j=

j

ICP ICP

I(t/I--+ I(t/I--+ C. ICP

cp' vj

and let CoO-) be the condition: R o S;;;

f-.

Then Co

I

satisfies all the assumptions we needed to make about a condition C to prove the existence of a fixed points satisfying C. Indeed it is obvious that any condition of the form R S;;; f- satisfies (ii) and (iii). To show that 1= (f-) satisfies Co provided f- does is a little more complicated. We lirst show that if f- satisfies Co' M is a context-sensitive model relative to f-. and cp a sentence of L o then for aB cECoh M

(I)

IfcpEB(c)then [CP]~.e= I,and

(2)

lf "I epEB(c) then [ep ]~.e = O.

The proof is by induction on the complexity of cp. Evidently (1) and (2) are satislied for atomic cp, (see the truth def.), and if they are satisfied for cp they are also satisfied for I cp. Suppose that cp has the form (Vv)t/I(v), that cECoh M and that cpEB(c). B(c)f- cp since f- satisfies Co' and so by Def. 1. (ix) and Def. 2. (iii) 111 (c. cp )ECoh M" Also, since R 4 S;;; f-, B(c) f- t/I (ä) for aB aE UM' So by induction hypothesis [t/I (ä)]~.e = 1 for each a EU M" It follows that [('l/v) t/I (v)]~.e = 1. Now suppose that IcpEB(c). Then cp, I ep E B(I11(C, { cp } )). From this it folIows, in virtue of Definition 2. ii, that I11(C, {ep} )EIncM" So [ep ]~.e = O. Next assume that cp has the form t/I-+ X. First let us suppose that epE B(c). Then t/I. t/I-+ XE B(111(C, { t/I } )). So. because R I S;;; f- XE B(I11(C, { t/I } )) and so on the strength of the induction hypothesis [X]~.m(c.{",}} = 1. So [ep]M.c= I. Finally suppose "lepEB(c). Then t/lEB(c) since R 2 S;;; f-. and so by induction hypothesis [t/I ]~.e = 1. Also B(I11(C, {t/I})) = B(c) by Def. 2. (iii) and Def. I. (ix). Since R 3 s;;;f-,IXEB(c)=B(I11(C,{t/I})). So by induction hypothesis [X]~l.m(c.(",}) = O. So [cp ]~.e = O. To establish that ~ (f- ) satisfies Co we must show that ~ (f- ) contains each ofthe relations R I - R 4 . Consider e.g. R I . To show that R I s;;; ~ we must show that for all M and cE C M' if. putting I11(C, {t/I, t/I --+ ep} ) = c'.c' E CohA{ then [ep ]~.c. = 1. But this now follows from the fact that since R I S;;; fB(c')f- wh ich I Iisted in section 15, in particular the schema of Modus Ponens. In this respect our theory agrees with the Fine- Thomason account. rather than with the many-valued treatment - according to which M P may yield concIusions whose values are below those of both of the corresponding premises. and thus is not a valid principle in the strict sense. The supervaluation account. however, manages to hold on to M P only at the cost of branding the conditionals Ptk) -> P(k + I), which play such a central part in the versions of the paradox we have here considered. as being less than perfectly true. The present theory indicates how one could avoid paying that price. But, as so many quest ions are still waiting to be answered, it is too early to assess whether this can be achieved without incurring even greater expense elsewhere. I cannot emphasize enough in this connection that virtually all metamathematical work in the area of which this paper has tried to produce a first, sadly incomplete, chart, is yet to be done. The scope for such work appears, from the modest efforts I have so far made in (his direction, almost unlimited; yet anyone who launches himself into the labyrinth of possibililies easily succumbs 10 doubts whether the work deserves to be

Vagueness THE

PARADOX

OF THE

401 HEAP

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done at all. For it appears less than likely that any reasonable logic could be found there. Understandable though such defeatism may be, it is, I believe, ulti­ mately without justification. This is because a ‘negative’ result - i.e. a proof that a semantic theory of the sort I have sketched cannot generate a reasonable lo g ic-is as important as a ‘positive’ result, which would consist in finding a logic that does not preposterously diverge from the canons of inference on which we rely in most of our scientific and nonscientific reasoning. To see the importance of the negative result, let us, once more, ask what precisely it is about the paradox that is so disturbing. In the preceding sections we found that the semantic principles which seem to lie at the root of the paradox can be combined in a formally consistent manner. However, the success of this formal exercise must be measured, in part at least, in terms of the distance between the concept of validity which it yields and the classical concept that we know to be adequate in connection with predicates of a less troublesome nature than those which have occupied us here. If we could establish that this distance cannot be reduced beyond a certain substantial minimum we would thereby confirm the depth of the problem which the paradox pinpoints. I should hasten to add that the predicament cannot be a question only of how small this distance can be made. Even if we found the classical consequence relation itself to be a fixed point, this by itself would do nothing to reduce the apparent contradictions to which a person can be reduced in the concrete instantiations of the paradox which we discussed earlier. Let us consider once again the experiment with the screen, sup­ posing that you, the subject of the experiment, have had the benefit of the present essay. You just qualified the /c-th square as green, and now you refuse to apply the predicate to the k 4- 1-st square. When I chal­ lenge this seemingly inchoate succession of verbal acts you retort that on acceptance of the statement ‘the /c-th square is green’ the context ceased to be coherent, so that there was no longer a sound basis for making judgements involving the predicate ‘green’. I would then ask: “But why did the context cease to be coherent just n o w T You might reply that contextual coherence is not the kind of concept that is ruled by the principle EOI. After all the notion of a ‘context’ is an artifact of linguistic theory, and so is, a fortiori, that of contextual coherence. And what could prevent us from stipulating that this predicate is not to function in the fashion of those terms whose peculiar features prompted its introduction into the machinery of linguistic description?

402

Vagueness

272

HANS

KAMP

I do not see any way of refuting this position ; but this is not to say that I am convinced of its being correct. Too much has been left unsaid about the role of context in linguistic explanation to see with any clarity whether the constraints on the concept of contextual coherence are compatible with the stipulation of which we just spoke. But let us assume for the sake of argument that your last reply is un­ assailable. Even then you could hardly claim to have justified your behaviour. For, although you may have shown that that behaviour is not formally inconsistent you have done nothing to show t hat it was correct. To show that you must give me a reason why the context stopped being coherent just now, and not a little earlier or a little later. Your only possible argument here is, it would seem, to contend that no such reason need be given - that the truth value gap of the context predicate ‘coherent’ can, like those of other predicates that are vague but not subject to EOl, be resolved in any one of a number of ways. How it is to resolved is a matter for decision. Sometimes such decisions must be made; and then somebody has to make them. This is what you just did. The justification of your decision lies, as it must in the case of any genuine decision, in the circumstance that a decision had to be made, rather than the superi­ ority of the option you chose over its possible alternatives. 20. This last discussion suggests that predicates subject to the conditions (i)-(iii) of section 6 need not be incoherent after all. This conclusion is clearly at variance with the pessimism which permeates the earlier mentioned papers of Dummett and Wright. Should we conclude that, at least as matters stand, that pessimism is not quite warranted? No, I think we should not. Dummett is concerned with two sorts of predicates, viz (a) the observational predicates which, according to certain doctrines, must figure in the simplest reports of what is observed; and (b) the predicates that would form the substance of strictly finitistic mathe­ matics if such mathematics were possible. And these predicates, even if it is true that they are not strictly incoherent in themselves, nevertheless behave in a manner that is incompatible with a fundamental aspect of mathematical and scientific language. It is the task of mathematics as well as that of science not only to reveal truths that are eternal, but also to develop languages in which these truths can be ‘eternally’ expressed expressed, that is, in a manner invariant under all change of context in

Vagueness THE P AR A D O X OF THE HEAP

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which these truths might be contemplated or communicated. For those who are committed to that enterprise our semantics does not provide a solution. For it vindicates these predicates by placing them within a linguistic framework of precisely the sort which it has been their acknow­ ledged task to expunge.

P O S T S C R IP T

On page 258 I revealed my concern to formulate the truth definition so that it would be evident that no sentence could come out as both true and false in one and the same coherent context. But as a matter of fact the truth definition of p. 259 does not guarantee this at all. It doesn’t because the clauses (v) and (vi) reduce the question of truth and falsehood of the conditional (p -* \p in c for certain cases to the question of truth or false­ hood of I/» in m(c, H,_1’), is false, so that the Tarskian biconditional as a whole is not true.7 The objection applies specifically to the half of (7) used in the argument, the right-to-left conditional. The

502

Vagueness

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TIMOTHY WILLIAMSON

inference from H. & Vagueness toVTrue(‘H. & is allowed to be valid, agueness Vagueness because truth-preserving, but the rule of Conditional Proof, according to which the validity of the inference from P and a possibly empty set X of other premises to Q implies the validity of the inference from X to P—>Q, is held to break down in the presence of vagueness, so that one could not pass to the corresponding conditional.8 Since (6) and (9) intuitionistically entail ->True(‘H. & ’), and all the argument needs is -i(H. & -|H |), the validity of the inference from the former to the latter must also be denied. The proposal involves the abandonment of intuitionistic logic as it is most commonly and most perspicuously formalized, as a system of natural deduction in the style of Gentzen. For then Conditional Proofjust is the intuitionistic introduction rule for For example, given the validity of the inference from ‘That is crimson’ to ‘That is red’, the claim that the logic of the corresponding fragment of English is intuitionistic would normally be understood as implying the derivability of the conditional ‘That is crimson —> that is red’. Putnam’s claim that the logic of a vague language is intuitionistic must therefore be taken in the following etiolated sense: all substitution instances of intuitionistic theorems in the language are valid, as is the rule of modus ponens. All the intuitionistic inferences made in this paper are derivable on that narrower basis. It may be observed in passing that the rejection of the inference from ->True(‘Q’) to -iQ undermines a celebrated argument of Putnam’s. For he has argued on the basis of semantic extemalism that our sentence ‘We are brains in vats’ is not true, and concluded that we are not brains in vats.9 The final step has the rejected form. Of course, one might attempt to supply extra premises in the presence of which that step would be valid. The challenge is to do so without also validating the step from -iTrue(‘H. & ->H. ,’) to — >(Hf & ->H. ,), and thereby rehabilitating the argument against the epistemic theory of truth. That challenge will not be pursued here. How are the supposed exceptions to bivalence in vague languages to be analysed? In reporting with apparent approval Wittgenstein’s remarks on truth at Philosophical Investigations §136, Putnam says: When we ourselves are willing to apply truth functions to a sentence note how Wittgenstein emphasizes in our language - we regard the sentence as true or false, as a genuine Satz.10

503

Vagueness

PUTNAM ON THE SORITES PARADOX

53

Call the conjunction of this thought with the claim that vague sentences are neither true nor false in borderline cases Propositional Failure. Given Propositional Failure, we should be unwilling to apply truth functions to vague sentences in borderline cases. One might suppose that Putnam endorses Propositional Failure. Unfortunately, Propositional Failure makes the retreat to intuitionistic logic pointless. If one considers all substitution instances of intuitionistic theorems in the vague language, they include many in which truth functions are applied to vague sentences in borderline cases, e.g., - H . ,, all of which are close to 1; on this view, (3) has a degree of truth close to 0. Intuitionism itself provides no reason to reject the Tarskian biconditionals. Any proof of True(‘Q’) amounts to a proof of Q, and vice versa ; given the intuitionistic semantics for , this observation amounts to a proof of True(‘Q’) Q. Putnam, of course, does not accept intuitionistic semantics; the present point is just that since the semantics validates both intuitionistic logic and the Tarskian biconditionals, the logic provides no explanation of their supposed failure. No principled basis for the rejection of the Tarskian biconditionals has been found to vindicate the choice of intuitionistic logic. Thus the objection to the new sorites paradox has not been sustained. If

Vagueness

PUTNAM ON THE SORITES PARADOX

505

55

intuitionistic reasoning is valid in a vague language, then (6), the epistemic account of truth, is false of that language. The truth of ‘j grains make a heap and i—1 grains do not make a heap’ does not imply that it would be justified under optimal conditions. Since this gap between truth and justification is exactly what defuses the sorites paradox, we have something like an epistemic account of vagueness, on which our ignorance of cut-off points does not imply their non­ existence. On such an account, classical logic is unproblematic for a vague language.12 Thus, as applied to vagueness, intuitionistic logic undermines itself: it is strong enough to require philosophical distinctions from the perspective of which it is unnecessarily weak. Someone deeply attached to the epistemic account of truth might seek to preserve it by abandoning intuitionistic logic in favour of a logic that was, at least in some respects, even weaker. Such attempts will not be considered here. Putnam himself is no longer so attached to an epistemic account of truth: [T]ruth is sometimes recognition-transcendent because what goes on in the world is sometimes beyond our power to recognize, even when it is not beyond our power to conceive.13 Why should that remark lose its validity when vagueness is in question? Why should the ceasing to be of a heap when one of i grains is removed not be something that goes on in the world beyond our power to recognize, although not beyond our power to conceive? Unfortunately, the exploration of that question lies beyond the scope of this paper.14 NOTES 1. See H.Putnam, ‘Sense, nonsense and the senses: An inquiry into the powers of the human mind’, Journal o f Philosophy 91 (1994): 445-517, at 503, 510-511. 2. For simplicity, the argument has been transposed to propositional logic from the predicate logic used in Putnam’s exposition. When no ambiguity results, quotation marks will be omitted or used as comer quotes. 3. See H. Putnam, ‘Vagueness and alternative logic’, Erkenntnis 19 (1983): 297-314, reprinted in his Realism and Reason: Philosophical Papers Volume 3 (Cambridge: Cambridge University Press), at 285-286 of the latter. It is suggested in S.P. Schwartz and W. Throop, ‘Intuitionism and vagueness’, Erkenntnis 34 (1991): 347-356, at 355, that the connection between vagueness and intuitionism was made by Max Black, in ‘Vagueness: an exercise in logical analysis’, Philosophy o f Science 4 (1937): 427-455, reprinted in his Language and Philosophy (Ithaca, N.Y., Cornell University Press, 1949), at 36-37 of the latter. However, Black treats intuitionistic logic merely as

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TIMOTHY WILLIAMSON

a non-classical logic more developed than his own non-intuitionistic non-classical logic of vagueness, not as a candidate logic of vagueness. 4. See S.L. Read and CJ.G. Wrignt, ‘Hairier than Putnam thought’, Analysis 45 (1985): 56-58. 5. See H. Putnam, ‘A quick Read is a wrong Wright’, Analysis 45 (1985): 203. For subsequent discussion of Putnam’s proposal see S.P. Schwartz, ‘Intuitionism and sorites\ Analysis 47 (1987): 179-183; G. Rea, ‘Degrees of truth versus intuitionism’, Analysis 49 (1989): 31-32; Schwartz and Throop, op. cit.; H. Putnam, ‘Replies and comments’, Erkenntnis 34 (1991): 401-424, at 413-414; C.J.G. Wright, ‘Is higher order vagueness coherentV , Analysis 52 (1992): 129-139, at 129; P. Mott, ‘On the intuitionistic solution of the sorites paradox’, Pacific Philosophical Quarterly 75 (1994): 133-150. 6. ‘Vagueness and alternative logic’, op. cit., at 280. 7. In ‘Sense, nonsense and the senses’, op. cit., at 511, Putnam proposes an example of a vague sentence for which a truth-value gap occurs and the Tarskian biconditional fails (‘The number of trees in Canada is even’). 8. In discussion, Putnam cited Richard Heck, ‘A note on the logic of (higher-order) vagueness’, Analysis 53 (1993): 201-208, on this point. Heck’s discussion concerns a determinacy operator, not a truth predicate. 9. See Reason, Truth and History (Cambridge: Cambridge University Press: 1981), at 7-8. 10. ‘Sense, nonsense and the senses’, op. cit., at 513. 11. The standard intuitionistic sentence operators are truth functions, because the usual truth tables remain correct for them; it is just that the list of cases in such tables can no longer be regarded as exhaustive. 12. An epistemic theory of vagueness is defended in Timothy Williamson, Vagueness (London: Routledge, 1994). In ‘Vagueness and alternative logic’, op. cit., at 274, Putnam describes as ‘heroic but misguided’ the related account of vagueness in Israel Scheffler, Beyond the Letter (London: Routledge and Kegan Paul, 1979). Such unargued dismissals of the epistemic view of vagueness are no longer adequate. 13. ‘Sense, nonsense and the senses’, op. cit., at 516. 14. An earlier version of this material was presented to a conference on the philosophy of Hilary Putnam at Utrecht in September 1994, at which Bob Hale, Charles Travis, Crispin Wright and, especially, Putnam himself made helpful comments.

Name Index Alston, W.P. 342 Aristotle 46, 47, 473, 474 Armstrong, D.M. 152 Austin, J.L. 43, 85 Beaufort, F. 266 Berkeley, G. 36, 136 Berlin, B. 165, 167, 168 Black, M. 221, 222, 223, 230, 233 Blackburn, S. 458-9 Brouwer, L.E.J. 474 Campbell, R. xvii, xviii, 85, 221-37 Cantor, G. 96, 99 Cargile, J. 71 Carnap, R. 369 Chastain, C. 375 Cresswell, M.J. 327, 378-9 passim Davidson, D. 42-4 passim , 89, 479-80 passim , 482 Donnellan, K. 86 Dummett, M. 90, 104, 173, 176, 183-4, 186, 272, 293, 246, 315, 367, 368, 402, 427, 428, 473, 474, 477 Edgington, D. xvii, xxi, 207-18, 307-14 Esenin-Volpin, A. 116 Eubulides 4, 16,31,71,271,272 Evans, G. 321-2 Evans, R. 159 Field, H. 15, 16 Fine, K. xxi, 55, 80-83 passim , 324, 336, 345, 366, 367, 383,421 Frege, G. 41,93, 99, 105, 109, 113, 114, 124, 136, 279 Gibbard, A. 469 Goodman, N. 115, 137-42 passim 186, 187, 369, 474, 485 Gupta, A. 432, 433 Halldén, S. 81, 87 Hardin, C.L. xv, 151-72, 445, 458 Hart, W.D. xviii, 253-76 Heck, R.G. xxi, 315-22

Hilbert, D. 91,95 Hintikka, J. 333 Hughes, G.E. 327 Humberstone, L. 334 Hume, D. 90, 140 Hyde, D. xxii, 341-7, 349-50 passim Judd, D.B. 164 Kamp, H. xvii, xxii, xxiii, 80, 355^-07, 460 Kant, I. 98, 474 Kay, P. 165, 167, 168 Kelly, K.L. 168 Körner, S. 87, 131 Kripke, S. xii, 46-9 passim , 86, 247, 362, 432, 435, 498 Land, E. 485 Leibniz, G.W. 98 Lewis, D.K. 15, 378, 413, 484, 485 Locke, J. 136 McGinn, C. 459 Marcus, R.B. 449 Morgan, C.G. 75 Newton, I. 257 Odegard, D. 85 Owen, G.E.L. 473, 474 Parikh, R. 483, 487, 489 Peacocke, C. xv, xvi, 173-93 Pelletier, F.J. 75 Peirce, C.S. 474 Phalén, A. 98 Plato 71, 473, 477 Platts, M. 176 Przelecki, M. 80 Putnam, H. xii, xxiv, xxv, 46-9 passim , 86, 247, 294, 361, 362, 473-90, 491, 492, 493, 495, 497-506 passim Quine, W.V. 5, 42-4 passim , 91, 104, 221, 418, 476-7 passim, 480-82 passim, 484 Raffman, D. xxiv 437-70

508

Vagueness

Ramsey, F.P. 197, 203, 204 Read, S. xxv, 491-3, 495, 499, 500 Rolf, B. xii, xiii, 71-102, 104, 344 Russell, B. 41, 78, 87, 93, 105, 109, 114, 141, 342 Ryle, G. 43, 85

Tarski, A. 97-8, 225, 433, 482, 483, 484, 499, 501 Thomason, R. 367 Tye, M. xxii

Sainsbury, R.M. xvi, xvii, xix, xx, 197-206, 279-94, 298-305 passim , 310-13 passim, 314, 341,343 Salmon, N. 413 Sanford, D.H. xii, xiii, 14, 61-70, 71, 75-8 passim, 79, 84, 88, 367, 400 Scheid, D. 12-13 passim Sellars, W. 242,485 Soames, S. 413 Sorenson, R.A. xiii, xiv, 103-5, 343, 344, 350-51

Vickers, D. 163

Unger, P. xii, xiii, 3-40, 53, 61-3 passim, 66, 71, 76, 77, 78, 85, 104, 105, 295, 478-9 passim

Stalnaker, R. 484

Wheeler, S.C. xii, xiii, 41-59, 61, 67-9 passim, 85-90 passim, 100, 104, 105, 478-9 passim Williamson, T. xviii, xxi, xxv, 239-51, 323-39, 497-506 Wittgenstein, L. 43, 85, 90, 112, 249, 502 Wright, C. xiv, xv, xvi, xx, xxi, xxv, 71, 90, 109^9, 173, 174, 176, 177, 282, 283, 285, 286-90 passim, 291, 295-305, 307-10 passim, 314, 315-19 passim, 320, 402, 437, 438,439,491-3,495,499, 500

Tappenden, J. xxiii, 409-35

Zadeh, L.A. 55

passim