Liars and Heaps: New Essays on Paradox

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Liars and Heaps New Essays on Paradox

edited by

JC Beall

CLARENDON PRESS
OXFORD

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Liars and Heaps

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3

Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sa˜o Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York ß the several contributors 2003 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2003 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 0-19-926480-5 ISBN 0-19-926481-3 (pbk.) 1 3 5 7 9 10 8 6 4 2 Typeset by Kolam Information Services Pvt. Ltd, Pondicherry, India Printed in Great Britain on acid-free paper by T.J. International Ltd., Padstow, Cornwall

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ACKNOWLEDGEMENTS

The papers in this collection grew out of a conference at Storrs, Connecticut. The conference was sponsored by the University of Connecticut Humanities Institute (UCHI), the MIT Department of Linguistics and Philosophy, the UConn Philosophy Department, and the University of Connecticut Research Foundation. Special thanks to Richard Brown (director of UCHI) for his support and guidance throughout the conference. Tim Elder (Head of Department, UConn Philosophy) was typically supportive throughout the conference, as was Shelly Burelle (UConn Philosophy); to them I give thanks. Michael Glanzberg not only contributed to this volume but was also a coorganizer of the conference; without his help neither the conference nor this volume would have existed. (Thanks, Michael.) Thanks also to Peter Momtchiloff (of Oxford University Press) for his efficient guidance in the publication of this volume; his patience, competence, and humor make the publication process enjoyable. Finally, I thank Katrina Higgins, who, despite being sick and tired of hearing about paradox, continues to be a source of great support. JC Beall Storrs, 2003

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CONTENTS

Notes on the Contributors

ix

Introduction JC Beall

1

PART I: SORITICAL PARADOXES

7

1. A Site for Sorites Graham Priest

9

2. Cut-Offs and their Neighbors Achille C. Varzi

24

3. Vagueness and Conversation Stewart Shapiro

39

4. Context, Vagueness, and the Sorites Rosanna Keefe

73

5. Vagueness: A Fifth Column Approach Crispin Wright

84

6. Semantic Accounts of Vagueness Richard G. Heck, Jr.

106

7. Higher-Order Vagueness for Partially Defined Predicates Scott Soames

128

8. Against Truth-Value Gaps Michael Glanzberg

151

9. Gap Principles, Penumbral Consequence, and Infinitely Higher-Order Vagueness 195 Delia Graff

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viii / Contents PART II: SEMANTIC PARADOXES

223

10. A Definite No-No Roy A. Sorensen

225

11. Reference and Paradox Keith Simmons

230

12. On the Singularity Theory of Denotation JC Beall

253

13. The Semantic Paradoxes and the Paradoxes of Vagueness Hartry Field

262

14. New Grounds for Naive Truth Theory S. Yablo

312

15. A Completeness Theorem for Unrestricted First-Order Languages Agustı´n Rayo and Timothy Williamson

331

16. Universal Universal Quantification Vann McGee

357

Index

365

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NOTES ON THE CONTRIBUTORS

JC Beall, Assistant Professor, Department of Philosophy, University of Connecticut. He is co-author (with Bas van Fraassen) of Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic (Oxford University Press, 2003) and co-author (with Greg Restall) of Logical Pluralism (Oxford University Press, forthcoming). Beall works mainly in philosophy of language and logic (especially issues of paraconsistency), and is currently working on a monograph on truth, falsity, and paradox (Oxford University Press, forthcoming); he has published in those areas in Analysis, the Australasian Journal of Philosophy, Mind, Nouˆs, Philosophia Mathematica, Philosophical Quarterly, and the Journal of Philosophical Logic. Beall likes to write and read short papers, and he also likes cats. Hartry Field, Professor, Department of Philosophy, New York University. His books include Science Without Numbers (Princeton University Press, 1980), Realism, Mathematics and Modality (Blackwell, 1989), and Truth and the Absence of Fact (Oxford University Press, 2001). Field’s main research areas include metaphysics, epistemology, philosophy of logic, philosophy of mathematics, philosophy of science, and philosophical logic; he has published on those topics in the Journal of Philosophy, Mind, Nouˆs, the Philosophical Review, and the Journal of Philosophical Logic. Michael Glanzberg, Associate Professor, Department of Philosophy, University of Toronto. Glanzberg works mainly in logic and philosophy of language; he has published articles on truth, paradox, the semantics– pragmatic boundary, and the nature of linguistic context in Mind and Language, Nouˆs, the Journal of Philosophical Logic, and Synthese. Delia Graff, Assistant Professor, Sage School of Philosophy, Cornell University. She co-edited (with Timothy Williamson) Vagueness (Ashgate, 2002). Graff works chiefly in the areas of philosophy of language, philosophical logic, epistemology, and metaphysics; she has published papers in those areas in journals including Mind, Philosophical Studies, and, among others, Philosophy and Phenomenological Research.

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x / Notes on the Contributors Richard G. Heck, Jr. Professor, Department of Philosophy, Harvard University. He edited Language, Thought, and Logic: Essays in Honor of Michael Dummett (Oxford University Press, 1997). Heck works mainly in philosophy of language, logic, and mathematics; he has published in those areas in journals including Analysis, the Journal of Philosophical Logic, Mind, Nouˆs, the Philosophical Review, the Philosophical Quarterly, Philosophers’ Imprint, Philosophia Mathematica, The Monist, and others. Rosanna Keefe, Lecturer, Department of Philosophy, University of Sheffield. She is the author of Theories of Vagueness (Cambridge University Press, 2000) and co-editor of Vagueness: A Reader (MIT Press, 1997). Keefe works primarily in philosophy of language and philosophical logic; she has published in those areas in Analysis, the Australasian Journal of Philosophy, Mind, and the Proceedings of the Aristotelian Society. Vann McGee, Professor, Department of Linguistics and Philosophy, MIT. He is the author of Truth, Vagueness, and Paradox (Hackett, 1991). McGee works chiefly in philosophy of language (truth, reference, conditionals), philosophy of mathematics, philosophical logic (provability logic), foundations of probability, and recursion theory; he has published in those areas in Analysis, the Journal of Philosophical Logic, the Journal of Philosophy, the Journal of Symbolic Logic, the Philosophical Review, Philosophical Studies, Philosophia Mathematica, Philosophical Issues, and, among other places, Philosophical Topics. Graham Priest, Boyce Gibson Professor of Philosophy, University of Melbourne, and Arche´ Professorial Fellow at the University of St Andrews. His published books include In Contradiction (Kluwer, 1987), Beyond the Limits of Thought (Cambridge University Press, 1995; Oxford University Press, 2002), and Introduction to Non-Classical Logic (Cambridge University Press, 2001). Priest works on many topics in metaphysics, the history of philosophy, and formal and philosophical logic, especially those topics connected with paraconsistency. He also works on anything else that captures his imagination. Priest has published in various journals, including Analysis, the Australasian Journal of Philosophy, Mind, the Journal of Philosophy, the Journal of Symbolic Logic, and the Journal of Philosophical Logic. Time not spent doing philosophy is most happily spent practising karatedo. Agustı´n Rayo, Arche´ Research Fellow at the University of St Andrews and, as of July 2004, Assistant Professor of Philosophy at the University of California, San Diego. Rayo works mainly in the philosophies of logic, mathematics, and

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Notes on the Contributors / xi language; he has published papers on those topics in various journals, including Analysis, Nouˆs, and the Journal of Symbolic Logic. Stewart Shapiro, O’Donnell Professor of Philosophy, Ohio State University, and Arche´ Professorial Fellow at the University of St Andrews. He is the author of Foundations Without Foundationalism: A Case for Second-Order Logic (Oxford University Press, 1991), Philosophy of Mathematics: Structure and Ontology (Oxford University Press, 1997), and Thinking about Mathematics: The Philosophy of Mathematics (Oxford University Press, 2000). Shapiro works chiefly in the philosophy of mathematics, logic, philosophy of logic, and philosophy of language; he has published in those areas in Mind, the Journal of Philosophy, the Journal of Philosophical Logic, and the Journal of Symbolic Logic. Shapiro is a workaholic who, in addition to work, likes to jog and listen to the Grateful Dead and the Incredible String Band. Keith Simmons, Professor, Department of Philosophy, University of North Carolina. He is the author of Universality and the Liar: An Essay on Truth and the Diagonal Argument (Cambridge University Press, 1993) and co-editor (with Simon Blackburn) of Truth (Oxford University Press, 2000). Simmons’s main areas of research are logic, philosophy of logic, and philosophy of language; his articles in those areas have appeared in various journals, including the Journal of Philosophical Logic, Nouˆs, and Philosophical Studies. He is currently working on Russell’s paradox and the paradoxes of definability, and on deflationism and truth. Scott Soames, Professor, Department of Philosophy, Princeton University. He is the author of Understanding Truth (Oxford University Press, 1999), Beyond Rigidity (Oxford University Press, 2002), and the two-volume Philosophical Analysis in the Twentieth Century (Princeton University Press, 2003). Soames’s main areas of research are philosophy of language and the history of twentieth-century analytic philosophy; he has published on those topics in various journals including the Journal of Philosophy, Linguistics and Philosophy, Notre Dame Journal of Formal Logic, Nouˆs, the Philosophical Review, Philosophy and Phenomenological Research, Philosophical Studies, and Philosophical Perspectives. Roy A. Sorensen, Professor, Department of Philosophy, Dartmouth College. He is the author of several books, most recently Vagueness and Contradiction (Oxford University Press, 2001) and A Brief History of the Paradox (Oxford University Press, 2003). Sorensen works mainly in philosophy of language and logic, and has published in those areas in various journals, including

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xii / Notes on the Contributors Analysis, the Australasian Journal of Philosophy, Mind, Nouˆs, the Philosophical Quarterly, and the Journal of Philosophical Logic. Achille C. Varzi, Associate Professor, Department of Philosophy, Columbia University. His books include An Essay in Universal Semantics (Kluwer, 1999), Parts and Places (with R. Casati; MIT Press, 1999), Theory and Problems of Logic (with J. Nolt and D. Rohatyn; McGraw-Hill, 1998), and Holes and Other Superficialities (with R. Casati; MIT Press, 1994). Varzi’s main research interests are in logic, metaphysics, and philosophy of language; his papers in those areas appear in several books and journals, including the Australasian Journal of Philosophy, the Journal of Philosophical Logic, Mind, Notre Dame Journal of Formal Logic, Nouˆs, Philosophy and Phenomenological Research, Philosophy, Philosophical Studies, and Philosophical Topics. Currently Varzi is an editor of the Journal of Philosophy and an advisory editor of The Monist and of Dialectica. Timothy Williamson, FBA, FRSE, Wykeham Chair of Logic, Oxford University. He is the author of Knowledge and its Limits (Oxford University Press, 2000), Vagueness (Routledge, 1994), and Identity and Discrimination (Blackwell, 1990), and co-editor (with Delia Graff) of Vagueness (Ashgate, 2002). Williamson’s chief research interests include philosophical logic, philosophy of language, metaphysics, epistemology, and philosophy of mind (especially externalism); he has published numerous articles in those areas in Analysis, the Journal of Philosophy, the Journal of Philosophical Logic, the Journal of Symbolic Logic, Mind, Notre Dame Journal of Formal Logic, Studia Logica, and other journals and collections. Crispin Wright, FBA, Wardlaw Professor, University of St Andrews, and Global Distinguished Professor at New York University. His books include Wittgenstein on the Foundations of Mathematics, Realism Meaning and Truth, Truth and Objectivity, Rails to Infinity, Saving the Differences, Frege’s Conception of Numbers as Objects, and (with Bob Hale) The Reason’s Proper Study. His numerous journal articles range over topics in epistemology, philosophy of language, metaphysics, philosophy of mind, metaethics, philosophy of mathematics, and philosophy of logic. He is founder and co-director of the AHRB research centre, Arche´, at the University of St Andrews. Stephen Yablo, Professor, Department of Linguistics and Philosophy, MIT. He works chiefly in metaphysics (which is MIT-speak for areas other than ethics). His articles in ‘metaphysics’ have appeared in Analysis, the Journal of Philosophy, the Journal of Philosophical Logic, Nouˆs, and the Philosophical Review.

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Introduction JC Beall

Philosophers work at the limits of language; that is where standard principles are challenged; that is where paradoxes are found. The task, at least with respect to natural language, is to learn from such limiting cases; the task is to figure out what the paradoxes teach us about the very language we speak. The papers in this volume address (for the most part) two sorts of paradox: semantic and soritical.1 Typical semantic paradoxes include truth-theoretic paradoxes (e.g. the Liar family, Curry’s), paradoxes of denotation (e.g. Berry’s, Richard’s, Ko¨nig’s), and, in some of its guises, Russell’s paradox (qua paradox about naive extensions).2 Typical soritical paradoxes (from the Greek sorites, meaning heap) involve a series of conditionals to the effect that an object y is 1 Two papers address ‘really unrestricted’ quantification, a topic connected not only to Russell’s paradox but also to ontological pursuits. 2 My own view is that, in the end, Russell discovered two paradoxes, one concerning the naive theory of extensions and the other concerning mathematical sets. Inasmuch as sets are constructed within and for mathematics, Russell’s set-theoretic paradox may be solved by mere stipulation— much as is done in ZF or the like. (Mathematical sets, by my lights, are constrained only by ‘whatever works’ for mathematics.) Extensions are a different matter; they are constrained by our ‘intuitions’ about language, much as truth, denotation, etc. are so constrained; mere stipulation will not resolve those paradoxes. That semanticists simply adopted the mathematicians’ sets to play the role of extensions does not mean that they can simply adopt the mathematicians’ stipulations concerning paradox; extensions, as said, are constrained in ways that (mathematical) sets are not. Whether, in the end, any of the mathematicians’ sets (ZF, etc.) will adequately serve the role of (semantic) extensions is an open question. While that question is relevant to some of the papers in this collection (notably Rayo and Williamson’s, Ch. 15), it is (regrettably) one that I must now set aside.

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2 / JC Beall (say) bald if x is bald and x differs from y by only one hair. Example (where m is a single millisecond): (1) Any 1-year-old girl is a child. (2) If x is a child at time t, then x is a child at t þ m. Pick your favorite 1-year-old girl, who by (1) is a child. By premise (2), your chosen girl will remain a child even after seventy (or more) years. While each of (1) and (2) appears to be true, the conclusion—that Granny is a child— appears to be false (or, at least, untrue). Each of the mentioned sorts of paradox has been around for a long time, and by now each is fairly familiar (and, so, I skip further review). But what do such paradoxes teach us about English (or whatever natural language we may happen to speak)? To that question there is no settled answer, but there are many answers; some of them are explored in this collection.

Common Themes Contextualism. A lesson that many philosophers have drawn from the Sorites paradox is that vague predicates—predicates that give rise to soritical paradox—are contextually sensitive; they change extension from context to context. Many of the philosophers represented in this volume discuss such ‘contextualism’ with respect to the Sorites, and some discuss it with respect to semantic paradox. Unified solution. Philosophers love simplicity and uniformity. While few philosophers have advanced a unified solution to both semantic and soritical paradoxes, the virtues of such a framework (whatever it might look like) are evident—treating both sorts of paradox as instances of the same basic phenomenon (whatever that might be). One avenue towards such unification may be contextualism: if (as a growing number of philosophers think) the semantic paradoxes teach us that ‘is true’, ‘denotes’, etc. are contextually sensitive, then one might combine such a view with a contextualist response to the Sorites paradox.3 Contextualism, however, is not the only route towards unification; perhaps, as Field (Chapter 13 in this volume) suggests, the two sorts of paradox are instances of an indeterminacy that calls for a non-classical but detachable conditional. 3 In his Understanding Truth (Oxford: Oxford University Press, 1999) Scott Soames advocates such a position, at least with respect to the Liar and sorites.

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Introduction / 3

The Contributions The papers, for the most part, come in pairs, with one of the papers either responding to or extending the topic of the other.4 Each of the authors does a nice job of clearly advancing her/his own thesis and arguments, and so there is little point in repeating those theses or arguments here. Instead, I will simply indicate the target topic of each (pair of ) paper(s).5

Soritical Paradox Graham Priest (Chapter 1) advances a novel (fuzzy) approach to identity, taking the lesson of the Sorites to require as much. In response Achille Varzi (Chapter 2) suggests not only that vague phenomena may require psychological (cognitive scientific) explanation, as opposed to philosophicalcum-logical, but also that a supervaluational approach to the semantics of vagueness is at least as promising as Priest’s novel fuzzy approach. Stewart Shapiro (Chapter 3) extends the work of Diana Raffman (see Shapiro’s citations) by advancing a pragmatic, contextual approach to vagueness, discussing ways in which the pragmatic facts yield semantic facts. Rosanna Keefe (Chapter 4) not only raises worries about pragmatic approaches in general; she also questions the extent to which Shapiro’s account can block the Sorites. Crispin Wright (Chapter 5) propounds a novel account of vagueness, one that has been overlooked by the (often thought to be exhaustive) trio of standard conceptions—semantic, in rebus, and epistemic. Wright sketches a broadly epistemic account of borderline cases while advancing agnosticism with respect to vague instances of bivalence (bivalence applied to vague 4 One exception is Roy Sorensen’s paper (Ch. 10), which, for reasons that Sorensen notes, neither responds to nor extends the topic discussed by Delia Graff. 5 NB: The division into ‘soritical’ and ‘semantic’ paradoxes is slightly misleading on two fronts. First front: some of the papers under ‘semantic’ (e.g. Field’s, Yablo’s, Glanzberg’s; Chs. 13, 14, 8) bear on soritical paradox as much as they do on semantic. That said, the term ‘semantic’ is appropriate; it marks a point of emphasis in the given set of papers. Second front: Rayo and Williamson’s paper, and also McGee’s (Chs. 15, 16), are connected more closely to what Ramsey would have classified as ‘logical paradox’ (at least inasmuch as Russell’s paradox is relevant to the issue of absolutely unrestricted quantification) than to semantic paradox—at least on Ramsey’s division. For present purposes, it may be best to ignore Ramsey’s ‘distinction’, which is controversial anyway.

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4 / JC Beall predicates). Richard Heck (Chapter 6) does not directly challenge Wright’s novel position; instead, he defends one of the standard conceptions of vagueness (namely, the semantic conception) against Wright’s arguments. Scott Soames (Chapter 7) expands on his earlier (partial predicates) approach to vague terms; specifically, he tackles the problem of ‘higher-order vagueness’. Michael Glanzberg (Chapter 8) responds to Soames’s position and others like it by challenging ‘gappy’ approaches to paradox, in general. Delia Graff (Chapter 9) advances novel and, if sound, wide-ranging arguments against (higher-order) ‘gap principles’. In turn, she challenges the extent to which supervaluationists can endorse classical logic.

Semantic/Logical Roy Sorensen (Chapter 10) turns the discussion towards semantic paradox; he discusses a novel paradox within a family of so-called (by Sorensen) no-no paradoxes. Keith Simmons (Chapter 11) applies his (contextual) ‘singularity theory’ to the paradoxes of denotation; he also presents a novel (infinite) paradox of denotation, and argues that his singularity theory handles it. JC Beall (Chapter 12) challenges Simmons’s position; he argues that Simmons’s position is either inconsistent or unwarranted. Hartry Field (Chapter 13) advances a novel response to the Liar paradox, one that seems to generalize across all other semantic paradoxes and also to the soritical paradoxes. Stephen Yablo (Chapter 14) raises some questions about Field’s approach and opens up novel variations. Agustı´n Rayo and Timothy Williamson (Chapter 15) present a novel approach to (absolutely) unrestricted quantification, one that avoids Russell-like paradox; they present a completeness theorem for their system. Vann McGee (Chapter 16) discusses not only the logical significance but also the philosophical import of Rayo and Williamson’s framework, especially with respect to robust ontological pursuits.

Requisite Background The papers in this collection cover a wide range of issues and presuppose an equally wide range of background. There is no easy way of providing this, at

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Introduction / 5 least not without making this introduction the longest ‘contribution’ in the collection. Fortunately, there are a few introductory sources to which one can go for at least much of the background:6 . On vagueness and the Sorites: . The introductory chapter in Rosanna Keefe and Peter Smith (eds.), Vagueness: A Reader (Cambridge, Mass.: MIT, 1987). . R. M. Sainsbury and Timothy Williamson, ‘Sorites’, in R. Hale and C. Wright (eds.), A Companion to the Philosophy of Language (Oxford: Blackwell, 1997). . On semantic paradoxes: . Mark Sainsbury, Paradoxes (Cambridge: Cambridge University Press, 1995). . Michael Clark, Paradoxes from A to Z (New York: Routledge, 2002). . For common non-classical logics: . JC Beall and Bas van Fraassen, Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic (Oxford: Oxford University Press, 2003). . Graham Priest, An Introduction to Non-classical Logic (Cambridge: Cambridge University Press, 2001). While the foregoing sources will not provide everything that one might want or need, they provide a start. 6 The exception is Rayo and Williamson’s paper (Ch. 15), which presupposes more technical background than the others. Perhaps the most useful approach to that paper (at least for those lacking the requisite background) is first to read Vann McGee’s contribution (Ch. 16) and some of the cited material in each of the two given lists of references.

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I SORITICAL PARADOXES

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1 A Site for Sorites Graham Priest

1. Introduction The Sorites is a very hard paradox; I think it is harder than the paradoxes of self-reference. A measure of this is the fact that the paradoxes of self-reference may be solved by abandoning the law of non-contradiction. This has seemed to some a very drastic solution. But even this cannot help solve the Sorites: Sorites arguments lead not just to contradictions that can be isolated, but to wholesale contradiction. Using Sorites arguments one can prove nearly everything. For example, we can prove that you are scrambled egg as follows. Let b0 be you, and suppose that there are n molecules in your body; let b0 , b1 , ..., bn be a sequence of objects each of which is obtained from its predecessor by replacing one molecule of you with a molecule of scrambled egg, so that bn is all scrambled egg. Let bi be the statement that you are bi . Then clearly b0 , and for any i, bi ! biþ1 . Hence by n applications of modus ponens bn : you are scrambled egg. I used to think that solving the Sorites paradox was an issue of logic; specifically, that one needed to give an account of the conditional that Versions of this paper were given at a meeting of the Australasian Association of Philosophy held at the University of Canterbury, Christchurch, New Zealand, in July 2002, at the conference ‘Liars and Heaps’ held at the University of Connecticut, Oct. 2002, and at seminars at the University of St Andrews, Dec. 2002. I am grateful to the many members of the audiences on those occasions for many helpful thoughts and criticisms. The commentator on the second of these occasions was Achille Varzi, whose reply also occurs in this volume. In fairness to Achille, I have not modified the paper in response to his interesting comments. It is not easy to hit a moving target; further discussion can take place elsewhere.

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10 / Graham Priest showed why the argument fails, though we are ineluctably drawn to it. But, this, I now think, is false. The Sorites phenomenon has nothing, at root, to do with conditionals, validity, or any other apparatus of the logician. The phenomenon arises at a much more fundamental level, one prior to the engagement of any logical paraphernalia. It arises simply because we are forced to recognize the existence of cut-off points where both common sense and philosophical intuition scream that there are none. Thus, the only thing left for a solution to do is to explain why we find the existence of a cut-off so counter-intuitive. This is the site at which a solution to the Sorites needs to be sought. The paper has three parts. The first explains why, I take it, we are stuck with cut-offs. The second formulates the beginnings of an explanation for this fact. The third discusses some issues that need to be addressed for this explanation to become a fully adequate solution to the Sorites paradox.

2. The Existence of Cut-Offs 2.1 The Forced March Sorites The existence of a cut-off of some kind is forced upon us by a version of the Sorites sometimes called the ‘forced march Sorites’, which is, in fact, very close to the original version of the argument.1 We can formulate this as follows. Consider the Sorites argument above. Let qi be the question ‘Is it the case that bi ?’ If asked this question, there is some appropriate range of answers. What these are exactly does not matter. They might be ‘yes’, ‘no’, ‘I don’t know’, ‘yes, probably’, ‘er . . . ’, or anything else. All that we need to assume is that an appropriate answer is justified by the objective state of affairs; specifically, by the nature of bi .2 Now, suppose I ask you the sequence of questions: q0 , q1 , ... Given any question, there may be more than one appropriate answer. For example, you might say ‘yes’; you might same ‘same answer as last time’ (having said ‘yes’ last time). All I insist is that 1 The term ‘forced march Sorites’ was a coined (as far as I know) by Horgan (1994, sect. 4). The version I give here is slightly different from, and, it seems to me, tougher than, the version he gives there. His version is a metalinguistic identity Sorites. In due course, it will become clear how to deal with this. For the original formulation of the argument, see Williamson (1994, ch. 1) and Keefe and Smith (1997, ch. 2). 2 The justification here is semantic, not epistemic. The answerer is personified simply to make the situation graphic.

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A Site for Sorites / 11 once you answer in a certain way you stick to that until that answer is no longer appropriate. Suppose that in answer to the question q0 , you answer a. This may also be justified in answer to q1 , q2 , and so on. But there must come a first i where this is no longer the case, or it would be justified in answer to qn , which it is not. Thus, for some i, bi
1 justifies this answer; bi does not. The objective situation therefore changes between bi
1 and bi in such a way. And this, of course, is where intuition rebels. How can a single molecule make a difference? And by changing the example, we can make the difference between two successive states in a Sorites progression as small as one pleases. The difference may therefore fall below anything that is cognitively accessible to us; but vague predicates just don’t seem to work like this. Of course, how we should theorize this cut-off is another matter. Different theorists do this in different ways. A cut-off may be theorized as a change from truth to falsity; a change from truth to neither truth nor falsity, or to both truth and falsity; a change from being 100 per cent true to less than 100 per cent true; a change from maximal degree of assertibility to less than maximal degree; and so on. But never mind the details. What the forced march Sorites demonstrates is that any solution must face the existence of a cut-off. It cannot disappear it. All that is left for a solution to do is to theorize the nature of the cut-off, and to explain why we find its existence so counterintuitive. This is the only form that a solution to the Sorites can take. Let us consider a couple of replies. Here is one. The existence of a cut-off point seems odd because of the apparently arbitrary nature of its location. Suppose that the correct answer in the forced march Sorites changed at every question. The arbitrariness, and so the oddness, would then disappear. How could this be? One possibility is that to answer the question I simply show you the object at issue—which is changing from stage to stage. Another is that an answer is of the form ‘It is true to degree r’—as in fuzzy logic—where r is a different real number every time. The response to the first suggestion is fairly obvious. Let us not quibble about whether responding by showing is linguistic. If it is, the language is not ours. The problem into which the Sorites leads us is posed by the use of our language with its vague predicates, questions, and answers. We want a solution that applies to that language. One response to the second suggestion is similar. Even though, in this, the response to the question is by saying, not showing, a language with an uncountably infinite number of replies is not ours. But I think that there are greater problems with this response. However

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12 / Graham Priest one conceptualizes degrees of truth, there are Sorites progressions where truth value does not change all the time. Thus, even if you were changed by replacing one molecule of your body with a molecule of scrambled egg, you would still be as you as you could be. You change more than that every morning after breakfast. Similarly, dying takes time, and so is a vague notion. But when your ashes are scattered to the four winds—and thereafter, if not before—you are as dead as dead can be. And if a correct answer to the relevant question does not change at every point, we face a counter-intuitive cut-off. A second reply is to the effect that the answerer may ‘‘refuse to play the game’’. Of course, if they do this for subjective reasons, such as the desire to be obstreperous, this is beside the point. They might, however, do so for a principled reason, namely that the rules of the ‘‘game’’ are impossible to comply with. They lead the answerer, at some point, into a situation where they cannot conform. Now, it would certainly appear that it is possible to comply with the rules at the start: the first few answers present no problems. But then we may simply ask them to play the game as long as it is possible. If the answer changes before this, the point is made. If, however, they stop at some point before this, it must be because the situation is such as to require them both to give and not give the same answer as before. This was not the case at the question before, so the semantic situation has changed at this point. The only other possibility is for the answerer to say that the game in unplayable right at the start. But this can only be because there is no appropriate answer they can give even in the first case—and presumably, therefore, in all subsequent cases. This is not only implausible; it means that even in the most determinate case there is no answer that can be given. We are led to complete and unacceptable semantic nihilism.

2.2 Epistemicism and Contextualism Of the solutions to the Sorites paradox currently on the market, very few address the counter-intuitiveness of the cut-off point explicitly. Perhaps the one that might be thought to do so most naturally is the epistemicism of Sorensen and Williamson.3 Sorensen and Williamson subscribe to classical logic. In effect, then, they take all predicates to be semantically precise ones. In a sense, there are no vague predicates. In a soritical progression there is therefore a precise cut-off where the sentences turn from true to false or vice versa. 3 See Sorensen (1988, esp. 189–216) and Williamson (1994, esp. chs. 7, 8).

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A Site for Sorites / 13 The distinctive feature of epistemicism is an attempt to explain why we find the existence of the cut-off point counter-intuitive in terms of features of our knowledge. We do not know where it is—indeed, there are reasons why we cannot know where it is: the location of the cut-off is in principle unknowable. This is why we find its existence counter-intuitive. It should be noted that, though Sorensen and Williamson deploy epistemicism in defence of classical semantics, it could be deployed equally in defence of any of the standard semantics for vagueness. As I have noted, they all entail the existence of cut-off points, and all, therefore, are in need of an explanation of why this is counter-intuitive. Epistemic considerations of the kind in question can be invoked. Williamson’s explanation of the in-principle unknowability of the cut-off point can be put in various different ways, but at its heart it depends on the fact that knowledge supervenes on an evidential basis. If two situations are effectively the same in the evidence that they provide (are relevantly similar) then I cannot know something about one but not about the other. In particular, then, if I know something about one such situation, that thing must be true in the other situation too—or, being false, I would not know it. Williams calls this the ‘margin of error principle’. In particular, suppose that b0 , b1 , ... , bn is a soritical sequence of statements. Suppose that bi and all prior members are true, that biþ1 and all subsequent members are false, and that one knows where the cut-off is, i.e. one knows that bi is true and that biþ1 is not. Since I know that bi is true, by the margin of error principle, biþ1 must be true too—which, ex hypothesi, it is not. Though epistemicism is in the right ball-park for an explanation, it does not stand up well to the cold light of inspection. A major worry is that the very phenomenon that explains why we cannot know where the cut-off point is undercuts its very existence. The meanings of vague predicates are not determined by some omniscient being in some logically perfect way. Vague predicates are part of our language. As a result, their meanings must answer in the last instance to the use that we make of them. It is therefore difficult to see how there could be a semantic cut-off at a point that is in principle cognitively inaccessible to us. Of course, cut-off points for crisp predicates may be inaccessible to us too in a certain sense. Thus, if an electron accelerates uniformly from rest to some velocity, there must be a precise instant at which it has half that velocity. Even if we know the terminal velocity in question, we may not be able to determine that instant, due to the limitations of our measuring instruments, etc. But the in-principle unknowability of the

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14 / Graham Priest cut-off points for vague predicates is quite different from this. In the case of the electron we know exactly what it is about the world that determines where the cut-off point resides, and why that particular fact settles the matter. In the case of the vague predicate, we have neither of these things. To suppose that such exists would appear to be a form of semantic mysticism.4 Worse, and crucially in the present context, it is doubtful, in the end, that epistemicism can explain why we find the existence of cut-off points so counter-intuitive. There are many things that we cannot know and whose existence we do not find puzzling in the same way at all. For example, there is a well-known model of the physical cosmos according to which the universe goes through alternating periods of expansion and contraction. In particular, the singularity at the big bang was just the end of the last period of contraction and the beginning of the current period of expansion. Suppose this is right. Then there must be many facts about what happened in the phase of the universe prior to the big bang—for example, whether there was sentient life. Yet all information about this period has been wiped out for us—lost in the epistemic black hole that is the big bang. Yet we do not find the existence of determinate facts before the big bang counter-intuitive. Indeed, we seem to have no problem imagining there to be such things, though they are and ever will be cognitively inaccessible to us. To explain why we cannot know the existence of something does not, therefore, explain why we find its existence counter-intuitive. Another account of vagueness that might be thought to lend itself to explaining why we find the existence of precise cut-off points counterintuitive is contextualism, of the kind proposed by Graff (2000). For present purposes, the pertinent features of the view are as follows. First, vague predicates are contextually dependent. This is plausible: ‘tall’ for a basketball player and ‘tall’ for a pygmy certainly have different extensions. Second: what is psychologically salient is part of the context. This seems plausible too. When we point in a general direction, and say ‘that’, we expect the hearer to take the referent of the demonstrative to be whatever is salient in that direction. Third, and most importantly here, is what Graff calls the similarity constraint: if, given a fixed context, two salient objects are relevantly similar with respect to a predicate, then it applies to both or neither. This, too, is not implausible: it is a localized version of the natural idea that vague predicates are tolerant with respect to small changes. Given these ideas, we might try to explain why we find the existence of a cut-off point counter-intuitive as 4 As Crispin Wright puts it his detailed critique of epistemicism (1995). See also Horgan (1994, sect. 5).

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A Site for Sorites / 15 follows. Whenever we look to find the relevant cut-off at a certain point, this thrusts those objects on either side of the point into salience. Thus, because of the similarity constraint, the cut-off point relative to that context is not there. In other words, wherever we look for the cut-off point it is not there. So we come to believe that there isn’t one. Again, one may have worries about this sort of account of vagueness quite generally. Can’t I have a very short Sorites, say with a handful of colour strips, where it is clear that the endpoints have different colours, and yet where every strip in the context is salient? After all, I may not be able to get a hundred strips into mental focus at the same time, but I would certainly seem to be able to get five or six. But setting this aside, it is not clear that contextualism really does explain why we find the existence of a cut-off counter-intuitive. Suppose that we look to find the cut-off at some point or other. The similarity constraint explains why it is not there. With respect to that context it must therefore be elsewhere—outside the area of salience. But the thought that there is a sharp cut-off point somewhere else is still as puzzling as before. The tolerance of vague predicates seems, after all, to be a general phenomenon, not simply localized to the area of salience. What could make that the cut-off point is therefore just as puzzling as before. At least as applied in the most obvious ways, then, neither epistemicism nor contextualism provides an explanation as to why we find the existence of a cut-off counter-intuitive. What could provide an explanation? Conceivably there could be many putative explanations. Maybe epistemicism or contextualism can be deployed in some other way. Maybe an appropriate explanation could be purely psychological: some deep psychological mechanism produces the illusion of cut-off freedom. And if we are lucky enough to come up with a number of viable explanations, we will have to determine which is the best. But at the moment, this is a non-issue; for presently we have none. In the next part of the paper I want to sketch one possible explanation. This is neither epistemic nor purely psychological, but logical.

3. An Explanation 3.1 Vague Identity The explanation is parasitic on fuzzy logic—though fuzzy logic is not, in itself, a solution to the problem, as I have already noted. The logic is harnessed to allow a certain explanation of the counter-intuitiveness of a

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16 / Graham Priest precise cut-off point. For the sake of definiteness, let us suppose that the logic is determined by the continuum-valued semantics of Łukasiewicz—though other fuzzy logics could also be employed. The details of the logic are well enough known for me not to have to repeat them here.5 I remind you of just one fact. For something to be acceptable, it does not have to have unit truthvalue. A value high enough will do. In practice, the context will determine some value 0 < e < 1, such that a value greater than or equal to e is sufficient to make a sentence acceptable. The inferences that preserve acceptability whatever value of e is chosen are precisely those whose conclusions are always at least as true as (the conjunction of) their premises.6 Now, let us start by recalling that there are Sorites arguments that depend not on modus ponens but on the substitutivity of identicals.7 Consider a soritical sequence of colour patches such that each is phenomenologically indistinguishable from its immediate neighbours, which begins with a pure shade of red and ends with a pure shade of blue. For the sake of definiteness, let us suppose that there are 100 such patches. Let ci be the colour of the ith patch. Clearly, c1 ¼ c1 , and for all 1  i < 100, ci ¼ ciþ1 . By a sequence of substitutions c1 ¼ c100 . This argument shows that identity itself must be a fuzzy predicate; identity statements must therefore have degrees of truth. An appropriate semantics for identity is as follows.8 The domain of quantification is furnished with a normalized metric. This is a map, d, from pairs of objects to real numbers satisfying the conditions: 0  d(x, y)  1 d(x, x) ¼ 0 d(x, y) ¼ d( y, x) d(x, y)  d(x, z) þ d(z, y). An identity statement, a ¼ b, has degree of truth 1
d(a, b), where a is the denotation of a, etc. The members of the domain, as furnished with the appropriate notion of identity, can be thought of as fuzzy objects, objects whose identity comes by degrees. 5 See e.g. Priest (2001, ch. 11). 6 See Priest (2001: 11.4.10). 7 See Priest (1991). 8 See Priest (1998). Note that that paper reverses the usual conventions concerning 1 and 0 as degrees of truth.

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A Site for Sorites / 17 It is not difficult to show that the transitivity of identity is not (globally) valid in these semantics. Indeed, a model of the colour Sorites is provided by the interpretation where: d(ci , cj ) ¼

ji
jj 100 .

The truth-values of c1 ¼ c2 and c2 ¼ c3 are 0.99; the truth-value of c1 ¼ c3 is 0.98. Hence the inference c1 ¼ c2 , c2 ¼ c3 ‘ c1 ¼ c3 is invalid. Although the transitivity of identity is invalid, there is a weaker notion of validity that it satisfies: local validity. Loosely, an inference is locally valid if its conclusion will be acceptable provided that its premises are true enough. A crucial property of locally valid inferences is that a chain of locally valid inferences is itself locally valid, though the degree of truth required by the premises becomes higher and higher the longer the chain is. Consequently, locally valid inference can be used a few times to make conclusions acceptable, but prolonged use is liable to end in something unacceptable. (Details of how to make these ideas formally precise can be found in Priest 1998.) Not only is the transitivity of identity invalid but locally valid; so is modus ponens.

3.2 Fuzzy Truth-Values With this background, we can now turn to the required explanation. The idea is, very simply, to take semantic values themselves to be fuzzy objects. Thus, the relevant domain of quantification is the set of real numbers between 0 and 1 (thought of as fuzzy real numbers), and we take the metric on these to be the standard distance metric jx
yj. To explain how things work, it will be easiest to take a simple example. Suppose that a0 , ..., a9 is a soritical sequence of sentences whose semantic values are as shown in the following table. I write the truth-value of ai as t(ai ). The second row tabulates the value of the sentence t(a0 ) ¼ t(ai ). ai

a0 a1 a2 a3 a4

t(ai ) 1 t(a0 ) ¼ t(ai ) 1

1 1

a5 a6 a7 a8 a9

1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0

0 0

0 0

Let us suppose that the cut-off for acceptability, e, is 0.7. As the forced march Sorites requires, there is a first sentence where t(ai ) is distinct from t(a0 ). In fact, this is a4 , since 4 is the first i for which

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18 / Graham Priest t(a0 ) ¼ t(ai ) is not acceptable (less than 0.7). So far, no surprises. But now, and crucially, why do we find the existence of such a cut-off counterintuitive? As is easy to check, the truth-values of t(a3 ) ¼ t(a4 ) and t(a4 ) ¼ t(a5 ) are both 0.8, which is greater than 0.7. Thus the value of a4 is identical with that of each of its neighbours! This is why we find the existence of the cut-off counter-intuitive! In fact, as is easy to check, for all 0  i  9, the value of t(ai ) ¼ t(aiþ1 ) is  0:8 (the bound being obtained at several places). Hence the value of the sentence 8i t(ai ) ¼ t(aiþ1 ) is 0.8 too. Hence, every sentence has the same truth-values as its neighbours. Note that we cannot use this fact to show that truth-values are identical all the way down the Sorites, since the transitivity of identity fails.9 The precise numbers employed in the preceding example are, of course, simply illustrative. The general point that they illustrate is that the truthvalue of the sentences must change eventually as we go down the sequence, despite the fact that it never changes as we go from each sentence to its neighbour. Unlike the epistemic solution to the Sorites, this does not fly in the face of common sense. It is exactly what common sense seems to tell us! What this solution gives us is an account of how the trick is turned. Not all sequences make a cut off-point counter-intuitive, of course. Thus, suppose we consider three stages of a person’s life: age less than 10, age 10–20, and age over 20. Let us ask at which stage they change from being a child to not being a child; the answer is in the second stage, and there is nothing counterintuitive about this. The idea of the solution just adumbrated is that when the existence of a cut-off point is counter-intuitive, the distribution of truthvalues and the relevant level of acceptability conspire to make it so in the way indicated.10

9 A little computation shows that the value of the sentence 9i(t(ai ) ¼ t(a0 ) ^ t(aiþ1 ) 6¼ t(a0 )) is less than 0.7. It might therefore be suggested that the model does not verify the existence of a cut-off point. The sense in which there is a cut-off point is that there is an i such that t(ai ) ¼ t(a0 ) is acceptable and t(aiþ1 ) ¼ t(a0 ) is not. To express this in the language we would need to extend it, as we could do, to express the thought that a sentence is acceptable. 10 Let me comment on one other feature of the present proposal. It is frequently objected to fuzzy logic that the supposition that a sentence has a real-valued truth-value—with all its infinite precision—is itself highly counter-intuitive. The theory of fuzzy truth-values can be seen as addressing this problem too. The truth-value of a sentence is a fuzzy real. If, suppose, it is 0.7, it may equally be 0.65 and 0.75.

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A Site for Sorites / 19

4. Further Issues 4.1 Degrees of Truth and Local Validity So much for the idea. In the final sections of the paper I will reflect on issues that need to be faced if this solution to the Sorites is to be articulated into something that is fully adequate. Suppose that one has a language with vague predicates, including identity, L1 . We may, in a metalanguage, give a theory, T, which provides an account of a continuum-valued semantics for the language (including the nontransitive semantics for identity). While the underlying logic of T would normally be left at an informal level, a natural enough assumption would be that this is classical logic. All this is orthodox and routine enough. In one sense, the preceding construction can be seen as a particular example of this. The language in question, L1 , has names for the sentences of some language, L0 (which might or might not be the same is L1 ), and for real numbers; it contains function symbols, including t and those naturally interpreted as expressing arithmetic functions; finally, it contains an identity predicate, and possibly various others. We may give a semantics for this language in the way just described. In a sense, this is all that is presupposed in the preceding considerations. However, as will be clear, the particular language, L1 , in question is not any old language. It is itself a language containing metalinguistic notions. Hence, one might reasonably expect that it be not only the language whose semantics is in question, but the vehicle for the metalinguistic theory about L0 itself. If we think of it in this way, we can certainly no longer take the logic of the metatheory to be classical logic: it must be a fuzzy logic of some kind. What ramifications does this have? The matter is far from straightforward. Part of the problem is that there are at least two distinct purposes for which metalinguistic reasoning may be engaged, though these are often not clearly distinguished. One is in an investigation of truth; the other is in an investigation of validity. Let us take these two matters in turn. One matter for which metalinguistic reasoning is employed is to articulate a notion of truth, and establish certain things as true or otherwise. In the present case, it is not truth simpliciter that is at issue, but degree of truth. Thus, we suppose that L1 contains axiom schemata such as:

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20 / Graham Priest (Neg) t(h:ai) ¼ 1
t(hai) where, now that we are being more precise about the syntax, h:i is an appropriate name-forming device. Given that we also have other axioms that specify the degrees of truth of atomic sentences (or allow us to infer them from more fundamental axioms concerning the denotations of their parts), we may infer the degrees of truth of more complex sentences. Now the crucial question: what underlying logic is required for this project? A full answer to this question can be provided only by working out the project in detail, for which this is not the place. But the amount of logic required would seem to be surprisingly little. Suppose, for example, that we have established that, for some particular a, t(hai) ¼ 0:5. The computation of the value of :a goes as follows: t(hai) ¼ 0:5 t(h:ai) ¼ 1
t(hai) t(h:ai) ¼ 1
0:5 1
0:5 ¼ 0:5 t(h:ai) ¼ 0:5

Already established Neg SI Arithmetic theorem TI

Here, SI and TI are, respectively, the substitution of identicals and the transitivity of identity. Though the argument deploys a somewhat minimal amount of logic, the most notable thing in the present context is that the argument uses inferential moves concerning identity that are not valid, notably TI. This is the first issue that needs to be faced. The problem is not, in fact, as desperate as it may at first appear. Even though TI is not valid, it is locally valid. This means that, though we cannot use it over ‘‘long distances’’, it is perfectly all right to employ short chains of inference involving it. (It should be noted that though SI—of which TI is a special case—is not valid, an appropriate form is also locally valid.11) This raises two questions. The first, and most obvious, is how short ‘short’ is. The answer to this depends on factors such as how true the initial axioms are, what degree of truth is acceptable, and so on. While these questions demand to be addressed in detail, it can be said that if things are set up right, the short may be very long—maybe longer than any chain of inference that anyone can construct in practice. The invalidity may not, therefore, be a practical problem. 11 See Priest (1998, sect. 6).

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A Site for Sorites / 21 Eventually, though—wherever eventually is—the acceptability of the arguments may give out. The second question is what to make of this. The obvious answer is that our (meta)theoretical machinery will cease to be sufficient to determine degrees of truth for certain sentences. Perhaps this is something that can simply be faced with equanimity. Maybe there just is no fact of the matter concerning the degrees of truth of such sentences. If these sentences are really long—say, longer than anything constructible in practice, and so transcending anything humanly meaningful—this is perhaps not implausible.

4.2 Kicking Away the Ladder The second purpose for which metalinguistic reasoning may be employed is in an investigation, not of truth, but of validity. A metatheory is used to formulate a semantic notion of validity for languages of a certain kind, and to investigate what inferences, couched in those languages, are and are not valid. In addition, we may hope to isolate a set of proof procedures and show them to be sound, and maybe also complete, with respect to the semantic notion. What of the means of inference employed in these investigations themselves? Suppose that the language of the metatheory is itself a language of the kind in question. Then if the metatheoretic reasoning is to be understood as giving an account of, or justification for, those inferences that are acceptable for languages of this kind, the inferences employed should not be ones that are demonstrably invalid. More generally, one might hope, one should be able to demonstrate the soundness of the class of inferences employed. This is the situation we are in if we take the language L1 as a fuzzy language in which an account of fuzzy validity is to be given. How do things stand with this? A definition of validity is easy enough to formulate. Sticking to the onepremise case, for simplicity, we can define the validity of an inference with premise a and conclusion b, V(haihbi), as follows: V(haihbi) $ 8u(u(hai)  u(hbi)). But what inferences should one be allowed to use to reason about this notion of validity? It is clear, for a start, that it is no longer to be expected that this is classical logic. The whole point of applying fuzzy logic to the Sorites is that inferences like modus ponens and TI are not valid. A better guess is that we take it to be fuzzy logic itself. Such a logic, however, is too weak. Establishing that various inferences are valid, let alone establishing general soundness (and

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22 / Graham Priest completeness) results, is going to require inferences such as TI, SI, and modus ponens. Merely consider, for example, the natural argument to show that the inference from a ^ b to a is valid: u(ha ^ bi) ¼ Min(u(hai), u(hbi) ) Truth condition Min(u(hai), u(hbi)  u(hai) ) Arithmetic theorem u(ha ^ bi)  u(hai) SI 8u(u(ha ^ bi)  u(hai) ) Universal generalization It is therefore clear that the arguments employed must include locally valid inferences as well. This raises the same kind of issue that we met in the last section. In particular, there is a question of the extent to which such inferences are acceptable in this context. It may also mean that certain inferences are neither demonstrably valid nor demonstrably invalid. Maybe, then, there is no fact of the matter here. But a new question is also posed. Is reasoning of this kind sufficient to show the validity (including local validity) of the very reasoning in question? That is, can the soundness of this reasoning be demonstrated employing the reasoning itself ? Moreover, but less crucially, can it also establish it to be appropriately complete. No doubt it can using classical logic. Can the classical ladder be kicked away? The question cannot be answered without detailed investigation, but the following comments are in order. As it is not uncommon for bright students to observe, demonstrating the validity of a principle of inference tends to employ the very principle in question (together with some fairly basic logical apparatus, such as modus ponens, etc.—which we have, in this case, though it is only locally valid). One would therefore expect that most sensible logics would be able to establish the validity of any inference it holds to be valid. This much, at least, ought to be possible in the case at hand. And this is enough to ensure that the logic is selfcoherent. Establishing a formal soundness theorem requires somewhat more by way of reasoning; for example, it requires that one can define the set of axioms and reason appropriately about its members. This requires a certain amount of set theory—at the very least the ability to define certain sets by recursion. And the corresponding completeness proof requires even more—that one can show the existence of various counter-models and demonstrate their properties. This does not necessarily require classical logic. Recall that there are intuitionistically valid soundness and completeness proofs for intuitionistic

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A Site for Sorites / 23 propositional logic.12 But constructing such proofs is not at all a routine matter. Witness the problems around the production of an intuitionistically valid completeness proof for intuitionistic predicate logic.13

5. Conclusion The issues that I have raised in the last part of the paper all need to be addressed to make the solution to the Sorites that I have suggested fully articulated. Some of these are distinctly non-trivial. For this reason, I do not claim that the solution I have suggested is right. It does, however, seem to me to be one that is both plausible and worthy of further investigation.

REFERENCES D u m m e t t , M. (1977), Elements of Intuitionism (Oxford: Oxford University Press). G r a f f , D. (2000), ‘Shifting Sands: An Interest-Relative Theory of Vagueness’, Philosophical Topics, 28: 45–81. H o r g a n , T. (1994), ‘Robust Vagueness and the Forced-March Sorites’, Philosophical Perspectives, 8: 159–88. K e e f e , R., and Smith, P. (eds.) (1997), Vagueness: A Reader (Cambridge, Mass.: MIT Press). P r i e s t , G. (1991), ‘Sorites and Identity’, Logique et Analyse, 135–6: 293–6. —— (1998), ‘Fuzzy Identity and Local Validity’, The Monist, 81: 331–42. —— (2001), Introduction to Non-Classical Logic (Cambridge: Cambridge University Press). S o r e n s e n , R. (1988), Blindspots (Oxford: Oxford University Press). W i l l i a m s o n , T. (1994), Vagueness (London: Routledge). W r i g h t , C. (1995), ‘The Epistemic Conception of Vagueness’, Southern Journal of Philosophy, 33: 133–59. 12 See e.g. Dummett (1977, esp. 214).

13 See Dummett (1977, sect. 5.6, esp. p. 259).

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2 Cut-Offs and their Neighbors Achille C. Varzi

I agree with Priest: the Sorites is a very hard paradox, possibly harder than the Liar.1 For the Liar can be isolated, whereas the Sorites is everywhere and can take us anywhere. And I agree that the paradox is so hard because it systematically imposes upon us the existence of unbelievable or otherwise unacceptable cut-off points. No solution can avoid explaining why this happens. But I have two points of disagreement with Priest’s account of the matter. The first concerns the emphasis that he places on the working of natural language. And the second, more important problem concerns the line of explanation that Priest offers of the reason why we are stuck with cutoff points—hence his tentative solution to the paradox.

1. Sorites Progressions and Linguistic Vagueness Let’s focus on the forced march Sorites, in the form given by Priest. We have got a series of questions q0 ...qn , each of the form ‘Is it the case that bi ?’, and the facts are such that the answer to the first question, a0 , is different from the answer to the last question, an . Given these facts, there is no way out: the series of answers must involve a cut-off point. No matter what form the answers take, and no matter how we feel about the relative indistinguishability of any two successive bi ’s, if a0 6¼ an there must be a first k > 0 such that a0 6¼ ak . Thanks to Matthew Slater for helpful exchanges on the topic of this paper. 1 Unless otherwise specified, all references are to Priest, Ch. 1 in this volume.

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Cut-Offs and their Neighbors / 25 Now, Priest takes the problems into which this scenario may lead us to be problems posed by the vagueness of our language. So where exactly does language play a role in this scenario? Not in the answers, for these can be anything we like, including straight Yes or No answers. Besides, as Williamson made clear a while ago, the answers are actually irrelevant since the problem concerns what one should believe, not just what one should say.2 So language must play a role in the questions, the qi ’s, hence in whatever bits of language we use to express the statements corresponding to the bi ’s. (The locution ‘Is it the case that . . . ?’ is ultimately redundant.) In Priest’s example, each bi is a statement to the effect that you are bi , namely, an object that comes from you, b0 , by i successive replacements of one molecule of your body with a molecule of scrambled egg. In other words, each bi is an identity statement of the form (1) b0 ¼ bi . The notion of replacement, we may suppose, is defined with precision, for the predicate ‘scrambled egg’ that occurs in our description of the procedure can be defined in precise terms. And we may also suppose that your body, b0 , is defined with similar precision, though in actual circumstances this may be a hard thing to do. Accordingly, we can safely assume that in each case the two terms flanking the identity predicate in (1) have a precise semantics. That leaves only the identity predicate itself; no other bit of language is involved in our questions. But don’t we know exactly what this predicate stands for? It stands for the identity relation. So either we say that there really are many, slightly distinct candidates for this relation, at least as we conceive of it when we use the identity predicate in ordinary discourse, or else language plays no role at all in the picture and the burden of the Sorites would be entirely on the identity relation itself. (We would then have a case of ontological vagueness.) Thus, when Priest says: ‘‘How can a single molecule make a difference? . . . [V]ague predicates just don’t seem to work like this,’’ he is making a claim that goes beyond the data. Vague predicates play no role in this scenario except for identity. At most we can say: ‘The identity predicate doesn’t work like this’. And we can say that only if we reject the thought that the burden of the Sorites is on the identity relation. Now, I am happy to reject that thought, so let’s put ontological vagueness to one side. The question is, what are we to make of the fact that the burden 2 Williamson (1994: 14).

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26 / Achille C. Varzi of the forced march Sorites is entirely on the identity predicate? One obvious response is that this is just a byproduct of the particular example under discussion. After all, in the scenario described by Priest the questions involve identity statements, but they could be questions of a different sort. For example, the qi ’s could be questions of the form ‘Is it the case that bi ?’, where each bi is a statement of the form (2) bi is a heap, with bi defined in the obvious way. In that case the reasoning above would show that the burden of the Sorites is on the predicate ‘heap’, not the identity predicate. So by generalization we would get to the desired point: generally speaking, the Sorites forces us to recognize the existence of cut-offs in the extensions of natural language predicates, whereas common sense and linguistic practice scream that there are no cut-offs. At this point, however, we could also go further in the generalization, and this is where I get my concerns. Why should the Sorites consist in a series of questions in natural language? Questions are stimuli of a peculiar sort, prompting specific reactions in the form of answers. But we can imagine a Sorites in which the stimuli come in a different format. Consider, for instance, the following variant of the scenario described by Priest. We have got a series of visual stimuli s0 ... sn , each in the form of a digitalized cartoon drawing. The first stimulus, s0 , is a picture of Snow White; the last stimulus, sn , is a digital picture of a monster; and each intermediate stimulus is obtained from its predecessor by changing just one pixel. So the whole series is a short movie where Snow White is gradually warped into a monster. You are a little kid and you are sitting in front of the video screen. At the beginning you react with joy. At the end you are scared. So your initial response, r0 , is different from your terminal response, rn . So there we are: there must be a first k > 0 such that r0 6¼ rk . Sooner or later there must be a change—a cut-off—in the way you respond to your visual stimuli. And yet the process is perfectly gradual; the difference between any two successive frames falls below anything that is cognitively accessible to you. We are forced to recognize the existence of a cut-off point where both common sense and psychological intuition scream that there is none. I doubt that we can say language plays any role in a scenario of this sort. Maybe concepts do—I’m not sure. But there is no reason to suppose that the relevant concepts are the intensions of linguistic predicates. And there are of course many such scenarios. For another example, consider a game-theoretic

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Cut-Offs and their Neighbors / 27 setup in which the series is construed as a centipede, i.e. as a game of take it or leave it. Money on the table accrues gradually. At the beginning of the game it is rational for you to leave the money on the table, for your opponent will do the same. At the end it is rational for you to take the money, for otherwise your opponent would take it instead. So there must be a cut-off point somewhere, a point during the game where the rational thing for you to do changes from the first type of response (leave the money) to the second (take the money). But of course the game can be construed so that the difference between any two successive stages is completely negligible for both players. For instance, we may suppose that the amount on the table accrues at the rate of just one penny each time. So there we are again. We are stuck with a cut-off that we cannot accept. And forget the fact that we can give a linguistic description of what goes on in the game. That would certainly allow us to blame it on the predicates that we use, for example on the predicate ‘amount of money that a rational player (i.e. a player following the standards of ideal rationality) ought to leave on the table’. That such a predicate is ultimately vague would be an interesting result by itself.3 But of course this is not the main story. At bottom, the problem is not one of language. It is one of rationality tout court. Besides, it is not difficult to imagine scenarios in which the soritical victims are non-linguistic creatures.4 A pigeon is trained via a regime of reward and punishment to peck to the right on the presentation of red stimuli, and to the left for other (clearly distinct) colors. Presented with a soritical series of stimuli of color patches, running from red to yellow with each patch pigeon-visually indistinguishable from its neighbor, the pigeon is bound to find itself stuck in the business of switching from right-pecking to leftpecking. Surely this has nothing to do with the vagueness of ‘red’, even if we could blame the trainer for scarcity of instructions. So I think Priest is right in pointing his finger on the cut-offs. That’s where we get into troubles—troubles that cannot be solved by playing around with our logical paraphernalia. But the cut-offs as such don’t seem to have much to do with language, either. The troubles they originate are deeper and more general and I doubt they can be handled any better by exercising our semantic paraphernalia. The troubles arise at the level of cognition broadly understood. And language is but one significant part of but one sort of cognitive system. 3 See Collins and Varzi (2000). 4 Thanks to the OUP reader for suggesting the following example.

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28 / Achille C. Varzi This is not to deny that a linguistic account can be useful, of course. In some cases a semantic theory can be worked out that helps us understand why we are stuck with unexpected cut-off points in the presence of vague predicates, just as a rational decision theoretic account (for example) can help explain why we are stuck with cut-offs in the centipede. Of the semantic theories presently on the market Priest only mentions epistemicism and contextualism in this connection; I think supervaluationism is also in the right ballpark—whether or not it stands up well to ‘‘the cold light of inspection’’. I’ll get back to that shortly. What I don’t think is that any such theory is in the right ball-park for a general explanation of the phenomenon. In fact I am not even sure that philosophy itself is in the right ball-park for a general explanation. At bottom it may well be a phenomenon that calls for a psychological account. It may well be that cognitive science and the behavioral sciences at large are the only good candidates for a general explanation of the phenomenon. And it may well be that this sort of explanation is part and parcel of a general explanation of why it is that we have such a hard time dealing with the changes in our environment, when the changes are gradual. Philosophy has little to say about this—let alone semantics. (This applies to other features of the Sorites paradox, too, such as the variability with which different subjects react to the same stimuli. In the Snow White scenario, for example, some children would start to get terrified as soon as Snow White’s nose starts turning red while others would not get scared until the process is almost over, but it is unreasonable to suppose that such variability reflects differences in their vague idiolects or conceptual apparatuses. This is something that we might only be able to explain by looking into the children’s individual psychophysiology. Ditto for the other cases.)

2. Semantic Explanations Having said this, let us put the business of a general explanation aside. Let us try and focus on the limited task of assessing the explanatory force of a semantic theory—a theory designed to explain the Sorites phenomenon at least inasmuch as it arises in the presence of vague predicates. Ideally, this would have to be a theory that follows from the big theory. But since the big theory is still too far away, we might as well try to work out our views bottom-up.

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Cut-Offs and their Neighbors / 29 What options are there? As I mentioned, Priest considers two options— epistemicism and contextualism—and he has misgivings about both. On this I am inclined to agree. Concerning the first option, Sorensen and Williamson have given us a sophisticated explanation of why it is impossible to identify the semantic cut-offs of vague predicates.5 But such an explanation does not amount to an explanation of why we are deeply disturbed by the thought that such cut-offs must exist. As Priest puts it, to explain why we cannot know the existence of something is not to explain why we find its existence counter-intuitive—and I agree with that. I also agree that the second option—contextualism—leaves the issue unresolved: the thought that there is a cut-off point in a soritical series is puzzling even if we convince ourselves that it must lie outside what Graff calls the area of ‘‘salience’’ set by the context.6 I do not, however, agree that these are the only two accounts on the market to be considered in this connection. On my reckoning, supervaluationism provides a valuable alternative—in fact a better alternative. For supervaluationism does offer an explanation. Supervaluationally, the reason why we are disturbed by the existence of the cut-offs is that we cannot pin them down in any way. Ordinarily, when we recognize the truth of an existential statement we can also recognize the truth of one of its instances. We recognize the truth of a statement such as (3) There is a number k > 0 such that k is greater than 32 but less than 42 because we can specify a value k > 0 that satisfies the condition (4) k is greater than 32 but less than 42 . This natural impulse to demand an instance whenever an existential statement is asserted (or a counterinstance whenever a universal statement is denied) is perfectly justified in the ordinary semantics for precise languages. It is not, however, a defining feature of the meaning of the existential quantifier. At least, it is a controversial issue whether it is a defining feature; it certainly isn’t in classical logic, for in classical logic existential statements are not constructive. In any event, there is no guarantee that the impulse is justified when it comes to vague languages. And this is precisely what the supervaluational account tells us. When it comes to statements asserting the existence of a cut-off point in the extension of a vague predicate, there is a 5 See Sorensen (1988: 189 ff.) and Williamson (1994, chs. 7–8).

6 Graff (2000).

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30 / Achille C. Varzi tension between our natural impulse to ask for an instance, on the one hand, and the impossibility to deliver an instance, on the other.7 We recognize the truth of a statement of the form (5) There is a number k > 0 such that bk
1 and not bk , where the bi ’s involve vague predicates, because we recognize that this statement is true for every precisification of those predicates. If it is true for every precisification of those predicates, then we can’t go wrong if we say that it is true simpliciter—or so goes the story. But we cannot recognize the truth of any instance of the form (6) bk
1 and not bk because no such instance is true for every precisification. It is true that there exists a k that marks the cut-off, but there is no k such that it is true of it that it marks the cut-off. This is disturbing, no question about that. Indeed it is because of this disturbing feature that many philosophers dislike the supervaluational account. (‘A true existential statement must have true instances!’, stamp the foot, bang the table.8) But I would rather say that the account deserves consideration precisely because of its ability to dispel this disturbing feature. Supervaluationism is in the right ball-park for an explanation of the Sorites phenomenon precisely because it tells us why we are disturbed by the existence of cut-off points. We are disturbed because we can recognize the truth of an existential without being able to recognize the truth of any instance—and this goes against our natural inclination to always demand a true instance.

3. Vague Identity and Fuzzy Truth-Values It is not my intention to insist further on the merits of supervaluationism. There may be other reasons why one could be dissatisfied with that theory— for instance, reasons having to do with the intuition that the vagueness of a predicate is part of its sense, which would make the notion of a precisification incoherent9—but this is not the place to take up such issues. I just wanted to illustrate my reasons for thinking that supervaluationism is in the right 7 The tension is diagnosed in McGee and McLaughlin (1995: 207) and Keefe (2000: 185). 8 See e.g. Tappenden (1993: 564). But the objection is a popular one. 9 See e.g. Fodor and Lepore (1996).

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Cut-Offs and their Neighbors / 31 ball-park for a semantic explanation of the Sorites. Let me now turn to Priest’s own account, as presented in the second section of his paper. As I said, I believe there is a problem with this account, even if we confine ourselves to the modest task of providing an explanation of linguistic vagueness. Let me first make a methodological remark. A great deal of Priest’s account focuses on the intuition that the identity predicate is not globally transitive, and that is what I want to focus on. But Priest’s working example is misleading in this regard. We are supposed to consider a sequence c0 ...cn of color patches such that each is phenomenologically indistinguishable from its immediate neighbors, and we are asked to agree with the soritical premise (7) For every k > 0, ck
1 ¼ ck . This cannot be right, for surely phenomenological indistinguishability falls short of identity. What we want to say is that if we take the identity predicate to express phenomenological indistinguishability, then (7) holds. Since this is a can of worms, however, I think it is better to bypass the issue altogether and work with an example that does not involve phenomenological issues, at least not explicitly. So I am going to stick to the initial example—the forced march Sorites determined by the series b0 ...bn that begins with you, b0 , and ends in scrambled egg, bn . Here we are stuck with a paradox in so far as the statement (8) For every k > 0, bk
1 ¼ bk is prima facie true. And unless you have reasons to accept ontological vagueness, this means that the identity predicate is vague. Now, Priest’s account consists of three basic claims. The first, which for the sake of the argument I shall embrace, is that the vagueness of the identity predicate must be cashed out in terms of a fuzzy semantics—that is, identity statements have degrees of truth. The second claim is that the appropriate fuzzy semantics for the identity predicate is defined by the following truth conditions, (9) t(a ¼ b) ¼ 1
d(a, b), where t is the (continuum-valued) valuation function, d is a normalized metric, and a, b are the denotation of a, b. I am going to accept this claim, too, for (9) seems to me to be the only reasonable option for a fuzzy semantics. The third claim is that the semantic values should themselves be treated as fuzzy objects, that is, objects about which we can make fuzzy identity statements. This is actually the main idea, and it is here that we find a sketch

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32 / Achille C. Varzi of how a fuzzy-theoretic apparatus can provide the required explanation. It is with this sketch that I have concerns. Take our soritical series, suppose for simplicity that n ¼ 9, and consider the semantic values in the following table (which correspond to Priest’s): bi

b0 b1 b2 b3 b4 b5 b6 b7 b8 b9


(bi ) 1

1

1 0:8 0:6 0:4 0:2 0

0

0

In a degree-theoretic semantics we can define, besides truth, 1, and falsity, 0, a third semantic condition with respect to which a statement can be classified. We can say that a statement is acceptable if its truth-value does not fall below a certain threshold, e, which in practice is determined by the context. Let us follow Priest in supposing that in the present case e ¼ 0:7. And let us follow Priest in taking the notion of a cut-off point to be defined with respect to acceptability rather than perfect truth. This amounts to saying that our responses to the relevant series of questions change in the relevant sense only when their numerical values fall below the threshold of acceptability. An acceptable identity statement is, after all, contextually acceptable even if its truth-value is not the unit. So a cut-off point is not to be thought of as the first k > 0 such that (10)
(bk ) 6¼
(b0 ) but, rather, as the first k > 0 such that (11)
(bk ) < e. Taking the metric to be the standard distance jx
yj, it is now easy to see how the proposed explanation works. With regard to the series consisting of the fuzzy answers to our soritical questions, i.e., equivalently, the series consisting of the identity statements (di )
(b0 ) ¼
(bi ),

(0  i  n)

the cut-off is given by i ¼ 4. But the existence of such a cut-off point is not backed up by a corresponding cut-off in the series consisting of the following identity statements, (si )
(bi
1 ) ¼
(bi ),

(0 < i  n)

i.e., effectively, the identity statements where each answer (here: the fuzzy truth-value of bi ) is compared to its neighbors. In fact, every identity

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Cut-Offs and their Neighbors / 33 statement in this second series gets a truth-value  0:8 and is therefore acceptable, including the special case t(b3 ) ¼ t(b4 ). So the explanation is this: Our answers will eventually hit the cut-off; and yet, given any two neighboring answers, it is acceptable to say that they have the same truthvalues. In other words, as Priest puts it, the acceptability of the bi ’s must drop eventually as we go down the series, despite the fact that it never drops as we go from each bi to its successor. One might be tempted to object that this answer depends too heavily on the availability of a fixed value for the threshold of acceptability, e. In order for the explanation to be satisfactory, we need an account of how the threshold value for e is determined, and all we are told is that this value is determined by the context. But if the context determines the threshold of acceptability—the objection goes—then it also determines the cut-off between truth and falsity in a classical, bivalent setting. After all, once we settle on a value for e we have a straightforward mapping of our fuzzy semantics into a bivalent semantics by matching acceptability with full truth: (12) f (bi ) ¼ 1 iff t(bi )  e. So if the context does the job with respect to e, then it should also do the job with respect to classical logic. This objection misfires, though. For Priest’s point is that the explanation does not come from the choice of e. It comes from the fact that given any reasonable choice for e, the truth-values of the (si ’s) are all higher than e and so these equations are all acceptable. But, of course, this property is not preserved by the map to classical bivalent semantics. With reference to our example, the equation (13) f (b2 ) ¼ f (b3 ) is true (hence classically acceptable) but the next equation, (14) f (b3 ) ¼ f (b4 ), is false (and classically unacceptable). There is another potential objection to Priest’s account that misfires. As Priest observes in a footnote (n. 10), the truth-values of his fuzzy semantics should be thought of, not as precise real numbers, but as fuzzy reals— otherwise we run into a familiar objection to fuzzy logic. (Assigning a precise truth-value between 0 and 1 is tantamount to ascribing infinite precision to vague statements.) If so, then one might be tempted to protest that whether a

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34 / Achille C. Varzi given statement falls below the acceptability threshold may itself be indeterminate in some cases. This is not the case in our example, but that is just because the example is oversimplified. In general the (fuzzy) distance between the truth-values of the soritical statements is much less than 0.2 and therefore there may be genuine indeterminacy with regard to their acceptability. This objection is also inappropriate, though. It misfires because the relevant indeterminacy is immaterial. The counter-intuitiveness of Sorites phenomena lies in the fact that there must be a cut-off, regardless of where exactly it is located in the soritical sequence. And Priest’s explanation is that regardless of which elements of the sequence fall below the acceptability threshold, e, any two neighboring statements in the sequence turn out to be acceptable. So much for bad objections. But here are two worries that I believe deserve consideration. The first concerns the fact that the proposed explanation depends heavily, if not on the precise choice of e, on the standards of precision that determine this choice. In the example under discussion, e is set at 0.7 and the explanation goes through because each si gets a value  0:8. But suppose the context was such as to determine a higher threshold, say e ¼ 0:9. In that case, only the first two and the last two si ’s would turn out to be acceptable and the explanation would break down. In fact, we would immediately get a cut-off at i ¼ 3. This would not be the same cut-off that we hit as we move along the series of the di ’s, but that is hardly a difference that can explain why we find the latter counter-intuitive. So what are we to say in cases like this? I suppose Priest would simply say that we are dealing with an implausible scenario: the threshold is just too high. After all, in the limit case where acceptability coincides with perfect truth (i.e. when e ¼ 1) we are back to a classical bivalent scenario, where no explanation is available at all. So, more generally, the explanation may fail when acceptability is dangerously close to perfect truth. This may be right. But the question remains of why 0.9 (or any other value greater than 0.8, for that matter) should be regarded as a dangerous threshold in this respect. More generally, inspection shows that the proposed explanation only works when the value of e is lower than the least value attached to the si ’s. But I don’t see how this constraint can be implemented without rendering the explanation itself dangerously circular. The second worry is more general. As it stands, Priest’s account relies on his third claim, to the effect that the basic fuzzy semantic framework must be supplemented by an account whereby the semantic values themselves are fuzzy objects. The worry is: What exactly is the cash value of this supplemen-

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Cut-Offs and their Neighbors / 35 tary move? Suppose we don’t do that. Then we could mimic Priest’s explanation of what goes on as follows. Relative to the soritical series consisting of the identity statements (bi ) b0 ¼ bi

(0  i  n)

we are forced to accept a cut-off point (in the given example: the cut-off given by i ¼ 4). But relative to the series consisting of the following identities (gi ) bi
1 ¼ bi

(0 < i  n)

there is no cut-off point. For every such statement is assumed to be acceptable, if not plainly true. Therefore, the acceptability of the bi ’s must drop eventually as we go down the series, despite the fact that the acceptability of the gi ’s never drops. End of the story. Would this be a good explanation of why we find the existence of the cut-off counter-intuitive? If it were, then the detour through the meta-identity statements proposed by Priest would be unnecessary. And if it were not a good explanation then the question is what makes Priest’s explanation any better. Here is why I don’t think this explanation would be a good one. It would not be a good explanation because it would simply amount to a redescription of the puzzle. After all, the Sorites phenomenon is puzzling precisely because the difference between any two neighbors of the given sequence is negligible whereas the difference between the first member and some later, distant member is not negligible. Small differences accrue—that’s exactly the problem. Of course a fuzzy logician can infer from this that the identity predicate is not globally transitive, but that doesn’t add much. Unless we tell a story that explains why this outcome is at odds with the intuition that identity is transitive, the claim that identity is not globally transitive simply reports the facts. I am sure Priest agrees with this, for otherwise he would have thought that his earlier attacks on the problem already had the explanatory force that he is after. (On such earlier accounts, the explanation traded on the difference between local and global validity. We mistakenly think that inferences that can be used over short distances—such as inferences exploiting the transitivity of identity—are reliable over long distances as well, which is not the case: in particular, identity is transitive locally but not globally.10) 10 See e.g. Priest (1998).

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36 / Achille C. Varzi So if this simplified explanation is not good, what exactly do we gain by making the extra step of treating the semantic values themselves as fuzzy objects? As far as I can see we don’t gain much. It is true that the gi ’s and the si ’s are distinct: in one case we have a series of identities concerning pairs of neighboring bi ’s (the objects obtained by replacing i molecules of your body with i molecules of scrambled egg); in the second case, corresponding to Priest’s strategy, the series of identities concern pairs of neighboring bi ’s (which are statements to the effect that you are identical with the object obtained by replacing i molecules of your body with i molecules of scrambled egg). And whereas we may suppose that the identities of the first series are all perfectly true, the identities of the second series need not be true: what matters is that they are acceptable. However, such differences are irrelevant when it comes to the requested explanation. We are going from one soritical series to another, and what we are told about the second case is exactly what the simplified account tells us about the first: in both circumstances we are dealing with a series of neighbor-identities (the gi ’s and the si ’s, respectively) all of which are acceptable despite the fact that the corresponding distance-identities (the bi ’s and the di ’s) eventually drop below the acceptability threshold. To put it differently, Priest’s semantics satisfies the following biconditionals: (15) bi is acceptable iff di is acceptable (0  i  n) (16) gi is acceptable iff si is acceptable (0 < i  n) i.e., effectively: (15’) b0 ¼ bi is acceptable iff t(b0 ) ¼ t(bi ) is acceptable (0  i  n) (16’) bi
1 ¼ bi is acceptable iff t(bi
1 ) ¼ t(bi ) is acceptable (0 < i  n) And since acceptability is what matters, it seems to me that this makes the detour proposed by Priest redundant. If it is redundant, it adds no extra value. If it adds no extra value, then the explanation that we get by taking the detour cannot be any better than the explanation that we get without taking the detour. And as we have seen, the explanation that we get without taking the detour is not good enough.

4. Conclusion My conclusion, therefore, is not too optimistic. I don’t think the novel account offered by Priest will take us closer to the heart of the problem.

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Cut-Offs and their Neighbors / 37 On my reckoning, the supervaluational account would still fare better, but never mind that. At this point I would rather be inclined to end on a note of methodology. We have got many semantic accounts. None of them seems capable of delivering the whole story, but many succeed in diagnosing at least some of the peculiarities of the Sorites phenomenon. So perhaps a good thing to do at this point would be to try and put the pieces together. On the other hand, because of my earlier remarks about the non-linguistic nature of the phenomenon in its most general form, I am not too optimistic that this strategy will take us very far, either. The Sorites is a deep and bewildering puzzle precisely because it arises at a deep and fundamental level, one that appears to be prior to the engagement of any logical and semantic paraphernalia. Unless we take that into serious account, I doubt we can achieve the sort of explanation Priest is after, which is why I think we are still stuck with the puzzle. In Herzog, Saul Bellow tells a story about a club in New York that most of us should find familiar.11 It’s that club where people are the most of every type. There is the hairiest bald man and the baldest hairy man; the shortest giant and the tallest dwarf; the smartest idiot and the stupidest wise man. And what do they do? On Saturday night they have a party. Then they have a contest. And if you can tell the hairiest bald man from the baldest hairy man—says Herzog—you get a prize. My impression is that if we entered the contest we might perform better than our fellow non-philosophers. At the end of the day, however, I suspect we would still be puzzled. We would still find ourselves looking at the prize with an overwhelming sense of incredulity—that sense of incredulity that we experience whenever we hit a cut-off.

REFERENCES B e l l o w , S. (1964), Herzog (New York: Viking Press). C o l l i n s , J. D., and A. C. Varzi (2000), ‘Unsharpenable Vagueness’, Philosophical Topics, 28: 1–10. F o d o r , J. A., and E. Lepore (1996), ‘What cannot be Evaluated cannot be Evaluated, and it cannot be Supervalued Either’, Journal of Philosophy, 93: 516–35. G r a f f , D. (2000), ‘Shifting Sands: An Interest-Relative Theory of Vagueness’, Philosophical Topics, 28: 45–81. 11 See Bellow (1964: 295–6).

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38 / Achille C. Varzi K e e f e , R. (2000), Theories of Vagueness (Cambridge: Cambridge University Press). M c G e e , V., and B. McLaughlin (1995), ‘Distinctions without a Difference’, Southern Journal of Philosophy, 33 (suppl.): 203–52. P r i e s t , G. (1998), ‘Fuzzy Identity and Local Validity’, The Monist, 81: 331–42. S o r e n s e n , R. A. (1988), Blindspots (Oxford: Clarendon Press). T a p p e n d e n , J. (1993), ‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’, Journal of Philosophy, 90: 551–77. W i l l i a m s o n , T. (1994), Vagueness (London: Routledge).

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3 Vagueness and Conversation Stewart Shapiro

‘I don’t know what you mean by ‘‘glory,’’ ’ Alice said. Humpty Dumpty smiled contemptuously. ‘Of course you don’t—till I tell you. I meant ‘‘there’s a nice knock-down argument for you!’’ ’ ‘But ‘‘glory’’ doesn’t mean ‘‘a nice knock-down argument,’’ ’ Alice objected. ‘When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean—neither more nor less.’ ‘The question is,’ said Alice, ‘whether you can make words mean so many different things.’ ‘The question is,’ said Humpty Dumpty, ‘which is to be master—that’s all.’ (Lewis Carroll, Through the Looking Glass)

This paper is a pilot for a longer work, which includes a model-theoretic semantics and a treatment of so-called higher-order vagueness. My biggest debt is to my colleague and friend Diana Raffman, whose work inspired the present account, and who continues to give me valuable advice. Thanks also to the members of a graduate seminar on vagueness that we gave together in the spring of 2002: Julian Cole, Sven Walters, Steven James, Jack Arnold, and Michael Jaworski. I am also indebted to the audiences at the ‘Liars and Heaps’ conference held at the University of Connecticut in the autumn of 2002, an Arche´ workshop and subsequent seminar at the University of St Andrews, the Hebrew University Logic Colloquium, and the Philosophy Colloquium at the University of Maryland. I have received valuable feedback from Carl Posy, Mark Sainsbury, Crispin Wright, Agustı´n Rayo, Patrick Greenough, Delia Graff, Barbara Scholz, Tim Williamson, Neil Cooper, Roy Cook, Rosanna Keefe, Brian Weatherson, Graham Priest, Michael Morreau, and Georges Rey.

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40 / Stewart Shapiro

1. Determinacy The present account of how vague terms arise in natural language, and how their meaning is determined, begins like that of Vann McGee and Brian McLaughlin (1994) (see also McGee 1991). It is a truism that the competent users of a language somehow determine the meaning of its words and phrases, by what they say, think, and do. Lewis Carroll’s Humpty Dumpty says that it is a question of ‘which is to be master’. When it comes to meaning, the proverbial we—the community of competent, sincere, honest, wellfunctioning speakers—are master. To be sure, I have no special insights to bring as to how words acquire their meaning. There is also a nice question concerning who, exactly, are the competent speakers of a natural language like English. Or, perhaps better, what is it to be a (competent) speaker of English? Who are ‘we’? Let F be a monadic (possibly complex) predicate in a natural language. McGee and McLaughlin (1994, §2) introduce a technical term ‘definitely’ as follows: ‘to say that an object a is definitely an F means that the thoughts and practices of speakers of the language determine conditions of application for . . . F, and the facts about a determine that these conditions are met’. In a note, they add that the word ‘determinately’ would be better, since it captures the underlying idea. I will use the latter term here. In general, when ‘a sentence is [determinately] true, our thoughts and practices in using the language have established truth conditions for the sentence, and the [presumably] non-linguistic facts have determined that these conditions are met’. McGee and McLaughlin insist that the ‘determinately’ operator is not, or may not be, compositional. It may be that a sentence of the form ‘F _ C’ is determinately true even if it is not the case that F is determinately true, nor is it the case that C is determinately true. More notably, it may be that a sentence F fails to be determinately true without :F being determinately true. With McGee and McLaughlin, I believe that this is the source of vagueness. An object a is a borderline case of a predicate F if Fa is ‘unsettled’, i.e. if a is not determinately F, nor is a determinately non-F. Since we are still at the level of truisms and definitions, nothing should be all that controversial so far. McGee and McLaughlin are free to define technical terms as it pleases them, and I am free to follow suit. One point of contention is whether or not ‘determinately’, so defined, is well-defined. I will have a bit more to say on this by way of further articulation (e.g. Section 6 below), but will just assume for now that we have hold of at least a vague

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Vagueness and Conversation / 41 notion here, which can give us a start on theorizing. Another point of contention is the thesis that determinateness is not compositional. I do not know how to defend this, and so will take it as a premise toward what follows. A more crucial point of contention is whether there are any borderline cases at all—as the term ‘borderline’ is defined here (in terms of the technical notion of determinateness). Famously, Timothy Williamson’s (1994) epistemicism maintains that the thoughts and practices of language users fix precise extensions, and complementary anti-extensions, for every predicate of the language. Consider, for example, the sentence ‘Josh arrived around noon’. According to Williamson, the established use of language determines a real number n such that if Josh arrived exactly n seconds after noon, then the sentence is true, but if he arrived n þ 0:0000001 seconds after noon, the sentence is false. Williamson would thus have little use for the above definition of ‘borderline’, since on his view, there simply are no ‘unsettled’ cases. Williamson defines the word ‘borderline’ in terms of what can be known, rather than in terms of truth conditions, as above. Here, of course, I insist on the above definition, noting that the property might be empty (but hoping that it isn’t).

2. Tolerance and Open-Texture Crispin Wright (1976, §2) defines a predicate F to be tolerant with respect to a concept f, ‘if there is . . . some positive degree of change in respect of f insufficient ever to affect the justice with which F applies to a particular case’. Vague predicates, or at least some vague predicates, are said to be tolerant. Wright’s language suggests that the issue concerns proper judgement, which I take to be an important insight. Consider the predicate ‘bald’, which is arguably tolerant with respect to number and arrangement of head hair. Suppose that two men h, h’ differ only marginally in the amount and arrangements of their head-hair. Wright’s principle seems to say that if someone competently judges h to be bald, then she must judge h’ to be bald too—either both are bald or neither are. Similarly, the predicate ‘red’ is tolerant with respect to indistinguishable, and perhaps small, differences in color. If two colored patches p, p’ are visually indistinguishable, and if someone judges p to be red, she must judge p’ to be red as well. As plausible as this interpretation of tolerance may seem, especially in the case of the colored patches, it leads to absurdity in the

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42 / Stewart Shapiro now familiar manner. In general, a Sorites series arises when we have a (prima facie) tolerant predicate P, and a series of objects running from a clear (or definite) instance of P to a clear non-instance of P, with each differing marginally from its neighbors. Perhaps we can agree on the following weaker principle of tolerance: Suppose a predicate P is tolerant, and that two objects a, a’ in the field of P differ only marginally in the relevant respect (on which P is tolerant). Then if one competently judges a to have P, then she cannot judge a’ to not have P. This, I submit, is the key to avoiding contradiction. Suppose that h and h’ differ only marginally in the amount and arrangements of their head-hair. It is compatible with the principle of tolerance that someone judge h to be bald and leave the bald-state of h’ unjudged (one way or the other). It does violate tolerance, in the present sense, if a subject judges h to be bald and decides to leave h’ unjudged. The observation here is that the subject does not violate tolerance if he has not considered the state of h’. The principle of tolerance demands that if this person were then asked, or forced, to judge h’, and does not change his mind about h, then he must judge h’ to be bald. Suppose that our subject judges h to be bald and we inquire about h’. Then he can satisfy the principle of tolerance by either judging h’ to be bald, or by judging h’ to be not bald and taking back his previous judgement that h is bald. I suggest that in the cases of interest, the meaning of the word ‘bald’ and the semantic and non-semantic facts allow this option. Let P be a tolerant predicate for which there is a Sorites series: a list a1 , ... , an , where it is determinate that Pa1 , it is determinate that :Pan , and for each i < n, the difference between ai and aiþ1 is insufficient to affect the justice with which P applies. As noted above, I take it as a premise (without much in the way of argument) that somewhere in the series there is a borderline case of P. There is a j such that Paj is neither determinately true nor determinately false. Since P is a predicate of a natural language like English, and aj is in its range of applicability, our ‘thoughts and practices in using the language have established truth conditions for’ the sentence Paj . My premise is that, despite this, the (presumably) non-linguistic facts have not determined that these conditions are met. Moreover, language users have established truth conditions for :Paj , and those conditions have not been met either. As far as the language has evolved to date, Paj is still open. As McGee and McLaughlin put it, the sentence is ‘unsettled’.

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Vagueness and Conversation / 43 To be sure, this is not to say that a given competent speaker, when asked about Paj (under normal conditions, whatever those are), realizes that the sentence is indeterminate, or has the phenomenological feel that there is nothing to say or that she can go either way. Some competent speakers may feel that way, but other competent speakers may be inclined to assert Paj , while still others might be inclined to assert :Paj —all under the same circumstances. The claim now is only that if Paj is borderline, then the meanings of the words, the non-linguistic facts, etc. do not determine a verdict, even if some individual language users do themselves competently assert a verdict in some circumstances. The presence of ‘unsettled’ cases is precisely the sort of situation that suggests a supervaluational semantics, much like the one provided by McGee and McLaughlin (and Fine 1975). The idea is to introduce a notion of an admissible sharpening, or precisification, of the language. In the case at hand, a sharpening of the language would contain a sharp (i.e. non-vague) predicate P
such that for all x in the range of P, if Px is determinately true, then P
x, and if :Px is determinately true, then :P
x. A sentence F is super-true if it comes out true under all admissible sharpenings of the language. So super-truth coincides with determinate truth, as that notion is defined above. I agree that the overall supervaluational framework is natural and helpful here—eventually—but there are features of the use (and thus the meaning) of vague terms that need to be incorporated into it. Since we encounter, and effectively and competently decide, borderline cases regularly, there is more to truth than super-truth, and there is more to validity than the necessary preservation of super-truth. Suppose, again, that a is a borderline case of P. I take it as another premise that, in at least some situations, a speaker is free to assert Pa and free to assert :Pa, without offending against the meanings of the terms, or against any other rule of language use. Unsettled entails open. The rules of language use, as they are fixed by what we say and do, allow someone to go either way in the borderline region. Let us call this the open-texture thesis. Wright (1987: 244) seems to endorse the open-texture thesis (at least parenthetically): ‘Borderline cases are . . . cases about which competent speakers are allowed to differ.’ And Sainsbury (1990, §9): ‘Given the nature of boundarylessness, semantics give freedom. There is some number of minutes such that the nature of the concept of a person, together with the nature of the world, makes it neither mandatory nor impermissible to apply the concept to a foetus of that age in minutes.’

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44 / Stewart Shapiro Open-texture is a more or less empirical claim about the proper use of vague terms in language. However, I have very little in the way of empirical evidence to offer for it from this armchair, beyond a few intuitions. That’s why I call it a ‘premise’. I realize that with the assertion of open-texture, I am leaving the comfort of consensus and truism (if it has not been left already). Some supervaluationists claim that if a is a borderline case of P, then a lies outside the extension of P, and so it is (determinately) incorrect to assert Pa. It is a platitude that one should assert only truths (of which, see Section 6 below). And, again, the supervaluationist idea is that truth is super-truth. Since, by hypothesis, Pa is not super-true, it is not correct to assert it—or so argue these supervaluationists. Similarly, those inclined toward a fuzzy approach claim that if a is a borderline case of P, then Pa is less than completely true. Since, strictly speaking, one should assert only full truths, these theorists hold that in strict circumstances, it is not correct to assert Pa, in agreement with our supervaluationists. The fuzzy logicians might add that even in less strict situations, one should assert only near truths, and some borderline cases are not near truths. In contrast, I would take the premise of open-texture to apply generally, to any borderline case, no matter how ‘far’ it is from a determinate one. I will not pause to offer arguments against these opposing views, or the intuitions that lie behind them, but will rest content to let the entire framework serve as a rival account. Let’s be clear as to what the open-texture premise is, and what it is not. Suppose that a is a borderline case of P. As above, the open-texture thesis does not entail that a given competent speaker will always be conscious of the fact that he can go either way. The open-texture thesis is that in some circumstances, he can, in fact, go either way without offending against the meaning of the terms, the non-linguistic facts, and the like. Second, even if a is a borderline case of P, it is not true that the rules for language use allow a speaker to assert Pa in any situation whatsoever. For example, one is not free to assert Pa if one has just asserted (and does not retract) :Pa. This would offend against logic (dialetheism notwithstanding). Similarly, one is not normally free to assert Pa if one has just asserted (and does not retract) :Pa’, where a’ is only marginally different from a. That would offend against tolerance.1 1 This is not to say that tolerance can never be violated. To follow an example from Sainsbury (1990), consider an art shop that has red paints on one shelf, marked ‘‘red’’, and orange ones on another, labeled ‘‘orange’’. The proprietor would not be judged incompetent concerning English if one cannot really tell the last red jar from the first orange one.

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Vagueness and Conversation / 45 In short, if a sentence F is determinate, then one can correctly assert it at any time and can never correctly deny it. If :F is determinate, then one can correctly deny F at any time, and can never assert it. If F is unsettled, then it depends on the conversational situation one is in. We need a mechanism to track the features of a situation that allow and disallow certain moves. For this, I turn to Lewis (1979).

3. Conversational Score It is often noted that the truth values of instances of vague predicates are often relative to a comparison class. A short professional basketball player would not be a short business executive. A person can be wealthy in some nation, or neighborhood, while someone else with exactly the same resources is not wealthy in other situations. Similarly, the truth value of an instance of a vague predicate can vary with an instance to which it is compared. An income may be paltry with respect to Bill Gates, but not so with respect to George W. Bush. Or, to follow Graff (2000, §3), a man can be bald with respect to Yul Brynner and not so with respect to Mikhail Gorbachev (even though the comparison class is the same in both cases—all men). The present program introduces another relativity, to a conversational context. This includes not only the environment of the conversation, what it is about, and the implicit or explicit comparison class or paradigms, but also what has already been said in the course of the conversation. For this purpose, I borrow the notion of a conversational score, or conversational record, invoked by some linguists interested in pragmatics. The record, or score, is a local version of common knowledge. David Lewis’s influential ‘Scorekeeping in a Language Game’ (1979) develops the idea in sufficient detail for present purposes. During a conversation, the score contains the assumptions, presuppositions, and other items implicitly or explicitly agreed to. For example, the conversational score contains the range of quantifiers like ‘everyone’ and the denotata of proper names like ‘Barbara’ and ‘Joe’. It also settles the relevant comparison class and/or paradigm cases for predicates like ‘tall’ (professional basketball players, executives, etc.) and ‘bald’ (Brynner, Gorbachev). The score also contains propositions that have been (implicitly or explicitly) agreed to, and are not up for dispute or discussion, at least for the moment. Among these are the ‘presuppositions’ to the conversation.

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46 / Stewart Shapiro The conversational record is a sort of running database. It is continually updated, in that items are put on the record and, notably here, removed from it in the course of a conversation. Items get removed when the topic changes, when what is agreed to or some presupposition comes into question, or when some of the participants change their minds about items that are on the record. Since the participants in a conversation may be mistaken about what has been agreed to and what has not, they may be mistaken about what is on the conversational score. Normally, this sort of thing gets cleared up in due course, if it turns out to be important. Lewis (1979: 345) delimits some features of the conversational database, most of which are relevant here: (1) . . . the components of a conversational score at a given stage are abstract entities. They may not be numbers, but they are other set-theoretic constructs: sets of presupposed propositions, boundaries between permissible and impermissible courses of action, or the like. (2) What play is correct depends on the score. Sentences depend for their truth value, or for their acceptability in other respects, on the components of the conversational score at the stage of the conversation when they are uttered . . . [T]he constituents of an uttered sentence—subsentences, names, predicates, etc.—may depend on the score for their intension or extension. (3) Score evolves in a more-or-less rule-governed way. There are rules that specify the kinematics of score: (3)

If at time t the conversational score is s, and if between time t and time t’, the course of the conversation is c, then at time t’ . . . the score is some member of the class S of possible scores, where S is determined in some way by s and c.

(4) The conversationalists may conform to directives, or may simply desire, that they strive to steer components of the conversational score in certain directions . . . (5) To the extent that conversational score is determined, given the history of the conversation and the rules that specify its kinematics, these rules can be regarded as constitutive rules akin to definitions.

Lewis points out that unlike most games, conversations tend to be cooperative. Indeed, that is often the point of having a conversation in the first place. ‘[R]ules of accommodation . . . figure prominently among the rules governing the kinematics of conversational score’ (1979: 347). The idea is that the conversational record tends to evolve in such a way that, other things equal, whatever is said will be construed as to count as correct. The record will be updated to make this so, if possible. Suppose, for example, that

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Vagueness and Conversation / 47 someone utters a sentence like ‘Harry no longer smokes unfiltered cigarettes’, which has the presupposition that Harry used to smoke unfiltered cigarettes. Unless someone objects, the presupposition goes on the record. To be sure, cooperation is ‘not inevitable, but only a tendency’, as Lewis puts it. And, of course, presuppositions can be retracted later. Lewis goes on to illuminate six diverse features of the semantics and pragmatics of natural language by using this notion of conversational score. The list includes definite descriptions, performatives, and a painfully brief treatment of our present topic, vagueness (1979: 351–4). That discussion begins by noting that with a vague predicate like ‘bald’, ‘nothing in our use of language’ fixes a sharp border between the bald and the non-bald. This much echoes the above McGee–McLaughlin theme, and like them, Lewis gestures toward the framework of supervaluation. Suppose that Fred is a borderline case of ‘bald’. Then whether the sentence ‘Fred is bald’ is true depends on ‘where you draw the line’. Relative to ‘some perfectly reasonable ways of drawing a precise boundary between bald and non-bald, the sentence is true. Relative to other delineations, no less reasonable, it is false.’ Philosophers and linguists cannot settle on a single, precise border, but ‘must consider the entire range of reasonable delineations’. Lewis says that if ‘a sentence is true over the entire range, true no matter how we draw the line, surely we are entitled to treat it simply as true’. In other words, simple truth is super-truth. But this is truth only in a strict sense. Lewis goes on to recognize a looser use of language: But also we treat a sentence more or less as if it is simply [i.e. super-]true, if it is true over a large enough part of the range of delineations of vagueness. (For short, if it is true enough.) If a sentence is true enough (according to our beliefs) we are willing to assert it, assent to it without qualification . . .

He notes that we usually do not get into any trouble with this loose use of language, although we can and sometimes we do—witness Sorites paradoxes. Lewis (1979) uses the notion of conversational score to resolve an issue that concerns his notion of ‘true enough’: When is a sentence true enough? Which are the ‘large enough’ parts of the range of delineations of vagueness? This is itself a vague matter. More important for our present purposes, it is something that depends on context. What is true enough on one occasion is not true enough on another. The standards of precision in force are different from one conversation to another, and may change in the course of a single

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48 / Stewart Shapiro conversation. Austin’s ‘France is hexagonal’ is a good example of a sentence that is true enough for many contexts, but not true enough for many others. Under low standards of precision it is acceptable. Raise the standards and it loses its acceptability.

To adapt Lewis’s example, suppose that someone says ‘France is hexagonal’, and gets away with it in a conversation. The rules of accommodation suggest that appropriately loose standards thereby go on the conversational record, since such standards are needed to make the assertion true (i.e. true enough). Thereafter, ‘Italy is boot shaped’ is also true enough, since that conforms to the same low standard (assuming that Italy is at least as boot shaped as France is hexagonal). But if someone later denies that Italy is boot shaped, insisting on the differences, then the standards are thereby raised, and ‘France is hexagonal’ is no longer true enough. So it comes off the record. Lewis then provides a clean, effective reply to Peter Unger’s (1975: 65–8) argument that nothing (or hardly anything) is flat. Unger claims, and Lewis agrees, that nothing can be flatter than something that is flat. So suppose someone says that Kansas is flat. This cannot be correct, since a level sidewalk is flatter than Kansas. But this sidewalk is not flat either, since a sheet of metal is flatter than that. And a sheet of metal is not flat, since . . . So we can only say that Euclidean planes are flat. No physical objects qualify. Clearly, Unger’s conclusion offends against language use. We use the word ‘flat’ in all sorts of contexts, and presumably we know what we are talking about. We also manage to communicate using the word. To echo Humpty Dumpty, who is master? To be sure, someone who says that Kansas is flat does not mean that it is perfectly flat. To use Lewis’s terminology, the speaker says that it is flat enough. When the remark about Kansas is made, the rules of accommodation require a relatively low standard, since Kansas does have small hills here and there (not to mention the curvature of the earth). Now suppose that later in the conversation, someone says that his sidewalk is flatter than Kansas, or complains that his sidewalk is not flat. According to Lewis, the truth of this assertion requires a higher standard for ‘flat’ (or ‘flat enough’) than had been established with the statement about Kansas. The rules of accommodation thus put a higher standard on the record. On this new standard, Kansas is not flat (enough). So once the standards are raised, ‘Kansas is flat’ is removed from the record.

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Vagueness and Conversation / 49

4. Conversation and Sorites Unfortunately, this is the extent of the treatment of vagueness in Lewis (1979), and I am not aware of any use of conversational score with respect to vagueness in his other writings (including his extensive writing on vagueness). In particular, Lewis (1979) does not invoke conversational score to deal with Sorites. Indeed, the only (possible) reference to the paradox is a remark that loose thinking sometimes leads to trouble. Given the framework he proposes, Lewis accepts the standard supervaluational resolution. Recall our premise of open-texture: borderline cases of vague predicates can go either way in some conversational contexts. So once a borderline case is ‘resolved’ in the course of a conversation, that information goes on the conversational record, and will remain on the record unless it is implicitly or explicitly retracted. To develop a fully general account of reasoning with a conversational score, we would need a paraconsistent logic, since it is possible for conflicting propositions to be put on a record, without the group retracting any of them. The participants may not realize they have contradicted themselves. Even if they do realize the contradiction, they may not want to retract the statements, since they remain attracted to each of them individually, and do not know which one(s) to give up. Such is paradox. The presence of a contradiction on the conversational record (whether detected or not) does not commit the group to every proposition whatsoever. Nevertheless, I take it that consistency is a regulative ideal for a conversational record, in the sense that the participants try to achieve consistency, retracting questionable items from the score as needed (Priest 1987 notwithstanding). For present purposes, however, we need not invoke paraconsistency, since our focus is on admittedly artificial cases in which consistency is easily enforced. Let us begin with a conversational version of a Horgan-style ‘forced march’ Sorites situation (see Horgan 1994). Suppose we have a series of 2,000 men lined up. The first is a mature Yul Brynner, who is clearly bald—he has no hair at all (or so we will assume). The last man is Jerry Garcia, in his prime, who we will take as a paradigm case of a man who is not bald. The hair of each man in the series (who has hair) is arranged in roughly the same way as his predecessor. After the first, each man differs from the one before by having only slightly more hair, perhaps imperceptibly more. Now suppose that the participants in a conversation start asking themselves about the baldness-state of each man in the series, starting with Yul

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50 / Stewart Shapiro Brynner, and they insist on a communal verdict in each case. As each question in the form ‘Is man n bald?’ is put, they are to provide a ‘yes or no’ answer. This forced bivalence is only a convenience. The situation would be essentially the same if we gave them the option to answer ‘borderline bald’, ‘unsettled’, ‘no fact of the matter’, etc. It would also be the same if we gave them the option to agree on silence. To paraphrase Raffman (1994: 41 n. 1), it isn’t merely that there is tolerance between (determinate) cases of baldness and (determinate) cases of non-baldness. There is tolerance between baldness ‘and any other category—even a ‘‘borderline’’ category’. Or Wright (1976, §1): ‘no sharp distinction may be drawn between cases where it is definitely correct to apply [a vague] predicate and cases of any other sort’. What matters for present purposes is that the participants in our conversation must answer (or refuse to answer) by consensus—whatever the allowed answers may be. If, at any point, they want to stop answering, they must agree to shut up. We do not allow them to stop the march by simply failing to agree on a verdict. So let us allow the simplifying assumption that only two answers are allowed: ‘bald’ and ‘not bald’. Being competent speakers of English, the conversationalists all agree that Yul Brynner is bald, that the second man is bald, etc. Eventually, they will move into the borderline area, and encounter cases whose baldness state is ‘unsettled’, as in Section 1 above. Again, the thoughts and practices in using the language have established truth conditions for statements about baldness, and truth conditions for non-baldness. In the borderline region, the non-linguistic facts have not determined that either of these truth conditions are met. Nevertheless, the conversationalists in this exercise will probably continue to call the men bald as they move through the borderline area—for a while. If they call man n bald, they will probably call man n þ 1 bald as well, since by hypothesis, they can barely tell the two heads apart in the relevant respect. This is all right, so far as their language competence goes. Given our opentexture premise, borderline cases can go either way (without offending against meaning and the facts), and the participants in this conversation are just going one way rather than the other as they enter the borderline area from this direction. Again, it does not matter whether they realize that they can go either way in any of these borderline cases. For some of the conversationalists and for some of the cases, it certainly won’t feel like they can go either way. Following their instructions, they just call ’em as they see ’em.

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Vagueness and Conversation / 51 And as they start to move into the borderline cases, they continue to see ’em to be bald—for a bit. This puts propositions like ‘Man 923 is bald’ and ‘Man 924 is bald’ on the conversational score (assuming that those are borderline cases). Now recall the principle of tolerance: Suppose a predicate P is tolerant, and that two objects a, a’ in the field of P differ only marginally in the relevant respect (on which P is tolerant). Then if one competently judges a to have P, then she cannot judge a’ to not have P. Let us assume that tolerance is in force in the present conversation. Arguably, it is part of the meaning of ‘bald’ that it is tolerant, and so there is usually no option to violate tolerance while being true to the meaning of the words of the language (but see note 1 above). In particular, if we were to present any adjacent pair from the series to the conversationalists at the same time, they would agree that either both of the men are bald or that neither are (or that both are borderline, etc., if we suspend the forced bivalence). This explains why they are likely to see man 924 as bald if they have just declared that man 923 is bald. Nevertheless, since the participants in the exercise are competent speakers of English, we can be sure that they will not blindly go through the entire series, and call #2000, Jerry Garcia, bald. That way lies madness (or at least incompetence). Eventually, a few of the participants will demur from calling one of the men bald. If this group finds themselves in a small minority, they will go along with the majority, to maintain the consensus—for a while. As the group proceeds through the borderline area, more and more of the participants will demur, or feel like demurring. At some point, enough of them will demur that the consensus on ‘this man is bald’ will break down. Given the ‘forced march’ instruction, where they must achieve consensus on each answer, the group will eventually agree to call one of the men ‘not bald’. Suppose this first happens with #975. Recall that they have just agreed, perhaps reluctantly for many of them, that #974 is bald, and so they put ‘Man 974 is bald’ on the conversational score. Of course, when they then said that #975 is not bald, they did not contradict themselves. Nor have they violated the tolerance principle. At that point, the tolerance applies in reverse—we take the contrapositive. In declaring man 975 to be not bald, they implicitly deny that man 974 is bald, and so ‘Man 974 is bald’ is removed from the conversational record. It is similar to what happens when any

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52 / Stewart Shapiro presupposition is challenged (or contradicted) in the course of a conversation. The event as described here is quite similar to the outcome in one of Lewis’s scenarios, recounted in the previous section. In that story, the participants in a conversation first agree to a ‘low’ standard when they accept ‘France is hexagonal’. Later, when they demur from ‘Italy is boot shaped’, the standard is raised, and so ‘France is hexagonal’ is implicitly removed from the record. Similarly, when the present conversationalists explicitly declare that #975 is not bald, they implicitly retract the statement that #974 is bald. In short, the conversational score is the device that enforces tolerance. Just as ‘Man 974 is bald’ comes off the record, so does ‘Man 973 is bald’; ditto for a few more of their recent pronouncements. We cannot and thankfully need not set a precise boundary as to how many sentences are removed from the record once they jump. Exactly what is and what is not on the conversational record is itself a vague matter. Typically, borderline cases of ‘what is on the record’ do not interfere with a conversation. If a question arises about a specific case—say #967—the participants can ask each other about that fellow, and so the status of that case can be made explicit. So let us continue the thought experiment by reversing the order of query. We assume that man #975 is the first ‘jump’, where our participants shift and deny that the man is bald. Suppose that we explicitly ask them about #974 again, after reminding them that they just called that man ‘bald’, and that they can barely distinguish #974 from #975 (if at all). Although I don’t intend to apply for a grant to confirm this empirical hypothesis, I’d speculate that they would explicitly retract that judgement, saying that #974 is not bald (and thus put ‘Man 974 is not bald’ on the record). The new consensus on the non-baldness of #975 will spread backward to cover #974. Suppose that we then ask them about #973. They would retract that judgement as well. Then we can ask about #972. Of course, they will not move all the way back down the series, and end up denying that #1, Yul Brynner, is bald. After all, he has no hair at all. At some point, they will jump again, and declare a certain fellow to be bald—suppose it is man 864. This again will result in the removal of certain items from the conversational record, such as the denial that man 865 is bald. If our conversationalists do not lose patience with us, we can then go back up the series, but the pattern is clear enough. They would declare men they encounter to be bald for a while, eventually jumping. In general, our conversationalists will move backward and forward through the

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Vagueness and Conversation / 53 borderline area. Tolerance is enforced at every stage, by removing judgements from the conversational record whenever a jump occurs. Notice, incidentally, that pending empirical research, there is no reason to think that the participants will always jump at the same place(s) as they move back and forth through the middle part of the series. We only know that come hell or high water, they will eventually jump, well before they call Jerry Garcia bald and well before they deny that Yul Brynner is bald—assuming of course that they remain competent speakers of English and do not go batty as a result of this experiment. We know this, since we are ourselves competent speakers of English. So how, exactly, is the Sorites paradox resolved? Consider the version that uses a single, inductive premise: for each i < n, if Pai then Paiþ1 . It is outright false that for each i < 2000, if our conversationalists judge man i to be bald, then they will in fact judge man i þ 1 to be bald. As they go through the forced march, sooner or later they will jump. And this jump does not undermine their competence as speakers of English. Indeed, if they did not jump, and went on to call man 2000, Jerry Garcia, bald, they would thereby display incompetence. Moreover, as we have seen, a jump does not violate tolerance, since it comes with a retraction of items from the conversational record. Something similar happens with the version of Sorites that avoids a single, inductive premise in favor of each instance of it. Whenever the conversationalists jump from ‘bald’ to ‘not bald’, they thereby undermine one of the premises in the long argument. Although the inductive premise is never true (in full splendor), notice that in the forced march version of the Sorites, the conversational score never contains what may be called a ‘strong counterexample’ to the inductive premise—so long as tolerance is in force. That is, there is never a number n, such that ‘Man n is bald’ and ‘Man n þ 1 is not bald’ are both on the record at the same time. If they jump to ‘not bald’ at n þ 1, then ‘Man n is bald’ is retracted, and removed from the score. In the scenario as envisioned, tolerance is enforced on the conversational record. This may just be a feature of the forced march, however. Suppose instead that the participants consider the men in the series in random order. Then they very well might judge man n to be bald and sometime later judge man n þ 1 to be not bald without explicitly or implicitly retracting the first

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54 / Stewart Shapiro judgement. Indeed, they might even judge man n to be not bald and later judge man n þ 1 to be bald—violating a penumbral connection. In fact, if they are not careful, they might even judge a man to be bald and a bit later judge the very same man to be not bald. In such cases, they have not noticed the violation of tolerance, penumbral connection, or consistency. But once the violation is pointed out, they will retract at least one of the judgements, and so one of the offending propositions will go off the record. One nice feature of the forced march is that tolerance, penumbral connection, and consistency are easily enforced. Not so in general. In the forced march, there will surely be cases m, where, for example, ‘Man m is bald’ is on the record and ‘Man m þ 1 is bald’ is not on the record. But that does not undermine tolerance, as that notion is defined above (Section 2). Since man m and man m þ 1 differ only marginally, the participants cannot judge them differently at the same time, but there is nothing to prevent them from judging one and not the other.2 It follows, incidentally, that the conversationalists cannot simultaneously judge every member of the series and remain competent in the use of the term ‘bald’, assuming again that tolerance remains in force. That is, the participants cannot have a judgement for every man on the conversational record all at once. If they did, they would either violate tolerance, or else they would call Yul Brynner not bald, or call Jerry Garcia bald. At any time, some of the baldness states will have to be left unjudged. That’s life. Perhaps they can get to a point where they have done a maximal amount of judging. Suppose, for example, that they judge every man from #1 to #974 to be bald, and they judge every man from #979 to #2000 to be not bald. Moreover, suppose that they cannot make a judgement about any of #975 through #978 without jumping—without retracting some of their previous judgements. Such is the nature of the series.

5. Raffman on Vagueness The present resolution of the Sorites paradox has the same structure as that of Raffman (1994, 1996), and I acknowledge a deep debt to that penetrating work. Raffman envisions a single subject who is marched through a Sorites 2 As noted in Sect. 2 above, it would violate tolerance if the conversationalists judge man n to be bald, and decide to leave man n þ 1 unjudged.

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Vagueness and Conversation / 55 series consisting of color patches. She is explicit, however, that her account is to apply to vague terms generally. To adapt her program to our series above, the subject would begin by calling the first few fellows bald, and continue that way a bit into (what I call) the borderline area. But she eventually jumps and calls one of the men not bald. Again, suppose that this first occurs at #975. Raffman says that at that point, the subject has changed to a different psychological state. In the new state, it is not true that, say, #974 is bald. Once again, it is a commonplace that many properties are relative to context. For example, a lightweight internal lineman among high school football players would be an extremely heavy jockey. That is what Raffman calls an ‘external’ relativity to a comparison class. For vague predicates generally, Raffman proposes that there is an additional, internal relativity, to a psychological state. Patches look red, or are red only relative to the state of a competent person judging them to look red. A man is bald, or not, only relative to the state of a competent person judging whether he is bald. Raffman argues that (something like the tolerance principle entails that) a competent subject cannot give different verdicts to ‘Man 974 is bald’ and ‘Man 975 is bald’ at the same time—and while in the same state. As above, if the subject is presented with any adjacent pair and asked to judge them both, she will agree that they are either both bald or neither bald: ‘at no time are adjacent [members] of the series simultaneously category-different’ (1994: 53). So it is never the case that an instance of the inductive premise is false. So long as the subject remains in the same state, she will not differ in her judgement concerning any adjacent pair of men in the series. Drawing on some psychological research, Raffman (1994: 50) proposes that when the subject jumps during the forced march, she moves into a new psychological state. There is a ‘category shift’ in which the judged category— ‘not bald’ in this case—‘spreads backward’ along a string of the preceding men in the series. In the new state, the extension of non-bald ‘expands backward, instantaneously’, to include some of the men that formerly fell in the extension of ‘bald’. To adapt Raffman’s narrative to the present example: My hypothesis is that, at the moment of judging [#975], the speaker undergoes a kind of Gestalt shift that embraces [#974] (and probably some of [his] predecessors) as well as [#975]. At the moment of shift to [‘not bald’], the speaker is disposed to judge both [#974] and [#975] (plus some [men] on both sides) as being [not bald], thereby allowing for a change in kind while preserving the effective continuity of the series.

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56 / Stewart Shapiro Intuitively speaking, a string of [men] shift their [baldness state] simultaneously, so that [#974] and [#975] never differ . . . at the same time. Like the duck-rabbit and Necker cube, these [men] can ‘go either way’: they can be ‘seen as [bald]’ or ‘seen as [not bald]’—now one way, now the other . . . If asked to reverse direction and retrace his steps down the series toward #1, the speaker would now judge [as not bald] some [men] that he previously judged [bald]. At some point, of course, he would shift back to [‘bald’]; for example he might judge [#974] through [#903] [not bald], but then undergo a Gestalt switch back to [‘bald’] at [#902]. And so forth . . . (Raffman 1996: 178) My claim is that whenever marginally different items are assigned incompatible predicates relative to the same external context, a Gestalt-like shift has occurred so that those predicates are assigned relative to different internal contexts. (Raffman 1996: 180)

The reference to gestalt shifts is suggestive. A subject’s perception of a Necker cube, for example, changes without any changes in the drawing itself. With a vague predicate, the judgement shifts with a small but perhaps imperceptible change in the item being judged.3 The relativity to psychological state allows for an distinction between a ‘categorical judgement’, which consists of a subject considering a single case, and a ‘comparative judgment’, where the subject deals with two (or more) instances at once. Raffman illustrates this, and the relativity, by envisioning a forced march in which the items are presented in pairs. Let us return to our series of 2,000 men. We first ask the subject, or the participants, to judge the pair <#1,#2>. Both are judged to be bald. Then we present the pair <#2,#3>. Both bald. Then <#3,#4>, and so on. Since the pairs are marginally different, perhaps indistinguishable, the subject(s) will always judge each pair alike. But, again, the subject(s) will eventually jump. They won’t go on to call Jerry Garcia and his predecessor both bald. Suppose that the pair <#887,#888> are judged to be bald, but the pair <#888,#889> are judged to be not bald a few seconds later. If we do not allow for a contextual shift of some sort—be it a change in psychological state, a change in the conversational record, or something else—the subjects will have contradicted themselves at that point. Since man 888 is in both pairs, he is judged to be bald and then judged to be not bald a moment later. There are not many options for interpreting this scenario. The theorist can conclude that vague predicates like ‘bald’ have some contradictory cases, or that any language (like English) with vague predicates is incoherent generally 3 Thanks to an anonymous referee.

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Vagueness and Conversation / 57 (following Dummett 1975). Or the theorist can find fault with the scenario. Perhaps there is something illegitimate in demanding a forced march through a Sorites series. I presume that Williamson (1994) would counsel this reply. According to this staunch epistemicist, the subject(s) should refuse to answer when they get near the (precise) border, despite their instructions. Since they do not and cannot know where the border is, they have no business venturing an opinion on the state of baldness of the fellows near that border. Or the theorist can claim that the subject(s) are incompetent, or have made some sort of mistake with their answers. But what sort of mistake? The subjects are told to call them as they see them. What did they do wrong? Or the theorist can follow Raffman and the present program and say that a shift of context has occurred. Man 988 is bald in one context and not bald in another. Raffman’s contextualist framework complements the present one. She sees the extensions of vague predicates as varying with the psychological states of (competent) speakers of the language, while I see the extensions as varying with conversational records among (competent) speakers. In one sense, her thought experiment is a special case of the present one. We might think of hers as a limiting case of a ‘conversation’ consisting of only one participant. The ‘conversational score’ would contain features of the psychological state of the subject. Of course, a conversation consists of its individual members, and Raffman’s notion of psychological state is intended to explain what happens in the mind of those individuals. The communal, conversational analogue of Raffman’s gestalt shift is what I call a ‘jump’, the combination of a breakdown of consensus among the members of the conversation, and the forging of a new consensus on a new verdict. A consensus breaks down only if enough of the individual participants demur from the communal judgement, and if Raffman is correct this happens when enough of them have made the relevant gestalt shift (and refuse to switch back even for the sake of conversational harmony). So if Raffman’s account can be sustained, it is more basic than mine. I have little to contribute to the articulation of the relevant notion of psychological state, nor to the psychological explanation of individuals as they judge vague predicates. The present account focuses instead of the communal aspects of language use, such as the role of language in public communication. Surely, it is individuals who communicate, and these individuals are in different psychological states at different times. I do not have much to say by way of a detailed

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58 / Stewart Shapiro account of how this works. What is the bridge between the different psychological states of individual communicators and the public side of communication? The cooperative features of conversations that Lewis (1979) sketches and, in particular, the defeasible drive for consensus, feed into the role of conversational record in handling borderline cases of vague predicates. They suggest the more communal sort of relativity that the present work focuses upon.

6. Determinacy Revisited, Truth, and Classical Semantics The time has come to further articulate, or refine, the notion of determinacy, to indicate the contextual elements that are relevant to its extension.4 Let F be a monadic (possibly complex) predicate in a natural language. Recall the previous definition of determinacy, quoted from McGee and McLaughlin (1994, §2): ‘to say that an object a is [determinately] an F means that the thoughts and practices of speakers of the language determine conditions of application for . . . F, and the facts about a determine that these conditions are met’. A first and hasty interpretation of this is that a is determinately an F if and only if the (linguistic) meaning of the predicate F and the non-linguistic facts determine or guarantee that Fa holds. In short, Fa is determinate in this sense if its truth supervenes on meaning and non-linguistic fact. Since English is a public language, it seems plausible that the meaning of its words is at least largely independent of the context of utterance (ambiguity and considerations like those in Davidson (1986) notwithstanding). Otherwise, we are left with the specter of Humpty Dumpty’s claim to Alice, and no real communication. And, of course, most non-linguistic facts (such as the amount and arrangement of hair on a person’s head) are independent of the context of utterance. So the present interpretation of determinacy would allow virtually no contextual elements into the notion. Suppose, for example, that Harry just went to see a baseball game, the only game in town at the time. No vagueness about that (or none that matters here). Still, the sentence ‘Harry went to the game’ would not be determinately true, since nothing about the meaning of the words and the relevant facts about Harry, the players, etc., determines 4 I am indebted to Agustı´n Rayo here.

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Vagueness and Conversation / 59 that this sentence is true. The (linguistic) meaning of the phrase ‘went to the game’ does not fix which game we are talking about. Similarly, on the present interpretation, the sentence ‘Everyone is present and accounted for’ would not be determinately true (in any context), since the meanings of the words by themselves do not fix who we are talking about. The sentence, ‘I am hungry’, would also fail to be determinately true in any context, since meaning and relevant facts do not determine the referent of ‘I’. McGee and McLaughlin did not have such an austere notion of determinacy in mind. Surely, ‘the thoughts and practices of speakers of the language’ conspire to fix some items in various contexts of utterance. In the first example, the members of the conversation would determine which game they are talking about, by what they say, do, and think. So ‘Harry went to the game’ would be determinately true, on the McGee and McLaughlin notion. Similarly, in a given context, the sentence ‘Everyone is present and accounted for’ would be determinately true or determinately false (unless there is some vagueness involved). The conversation would fix the extent of ‘everyone’, and this information would be on the conversational score. If a hungry speaker says ‘I am hungry’, then what she says is determinately true. I conclude, then, that at least some contextual elements should be included in the analysis of determinacy. In particular, when it comes to vagueness, determinacy is sensitive to the comparison class, the paradigms, or the comparison cases. Suppose, for example, that during a conversation about professional basketball players, somebody says that a player who is six feet one inch tall is short. That sentence is determinately true (in that context). The conversational record would indicate the comparison class, and given that class, the sentence is true. So far, I presume that everything is in line with the notion of determinacy invoked by McGee and McLaughlin (1994). However, I suspect that I am about to part company with them. I propose that the borderline cases of a vague predicate that have been decided in the course of a conversation not be included in what fixes determinate truth. Suppose that the relevant comparison class, paradigm cases, or comparison cases for ‘baldness’ are fixed. Assume that Harry remains a borderline case. According to the foregoing open-texture thesis, in some situations, speakers are free to assert ‘Harry is bald’ without undermining their competence, and they may likewise assert ‘Harry is not bald’ in some situations. Unsettled entails open. So suppose that someone asserts ‘Harry is bald’, in the course of a conversation, and this

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60 / Stewart Shapiro assertion goes unchallenged. As above, the sentence (or proposition) ‘Harry is bald’ goes on the conversational score. One might take it that ‘Harry is bald’ would then be determinately true in the context after the utterance, at least until the assertion is retracted (in which case the context changes). After all, once the utterance is accepted onto the record, the thoughts and practices of (those) speakers have fixed a truth value for this sentence. Nevertheless, I propose that ‘Harry is bald’ is not determinately true in the context of this conversation. As noted in Section 5 above, Raffman (1996) distinguishes the ‘external’ context of a judgement (concerning a vague predicate), which fixes the comparison class and the like, from the internal context, consisting of the subject’s psychological state. Just about everyone accepts that the extensions of vague predicates can vary with the external context. Raffman’s thesis is that these extensions also vary with the internal, psychological context. Once again, I invoke a similar distinction here, in the broader perspective of public conversations. The present proposal is that ‘external’ factors—comparison class, paradigm cases, comparison cases, etc.—are included in what fixes determinate truth and determinate falsehood, but internal factors, and in particular, the decided borderlines cases, do not modify the extension of determinate truth and determinate falsehood. When it matters, I will use the phrase ‘e-determinate’ to indicate the present notion, and distinguish it from the broader McGee–McLaughlin notion, where all contextually fixed factors, including decided borderline cases, are included. To mark the contrast, I’ll use the word ‘established’ for the broader McGee–McLaughlin notion. Since the word ‘determinacy’ is a term of art, I am free to define it as I please, so long as the definition is coherent (even if vague). Nevertheless, the reader deserves a word of justification for my notion of e-determinateness. One advantage of the broader notion of ‘established’ is that it allows, or may allow, the imposition of ordinary classical semantics. That is, within a fixed context, one can coherently maintain that vague predicates have precise extensions, and complementary anti-extensions. Raffman, for example, uses counterfactual conditionals to fix the extensions. Consider, for example, our paradigm Sorites series, consisting of 2,000 men ranging from Yul Brynner to Jerry Garcia. Fix what Raffman calls the external context, and consider a single competent subject in a particular psychological state s. Then a given man m is in the extension of ‘bald’ in s if it is the case that the subject would judge m to be bald when in state s, and m is in the anti-extension of ‘bald’

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Vagueness and Conversation / 61 otherwise. To be sure, at any given time, neither the observing subject nor an outside examiner can learn the baldness statuses of all of the men relative to the psychological state s, or at least not in the straightforward manner. As noted, for some of the men in the series, if the subject actually did judge them to be bald, or not bald, she would no longer be in the state s. Some judgements trigger a shift to a new psychological state. Once that happens, we are no longer able to determine the status of the other men in the original state s (unless we could get the subject back into s). Nevertheless, on Raffman’s view, for each external context and each psychological state, there is a fixed and precise extension for each vague predicate in that state, fixed by the indicated conditionals, most of which are counterfactual. Raffman’s proposal might be imported full-blown into the present conversational context. Let us define a man in our series to be in the extension of ‘bald’ at a given stage in a conversation, if the conversationalists would judge him to be bald, if they were asked that question. In the limit, however, we can actually determine the status of only a few of the men that way. Once we ask the conversationalists about any of the men in the borderline area, and get an answer, we thereby change the context, since the conversational record changes at that point. We are now at a new stage in the conversation, and all bets are off concerning the original stage. Nevertheless, the status of each man in the series is fixed at any given time by how the subjects would respond, if queried about that man. Another prominent contextualist, Delia Graff (2000), holds that at any given time, there are sharp borders in the Sorites series. The problem is that the borders shift with the focus of the subjects. The reason a subject never sees or notices the border is that it is never located at the place where she is looking—the border is never (and can never be) salient. Such is the nature of contextually determined vague predicates. Graff’s view is an epistemicism of sorts. With the imposition of precise extensions (and complementary antiextensions) in each context, the foregoing contextualists thus invoke an ordinary, classical model-theoretic semantics. With that comes classical logic, a neat package. To accomplish this, however, the Raffman framework assumes that the counterfactuals are all well-defined, and each has a unique truth value. In particular, there must be a fact of the matter concerning how the subject would respond in the given psychological state, if queried about a given case. If there is any vagueness, or indeterminacy, in the counterfactuals, the

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62 / Stewart Shapiro imposition of classical semantics might fail. The same goes for the potentially similar treatment in the conversational context. There must be a fixed fact of the matter concerning how the conversationalists would respond to each query. Similarly, Graff’s imposition of classical semantics stands or falls with the assumption that at any given time, there are indeed precise, but unknowable, borders that lie outside of the range of consideration. I do not intend to challenge these assumptions or premises of my fellow contextualists. Let me just register skeptical agnosticism concerning the presuppositions, and then concede, for the sake of argument, that we can indeed achieve classical semantics, and classical logic, if we invoke the broad notion of determinacy (i.e. establishment). That is, I assume for the sake of argument that if the external context and the internal psychological state are both held fixed, then we can and should reason classically. Note, however, that the relevant internal context can change very rapidly. In Raffman’s case, it is an empirical question exactly when, and how often, a given subject’s psychological state changes. At a minimum, the state changes whenever one of the counterfactuals if asked about man #i, the subject would respond ‘bald’ changes its truth value. It is not implausible that this happens every moment, or every few moments (given how delicate things can get in the borderline area, see § 8 below). In the present conversational framework, the context changes every time something is added to the score, which happens every time something not already established is said. For Graff, the extension of vague predicates changes every time the subject’s attention shifts more than a little. This is not a complaint against classical logic. It is a commonplace that one can reason classically only so long as the references of the terms and the extensions of the predicates in an argument remain fixed. We remind introductory students of this, noting the ancient fallacy of four terms and the like. On the views under discussion, the extensions change with every change in internal or external context, and we can reason correctly only if we are aware of this. This observation also explains why the Sorites reasoning seems so tempting, and it indicates how to resist the key premise. However, the constant shifting of context puts a damper on the appropriateness of classical logic when reasoning with vague predicates (if indeed it is appropriate). The supposedly precise extensions of the predicates can change in the very act of

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Vagueness and Conversation / 63 our considering them as we go through an argument, trying to reason in an ordinary context. We might call this a Heraclitus problem. The river changes every time we step into it. The extensions of vague predicates change momentarily, right before our eyes, even as we are considering the premises of an argument. As noted, I will not mount a challenge to the assumptions that lead to the imposition of classical model-theoretic semantics and classical logic. Nevertheless, I am interested in a notion of ‘context’ and of correct reasoning therein, where the shifts are not so erratic. It seems to me that it is plausible to hold the external context fixed for a period of time, since that does not change quite so rapidly, nor need it change without notice. Again, the external context fixes the comparison class, the paradigm cases, and/or the comparison cases (as well as the extensions of proper names and pronouns like ‘everyone’). I submit that this is the normal background of reasoning with vague predicates, and I am interested in the norms of correct reasoning in such contexts. Let the actual extensions of vague predicates vary within such a context, as we reason about them. For this purpose, the current notion of e-determinacy is an interesting and important one for the logic of vague predicates. Unfortunately, this can be pursued in detail only on another occasion. It is clear that e-determinate truth is not the same thing as truth. There is no analogue of the supervaluationist slogan that truth is super-truth. Suppose, again, that external factors are fixed and that Harry remains a borderline case of baldness. And suppose that a given competent subject judges Harry to be bald, in a given conversational context. Then it is true in that context that Harry is bald. Nevertheless, it is not e-determinately true that Harry is bald in that context. As far as e-determinacy goes, Harry remains a borderline case. There is, I presume, a norm of conversation that one should only assert truths (other things equal, of course).5 The norm is not that one should only assert e-determinate truths. That would preclude us from deciding any borderline cases, even temporarily. The norm is that one should only assert truths, and on the present account, truth varies with context. If Harry is borderline bald, and nothing on the conversational score (thus far) precludes judging Harry to be bald (e.g. it is not the case that ‘Harry is not bald’ is on the score), then a speaker cannot help satisfying the norm in sincerely asserting (or denying) Harry’s baldness. The acceptance of ‘Harry is bald’ 5 I am indebted to Patrick Greenough here.

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64 / Stewart Shapiro onto the conversational score thereby makes the sentence true in that context, and thus automatically satisfies the norm. Of course, this holds only for items in the borderline area. Anyone who calls Jerry Garcia bald or denies that Yul Brynner is bald has violated the norm.

7. What does all this Conversational Stuff have to do with Semantics and Logic? A common, perhaps natural, response to the present program (and to Raffman’s) is that it focuses on the pragmatics (or psychology) of the use of vague terms, and has nothing to do with semantics, and thus nothing to do with logic.6 The underlying objection seems to be that the correct resolution of the Sorites paradox should turn exclusively on the meaning of the terms. Logic flows from meaning. This presents a false dilemma. Raffman (1994: 43) writes that her ‘story is at bottom a psychological one, resting on a hypothesis about the mental representations that underlie our usage of vague words’ and so ‘where vague predicates are concerned, logic and semantics are more intimately entwined with psychology than might have otherwise been supposed’. The present story turns more on the pragmatics of conversations involving vague terms, and so I would conclude instead (or, better, in addition) that where vague predicates are concerned, logic and semantics are intimately entwined with pragmatics. Along similar lines (perhaps), Wright (1987: 277) proposes that we seek a semantics according to which linguistic competence is understood ‘on the model of a practical skill, comparable to the ability to . . . ride a bicycle’. Of course, unlike bicycle riding, we exhibit the practical skill of language mastery in concert with each other (more or less). Otherwise, there would be no communication, which is, after all, the point of this enterprise. The meanings of vague terms are intimately tied up with the proper display of this skill. As Humpty Dumpty ought to have put it, competent users of the language are its masters. I submit that the correct use of vague terms is bound up with psychology and pragmatics. The notion of open-texture (see Section 2 above), 6 Much of the argument in this section must be heavily revised, due in large part to the criticisms brought in Rosanna Keefe’s contribution to this volume (Ch. 4), and due to conversations with Diana Raffman and Crispin Wright.

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Vagueness and Conversation / 65 which is shared between the present account and Raffman’s (and Wright’s), shows how this is so. This requires at least some revisions of some widely held views on semantics, meaning, and extensions. Raffman (1994: 69–70) articulates the following biconditional: (B) [A]n item lies in a given category if and only if the relevant competent subject(s) would judge it to lie in that category.

See also Wright (1976, 1987). Although Raffman only invokes (B) in the context of vagueness, I would think that for a vast range of predicates—vague or not—in public languages, the thesis is in the neighborhood of a truism. A piece of metal is gold if and only if relevant competent subjects would judge it to be gold (if asked); an animal is a marsupial if and only if relevant competent subjects would judge it to be a marsupial; a natural number is prime if and only if relevant competent subjects would judge it to be prime. And a man is bald if and only if relevant competent subjects would judge him to be bald. My suggestion that many cases of (B) are true depends crucially on what it is to be a ‘relevant competent subject’. The claim, so far, is only that in a lot of cases, it is possible to specify what it is to be a relevant competent subject in such a way that (B) comes out largely true. Moreover, the ‘if and only if’ in (B) is only a material biconditional. No causal or semantic connections are postulated. In each case, even after the class of relevant competent subjects is specified, so that (B) is largely true (when it is), there is a question as to which is the chicken and which the egg. It is an instance of what Wright (1992: 108–40) calls the Euthyphro contrast. To adapt one of Wright’s examples, consider the following instance of (B): A story is funny if and only if competent subjects would judge it to be funny. In this case, competent subjects are those with normal senses of humor, and I presume that with this stipulation, there is no controversy over the truth of this instance of (B). It is plausible that in this case, the judgements of these competent judges are somehow constitutive of funniness. What makes a story funny is that people tend to find it funny. Plausibly, then, humor is responsedependent, or, to paraphrase Raffman, judgement-dependent. Let us call this the Euthyphro reading of (B).

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66 / Stewart Shapiro Contrast this with another case of (B): A number is prime if and only if competent subjects would judge it to be prime. In this case, the relevant competent subjects are those good at arithmetic, or perhaps (some) mathematicians. With this stipulation, the material biconditional is largely true, at least for smallish numbers. But in this case, of course, it is not the judgements of these competent subjects that make the number prime.7 A number is prime if it has exactly two (distinct) factors: itself and 1. A subject is competent here only if she gets it right most of the time, other things equal. In this case, competent judges track the truth; their judgements do not constitute it. Call this the Socrates reading of (B). A Euthyphro reading of an instance of (B) is a thesis about the extension of the predicate in question (‘funny’). A Socrates reading is a thesis about what it is to be competent. To be sure, I do not claim that this distinction is a sharp one, nor that it is an all or nothing matter. I presume that many predicates have a mixture of Socratic and Euthyphronic factors, and perhaps there are no, or very few, pure cases of either. Recall our current example: A man is bald if and only if competent subjects would judge him to be bald. In this case, the specification of ‘competent’ is rather broad. A subject is competent if she understands the language, and is accurately perceiving the man in question under normal conditions (so she can see at least roughly how much hair each of them has and how it is arranged). Raffman (1994: 70) writes that thesis ‘B is true with respect to borderline cases because our competent judgements of borderline cases determine their category memberships; conversely B is true with respect to clear cases because clear cases determine what counts as competent judgement.’ In present terminology, if the predicate is vague, then (B) is at least largely Socratic for clear cases. If someone asserts that Yul Brynner is not bald, or that Jerry Garcia in his prime is bald, then we would regard her as at least temporarily or locally incompetent—either as not understanding the meaning of ‘bald’, or not looking carefully, or misperceiving. But when it comes to borderline cases, (B) is to be read with Euthyphro. Since, by open-texture, borderline cases can go either 7 What is at stake here is the objectivity of the relevant discourse (see Wright 1992). Readers sympathetic to (semantic) anti-realist accounts of mathematics should pick another example. The present distinction is more complicated, if available at all, for anti-realists about all discourse.

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Vagueness and Conversation / 67 way, the judgements of otherwise competent subjects determine whether the man is bald—in the relevant conversational (or psychological) context.8 As Raffman (1994: 44) puts it early on, ‘an adequate treatment of vague predicates and their Sorites puzzles must appeal to the character of our judgements about the items in the series’. Every vague predicate is judgementdependent in its borderline area. In any case—and on either reading—thesis (B) is not a statement about the meaning of the predicate in question (whether it is vague or not). Raffman (1994: 58) writes that the ‘sorites . . . is solved independently of any particular meaning analysis of the predicate . . . On the contrary, all that is required to solve the puzzle is a claim about the correct application or extension . . . of the predicate at issue.’ A bit later, she elaborates: I do not claim that the meaning analysis or intension of a vague predicate includes a judgmental element. For instance, I do not claim that in calling an object red one means or is saying, in either the ‘speaker’ or ‘semantic’ sense, that the object is merely red-relative-to-me-now or red-relative-to-such-and-such-a-context . . . Rather, I claim that the extension of ‘red’—the class of objects that satisfy the predicate—is always relativized to certain psychological (and nonpsychological) contexts. The sorites is a puzzle about the correct application of vague predicates, and that is all my story addresses. (Raffman 1994: 66)

The first entry in Random House Webster’s Unabridged Dictionary for ‘bald’ reads ‘having little or no hair on the scalp’.9 There is no reason to challenge this definition, which makes no mention of—or even implicit reference to—the judgements of competent uses. Clearly, however, what counts as having little hair on the scalp depends on what Raffman calls the ‘external’ context. Is it a sales meeting of a company that makes shampoo, or a sales meeting of a company that makes toupees, or a sales meeting of a company that makes sunscreen? And if the foregoing account is correct, what counts as having little hair can also vary with the shifting internal context of a conversation (or, on Raffman’s account, it varies with psychological state). When Alice objected to Humpty Dumpty’s claim that his words mean just what he chooses them to mean, he replied that it is a question of who ‘is to be master’. Surely, our sympathies here are with Alice. Individual speakers are not ‘masters’ of the meanings (or intensions) of the words they use, 8 To be sure, ‘borderline bald’ and ‘competent speaker’ are both vague. Thus, it may not be determinate in a given case whether (B) is to be read with Socrates or with Euthyphro. 9 The relevant entry in the Oxford English Dictionary reads in part ‘having no hair on some part of the head where it would naturally grow’.

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68 / Stewart Shapiro independently of the thoughts and actions of the wider community of language users (up to the compelling points in Davidson 1986). But in the borderline region, individual, competent speakers are indeed ‘masters’ of the extensions of vague terms. Humpty Dumpty is right in this limited domain.

8. Underground in the Borderline Area Recall the Sorites series introduced in Section 4 above, which consists of 2,000 men ranging from Yul Brynner to Jerry Garcia in his prime. We envisioned a forced marched conversation through this series, starting with Yul Brynner. Given their instructions, and assuming their competence in English (and normal observation), the participants to the conversation will eventually jump and call one of the men not bald. Again, suppose that this first happens with man 975. If we were to run the same series with the same group of English speakers (in the same order) on another occasion, they might jump at a different place, say #967 or #984. They would almost certainly jump at a different place if they were marched through the series in the reverse order, starting with Jerry Garcia, and moving down. Presumably, something causes the jumps when they occur. The events are too macroscopic for quantum randomness to be involved (or are they?). The point here is that so long as the man being judged for baldness is in the borderline area, the meanings of the word ‘bald’, the external contextual factors, and the non-linguistic facts (about the number of and arrangements of his hair) do not determine a correct response. So as far as the normativity of meaning goes, they can go either way. Raffman (1996) emphasizes that the exact location of the jump carries ‘no normative force’; it underwrites ‘no distinction between correct and incorrect usage’. A different group, or the same group on another occasion, that jumps at a different place is not thereby in error. Raffman elaborates further: ‘there is no reason to [jump], hence no justification for [jumping], at any particular #n in the series (as opposed to #(n  1) or #(n þ 1)). If there were a reason, then either the predicate in question would not be vague or the differences between adjacent items in the series would not be marginal in the sense required to generate a paradox.’ By ‘reason’, Raffman presumably means a ‘semantic reason’. It is virtually analytic that if semantic rules (and non-linguistic facts) always determine when the jump is supposed to occur, then the predicate is not vague after all—there are no borderline cases.

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Vagueness and Conversation / 69 The situation is (vaguely) reminiscent of the old saw about Buridan’s ass. The animal finds itself exactly midway between two bales of hay. Since he does not have any reason to walk toward the first bale (as opposed to the second), he does not approach that bale. Similarly, he does not have any reason to go to the second bale (as opposed to the first). So the poor animal remains where it is, and starves to death. An unfortunate and irrational outcome. He really should have gone one way or the other, and it does not matter which. Similarly, our conversationalists do not have a semantically compelling reason to jump at man 974, nor do they have a semantically compelling reason to jump at #975, nor do they have a semantically compelling reason to jump at #976, etc. But they are required to jump somewhere, and they become semantically incompetent if they do not. Raffman (1994: 46) writes: Now one thing we know is that at some point on each (complete) run of judgements along the series . . . a [jump] just does occur: the subject’s slide down the slippery slope is broken . . . Just where the shift does occur on any given run will depend on a constellation of factors, including the direction in which the subject proceeds along the series, where in the series he begins his judgements, his perceptual state at the time, and so forth . . . But occur it does, on every run.

It is straightforward to adapt these observations to the present, social conversational context. In a given case, the exact location of the jump is a ‘brute mechanical’ matter (Raffman 1994: 65). Presumably, it will depend on how tired or attentive the subjects are, what else is on their minds at the time, their emotional state, and countless other factors. To be sure, these factors have nothing to do with the amount and arrangement of hair on the heads of the men in the series, and so are irrelevant to the meaning and thus the proper application of ‘bald’. But they are most relevant to its extension in the given context. They are relevant to what counts as having little or no hair on the scalp, in the given context. Raffman (1994: 53) adds that what ‘will or will not trigger the [jump] is not something to which we, as judging subjects, have access; so far as the subject is concerned, the [jump] simply occurs’. Presumably, the members of the conversation can feel the shift coming as they march through the Sorites series. When they move through the borderline area, they will find themselves less and less comfortable with their judgements that the men are bald, and the group will find it harder and harder to maintain consensus on

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70 / Stewart Shapiro baldness. The jump is not a blind process, in the sense that they have no idea when it will occur. After all, they are in charge of the situation—masters, as Humpty Dumpty would put it. But the jump is blind in the sense that the subjects need not be aware of the factors that actually trigger the shift—i.e. when enough is enough. An adaptation of an example from Wright (1987) illustrates the situation. Consider a digital tachometer connected to a motor. This is a ‘brute mechanical’ device if anything is. Suppose the device registers in units of 10 rpm, and suppose that the motor is running smoothly 300 rpm. The tachometer dutifully reads ‘300’. Now we slowly increase the speed of the motor in increments of 0.1 rpm. Of course, the tachometer will jump to 310 somewhere. Say it happens when the motor hits 306.2 rpm. If we speed the motor up a bit more, and then start slowing it down, the tachometer will jump back to 300 somewhere. But there is no reason to think that this jump back will occur at 306.2. In a typical case, the tachometer will keep its holding at 310 for a bit, and not jump back to 300 until later, say 305.7 rpm. If we did the experiment on another occasion—the next season, say—we might get slightly different results. It might first jump to 310 at 306.4 rpm and it might jump back to 300 at 305.8. On each occasion, the exact location of the tachometer’s jump from 300 to 310, or back, depends on a number of factors, such as the ambient temperature, relative humidity, wear and tear on the parts of the tachometer, fluctuations in the power source, etc. Of course, these things have nothing to do with the speed of the motor, and so are irrelevant to the correctness of the reading. For the practical purposes at hand, this does not matter. So long as the seemingly random fluctuations affect the reading only in the ‘borderline’ area—say, roughly 302.5 to 307.5 rpm—they will not keep the tachometer from operating correctly. When the motor is running at or near a ‘clear’ case, say 309.5 rpm, then these fluctuations do not affect the reading. The analogy with our conversationalists on a forced march is straightforward. When they are in the borderline area, they can go either way without impugning their competence, and which way they go in that area depends on a number of factors irrelevant to the baldness state of the men in the series. When they are considering a man who is not in the borderline area, then these side factors do not affect them, and they judge correctly (if they are competent and observing properly). With the tachometer and the Sorites series for baldness (or redness), the exact location of the jumps does seem to be completely arbitrary, in that it

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Vagueness and Conversation / 71 really does not matter at all where the jumps occur (so long as they occur in the borderline area). With some Sorites series, however, the jump point is not arbitrary. Consider a Sorites that goes from a spatially separated sperm–egg pair to a resulting 2-year-old baby developed from that pair, with increments of 0.01 seconds. The location of the borderline of the predicate ‘person’ or ‘human being’ (or ‘being with human rights’) has obvious moral ramifications. People who put the border in one place rather than another are not acting arbitrarily. They may well have compelling reasons for what they do. The point here is that there are no linguistic reasons for putting the border in one place rather than another. Presumably, both sides of the abortion debate are speaking the same language. If I consistently put the border in the region of viability, and a mob comes and burns down my house as a result, they will not be accusing me of linguistic incompetence (whatever else they may accuse me of ).10 Returning to our paradigm cases, for both the conversation about baldness and the tachometer, the jump point is a ‘brute mechanical’ process, and there is no principle of tolerance for it. Nature does allow sharp jumps (or at least apparent discontinuities). Raffman (1994: 56) writes that ‘we have no intuition that if [#874] does not trigger . . . a [jump] then neither will [#875]’, just as we have no intuition that if raising the motor to 306.1 rpm does not cause the tachometer to jump to 310, then neither will 306.2 rpm. To be sure, if our conversationalists jump too close to the clear cases, their competence as speakers of English (or their powers of observation) is compromised. Similarly, if our tachometer jumps at the wrong place—say, it reads 310 when the motor is running at 300.2 rpm—its accuracy will be compromised, and possibly also its usefulness (depending on what it is to be used for). Although there is no principle of tolerance in the operation of a mechanical device, the usefulness of such a device is a context-dependent, and probably also a judgement-dependent matter. How accurate the tachometer need be to be useful is a vague matter, and a principle of tolerance does hold for that. So how close to the border is ‘too close’ for competent speakers to jump? How inaccurate can the tachometer be without being ‘too inaccurate’ for a given purpose? Since these are also vague matters, one might think that we have just postponed (or moved) the problem of this work. This raises a problem in the neighborhood of higher-order vagueness, which is a topic for another day. 10 I am indebted to Carl Posy here. Notice that in any case, there will still be some arbitrariness in the jump point. Viability and, to a lesser extent, conception are themselves vague matters.

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72 / Stewart Shapiro

REFERENCES D a v i d s o n , D. (1986), ‘A Nice Derangement of Epitaphs’, in Richard Grandy and Richard Warner (eds.), Philosophical Grounds of Rationality (Oxford: Oxford University Press). D u m m e t t , M. (1975), ‘Wang’s Paradox’, Synthese, 30: 301–24; repr. in Keefe and Smith (1997: 99–118). F i n e , K. (1975), ‘Vagueness, Truth and Logic’, Synthese, 30: 265–300; repr. in Keefe and Smith (1979: 119–50). G r a f f , D. (2000), ‘Shifting Sands: An Interest-Relative Theory of Vagueness’, Philosophical Topics, 28: 45–81. H o r g a n , T. (1994), ‘Robust Vagueness and the Forced-March Sorites Paradox’, Philosophical Perspectives, 8: Logic and Language, 159–88. K e e f e , R., and P. Smith (1997), Vagueness: A Reader (Cambridge, Mass.: MIT Press). L e w i s , D. (1979), ‘Scorekeeping in a Language Game’, Journal of Philosophical Logic, 8: 339–59. M c G e e , Vann (1991), Truth, Vagueness, and Paradox (Indianapolis: Hackett). —— and B. McLaughlin (1994), ‘Distinctions without a Difference’, Southern Journal of Philosophy, 33 (suppl.), 203–51. P r i e s t , G. (1987), In Contradiction: A Study of the Transconsistent (Dordrecht: Nijhoff). R a f f m a n , D. (1994), ‘Vagueness without Paradox’, Philosophical Review, 103: 41–74. —— (1996), ‘Vagueness and Context Relativity’, Philosophical Studies, 81: 175–92. S a i n s b u r y , R. M. (1990), ‘Concepts without Boundaries’, Inaugural Lecture, pub. King’s College London, Department of Philosophy; repr. in Keefe and Smith (1997: 251–64). U n g e r , P. (1975), Ignorance (Oxford: Oxford University Press). W i l l i a m s o n , T. (1994), Vagueness (London: Routledge). W r i g h t , C. (1976), ‘Language Mastery and the Sorites Paradox’, in G. Evans and J. McDowell (eds.), Truth and Meaning: Essays in Semantics (Oxford: Oxford University Press) repr. in Keefe and Smith (1997: 151–73). —— (1987), ‘Further Reflections on the Sorites Paradox’, Philosophical Topics, 15: 227–90; repr. in Keefe and Smith (1997: 204–50). —— (1992), Truth and Objectivity (Cambridge, Mass.: Harvard University Press).

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4 Context, Vagueness, and the Sorites: Comments on Shapiro Rosanna Keefe

1. Introduction Shapiro and Raffman have, together, told an interesting and plausible psychological and pragmatic story about many of our judgements involving vague predicates. Shapiro explains how, if we focus on changes in conversational context (prompted by changes of score, in Lewis’s sense), then the fact that, when taken through a Sorites series, subjects will at some point jump from applying the Sorites predicate to denying that it applies, does not mean that those subjects are violating a plausible principle governing judgement, a principle that he labels the tolerance principle. In this paper I am particularly concerned to take up a question Shapiro himself addresses: ‘What does this have to do with semantics and logic?’ Is there a (relatively simple) way to get from facts about our judgements to facts about truth-conditions of the sentences expressed? In particular, what can facts about how subjects respond to members of a Sorites series tell us about the true classifications along that series? First, I will briefly explore some features of, and possible problems with, the use of conversational score (in Section 2). Then I’ll take up the question of the bridge between the pragmatic and the semantic. And finally, in Section 4, I will ask what response to the Sorites paradox Shapiro’s view can offer us. I would like to thank Dominic Gregory and Stewart Shapiro for discussion of the issues and of this paper.

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74 / Rosanna Keefe

2. The Pragmatic Story Shapiro, like Raffman, considers in detail the classifications of subjects who are faced with the members of a Sorites series for F in turn, and made to classify each of them as F or not-F. This kind of ‘forced march’ case is of particular interest, since the danger seems to be that the subject will end up classifying one member of the series as F and the very next one as not-F even though the two things seem to that subject to be exactly the same in the relevant respects. For, if they never do this, they will be led to an absurd misclassification of the late members of the series. But if they do, their behaviour will threaten various compelling principles about judgement. For example, Shapiro offers the following principle, which he calls a principle of tolerance: Suppose a predicate P is tolerant, and that two objects a, a’ in the field of P differ only marginally in the relevant respect (on which P is tolerant). Then if one competently judges a to have P, then she cannot judge a’ to not have P.

The challenge is thus to explain either how such a principle of tolerance, and other related principles, could fail to be true, or how the subject’s jump from F to not-F through their Sorites series does not really refute such principles after all. Shapiro’s strategy is basically the second of these. The principle of tolerance is not threatened (at least, not in the same way) if the context in which the subject judges that Fa is a different context from that in which she judges not-Fa’. Raffman’s response similarly takes this line. For Raffman, the context of those judgements changes because of a psychological change in the subject: things appear differently when the first judgement is made from how they appear when the second is made, and now it is no longer the case that the first judgement holds for that new context (e.g. what had seemed red now seems not-red). Shapiro concentrates instead on a shift in conversational context. The idea is to use Lewis’s account of how conversational context can change with a ‘change in score’ when different presuppositions are introduced into the conversation, or different standards for the application of predicates are imposed (e.g. suppose in this context this desk is considered to be flat—we could deny that it is flat, but that would be to change the score). We can then see that if, through a change in conversational score, the standards for F-ness change between the point at which the subjects judge a to be F and the point

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Context, Vagueness, and the Sorites / 75 at which they judge a’ not to be F, then there is an important sense in which those two judgements are not made together—the former judgement is no longer operative by the time the latter is made—and so the tolerance principle is not applicable. If the score is changed once a’ is judged not-F, then relative to the new score, a could become not-F after all. In both Raffman’s and Shapiro’s scenarios we get a phenomenon of ‘backward spread’, where the new classification of one case (e.g. as not-F in a series of Fs) is accompanied by the reclassification in the same way (as not-F) of cases that had previously been judged the other way (as F). The point at which there is a jump in the classification doesn’t turn out to be a point that violates the appropriate version of principles such as Shapiro’s tolerance principle. Let’s look in a little more detail at the role of conversational score, especially in relation to the tolerance principle. ‘The conversational score is the device that enforces tolerance,’ Shapiro writes (Chapter 3, p. 52). But if we were to maintain the principle of tolerance in its general form by appeal to conversational score, there would need to be a change of conversational score whenever there was a jump in judgement. In other words, if there were cases where there is a jump in judgement without the required change in score, then tolerance would fail. And, I suggest, there will be such cases. To argue this, in detail, we would need to know more about what is required for a change in conversational score. One suggestion is that agreement (perhaps tacit agreement) among participants in a conversation is needed for change in conversational score. Perhaps Shapiro has this in mind, since in setting up the forced march case, he describes the scenario in such a way that consensus is reached at the point of the jump: everyone agrees to change from Fs to not-Fs (p. 51 above). But things would rarely go this smoothly if people were faced with the forced march. Conversational participants would typically jump at different points. They might reach consensus by way of a kind of stipulation if it were important to do so. But if they reach consensus in this kind of way (e.g. stipulating a boundary for legal purposes), the phenomenon of backward spread will not then take hold. For they will not see the classification of something as not-F as a reason to classify previous items as not-F as well. Moreover, there will be plenty of situations in which they won’t reach consensus at all, and in such scenarios appeal to conversational score looks to be no help in saving Shapiro’s tolerance principle.

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76 / Rosanna Keefe Perhaps, then, we should drop the need for consensus and consider an individual speaker facing the forced march to bring about the change in score automatically (and without the need of other conversationalists’ consent) when they change from judging members of the series as Fs to judging them as not-Fs. One problem with this suggestion is that speakers seem to be able to change categories through a Sorites series without instituting a change in score. I suggest that typically, when faced with the forced march scenario, speakers will jump at some point without intending that point to have the significance that a change in conversational score would give it. They are forced to make judgements on the members of the series and will not typically be in the situation of dictating new ranges of applicability. If asked whether they now intend a different extension of ‘bald’, surely a common response will be to deny that they do. They have jumped at that point because they had to jump somewhere. Moreover, the subject will typically be alert to the fact that imposing backward spread will only lead to the same problem in the other direction, so they won’t be tempted to think that letting the classification spread backwards will help. But, it might be replied, perhaps someone, or some people, can change conversational score without actually intending to. Though in some Lewistype cases change of score is explicit and/or consciously affected, not all plausible cases are like this. There will, instead, be surrounding (behavioural) evidence that such a change was intended: in many of Lewis’s cases of changing score, it is easy to see, for example, how the speaker would be saying something absurd if no score-change had taken place. But in Shapiro’s case, there are no such grounds. At least, it is begging the question to consider it absurd for a subject to change from Fs to not-Fs through as series without instituting backward spread. There is (or at least need be) no evidence for change of score in our Sorites cases. Moreover, to say that someone can change the score without realizing it is one thing; it is another to say that they can do so even when they actually intend not to change the score. When speakers facing the forced march emphasize the arbitrariness of the point they chose to change at, they are likely to deny that anything like a change of score has occurred. To summarize. I suggest that change of score does not usually play the role in speakers’ judgements that Shapiro needs it to play. Though it may be legitimate to change score in the described way, if this is rarely what speakers

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Context, Vagueness, and the Sorites / 77 do in response to vague predications, this fact will not be sufficient to preserve Shapiro’s tolerance principle.

3. From the Pragmatic and Psychological to the Semantics Shapiro starts from a pragmatic (perhaps also psychological) story about the dynamics of conversation when confronted with Sorites series or borderline cases. Raffman proposes a psychological story about the internal states of subjects in such contexts. Can we get from stories of these kinds to an account of the semantics of vague predicates? Shapiro hopes so. In essence, the most straightforward way to make such a transition will be to take what subjects judge—given their psychological states and the conversational context etc.—to determine what is actually true in those contexts. This is, indeed, the route Shapiro and Raffman take. It is reasonable, for example, to think that if you change the score on ‘flat’—thereby raising the standards for what counts as ‘flat’—and this meets with no objection from the other conversational participants, then in attributing flatness to something that fails to meet the new standards, I would say something false. So, in at least some cases of changing score, the thought that it establishes truths is not unreasonable. (When the score involves shared presuppositions, things cannot be this simple, however, for those presuppositions might just happen to be false.) Shapiro endorses Raffman’s principle: (B) [A]n item lies in a given category if and only if the relevant competent subject(s) would judge it to lie in that category.

This suggests that if a competent subject judges members of a Sorites series in such a way as to draw a sharp boundary between the Fs and the non-Fs, then some such boundary does indeed exist. Regarding the Sorites inductive premiss, (B) would also seem to bridge the gap between the observation that subjects don’t always judge a member of the series in the same way as the previous one and a claim that that premiss is false. Similarly, Shapiro’s thesis about open texture—that we are permitted to judge borderline cases either way—will, together with (B), yield semantic consequences about the classifications of those cases.

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78 / Rosanna Keefe In short, principle (B) swiftly offers justification for looking at pragmatic– psychological phenomena concerning how speakers actually behave and judge, even if we are primarily interested in the logic and semantics of vague language. I shall shortly raise a number of problems for and questions about principle (B) itself. First, observe that the following fact is a threat to the attempt to use the principle to get from pragmatic–psychological data to semantics: the same linguistic behaviour could be appropriate to different semantic theories. For example, if there is no fact of the matter about whether Fb, then it may be appropriate to judge either way, or to abstain from judging, but that behaviour is equally appropriate if we simply cannot know either way, or, perhaps, if Fb has some intermediate truth value. So, it looks unlikely that we will be able to draw significant conclusions about the semantics from the pragmatic– psychological facts alone. Nonetheless, those facts may still play a more modest role. But is (B) a promising principle to use to draw out their relevance? Here is one question. Does principle (B) allow for the possibility of subjects disagreeing over the classification of borderline cases? Suppose X and Y are both competent subjects in conversation with one another (so, presumably, in the same conversational context) and X claims that Fb while Y claims that not-Fb. It then seems that b lies both in the category and outside it. Shapiro could respond by maintaining that the subjects must, after all, be speaking within different contexts (and that b is F relative to one of those contexts and not relative to the other). But they may be discussing the matter with one another and openly disagreeing, in which case it seems that the conversational context must be the same. Moreover, such a response threatens to be ad hoc. It is not reasonable to postulate new contexts at will, just to preserve consistency with one’s theoretical principles. Also ad hoc would be a response that declares that at least one of the subjects must fail to be competent in this scenario. Aside from problems concerning disagreements, we should be suspicious of a principle like (B) when, for example, we observe how quickly it would dispose of a theory like Williamson’s epistemicism. On that view, items fall into categories without our knowing it and without our being at all inclined to judge that they do. The point at which subjects happen to jump from classifying things as F to classifying them as not-F is extremely unlikely to correspond to the point at which the epistemicist’s boundary between Fs and not-Fs falls.

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Context, Vagueness, and the Sorites / 79 Shapiro describes (B) as ‘in the neighborhood of a truism’ (p. 65 above); e.g. if someone judges Fa when a is not F, they are not competent in the relevant way. But this surely imposes too strict a demand on competence. Even less plausible is the ‘only if ’ direction of the biconditional (B), which implies that the subject is not competent unless they would always make the (relevant) correct judgements about F. Competence is compatible with some errors and ignorance. Even if we don’t agree with Williamson, we might think that there are some Fs that aren’t known to be F, and so that wouldn’t be judged as F. And note that to give up on (B) is not to give up on what we might find compelling in the Humpty Dumpty thesis that we are masters of our words. Williamson, for example, endorses the view that it is use that determines meaning—it is just that the relationship between speakers’ use (and judgements) and the extensions of predicates is much more complex than that given by (B). Let us, next, look more carefully at the policy of using (B) to get from the pragmatic story about forced march Sorites paradoxes to semantics of the vague predicates involved in the paradoxes. Is it reasonable to draw any significant conclusions from the response subjects are driven to make when they are marched through a Sorites series and forced to judge each case either one way or the other? A subject will typically consider the point of the jump to be an arbitrary one—this seems to be to deny that it has any semantic significance. Are we to consider the subject to be mistaken about their control over language? This seems an unreasonable conclusion. I suggest a couple of analogies. First, suppose you make subjects respond with ‘yes’ or ‘no’ to questions involving unfulfilled presuppositions; e.g. you ask them ‘Have you stopped f-ing?’ when they’ve never f-ed. They may be reluctant to answer yes or no—both answers are misleading—but they may nonetheless choose one of those answers when forced. Surely their choice in that situation should not be taken as deeply significant, nor as helping to illuminate the semantics of sentences involving unfulfilled presuppositions. (Even if subjects typically made the same choice, this would not make their arbitrary choice any more significant.) Second analogy: reading too much into the response to forced march paradoxes seems rather like forcing someone to guess the weight of something and then taking that guess to reveal that the subject believes that the weight is exactly that. Shapiro briefly considers alternative responses to the forced march scenario in order to argue that appeal to a shift in context is the best approach to take. In particular, he considers the theorist who claims that ‘the subject(s)

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80 / Rosanna Keefe are incompetent, or have made some sort of mistake with their answers’. In reply, he writes, ‘But what sort of mistake? The subjects are told to call them as they see them. What did they do wrong?’ (p. 57) Now, I agree that the charge of incompetence is unreasonable if subjects have been made to judge one way or another and have done exactly that. On the other hand, they can, surely, have been mistaken in their judgements (without being incompetent). What they did wrong was perhaps unavoidable given what they were asked to do (namely, make a judgement on every case), but nonetheless, this is no reason to expect their judgements to be right. In short, it isn’t clear that we can use predictions about the forced march Sorites to illuminate the semantics of the predicates involved, especially given a typical subject’s insistence on the arbitrariness of many of their judgements through the series. Moreover, we need to consider the fact that the behaviour inevitably exhibited by a subject facing the forced march series is not typical of subjects’ responses to Sorites in general. What about when judgements aren’t made through the series? Consider, for example, the kinds of conversational contexts where the Sorites premise is considered in the abstract and found compelling. It isn’t clear how facts about conversational score can shed any light on this. Or suppose you present someone a whole Sorites continuum and ask them to show the point of the left-most non-red patch, where they are obliged to respond by picking a patch arbitrarily. We can’t then excuse the subject because they changed score or context between pairs while maintaining the rule of judging pairs the same: that way out is not available with such a scenario. Now, Shapiro, unlike Raffman, does not make reflections on forced march Sorites do all the work; there is also his open-texture thesis, stating that subjects are allowed to go either way in the classification of borderline cases. A number of the same problems (and possibly some new ones) face this thesis, however. The suggestion that borderline cases can be decided either way—a close relative of the thought that we can stipulate boundaries to vague predicates— seems reasonable when we consider certain contexts. For example, for legal purposes a stipulated boundary to ‘child’ is not only accepted, but desirable, and faced with a practical task of sorting and storing paints of a range of colours, a stipulated boundary to ‘red’ will help. But, I suggest, such situ-

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Context, Vagueness, and the Sorites / 81 ations are in the minority. Much of the time it would be misleading, at best, to stipulate boundaries or call things red that are in fact borderline, and in many circumstances it would certainly not be legitimate. If I choose to describe as orange something that is in fact borderline red–orange, you will reasonably believe that it is orange and not borderline and you could easily be led astray. Yet, Shapiro needs a general thesis according to which it is always acceptable to decide borderline cases one way or the other, if there are to be any logical and semantical consequences (even assuming (B) ) and that thesis looks highly questionable. Moreover, it is not even clear that we can’t do something similar in classifying definite cases to what Shapiro has identified as legitimate with borderline cases. A boundary to ‘red’ among the definitely red things might in some cases be reasonable (‘red’ becomes ‘bright red’ in the context, e.g., perhaps, in sorting paints again). There, we might say that it can be pragmatically appropriate to call something not red in that context, though what is said then is strictly false. Perhaps similarly, then, when we do decide borderline cases one way or the other, those predications remain neither true nor false (even in that context). So it is not clear that a feature which specifically applies to borderline cases has been identified here. In short, the phenomenon identified as open-texture is rarer than Shapiro needs it to be, may not even be distinctive of vague predicates, and may have no semantic significance. In this section I have explored the move from a story about pragmatics– psychology to a theory of semantics. I have argued that a principle like (B), which would validate the move, is not defensible. It is, anyway, not clear exactly how a principle of this kind would yield a theory of vagueness or a solution to the Sorites paradox, especially since, at best, it would need to focus exclusively on the very unnatural cases of forced march Sorites.

4. The Sorites Paradox In this section I consider whether we can use the distinctive resources of Shapiro’s approach to solve the Sorites paradox. Now, it is not altogether clear from his paper what Shapiro’s solution to the paradox is intended to be. He appears to be declaring the inductive premiss false because it has a false instance, namely the instance of the

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82 / Rosanna Keefe jump. For, he observes that it ‘is outright false that for each i<2000, if our conversationalists judge man i to be bald, then they will in fact judge man i þ 1 to be bald’ (p. 53) and he goes on to conclude that ‘the inductive premise is never true (in full splendor)’ (p. 53). Indeed, given principle (B), that claim about the conversationalists’ judgements would seem to guarantee the falsehood of the premiss. Raffman similarly, says: ‘Now we see how shifts can occur in an effectively continuous series. Now, that is, we see how [the inductive premiss] could be false.’ In other words, for Raffman, the jump does falsify the premiss. So, the story could be taken to show how the inductive premiss of the Sorites is false, thereby solving the paradox in the popular way of refusing to endorse that central premiss. But I have several worries about this. First, a point closely related to my earlier discussion: this response may rely too much on considering Sorites paradoxes in forced march contexts. That speakers will jump in classification when forced to judge one way or another in a Sorites series is not necessarily good grounds on which to draw conclusions about the Sorites paradox in general. Putting this point aside, the described response is still puzzling. The problem is with sustaining both the treatment of the Sorites inductive premiss, taken as false, and the line on the tolerance principle, which Shapiro seeks to preserve. Matching the explanation of how tolerance is not violated, the story about changes in conversational score might be thought to explain how the inductive premiss can remain true (when read in the right way to forbid cheating by change of context), despite the jump in classification from F to not-F at some specific point. Backward spread would prevent the instance of the jump counting as a falsifying instance of the premiss, for the object prior to the jump would no longer count as F. This raises the question how far the backward spread goes. If it were taken to spread all the way back through the series, then tolerance could be maintained and the premiss held as true. But surely that is highly implausible: indeed Shapiro dismisses this possibility. So, the suggestion that the role of the contextualist account is to show how the Sorites premiss can in fact be true is not promising. It can’t avoid the usual problems of accepting the Sorites inductive premiss—namely the conclusion that the hairiest man in the sequence is bald. Shapiro suggests that the extent of the backward spread is a vague matter. There is no precise point to end the spread. But if this is a legitimate response about the boundaries to the cases of F when backward spread has occurred, why is it not a legitimate response more generally regarding the boundaries of

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Context, Vagueness, and the Sorites / 83 F-ness? In other words, the thought here seems to be that the course of the conversation, plus linguistic meaning etc., determines only a vague quantity of backward spread. But why then can’t we say that those factors determine only a vague boundary and cease to appeal to backward spread at all? To summarize: it isn’t clear to me how the Sorites is to be solved by calling upon Shapiro’s story about context change. And other responses to the paradox that are suggested by the approach are also problematic.

REFERENCES Ru f f m a n , D. (1994), ‘Vagueness without Paradox’, Philosophical Review, 103: 41–74. Sh a p i r o , S. (2003), ‘Vagueness and Conversation’, This Volume: 39–72.

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5 Vagueness: A Fifth Column Approach Crispin Wright

1. The Vagueness Trilemma Anyone must agree that vagueness pervades the lexicon of natural languages: almost everything we say is expressed in vague vocabulary. It is a little more controversial, but presumably true, that this is unavoidable: that a language stripped of vague expressions would suffer not merely in point of usefulness— often a vague judgement is exactly what we need—but in its very expressive power. (We need concepts, for instance, of rough impressions, of casual appearances, and of circumstances in which a precise predication—say, ‘‘is more than six feet tall’’—may justifiably be made on the basis of rough-andready observation; and we need to be able to express these concepts.) This makes vagueness a topic of central philosophical significance in at least two ways. Both the philosophy of language—in so far as it is concerned with what it is, at the most general level, to have mastery of a natural language—and the metaphysics of the relationship between natural languages and the world they serve to represent must demand an understanding of the nature of vagueness. While some of the problems raised by vagueness were formulated in antiquity, it received only occasional and unsystematic attention from analytic philosophy until the mid-1970s. Since then there has been an explosion of attention and publication.1 The tendency of the magnified effort, however, 1 The 1975 Synthese special number on the topic was a crucial spur.

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Vagueness: A Fifth Column Approach / 85 is not to inspire a sense that things are moving interestingly in concert towards an enhanced understanding. Rather, it has been towards fragmentation. The contemporary context is one in which, for each of the principal views proposed—indeed, as it can seem, for each of the possible views—a significant constituency of philosophers in the field are opposed to or sceptical about it, and are so for powerful and fundamental-seeming reasons. We would like to understand what is the nature (or natures) of vagueness and why so much of natural language is vague. And we would like to understand why there is not really—as presumably there is not—any genuine paradox at the heart of vagueness. But we seem to be light years from understanding these matters. The problematic character of vagueness surfaces immediately in the difficulty in saying what the basic phenomenon—the occurrence of ‘borderline cases’—consists in. Vagueness manifests itself, of course, in our hesitancy or unwillingness to make judgements in such cases, and in conflicts among (hesitantly made) judgements—either those of several thinkers, or of a single thinker at different times. But these manifestations are hardly distinctive; they may be present with precise judgements as well. We also express our recognition of vagueness in the clumsy intuitive rhetoric of ‘‘No fact of the matter’’, ‘‘It is and it isn’t’’, and ‘‘It neither is nor isn’t’’, etc. But this rhetoric is of dubious coherence: the latter two locutions, besides being inconsistent with each other, are also internally inconsistent, while there being ‘‘no fact of the matter’’, if that is ever a fact, would seem sufficient to ensure that a targeted judgement is not true—and hence that it should be acceptable to deny it. So there is a very basic problem even in giving a characterization of the phenomenon to be explained. The lack of such a characterization has not, however, discouraged philosophers from taking sides among what seem to be the only possible kinds of view. Each of the three following contrasting conceptions stands out in the recent concentration of work. Perhaps the most intuitive approach is that vagueness is a semantic phenomenon—something which originates in shortfalls, as it were, in the meanings we have assigned to expressions. On this type of view, a vague statement is one whose meaning is akin to a partial function and somehow fails in certain cases, even in conjunction with the relevant facts, to determine whether it should count as true or as false. The rules for the use of ‘‘tall’’ prescribe, for instance, that a man who stands 6’ 2’’ is tall, and that one who stands 5’ 6’’ is not, but fail to prescribe for a man of 5’ 10’’. The actually relevant facts—that

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86 / Crispin Wright the man stands 5’ 10’’—do admit of perfectly precise description. There is no vagueness in the matter to be described. The vagueness consists in the failure of the semantic rules for ‘‘tall’’ to cater for certain kinds of precisely describable situations. This contrasts with the view that vagueness originates in rebus, in objective indeterminacies in the items which we use language to describe. ‘‘Morocco extends more than 150 miles east of Agadir’’ is prima facie indeterminate in truth value, but here the source of the indeterminacy, it may be claimed, lies not with the language but with what is being described. The predicate ‘‘ . . . extends more than 150 miles east of Agadir’’ is precise enough, nor is it indeterminate what ‘‘Morocco’’ refers to—it refers to the sovereign territory of Morocco. Rather, the indeterminacy is in the real extent of that territory. Morocco itself—that very territory—lacks sharp boundaries (rather like a shadow that is blurred on one side). These two views agree that certain kinds of meaningful expression, featuring in some quite definitely true statements and some quite definitely false ones, may also occur in meaningful statements which are borderline— statements which they see as challenging the principle of Bivalence, that every statement is determinately true or false. The third player in the contemporary debates is the view that vagueness, properly understood, actually presents no challenge to the principle of Bivalence, whether semantic or worldly in origin. Rather those aspects of our linguistic practice which we take to reflect indeterminacy are better seen as flowing from our own (unavoidable) ignorance of what are in fact sharp thresholds to the correct application of our expressions. On this—Epistemicist—proposal, vague statements should be conceived as perfectly determinate in truth value, though what truth values they possess we do not know. This seemingly fantastic suggestion has been worked out in depth by Tim Williamson and others, and is supported by a developed account of the putative barriers to knowledge in borderline cases. These proposals are not, to be sure, inconsistent with each other except in so far as they aspire to be comprehensive. One might in principle be eclectic, reserving the semantic approach for some examples, for instance, while taking an epistemic view of others, though it is not obvious how such an eclectic stance might be motivated. What can seem hard to see is how there could be any other—fourth—type of view. For presumably—so one might reason—borderline cases either present genuine indeterminacies or they do not. The Epistemicist thought, for its part, is that they do not. But if they

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Vagueness: A Fifth Column Approach / 87 do, then presumably the indeterminacy is sourced either in the semantics of the statements in question or in their subject matter. So it can easily seem as if we must, in the end, go for one of these three types of position. The topic is all the more perplexing, accordingly, for the fact that each of the three seems problematical and unhappy in serious ways. A crux for any conception of vagueness is how it copes with the Sorites paradox. A sound conception of vagueness must not merely block the paradox but—so I should contend—block it in a way that acknowledges it as a reductio of the major premiss and accordingly incorporates a diagnosis both of that premiss’s plausibility and of the error that it nevertheless contains. Of the three proposals canvassed, only Epistemicism resolves the paradox directly. It simply denies the relevant major premisses, insisting that there are sharp but unidentifiable thresholds to, e.g., colours, heaps, and the territory of Morocco. What, though, is Epistemicism’s story about the plausibility of those premisses? Why, according to the epistemic conception, are we tempted to suppose that, e.g., if a man with merely n hairs of normal thickness, etc., is bald, so is one with n þ 1? Well, since that supposition would not be tempting in the least if we knew that ‘‘bald’’ actually presented a sharply demarcated property, part of the explanation, the Epistemicist may say, is our ignorance about the real semantic nature of vague expressions. But that (alleged) ignorance doesn’t explain enough. If we were merely receptive to the possibility that vague expressions present sharply demarcated properties, the temptation would already be gone. So the Epistemicist’s explanation has to account for (what it must view as) our prejudice against that possibility—it has to explain our succumbing to (what it must view as) the illusion that vague expressions do not have the kind of semantic depth that, e.g., expressions for natural kinds, as commonly construed, possess: the illusion that the nature of the property ascribed by a vague predicate is often fully available to a thinker just in virtue of understanding that predicate. I am not aware that Epistemicists have addressed this need in any very convincing way. The other two, indeterminist types of view need some supplementary proposal about how and why Sorites reasoning breaks down. Among philosophers favouring a semantic view, the most widely received such proposal is that the truth conditions and logical powers of vague statements are subject to a broadly supervaluational analysis. A vague statement is true, on this proposal, just if it would be true if all the expressions in it were made perfectly precise, but in a fashion which respected the range of cases where, vague as they are, they nevertheless definitely do or don’t apply. Thus ‘‘Jones is tall’’ is

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88 / Crispin Wright true just in case it would be true under every such admissible way of making ‘‘tall’’ perfectly precise—every way consistent with respecting the cases where it is already definitely correct to describe someone as ‘‘tall’’ or as ‘‘not tall’’. Since, were language to be made totally precise, there would be the sharp thresholds which the Epistemicist believes there already actually are, the semantic-cum-supervaluational proposal can similarly resolve Sorites paradoxes by simple denial of their major premisses. Supervaluations, however, need not be the exclusive property of the semantic view. A supporter of vagueness in rebus may adapt the proposal by fashioning a suitable notion of what it is for an entity—object or property— to be a precise counterpart of a vague one. For instance, a precise counterpart of Morocco may be taken to be any sharply bounded land-mass whose being completely contained within Morocco is a vague matter. And now ‘‘Morocco is larger than Senegal’’ may be reckoned true just in case every precise counterpart of Morocco is larger than Senegal. The wide reception of supervaluational semantics for vague discourse is no doubt owing to its promise to conserve classical logic in territory that looks inhospitable to it. The downside, of course, rightly emphasized by Williamson and others, is the implicit surrender of the T-scheme.2 In my own view, that is already too high a cost. And there are additional concerns about the ability of supervaluational proposals to track our intuitions concerning the extension of ‘‘true’’ among statements involving vague vocabulary: ‘‘No one can knowledgeably identify a precise boundary between those who are tall and those who are not’’ is plausibly a true claim which is not true under any admissible way of making ‘‘tall’’ precise. But in any case the proposal that truth and valid inference among vague statements operate supervaluationally comes—at least in the context of a background indeterminism about vagueness—completely out of the blue. The conception of indeterminacy shared by the semantic and in rebus approaches is one of a situation in which, whether for reason of shortfall in meaning or lack of definition in the world, a statement fails either to represent or to misrepresent reality. Yet supervaluationism insists that truth and valid inference among vague statements operate as if there was no such indeterminacy, as if we had to deal only with fully precise concepts and definite situations. Such an approach is open to the complaint that it changes the subject, rather than helps to account for it. At the very least, it should be no less legitimate on any indeterminist view to seek 2 Cf. Williamson (1994: 162).

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Vagueness: A Fifth Column Approach / 89 a semantics and proof theory for vague claims which treat the challenge they pose to Bivalence as a challenge to classical logic too—there seems to be nothing to be said in favour of the idea that supervaluations get the logic of vagueness right. But in that case there is nothing to be said in favour of the idea that the solution they let the indeterminist provide to the Sorites paradox is anything but adventitious. Actually, the very idea of semantic indeterminacy as an account of the constitution of vagueness in general3 is much more difficult to make sense of than philosophers have generally acknowledged. The basic idea—of the existence of a range of cases which we, as it were, lack any instruction how to describe—is intelligible enough. But—to indicate just one difficulty—it is a generally accepted datum of the problem that, in a wide class of examples, the distinction between the borderline cases and those which we have a mandate to describe as, e.g., ‘‘heaps’’ is not a sharp one. This is the phenomenon of ‘‘higher-order’’ vagueness: in the gradual transition from heaps to agglomerations of sand too small to count as heaps, there does not seem to be a sharp threshold between the heaps and the indeterminate cases, nor between the latter and the non-heaps; rather the region of indeterminacy has its own (pair of) fuzzy borderlines. So now consider an agglomeration X in the borderline between the heaps and the initial range of indeterminate cases. The latter are conceived as being such that there is no mandate to describe them as heaps and no mandate to describe them as non-heaps. So on the simple semantic view of indeterminacy—that it is a matter of lack of any semantic mandate—X should be such that there is no mandate to describe it as a heap, no mandate to describe it as a non-heap (since it is less of a nonheap than things—the initial range of indeterminate cases—there is already no mandate to describe as non-heaps) but also no mandate to describe it as borderline. Which is to say: no mandate to describe it as we just did—as being such that there is no mandate to describe it as a heap and no mandate to describe it as a non-heap. That result—that X fits a certain description which there is no mandate to describe it as fitting—commits the semantic theorist to a version of Moore’s paradox, and raises a serious question whether any coherent characterization of vagueness, as conceived by the semantic view, is possible at all. This point has not been widely grasped. As for the idea of vagueness as an indeterminacy situated in rebus, this— even if locally arguable for items such as Morocco and Mount Everest—is 3 As opposed to some special cases.

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90 / Crispin Wright manifestly unintuitive for the general case. Nothing in the imagery of blurred shadows helps us understand the vagueness of quantifiers, like ‘‘many’’ or ‘‘about twenty’’, and it strains credulity to suppose that in our use not merely of basic vague predicates like ‘‘tall’’, but also of vague compounds like ‘‘very tall’’, ‘‘unusually tall’’, ‘‘quite unusually tall’’, and so on, we merely respond to objectively vague properties put up by the world. But there is a more basic—and again, neglected—difficulty arising with the conception of indeterminacy as a worldly situation in the first place. To try to conceive of the indeterminacy in truth value of a statement of a type which, in different circumstances, could be true as originating in the character of the relevant prevailing states of affairs must commit one, it seems, to thinking of those states of affairs as different in kind from and incompatible with the obtaining of the kind of state of affairs which would make the statement true. But then we seem to have, not a situation of indeterminacy, but one in which the world is inconsistent with the truth of the statement—so one in which it is determinately untrue. (This, of course, is a generalization of the thought which inspired Gareth Evans’s much discussed argument for the impossibility of vague identity in re.) It is a major question whether it is possible to have, even for quite local instances, a genuinely in rebus conception of vagueness which does not, in effect, lose hold of the idea of indeterminacy and degenerate into the thought that there are more ways for statements of certain kinds to be untrue than are catered for by standard grounds for denying them. Epistemicism, finally, for all its theoretical simplicity, is—at least in the present state of our understanding—open to the charge that it makes an utter mystery of the semantics of vague expressions. We have no conception of what would constitute the relationships between vague expressions and the particular sharply demarcated entities—objects, properties, functions— which, according to Epistemicism, are somehow established, beyond our ken, as their semantic values; nor do we have the slightest independent reason— independent, that is, of the problems encountered by the opposed views—to believe that such associations exist. There are important subsidiary issues concerning just how effective the explanations offered by Epistemicists are for our putative (ineluctable) ignorance of these matters. But the major concern for any proponent of Epistemicism must be whether there is any real likelihood that it will ever be possible satisfactorily to redeem the hostages the view holds out to the theory of meaning and reference. Each of the three broad, collectively seemingly exhaustive conceptions of vagueness is thus open to misgivings radical and immediate enough to

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Vagueness: A Fifth Column Approach / 91 provide in effect for another paradox. We might call it the Vagueness Trilemma: none of the three possible views seems to bear serious scrutiny. Maybe some of the objections that beset the three alternatives can, with resource, be assuaged. But the solution I shall here pursue is to make a case that the three alternatives are not exhaustive.

2. What are Borderline Cases? Let’s begin by scrutinizing a little further the notion of a borderline case. There is no reason to deny that one kind of borderline case does more or less fit the semantic indeterminist model. Such examples are borderline cases of the application of a concept associated with sufficient conditions, with necessary conditions, but for which no condition is explicitly acknowledged as both necessary and sufficient. John Foster4 used to work with the example of ‘pearl’. It suffices for something to be a pearl that it be an approximately spherical object of a certain distinctive appearance and constitution, naturally produced within an oyster in a certain kind of way. It is a necessary condition for something to be a pearl that it be an approximately spherical object of that distinctive appearance and constitution. But the ordinary understanding—or let’s so suppose—leaves it open whether natural occurrence within an oyster is necessary for pearlhood. In that case it is objectively indeterminate—the rules for the use of ‘pearl’ leave it open—whether a pearl-like object meeting the first condition but synthesized in a vat of chemicals is a pearl or not.5 Of course what’s salient about such cases is that the distinction between the determinate instances and non-instances and the indeterminates is itself determinate; so the problem I canvassed earlier does not arise. Our interest, though, is in the types of vagueness associated with susceptibility to a Sorites series—vagueness associated with gradual change in a relevant parameter of degree (one associated with a significant comparative, ‘is more/less f than’) and where the distinction between the definite cases and the borderline cases is itself, at least prima facie, vague. How should we conceive of the borderline cases in this—the intended and crucial—range of examples? For ease of exposition, let’s restrict our attention to the case of vague (monadic) predicates, and focus not on the objects that are borderline for such a predicate, F, but on the associated propositions that such an object 4 In graduate classes in Oxford long ago.

5 Cf. Williamson (1990: 107) on ‘dommal’.

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92 / Crispin Wright is/is not F. Let a verdict be a judgement that such a proposition is true or that it is false. What is the correct account of the status of the propositions which these verdicts concern? Third Possibility is the generic view that such propositions have some kind of third status, inconsistent with each of the poles (truth and falsity.) Examples of Third Possibility are the claims that propositions in question lack a truth value, that they have some unique third truth value, and that they have one among a number of values intermediate between the two polar values (or, in a more sophisticated version, the two vaguely bounded clusters of polar values). If any form of Third Possibility is correct, then the verdicts associated with the propositions in question are determinately incorrect—indeed, there seems no reason to deny that they are false. Third Possibility entails but is not entailed by Verdict Exclusion. Verdict Exclusion says that no verdict about such a proposition is knowledgeable6— that the correct stance about borderline propositions is one of agnosticism. According to Verdict Exclusion, one ought, all things considered, to offer no verdict about a borderline case and to have no opinion which could be expressed in such a verdict. (Notice that Williamson’s version of Epistemicism would enforce Verdict Exclusion—if the arguments from margins of error to the impossibility of knowledge about the F-ness of borderline cases are accepted—but, in view of the endorsement of Bivalence, would reject Third Possibility.) Although a great deal of work in the field has been informed by an acceptance of Third Possibility or, more modestly, of Verdict Exclusion, I think these views—or, more specifically, the notion that we are warranted in holding either of these views—is very difficult to sustain. The manifestation of vagueness, in the kinds of case we are concerned with, is not a consensus on certain cases as borderline—not if that is to be a status which undercuts both polar verdicts. Rather, the impression of a case as borderline goes along with a readiness to tolerate others’ taking a positive or negative view—provided, at least, that their view is suitably hesitant and qualified and marked by a respect for one’s unwillingness to advance a verdict. I do not deny that psychological laboratory experimentation might actually disclose that large numbers of otherwise competent subjects would converge on regarding certain colour chips, for example, as borderline between red and orange and on certain photographs of balding men as borderline between bald and not-bald. What I 6 Or if that is different, that no such verdict is warrantable.

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Vagueness: A Fifth Column Approach / 93 am saying is that the existence of such a convergence is empirical and its occurrence, if it occurs, is left entirely open, as far as vagueness of the relevant concepts is concerned. What the vagueness of those concepts does not leave open is, first, that there should be no (stable) convergence among competent subjects about a threshold and, second, that each competent subject should offer progressively less confident verdicts, eventually entering a range where any verdict is uncomfortable, and then later a range where confidence is gradually restored. Those points are consistent with there proving empirically to be no cases for which there is a convergence on unwillingness to issue any verdict. The central manifestation of borderline cases is not a convergence on such unwillingness, but—always among competent judges—in weakness of confidence in such verdicts as are offered, in their instability, and in the unwillingness of some to endorse any verdict. One would also expect this same pattern—a mix of tentative and perhaps conflicting verdicts and unwillingness to return one—to be replicated within the judgements of a single competent subject about a single case made on a number of separate occasions. These reflections are, to stress, strictly inconsistent neither with Third Possibility nor, therefore, with Verdict Exclusion. What they are inconsistent with is our knowing that either of those proposals correctly characterizes borderline cases—or better, if someone insists that either is a correct characterization, with there being any definite (known) borderline cases in the sense of that characterization. Since we should take it that, however borderline cases should be characterized, it is a datum that vague concepts give rise to them, we should conclude that both Third Possibility and Verdict Exclusion are misdirected accounts. When, in a Sorites series, I reach a range of cases about which I am reluctant to give a verdict, it does not convict you of incompetence if you are not so reluctant provided your willingness to take a view is appropriately qualified and it is wholly understandable to you that others may not share it. To regard a case as borderline is not to regard it as having a status inconsistent with either polar verdict, but to feel that one cannot knowledgeably endorse a polar verdict. And that much is consistent with recognizing that other, competent judges may, tentatively, feel able to do so. My impression that a case is borderline is not defeated if they do so. But it is sustained by others’ recognition that we are within a region where divergences of this kind among competent judges are to be expected. Just for that reason, my impression that the case is borderline is not an impression that the case has a status inconsistent with the correctness of a verdict. Nor is

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94 / Crispin Wright my having that impression a commitment to regarding any verdict as nonknowledgeable. If it were, then in regarding the case as borderline, I’d be committed to regarding anyone who advanced a verdict, however qualified, as strictly out of order—as making an ungrounded claim and performing less than competently. But that they are doing anything of that sort is just what I don’t know. Against Third Possibility and Verdict Exclusion as characteristic of borderline cases, I therefore wish to set the following contrary thesis of Permissibility: with the kind of vague concepts with which we are concerned, a verdict about a borderline case is always permissible; it’s always all right to have a (suitably qualified) opinion. And this permissibility is not a matter, merely, of its being excusable to have a (mistaken, or unwarranted) view, as it would have to be if Third Possibility, or Verdict Exclusion, were respectively correct. Rather it is a matter of its being consistent with everything one knows, when one competently takes a case to be borderline, that a verdict about that case is correct and that one who advances it does so warrantedly. Permissibility is meant to encapsulate the idea that regarding a case as borderline is reaching a point where one’s springs of opinion have so weakened that one is unable to reach one or, at best, any opinion one reaches is weak and unstable; and if the thinker is a competent judge, then it will go with this predicament that it is understandable and consistent with her competence that she be in it. But that it is understandable and expectable that she and others get into such a predicament is not something which empowers her to reject the veracity of a verdict, or the competence of those whose verdict it is. Both Third Possibility and Verdict Exclusion have it that the recognition of a borderline case is the recognition of a case of a certain kind of respectively ontological, or epistemic status. Against that, Permissibility maintains that to regard X as a borderline case of F is neither to recognize that there is no correct polar verdict about ‘‘X is F’’, nor that no such verdict can be knowledgeable. Rather it is, first and foremost, a failure to come to a view. And failure to come to a view, it goes without saying, is in general quite consistent with there being a true view; and with someone who holds it doing so knowledgeably. I am under no impression that these sketchy remarks can stand without further refinement and elaboration. But I do contend that they take us in the right direction. A correct account of the kind of vagueness in which we are interested must start not from the idea of our recognition of some sort of third status or epistemic impasse but rather from the idea of a failure of judgement—an inability of (significant numbers of) competent judges to

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Vagueness: A Fifth Column Approach / 95 come to a view in what we conceive as the best, or anyway good enough, background circumstances for the formation of the type of view in question. And however the account proceeds to elaborate that starting point, it will therefore be inconsistent with both the semantic and the in rebus types of indeterminist view, each of which goes with the idea that borderline cases have a status incompatible with truthful, let alone knowledgeable verdicts about them.

3. Reconfiguring the Range of Options According to the Epistemic conception of vagueness as ordinarily understood, a borderline case of a vague predicate is one where we remain ignorant whether or not the predicate applies even when background conditions obtain which suffice for knowledge in clear cases. No indeterminist need contest that—indeed nobody should contest it. What is controversial is what more should be said. The indeterminism of both the semantic and in rebus views involves adding Third Possibility, and so rejecting Bivalence while accepting Verdict Exclusion. Epistemicism, by contrast, at least in the hands of its principal proponents hitherto, insists on Bivalence but rejects Third Possibility while accepting Verdict Exclusion. The considerations of the preceding section, if sound, suggest that we should not accept either Third Possibility or Verdict Exclusion. So what should we think? Third Possibility and Bivalence are inconsistent with each other; and Third Possibility entails Verdict Exclusion. So there are actually five prima facie coherent options: I Indeterminism

Third ACCEPT Possibility

II Exclusive Epistemicism

III Pessimism

IV Non-exclusive Epistemicism

V Agnosticism (Intuitionism)

NOT ACCEPT NOT ACCEPT NOT ACCEPT NOT ACCEPT

Verdict Exclusion

ACCEPT

ACCEPT

ACCEPT

NOT ACCEPT NOT ACCEPT

Bivalence

NOT ACCEPT

ACCEPT

NOT ACCEPT ACCEPT

NOT ACCEPT

Note that non-acceptance is here to be construed as a stance consistent with agnosticism about the principle in question—it is implied by, but need not involve rejection of, that principle in the sense involved in a willingness to

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96 / Crispin Wright contradict it. The Epistemicist options seem worth dividing into two— columns II and IV—because as a number of commentators have remarked, it is by no means obvious that, pace Williamson, Epistemicism must accept it as a datum, calling for explanation, that the determinate truth values imposed on borderline cases by Bivalence are beyond all possibility of knowledge. The most salient aspect of the table, however, is the point, obvious enough, that in rejecting Indeterminism, we have not—or not yet—committed ourselves to Epistemicism as ordinarily understood. There remains the Agnostic option, and its Pessimistic variation: the positions marked by non-acceptance— columns III and V—of both Third Possibility and Bivalence. If either of these positions can be supplied with a coherent philosophical motivation, we may have a way out of the overarching trilemma and an improved perspective on the entire set of issues. It is the non-pessimistic version of the view that looks to have the better prospects. The reason is that a wide class of vague expressions seem to be compliant with an intuitive version of Evidential Constraint: if someone is tall, or bald, or thin, that they are so should be verifiable in normal epistemic circumstances. Likewise if they are not bald, not tall, or not thin. But any predicate F which, under feasible cognitive circumstances C, satisfies the following pair of conditionals: (i) C ! (Fa ! it is feasible to know [Fa]) (ii) C ! (:Fa ! it is feasible to know [:Fa]) will be such as to give rise to contradiction, modulo the realization of circumstances C, if Verdict Exclusion is accepted. The view I suggest we consider is accordingly the view that concerning borderline cases, we should accept none of Third Possibility, Verdict Exclusion, and Bivalence. This is a pretty thoroughgoing agnosticism. The preceding section explored motives for circumspection about Third Possibility and Verdict Exclusion. What we now require, accordingly, are motives sufficient to refuse Bivalence consistent with that circumspection. Of course that combination, marked by their acceptance of the double-negation of the Law of Excluded Middle but refusal of the law itself, is exactly the Mathematical Intuitionist trademark.7 But what motivation is there for it in the present setting? In the 7 The possible utility of intuitionistic distinctions in a philosophical treatment of vagueness was first proposed by Putnam (1983). I myself was among the original critics of his proposal (Read and Wright 1985). It was only much later that I realized how the principal objections of that note might be answered.

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Vagueness: A Fifth Column Approach / 97 concluding section I’ll outline three possible lines. But before that, let’s review how the column V position may address at least two of the challenges that confront any satisfactory treatment of vagueness.

4. The Misconceived Conditional and the Sorites Any satisfactory treatment of vagueness must, at a minimum, (i) say what is wrong with the following conditional (9x)(Fx & :Fx’) ! F is not vague (the misconceived conditional8) and (ii) solve the Sorites paradox. The tasks are of course interrelated. The classic formulation of the Sorites presents an inconsistent triad {F0, :Fn, (8x)(Fx ! Fx’)} ) L over a suitably ordered finite series. Naturally this is only a paradox because all the premisses seem well-motivated. But there is a tendency for the motivations for the major premiss (8x)(Fx ! Fx’) —what I once called the Tolerance Principle for F—to be local to the choice of F, and not obviously owing to its vagueness. If one is running a Sorites for F ¼ ‘‘looks red’’, for example, the thought can seem compelling that if x and x’ look absolutely similar—as they may do even when one matches something which the other does not—then if either looks red, both will. But while predicates of phenomenal colour are certainly vague, this argument to motivate the major premiss of the Sorites has more to do with phenomenality (and, of course, presumably involves a misunderstanding of it) than with vagueness. A similar thought would extend to any predicate justifiably applied on the basis of casual observation. If x and x’ are sufficiently similar, then casual observation will detect no difference between them. But then the case for saying that either is F will be perfectly matched by the case for saying the other is. These are good paradoxes and certainly need a solution. But a Sorites paradox of vagueness, properly so regarded, must appeal to the very vagueness of the targeted expression in the motivation for the major premiss. And how that may be done is exactly what the misconceived conditional 8 Prominent in Timothy Chambers’s (1998) objections to Putnam.

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98 / Crispin Wright brings out. Surely, the thought is, if there is a sharp cut-off point in the series in question—a last F case immediately followed by a first non-F one—then F is after all precise—at least in that series—rather than vague, just as the conditional says. So now, contraposing, there is no sharp cut-off if F is vague. The sting—the No Sharp Boundaries paradox—is then this entailment: {F0, :Fn, :(9x)(Fx & :Fx’)} ) L —that the inconsistency remains even after the major premiss is taken in a form which seems just to be a description of F’s vagueness. So now we have a paradox of vagueness as such. And to resolve it must involve finding something amiss with the misconceived conditional.9 Both standard Epistemicism and its column IV relaxation are in no difficulty in doing so. The misconceived conditional will fail because vagueness is not, epistemically conceived, a matter of lack of sharp boundaries. Bivalence will enforce its antecedent for precise and vague expressions alike, whereas the consequent will of course fail for the latter. Indeterminism will likewise have no difficulty if allied to supervaluations: the antecedent of the misconceived conditional will be (super-)true even when F is vague.10 But what can the column V theorist say? The column V theorist cannot object to the conditional in the fashion of Epistemicism, on the ground of true antecedent but false consequent, since her position precludes her taking the antecedent to be true. For suppose the theorist has somehow motivated an agnostic stance with respect to Bivalence as applied to simple predications of a vague F over the objects featuring in a Sorites series for F. We do not, she has persuaded us, know of any sufficient reason for the view that each such predication results in a proposition such that either it or its negation is true. Then we must also take it that we have no sufficient reason for accepting the antecedent of the misconceived conditional. For if we had, then—since F-ness monotonically decreases, as it were, 9 Or at least with its contrapositive. 10 Matters are less straightforward for non-supervaluationist indeterminism. One thought would be that the misconceived conditional will be harmless in that framework since classical reductio—needed in the derivation of the paradox from :(9x)(Fx & :Fx0 ) will have to be qualified to allow for Third Possibility. However, we are presumably at liberty to introduce a wide negation whose application to any statement produces a truth just in case that statement has some value other than truth. This negation should sustain classical reductio. The impression that the kind of vagueness we are concerned with precludes any sharp thresholds in the kind of series in question between truth and any other kind of status will then—apparently—suffice for the misconceived conditional with negation so construed.

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Vagueness: A Fifth Column Approach / 99 in the series in question, we should know that it consisted in an initial segment of F cases followed immediately by a remainder of non-F ones— and then we’d know Bivalence held over the series of propositions in question, contrary to hypothesis. How then is the column V theorist to fault the misconceived conditional? Well, what she can observe is simply that we certainly also have no sufficient reason for affirming the negation of the antecedent of the misconceived conditional—the paradox itself rules that out. So we should be agnostic—openminded—about the antecedent. But the consequent—that F is not vague—is false by hypothesis. Since no thinker can rationally accept a conditional with a consequent she knows is false but an antecedent about which she ought to keep an open mind, the misconceived conditional is unacceptable in any case, since it is epistemically open that it is false. So the column V theorist may rationally refuse to accept it. What, more generally, of the Sorites paradox? What exactly is the solution the column V theorist may propose? Well, simply that there is no obstacle to treating the Sorites reasoning as just what it appears to be, a demonstration that the major premiss is false. That is an unsatisfactory proposal only if there is strong independent motivation to regard that premiss as true. But that motivation was the thought that the very vagueness of F should suffice for the truth of the major premiss, taken in the form: :(9x)(Fx & :Fx’). And that thought, with the misconceived conditional, is now rejected. For the column V theorist, and any agnostic about Bivalence for the relevant range of statements, recognition of the vagueness of F has to be consistent with agnosticism about the existence of a sharp cut-off in the series in question; that is, consistent with open-mindedness about the truth of (9x)(Fx & :Fx’). The Sorites reasoning itself enforces the denial of the major premiss—that is really only common sense, shared by virtually all responses to the problem. A solution has to consist in explaining why that premiss is under-motivated by the phenomenon, why F’s vagueness does not enforce it. And the explanation offered by the broadly epistemic conception of vagueness which I have been advocating—and which may, so far as I can see, be quite comfortably accepted by the more orthodox Epistemicism of columns II and IV—is that the recognition of borderline cases is the recognition of a range of phenomena—the drying up of the ‘springs of opinion’ for a significant class of competent judges, the occurrence of gentle disagreement among tentative views on the part of others, etc.—which broadly are about us and which entail nothing about the actual distribution of instances and

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100 / Crispin Wright counter-instances of F within the relevant range of cases, a fortiori do not entail that there is no case which is F whose immediate successor is not. What is true is that, in the presence of the phenomena noted, we have no clear conception of how a threshold, should there be one, might be identified. But lack of a clear conception of how something might be known is not a sufficient reason for saying it cannot be known (even if we are disposed to grant that it would follow from the latter that it couldn’t be true). For these reasons, theorists of each of columns II–V—an unholy alliance, no doubt—may unite in agreeing that and why the major premiss for the No Sharp Boundaries paradox is poorly motivated by the phenomenon, and that the paradox may be taken as a simple reductio of that premiss. There remains the discomfort—for all but those inclined to Epistemicism proper—of the apparent implication of a sharp threshold in all Sorites series. That implication may be avoided if broadly intuitionistic restrictions allow us to refuse the transition from ::(9x)(Fx & :Fx’) —established by the reductio—to (9x)(Fx & :Fx’). Such restrictions will be well motivated, as we have seen, if there is indeed a strong case for agnosticism about Bivalence over the relevant range of predications of F, for—again— (9x)(Fx & :Fx’), taken as the statement of the existence of a sharp cut-off,11 cannot be regarded as known to hold in the series in question unless Bivalence is. What is that case?

5. Motivations for Agnosticism about Bivalence concerning Vague Predications First Motivation Suppose we are working in a discourse which we regard as subject to the principle of Evidential Constraint: 11 —rather than, e.g., read supervaluationally.

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Vagueness: A Fifth Column Approach / 101 (EC) P ! it is feasible to know that P.12 And suppose that we think we know that Bivalence holds over the discourse. Then we ought to think that we know, for each proposition expressible in the discourse in question, that the disjunction It is feasible to know P or it is feasible to know not P holds. But maybe we are uncomfortable about that—suppose, for instance, the discourse is number theory and P is Goldbach’s conjecture. Do we have any sufficient reason to think that a proof is available one way or the other? If we think not, and agree with the Mathematical Intuitionists, for whatever reasons, that truth in number theory is to be explicated in terms of provability—and hence that EC holds locally—we should be uncomfortable about accepting Bivalence. For we do not seem to have warrant for certain claims which, if Bivalence was warranted, we would have warrant for. Goldbach’s conjecture is—for us, in our present state of information—an example of a kind of statement I have elsewhere called a quandary.13 A statement P presents a quandary for a thinker T just when the following conditions are met: (i) (ii) (iii) (iv)

T does not know whether or not P T does not know any way of knowing whether or not P T does not know that there is any way of knowing whether or not P T does not know that it is (metaphysically) possible to know whether or not P.

The suggestion, then, is that, if P is a quandary for T, then the claim that It is feasible to know P or it is feasible to know not P is unwarranted for T. So if P belongs to a range which we regard as subject to EC, Bivalence is unwarranted as applied to P and other statements in the same case. Note that the clauses for quandary did not include undecidability: (v) T knows that it is impossible to know whether or not P. 12 The modality involved in feasible knowledge is to be understood, of course, as constrained by the distribution of truth values in the actual world. Feasible knowledge is factive: the range of what, in the intended sense, it is feasible for us to know goes no further than what is actually the case. 13 See Wright (2001).

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102 / Crispin Wright Goldbach is not in that situation. And, on pain of contradiction, no statements which are subject to EC can be. Let F be a vague atomic predicate and consider a range of predications, Fa, :Fb, ... , made under the best possible circumstances for assessing their truth values. In a very wide class of cases—at least, this holds of all the standard examples of Sorites-prone predicates—it is plausible that such predications are subject to EC. The conditions of being red, not being red, looking red, not looking red, being bald, not being bald, being tall, not being tall, etc., are all such that, under the best circumstances, they show. But even under the best circumstances, such concepts may present borderline cases. The key ingredient in the first line of motivation is then that borderline cases are a subclass of quandary. Borderline cases are cases where, for some significant number of competent judges operating under good enough conditions, the springs of opinion run dry. If, as is plausible, we may legitimately add to the characterization of the epistemic impasse in which they find themselves, a failure to know even whether a knowledgeable opinion about such a case is metaphysically possible, then both ingredients—quandary and evidential constraint—necessary to transpose to vague statements the intuitionistic reservation about Bivalence for number theory are in place.

Second Motivation It is interesting that a related line of thought can proceed without actual endorsement of Evidential Constraint, just on the basis of a sympathetic agnosticism about it—one which reserves the possibility that it might emerge as correct. The argument would be this. Suppose we are so far open-minded— unpersuaded, for instance, that any considerations so far advanced in favour of regarding simple colour predications on available, visible objects as subject to EC are compelling, but sufficiently moved to doubt that we know that their truth is in general subject to no form of evidential constraint. Suppose we are also satisfied that their vagueness deprives us of any grounds for thinking that we can in principle decide any such statement. The key question is then this: are we in a position where it is rational to leave epistemic space for our coming to be rationally persuaded of EC for these statements by considerations which would not improve our abilities to verify or falsify them? If the possibility of such considerations is epistemically open, then it

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Vagueness: A Fifth Column Approach / 103 must be that our (presumably a priori) grounds for Bivalence are already less than compelling—for what is open is precisely that we advance to a state of information in which EC is justified and yet in which borderline cases continue to present quandaries. And then the first line of motivation will kick in. But in that case we should recognize that Bivalence already lacks the kind of support that a fundamental metaphysical principle, and especially one which is supposed to ground a fundamental logical principle, should have—for that should be support which would be robust in any envisageable future state of information.

Third Motivation Neither of the foregoing lines of thought, however, is available to a theorist who holds that mere quandaryhood under-characterizes borderline cases— that at least in (as it were, central) borderline cases we know there is no knowing P and no knowing :P. This is clause (v) above—in effect, the Verdict Exclusion view of (some) borderline cases. Verdict Exclusion is, to stress, inconsistent with Evidential Constraint. If we think we know now that Verdict Exclusion holds, we should reject not merely arguments which assume EC but arguments which assume agnosticism about it. (We have therefore so far seen no motivation for the Pessimism of column III.) The most basic reasons, however, for incredulity about Bivalence in this context are quite independent of EC. Critics of standard Epistemicism14 have generally fastened onto its perceived hostages to semantic theory. If there really are the sharp boundaries to the application of vague expressions in which the Epistemicist believes, then each vague predicate, e.g., is associated (in any given context of use) with a property as its semantic value whose extension is absolutely definite. But where is the theory that tells us what constitutes these associations? What makes it the case, for example, that my use of ‘‘tall’’ as a predicate of human males denotes the property it does—say: being more than 5’ 10:327’’ tall—and can any such alleged association be reconciled with the supervenience of meaning on use? (How, for instance, would my use of ‘‘tall’’ have differed if its association had been with being more than 5’ 10:326’’ tall instead?)

14 For instance, Wright (1995) and Schiffer (1999).

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104 / Crispin Wright These are searching questions. In response to them, Williamson, for one, has tended to reply (uncharacteristically weakly, it seems to me) that reference is a notion of which we lack an adequate philosophical account in any case—that his view ‘‘has not been shown to be inconsistent with anything taught by the theory of reference’’.15 That is like defending the claim that the lifespan of the human person is standardly about one day—that we cease to exist in sleep, to be replaced by another centre of consciousness with the same range of seeming-memories, etc., on waking—by appeal to the unclear and vexed nature of the concept of personal identity. Sure, reference—and personal identity—are philosophically perplexing notions. But that is not to say that they are in such bad shape that no (consistent) view involving them can reasonably be discounted. If someone wanted seriously to maintain the sleep-replacement hypothesis, they would first owe an explanation of how the notion of personal identity allows it as a genuine possibility—it is insufficient to say that, in the present state of unclarity of that concept, we cannot rule the hypothesis out. It is no different with Epistemicism and reference. In particular the one reasonably clear model (or type of model) we have of how the property presented by a predicate may not be transparent to those who fully understand that predicate—the model of lay natural kind terms like ‘water’ and ‘heat’ owing to Kripke and Putnam—seems to have no relevant bearing on vague expressions in general. I myself see no reason to expect that we shall ever have a generally satisfactory theory of reference—especially predicate reference—which discharges Epistemicism’s debts. To the contrary, I believe we never shall. But let that opinion pass. The question is: can anyone at all justifiably take themselves to know that Bivalence is good for vague sentences? If it is, each vague expression is associated with a sharply bounded semantic value of the kind appropriate to it, a sharply bounded property, relation, function, or whatever. Grant that our so far articulated philosophical understanding of the determination of semantic value does not put us in position to rule that out (even if we regard it as outlandish). The question is: can anyone, even the most rampant Epistemicist, put her hand on her heart and say that she knows that such is indeed the situation—that the required semantic associations really are in place? Williamson’s defensive point was: well, you cannot rule it out. But we can grant that and still quite rightly be agnostic about the matter. And if we are, we should be agnostic about Bivalence too. 15 Williamson (1996: 43).

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Vagueness: A Fifth Column Approach / 105

6. Conclusion Let me close, then, by insisting on something once regarded as obvious: that no one, in our present state of understanding of these matters, can reasonably take anything but an agnostic view of Bivalence as applied to vague statements. If what I have been saying is right, the consequences of this claim are interesting and liberating. At the least, we need not worry about the Sorites, for it is disarmed in any case. Crucial remaining issues include: to refine the characterization of the kind of broadly epistemic conception of borderline cases that I have suggested, and to address the need for an account of how—if not by mysterious associations with sharp semantic values—the extension of vague expressions should be conceived as determined. It is here that I think there may be a role for notions of response-dependence. But that is for another occasion.

REFERENCES C h a m b e r s , T i m o t h y (1998), ‘On Vagueness, Sorites, and Putnam’s ‘‘Intuitionistic Strategy’’ ’, The Monist, 81: 343–8. P u t n a m , H i l a r y (1983), ‘Vagueness and Alternative Logic’, in Putnam, Realism and Reason (Cambridge: Cambridge University Press). R e a d , S t e p h e n , and C r i s p i n W r i g h t (1985), ‘Hairier than Putnam Thought’, Analysis, 45: 56–8. S c h i f f e r , S t e p h e n (1999), ‘The Epistemic Theory of Vagueness’, Philosophical Perspectives, 13: 481–503. W i l l i a m s o n , T i m o t h y (1990), Identity and Discrimination (Oxford: Blackwell). —— (1994), Vagueness (London: Routledge). —— (1996), ‘Wright on the Epistemic Conception of Vagueness’, Analysis, 56: 39–45. W r i g h t , C r i s p i n (1995), ‘The Epistemic Conception of Vagueness’, Southern Journal of Philosophy, 33, suppl.: Vagueness, ed. Terence Horgan, 133–59. —— (2001), ‘On Being in a Quandary: Relativism, Vagueness, Logical Revisionism’, Mind, 60: 45–98.

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6 Semantic Accounts of Vagueness Richard G. Heck, Jr.

Crispin Wright’s reflections on vagueness have, for nearly thirty years now, been a touchstone for all serious work on it. His most recent efforts, beginning with the paper ‘On Being in a Quandary’1 and continued in this volume,2 seem to me extremely important. What I want to do here, however, is not to try to evaluate his new position, nor even to engage it directly, but instead to defend one of the more traditional views he criticizes, namely, the view that vagueness is a ‘semantic’ phenomenon. I am not at all sure this sort of view is right, but I think we need a better sense than we presently have of what might be wrong with it if we are to discover a better view. I will begin, however, by discussing another question that has obviously exercised Wright a great deal, namely, what is really wrong with Epistemicism and other views that are committed to the retention of bivalence even for vague predicates. Remarks bearing some non-trivial relation to the following were made in response to Crispin Wright’s paper ‘Vagueness: A Fifth Column Approach’, published in the present volume (Ch. 5), and delivered at the conference whose proceedings this volume is. Much thanks to JC Beall and Michael Glanzberg for organizing a terrific conference, from which I learned a great deal. Many of the ideas expressed in this paper were developed in graduate seminars on vagueness given at Harvard University in fall 1997 and then again in fall 2001. Thanks to all those who attended for their participation, but especially to Delia Graff, who attended the first of these and with whom I have had many helpful discussions on these topics over the years. Thanks also to an anonymous referee for Oxford University Press, whose objections improved the paper. And, finally, thanks as usual to Jason Stanley, for discussions over the last couple years, especially concerning various versions of contextualism. 1 See Wright (2001). 2 Ch. 5; references to this paper are included in the text.

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1. Do Vague Predicates have Determinate Classical Extensions? Wright worries that Epistemicism has an unpaid, and unpayable, metasemantic debt: Williamson owes an account of what fixes the ‘sharp boundaries’ of (particular uses of) vague predicates. Many have voiced this concern, but, to the best of my knowledge, no principled reason has yet emerged to believe that no such account will be forthcoming. In the present paper, as elsewhere in his writings—and Wright is not unusual in this regard—we find little more than a profession that Wright is unable to see even the shape of a decent account (pp. 104–5). I can’t see it either, but I’d still like to try to do better, to offer (what I take to be) a principled reason to doubt that vague predicates have determinate classical extensions. Most reflections on this problem begin (and often end) with the thought that it must be our use of ‘heap’, ‘red’, and ‘chair’ that, somehow or other, fixes their extensions. Williamson does a fine job arguing, however, that, if what is intended here is some sort of supervenience thesis, then it poses no threat to Epistemicism.3 So more has to be said. A second thought might be this one. By hypothesis, our linguistic dispositions leave certain sorts of cases undecided, namely, the cases that we call borderline cases—these, obviously, being the ones that cause all the trouble. And it is hard to see what, if not our linguistic dispositions, is supposed to decide them. But Williamson could simply dismiss this sort of concern, too. It is based upon a conception of how the extension of a predicate is fixed by its use that is far too naive to take seriously. The use of a predicate never fixes its extension by deciding every case directly. Still, the question remains: What does fix it? I’ve heard it suggested that we should just ‘split the difference’: take the boundary between the red and the not-red to lie exactly between the things our use of ‘red’ definitely determines should be so-called and those it definitely determines should not. But, for one thing, that’s uncomfortably ad hoc, and, for another, it is more easily applied to ‘red’ than it is to ‘chair’. I’ll raise a deeper worry about it later. In one of his discussions of this matter, Williamson writes, regarding the predicate ‘heap’, that its extension is not so hard to determine as one might think.4 The Oxford English Dictionary says, amplifying a bit, that a heap is ‘‘[a] collection of things lying upon one another so as to form an elevated mass’’ 3 See Williamson (1996, sect. 7.5). 4 Williamson (1996: 213). The argument originates with Hart (1991–2).

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108 / Richard G. Heck, Jr. and in which some of the objects are completely and stably supported by others. Not much reflection is needed to see that the minimum number of grains of sand needed to construct a heap is therefore four: three on the bottom, and one sitting on top of them. It’s easy to get the sense Williamson’s tongue must have been firmly in cheek when he made this suggestion, and he is clear enough that he does not expect to be able to make similar remarks about every vague predicate. I am going to suggest, however, that his inability to do so is, ultimately, what underlies the impossibility of his answering the meta-semantic question what fixes the extension of a vague predicate. Wright remarks toward the end of his paper that ‘‘the one reasonably clear model . . . we have of how the property presented by a predicate may not be transparent to those who fully understand that predicate [is] the model of lay natural kind terms like ‘water’ . . . ’’ (p. 104). The thought here is simple and, I think, compelling. Whether something is water is not decided entirely by our linguistic dispositions, that is, by our responses to putative samples of water. It is not even by how we would respond in epistemically ideal conditions. Our use of the term ‘water’ is ‘focused’ upon a particular property, and the world itself decides whether a particular bit of stuff has or fails to have that property. How we react to putative samples of water is, of course, part of what focuses our use of the term upon the property of being water. And so, in that sense, there is no threat here to the intuitive thought that use determines meaning. But, at the same time, we can easily imagine ourselves being, even under ideal epistemic circumstances, wrong about whether something is water, and for that reason it could well be, and remain, a mystery just where the boundary between what is and is not water lies. Williamson’s remarks about the word ‘heap’ apply this model to it. What our linguistic dispositions leave unresolved may be resolved by the world itself, for there is a particular property on which our use of the term ‘heap’ focuses: our day-to-day use of the term—although it does leave borderline cases unresolved, in one sense—is focused upon a particular property, and the world itself decides whether a given object has it. One might speculate, then, that something similar should be true of color words, like ‘red’: perhaps there is a particular property (be it physical, dispositional, or what have you) on which our use of this term is focused. Then, once again, the world itself could be left to decide, of any given object, whether it had that property, and our ignorance about where the boundary between what is and is not red lay would be no more puzzling that our ancestors’ ignorance about where the boundary between what is and is not water lay.

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Semantic Accounts of Vagueness / 109 There may be some cases to which this model can appropriately be applied. But the strategy will not generalize to all cases of vague predicates. (Williamson does not suggest that it will.) So far as I can see, there is no mind-independent property that our use of the term ‘chair’ might plausibly be thought to pick out: chairs form not a natural kind but an artifactual one, and they have no real essence, but only a nominal one. Even the case of ‘red’ is problematic. Color words like ‘scarlet’ and ‘vermilion’ are all the more so. Worse yet, most of these—and certainly such terms as ‘flat’—exhibit context-dependence, as Diana Raffman, Delia Graff, and others have rightly emphasized. The fact that such terms can be used in different contexts to pick out different properties makes it obvious that there is not going to be a property that does for our use of ‘flat’ what the essence of heaphood might do for our use of ‘heap’. I began this discussion by quoting Wright’s suggestion that natural kind terms provide us with ‘‘the one reasonably clear model . . . we have of how the property presented by a predicate may not be transparent to those who fully understand that predicate’’. We have, however, come close now to a stronger conclusion. As I said earlier, our use of the term ‘water’—whatever precisely might be meant by ‘use’ here—does not decide directly how the term should be applied in every case: it is not as if every putative sample of water has been examined and determined either to fall within or without the extension of the predicate. We might then ask why, even in the case of a term like ‘water’, we should believe that every such sample is either one to which the predicate should be applied or one from which it should be withheld. The answer, it seems to me, is again that, although our use of the predicate certainly does not decide directly, of every such sample, whether the predicate should be applied to it, our use of the predicate does focus upon a particular property, and the world then decides whether the sample has the property or not. Our use therefore decides indirectly whether the predicate applies to a given sample: Our use focuses the predicate on the property, and the world decides whether the sample has the property. Imagine now a world, otherwise like our own, circa 1500, but in which there is no water—one in which there is, moreover, no XYZ nor any other natural kind of stuff that fills the lakes and rivers. In this world, what fills the lakes and rivers is a motley collection of fluids having little more in common, chemically speaking, than the things we call ‘fabric’. Now, supposing the use of the term ‘water’ in this world to be pretty much the same as it was in our world circa 1500, does this term have a determinate classical extension in this

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110 / Richard G. Heck, Jr. world? There is a temptation to say it would: it would have applied to all the odorless, colorless liquids, or something along those lines. But I see no particular reason to believe that. In this case, our linguistic dispositions certainly would not decide every case directly. There would be plenty of putative samples of water that were, in some ways, very similar to the other things we call ‘water’, and in other ways not quite like them, and there would be no chemical or other natural basis on which to draw a boundary. It seems to me, then, that the lesson of these sorts of examples is not just that the extension of such terms as ‘water’ is fixed by the world—so that, had the world not been cooperative, it would have been fixed entirely by our linguistic dispositions. It is that only the world can fix a determinate classical extension for a predicate. It would take more space than I have here to establish this claim, not to mention more and better arguments than I have available. But the basic idea is fairly simple and, I think, plausible. If we consider the matter just in terms of the possible extensions for the term ‘water’, then it is very difficult to see how one rather than another of the possible extensions could be uniquely determined as the extension of this term. But our use of the term ‘water’ is, as I put it before, focused on the property of being water, and the world itself is then left to decide what has this property. If properties were as common as extensions, of course, then reference to them wouldn’t help. But they aren’t. The property of being water is the chemical property of being H2 O, and such properties are, so to speak, sparse. It is, for that reason, much easier for our use of the term ‘water’ to focus upon a specific property than it is for our use of the term to pick out a particular extension. It is only because the property of being H2 O stands out from the crowd, as it were, that our use of the term focuses upon it: our use of the term is, as it were, drawn to this property, because it is the only one in the neighborhood. If, as in the example mentioned above, the word ‘water’ did not pick out a natural kind, there simply wouldn’t be a natural property on which our use of it might focus, and its extension would be left indeterminate.5 What I am suggesting can be put this way. To ask why vague expressions fail to have determinate classical extensions is to ask the wrong question. The great mystery is how any expression ever comes to have a determinate classical extension. And so far as I know, the only plausible resolution of this mystery appeals essentially to the idea that the candidate extensions are 5 The intuitions I am exploiting are used to great effect in Lewis (1984).

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Semantic Accounts of Vagueness / 111 sufficiently scarce that our use of a given predicate may, again, be drawn to one or another of them—the candidate extensions being the extensions of properties, these themselves being scarce. If the set of candidate extensions for the predicate were, on the contrary, plentiful, then there would be nothing in our use of the predicate that could distinguish one of them from all the others. Let me say again that the preceding is not really intended as an argument. It has, rather, the status of a conjecture; one, I think, that has some plausibility. But a similar line of thought is, it seems to me, particularly compelling in the case of vague predicates that exhibit context-dependence. Consider, for example, a particular utterance of ‘He is tall’. For concreteness, assume a degree-theoretic account of adjectives of this sort. Such adjectives are associated with scales—in this case, a scale of height—and the contextdependence of such adjectives is explained in terms of the fact that ‘tall’ means roughly: of a degree of height greater than d, where d is a contextually determined degree of height. Context is thus obliged, in any given case, to fix a point along the scale that will divide the tall from the not-tall.6 So the question is: What reason do we have to suppose that, in typical cases, context is always sufficient to fix a particular degree of height dividing the tall from the not-tall? None of the degrees is, in any way, more ‘natural’ than the others. So the job context must do is very difficult indeed. I simply see no reason to suppose that ordinary contexts fix unique such degrees, nor even that they fix the degrees precisely enough to decide, of every object in some contextually relevant domain, whether it counts as tall or not. What I suspect, rather, is that context restricts the set of degrees as far as is needed for conversational purposes and that further such restrictions are negotiated as they become necessary.7 It is, indeed, hard to see why context ought to be expected to do any more than that. There is no a priori reason to suppose that every context is sufficient to fix a precise such degree—any more than there is a priori reason to suppose that every context is sufficient to fix the referent of the demonstrative ‘he’. If so, however, we should ask whether such an insufficiency, if it were to arise in a particular case, would have to frustrate the purposes of the conversational participants. In the case of ‘he’, the answer is that it typically would: if there is 6 On Graff ’s view, vague predicates are not really context-dependent, but the difference does not really matter for present purposes. See Graff (2000). 7 The ideas I am expressing here thus have some similarities to those expressed by Jamie Tappenden in his (1993).

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112 / Richard G. Heck, Jr. no fact of the matter about to whom the speaker’s use of the word ‘he’ refers, then, presumably, one would expect no agreement among the conversational participants about to whom it refers, and that will lead to communication breakdown. But, in the case of ‘tall’, so far as I can see, no such breakdown need be expected. If context is insufficient to decide whether, say, Bob counts as tall in it, then, if it matters whether Bob counts as tall, there will be a problem. But it need not matter, and there need be no problem. At this point there is a natural objection. Consider some vague predicate, say, ‘chair’, and suppose what I’ve said so far is correct: there are many classical extensions this predicate might have consistent with its use. Then the thought is that we should simply take the extension of ‘chair’ to be the intersection of these various possible extensions, that is, the set of objects of which it is determinately true, so that it is false of any object of which it is not determinately true. The idea that we should ‘close off ’, as I’ll put it, borrowing the term from Kripke, is a common one.8 The suggestion is a non-starter, however, at least if it’s offered against the background I’ve just outlined. Suppose we have some independently motivated theory of what fixes the reference of a predicate and that, according to that theory, there are different extensions the predicate might have, consistently with the facts concerning its use. To suggest, in this context, that we just close off would be to abandon the theory I’m assuming we have. The closed-off extension may well be one of the possible extensions for the predicate, but, by hypothesis, the facts concerning the use of the predicate do not determine that the closed-off extension is the extension of the predicate. Closing-off is just an ad hoc construction of no semantic relevance. The same is true of the suggestion mentioned earlier, that we should just ‘split the difference’. As it happens, the actual closing-off construction in Kripke illustrates this point. Let us recall how it works. On Kripke’s theory, the ‘use’ of the truthpredicate is completely captured by four rules of inference, the so-called T-rules: A ‘ T(uAv) T(uAv) ‘ A :A ‘ :T(uAv) :T(uAv) ‘ :A 8 For the terminology, see Kripke (1975). The suggestion is made explicitly in Williamson (1996: 208). Even Wright seems sympathetic with it, using something similar as an objection to what he calls Third Possibility. See p. 92 ff. above.

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Semantic Accounts of Vagueness / 113 The proof of the fixed point theorem then shows that there is an extension the truth-predicate could have, consistent with these rules and, indeed, that there are infinitely many such extensions. One might think, of course, that there is more to our ‘use’ of the truth-predicate than just these four rules— something that has the effect of requiring truths to be grounded, for example—so that a unique extension is determined from among the fixed points, after all. But the suggestion of interest to us is another one Kripke makes: that we ‘close off ’ to a classical extension by taking the (classical) extension of the truth-predicate to be the intersection of its extensions at the various fixed points—which, since there is a minimal fixed point, is just to take its (classical) extension to be its extension at the minimal fixed point. I remember George Boolos defending this view in a class on truth. ‘‘So’’, said George, ‘‘the liar sentence is not true. Neither is its negation.’’ I remember myself objecting, ‘‘George, you have just uttered the liar sentence! And you have simultaneously said that it is not true.’’ George responded by asking me if I was calling him a liar. ‘‘Is that a moral criticism?’’, he asked, grinning broadly. I didn’t know what to say then, but I do know now, namely, that the closing-off construction begins with the thought that the extension of the truth-predicate is fixed, if by anything, then by the T-rules. To close off is simply to abandon that idea. Indeed, it is to deny that the T-rules are even valid (and that is what was fueling my objection). Closing-off thus leaves the original construction of the minimal fixed point—the construction that determines the extension of the truth-predicate—unmotivated. Why should we care about the extension of the truth-predicate at the minimal fixed point if the T-rules are not even valid?9 I do not say that closing-off, in the case of vague predicates, would suffer from exactly this problem, though it might, in some cases: the intersection of all the admissible classical extensions could, in some cases, prove to be something an independently motivated theory of what fixes reference actually determined not to be the extension of the predicate. But, in most cases, the problem will not be that bad. Nonetheless, closing-off will always be ad hoc. To ‘close off ’ is, once again, to take the intersection of a certain class of possible extensions for a predicate. So the closing-off construction begins with some set of extensions that our use of the predicate does not rule out as possible extensions. If so, however, then, by hypothesis, there are extensions the 9 There is an independent way to get what is, in effect, the same theory as one would get by closing off: the theory in question is KF, the Kripke–Feferman theory. It may have some independent motivation, but my point here is simply that it needs some independent motivation.

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114 / Richard G. Heck, Jr. predicate could have, consistent with our use of it, other than the one delivered by the closing-off construction. But if the closing-off construction is what determines the extension of the predicate, then those other extensions aren’t really extensions the predicate could have, consistent with our use of it, and it is unclear what their status is.

2. Indeterminacy and the Appeal of the Sorites I have argued, then, that typical vague predicates—certainly ‘chair’ and ‘tall’, though maybe not ‘heap’—fail to have determinate classical extensions. You will note that I have been careful not to say that their extension is indeterminate. If we understand indeterminacy as Supervaluationism, for example, would understand it, then it does not follow from what I have argued that the extensions of vague predicates are indeterminate. Wright’s view is, obviously, consistent with what I have been arguing, but he in particular would not want to make this claim. What I have argued could thus be put, neutrally, as follows: Although there are some classical extensions that a given vague predicate could not have, there will be many that it could have, consistent with the facts involving its use, even if use is conceived, as I think it should be, broadly, as involving relations with the world. There are, obviously, a variety of ways to respond to this observation. One would be to regard vague predicates as having indeterminate extensions: a view of this sort is what Wright calls a semantic view of vagueness. Another view, Wright’s own favored view, is that the classical model breaks down here. The classical notion of an extension (more generally, classical semantics) is simply inapplicable to the case of vague predicates; something like the intuitionistic notion of a species (more generally, intuitionistic semantics) is needed here. As it happens, this view is one I’ve also suggested, though in connection with Evans’s argument that there are no vague objects, not the Sorites.10 But I’ve recently begun to wonder, myself, whether Supervaluationism is quite as lifeless as it is routinely claimed to be. In the remainder of this paper, then, I’d like to try to motivate a version of Supervaluationism and to defend it against Wright’s criticisms. Before I begin, I want to emphasize one point. One sometimes gets the sense from the literature that what we need is the solution to the Sorites 10 See Heck (1998).

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Semantic Accounts of Vagueness / 115 paradox or the correct theory of vagueness. But what we regard as vague predicates, or even Sorites-susceptible predicates, may not form a semantic kind. If not, then different accounts will be needed of what is responsible for the apparent vagueness from which such predicates suffer—the presence of borderline cases, if that is what is distinctive of vagueness. One might hold, on broadly methodological grounds, that a unified theory would be preferable if one were available. But if one considers the full range of vague expressions, from adjectives like ‘tall’ to common nouns like ‘chair’ to verbs like ‘to shout’, then I see no a priori reason myself to expect a uniform account. I wish, then, to defend the coherence of Supervaluationism not because I think it the solution to the Sorites paradox, but because I see no reason to doubt that semantic vagueness is a real phenomenon. Indeed, I think it likely, as I said above, that many uses of context-dependent vague predicates are semantically vague. Wright suggests that semantic treatments of vagueness begin with the thought that vagueness ‘‘originates in shortfalls . . . in the meanings we have assigned to expressions’’ (p. 85). On my view, that is almost right. For emphasis, the view should be that some vagueness originates in semantic insufficiency. A more important point is that the shortfall need not concern the meaning of the expression, if by that is meant something at the level of Fregean sense. The view is that there is an indeterminacy regarding the extension of the predicate. But there is no obvious reason to suppose that this indeterminacy need be due to some indeterminacy regarding the predicate’s meaning, in the intuitive sense. Failure to appreciate this point is, I think, behind at least some objections to Supervaluationism.11 Semantic indeterminacy, however, is not vagueness, for the mere fact that it is indeterminate what the extension of a predicate is is plainly no reason to expect that it will give rise to the sorts of paradoxes to which vague predicates typically do. One should not, however, overreact to this observation. It does not give us reason to reject indeterminacy as a fundamental part of the theory of vagueness (at least for some expressions—a qualification I shall henceforth drop). Minimally, to be sure, a story needs to be told about what the indeterminacy of vague predicates has to do with the Sorites paradox. That question is equivalent, of course, to the question what vague predicates’ indeterminacy has to do with the appeal of the crucial assumption that, say, if one thing is red, anything pairwise indistinguishable from it is also red. 11 I think, in particular, that it lies behind the objections in Fodor and Lepore (1996).

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116 / Richard G. Heck, Jr. What is the appeal of that premise? In his earliest writings on this topic Wright suggests that it originates in a conception of the kinds of grounds one could have for judging that something is red. One would ordinarily suppose that whether something is red is the sort of thing one can tell by looking. If so, then surely you can’t have two things you can’t tell apart by looking one of which is red and the other of which is not. But it seems to me that the basic intuition here doesn’t really support that claim. Rather, the basic intuition is that a certain combination of views is irrational: one cannot simultaneously hold that a is red, that b is not, and that a is pairwise indistinguishable from b.12 That is already enough to get something like Sorites reasoning started. Faced with a chip I agree is red and another I agree is pairwise indistinguishable from it, it is easy to feel some compulsion to say that the second chip must also be red. After all, it would be irrational to say it wasn’t. But, in fact, that move can be resisted. One might simply have no view about whether the next patch is red, and there’s nothing irrational about that. One might object that it is irrational to hold that a is red and that b is pairwise indistinguishable from a, but to refuse to commit oneself to the claim that b too is red. After all, one might say, one has the very same evidence that b is red as that a is red. I do not deny that the objection has some force, but it seems to me that one does not (or at least need not) have the same evidence that b is red. Why might one think otherwise? The evidence one has is simply how a and b look. But, one might suppose, if a and b are pairwise indistinguishable, then surely they look the same. But, of course, this move must be resisted, lest we find ourselves committed to the conclusion that all patches look the same. More positively, though, the evidence one has that a is red will, in a typical case, be how it appears to one: that is, one’s evidence is one’s perceptual experience of the patch—more precisely, what one’s experience of it represents about its color. We can identify one’s evidence that the patch is red, then, with the representational content of one’s perceptual experience of the patch (so far as its color is concerned). I see no reason to suppose that, if patches a and b are pairwise indistinguishable, then the representational content of one’s perceptual experience of a and of b (so far as their color is concerned) cannot differ. One might not realize that the content differed in the two cases, but that is another matter.13 12 For a similar observation, see Raffman (1994). As will be clear, I end up making something rather different of the observation, which, in its present form, seems to have been in the air. 13 I suppose one might feel some threat to first-person authority here, but I don’t think we have any first-person authority about the contents of perceptual states, in the relevant sense.

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Semantic Accounts of Vagueness / 117 Sorites-type paradoxes can be resolved by distinguishing between their major premises—for example: if a patch is red, then any patch pairwise indistinguishable from it is also red—and what we might call an expression of the quasi-tolerance14 of the relevant predicate: if one regards a patch as red, and regards another as pairwise indistinguishable with it, one cannot rationally regard the latter as not being red. The quasi-tolerance of ‘red’ derives from its observationality: other observational predicates can therefore be expected to satisfy similar principles. Non-observational predicates that are susceptible to Sorites-type reasoning will satisfy similar principles, but these will derive from other aspects of their use.15 This interpretation of the appeal of the Sorites paradox is available to philosophers with many different views of vagueness. It is, for example, similar in feel to the margin of error principles deployed by Williamson.16 But there is an important difference, one that can be revealed by considering a version of Williamson’s example of the crowd. Say I’m seated at Fenway Park and someone asks me how many people are there for the ballgame. To hold any view, on the basis of a glance at the crowd, about exactly how many people were present would be unjustified. But it isn’t as if any view about exactly how many people were present would be unjustified on any possible ground. On the contrary, one could lock the doors and count everyone and then one could determine exactly how many people were there. In the case of vagueness, though, the intuition is different. It isn’t just that one can’t know on the basis of casual observation that one patch is red and one pairwise indistinguishable from it is not. This combination of views seems irrational whatever its basis. And surely Williamson would agree: he thinks it is impossible for us to know where the boundary is between what is red and what is not. But the example of the crowd and the margin of error principles it generates are insufficient to ground that conclusion. It’s easy enough to see why the margin for error required for knowledge of color is greater than pairwise indiscriminability if the ground on which one makes one’s judgement is observational. If I judge that patch A is red and do so on the basis of how it looks to me, then, plausibly, I cannot know that patch A is red if I would have been wrong had it been indiscriminably different. 14 The notion is similar to but different from the notion of tolerance that plays an important role in Wright’s earliest published reflections on vagueness, for which see his (1976). 15 For the classic discussion of how such principles derive from aspects of our use of certain predicates, see Wright (1976). 16 See Williamson (1996, sect. 8.3). As mentioned above, similar notions are found in Graff (2000) and Raffman (1994).

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118 / Richard G. Heck, Jr. But if there is a sharp boundary between what is red and what is not, why can’t there be some other way to know where it is? If one were to judge on some other ground, the margin of error principle just stated would not apply. One might reply that, since the location of the boundary is fixed by facts about casual observation, if its location can’t be determined by casual observation, then it can’t be determined at all. But that simply doesn’t follow. How one might find out whether something is red is no more restricted by what fixes the extension of ‘red’ than how one might find out whether something is a virus is restricted by what fixes the extension of ‘virus’— whatever that is. Contextualists offer a different sort of answer. On Graff ’s view, for example,17 if one sets out to find the boundary between what is and is not red, then, as one shifts one’s gaze from one part of a Sorites series to another, the extension of the predicate ‘red’ itself shifts, in such a way that the boundary moves away from where one is looking. So even if, say, the boundary at time t0 were between patches p237 and p238, if one were to look at these patches at time t1 , it would move to, say, between p125 and p126 . And if one looks there, it will move again. There is, as Graff puts it, always a boundary, but it will never be where one is looking for it. But this view too simply fails to answer the question why we cannot locate the boundary between what is and is not in the extension of ‘red’ as that term is used at time t0 . Suppose I say, at t0 : Some of these patches are red; call them the reddies. I might then ask which is the last of the reddies. By hypothesis, it is between patches p237 and p238 . Graff tells us that, if we look at these patches, the boundary between what is and is not in the extension of ‘red’, as we would then be using it, would shift, so that it will not, at time t1 , be true to say that p237 is red and p238 is not.18 But the question is not whether we can then say 17 I include Graff here, since, though her view is not, ultimately, contextualist, it is similar in relevant respects. See also Raffman (1994) and Stewart Shapiro’s contribution to this volume (ch. 3). 18 There are some subtle questions here about how exactly this point should be expressed on Graff ’s view. It seems uncomfortable to say that a patch whose color has not changed is no longer red at t1 but was at t0 , on the ground that our interests have changed. Something like that is, however, what Graff ’s view entails, though temporal rigidification on our interests may allow her to avoid a direct conflict with semantic intuitions, since she could then deny that we can truly say, in natural language, something like: ‘Although this patch is now red, in five minutes it will not be, even though its color will not change’. A similar problem arises with modal contexts: ‘If we had had different interests, this patch would not have been red, even though its color would not have changed’. Modal rigidification will allow Graff to avoid that consequence, but it causes other problems, for which see Stanley (forthcoming).

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Semantic Accounts of Vagueness / 119 that one is red and the other isn’t. The question is why we cannot locate the last of the reddies. Maybe the extension of the word ‘red’ as we would then be using it would indeed shift, but the point does not seem relevant. There is no such shift in the extension of ‘the reddies’. The point here can be put somewhat differently. Contextualists explain the appeal of Sorites reasoning as follows. When we consider any particular Sorites conditional, say (C125 ) If p125 is red, then p126 (from which it is pairwise indistinguishable) is also red, principles that govern how context fixes the extension of ‘red’ guarantee that the conditional will be true, for those principles imply that no two salient color patches whose pairwise indistinguishability is salient can be differently characterized in respect of color. But there is a boundary between that to which ‘red’ applies and that to which it does not—say, between p237 and p238 . But when we come to consider (C237 ) If p237 is red, then p238 (from which it is pairwise indistinguishable) is also red, the same principles governing how context fixes the extension of ‘red’ will guarantee that it is true. The extension of the term ‘red’, however, will have shifted. Thus, the Sorites premise (C) For every i, if pi is red, then piþ1 is also red has a certain kind of appeal: Whenever we consider one of its instances, (Ci ), the rules that govern how the extension of ‘red’ is fixed by context will guarantee that instance’s truth. But the conditional itself is false. To think otherwise is simply to equivocate. The difficulty is that this strategy does not generalize. Confronted with the sequence of patches, I might say:19 Some of these patches are red. I wonder which of them is the last? It seems hard to imagine that there could be a last one. After all, p1 is one of the red ones. But since p2 is pairwise indistinguishable from it, surely it too is one of them. But p3 is pairwise indistinguishable from it, so surely 19 This observation was inspired by an argument due to Jason Stanley, for which, see his (forthcoming). His argument makes use of VP ellipsis and so does not apply to Graff’s view, but only to more straightforwardly contextualist views like Raffman’s or Soames’s.

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120 / Richard G. Heck, Jr. it too is one of them. And p4 is pairwise indistinguishable from it, so surely it too is one of them. Etc. etc. Contextualists have no explanation of the appeal of this reasoning: the use of the anaphor ‘them’ fixes the collection of objects that is at issue, much as the introduction of the term ‘reddies’ did above, so that no contextual shift can occur as we proceed through the series. In principle, to be sure, one could respond that this particular reasoning is not as compelling as the usual sort of Sorites reasoning, or perhaps insist that it is compelling only because we confuse it with ordinary Sorites reasoning. But I see no plausibility in such responses.20 These might seem fiddly technical objections meriting fiddly technical responses. In fact, however, they go to the heart of the contextualist strategy. Contextualism is the view that what we experience as vagueness is a form of context-dependence.21 And the first-blush response that almost everyone seems to have to it is: OK, fix the context; the extension of ‘red’ in that context is still vague. The objections just sketched simply refine this intuition. I suggest that they show that no view that attempts to disarm the Sorites by appealing to context-driven changes in the extension of the relevant predicate can succeed. The Sorites reasoning is just as appealing when one nails the extension down as it is when one allows it to vary.22 Now, as I said above, the fact that vague predicates lack determinate classical extensions cannot, by itself, explain the appeal of the Sorites premise. But I think we can now see that to think it did would be to get things precisely backwards. It is not the indeterminacy of ‘red’23 that explains the appeal of the Sorites premise but the appeal of the Sorites premise—the quasitolerance of vague predicates—that explains, at least in part, why ‘red’ lacks a determinate classical extension. That one cannot rationally regard a patch as red and another pairwise indistinguishable from it as not red implies that there will be borderline cases—cases in which neither the view that an 20 Another possibility would be to insist that ‘them’ is a pronoun of laziness, or that the reasoning is only appealing if it is. I see no plausibility there, either, however, and, in any event, that response is not available to the version of the argument using the term ‘reddies’. 21 Again, Graff ’s view should not really be so described, but the differences do not matter for present purposes. 22 The point is not, of course, that ‘red’ and the like are not context-sensitive. The point is that their context-sensitivity cannot be used to disarm the Sorites. 23 Indeterminacy, that is, in the extension it has when uttered on a particular occasion. I’ll drop this sort of qualification henceforth, when it’s not needed.

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Semantic Accounts of Vagueness / 121 object is red nor the view that it is not red is rationally compulsory (or, perhaps, even rationally defensible). If so, then our ordinary use of ‘red’ will fail to resolve at least some such cases directly. I emphasized above that it does not immediately follow that our use of ‘red’ leaves such cases completely unresolved: remember ‘water’. But absent a natural kind or property to resolve for us what our own usage does not, ‘red’ will lack a determinate classical extension. Let me say that again, because the point seems to me to be important, and as yet unappreciated: It is not because vague predicates have borderline cases that Sorites reasoning involving them is appealing. On the contrary, it is because Sorites reasoning involving vague predicates is so appealing, and because there is no natural kind or property to fix their extensions, that they have borderline cases. To the question what vagueness is we might therefore answer: quasi-tolerance, in the absence of a natural kind or property that might fix a classical extension. One other point. One sometimes gets the sense that, on semantic views of vagueness, vagueness is due to laziness, so that the extension of ‘red’ is indeterminate because we have not bothered to fix a determinate extension for it, though we could, if we wished to do so. I hope the foregoing corrects this misimpression. The indeterminacy, if such there is, in the extension of the predicate ‘red’ is due, on my view, to its observationality. As Wright long ago suggested,24 then, there is a sense in which we could resolve this indeterminacy only if we sacrificed something important about our use of this word, namely, its observationality. Other cases may be different.

3. In Defense of Supervaluationism If vague predicates lack determinate classical extensions, then there will be no classical semantics for a language containing vague predicates. There are, to be sure, plenty of ways to proceed. But it is natural at this point, or so it seems to me, to suggest that we try to provide such a semantics in terms of what is already available to us, namely, a set of extensions that ‘red’ could have, consistently with our use of it. So consider an utterance of ‘That is red’. When shall we regard it as true? Well, surely, if it would have been true (false) no matter which of the various possible extensions for ‘red’ it might have had, it 24 See, again, Wright (1976).

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122 / Richard G. Heck, Jr. is hard to call it false (true). Conversely, if there are extensions ‘red’ could have, consistently with our use of it, according to which this utterance would not have been true (false), it is hard to call it true (false). But that is just to say that the utterance is true (false) iff it is true (false) no matter which of the possible extensions we suppose ‘red’ to have, and that is the basic idea behind Supervaluationism. It therefore seems to me—contrary to Wright’s remark that this idea ‘‘comes . . . completely out of the blue’’ (p. 88)—at least a wellmotivated position. More would need to be said, of course, to motivate Supervaluationism’s distinctive logical theses, in particular, to motivate its treatment of the logical connectives. But we may set that issue aside. The resolution of the Sorites paradox offered above does not depend upon the Supervaluational treatment of the logical connectives: not only is it compatible with various treatments of those connectives, it is, as I mentioned, compatible with epistemicist and contextualist treatments, as well. The question how the logical connectives should be handled is an interesting and important one, as is the question whether we should regard the Sorites premise as false (as having a true negation) or merely as untrue, but both of these are less central than the question how we can coherently refuse to accept the Sorites premise and what its attractions were, in the first place. I do not say that Supervaluationism, whether in its familiar classical form or in some other form, is the right view. But I am sure that it is not as flawed as most people seem to think it is. Let me close by defending it against some of the objections Wright brings against it. First, Wright endorses Williamson’s criticism that Supervaluationism ‘‘implicit[ly] surrender[s] . . . the T-scheme’’. Speaking for myself, however, I see no force in this objection. For one thing, I take the T-scheme to be unmotivated, at least in so far as its content exceeds that of the T-rules, mentioned earlier, which Supervaluationism can happily endorse. But more seriously, in the context of any non-classical semantics, it must at the very least be an open question whether the truth-predicate and the biconditional behave in such a way that the T-scheme is validated. Even if one assumes that u‘S’ is truev and S must always have the same truth-value, it does not follow that u‘S’ is true iff Sv is always true, unless one makes additional assumptions about the semantics of the biconditional. I see no reason to assume either that u‘S’ is truev and S must always have the same truth-value or that a suitable biconditional will always be present in the language or definable if it is not. If, as deflationists hold, there were no way to explain what one means by ‘true’ except by

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Semantic Accounts of Vagueness / 123 appealing to the T-scheme, that would be one thing. But it wants argument that that is the situation in which we find ourselves. Second, and much more seriously, Wright raises the specter of higherorder vagueness. As he puts the point, ‘‘it is a generally accepted datum of the problem that, in a wide class of examples, the distinction between the borderline cases and those which we have a mandate to describe as, e.g., ‘heaps’ is not a sharp one’’. This is, of course, a familiar and serious problem, and Wright has a clever new argument that purports to show it leads to an incoherence in Supervaluationism (p. 89). The argument is as follows. On the semantic view of indeterminacy, for an object to be on the borderline between the heaps and the non-heaps is for there to be ‘‘no semantic mandate’’ either to describe it as a heap or to describe it as a non-heap. So consider something that is on the borderline between the heaps and the things that are on the borderline between the heaps and the non-heaps— something that would exhibit the second-order vagueness of ‘heap’. Then— applying the idea of vagueness as indeterminacy, that is, lack of semantic mandate—we should have no mandate to describe it as a heap, no mandate to describe it as within the first-order borderline—that is, as being on the borderline between the heaps and the non-heaps—and, of course, no mandate to describe it as a non-heap. But something we have no mandate to describe either as a heap or as a non-heap is, on this view, something on the borderline between the heaps and the non-heaps. So it ‘‘fits a certain description which there is no mandate to describe it as fitting’’, and that commits someone who takes vagueness to be semantic indeterminacy to ‘‘a semantic version of Moore’s paradox’’ (p. 89). I think that overstates the case, however. The object we are ‘considering’— the one that allegedly exhibits the second-order vagueness of ‘heap’—is not an object we have actually identified as such, but just one we are supposing exists. If we had identified a particular object as such, then we would indeed have an object we were committed to saying was (i) on the first-order borderline but (ii) something we had no mandate to describe as being on the first-order borderline, and that would be a semantic version of Moore’s paradox. But we have identified no such object. Perhaps the Supervaluationist could deny that we can identify any such object without denying that such objects exist. But the idea behind Wright’s objection can be restated in a way that makes it quite powerful. An object exhibiting the second-order vagueness of ‘heap’ is one that is on the borderline between the heaps and the things on the borderline between the heaps and the non-heaps. But then, by

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124 / Richard G. Heck, Jr. hypothesis, it is not (definitely) a heap, so it would therefore seem as if it must be on the borderline between the heaps and the non-heaps. If so, there can be no second-order vagueness. I think this argument too can be met, but let me not pursue that issue now.25 The more important point is that this argument—like most of the discussion of higher-order vagueness in the extant literature, including my own previous discussions—assumes that the boundary between the heaps and the things on the borderline between the heaps and the non-heaps is not just seemingly vague but really vague, that is, that it is vague in the same sense that the boundary between the heaps and the non-heaps is vague. Wright makes this assumption when he supposes that the semantic theorist is committed to explaining the vagueness of the second-order boundary just as she explained the vagueness of the first-order boundary: as a matter of semantic indeterminacy. That can be denied, and I hereby deny it. Imagine yourself in possession of the Philosophers’ Grail: a solution to the problem of intentionality, a theory of what determines reference that is known to be correct. Suppose further, as I have also suggested, that this theory implies that vague predicates lack determinate classical extensions. Instead, there are some extensions that, e.g., ‘red’ might have which are consistent with our use of it and some its having which is not consistent with our use of it. This distinction is perfectly precise, or at least it can be supposed to be with no threat to any of the remarks I have made to this point and certainly without any threat to Supervaluationism. Of course, not being in possession of the Grail, we have little idea where the boundary lies. But that isn’t vagueness. It’s just ignorance. The temptation, obviously, will be to reply that if one is going to make this move here, one ought to have made it earlier. Why not just say that there is a sharp boundary between the heaps and the non-heaps, but that we just don’t know where it is? But again, from the perspective of the view I’ve been developing, the cases are not analogous: I’m suggesting that the correct theory of what determines reference tells us that there is no sharp boundary between the heaps and the non-heaps (because ‘heap’ has no determinate classical extension) and that it also tells us which extensions are consistent with our use of ‘heap’ and which are not. There is nothing ad hoc about the refusal to go epistemic at one point but not at the other, if that is in fact what the 25 What is needed to meet the argument is greater sensitivity to the logical principles governing higher-order vagueness, for discussion of which, see Heck (1998).

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Semantic Accounts of Vagueness / 125 correct theory says.26 I don’t think there’s even anything very counterintuitive about this combination of views. I argued above that the basic intuition about vague predicates, the one that drives the Sorites, is that they are quasi-tolerant. I argued further that the quasi-tolerance of vague predicates is ultimately responsible for their lacking determinate classical extensions. But what is the analogue of quasi-tolerance for higher-order vagueness? The quasi-tolerance of ‘red’ consists in its being irrational simultaneously to hold that a is red, that a is pairwise indistinguishable from b, and that b is not red. For ‘red’ to be second-order quasi-tolerant would thus seem to be for it to be irrational simultaneously to hold that a is red, that a is pairwise indistinguishable from b, and that b is borderline. Well, would that be irrational? As I said above, it certainly is not irrational just to have no opinion about whether b is red. But that is presumably weaker than holding that b is borderline. So perhaps it is true that it would be irrational to hold that a is red, that a is pairwise indistinguishable from b, and that b is borderline on the sorts of grounds one typically has, namely, observational ones. But I see no reason to suppose that it would be irrational on any possible grounds: if we possessed the Grail, we might have very good grounds indeed. Even if such a combination of views were irrational, on any possible grounds—even if there is real vagueness, rather than ignorance, about where the border between the definitely red and the borderline lies—it is far from obvious that the same reasoning can be used to establish the existence of third-, fourth-, and higher-order vagueness, through all the finite orders. The matter needs further consideration, at least. Third, last, and most interestingly, Wright argues that Supervaluationism is extensionally inadequate, assigning the wrong truth-value to certain sentences involving vague predicates: [T]here are additional concerns about the ability of supervaluational proposals to track our intuitions concerning the extension of ‘true’ among statements involving vague vocabulary: ‘No one can knowledgeably identify a precise boundary between those who are tall and those who are not’ is plausibly a true claim which is not true under any admissible way of making ‘tall’ precise. (p. 88)

Let alone, obviously, under all. There are various things one might say about this, but the best seems to me to be to acknowledge the problem and attempt to relocate it. The problem might seem to have to do with knowledge about 26 Maybe it doesn’t say that. But it is not ad hoc to suppose it does. As I argued above, there is in fact some reason to suppose that is how things are.

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126 / Richard G. Heck, Jr. where the boundary lies. But its real source is in the presence of psychological vocabulary. Consider this statement: (1) Bill believes that John is tall. Is that true if, and only if, Bill believes that John is F, for every acceptable way, F, of making ‘tall’ precise? One can go some way towards defending the idea that it is. Bill will believe all those things, one might say, if, and only if, he believes John is in the intersection of all the possible extensions for ‘tall’. So that seems OK. But now consider: (2) Bill does not believe that John is tall. Is that true if, and only if, Bill does not believe that John is F, for every acceptable way, F, of making ‘tall’ precise? That would make its truth depend upon Bill’s not believing for any acceptable way of making ‘tall’ precise that John is F. It follows that neither (1) nor (2) need be true: if Bill believes that John is F for some but not all acceptable F, then neither will be. So such statements as (1) are, it would seem, vague, not because of any vagueness in ‘believes’ (though such there may be) but simply because of the vagueness in ‘tall’. That may or may not be a comfortable position. But it is not one to which Supervaluationism is committed. (1) should be taken to be true if, and only if, Bill stands in the believing-relation to the proposition that John is tall. And what these examples really show is that Supervaluationism needs an account of which proposition that is. That is to say, in its present state of development, Supervaluationism offers us at best an account of the truth-conditions of simple statements involving vague vocabulary. It has not yet offered us any account of the meanings of those statements. That, it seems to me, is the real challenge facing Supervaluationism. But I know of no reason to suppose it cannot be met, though I certainly do not myself know how (or that) it can be met. Of course, one can simply stipulate that, say, a vague predicate contributes to the proposition expressed by a sentence containing it the set of its possible extensions, where propositions are conceived as structured. But that is not very illuminating. What one wants to know is what is involved in believing the resulting proposition, and, so far as I know, nothing has ever been said about this question. That is, to some extent, a result of the fact that vagueness is too often treated as a linguistic phenomenon. But it is not fundamentally a linguistic phenomenon. My belief that my pen is red suffers from vagueness as much as, and apparently in the much same way as, my

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Semantic Accounts of Vagueness / 127 utterance of ‘My pen is red’ does. Perhaps surprisingly, it is a virtue of semantic accounts of vagueness, such as Supervaluationism, that they extend smoothly from language to thought: my concept of red may fail to have a determinate classical extension for much the same reason that my word ‘red’ has no determinate classical extension. Wright’s view shares this virtue. So does Epistemicism. But not all theories of vagueness do. Contextualism, for example, does not. It is not at all clear that we have a single concept of red exhibiting the same sort of context-dependence exhibited by our word ‘red’.27 On the contrary, the belief I express when I say ‘My pen is red’ is one I can retain through changes in the context that may force me to express this belief differently.

REFERENCES F o d o r , J e r r y , and E r n e s t L e p o r e (1996), ‘What cannot be Evaluated cannot be Evaluated, and it cannot be Supervalued Either’, Journal of Philosophy, 93: 516–35. G r a f f , D e l i a (2000), ‘Shifting Sands: An Interest-Relative Theory of Vagueness’, Philosophical Topics, 28: 45–81. H a r t , W. D. (1991–2), ‘Hat-Tricks and Heaps’, Philosophical Studies (Dublin), 33: 1–24. H e c k , R i c h a r d (1998), ‘That there might be Vague Objects (So far as Concerns Logic)’, The Monist, 81: 277–99. K r i p k e , S a u l (1975), ‘Outline of a Theory of Truth’, Journal of Philosophy, 72: 690–716. L e w i s , D a v i d (1984), ‘Putnam’s Paradox’, Australasian Journal of Philosophy, 62: 221–36. R a f f m a n , D i a n a (1994), ‘Vagueness without Paradox’, Philosophical Review, 103: 41–74. S t a n l e y , J a s o n (forthcoming), ‘Context, Interest-Relativity, and the Sorites’, Analysis. T a p p e n d e n , J a m i e (1993), ‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’, Journal of Philosophy, 90: 551–77. W i l l i a m s o n , T i m o t h y (1996), Vagueness (New York: Routledge). W r i g h t , C r i s p i n (1976), ‘Language-Mastery and the Sorites Paradox’, in Gareth Evans and John McDowell (eds.), Truth and Meaning: Essays in Semantics (Oxford: Oxford University Press). —— (2001), ‘On being in a Quandary: Relativism, Vagueness, Logical Revisionism’, Mind, 110: 45–98. 27 Graff ’s version of the view fares better here, since, on her view, our concept of red is not context-dependent but relational.

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7 Higher-Order Vagueness for Partially Defined Predicates Scott Soames

1. Background In this paper I will talk about a perplexing problem that arises for the theory of vague and partially defined predicates that I sketched in my book Understanding Truth, and which can, I think, be expected to arise for other theories that employ partially defined predicates.1 The problem is that of making sense of so-called higher-order vagueness. This problem is often regarded as the chief difficulty facing analyses which treat vague predicates as partially defined. Although I can’t claim to have solved the problem, I will argue today that it is more tractable than it is often taken to be. I begin by rehearsing the basic framework that I presuppose. The central idea is that vague predicates are context-sensitive and partially defined. To say that a predicate P is partially defined is to say that it is governed by linguistic rules that provide sufficient conditions for it to apply to an object, and sufficient conditions for it to fail to apply, but no conditions that are both individually sufficient and jointly necessary for it to apply, or fail to apply. Because the conditions are mutually exclusive, but not jointly exhaustive, there will be objects not covered by the rules for which there are no possible grounds for accepting either the claim that P applies to them, or the claim that it does not. P is said to be undefined for these objects. Its extension Thanks to Alexis Burgess for comments on an earlier draft. 1 Scott Soames, Understanding Truth (New York: Oxford University Press, 1999).

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Higher-Order Vagueness / 129 is the collection of things to which it applies, and its antiextension is the collection of things to which it doesn’t apply. The system is disquotational in that for any name n, we accept the statement 9‘P’ applies to n; just in case we accept 9 ‘Pn’ is true;, which we accept just in case we accept 9 Pn;.1 When P is undefined for the referent of n, we do not accept 9Pn;, 9‘Pn’ is true;, or 9‘P’ applies to n;, nor do we accept the negations of these claims. We regard it as a mistake to do otherwise, since (i) none of these claims is a necessary consequence of the set of underlying non-linguistic facts together with the rules of the language governing the expressions they contain, and (ii) given the rules governing the predicates, even one who was omniscient about all nonlinguistic facts would have no grounds for accepting them. A distinction is made between the extension of P and its determinate extension, the latter being the set of objects o, such that the claim that P applies to o is a necessary consequence of the rules of the language plus the set of underlying non-linguistic facts. This distinction results from the fact that there are some objects o, such that the claim that o is not in the determinate-extension of P is true, whereas the claim that o is not in the extension of P is to be rejected because the predicate 9is in the extension of ‘P’; is, like the predicate P, undefined for o. Similar remarks apply to the distinction between the antiextension and the determinate-antiextension of P. Corresponding to these distinctions, there is also a distinction between truth and determinate truth.2 In addition to being undefined, vague predicates are also context-sensitive. Given such a predicate P, one begins with a pair of sets. One, the default determinate-extension of P, is the set of things to which the rules of the language, together with the underlying non-linguistic facts, determine that P applies. The other, the default determinate-antiextension of P, is the set of things to which the rules of the language plus the underlying facts determine that P does not apply. For all objects o, P is undefined for o just in case o is in neither of these sets. Since these sets don’t exhaust all cases, speakers have the discretion of adjusting the extension and antiextension so as to include initially undefined cases. Often they do this by explicitly predicating P of an object o, or by explicitly denying such a predication. When a speaker does this, and other conversational participants go along, the extension (or antiextension) of the predicate in the context is adjusted so as to include o, plus all objects that bear a certain relation of similarity to o.

2 See ibid., ch. 6.

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130 / Scott Soames We can illustrate these points with the help of an example. The model is clearest and most intuitive with simple observation predicates like blue. A characteristic feature of these predicates is that we learn them not by being given verbal definitions, but by being given clear and obvious examples of things to which they apply and things to which they don’t. We are told when presented with some reasonable range of objects this is blue and that is not—or, if we are not explicitly told, we note that there are certain objects that everyone we encounter seems ready to call blue and other objects that everyone we encounter seems ready to characterize as not blue. These learning experiences give rise to beliefs about conditions for proper application of the predicate. Think of it this way: People say of a certain object that it is blue. We observe the object, which is perceptually represented to us as being a certain shade of color. Call this shade B1. They say of a different object that it is not blue. We observe that object, which we perceive to be of a different shade—call it NB1. On the basis of experiences like these, we form the belief (which virtually everyone we encounter seems to share) that objects of the first shade—B1— are objects to which blue applies, and objects of the second shade—NB1—are objects to which the predicate does not apply. We may idealize this situation by saying that we first entertain, and then come to accept, the hypothesis that the following pair of rules governs the application of the predicate blue in the language. Blue 1 (a) If an object is B1, then blue applies to the object. (b) If an object is NB1, then blue does not apply to the object. In saying that the agent first entertains, and then comes to accept, the hypothesis that these rules govern the predicate in the language of his community, I don’t mean that the agent formulates these rules in words. Most likely the agent has no words, at least no non-indexical words, that stand for these specific shades. Rather, he comes to accept the propositions expressed by (a) and (b). Other learning experiences with blue lead the agent to accept other pairs of rules, involving different shades, as governing the predicate, as well. At some point in this process, the agent is counted as having successfully learned the meaning of the word, as it is used in his linguistic community. At this point, the agent will have accepted a set of rules Blue 1—Blue n, the (a) versions of which provide sufficient conditions for blue to apply to an object and the (b) versions of which provide suffi-

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Higher-Order Vagueness / 131 cient conditions for blue not to apply to an object. However, although these conditions will be mutually exclusive, the requirement that they be mutually agreed upon and generally adhered to by the overwhelming majority of speakers, no matter what the context, will ensure that they are not jointly exhaustive. Since there will be shades of color, and objects having those shades, about which the rules say nothing, the rules do not provide a set of conditions which are individually sufficient and jointly necessary for blue to apply to an object, or for it not to apply. This illustrates the partiality of the predicate in the language of the speaker’s community. Context-sensitivity results from the fact that speakers have the discretion to apply the predicate blue, or its negation not blue, to objects for which it is undefined by the rules of the language. Often they do this by asserting that some contextually salient object is blue, or that it isn’t. Consider the positive case. If the other conversational participants accept the characterization of the object o as blue, then the extension of the predicate in the context is adjusted to include o and all objects that bear a certain relation of similarity to it. In general, the relation involved in these contextual adjustments is determined by the meaning of the predicate together with the intentions of speakers and hearers in the context. Putting aside various complications, let us suppose that when an agent characterizes as blue an object o for which the predicate is initially undefined, he adopts a contextual standard that counts o, all objects uncontroversially regarded to be bluer than o, as well as all objects that are pairwise indiscriminable from o by ordinary observation in good conditions, as being blue as well. Let Bc be a particular shade that applies to precisely this class of objects. We may then characterize what has happened in the context as a result of the speaker’s predicating blue of o: as a result of doing this, the speaker has adopted a rule governing blue in the context that contains the following condition for positive application of the predicate. If an object is Bc, then blue applies to it. Although not a rule of the language governing the predicate, this rule is one that speakers are free to adopt at their discretion in particular contexts of utterance. We have now illustrated both parts—partiality and context-sensitivity—of the semantic analysis of vague predicates that I will presuppose in what follows. In my opinion these two features of the analysis naturally go

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132 / Scott Soames together, and are mutually reinforcing. Given an analysis that posits one, we can find substantial reasons for adopting the other as well.3

2. Consequences for the Sorites Paradox This brings me to the Sorites paradox. Since semantic theories of vagueness are often judged by the solutions they provide to the paradox, I will say a few words about this. Although all Sorites predicates are vague, not all vague predicates are natural Sorites predicates, with application conditions based on the position of objects in a more or less single and unified underlying continuum. Since the semantic analysis of vagueness is intended to apply to all vague predicates, it should be motivated to a substantial degree by considerations independent of the Sorites. Any light it sheds on the paradox is an extra benefit. In the case of the analysis I advocate, there are two general consequences that the model has for the paradox. First, the fact that vague predicates are partially defined means that the semantic categorization imposed on the world by such a predicate will include more than two categories. There may well be sharp and precise lines dividing the objects in different categories, but typically these lines are not properly characterized as separating objects to which the predicate applies from those to which it does not apply. Second, context-sensitivity tells us that the lines are movable. When one looks closely at the mechanisms by which these lines are adjusted in conversational contexts, one finds that in many cases the mechanism makes it practically impossible to display them; any attempt to display the precise line dividing objects to which the predicate applies (or doesn’t apply) from objects for which it is undefined has the effect of moving the line elsewhere. This constant and elusive movement creates the illusion that there are no sharp lines to be drawn. For example, let us assume that the predicate blue is partially defined, with a default determinate-extension and a default determinate-antiextension, plus a range of objects for which, absent temporary conversational adjustments, it is undefined. Suppose further that the conventions governing the predicate include constraints on how its extension and antiextension may be adjusted within this range. In particular, it is accepted that, typically, one who 3 This is argued in the final section of my ‘Replies’ in the symposium on Understanding Truth in Philosophy and Phenomenological Research, 65/2 (Sept. 2002).

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Higher-Order Vagueness / 133 explicitly characterizes something x as blue on the basis of ordinary perceptual evidence is, all other things considered equal, committed to a contextual standard that counts all objects that look bluer than x, plus objects perceptually indistinguishable in color from x (when paired with x and viewed together) as blue. Finally, suppose that two stimuli can be perceptually indistinguishable in this sense even though they differ slightly in the physical characteristics that cause them to look blue. Given this supposition, we can construct a sequence connecting x1 —which definitely looks and is blue—to xn —which definitely is not, and does not look, blue—in which any two adjacent items in the sequence are (pairwise) perceptually indistinguishable in color. When an agent characterizes an object xi —for which the predicate is initially undefined—as blue, the (determinate) extension of the predicate is adjusted to include xi , all earlier items in the sequence, plus xiþ1 —which is perceptually indistinguishable in color from xi . As a result of this adjustment there is now a sharp line between xiþ1 and xiþ2 separating items to which the predicate determinately applies (in the context) from items for which it remains undefined. However, if one attempts to display this line, by showing the agent xiþ1 and xiþ2 together, and asking him to characterize them, he will, quite properly, resist the invitation to treat them differently. For if he now explicitly endorses his previously implicit commitment to counting xiþ1 as blue, then his assertion that xiþ1 is blue will have the immediate effect of adjusting the contextual standards so as to count xiþ2 as blue as well. By focusing on and making judgements about what had been the line separating objects to which the predicate (determinately) applied from those for which it was undefined, the agent has imperceptibly moved the line, thereby engendering the illusion that there was no sharp and precise line in the first place. In my opinion, this analysis has illuminating implications for different versions of the Sorites paradox. For example, in dynamic versions of the paradox an agent presented with a Sorites sequence about which he is asked to make judgements can easily be pressured into making a series of positive claims 9 x1 is F;, 9x2 is F;, . . . ,9 xi is F; that comes to an end when he refuses to go further, and either assents to a negation 9 xk isn’t F; or refuses to make any judgement at all. At this point, pressure can be generated in the opposite direction, with the result that the agent will dissent from, or withhold judgement on, sentences 9 xj is F; to which he previously assented. The semantic model of vague predicates just sketched indicates how and why this pressure is generated, and explains why such an agent need not be viewed

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134 / Scott Soames as contradicting himself or going back on something he originally asserted. He need not be seen as having done these things because the different judgements he makes change the extension and antiextension of the predicate in such a way that the proposition expressed by 9xj is F ; when he assents to it differs from, and is compatible with, the proposition it expresses when he dissents from or withholds judgement about it. The semantic analysis also points to a useful lesson: although there is something about the meanings of many vague predicates that resists drawing stable boundary lines for applying them, the semantic rules governing such predicates are coherent as they stand, and there is no compelling practical or theoretical need for stable boundaries.4 In addition, the analysis provides the basis of rejecting the major premise MP of a generalized version of the Sorites paradox, while also explaining the deceptive plausibility it enjoys by virtue of its association with the more plausible and defensible premise MP* that arises directly from the rules for adjusting the extension of the vague predicate—in this case blue. (The background for the paradoxical argument includes the claim that there is a sequence S starting with something B that definitely looks and is blue, and ending with something NB that definitely is not and does not look blue. Moreover, for all members si and siþ1 of S, si is perceptually indistinguishable in color from xiþ1 to competent observers in good light under normal conditions.) (MP) For any two colored items x and y that are perceptually indistinguishable in color to competent observers in good light under normal conditions, x and y look to be and are of the same color. Hence for each si , siþ1 is blue, if si is blue. (MP*) For any two colored items x and y that are perceptually indistinguishable in color to competent observers in good light under normal conditions, a person who characterizes blue as applying to x, (in such circumstances) is, all other things being equal, committed to a standard that counts blue as applying to y as well. Hence for each si , siþ1 is counted as blue, if si is explicitly characterized as blue. By allowing us to distinguish the roughly correct MP* from the incorrect Sorites premise MP, the semantic model that I here presuppose is capable of 4 See Soames, Understanding Truth, ch. 7.

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Higher-Order Vagueness / 135 dispelling important and widespread confusions about standard versions of the Sorites paradox.5 Nevertheless, I don’t regard the semantic model as providing a complete solution to the Sorites. As we will see, the model remains vulnerable to certain strengthened, revenge versions of the paradox, when we take higherorder vague predicates into account. It will be evident from the way these versions arise that there is a limit to how far one can go in defusing them by appealing to my semantic model. Even if the model is more or less correct, as I believe it to be, and even if it tells us important things about the Sorites, as I believe it to do, there remains a fundamental mystery brought out by the Sorites that the model does not resolve or illuminate. But that is getting ahead of ourselves. Higher-order vagueness can seem to be a perplexing problem for my semantic account from the very beginning. The central problem arises from treating vague predicates as partially defined. There is a natural line of reasoning arising from this characterization that makes it difficult to see how there could be any higher-order vagueness in the first place. What is the problem?

3. The Prima Facie Problem of Higher-Order Vagueness Let P be a vague predicate that is undefined for objects that are in neither its default determinate-extension nor the default determinate-antiextension. Let 9 is determinately P; apply to an object o just in case o is in the determinateextension of P.6 This predicate applies neither to any object for which P is undefined nor to any object in its determinate-antiextension. Is it partially defined? There is reason to think that it can’t be. In giving the analysis of P, we specified three and only three relevant categories of objects—those to which P determinately applies, those to which it determinately fails to apply, and those for which it is undefined. If these categories are jointly exhaustive, then 9 is determinately P; is totally defined, and so cannot be either vague or partial.

5 See ibid., ch. 7. For objections and a reply, see Timothy Williamson, ‘Soames on Vagueness,’ and my ‘Replies’. 6 The just in case connective is used to form biconditionals that always have truth values when its arguments are undefined, or are otherwise such that we must reject assignments of truth values to them. (This is not a definition.) See Understanding Truth, ch. 6, for further discussion.

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136 / Scott Soames That sounds like a problem. The reason it is a problem is not, in my opinion, that there couldn’t be vague predicates for which the relevant higher-order predicates were totally defined. It seems to me that we could, if we wanted, introduce an artificial predicate P which was both contextsensitive and partially defined, for which the higher-order predicate 9is determinately P; was totally defined. On my view, P would then be vague, even though it would not give rise to higher-order vagueness. We could introduce P with this result, provided (i) that it was fully determinate what the rules governing our new predicate P were, and (ii) that these rules did not contain other vague or partially defined concepts, and so were not themselves vague or partial. Thus, higher-order vagueness is not a sine qua non for vagueness. However, the cases in which higher-order vagueness doesn’t arise are special, and different from what we find with ordinary predicates like bald, red, poor, and young. The problem is that higher-order predicates—9is determinately P;—corresponding to ordinary vague predicates like bald and red also appear to be vague. Not only is there no sharp and precise line dividing the objects to which red or bald apply from the objects to which they don’t, there also seems to be no sharp and precise line dividing (i) the objects to which it is determined, by the rules of the language and the underlying non-linguistic facts, that these predicates apply from (ii) the objects for which this is not determined.7 Thus, it would seem that the predicates is determinately bald and is determinately red are themselves partial. If they are also context-sensitive (which I will here assume), then they too should count as vague. This means that analyses of ordinary predicates like bald and red that treat them as partial and vague must explain how and why the higher-order predicates corresponding to them are also partial, and vague. How might this be done?

4. Proposed Explanation 4.1 The Idea Think again about bald. There are some individuals to which it determinately applies, others to which it determinately does not apply, and still others for which it is indeterminate whether or not it applies, and so is undefined. 7 Here and in what follows, I will presuppose the default settings of ordinary vague predicates when talking about objects to which they apply, don’t apply, or are undefined—

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Higher-Order Vagueness / 137 Although these three categories are mutually exclusive, we should not assume that they are jointly exhaustive; after all, there may be individuals o such that we can find no possible basis for asserting that the predicate x is determinately bald applies to o, that it doesn’t apply to o, or even that it must either apply or not apply to o. The reason for this is that there may be no possible basis to assert either (i) that the claim that bald applies to o is a necessary consequence of all non-linguistic facts about o plus the rules of the language governing bald, or (ii) that the claim that bald applies to o is not a necessary consequence of all non-linguistic facts about o plus the rules of the language governing the predicate bald, or (iii) that one of these claims about necessary consequence must be true. How could this be? We may think of the rules governing bald as being of the sort indicated by the pair Bpos and Bneg , where so and so and such and such in the antecedents of the two conditionals are mutually exclusive, but not jointly exhaustive.8 Rules governing bald (Bpos ) For all o, if o is so and so, then ‘bald’ applies to o (and so o is bald). (Bneg ) For all o, if o is such and such then ‘bald’ doesn’t apply to o (and so o isn’t bald). The rules governing the predicate determinately bald are the rules governing bald plus the rules Dpos and Dneg governing determinately.9 Rules governing determinately (Dpos ) For all o, if o is such that the claim expressed by 9 ‘P’ applies to x; relative to an assignment of o to ‘x’ is a necessary consequence of the unless special contextual standards are explicitly indicated. Thus, when P is such a predicate, 9 is determinately P; will standardly be taken to apply to o just in case o is in the default determinate-extension of P. Of course, when the determinate-extension of P is contextually adjusted, the extension of 9is determinately P; is also adjusted. But such cases will concern us only when explicitly indicated. 8 All instances of these rules are assumed to have definite truth values. 9 In giving the rules for determinately, I have simplified matters to focus on the most dramatic and important case—uses of a vague predicate in contexts in which it carries its default determinate-extension and antiextension. In such a context for an object o to be determinately bald is for the claim that ‘bald’ applies to o to be a necessary consequence of the rules of the language governing ‘bald’ plus the underlying facts. In a context in which speakers have already exercised their discretion by adjusting the extension of ‘bald’, for o to be determinately bald is for the claim that ‘bald’ applies to o to be a necessary consequence of the rules already in force in the context plus the underlying facts.

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138 / Scott Soames set of all non-linguistic facts about o plus the rules of the language governing P, then 9determinately P; applies to o (and the claim expressed by 9x is determinately P; relative to an assignment of o to ‘x’ is true). (Dneg ) For all o, if o is such that the claim expressed by 9‘P’ applies to x; relative to an assignment of o to ‘x’ is not a necessary consequence of the set of all non-linguistic facts about o plus the rules of the language governing P, then 9 determinately P; does not apply to o (and the claim expressed by 9x is not determinately P; relative to an assignment of o to ‘x’ is true). These rules are sensitive to three things: (i) the set of all non-linguistic facts about o, (ii) the rules of the language governing P, and (iii) the relation of necessary consequence. I will take (i) and (ii) to be sets of propositions, and (iii) to be a relation holding between sets of propositions and individual propositions, which, when applied to propositions that are precise and nonvague, is itself precise and well defined. In order to simplify the discussion, I will further assume that the propositions in (i) are all fully defined, precise, and true—no vagueness allowed here. However, no such assumption will be made in the case of (ii). If there is any vagueness about what the rules of the language are, or if there is any vagueness in something which definitely is a rule of the language, then this may affect the results achievable by applying Dpos and Dneg . With this in mind, we return to our question How can the conditions in the antecedents of Dpos and Dneg be seen as anything other than jointly exhaustive when P is the predicate ‘bald’? The answer is that whether or not we can establish or correctly accept the claim that these conditions are jointly exhaustive depends on whether or not we can establish or correctly accept the claim that for each potential rule R of the form , it is determinate whether or not R is a rule of the language governing bald. The crucial point is that we cannot do this. Graph G1 of the baldness continuum illustrates this point. (G1)

bald

?

undefined ? not bald bald xxxxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxx Region 1 Region 2 Region 3 Region 4 Region 5

Region 1 consists of individuals who would be judged to be clearly bald by virtually every competent speaker, provided the speaker were fully apprised

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Higher-Order Vagueness / 139 of the relevant facts about them, for example, by observing them in normal conditions. There is no serious question about these individuals; they are bald. Similarly, there is no serious question about those rule candidates Bpos that classify only members of region 1 as individuals to which bald applies; such candidates are included in the rules of the language governing bald. Region 2 consists of individuals about whom there is moderate uncertainty or disagreement. Most competent speakers would judge these individuals to be bald, and few if any would confidently characterize them as not bald, but a significant number would be uncertain whether they qualify as bald, and would be somewhat reluctant to pronounce judgement on them. This region of individuals gives rise to undefinedness in the predicate 9is a rule of the language governing the predicate ‘bald’;. Rule-candidates Bpos that classify all members of region 1, some members of region 2, and no members of any other region as individuals to which bald applies are rules for which the predicate is a rule of the language governing ‘bald’ is undefined. Region 3 contains paradigmatically borderline cases of baldness. There is great uncertainty and variation among speakers, and across time, regarding whether they classify individuals in this region as bald or not bald; and often they may be reluctant or unwilling to classify these individuals as either. Rule candidates Bpos that classify bald as applying to some individuals in region 3, as well as rule candidates Bneg that classify bald as not applying to these individuals, are not rules of the language governing bald. They may be rules that speakers have the discretion to adopt in particular conversational circumstances; however, they are not rules that are constitutive of the language itself. Regions 4–5 are mirror images of regions 1–2, with not bald replacing bald and Bneg replacing Bpos . On this way of looking at things, the rules of the language governing bald include many different pairs of positive and negative conditionals, even though one pair may subsume many others—i.e. cover every case that the others do, and more. So what are these rules? The rules governing bald include pairs of rules < Bpos , Bneg > in which the antecedent of Bpos applies only to individuals in region 1 and the antecedent of Bneg applies only to individuals in region 5. The rules of the language governing bald do not include any pair in which either the antecedent of Bpos applies to individuals outside regions 1 and 2, or the antecedent of Bneg applies to individuals outside of regions 4 and 5 (though speakers may choose to adopt these rules in particular contexts). Any pair < Bpos , Bneg > in which either (i) the antecedent of Bpos applies to individuals in region 2 (but none in regions

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140 / Scott Soames 3–5), while the antecedent of Bneg applies only to individuals in regions 4 or 5, or (ii) the antecedent of Bneg applies individuals in region 4 (but none in regions 1–3), while the antecedent of Bpos applies only to individuals in regions 1 or 2, is such that we can draw no conclusion regarding whether or not it is a rule of the language governing bald. Speakers can decide to be guided by these rules in particular conversations, but, if they do, there will be individuals o2 in region 2, or o4 in region 4, such that we can establish no correct answer to the question Are speakers’ classifications of o2 as bald, or o4 as not bald, correct because they are consequences of the facts about these individuals plus the rules of the language governing ‘bald’, or are they correct because in making these classifications speakers have exercised their option of adopting extensions of the rules of the language? If this is right, then there is reason to resist the claim that is determinately bald is a totally defined predicate. The basis for the resistance is that for some rules there is simply no saying whether or not they are rules of the language governing bald. Let R be the class of such rules. For certain objects o—namely those in region 2 of G1—the question of whether the claim that bald applies to o is, or is not, a necessary consequence of the rules of the language governing the predicate can be answered only by assuming that certain members of R are rules of the language, or by assuming that they aren’t. Since neither of these assumptions can be established, there is no possible justification for accepting them; thus, we should reject both the claim that these objects are determinately bald and the claim that these objects are not determinately bald, just as we rejected both the claim that they are bald and the claim that they are not bald. So, we reject the claim that determinately bald is totally defined. The reason for this is that the rules governing the predicate 9determinately P; make use of the predicate 9is a rule of the language governing ‘P’; which cannot be seen as total, when P is an ordinary vague predicate like bald. In saying this, I recognize that the picture I have sketched is incomplete, and that there are unfinished tasks that need to be pursued. Certainly, one would like more informative descriptions of different regions in the graph, including (non-circular?) explanations of crucial concepts—like that of being a competent speaker—employed in giving those descriptions. There is also the issue of locating vagueness in the descriptions of these regions, and exploring the sources and consequences of such vagueness. Despite these unresolved matters, I am not convinced that there is any irresolvable mystery here. As far as I can tell, the predicates bald, determinately bald, and is a rule of the language

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Higher-Order Vagueness / 141 governing ‘bald’ do fit the broad-brush picture I have sketched. Let us try to fill out that picture a little further.

4.2 Iterating ‘Determinately’ The points we have made so far are visually represented by the graphs G1 for bald, G2 for determinately bald, and G3 for determinately not bald. (The question marks indicate that it is so far an open question how individuals in the region should be characterized.) (G1)

(G2)

(G3)

bald

?

undefined ? not bald bald xxxxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxx xxxxxxxxxxxxxxx Region 1

Region 2

Region 3

Region 4

Region 5

det. bald

?/ undefined not det. bald det. bald xxxxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxx xxxxxxxxxxxxxxx Region 1 Region 2 Regions 3, 4, and 5

not det. not bald

det. not bald ?/undefined det. not bald xxxxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxx xxxxxxxxxxxxxxx Region 5 Region 4 Regions 1, 2, and 3

We have rejected the claim that determinately bald is totally defined. Should we accept the claim that it is partial (and presumably vague as well)? If so, do partiality and vagueness go even higher? Consider again the graphs and the question marks they contain. We know that the question marks in regions 2 and 4 of G1 do not indicate that bald is undefined for individuals in the regions. But, for all we have said up to now, the question marks in region 2 of G2 and region 4 of G3 might represent individuals for which the predicates is determinately bald and is determinately not bald are, respectively, undefined. Suppose this is so. We can then use (i) and (ii) to establish that the predicate is determinately determinately bald is totally defined. (i) Just as the individuals of whom it can properly be said that they are determinately bald are the same as the individuals of whom it can properly be said that they are bald, so the individuals of whom it can properly be said that they are determinately determinately bald

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142 / Scott Soames are the same as the individuals of whom it can properly be said that they are determinately bald. Thus, the initial section of the graph for determinately determinately bald is the same as the initial section of the graph for determinately bald. (ii) Just as the individuals of whom it can properly be said that they are not determinately bald include all and only those of whom it can properly be said either that they are not bald or that it is undefined whether or not they are bald, so the individuals of whom it can properly be said that they are not determinately determinately bald include all and only those of whom it can properly be said either that they are not determinately bald or that it is undefined whether or not they are determinately bald. So, if we accept the claim that the predicate determinately bald is undefined for every individual in region 2 of G2, and hence that every individual in that region is one of which it can properly be said that it is undefined whether or not that individual is determinately bald, then we get the graph G4 for determinately determinately bald, which is the graph of a totally defined predicate. (G4)

det. det. bald

not det. det. bald

xxxxxxxxxxxx xxxxxxxxxxx xxxxxxxxxxxxxxx xxxxxxxxxx xxxxxxxxxxxxxx Region 1 Regions 2, 3, 4, and 5

That is a surprising result. How is that when we start with bald and add determinately we get a predicate which cannot correctly be characterized as total, whereas when we start with that predicate and iterate determinately, we do get a totally defined predicate? The answer is that we have made a mistake. The crucial assumption, used in (ii), is that every individual o is either determinately bald, not determinately bald, or such that the predicate determinately bald is undefined for o. How do we know that? If at an earlier stage—in moving from bald to determinately bald—we had started with the assumption that every individual o is either bald, not bald, or such that bald is undefined for o, we would have reached the conclusion that determinately bald was totally defined—which we certainly did not. But if we didn’t make that assumption in the previous case, in moving from G1 to G2, why should we make the corresponding assumption in this case, in moving from G2 to G4? The issue concerns the regions in the graphs labeled with question marks. All we know so far is that when, in G1, o is an individual in one of these regions, we reject

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Higher-Order Vagueness / 143 the claim that o is bald, we reject the claim that o is not bald, and we reject the claim that the predicate bald is undefined for o—all for the same reason, we see that it is impossible in principle to justify these claims. This being so, we need to clarify the status of the regions in the other graphs presently marked ‘?/undefined’. The individuals in region 2 are undefined for determinately bald just in case those individuals are not determinately determinately bald, which will be so just in case determinately determinately bald is a totally defined predicate. How do we evaluate the claim that it is such a predicate? The first thing to notice is that the rules governing determinately determinately bald are the same as the rules governing determinately bald; they are the rules governing bald plus the rules Dpos and Dneg governing determinately, given earlier. In the present case, we apply the rules twice, once letting P be the predicate bald, and once letting P be the predicate determinately bald. The reasoning is given in (i) and (ii), and the results are summarized in (iii). (i) Suppose we are given that the claim that bald applies to o is a necessary consequence of the rules governing bald plus the underlying nonlinguistic facts about o. Then, using Dpos we derive that determinately bald applies to o (and hence that o is determinately bald). Since the rules governing determinately bald—namely, the rules < Dpos , Dneg > governing determinately plus the rules governing bald—include the rules used in the foregoing derivation, this means that the claim that determinately bald applies to o is a necessary consequence of the rules governing determinately bald plus the underlying non-linguistic facts about o. But then, using Dpos again, we get the result that determinately determinately bald applies to o, and hence that o is determinately determinately bald. (ii) Suppose we are given that the claim that bald applies to o is not a necessary consequence of the rules governing bald plus the underlying non-linguistic facts about o. Then, using Dneg we derive that determinately bald does not apply to o (and hence that o is not determinately bald). Since the rules governing determinately bald include those used in the foregoing derivation, this means that the claim that determinately bald does not apply to o is a necessary consequence of the rules governing determinately bald plus the underlying non-linguistic facts about o. But then, given the consistency of these rules (with the underlying non-linguistic facts), we conclude that the negation of

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144 / Scott Soames that claim—namely, the claim that determinately bald applies to o is not a consequence of the rules governing determinately bald plus the underlying non-linguistic facts about o. Finally using Dneg again, we get the result that determinately determinately bald does not apply to o, and hence that o is not determinately determinately bald. (iii) When we are not given either (i) that the claim that bald applies to o is a necessary consequence of the rules governing bald plus the underlying non-linguistic facts about o, or (ii) that the claim that bald applies to o is not a necessary consequence of the rules governing bald plus the underlying non-linguistic facts about o, we cannot use the rules < Dpos , Dneg > governing determinately to get any result. We conclude that the rules of the language together with the underlying non-linguistic facts give us the same results for determinately bald and determinately determinately bald. Since it is impossible to justify the claim that either predicate is totally defined, we reject this claim, and for any individual o, we accept the claim that o is (is not) determinately bald just in case we accept the claim that o is (is not) determinately determinately bald. The iteration of determinately does nothing.

4.3 What Not to Say We have rejected the claim that determinately bald and determinately determinately bald are totally defined predicates. Are they partially defined? There is reason not to say this. I have said that partially defined predicates are those that are undefined for some objects, and that totally defined predicates are those that are not undefined for any object, where by undefined I have meant the following: Undefinedness. P is undefined for o just in case the rules of the language governing P together with the underlying non-linguistic facts about o do not determine either that P applies to o or that P does not apply to o— which in turn holds just in case neither the claim that P applies to o nor the claim that P does not apply to o is a necessary consequence of the rules governing P together with the non-linguistic facts about o (i.e. just in case neither 9 x is determinately P; nor 9 x is determinately not P; expresses a truth relative to an assignment of o to ‘x’). Given this, one cannot correctly say that determinately bald is undefined for o. For if one does say this, one must then admit that o is not determinately

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Higher-Order Vagueness / 145 determinately bald. But that conflicts with what we have just found— namely, that just as we must reject, as unjustifiable, the claim that o is determinately bald, without accepting its negation, so we must reject the claim that o is determinately determinately bald, without accepting its negation. So is determinately bald undefined for o or not? Since neither claim can be justified, we have no option but to reject both. A similar result holds for (i) the claim that determinately bald and determinately determinately bald are partially defined predicates, and (ii) the claim they are not. Since our characterization of what it is to be a partially defined predicate requires the predicate to be undefined for some objects, we must reject the claim that these are partially defined, while continuing to reject the claim that they are totally defined. What can we positively assert about these predicates, and about the regions on the graphs for them that are labeled with question marks? As for the predicates, though they cannot correctly be characterized as partial in the original sense, they can be characterized as partial in a weaker and extended sense.

4.4 What we Can Say: Weak Partiality A predicate P is weakly partial just in case there are some objects o such that, no matter how much information one is given about the rules of the language and the underlying non-linguistic facts, one cannot correctly accept either the claim that P applies to o or the claim that P does not apply to o (or the claim that either P applies to o or it doesn’t). Ordinary, partially defined predicates like bald are weakly partial, as are the corresponding higherorder predicates formed by attaching one or more occurrences of determinately to them. The difference between partiality and weak partiality can be illuminated by considering the contrast between regions 2 and 3 on the graph G1 for bald. We consider a pair of claims—the claim that bald applies to o2 , and the claim that bald applies to o3 —where o2 and o3 are individuals in regions 2 and 3, respectively. Neither claim can be accepted because neither can be justified. But the reasons for the lack of justification are different in the two cases. In both cases, in order to justify the claim that the predicate applies to the object one has to establish the premise that there is a rule of the language governing bald which characterizes the predicate as applying to the object. In the case of o3 we can refute this needed premise. In the case of o2 we can neither refute it nor establish it. What the cases have in common is that since the needed premise can’t be established, one in possession of all the facts cannot be

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146 / Scott Soames justified in accepting the claim that the predicate applies to the object, even though in neither case can one be justified in accepting the negation of that claim either. Genuinely partial predicates always include cases like o3 ; predicates which are only weakly partial include cases like o2 , but none like o3 . As for the regions on the graphs labeled with question marks, let us take region 2 of the graph G1 for bald as a representative example. Let o be an individual in this region. We can’t correctly say that bald is undefined for o because there are pairs < Bpos , Bneg > which are candidates for being rules of the language governing bald according to which bald does apply to o—where candidates are rules which we cannot show not to govern the predicate in the language. Since we can’t show this, we cannot correctly say that bald is undefined for o. Of course, we also cannot correctly say that bald applies to o, because there is no pair of rules < Bpos , Bneg > which characterize bald as applying to o that we can show to be rules of the language that do govern the predicate. It is helpful in summarizing this situation to introduce the notion of a predicate P being undefined for an object o relative to a rule R. Relative Undefinedness. P is undefined for o relative to a rule R: iff neither the claim that P applies to o nor the claim that P doesn’t apply to o is a necessary consequence of R plus the set of underlying nonlinguistic facts about o. P is defined for o relative to R just in case P is not undefined for o relative to R. Absolute undefinedness is defined in terms of relative undefinedness. Absolute Undefinedness. P is undefined for o iff (i) for all rules R which are such that we can, in principle, establish that R is a rule of the language governing P, P is undefined for o relative to R, and (ii) there is no rule R which is a candidate for being a rule of the language governing P, relative to which P is defined for o. (A candidate is a rule which we cannot, in principle, show not to be a rule of the language governing the predicate.) In the presence of natural background assumptions—e.g. the assumption that if two rules are such that they should both be accepted as rules of the language, then they don’t give conflicting characterizations of whether a predicate applies to any object—this definition gives the same results as the characterization of undefinedness given earlier. With this in mind, we can characterize each individual o in region 2 of the graphs as follows:

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Higher-Order Vagueness / 147 Individuals o in Region 2 (i) Every rule R which is such that we can establish that R is a rule of the language that governs the predicate bald is such that bald is undefined for o relative to R. (ii) Nevertheless, there remain candidates for being a rule governing bald which characterize bald as applying to o. (iii) For these reasons, we cannot establish, or correctly accept, any of the following claims: that bald applies to o, that bald is undefined for o, that determinately bald applies to o, that determinately bald does not apply to o, that determinately bald is undefined for o (ditto for determinately determinately bald). (iv) It is the case, however, that o is not determinately not bald. (See G3.) We have now distinguished predicates which are merely weakly partial from predicates which are (also) partial in the original sense. Ordinary vague predicates like red and bald are partial without qualification. Higherorder predicates built from them using the determinately operator are weakly partial (and correspondingly weakly vague). Is this the end of the story? Is there anything more to say about higher-order vagueness for partially defined predicates? I suspect there is.

4.5 Superundefinedness, Superdeterminateness, and Sharp Lines Call the individuals in regions 2 and 4 of G1 superundefined, meaning by this that they are individuals of whom we cannot, in principle, establish that bald applies to them, that bald doesn’t apply to them, or that bald is undefined for them, no matter how much information we are given. Since we cannot establish any of these claims, we cannot justifiably accept them. More precisely, we cannot accept them while maintaining that in so doing we are not exercising our discretion by contextually changing the conversational standards governing the predicate bald. Call objects that have this status objects for which the predicate ‘bald’ is superundefined. More generally, when an object o has this status for an arbitrary predicate P, we say that P is superundefined for o. With this definition in place, it seems plausible to suppose that for any predicate P and object o, either (i) P applies to o, (ii) P does not apply to o, (iii) P is undefined for o, or (iv) P is superundefined for o. These categories really do seem to be jointly exhaustive. Supposing that they are, we

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148 / Scott Soames may introduce an operator which attaches to a predicate P to form a totally defined predicate 9 superdeterminately P;. Superdeterminately Predicates. The predicate 9superdeterminately P; applies to an object o just in case it is not the case either that (i) P does not apply to o, or that (ii) P is undefined for o, or that (iii) P is superundefined for o. Would it be a bad result if there really turned out to be such predicates? I don’t see that it would. The point of our discussion of higher-order vagueness for partially defined predicates has not been to avoid drawing sharp lines between all categories of objects to which one might think of applying a vague predicate. The point has been to accommodate what appears to be the genuine sense in which the higher-order predicate 9determinately P; is vague (more precisely, weakly vague) when P is an ordinary vague predicate like bald, or red. We have done that. As for sharp lines, the important questions are If they exist, what do they separate? and How do they arise? The lines I have been concerned with arise from the nature of contextual theories—theories that hold that there is a range of discretion within which speakers may acceptably adjust the contextual standards of what counts as red, bald, and the like. Since there are limits to the range of discretion that speakers have, there must be some items for which the rules of the language allow no discretion. For example, there must be some items for which any characterization conflicting with the characterization that the predicate applies to them is incorrect, no matter what the context. Let us focus on this class of items, and the line separating them from the next class of items. This is the line between regions 1 and 2 in the graph G1 for bald. Individuals in region 1 are such that it is determinate that bald applies to them; hence, speakers have no option to characterize them in any other way. Since we know that individuals in region 2 are not determinately not bald, we know that one can correctly characterize the predicate as applying to them. However, if one does characterize bald as applying to these individuals, we can’t say whether the rules of the language governing the predicate leave one any discretion to do otherwise. We may put this by saying that the individuals in region 2 are such that it is always correct to characterize bald as applying to them, but we cannot say whether the reason this is correct is because the rules of the language determine this characterization, or because in characterizing the predicate as applying to these individuals one is adopting a contextual standard that makes it correct. The line between these individuals

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Higher-Order Vagueness / 149 in region 2 (which may always correctly be said to be bald) and the individuals in region 1 (which may also always be correctly said to be bald) may very well be sharp. However, it is a line which, by its very nature, one would not expect speakers to notice. Hence, it is no embarrassment to the theory that they don’t.

4.6 Implications for the Sorites If I am right, then semantic models of vague predicates as both partial and context-sensitive do not allow one to avoid the conclusion that the meanings of these predicates impose classifications of individuals in their domains of potential application into sharply defined categories. Because of this, strengthened versions of the Sorites paradox can be constructed exploiting this fact. A Strengthened Sorites Argument A man with no hair is superdeterminately bald. For all x, if x is superdeterminately bald, then a man with one more hair is too. So everyone is superdeterminately bald. Because of this one might wonder whether in using the semantic model I have defended we have made any progress in defusing the paradox. In my opinion we have, though we certainly have not fully resolved it. The puzzle that remains is how the linguistic behavior on which the semantics of our language supervenes results in such fine-grained classifications of the objects in the domains of our predicates. This is a problem for all theories of vague terms, and nothing I have said constitutes an answer to it. However, if I am right about the semantics of these terms, then, it seems to me, these fine-grained classifications turn out to be less paradoxical and problematic than they were before. In particular, they do not pose the threat to our notion of linguistic competence that would be posed by a sharp, finegrained bifurcation of the domain into objects to which a predicate definitely applies and those to which it definitely does not apply. The distinction between truth and falsity, or truth and untruth, is very important to speakers; and the norms of language use presuppose that we are able to closely track the truth. One lesson that has sometimes been drawn from traditional versions of the Sorites is that in order to avoid absurdity, we must embrace a semantic theory that distinguishes between those objects of which

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150 / Scott Soames a predicate is true and those of which it is not true in such a precise and finegrained way that we can no longer view ordinary speakers who understand the predicate as competent to make the distinction, or as able to track the truth of statements made using it. That is paradoxical. How can a distinction based on meaning that is so important to language use be opaque to fully competent speakers who understand the meanings of their words? If the meaning of an ordinary predicate imposed a precise, fine-grained classification between objects to which it applied and those to which it did not, wouldn’t fully competent speakers know this, and be able to locate the boundary with a high degree of accuracy? The virtue of the semantic account I have sketched is that it does not provoke these questions.10 The distinction between truth and falsity is important enough to speakers that we expect an account of meaning (which is grasped by competent speakers) to classify statements into those categories in ways that fully competent speakers in possession of all relevant non-linguistic facts are able to approximate. By contrast, the sharp distinction between (i) statements the truth of which are determined by the rules of one’s language together with non-linguistic facts and (ii) statements for which there is no saying whether their truth is so determined or whether their truth results from the exercise of speaker discretion in adjusting the boundaries of context-sensitive predicates is a highly theoretical one, of which speakers need have no clear and precise pretheoretical grasp. Since their shaky grasp of this distinction in no way impugns their competence, it is not paradoxical. Although all sharp, finegrained distinctions imposed by the semantics of vague predicates are theoretically puzzling, they need not be paradoxical. 10 More precisely, it doesn’t provoke these questions for ordinary predicates like red and bald. Although related questions may arise for technical predicates, like superdeterminately bald, the sharp distinctions between things to which these predicates apply and those to which they don’t are defined in terms of the theoretically less troubling distinctions corresponding to the ordinary vague predicates they arise from.

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8 Against Truth-Value Gaps Michael Glanzberg

Many things are neither true nor false: shoes and ships and sealing wax, to name a few. But these things are neither true nor false because they are not the kinds of things that can be either. There are also some things that are apt for being true or false. Preferences vary on exactly what these things are. Common candidates include utterances, interpreted sentences paired with contexts, and propositions. Can there be something that is apt to be true or false, but fails to be either? This is the question of whether there are substantial truth-value gaps. It has been a persistent idea in the philosophy of language that there are substantial truth-value gaps. This view was held, at some moments, by Strawson1 and by Frege.2 More recently, Scott Soames3 has presented an Thanks to JC Beall, Alex Byrne, Jason Decker, Tyler Doggett, Paul Elbourne, Adam Elga, Warren Goldfarb, Delia Graff, Richard Heck, Charles Parsons, Mark Richard, Susanna Siegel, Jason Stanley, Judith Thomson, Carol Voeller, Brian Weatherson, Ralph Wedgwood, Steve Yablo, Cheryl Zoll, and an anonymous referee for valuable comments and discussions. Versions of this material were presented in my seminar at MIT in the fall of 2000, and at the University of Maryland Baltimore County. Parts of this paper also derive from my comments on a paper of Scott Soames at the ‘Liars and Heaps’ conference at the University of Connecticut in the fall of 2002. I am grateful for the help of these audiences, and especially to Professor Soames. 1 The case of Strawson is somewhat complicated. I believe there is good reason to see the view of Strawson (1952) as advocating substantial truth-value gaps. However, there is also good reason to see the view of Strawson (1950) as advocating a different position, not requiring substantial gaps. Strawson (1954) at least appears to refer to both, without mentioning the differences. 2 Under the obvious reading, Frege (1892) endorses substantial truth-value gaps. Some interpreters of Frege, notably Evans (1982), have raised questions about whether this position is consonant with the rest of Frege’s views. As Evans reminds us, Frege did not always hold it, as is shown by the unpublished and contentiously dated ‘17 Key Sentences on Logic’ (Frege, n.d.). 3 Soames (1989), developed further in Soames (1999).

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152 / Michael Glanzberg argument in favor of the view, and has applied it to some issues related to semantic paradoxes and to vagueness. In its own right, the question is deeply involved with some of the very basic issues in the philosophy of language: content, assertion, and truth. In this essay I shall argue that there are no substantial truth-value gaps. There are some phenomena that appear like gaps, but they are importantly different. There are faux gaps, as I shall call them, but no substantial gaps. In particular, attention to the role of context dependence, and the ways in which utterances of meaningful sentences can fail to express propositions in some contexts, provides a rich theoretical basis for explaining away apparently substantial truth-value gaps as merely faux gaps. My strategy will be as follows. I shall first argue that an attractive picture of the relation of truth bearer to truth value leads to a standard that needs to be met by any account of truth values. The standard is easily met by views that do not admit truth-value gaps, but I shall argue we have no idea how a view that admits truth-value gaps could meet it. This shows, from quite general principles, that truth-value gaps are unmotivated, and indeed appear to be in conflict with some wellmotivated principles. I shall then argue that over and above these general considerations, we will not find any specific phenomena that require truthvalue gaps. Any phenomenon whose explanation might appear to require substantial truth-value gaps can be adequately explained in other ways, by appeal to the right kind of faux gap. I shall pay particular attention to the ways in which context dependence can introduce faux gaps. My case against gaps is thus that they are poorly motivated, it is mysterious how they can be compatible with some attractive general principles, and they are useless anyway. In the end, a full exposition of the last part of the argument would require survey of a huge range of linguistic phenomena, and I do not have the space to do more than a small sample of this. However, I shall suggest that the reasons that truth-value gaps appear unnecessary in the cases I shall consider point to some widely applicable strategies for avoiding them. Strictly speaking, this amounts only to partial evidence, but I believe the combination of lack of motivation and some general ways to avoid gaps makes it compelling all the same.

1. A Framework In this section I shall describe some background machinery that will constitute a framework for the rest of the discussion. I shall use this framework to

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Against Truth-Value Gaps / 153 sharpen the question of whether there are substantial truth-value gaps, and to survey some of the motivations for thinking that there are. My arguments that there are not will begin in the following section. Much of our focus will be on assertion. Assertions are speech acts: actions taken by speakers which express contents. For our purposes, we may think of the contents of assertions as given by truth conditions. This is a significant idealization, and simply evades some delicate issues about the nature of content. For our concerns with truth values, however, it is harmless.4 I shall call the truth-conditional contents of assertions propositions, which I shall take to be the primary bearers of truth. I should note, however, that in characterizing propositions by way of assertions, my stance towards the ontology of propositions is quite neutral. I find introducing them convenient, but I invite those who prefer to talk about speech acts themselves, or sentences and contexts, to carry out the needed transformation of my terminology into theirs. We may think of propositions as sets or collections of individual conditions for truth. I shall sometimes resort to common terminology and call these possible worlds. However, this must be understood as mere terminology. Much of what is to follow is, indirectly, an investigation of the nature of truth conditions, so it would not do to build into our framework any strong commitments about their relation to modality. Assertions are actions, but they involve the deploying of interpreted sentences in contexts.5 It will thus be convenient sometimes to make reference to a two-stage semantic theory of the sort articulated by Kaplan (1989). Such a theory assigns to sentences and other linguistic items characters: functions from contexts to intensions. Intensions, in turn, are functions from worlds to extensions. Very roughly (very!), we may think of characters as modeling whatever meanings are assigned by linguistic rules. 4 As an anonymous referee pointed out, this idealization does suppress a great deal about how and when propositions are expressed by utterances, as well as more standard issues of whether belief contents are more fine-grained than truth conditions. For instance, it suppresses questions about how certain aspects of sentence structure, sometimes called ‘information structure’ may affect whether a proposition has been expressed. (I have examined some of these issues in Glanzberg 2002. For additional discussion see von Fintel forthcoming.) It also suppresses questions of the role of certain presuppositions or conventional implicatures. (I have discussed this in Glanzberg forthcoming a. For additional discussion, see Barker 2003.) 5 I will sometimes talk about asserting a sentence 9 s;, and sometimes about asserting a proposition p, letting context and notation for sentences and propositions disambiguate. These may be taken as shorthand for ‘making an assertion using sentence 9s;’ and ‘making an assertion with content p’.

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154 / Michael Glanzberg Questions about substantial truth-value gaps are questions about the truth values of propositions. There have been quite a few different ideas about just what constitutes a truth-value gap. Some have suggested that it is a third truth value that could be assigned to propositions, over and above the values true and false. Others have proposed that it is rather for a proposition not to have a truth value. Others still have suggested it is a status in which any claim about truth value is inappropriate.6 For our purposes, it will be useful to abstract away from these differences. The pro-gaps view, in any of its forms, proposes that a proposition amounts to a division of the set of all worlds into three classes: those in which the proposition is true, those in which it is false, and the rest. We may think of such a proposition as given by two sets of worlds: one of worlds in which it is true, and one of worlds in which it is false, where these sets need not exhaust the set of all worlds. By shifting from describing the pro-gaps position in terms of three values to describing it in terms of two sets, we simply avoid the dispute over how to characterize the third value. It is irrelevant for our concerns whether the two sets are generated by a threevalued function, or a partial function, or in some other way. According to the no-gaps view, a proposition amounts to a division of the set of all possible worlds into two classes: those in which the proposition is true, and those in which it is false ( ¼ not true). We may think of this as a two-valued total function on worlds, or simply as the single set of worlds in which the proposition is true. I shall thus describe the no-gaps view as the oneset view (two values but one set). In contrast, the pro-gaps view is the two-set view (three values, or two values and in some cases no value, etc.; but two sets). The fundamental point of contention between no-gaps and pro-gaps views is whether or not the one-set view is always adequate for modeling the contents of assertions. If it is, then there are no substantial truth-value gaps. If it is not, then the two-set view is generally right. If the two-set view is indeed generally right, then there must be cases where the two sets of a proposition do not exhaust the domain of all worlds. These cases would be substantial truth-value gaps. 6 The last position seems to be that of Soames (1999), though I am not sure if it is really different from the second option. Much of the literature does not discriminate between the first two options (cf. the discussion of the Kleene tables in Kripke 1975). Against certain background assumptions, it is possible to distinguish them. (If, for instance, we think of truth values as assigned by some computational process, then there is a difference between the process converging to a third value, and the process diverging.) However, in many settings the formal differences between the two are trivial (cf. the survey articles by Blamey 1986 and Urquhart 1986).

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Against Truth-Value Gaps / 155 Before launching into my arguments in favor of the one-set view over the two-set view, let me pause to consider some reasons one might favor the twoset view. The traditional justification for gaps proceeds by way of some examples where we are inclined to think a content has been expressed, but are disinclined to call it either true or false. As such examples appear to present us with propositions, but no truth values, they appear to demonstrate the existence of substantial truth-value gaps. I doubt there are any entirely uncontroversial examples. Historically, Strawson offered as examples cases of the failure of the presuppositions of definite descriptions. Suppose there is no lodger living next door to you, and a salesman says to you: (1) The lodger next door has offered me twice that sum.7 In a similar vein, Frege (1892) offered examples of fictional names, as in: (2) Odysseus was set ashore in Ithaca while sound asleep. In both cases, the theoretical issues are quite involved, and final judgments on the examples may well depend on how they are resolved. Russellians will find (1) implausible, and many views of fictional names give truth values to examples like (2). These days many find examples of borderline cases of vagueness more convincing. Let John be borderline bald and consider: (3) John is bald. It is a tempting idea that an utterance of this sentence is in no way defective, and so expresses a proposition; yet because John is a borderline case, the proposition can have no truth value. However, the claim that vagueness leads to truth-value gaps is controversial. Such theorists as Williamson (1994) and Graff (2000) deny that it does. More importantly, vagueness is a difficult and puzzling phenomenon. The Sorites paradox makes it count as a genuinely hard case. Drawing the conclusion that there are truth-value gaps because there is vagueness is thus a theoretically contentious claim, as drawing general conclusions from hard cases always is. It would be far preferable to find independent justification for truth-value gaps, and then apply them to hard cases like vagueness, or the Liar paradox. This might offer some explanation of the hard cases. 7 This example is from Strawson (1954). In that discussion, Strawson also raises the question of whether this example might be treated along the lines I suggest in Section 4.1. The paper appeals to Strawson (1952) for the characterization of truth-value gaps, but it also appeals to Strawson (1950) for examples, presumably meaning the infamous King of France.

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156 / Michael Glanzberg For this reason, some authors have sought to abstract away from some of the difficulties of vagueness cases, with the idea of identifying what it is for a predicate (vague or not) to induce truth-value gaps. A good example is the discussion of partial predicates of Soames (1989, 1999). Partial predicates are predicates which wind up being assigned by semantic rules partial extensions (relative to context and world). A partial extension for a predicate 9P; is a pair of sets of individuals: an extension Pþ of individuals of which the predicate holds, and an anti-extension P
of individuals of which the predicate does not hold. These two sets need not exhaust the domain of all individuals, so we have the analog for predicates of the two-set view of propositions. More fully, 9P; will be assigned, for each context, a function from worlds w to extensions Pwþ and anti-extensions Pw
in those worlds. The result for an atomic sentence of the form 9Pt; is a two-set proposition, consisting of h{w j t 2 Pwþ }, {w j t 2 Pw
}i. So long as there are some worlds where t is not in either Pwþ or Pw
, we have a genuine two-set, partial proposition. Soames (1989, 1999) gives us a scenario where we might see such a predicate as being introduced into the language. We are confronted with a roomful of people who fall into two groups: group (A) consists of people who are quite short, let us say four feet tall and under; while group (B) consists of people who are on the short end of average height, let us say five feet tall and over. We then lay down two stipulations governing a new word ‘smidget’: (4a) Anyone at least as short as someone in group (A) is a smidget. (Anyone of height four feet or less.) (4b) Anyone at least as tall as someone in group (B) is not a smidget. (Anyone of height five feet or more.) According to Soames, these rules introduce a partial predicate, with extension fixed by rule (4a) and anti-extension fixed by rule (4b).8 Now consider Mr Smallman, who is exactly four feet six inches tall. He falls into neither 8 It is unclear if there are any naturally occurring examples of predicates like ‘smidget’, aside from the controversial vagueness ones. Soames’s own position is that the truth predicate is one, but that conclusion is only correct if there are already other partial predicates which may appear in the scope of a truth predicate. As is suggested by Tappenden (1993), some legal concepts might provide examples. I shall discuss these more thoroughly in Section 5. There are some related phenomena. There are many pairs of contrary predicates, often arising from some kind of negative affix, such as ‘happy’–‘unhappy’ and ‘honest’–‘dishonest’ (cf. Horn 1989). Many gradable adjectives come in contrary pairs, such as ‘tall’–‘short’ and ‘happy’–‘sad’ (cf. Bierwisch 1989). But these do not provide single predicates with partial interpretations. There are also well-known examples of sortal predicates. When applied outside of their sorts, these tend to generate what philosophers call category mistakes, and linguists sometimes call selectional

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Against Truth-Value Gaps / 157 group (A) or (B). He hence falls into neither the extension nor the antiextension of ‘smidget’ in the world just described. According to Soames, the sentence ‘Mr Smallman is a smidget’ thus expresses a proposition which is neither true nor false in this world.9 Soames (1999 and Chapter 7 in this volume) goes on to apply the notion of truth-value gap so explained to vagueness cases. I agree with Soames that it does distill the essence of how a predicate could generate truthvalue gaps. In arguing against this model of gaps, I shall thus be offering indirect support for any view of vagueness that does not require gaps, and raising a question for any view that does. Though I have stressed that all the common examples of truth-value gaps are controversial, they do indicate a case in favor of gaps. The examples point to a pattern of usage. They provide a range of cases where we find it decidedly odd to call a claim true or to call it false, while it equally sounds odd to hold that nothing was said by the utterance. Thus, the examples may be taken to suggest that our ordinary usage indicates the presence of gaps. Moreover, it might be argued, this usage is easily explained. Many-valued or partial logic provides a ready theory for gappy propositions. The ‘smidget’ model given by Soames additionally provides an explanation of how (at least some) gaps arise. I do agree that there is something right about the intuitions associated with examples (1–4); something that in the end must be explained. But I do not think they can provide us with instances of substantial truth-value gaps. The version of the no-gaps view I shall defend may be put as follows: The one-set view is always adequate for modeling the contents of assertions. In the next section I shall begin to explain why I hold this position in spite of the violations. But as has been observed since Chomsky (1965), these often appear to exhibit some kind of ungrammaticality. Many current versions of thematic role theory bear this idea out. Grammaticality aside, the claim that these are examples of substantial truth-value gaps does not seem very plausible. 9 Strictly speaking, Soames insists that it is not correct to say that (the proposition expressed by) ‘Mr Smallman is a smidget’ is not true. Indeed, it seems that the only assessment pertaining to truth we can make for an assertion of this sentence by Soames’s lights is the silent stare. I do find it a bit puzzling that we should have an assertion which is manifestly incorrect, but we are not allowed to say so. But more importantly, as I mentioned above, the two-set model is insensitive to this rather delicate point. When we catalog the worlds in which the claim is true and those in which it is false, we get two sets which do not exhaust the domain of all worlds, regardless of exactly what the correct way to describe the worlds in neither set might be. With this in mind, we may continue to speak of worlds in which a proposition is neither true nor false as shorthand for ‘worlds which fail to fall into either set of the two-set model’, without taking issue with the substance of Soames’s position.

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158 / Michael Glanzberg examples and intuitions to the contrary. For now, let me clarify what my position entails. As a piece of mathematics, there is nothing incoherent about the idea of two sets of worlds. Those happy with the existence of worlds and sets should grant this. There is thus a sense in which we may grant that there are such things as gappy propositions. My claim is simply that this is mere mathematics; these things cannot be the contents of assertions. More importantly, I shall grant that there may be some uses for two-set apparatus in describing language. I shall not challenge the idea that there may be reason to describe certain terms and sentences by way of characters that yield two-set-based values. What my thesis requires is that there be no use for the extra set in describing the content of an assertion. In the case of a sentence which might have a two-set-based character, in a given context, either the gap between the two sets must somehow be closed in the process of an assertion expressing a proposition, or the attempt at assertion fails and no proposition is expressed. Genuine assertion is assertion of a one-set proposition.

2. The Dummettian Challenge A moment ago I admitted that there is something right about the pro-gaps intuition associated with examples like (1–4). At the very least the pattern of usage identified is a fact. There are some cases where we are in fact disinclined to apply the predicates ‘true’ or ‘false’. Yet I insist that there are no gaps. Why? I shall argue below that once we take seriously the idea that a proposition is the content of an assertion a speaker might make, the pro-gaps intuition loses its force. Indeed, once we think about matters in these terms, there is a strong countervailing intuition against truth-value gaps. There seems to be no more sense to be made of a proposition that is neither true nor false than there is sense to be made of an assertion that neither succeeds nor does not succeed in saying something correct. There is none. Let us call this the no-gaps intuition. The no-gaps intuition is not by itself enough to constitute an argument against truth-value gaps. But, I suggest, it does provide a challenge to any progaps theory. Spelling out the intuition will uncover a standard which must be met by any account of truth values. It also points to reasons why the no-gaps view can easily meet the standard, while it remains mysterious how a progaps view could. This presents a challenge to any defender of gaps, to explain

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Against Truth-Value Gaps / 159 how they can meet the standard. In this section I shall develop the challenge. In the following ones I shall argue that the challenge cannot be answered, and so there are no truth-value gaps. In extracting a standard from the no-gaps intuition, I am essentially following an observation made long ago by Michael Dummett. He remarked: We cannot in general suppose that we give a proper account of a concept [truth or falsehood] by describing those circumstances in which we do, and those in which we do not, make use of the relevant word, by describing the usage of that word; we must also give an account of the point of the concept, explain what we use the word for. (Dummett 1959: 3.)

Taking this cue from Dummett, I shall spell out the no-gaps intuition in the following argument: (i) Speech acts, including assertions, are moves within a practice of using language which is (partially) rule-governed and is something in which agents can engage intentionally. As such, speech acts have intrinsic purposes (or points, as Dummett puts it). (ii) The intrinsic purpose of assertion is to convey the information that something is the case, i.e. to assert 9 s; is to convey the information that s. (iii) Combining (ii) with the idea that propositional contents encapsulate truth conditions implies a form of the ‘truth-assertion platitude’ for the intrinsic purpose of assertion: the intrinsic purpose of assertion is to assert that truth conditions obtain. (iv) The truth of a claim is thus fundamentally a matter of a purposive act achieving its intrinsic purpose. Conclusion (iv) substantiates the Dummettian idea of looking for the point of the concept of truth. Assessing for truth is a matter of assessing a purposive act for success. We may thus think of truth as itself having a point or purpose, in so far as it is correctly applied exactly when a purposive act achieves its purpose. The same may be said for truth values. Classification for truth is assignment of truth value. Hence, truth values likewise inherit points from the practice of assertion. Any assignment of truth value amounts to an assessment of whether a purposive act has achieved its purpose. This leads to the Dummettian challenge: Any notion of truth value must explain the point of classifying a proposition as having that value in terms of the intrinsic purpose of the speech act of assertion. In particular, any

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160 / Michael Glanzberg account of truth-value gaps must explain, in terms of the intrinsic purpose of assertion, the classification of neither true nor false (or lacking a truth value, etc.). Just such an account is missing from the traditional examples, and I shall argue in Section 3, missing from Soames’s partiality example as well. In the remainder of this section I shall spell out and defend the steps in the argument for the Dummettian challenge.10 In doing so, we will be looking closely at the practice of assertion, and its relation to notions like truth. I hope the reason for doing so is by now clear. I am arguing that though the pro-gaps examples like (1–4) have some intuitive appeal, they do not go deeply enough to explain what a truth-value gap really is. In looking at assertion and truth, I propose to look deeper than they do. When we do, I shall argue, we find that we cannot make sense of substantial truth-value gaps. After this argument has been given, I shall return to the issue of how to make sense of the appeal of the examples.

2.1 Speech Acts and Practices Language use in general, and assertion as a crucial kind of language use, is purposive behavior. In making an assertion, or any other speech act, one is trying to accomplish something. But, furthermore, speech acts have intrinsic purposes, independent of the purposes speakers may have in engaging in them.11 The purpose of assertion is to describe something, or convey information; the purpose of a command is to give an order; the purpose of a verdict is to convict or acquit, etc. In the language of speech act theory, speech acts are eo ipso illocutionary acts.12 They all have in common that their purpose is to say something, which in various ways constitutes doing something. The purposes of a speaker may be at odds with the purpose of the speech act they perform. Typical examples of this come from the manipulation of 10 My discussion draws heavily on that found in Dummett (1959), as well as other Dummettian ideas, especially about the relation of language use to practices like games (cf. Dummett 1976, 1978). 11 I do not want to put too much metaphysical significance on the notion of intrinsicness. I need the distinction between purposes an act may acquire in virtue of an agent bringing that purpose to the act, and the purpose the act has in virtue of being the kind of act it is. I call the latter intrinsic. 12 Echoing Austin (1975: 98). Unlike some readers of Austin, I am not supposing that to call an act illocutionary is to liken it to the highly conventionalized speech acts it is natural to call ‘performative’. The more clearly linguistic acts—assertions, commands, and questions—are illocutionary acts.

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Against Truth-Value Gaps / 161 the purposes of speech acts themselves. A spy may have as his purpose not to convey information, and even to convey disinformation, but may surmise in a particular case that making an assertion, even a true one, could have this effect. The spy, for instance, says ‘The microfilm is hidden in the lacquer box’ with the purpose of causing us to doubt his sincerity, and so take him not to have given reliable information. He does so precisely by performing the action of presenting the information that the microfilm is in the lacquer box, but in a circumstance where he expects us to respond to his doing so in such a way as to cause us to doubt his sincerity. Having an intrinsic purpose is a feature of a wide range of actions, but not all. Natural actions, such as moving one’s hand, seem not to have intrinsic purposes. To have purposes, they need to acquire them from the agents who perform them. But types of actions which are created by practices (in the sense of Rawls 1955) typically do have intrinsic purposes. Clear examples are to be found in games, as both Rawls and Dummett have stressed. Hitting a ball may have no intrinsic purpose, but batting does. Its purpose is to hit the ball in whatever way produces home runs. Shoeless Joe Jackson may have an ulterior purpose of his own when he is at bat, but it is still the case that the purpose of batting is to produce home runs. In the case of games, the purpose of a move in a game is fixed by the game’s more general purpose: to win. Other practices may have more complex purposive structures. Moves in practices, be they moves in games or speech acts, have intrinsic purposes because of two interconnecting features of practices. Practices are (in part) constituted by their rules, and they are things in which agents can engage intentionally. A practice creates certain sorts of moves within it, in part by laying down some rules. Hitting a ball with a stick is not batting but for the rules of baseball. Moving a crowned piece of wood and a crenellated piece of wood is not castling but for the rules of chess. As they are created by practices, these moves may be endowed with intrinsic purposes by the practices that create them. At the same time, if the moves in a practice had no such intrinsic purposes, it would be impossible for an agent to engage in the practice intentionally. Consider a person who understands all the rules of chess, but does not understand that one plays a game generally to win, and that one makes moves to further this end. Such a person would be unable to play chess—to make moves in a chess game—intentionally. They could no doubt move pieces around the board, and they could tell you if those moves were or were not in accord with the rules. But they would have no idea what they were

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162 / Michael Glanzberg doing when they did so. They would have no idea what it is to make a move in chess. No more would they have any idea what could be the result of such a move, nor what could be a reason for making one. They could thus not make chess moves intentionally.13 They might have some private purpose that would give them some reason to want to move pieces around the board, perhaps in accord with some rules, but this would not be to play chess. The intrinsic purposes of moves in practices are features of the moves themselves, provided by the practices that create them. As I mentioned, they must be distinguished from the purposes an agent may have in engaging in the practice. They must also be distinguished from the attendant aim the agent will have of making a move at all. A batter is, of course, trying to bat. He may fail to do so. But if he manages to bat, he does something which itself has a purpose: leading to home runs. If he does not understand that this is what batting is for, he can hardly be described as intentionally batting, or even intentionally playing baseball. In some limited but important respects, we may say that language use is like a game.14 It is a practice in being a (partially) rule-constituted activity in which people can engage intentionally. As with other such practices, an agent could know all the rules of language—syntactic, semantic, and phonological—but, in ignorance of the intrinsic purposes of the moves so constituted, be unable to engage in the practice intentionally. For instance, someone who did not understand that the point of asserting is to convey information about how the world is would be unable to form an intention to assert something, as opposed to its negation, in a given situation. The best 13 It should be clear that I mean ‘intentionally’ and not ‘intentionality’. It might have been natural to talk about agents engaging in a practice rationally, but that is slightly too strong. It is possible for agents to act intentionally but violate the constraints of rationality. 14 Some aspects of the comparison of language to practices like games are highly controversial, so I should stress that my claim is limited in scope. I am only claiming that there is a practice of language use. This does not require that the rules of language be exactly like the rules of a game, or that linguistic competence be like knowing the rules of a game. It is entirely compatible with the conclusions I draw here that up to the level of engaging in the practice of using language, linguistic competence is a matter of the state of a specialized language faculty governed by the principles of universal grammar, as Chomsky has it (e.g. Chomsky 1986). This is not to say that my claim is entirely trivial, however. It is a non-trivial requirement that whatever the rules of a language are like, and whatever it is for a person to have linguistic competence, they must be such as to be able to contribute to an intentional practice at the level of language use. I am suggesting that this is compatible with at least a great deal of the Chomskian picture of language. (For additional discussion, sympathetic to a substantial role for the practice of language use but sensitive to Chomskian concerns, see Higginbotham 1989.)

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Against Truth-Value Gaps / 163 they could do would be to mouth some rule-governed noises. This would be the case even if they understood how the rules associate sentences with propositions in contexts, and how negation changes a proposition to its contradictory. They would still not understand what they would be doing in asserting either, and so could not form the intention to do one rather than the other. They would not know whether the point is to say what holds or what does not, and thus they would not have an adequate understanding of assertion to form the needed intentions.

2.2 Truth and the Intrinsic Purpose of Assertion So far we have established that as moves within a practice, uses of language like assertion have their own intrinsic purposes. We have established step (i) of the argument at the beginning of Section 2. To see how this relates to truth values, we need to move on to the next steps, and look more closely at what the intrinsic purpose of assertion is. This will establish a relation between the act of assertion and its propositional content, which will in turn provide a constraint on what truth values may be like. In step (ii) we observed, seemingly almost trivially, that the intrinsic purpose of assertion is to convey the information that something is the case. Though I shall not attempt anything like a full analysis of assertion, we need to fill this idea in somewhat. Our observation was that the purpose of asserting ‘The cat is on the mat’ is to convey the information that the cat is on the mat. This is not really a triviality at all. Let me stress, we are not talking about an agent’s attendant aim of making an assertion at all—of making a move in the practice. We are talking about the intrinsic purpose of assertion itself— the intrinsic purpose of the very move in the practice. It is easy to imagine a practice of using a language governed by nearly the same rules as ours where assertion has a different intrinsic purpose. A child’s game could easily have the purpose of asserting ‘The cat is on the mat’ by saying that the cat is not on the mat, for instance. The that-clause in our characterization of the purpose of asserting reports a proposition. Thus, we may conclude that the intrinsic purpose of assertion is somehow given in terms of the proposition—the content—expressed by an assertion. How? We have already observed that the purpose is not adequately described as simply determining a collection of truth conditions. This does not tell us what the speaker is to attempt to do with these conditions. Spelling out the idea of conveying information a little further, we must add that the

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164 / Michael Glanzberg purpose is to say that the truth conditions given by the proposition expressed obtain. In conveying the information that the truth conditions of ‘The cat is on the mat’ obtain, the assertion of this sentence describes the world as being some way—namely, one in which the cat is on the mat. The obtaining of truth conditions thus provides the intrinsic purpose of assertion. We have thus established step (iii) of the argument. This is an analysis neither of the notion of assertion nor of truth conditions. But it does point out a fundamental connection between truth and assertion. The intrinsic purpose of assertion is to say that the truth conditions expressed obtain. This purpose is achieved just when the proposition expressed is true. Thus, as it is often put, truth is the point of assertion. I have defended a particularly strong version of this claim. It is not merely that one who asserts should aim to assert what is true. It is, rather, that the intrinsic purpose of assertion is given by the concept of truth conditions obtaining, i.e. of truth. As I mentioned, intrinsic purposes are purposes of moves in practices themselves. Hence, truth is the intrinsic purpose of genuine assertion: a genuine move in the practice of language use. With an eye towards the issues that will occupy us in Sections 4–6, this must always be distinguished from the attendant aim the agent will have, of making an assertion at all, as well as from any further purposes the agent may have in engaging in the practice. Failure to make a genuine assertion is not the failure of a move to achieve its intrinsic purpose; it is the failure for there to have been a move made at all.15 Once we see that truth is the intrinsic purpose of assertion, we may extract the consequence for the nature of truth values given in the conclusion (iv) of the argument above. Assessing a proposition for truth is assessing whether the 15 The observation that there is some such relation between truth and assertion has become quite common. Other examples, over and above the seminal work of Dummett, include Wiggins (1980), Davidson (1990), and Wright (1992). Similar ideas appear in much of the literature on truth and non-cognitivism in ethics, such as Wedgwood (1997). In an interesting paper Williamson (1996) argues that the unique constitutive rule of assertion is that one must assert that s only if one knows that s. He rejects a similar rule requiring only truth on grounds that it does not explain ‘‘the evidential norms for assertion’’ (Williamson 1996: 501). From my perspective, Williamson’s arguments seem to bear on a different issue from the one under investigation here. They appear to address the question of what rules control an agent’s engagement in a practice, rather than the nature of the acts created by the practice itself. Nonetheless, Williamson’s claim that the knowledge rule is the sole constitutive rule of assertion seems to put him at odds with my view. A more thorough discussion of this will have to wait for another occasion. (I do agree with the observation from Evans 1982 about the relation between assertion and the transmission of knowledge, which Williamson extends.)

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Against Truth-Value Gaps / 165 intrinsic purpose of assertion has been achieved. Though we can certainly make sense of truth outside of actual acts of assertion (as with all the better things we might have said), the truth of a proposition still comes down to the matter of whether an assertion of it—a genuine move in the practice— would have been successful in achieving its intrinsic purpose. Assessing for truth in a given world comes down to assessing whether an assertion would have been successful had things been some given way. Assessing for truth is assigning a truth value. Hence, what truth values there are and how they are assigned is fixed by the intrinsic purpose of assertion, and what ways there are of achieving or failing to achieve it. This gives us a standard that any account of truth values must meet: it must show how the truth values can be understood as evaluations of assertions for success or failure in achieving their intrinsic purpose. (This bears out Dummett’s suggestion that we need to ask after the point of the concept of truth.) This standard issues in the Dummettian challenge: to show how any account of truth values meets the standard. The challenge stands for any theory; but in particular, it is a challenge to any theory involving truth value gaps, to show how the classification ‘gap’ amounts to a distinct assessment of an assertion for achieving or failing to achieve its intrinsic purpose.

3. The Prima Facie Case and Partial Predicates In Dummett’s hands, this challenge was already enough to turn aside the kind of argument based on mere appeal to intuitions about examples like (1–3). We may grant that it appears odd to call certain assertions true or false; but this is not enough to meet the challenge, for it fails to tell us anything about what the point of the classification neither true nor false is. It fails to tell us how to understand this classification as an assessment of whether the assertions have achieved their intrinsic purpose. Furthermore, we may combine the Dummettian challenge with the nogaps intuition I mentioned at the beginning of Section 2, to begin to build a case against truth-value gaps. The intuition amounts to the observation that the no-gaps view has no difficulty in meeting the Dummettian challenge. We simply observe that the value true corresponds to the intrinsic purpose of an assertion being achieved, and false corresponds to it failing to have been achieved. It appears evident that these are the only ways an assertion can be assessed for whether it has achieved its intrinsic purpose. It either has or has

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166 / Michael Glanzberg not done so. Hence, true and false exhaust the truth values, as the one-set view has it. (There is, of course, the additional possibility that a speaker could fail to express a proposition—fail to make a genuine move in the practice that has an intrinsic purpose at all. But this cannot introduce substantial truth-value gap, as I shall discuss in Section 4.) More cautiously, we may note that it is at the very least mysterious what other alternatives for assessment there could be, and thus mysterious how things could be the way the two-set view requires. This is what I shall call the prima facie case against truth-value gaps: a one-set, gap-free theory has no trouble meeting the Dummettian challenge, while it is at best mysterious how any two-set, pro-gaps theory could meet the challenge. The prima facie case is just that: prima facie. We need an argument that no two-set theory could meet the Dummettian challenge. Ultimately, as I said, this will come down to investigating a range of examples. In this section I shall consider some general reasons why attempts by two-set theories to answer the Dummettian challenge fail, including Soames’s model of partial predicates. In the next section I shall turn to some more specific strategies for turning aside potential examples.

3.1 Conditional Bets The prima facie case relies on the idea that assessment for whether a purpose has been achieved is a binary matter: it either has or it has not. This is in a way too quick, for there are practices in which assessment for achieving intrinsic purposes is more complex. Dummett (1959, 1978) considered a model for practices in which assessment can have a ternary structure, just as a two-set view would have it. In arguing that assertion cannot be understood along the lines of this model, he thereby bolstered the prima facie case. Dummett’s model is a practice of making conditional bets. Most bets are conditional in the following way. We bet on a match between teams X and Y. You bet $10 that team X will win, while I bet $10 that team Y will win. If I am correct, I get $20—my $10 and yours as well. Conversely, if you are correct, you get $20. But there is a genuinely distinct third possibility. If the match is rained out, or there is a tie, then the bet is null. In this case, neither of us wins, and we each get back our original $10. This is a demonstrably different outcome, as it results in a different pay-out. We may describe this bet as conditional, as it is conditional on the match being played to a victory that one or the other of us wins. But no winner does not mean no bet was made.

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Against Truth-Value Gaps / 167 A genuine move in the practice of betting was made, just one that has a distinct outcome from either of us winning. This sort of move in the practice of betting has an intrinsic purpose with a ternary evaluation structure. Success in achieving the purpose is still just success, but the practice itself makes it the case that there are two different ways to fail to achieve the purpose. A better can fail by losing to someone else, or by the condition for there being a winner not being satisfied. The practice ensures that these are different, and marks it by assigning them different consequences. The condition for winning not being satisfied precludes either party in the bet from winning, so neither achieves their purpose. This outcome might thus be usefully compared with a status of being neither true nor false. The two-set view might take heart from the model of conditional bets. It is a practice which provides a ternary evaluation structure that works much like the two-set view needs for assertion. But as Dummett observed, we have no way to apply this model against the prima facie case. To do so we would have to make sense of a notion of assertion along the lines of the conditional bet model. This would have to be a speech act that is fundamentally like an assertion, is assessed as an ordinary assertion of its consequent if its antecedent condition holds, but is assessed as if no assertion were made if the antecedent fails. It would not be sufficient to apply the model to appeal to cases where a speaker might assert that some appropriate sort of dependence holds between antecedent and consequent, or to the act of forming an intention to assert only upon some antecedent being satisfied. Nor would it be sufficient to appeal to cases where a speaker attempts to assert but fails— fails to make a genuine move in the practice—because some condition does not hold. Rather, what is needed to apply the model is a genuine speech act which is like an assertion upon some condition holding, but otherwise as if no genuine move had been made. I doubt we can really make sense of such a speech act as it is described. The joint requirement that it both be a genuine speech act, but in some cases as if no speech act were made, is nearly incoherent. As Dummett observed, the best we can do is to understand it as an act which would be assessed as incorrect—failing to achieve its intrinsic purpose—just in case the antecedent condition were true but the consequent false. But this is indistinguishable from the ordinary assertion of the material conditional. The practice of language use, unlike that of betting, does not give us the resources to describe the needed kind of conditional assertion: one which would explain the status

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168 / Michael Glanzberg of neither true nor false as an assessment for whether or not an assertion has achieved its intrinsic purpose. Thus, the model of the conditional bet does not really provide any help against the prima facie case. The Dummettian challenge still goes unanswered.16

3.2 Partial Predicates In looking at conditional bets, we have now seen that some general considerations of what a model for truth-value gaps might be like fail to provide a response to the prima facie case. But it remains to be seen why partial predicates, like the one in example (4), cannot provide a model. Unlike a mere appeal to intuition about examples like (1–3), (4) attempts to offer an explanation of why there is a truth-value gap. Even so, I shall argue in this subsection that partial predicates offer no help in meeting the Dummettian challenge.17 A partial predicate theory proposes to explain the source of a two-set model as follows. Take an atomic sentence 9 Pt; where 9P; is a partial predicate. According to the theory, 9Pt; expresses a proposition that is true in a world w if t 2 Pwþ , and 9Pt; expresses a proposition that is false in w if t 2 Pw
. If for some w, t 62 Pwþ and t 62 Pw
, then the proposition expressed by 16 I hasten to note that I am only endorsing Dummett’s argument in so far as it shows that we cannot appeal to a notion of conditional assertion to answer the Dummettian challenge. I do not mean to say anything substantial about conditionals per se. I am not endorsing the material conditional analysis of natural-language conditionals, nor am I maintaining that speech acts involving conditionals must be ordinary assertions (and so have truth conditions). I do not insist that there are no speech acts which could reasonably be called conditional assertions, nor that these could not be usefully applied to the analysis of conditionals. I only claim that this will not give us a way to answer the Dummettian challenge. I am inclined to think that objections to Dummett’s argument, such as those in Edgington (1995), take him to be saying something much more substantial about conditionals, but matters of interpretation aside, I do not believe they bear on my highly limited claim. I am aware of one attempt, by Belnap (1970), to provide a somewhat more formal model of a related notion of conditional assertion. Perhaps due to his insistence on ‘‘depragmatizing’’ the notion, however, he does not really seem to address the matter that is at issue here. The comments by Dunn (1970) and Quine (1970) bear this out. 17 Dummett’s original argument was targeted at the notion of logical presupposition advocated by Strawson (1952) (and probably by Frege as well), as opposed to the notion of what Soames (1989) calls expressive presupposition, advocated by Strawson (1950). Soames rightly points out that the model of partial predicates does not rely on the idea of logical presupposition. Nonetheless, I shall argue that the Dummettian argument applies to it just the same.

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Against Truth-Value Gaps / 169 9 Pt; is neither true nor false in w.18 Soames’s own example attempts to show further how we can get a predicate like 9P; by laying down certain sorts of rules. None of this really helps to answer the Dummettian challenge at all. It is true that the semantics of 9 P; can generate distinct values corresponding to t 2 Pwþ , t 2 Pw
, and neither. But to answer the challenge, we would have to explain how these outcomes correspond to success or lack of success in an assertion achieving its intrinsic purpose. Simply introducing partial rules, and describing three outcomes in terms of them, does nothing to address this. At best, the appeal to partial predicates causes the challenge to reemerge with more details about subsentential structure. Asking about the intrinsic purpose of asserting a sentence like 9Pt; may be put as a question about the intrinsic purpose of predicating P of t. An answer to the challenge, in this special case, would be an explanation of how the intrinsic purpose of predicating P of t generates a ternary evaluation scheme. Simply giving a partial semantics for 9 P; does not even attempt this. Furthermore, we can repeat the prima facie case with similar attention to subsentential structure. The purpose of asserting 9Pt; is to describe t as having property P. This purpose has been achieved if t does bear P, otherwise it has not. This indicates a binary evaluation scheme. We can no more make sense of a third classification, corresponding to t neither bearing nor not bearing P, than we could make sense of an act neither succeeding nor not succeeding in achieving its purpose, just as we observed for truth and assertion. This leaves the point of the assessment t 62 Pwþ and t 62 Pw
just as mysterious as the point of the assessment neither true nor false was according to the prima facie case. The appeal to partial predicates does not advance the cause of truth-value gaps on this front at all. I believe that the mistake Soames makes is to emphasize the rule-governed aspects of language but forget the purposive aspects of language use. As we observed in Section 2.1, just stipulating some rules is not automatically enough to provide an intrinsic purpose for a move in a practice. Asserting a sentence like 9Pt; is to classify t as having P. Thus, if asserting such a sentence is to have an intrinsic purpose, there must likewise be an intrinsic purpose to classifying by P. A genuine predicate must provide an intrinsic purpose for classifying according to it. The ‘smidget’ rules do not by themselves provide such a purpose. 18 As I mentioned in note 9, Soames’s preferred description of the case is that we may not say anything about the truth status of the proposition. As I mentioned there, this difference is not important for the issue at hand, as it still leads to a two-set view.

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170 / Michael Glanzberg Compare the ‘smidget’ case with another. Children on the playground often play games that involve calling each other made-up names. It is familiar dialog: ‘You’re blynj . . . No, I’m not, you are’. Terms like ‘blynj’ can have rules much like the ‘smidget’ rules. As used by one group of children, it can be governed by the rules that you call Alice and Bill blynj, you say that Charlie is not blynj, and the rules are that you don’t say anything about Debbie at all. (We don’t play with her!) Thus, the rules of the playground assign to ‘blynj’ an extension and an anti-extension. Nonetheless, ‘blynj’ remains a nonsense word. In spite of the rules, calling something ‘blynj’ does not amount to trying in any way to describe something. No proposition is expressed in saying ‘Bill is blynj’. Nothing truthevaluable is presented in the act. Indeed, such acts have no illocutionary force at all. That is just not the kind of point the game has. For whatever reason, the children find it fun to say these things. That is the point. This may reflect in some way their opinions about the other children, but they are not reporting these opinions in propositions expressed. They are not trying to describe the other children. They are trying to have fun, and that is not something which is assessed in terms of truth value. We can say that the children ought not to say that Bill is blynj, perhaps because it upsets Bill. But this is not to offer the kind of practice-internal assessment we would need. It is just to say that the children should not play the game they are playing. We might introduce the genuine predicate ‘is called ‘blynj’ in this game’, but that is not in any way a move in the game, and furthermore, is not a partial predicate at all. Like the ‘blynj’ rules, for the ‘smidget’ rules to introduce a genuine predicate, they would need to provide a point, an intrinsic purpose, for classifying according to them. We do not usually need to specify such a purpose explicitly. We do not usually need to specify the intrinsic purpose of playing a game, as it is obvious to anyone who understands games in general once they are told the winning state of the game. Likewise, we do not usually need to specify the purpose of using a predicate, as it is transparently to describe an object as having the property expressed. But if we were to insist on allowing for truly partial predicates, this would not be enough. If confronted with Mr Smallman, we would be left asking what the point of applying the predicate ‘smidget’ to him might be. Following out the prima facie case, it seems that any way of answering this question would classify him as a smidget or not one. If the purpose was to classify people for being particularly short, we would simply check the (4a) rules and conclude Mr Smallman is not a

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Against Truth-Value Gaps / 171 smidget. If the purpose was to classify people for not being particularly tall, we would conversely check the (4b) rules and conclude Mr Smallman is a smidget. If we cannot give either answer, it is mysterious what the purpose of applying the predicate is. I thus conclude that the machinery of partial predicates by itself does not provide a way to overcome the prima facie case against truth-value gaps. To anticipate some of what is to come next, let me stress that this is not to conclude that a predicate like ‘smidget’ is incoherent. It only shows that if it is to be used to make a genuine assertion, that assertion must express a one-set proposition. How this could be done with partial semantic values like that of ‘smidget’ will be examined in Section 4.2. We have seen that neither the conditional bet model nor the partial predicate model provides a way to meet the Dummettian challenge. How a two-set view could meet the challenge remains mysterious. Moreover, our investigation has shown that two of the more likely models for meeting the challenge fail to make any progress at all. This gives us good reason to doubt there is any way the two-set view could meet the challenge. As the one-set view can easily meet the challenge, we have good reason to hold the one-set view over the two-set view.

4. Closing Gaps We now have some solid reasons for rejecting truth-value gaps. We have seen that any two-set theory must meet the Dummettian challenge, and we have seen that upon further investigation, it is mysterious how they could. We have reason to doubt that anything will outweigh the prima facie case. This is not yet conclusive, however. First of all, we have not yet faced directly the intuitions that originally seemed to favor gaps. Instead, we have worked at a very high level of abstraction, and at that level rejected some general models of gaps. This shows that it is advisable for the pro-gaps view to shift its approach. Rather than look for a way to answer the challenge at such a high level of generality, it will be better for the pro-gaps view to return to specific examples of phenomena which can allegedly only be explained by appeal to gaps. The pro-gaps view could then argue that because such phenomena are present, the practice must make room for gaps. The view could also insist that whatever explanation of the point of gaps may be needed will follow from an explanation of the phenomena. The prima facie case is

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172 / Michael Glanzberg wrong, the pro-gaps view would thus argue, because it failed to look at the details. I shall now attempt to argue that this approach will not succeed either. Ultimately, this is a tall order, for I believe that a full elaboration of the argument would require detailed investigation of a huge range of naturallanguage phenomena. The details do matter here, and I shall not be able to look at very many of them. Instead, I shall offer two strategies for showing that phenomena purported to require truth-value gaps do not really require them. I shall apply these to the examples we have considered, especially that of partial predicates. This will, I hope, lend credence to the tentative conclusion that any phenomenon apparently requiring gaps can be explained some other way. It then follows that the prima facie case is correct, and there are no truth-value gaps.

4.1 The Gap-Closing Strategy Some potential examples of truth-value gaps might appear to call for a systematic gap-closing strategy. Fregean examples of reference failure, such as (2) on its Fregean interpretation, may be addressed by such a strategy. These examples are supposed to provide non-referring names, and assertions of sentences containing them which express propositions. (I am not agreeing that we can have such a case, but only considering what might follow if we do.) The Dummettian challenge already turns aside the intuitive judgment that these propositions are neither true nor false. It suggests that in so far as a proposition is expressed in an example like (2), it must be false. However, the pro-gaps view might seek to provide an argument that this is not an acceptable answer. Such an argument might be based on interactions with negation. Suppose we have a non-referring term 9 n;, and an assertion of a sentence 9Fn; which expresses a proposition. It is evident that the proposition expressed cannot be true. But it might be held that the proposition expressed by 9:Fn; cannot be true either. If reference failure leads to nontruth, it might be said to do so in either case. But if our only other option is false, then we have the propositions expressed by 9Fn; and 9:Fn; in the same context both false. This is unacceptable, as it conflicts with the fundamental principle that negation must map false to true. It might be argued that truth-value gaps are precisely what we need to avoid this problem. With truth-value gaps, the problem can be solved by appeal to the three-valued table for so-called internal negation:

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Against Truth-Value Gaps / 173 p

:p

T F *

F T *

(We may think of * as a third value, or no value, or in any other way compatible with the two-set view.) If the proposition expressed by 9 Fn; is assigned *, this table shows that the proposition expressed by 9:Fn; should also be assigned *, without contradicting any principles of negation. Thus, having gaps appears to solve the problem. Dummett (1959) observed that we do not need substantial truth-value gaps to make use of this solution. Hence, even if it is the only solution available (which may already be doubted), it still does not give us a reason to require truth-value gaps. Generally, Dummett provides us a way of making use of many-valued logic without admitting substantial truth-value gaps. We simply think of the additional values as distinguishing different ways of being false (or true, if we like). We keep track, for instance, of false-as-the-result-ofreference-failure as opposed to false-as-the-result-of-not-bearing-a-property by distinguishing F and *. We may use this distinction to explain the ways propositions embed under operations like negation, by providing tables like the above. But in doing so, we may still insist that F and * are both ways of being false. So, in assessing a claim, we count both as failure to achieve its intrinsic purpose. We may keep to the twofold evaluation scheme provided by the prima facie case, and still use all the resources of manyvalued logic.19 This amounts to a kind of gap-closing strategy. Gaps are closed by making potential gap values into subspecies of one of the genuine truth values. If there are real examples of Fregean reference failure, as (2) is supposed to be, this strategy may be applied to them to show they do not lead to substantial truth-value gaps. The strategy might be applied more widely. It could be 19 Dummett (1959, 1976) describes this as the distinction between assertoric content and ingredient sense. At the level of the contents of assertions, we still have a binary evaluation scheme, but at the level of describing the ways propositions embed—provide ingredients to other propositions—we may use a ternary scheme. Formally, this is just the idea that in a many-valued logic the consequence relation is determined by which truth-values are designated. Though we have many values, we have two classes: designated values and non-designated values. In the standard three-valued logics, the value T is designated, while both F and * are undesignated. Truth in the sense of the point of assertion becomes the idea of designated value.

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174 / Michael Glanzberg applied to the ‘smidget’ case, but I shall suggest below that there is a more refined strategy that is appropriate for partial predicates. Regardless, this sort of gap-closing technique gives us a flexible way to account for logical phenomena without undermining the prima facie case. It shows one way to have the benefits of gaps, including many-valued or partial logic, without really needing substantial truth-value gaps themselves.

4.2 Partial Predicates Revisited: The Contextual Gap-Obviating Strategy The gap-closing strategy gives us good reason to doubt we will find examples that require truth-value gaps in global interactions between propositions, like those described by logical operators. But what of examples based on partial interpretations, as in (4) and (allegedly) (3)? Above, I argued that partial predicates on the model of (4) do not provide an answer to the Dummettian challenge. It might be replied, however, that there is no other way to explain the kind of behavior they present than to appeal to truth-value gaps. Thus, it might be insisted, there must be gaps, puzzling or not. In this subsection I shall show that this argument cannot work. I shall introduce a second strategy for avoiding substantial truth-value gaps, based on the effects of context. I shall show how we can use this strategy to incorporate the kinds of rules given in Soames’s example (4) in a way that does not lead to substantial gaps. We can explain the puzzling behavior of this sort of example without them. Let us consider again the kinds of rules that might lead to partial predicates; but this time, paying more attention to the role of context. The natural setting in which we lay down rules like those of (4) is one in which the contextually salient people fall into two groups. In the ‘smidget’ case (4), the contextually salient people fall into group (A) of very short people, and group (B) of people not particularly tall.20 We use this division to introduce the predicate ‘smidget’. In this context, this makes perfect sense. But observe, in this context, dividing the salient people into two groups is precisely and exhaustively to classify them. Those that are not in group (A) are in group (B), and vice versa. Relative to such a context, we may use the term ‘smidget’ with the usual purpose of predication: to classify according to whether 20 Soames explicitly says we have ‘‘assembled two groups of adults in the room’’ (Soames 1999: 164). I assume that we do so in a way that makes only them contextually salient, in the case where the introduction of ‘smidget’ seems to work.

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Against Truth-Value Gaps / 175 objects bear some property or fail to bear it. We need say nothing more specific about the point of using this predicate, as in this context, partiality plays no role. It is tempting to abstract away from the specifics of context to conclude that we have thereby introduced a functionally partial predicate. The predicate ‘smidget’ only works as if total because of highly specific aspects of the context. But the rules of (4) appear to work in other contexts as well. They describe an extension (people of height four feet or less), and an antiextension (people of height five feet or more) even in contexts where these groups do not exhaust the contextually salient individuals. As Soames himself notes (Soames 1999: 165), in introducing ‘smidget’ this way we have provided ourselves with an evidently useful, and seemingly meaningful term. As it is a term whose rules introduce an element of partiality, why have we not thereby introduced a partial predicate? We have indeed introduced a term governed by partial rules. When we write out the character of the term ‘smidget’, we will have to build in some partial extensions for it to follow the rules we have laid down. But I maintain that we may accept such terms without thereby accepting substantial truthvalue gaps. Compare the smidget context with a slightly different one. We gather the people in groups (A) and (B) in a room, but also place Mr Smallman in the middle of the room. In such a context, simply pointing to the two groups (A) and (B) would just raise the question of where to put Mr Smallman. If we refuse to answer the question, we are back to requiring some special story about how we are really classifying the individuals, and how we are really introducing a genuine predicate by doing so. The defense of the prima facie case has given us reason to doubt there can be any such story. Whether or not an attempt to deploy a predicate like ‘smidget’, governed by partial rules, is successful depends on the context in which the attempt is made. The examples we have considered suggest that an attempt may be successful only in a context where somehow, in spite of the partially in the rules, a one-set proposition is expressed. The attempt may be successful only if the context obviates the gap between the extension and the anti-extension by rendering it contextually irrelevant. We thus have a contextual strategy for closing gaps. We may allow partial predicates, or other sorts of terms with rules that introduce some partially. In a given context, an assertion involving such a term will only express a proposition—only be a genuine assertion—if the context obviates the gap by rendering it contextually irrelevant. Only if the context allows for the

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176 / Michael Glanzberg partial rules to produce a one-set proposition is a proposition expressed at all. This is entirely consonant with the prima facie case. Any genuine move in the practice, any assertion, will express a one-set proposition. When we appeal to two-set, partial apparatus, we do so only as a way to extract a one-set proposition in favorable contexts. We need not tell any additional story about how a two-set classification relates to the intrinsic purpose of assertion, for these sets will not figure in the contents of assertions. Genuine evaluation for truth—assessment of assertions—is just as the prima facie case has it. Let me offer some suggestions for how context may have this kind of effect. My goal is to make the contextual strategy plausible, rather than to develop a particular approach in full. To fix some details, let 9P; be a ‘smidget’-like predicate. 9P; has an intension [P]: w ! hPwþ , Pw
i, where for some worlds, Pwþ and Pw
do not exhaust all the individuals in that world. We normally expect a predicate introduced by rules like the ‘smidget’ rules to have a constant character. It does not behave like an indexical, which changes its intension from context to context. So if we like, we can think of the character of 9P; as the constant function from worlds to [P]. Granting that 9 P; has a welldefined character crudely reflects that 9 P; is a well-defined, meaningful term. When we look for ways to obviate the gap between Pwþ and Pw
, it will not suffice to look only at the world in which an assertion takes place. Even if in that world there is nothing contextually relevant in the gap, there will be many similar worlds where this is not the case, and they could produce a gappy proposition. In the ‘smidget’ case, suppose among the people in group (B) is Ms Tall, who is exactly five feet tall. It is no doubt possible for her to be a few inches shorter. Now consider the sentence (5) Ms Tall is a smidget. In a world w where she is six inches shorter, the character of (5) produces a gap. To obviate the gap, we have to rule this situation out, even if it is only a possible variant on the context. To see how context might do so, we may make use of (a somewhat simplified version of) the account of assertion in Stalnaker (1978). I have described the intrinsic purpose of assertion as to convey information, or to say that the world is some way. This requires providing a proposition, which gives the conditions that are in accord with the world being that way. A genuine assertion thus divides the possible world into those that are in accord with what we are saying, and those that are not. But as Stalnaker stresses, such an act takes place against a background of information. To convey information,

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Against Truth-Value Gaps / 177 one need only provide it relative to a background of shared information: a background of pragmatically presupposed propositions. This information fixes a collection of worlds that are in accord with the information presupposed, which Stalnaker calls a context set, and which we may think of as the set of contextually salient worlds. To successfully make an assertion, we need only divide the worlds in the context set. How do we divide these worlds? Very roughly, by relying on the characters of the sentences asserted. For a given sentence 9 s;, the character of 9 s; is a function from worlds to intensions. For any well-formed context, the character of 9 s; will have the same value on all the worlds within the context set. Thus, for a given context set C, the character of 9 s; may be assumed to provide an intension [s]C which maps worlds to truth values. So long as [s]C has the value true or false in each world in C (puts it in one set or the other, on the two-set model), it divides the worlds in C as needed. We may now apply this to a partial predicate like 9 P;. For a given context set C, [P]C is a function from worlds to extensions. So long as for any world w 2 C, Pwþ and Pw
exhaust the individuals in that world, for the purposes of dividing the worlds in C, 9P; behaves just as if it were a total predicate. Thus, for making assertions in C, nothing additional needs to be said about the intrinsic purpose of classification. This idea may be refined in many ways. For instance, contexts generally provide domains of salient individuals, not always taken to be all the individuals in the contextually salient worlds. We may thus think of a context as providing not only a context set C, but also a domain of individuals IC , presumably a subset of the individuals in each world in C. In such a context, genuine assertions must always be about the individuals in IC . Thus, so long as a predicate like 9 P; divides IC totally (for each w 2 C, (Pwþ [ Pw
)\IC ¼ IC ), 9 P; behaves contextually like a total predicate, and we may successfully use it without contradicting the prima facie case. The context I sketched for the introduction of the term ‘smidget’ might be taken to be like this. We are confronted with two groups of individuals, who comprise the set of salient individuals IC of the context. So long as we presuppose enough about their heights, the context set will be restricted to worlds which share the property of the actual world, of the established extension and antiextension of ‘smidget’ exhaustively classifying the salient individuals of those worlds. Using techniques like this, we may do justice both to the idea that the kinds of stipulations like those for ‘smidget’ do seem to lay down perfectly

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178 / Michael Glanzberg good rules, and to the prima facie case’s conclusion that we cannot understand the intrinsic purpose of using a functionally partial predicate. We may have both at the same time. We may have rules that produce partial extensions, and these may well induce correspondingly partial intensions and characters. We may grant that these rules characterize expressions that are meaningful, and may figure into fully grammatical or otherwise wellformed sentences. But we may still insist that successful use of such an expression is restricted to contexts where the gaps introduced by the rules are made irrelevant by the context. We thus understand the intrinsic purpose of using a predicate along the binary lines upon which the prima facie case insists. Once we see how this strategy works, we can see how there may be much more sophisticated contextual gap-obviating strategies as well. We have so far considered only predicates like ‘smidget’, which have their extensions and anti-extensions fixed by context-independent rules. Consider instead the predicate ‘lidget’, which is introduced in a manner similar to that of ‘smidget’. In the same setting as that for ‘smidget’, with the two groups (A) and (B), suppose there is also a contextually salient standard of similarity. Such a standard might be determined by our interest in who could be a basketball player, or a jockey. We then say: (6a) Anyone like a member of group (A) is a lidget. (6b) Anyone like a member of group (B) is not a lidget. As the comparison ‘like’ is going to be context-dependent, we may expect this predicate to have a non-constant character. Now consider again Mr Smallman. A context which fixes a contextually salient standard of similarity by way of our interest in choosing basketball players may well be one which fixes that Mr Smallman is a lidget in all contextually relevant worlds. Likewise, a context which fixes a standard of similarity by way of our interest in choosing jockeys may well fix that Mr Smallman is not a lidget in all contextually relevant worlds.21 The details will depend on a plausible analysis of the term ‘like’, and its relation to context, so I shall not attempt to provide them here. My point is that we need not see context simply as working to rule out individuals as not salient. It may also work with the rules that fix the extensions and anti-extensions of a term to help extend them to cover all the contextually salient individuals. 21 This example is similar to one found in Graff (2000).

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Against Truth-Value Gaps / 179 The result is that if there really are predicates like ‘smidget’ or ‘lidget’, they may be handled by a contextual strategy that obviates the gaps built into their semantics. This has the result that these predicates will carry substantial pragmatic presuppositions, which can lead to failure to express propositions in some contexts. I thus propose to assimilate some of the more troubling cases of gaps to the already familiar phenomenon of pragmatic presupposition failure.22 We now have seen two strategies for accommodating phenomena that may seem to generate truth-value gaps without them. The gap-closing strategy, introducing additional values but counting them as subspecies of one of the genuine values, allows us to use the techniques of many-valued logic without admitting substantial truth-value gaps. This strategy may be used to avoid examples of gaps like (2). The contextual gap-obviating strategy allows us to use rules which introduce partiality into semantic values, but still not admit two-set propositions as the contents of assertions. This strategy may be used to avoid examples of gaps like (4). (Those who find (1) at all convincing may apply either strategy.) This leaves the vagueness examples like (3), which are our next task.

4.3 Vagueness and Determinateness In Section 1 I suggested that vagueness examples like (3) are sometimes taken as the best case for truth-value gaps. In this subsection I shall discuss why the contextual gap-obviating strategy, together with the prima facie case, give us good reason to reject this case. I shall briefly also discuss some related issue about vagueness, determinateness, and gaps. It is commonly recognized that vague predicates are highly contextdependent. This invites explaining away cases like (3) as faux gaps by the 22 There are a number of further issues related to these sorts of presuppositions to be explored. Let me mention a couple. Like most presuppositions, it appears these can be satisfied by conditionalization, as well as by features of the extra-linguistic context. So, for instance, if Mr Almost is just a fraction of an inch above four feet tall, then ‘If everyone were either shorter than four feet or taller than five, then Mr Almost would be a smidget’ appears to express a proposition, presumably a true one. This behavior is common to presuppositions, as has been stressed by van der Sandt (1992). We similarly see ‘If France had a king, the King of France would be a Bourbon’ and ‘If Nader had voted for Bush, he would have regretted it’. I should also mention that I do not think presupposition failure is always sufficient for expression failure. This raises a number of complications, which I do not have space to address here. I have investigated some of them in Glanzberg (2002, forthcoming a).

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180 / Michael Glanzberg kinds of context-dependent mechanisms towards which I gestured in the discussion of (6). A good theory of vagueness along these lines requires much more development, but promising work along these lines has already been done by Raffman (1994, 1996) and van Deemter (1996). A great deal of work on the linguistics of gradable adjectives supports this work, including Klein (1980) and Kennedy (1997). My own preferred context-dependent approach to vagueness is the ‘interest-relative’ theory of Graff (2000). As Graff makes clear, though much of this work does not require bivalence, it equally does not require truth-value gaps.23 In light of this, a methodological remark is in order. To helpfully apply a device like truth-value gaps to highly problematic cases like vagueness or like the Liar paradox, what is really needed is an independent idea of what truthvalue gaps are, which could be applied to explain what is happening with these problematic cases. I applaud such theorists as Soames and Tappenden (1993) for attempting to do so. But the arguments I have given here show why I do not think their attempts, or indeed any account of truth-value gaps, can really do the job. Indeed, I believe the arguments given here show truth-value gaps to be mysterious to the point of being unintelligible. This renders their application to vagueness unhelpful, and theoretically costly. This conclusion is to work in tandem with more specific developments, such as the contextdependence approaches to vagueness I just mentioned (or the epistemicist approach to vagueness of Williamson 1994). The lack of independent justification for truth-value gaps strengthens the case made by such work, while the availability of bivalent approaches to problems like vagueness strengthens the case I have made here.24 There is much more to be said about vagueness, but I think this is sufficient to turn aside the worry that vagueness itself simply establishes the existence of truth-value gaps. Though I shall not say very much more about this matter, there is one further issue that is worth examining, to bolster the case that bivalent approaches to vagueness have the required theoretical strength to succeed. The issue I have in mind is the relation of gaps to determinateness. 23 Graff makes clear that though vague predicates are context-dependent, her theory is not simply a context-dependence theory. I am inclined to agree that there is more to vagueness than context dependence. But all I need to establish here is that the work that might have been done by truth-value gaps in an account of vagueness can be done by context dependence as well. Indeed, it would be enough to show it could be done by some boundary-shifting mechanism, be it context or something else. 24 The same may be said for the non-gap-based approach to the Liar paradox I have offered in Glanzberg (2001, forthcoming b).

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Against Truth-Value Gaps / 181 There is a sense in which we take borderline cases of vagueness, such as (3), to be somehow indeterminate. In contrast, we take clear non-borderline cases to exhibit some kind of determinateness. A person with no more than two hairs on their head would be determinately bald. It is tempting to say that this contrast is what requires truth-value gaps for vague predicates. The status of being indeterminate, it might appear, is the status of a proposition lacking a truth value, i.e. it is precisely the status of substantial truth-value gap. Similarly, it might be supposed that an adequate semantics for operators like determinately requires a gap-based framework.25 At the very least, it stands as a challenge to explain how determinateness operators work without appealing to gaps. I shall briefly indicate how a context-dependence approach might go about accounting for determinateness. Again, I shall be quite sketchy, as I mostly need to establish, in light of the prima facie case against gaps, that we are not left helpless to address the difficult phenomena of vagueness. The basic strategy I shall investigate is to assimilate phenomena of determinateness and indeterminateness to those of situations of ignorance of context. Let me begin with an example. Suppose you join a conversation late, and so are not up to date on all the relevant features of the context. You hear an assertion of (7) It has two axles. You may have no idea what is being asserted, as you have not yet gleaned what the contextually determined referent of ‘it’ is. Ignorance of this aspect of context can make for ignorance of the proposition expressed by this assertion (assuming there is one). But, in some cases, we may work around this. Suppose you know enough about the context to conclude that ‘it’ refers to a car. Then, you have reason to conclude that what was said was true, even if you do not know precisely what proposition it is. How can you do this, without access to the propositional content? One model is this. You have enough information to conclude that any context compatible with what you do know about the context, including that ‘it’ 25 An important example of this is given by the application of partial predicates to vagueness by Soames (1999; Ch. 7 in this volume). It is also typical of supervaluational approaches, such as Fine (1975). A supervaluationist approach to determinateness is also to be found in McGee and McLaughlin (1995), but they also posit a disquotational and bivalent notion of truth, over and above determinate truth (in their terminology, ‘‘definite truth’’, which they identify as a correspondence notion). Determinateness is, as has been much discussed, closely connected with the difficult issues of higher-order vagueness.

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182 / Michael Glanzberg refers to a car, is one in which the proposition asserted is true. We can think of what you know about the context as placing constraints on contexts. Any context meeting these constraints is one in which the proposition expressed in that context is true. (This should have a familiar ring—it is reminiscent of (though not the same as) what supervaluationists say about supertruth.) Now, consider a case where vagueness is more clearly at work. Suppose in this case that you have learned which car is in question. Suppose that someone says: (8) It is expensive. I shall assume, following my brief remarks above, that vague predicates are highly context-dependent.26 In this case, I shall assume that ‘expensive’ is sensitive to, among things, a contextually determined standard of expensiveness. Let us suppose, in this case, you do not know what the standard of expensiveness at work in the context is, again because you are ignorant of some features of the context. But, suppose you know something about the car, say that it is a Rolls-Royce, and that the salient individuals in the context are all philosophy professors. Then, without knowing the precise standard of expensiveness, you may well still be able to conclude that the proposition expressed is true. You may be able to conclude this, assuming that what you know about the context restricts the standards to those in which a RollsRoyce is indeed expensive.27 In this case, we may say that (8) is determinately true. It is true relative to any context within a small distance from the current context (where this distance is set by other features of the context). The idea is that it is determinately true, as its truth is insensitive to the very precise settings that the context may have made. Determinateness is insensitivity at the margins. Ignorance of context does not occur only when speakers join a conversation late, nor is it something that only some participants in a conversation may experience. Cases of common ignorance of context can be built upon demonstrative reference. Suppose I make a demonstration, such that it is evident to all concerned that I succeeded in picking out a unique individual, but no one knows just what that individual is. Then we are all ignorant of a 26 I should stress that there is much in Soames’s account of vagueness with which I agree. Especially, Soames (1999; Ch. 7 in this volume) emphasizes the context dependence of vague predicates. Though I take issue with his reliance on a two-set framework, important parts of his theory can be implemented in a gap-free, one-set environment. 27 This discussion is indebted to that of Graff (2000), which highlights the importance of ignorance of context for questions of vagueness.

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Against Truth-Value Gaps / 183 feature of the context. (Artificial examples may make this vivid. We can construct a case in which we have one man whose identity is obscured by a screen, and pointing to his shadow, say ‘that man’.) One idea for defining determinateness operators, without making use of gaps, is to make use of the ignorance of context phenomena we have just seen. So, we might very tentatively say that ‘determinately s’ expresses a truth just in case in any nearby context, ‘s’ expresses a truth. In the intensional framework I have been using, this might be put as: (9) [Determinately s]C ¼ {wj(8C’ near C)w 2[s]C0 }. As I have sketched it, we should think of a context C’ as being near C if C’ is compatible with what speakers know about context C. Hence, we get nontrivial determinately operators just when we have ignorance of context.28 We can detect the non-trivial aspects of determinateness by looking at speakers’ assertoric behavior. Suppose we have a case in which we are ignorant of the exact contextual settings of some vague term, say again, ‘expensive’. (Say we have screened off some aspects of the comparison class, for instance.) Consider three sentences: (10a) m is expensive. (10b) m is not expensive. (10c) m is not determinately expensive. In some contexts, for some m, we might well find we cannot assert either (10a) or (10b). Even if bivalence holds, we might simply have no grounds for asserting either. But in such a context, we might still be able to assert (10c). If we know there are some nearby contexts in which ‘m is not expensive’ holds, we could assert this.29 28 The determinateness operator I have defined here has one feature that bears additional mention. It is not, in the terminology of Kaplan (1989), an operator on content. This may be seen in (9), where we need to make reference not just to [s]C but also to [s]C0 . The input to ‘determinately’ is not just the proposition expressed by ‘s’ in the context of utterance, but what ‘s’ would express in other contexts as well. Operators like this are sometimes held to be problematic (Kaplan called them ‘monsters’). But I am inclined to see the complexity they introduce as a way to gain some insight into the clearly difficult matter of higher-order vagueness. 29 The theory of Soames (Ch. 7 in this volume) provides for a similar result, in a two-set framework. In that framework, (10a) has gap status, and the determinateness operator thus allows speakers to ‘assert into the gap’. (There is some complication here, as Soames distinguishes extensions and anti-extensions from determinate extensions and anti-extensions. But where they diverge, we cannot report the difference.)

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184 / Michael Glanzberg I should stress that my sketch of a semantics needs to be filled in, and could be filled in by a number of different ways. My own suggestion, in terms of ignorance of context, makes ‘determinately’ in a way epistemic. It is non-trivial just in those cases where we are ignorant of precisely what is expressed by an utterance in a context, but do know enough about the context to determine a range of options. Thus, it is to be understood as an aspect of speakers’ knowledge of speech situations. But formally, the role ignorance of context plays is to place constraints on contexts which define a nearness relation. C’ is near to C if C’ meets some constraints relative to C. Constraints could be developed which are less epistemic in nature, which could yield less epistemic accounts of determinateness than the one I have briefly outlined.30 This hardly settles the question of how to understand determinateness. But it does show that there are resources available that do not rely on two-set views. On my suggestion, determinateness results from stability of truth across small changes of context. Indeterminateness results from lack of such stability. This can be the result of switching truth values across changes of context, even if in each context, the predicate in question exhaustively partitions the domain of (contextually salient) individuals. This does not by itself suffice to reject gaps; the very same suggestion could be made in a twoset framework. But as before, we see how the contextual gap-obviating strategy works in tandem with the prima facie case to argue against truthvalue gaps. We have now seen several strategies for replacing substantial truth-value gaps with faux gaps. Among the faux gaps are subdivisions like * of the genuine truth values, and failures of a well-defined character to express a proposition in some contexts due to partiality. My claim is that we can always explain any Soames achieves this result by introducing pairs of rules for the determinateness operator. The important rule is the negative one, which says that if ‘s’ is not a necessary consequence of the rules of language and the facts, then ‘Not determinately s’ holds. In ‘Mr Smallman’ cases, according to Soames, we cannot assert ‘s’ or ‘:s’, but we can then assert ‘:Ds’ (using ‘D’ for ‘determinately’), and indeed, we have ‘D :Ds’. Soames argues that in fact ‘D’ is partial, making room for higher-order vagueness. This is so, because vague predicates will be defined not just by partial rules, but by families of partial rules which themselves may be vague. Hence, it can be indeterminate whether the negative rule for ‘D’ applies in some cases. On my proposal, we would expect indeterminacy about determinacy just in case speakers are ignorant about features of context which determine the nearness relation. 30 I suspect that the theory of McGee and McLaughlin (1995) could be recast in this form, though they talk about models satisfying constraints, not contexts. Basically, the idea I am entertaining combines elements of their view with those of Graff (2000). I’m not at all sure either would be happy with the combination.

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Against Truth-Value Gaps / 185 apparently gap-requiring phenomenon with a faux gap instead. We have seen how this may done in some important cases, and seen why doing so leaves room for phenomena of vagueness and determinacy. Of course, we have not examined anything like the range of phenomena required to fully substantiate this claim. Yet the strategies we have examined are extremely flexible, so I hope they do enough to make the claim appear plausible. These strategies combine with the prima facie case to make a strong argument against substantial truth-value gaps. From the prima facie case we have general theoretical reasons to see gaps as dubious. From the gap-closing strategies we have some reason to believe we can make do without them anyway. I believe this warrants, with due caution, the conclusion that there are no such things.

5. Pragmatics and Semantics The contextual gap-obviating strategy works on the border between semantics and pragmatics. Mapping this border can be quite complicated, and I shall pause briefly to comment on how the preceding arguments contribute to this task. The Dummett-inspired arguments of Sections 2 and 3 work at the level of what we might call Gricean (or Stalnakerian) pragmatics: pragmatics that concerns itself with what speakers do in using language as an intentional activity. This stands in sharp contrast to the levels of syntax or semantics.31 As I said in note 14, I wish to remain neutral as to whether syntax or semantics describe aspects of a language faculty in Chomsky’s sense, but it does seem that whatever they are, they are not much like the level of Gricean pragmatics. Most importantly for us, concepts like intrinsic purpose should not be expected to apply to them.32 We thus have an interface phenomenon: semantics (whatever its nature) must provide inputs to Gricean pragmatics, and so must be able to interface with pragmatics. 31 We should include here the semantics of indexicals and other context-dependent terms. We thus might describe this as Kaplanian semantics. Of course, there have been a great many developments since the work of Kaplan (1989). 32 Chomsky himself has been notoriously skeptical about including semantic principles pertaining to truth and reference in the language faculty (e.g. Chomsky 1992), though by the same token, Chomsky repeatedly notes that a great deal of what is often called semantics is included. In contrast, Larson and Segal (1995) sketches a view of the language faculty which includes a semantics module, whose rules crucially involve reference.

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186 / Michael Glanzberg Such interfaces can, and do, have complex structures. The gap-closing strategies I surveyed in Section 4 may be understood not only as ways to avoid truth-value gaps, but as illustrations of the structure of the semantics– pragmatics interface. The arguments of Sections 2 and 3 show that at the level of pragmatics, one-set gap-free propositions are required. I granted in Section 4.2 that there is no reason to disallow partial extensions, and corresponding characters, at the level of semantics. Assuming there are such terms, the semantics level may be able to produce two-set proposition-like objects, which are passed on to the level of pragmatics. Pragmatics must then do something with this input which, of necessity, results in a one-set proposition. One option is simply to ignore one of the sets. This amounts to the gapclosing strategy of Section 4.1. But I suggested in Section 4.2 that pragmatics can make much more subtle use of two-set input. The basic idea of appeal to sets of contextually salient worlds is that pragmatics might provide a way to reconstrue what starts out as two-set input as one-set output. The ‘lidget’ example (6) suggests that even more might be done. It suggests that pragmatics might be able to produce one-set propositions that are not exactly the same as either set of a two-set input, but derived from them in some rather complex ways. I believe that there are a great many examples of this sort of interface phenomenon in language.33 They may also be found in our ordinary applications of concepts. Consider law, for instance. It is a safe assumption that certain legal predicates have only extensions and anti-extensions fixed by statute and precedent. We may assume the law works with some partial 33 Let me briefly mention a couple of examples. One more of the semantics–pragmatics relation is provided by words like ‘even’, which have presuppositions and implicatures that do not appear in the propositions they express (e.g. Rooth 1985, 1992). Some recent thought about the syntax–semantics interface also posits information on the syntax side that is not present in semantic interpretation, for instance, the copy theory of movement (Chomsky 1993; Fox 2000). A similar relation is seen between acoustics and phonetics from the well-known phenomenon of categorical perception. This last example is strikingly similar to the sort of gap closing I discussed in Section 4. It is known that listeners are able to make extremely fine discriminations among sounds; yet when presented with speech sounds, they classify them into phonetic categories, and cannot discriminate among them. For instance, when presented with an evenly spaced range of sounds associated with an [æ], speakers perceive it as divided sharply into the three categories [bæ], [dæ], and [gæ], and cannot discriminate among sounds with these categories. Nonetheless, when presented with the same sounds in isolation from the vowel sound, speakers can discriminate between them, but do not perceive them as speech. (Here I follow Kenstowicz 1994. The experiments supporting these claim are attributed to Liberman. Thanks to Cheryl Zoll for pointing out this example.)

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Against Truth-Value Gaps / 187 concepts. But in application of the law, gaps are generally closed. In criminal law, for instance, it is impossible to be neither guilty nor innocent, and there is a general principle that if one fails to be guilty, one is innocent (called the ‘rule of lenity’). Thus, in application to criminal law, a gap-closing strategy like that of Section 4.1 is applied. In constitutional law, or any area where rulings are prospective, the courts may make more refined decisions about how gaps should be closed in some situations. They thus use a strategy more like that of Section 4.2.34 Let me conclude this section with one final remark about the relation between semantics and pragmatics, concerning the notion of denial. The arguments of Sections 2 and 3 do imply that the nature of such a speech act is quite limited. They show that there cannot be a sui generis speech act of denial that is on par with but distinct from assertion. Any such act, I conclude (with Frege 1918), would be simply the assertion of a negation. This does not imply that there is nothing that could reasonably be called a speech act of denial. There is certainly the phenomenon of rejecting an assertion but not asserting the negation of its contents. This can be done in a huge range of ways: the assertion can be rejected as unfounded, as inappropriate, or as misleading (generating a false implicature). As Horn (1989) has stressed, one can even reject the way a sentence is pronounced. I am willing to describe these sorts of cases as acts of denial, but I am doubtful that they really provide instances of a distinct speech act. They are certainly not assertions of negated sentences, but it seems to me that they are rather ordinary assertions about something else. They are about the sentence asserted in a prior speech act, or some aspect of the act of asserting it. I thus follow Horn (1989) in seeing these as instances of metalinguistic negation, rather than as a distinct speech act of denial.

6. Assertion Failure In the preceding sections I took pains to point out that evaluation for truth is evaluation of an assertion—a move in the practice of using language—for success. Falsehood, I claimed, amounts to lack of success. But in light of context-dependence issues, it must be stressed that truth and falsehood amount to success and failure of genuine moves in the practice of using 34 Here I take issue with Tappenden (1993).

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188 / Michael Glanzberg language. These are values that apply to propositions, and as I argued above, they must be understood in terms of genuine moves in the practice that could express these propositions. There is an additional category of no genuine move being made at all: failure to make an assertion. Failure to make an assertion is not the failure of not achieving the intrinsic purpose of the move. It is not making a genuine move at all. This is a kind of failure, of course. As I mentioned in Section 2, it is the failure of an agent to achieve their attendant aim in attempting to engage in the practice. More importantly for our current concerns, it is also failure to express a proposition. The result is thus a faux gap rather than a substantial one. Failure to express a proposition may sometimes look like the expression of a gappy proposition if we do not pay enough attention to context. But in terms of the way attempted moves in the practice are evaluated, they are entirely different. It may be objected that the situations I require to be failure to express a proposition, failure to make a genuine move, cannot really be so. Some of the arguments of Section 4.2, like those related to example (6), suggest that the extent of such failure may not be all that great. The more sophisticated the contextual gap-obviating devices, the less widespread assertion failure will have to be. But I do have to admit there will be some cases of it. In this section I shall defend my view against this objection, and consider more generally the nature of the phenomenon of expression failure. To fix an example, let us suppose we are in a context in which nothing can be done to obviate the question of whether or not Mr Smallman is a smidget. I am committed to holding that in such a context, an assertion of ‘Mr Smallman is a smidget’ fails to express a proposition. Soames (1999) advances an argument against my position. I believe it is mistaken, and seeing what is wrong with it will help bolster the intuitive appeal of my view. First, Soames asks us to compare two sentences.35 Consider again Ms Tall, who is five feet, and so ruled not a smidget by the rules. Compare: (11a) Mr Smallman is a smidget. (11b) Ms Tall is a smidget. Soames proposes that an assertion of (11b) expresses a proposition. If it does, he notes, it must somehow take into account the world in which Ms Tall’s height is just enough different to make her the same height as Mr Smallman. Hence, it does for this circumstance whatever an assertion of (11a) needs to do 35 I have modified Soames’s phrasing to agree with that of example (5).

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Against Truth-Value Gaps / 189 for Mr Smallman as he actually is. Soames concludes that as there is a proposition expressed by (11b), there must be one expressed by (11a) as well. This argument goes wrong with its first premise, that (11b) expresses a proposition. This is an equivocation on contexts. There are certainly contexts in which an utterance of (11b) could express a proposition, and the ones that naturally come to mind are so. If we just add to the original context in which ‘smidget’ was introduced the fact that Ms Tall is in group (B), we get one. However, this does not imply that in a context in which Mr Smallman or other individuals within the gap are contextually salient, an assertion of (11b) expresses a proposition. The arguments I presented above show that in these contexts (supposing we have no other way to contextually obviate the gap), an assertion of (11b) will no more express a proposition than will an assertion of (11a). The best Soames’s argument does is to further articulate why this is so. It might be replied that it is just intuitively obvious that an utterance of (11b) expresses a proposition in either sort of context. Why? I think the basic reason might be that it appears the speaker successfully conveys some information about Ms Tall’s height. It is worth pausing to note how this relates to the idea of expressing or failing to express a proposition. It is fairly obvious that attempted speech acts can have some of their intended effects, their perlocutionary effects (in the terminology of Austin 1975), even if they fail to achieve their illocutionary purposes, or fail to be genuine speech acts at all. You might intend to stop me from walking into the path of an oncoming car by giving me a warning. If your attempt to warn me fails, because fear stops you from making any more than an odd croaking noise, I might still on the basis of your doing so stop in my tracks. It is only a little less easy to see that in some cases, one can convey information without succeeding in making a genuine assertion. You might convey the information that you have a sore throat by making the same odd croaking noise. You might even be trying to assert that you have a sore throat, but be frustrated by the very condition. To take another example, let us suppose, as I think is relatively common, that failed uses of demonstratives lead to expression failure. Suppose I have nothing to drink with me, and someone walks up to me and says: (12) That looks very thirst-quenching. I would love to have some of it. The speaker fails to express a proposition (or two). But they clearly convey the information that they are thirsty. The same information could be

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190 / Michael Glanzberg conveyed by the speaker walking up to an empty water-cooler and looking sad, though. No proposition is needed. With this in mind, let us look again at the failed assertion of (11b). It does convey information about Ms Tall, but that is not adequate grounds to hold that a genuine assertion has been made. Though we learn something about Ms Tall, we are not able to contextually fix what property is being predicated of her. Hence, the attempt at assertion fails. To fail to express a proposition is not to fail to do anything at all. In many cases, it is to do something: something which can have all kinds of consequences and convey all kinds of information. But to express a proposition is not just to do something, or something which makes information available. Practically any act can do that. To express a proposition is to do something in particular. The account offered in Section 2 helps to explain what. It is to perform a purposive act with a particular kind of success condition. Why prefer this act to any old presentation of information? This is too large an issue to deal with thoroughly, but the success conditions for assertion give some form to the idea of describing the world as being some particular way. Merely making some information available is a much looser standard. (12) makes information available, but in a way that fails to describe the world in any one particular way, as it fails to fix the thing of which properties are being predicated. Likewise, suppose I say in a circumstance empty of lime trees: (13) THAT lime tree is yellow. I make available some information: that I somehow think there is a lime tree, that I somehow apply yellowness. If you pay attention to the stress indicated by the capital letters, you will also get the information that I somehow think it surprising for a lime tree to be yellow. But I do all of this without describing any particular thing as being any particular way. I do all of it without expressing a proposition. To exaggerate the situation, suppose I stop speaking part way through, so I only say: (14) THAT lime tree is . . . Certainly no proposition is expressed here. I make much less information available than I did with (13), but the information that I find it surprising for a lime tree to be something is still accessible. Likewise, in certain contexts, (11b) can fail to express a proposition by failing to fix what property is being predicted of Ms Tall, yet still convey information.

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Against Truth-Value Gaps / 191 Finally, let me turn to a related issue: belief. Just as it might have seemed plausible that an utterance of (11b) expresses a proposition in any context, it might be attractive to hold that anyone misinformed about Mr Smallman’s height could believe that Mr Smallman is a smidget. As Soames notes, in so far as belief is an attitude towards a proposition, it appears there must be a proposition to be believed.36 We may reply to this argument the same way we replied to the previous one. Failing to be in a belief state, in the specific sense of failing to be in the belief relation to a proposition, is not to be in no state at all. It is often to be in a state that has some information value. This is a familiar idea in the case of singular thoughts, such as those reported by demonstratives. In reporting on the mental state of someone claiming (13), we cannot quite say they believe that THAT lime tree is yellow; rather we have to say that they seem to believe, or have the illusion of belief, that there is a lime tree present, and it (in contrast to others) is yellow. This is not a genuine belief state, but it is a state such that we can conclude a great deal about the subject. They think (mistakenly) that there is an object present, that it is yellow, and so on.37 Likewise, I must say that the person who says ‘Mr Smallman is a smidget’ does not have a genuine belief. They are in a mental state that provides us with a great deal of information, including information pertaining to the height of Mr Smallman, but that is not enough to conclude that there is a genuine belief of a proposition reported by the claim.

7. Conclusion I have argued that there are no substantial truth-value gaps. The argument works at two levels. At a high level of abstraction, I offered the Dummettian challenge, and then argued that the challenge could not be met by pro-gaps views. At a somewhat less high level of abstraction, I argued that we can explain away apparently gap-requiring examples by some alternative 36 There is some debate over whether or not the objects of attitudes and the contents of assertions should be exactly the same things or not. However, at the coarse-grained level at which we are examining propositions, we may safely ignore this worry. 37 The literature on this idea is quite large, much larger than that for assertion. Perhaps the classic source for the idea of illusion of belief is Evans (1982), developed by McDowell (1982, 1986). Some related ideas are discussed from a more Kaplan-inspired point of view by Braun (1993).

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192 / Michael Glanzberg strategies. I conclude that substantial truth-value gaps are poorly motivated, in conflict with some attractive general principles, and useless. There are none.

REFERENCES A u s t i n , J. L. (1975), How to Do Things with Words, ed. J. O. Urmson and M. Sbisa`, 2nd edn. (Cambridge, Mass.: Harvard University Press). B a r k e r , S. (2003), ‘Truth and Conventional Implicature’, Mind, 112: 1–33. B e l n a p , N. D., Jr. (1970), ‘Conditional Assertion and Restricted Quantification’, Nouˆs, 4: 1–12. B i e r w i s c h , M. (1989), ‘The Semantics of Gradation’, in M. Bierwisch and E. Lang (eds.), Dimensional Adjectives (Berlin: Springer). B l a m e y , S. (1986), ‘Partial Logic’, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. ii (Dordrecht: Kluwer). B r a u n , D. (1993), ‘Empty Names’, Nouˆs, 27: 449–69. C h o m s k y , N. (1965), Aspects of the Theory of Syntax (Cambridge, Mass.: MIT Press). —— (1986), Knowledge of Language (New York: Praeger). —— (1992), ‘Explaining Language Use’, Philosophical Topics, 20: 205–31. —— (1993), ‘A Minimalist Program for Linguistic Theory’, in K. Hale and S. J. Keyser (eds.), The View from Building 20 (Cambridge, Mass.: MIT Press). D a v i d s o n , D. (1990), ‘The Structure and Content of Truth’, Journal of Philosophy, 87: 279–328. D u m m e t t , M. (1959), ‘Truth’, Proceedings of the Aristotelian Society, 59: 141–162; repr. in Dummett (1978). —— (1976), ‘What is a Theory of Meaning? (II)’, in G. Evans and J. McDowell (eds.), Truth and Meaning (Oxford: Oxford University Press); repr., in Dummett (1993). —— (1978), Truth and Other Enigmas (Cambridge, Mass.: Harvard University Press). —— (1993), The Seas of Language (Oxford: Oxford University Press). D u n n , J. M. (1970), ‘Comments on Belnap’, Nouˆs, 4: 13. E d g i n g t o n , D. (1995), ‘On Conditionals’, Mind, 104: 235–323. E v a n s , G. (1982). The Varieties of Reference (Oxford: Oxford University Press). F i n e , K. (1975), ‘Vagueness, Truth and Logic’, Synthese, 30: 265–300. F o x , D. (2000), Economy and Semantic Interpretation (Cambridge, Mass.: MIT Press). F r e g e , G. (1892), ‘U¨ber Sinn und Bedeutung’, Zeitschrift fu¨r Philosophie und philosophische Kritik, 100: 25–50; references are to the translation as ‘On Sense and Meaning’ by M. Black, repr. in Frege (1984).

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Against Truth-Value Gaps / 193 —— (1918), ‘Die Verneinung. Eine logische Untersuchung’, Beitra¨ge zur Philosophie des deutschen Idealismus, 1: 143–57; references are to the translation as ‘Negation’ by P. Geach and R. H. Stoothoff, repr. in Frege (1984). —— (1979), Posthumous Writings, ed. H. Hermes, F. Kambartel, and F. Kaulbach, trans. P. Long and R. White (Chicago: University of Chicago Press). —— (1984), Collected Papers on Mathematics, Logic, and Philosophy, ed. B. McGuinness (Oxford: Blackwell). —— (n.d.), ‘17 Kernsa¨tze zur Logik’, previously unpub. MS, trans. as ‘17 Key Sentences on Logic’, in Frege (1979). G l a n z b e r g , M. (2001), ‘The Liar in Context’, Philosophical Studies, 103: 217–51. —— (2002), ‘Context and Discourse’, Mind and Language, 17: 333–75. —— (forthcoming a), ‘Presuppositions, Truth Values, and Expressing Propositions’, in G. Preyer and G. Peter (eds.), Contextualism in Philosophy: On Epistemology, Language and Truth (Oxford: Oxford University Press). —— (forthcoming b), ‘A Contextual-Hierarchical Approach to Truth and the Liar Paradox’, Journal of Philosophical Logic. G r a f f , D., (2000), ‘Shifting Sands: An Interest-Relative Theory of Vagueness’, Philosophical Topics, 28: 45–81. H i g g i n b o t h a m , J. (1989), ‘Knowledge of Reference’, in A. George (ed.), Reflections on Chomsky (Oxford: Blackwell). H o r n , L. R. (1989), A Natural History of Negation (Chicago: University of Chicago Press). K a p l a n , D. (1989), ‘Demonstratives’, in J. Almog, J. Perry, and H. Wettstein (eds.), Themes from Kaplan (Oxford: Oxford University Press); first publication of a widely circulated MS dated 1977. K e n n e d y , C. (1997), ‘Projecting the Adjective: The Syntax and Semantics of Gradability and Comparison’, Ph.D. diss., University of California, Santa Cruz. K e n s t o w i c z , M. (1994), Phonology in Generative Grammer (Oxford: Blackwell). K l e i n , E. (1980), ‘A Semantics for Positive and Comparative Adjectives’, Linguistics and Philosophy, 4: 1–45. K r i p k e , S. (1975), ‘Outline of a Theory of Truth’, Journal of Philosophy, 72: 690–716. L a r s o n , R., and G. Segal (1995), Knowledge of Meaning (Cambridge, Mass.: MIT Press). M c D o w e l l , J. (1982), ‘Truth-Value Gaps’, in L. J. Cohen, J. Ło´s, H. Pfeiffer, and K. P. Podewski (eds.), Logic, Methodology, and Philosophy of Science VI (Amsterdam: North-Holland). —— (1986), ‘Singular Thought and the Extent of Inner Space’, in P. Pettit and J. McDowell (eds.), Subject, Thought, and Context (Oxford: Oxford University Press). M c G e e , V., and B. McLaughlin (1995), ‘Distinctions without a Difference’, Southern Journal of Philosophy, suppl., 33: 203–51. Q u i n e , W. V. (1970), ‘Comments on Belnap’, Nouˆs, 4: 12. R a f f m a n , D. (1994), ‘Vagueness without Paradox’, Philosophical Review, 103: 41–74.

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194 / Michael Glanzberg R a f f m a n , D. (1996), ‘Vagueness and Context-Relativity’, Philosophical Studies, 81: 175–92. R a w l s , J. (1955), ‘Two Concepts of Rules’, Philosophical Review, 64: 3–32. R o o t h , M. (1985), ‘Association with Focus’, Ph.D. diss., University of Massachusetts at Amherst. —— (1992), ‘A Theory of Focus Interpretation’, Natural Language Semantics, 1: 75–116. S o a m e s , S. (1989), ‘Presupposition’, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. iv (Dordrecht: Kluwer). —— (1999), Understanding Truth (Oxford: Oxford University Press). S t a l n a k e r , R. C. (1978), ‘Assertion’, in P. Cole (ed.), Pragmatics, vol. ix of Syntax and Semantics (New York: Academic Press); repr. in Stalnaker (1999). —— (1999), Context and Content (Oxford: Oxford University Press). S t r a w s o n , P. F. (1950), ‘On Referring’, Mind, 59; repr. in Strawson (1971). —— (1952), Introduction to Logical Theory (London: Methuen). —— (1954), ‘A Reply to Mr. Sellars’, Philosophical Review, 63: 216–31. —— (1971), Logico-Linguistic Papers (London: Methuen). T a p p e n d e n , J. (1993), ‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’, Journal of Philosophy, 90: 551–77. U r q u h a r t , A. (1986), ‘Many-Valued Logic’, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. ii. (Dordrecht: Kluwer). v a n D e e m t e r , K. (1996), ‘The Sorities Fallacy and the Context-Dependence of Vague Predicates’, in M. Kanazawa, C. J. Pin˜o´n, and H. de Swart (eds.), Quantifiers, Deduction, and Context (Stanford, Calif.: CSLI Publications). v o n F i n t e l , K. (forthcoming), ‘Would you Believe It? The King of France is Back!’, in A. Bezuidenhout and M. Reimer (eds.), Descriptions and Beyond (Oxford: Oxford University Press). v a n d e r S a n d t , R. A. (1992), ‘Presupposition Projection as Anaphora Resolution’, Journal of Semantics, 9: 333–77. W e d g w o o d , R. (1997), ‘Non-cognitivism, Truth, and Logic’, Philosophical Studies, 86: 73–91. W i g g i n s , D. (1980), ‘What would be a Substantial Theory of Truth?’, in Z. van Straaten (ed.), Philosophical Subjects: Essays Presented to P. F. Strawson (Oxford: Oxford University Press). W i l l i a m s o n , T. (1994), Vagueness (London: Routledge). —— (1996), ‘Knowing and Asserting’, Philosophical Review, 104: 489–523. W r i g h t , C. (1992), Truth and Objectivity (Cambridge, Mass.: Harvard University Press).

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9 Gap Principles, Penumbral Consequence, and Infinitely Higher-Order Vagueness Delia Graff

Philosophers disagree about whether vagueness requires us to admit truthvalue gaps, about whether there is a gap between the objects of which a given vague predicate is true and those of which it is false on an appropriately constructed Sorites series for the predicate—a series involving small increments of change in a relevant respect between adjacent elements, but a large increment of change in that respect between the endpoints. There appears, however, to be widespread agreement that there is some sense in which vague predicates are gappy which may be expressed neutrally by saying that on any appropriately constructed Sorites series for a given vague predicate there will be a gap between the objects of which the predicate is definitely true and those of which it is definitely false. Taking as primitive the operator ‘it is definitely the case that’, abbreviated as ‘D’, we may stipulate that a predicate F is definitely true (or definitely false) of an object just in case ‘DF(a)’, where a is a name for the object, is true (or false) simpliciter.1 This yields the following conditional formulation of a ‘gap principle’: I am grateful to Alexis Burgess, Michael Fara, Gilbert Harman, and Harold Hodes for helpful discussion of the topics in this paper, and also to audiences at the University of Connecticut, Oxford University, and the University of Rochester where this material was presented. 1 Throughout, single quotes are used for ‘scare’ quotation, direct quotation, name-forming quotation, and also, instead of corner quotes, for ‘quasi-quotation’.

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196 / Delia Graff (DF(x) ^ D:F(y)) ! :R(x, y): Here ‘F’ is to be replaced with a vague predicate, while ‘R’ is to stand for a Sorites relation for that predicate: a relation that can be used to construct a Sorites series for the predicate—such as the relation of being just one millimetre shorter than for the predicate ‘is tall’. Disagreements about the sense in which it is correct to say that vague predicates are gappy can then be recast as disagreements about how to understand the definitely operator. One might give it, for example, a pragmatic construal such as ‘it would not be misleading to assert that’; or an epistemic construal such as ‘it is known that’ or ‘it is knowable that’; or a semantic construal such as ‘it is true that’. Those who think that the gappiness of vague predicates amounts to a gap between truth and falsity will also accept a formulation of the gap principle as an argument schema without a definitely operator. Since they believe that a verifier and a falsifier for a given vague predicate cannot be adjacent elements of a Sorites series for that predicate, they will accept, for example, that it is not possible for it to be true that x is tall and that y isn’t unless it’s true that x is more than one millimetre taller than y; that from x’s being tall, together with y’s not being tall, it follows that x is not just one millimetre taller than y. Schematically: F(x) ^ :F(y) : ;:R(x, y) In what follows I argue that acceptance of gap principles creates problems for those who maintain that the gappiness of vague predicates amounts to a gap between truth and falsity. In particular, I argue first that ‘higher-order’ gap principles, in their conditional formulation, lead to contradiction when the definitely operator is given a semantic construal; and second, that supervaluationists who accept gap principles in their formulation as arguments must concede that the sense in which they endorse classical logic is more qualified than is typically advertised.

1. The Paradox of Higher-Order Gap Principles Some say that a predicate is vague just in case it has, or could have, borderline cases, where what it is to be a borderline case of a vague predicate such as ‘tall’ (for example) may be expressed using the definitely operator: to be borderline

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Gap Principles, Penumbral Consequence / 197 tall is to be neither definitely tall nor definitely not tall.2 I prefer to say that a predicate is vague just in case it has fuzzy boundaries of application on an appropriately constructed Sorites series for the predicate. The metaphor of fuzzy boundaries can be understood in at least two ways, which are themselves in turn metaphorically expressed. On one understanding, a predicate has fuzzy boundaries of application just in case it has a grey area of applicability. On this understanding, to have a fuzzy boundary of application just is to have borderline cases. On another understanding of the metaphor of fuzziness, a predicate has fuzzy boundaries of application along a Sorites series when there is an apparently seamless transition along the series from cases where the predicate applies to cases where it doesn’t apply. On this understanding, fuzziness is associated with susceptibility to paradoxical Sorites reasoning. I regard fuzziness understood as seamless transition as equally well described using Crispin Wright’s (1975) notion of tolerance—the applicability of vague predicates seems to us tolerant of small differences or changes yet not always tolerant of large ones, which is paradoxical since small changes accumulate. I prefer to think of fuzziness qua seamless transition in terms of Mark Sainsbury’s (1991a) notion of boundarylessness: although a vague predicate clearly does apply to some initial segment of its Sorites series but clearly doesn’t apply to some final segment, the transition from the former segment to the latter seems boundaryless; the transition seems to occur along the series while seeming not to occur at any point in the series. Crucially, the two conceptions of fuzziness come apart. There could seem to be a grey area of applicability on a Sorites series without an appearance of boundarylessness, since a grey area may itself appear sharply bounded. Conversely, there could in principle be an appearance of boundarylessness without there seeming to be any grey area at all. The metaphor of fuzziness may obscure this latter point, since it seems that a fuzzy boundary must occupy an extended region, and hence that to lie within this region is to be a borderline case. To that extent, the metaphor of fuzziness is not entirely apt. The positing of a gap between a vague predicate’s definite cases and its definite non-cases often seems, in theories of vagueness, to serve two functions, corresponding to the two conceptions of fuzziness just outlined. Consider the case of the predicate ‘tall (for a man)’. On the one hand, there are men who seem to fall in the grey area—when asked whether they are tall, we feel that some kind of hedged answer would be most 2 Although I do not think that gradable adjectives such as ‘tall’ have predicate-type semantic values, I regard it as harmless in this context to call them predicates.

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198 / Delia Graff appropriate; and moreover it seems that no further information would make us feel more confident in answering simply ‘yes’ or ‘no’. These men falling in the grey area are classified as not definitely tall but not definitely not tall either. On the other hand, when confronted (in our imagination) with a Sorites series for the predicate, we find ourselves unable to locate any point marking the transition from the tall to the not tall. The proposed reason for this inability is that between the men who definitely are tall and those who definitely are not are some men who are neither. Thus a theory of definiteness on this picture is required simultaneously to be both a theory of what it is to fall in the grey area, i.e. cause us to hedge, and a theory of our inability to locate a boundary. But we are no more able to locate a point marking a transition from the definitely tall to the not definitely tall than we are able to locate one marking a transition from the tall to the not tall. The push to accept higher-order vagueness comes from requiring that the very same explanation be given for this inability as was given for the first: just as we cannot locate a boundary dividing the tall from the not tall because there are men who do not fall definitely into either category between the men who fall definitely into one or into the other, we cannot locate a boundary dividing the definitely tall from the not definitely tall because there are men who do not fall definitely into either of these categories between the men who fall definitely into one or into the other. So although there is no obvious phenomenon of second-order hedging—anyone indecisive about whether to be indecisive about whether a given man is tall just is being indecisive about whether the man is tall; the claim that a given man is tallish-ish is not clearly even sensible—a theory of definiteness is now on this picture required to accommodate there being not only first-order borderline cases, men who are neither definitely tall nor definitely not tall, but also ‘second-order borderline cases’, men who are neither definitely definitely tall nor definitely not definitely tall. The problem does not stop there, for we also cannot locate a boundary between the definitely definitely tall and the not definitely definitely tall (assuming, for the sake of argument, that we can understand what it would be to fall into these categories). So in addition to our first-order gap principle (in conditional formulation) for ‘tall’, we also have many higher-order gap principles. (DT(x) ^ D:T(y)) ! :R(x, y) (DDT(x) ^ D:DT(y)) ! :R(x, y)

Gap Principle for ‘T(x)’ Gap Principle for ‘DT(x)’

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Gap Principles, Penumbral Consequence / 199 (DDDT(x) ^ D:DDT(y)) ! :R(x, y) .. .

Gap Principle for ‘DDT(x)’ .. .

I will assume that we have classical equivalences governing the interaction of the conditional with conjunction and negation, so that the following, for example, are equivalent: (j ^ c) ! :y and y ! (c ! :j).3 By ‘equivalent’, I mean must have the same truth value, where for the purposes of this definition, a lack of truth value is stipulated to be a truth value. The equivalences in question are accepted by standard gap theorists, such as supervaluationists and also those who adopt the Kleene strong three-valued truth tables. Now, using prime symbol notation to denote the successor of a term in a Sorites series (following Crispin Wright), we may represent our gap principles schematically as follows, extended to cover any number of iterations of the definitely operator, as indicated by superscripts. Gap Principle for Dn F(x): Equivalent formulation:

DDn F(x) ! :D:Dn F(x’) D:Dn F(x’) ! :DDn F(x).

Some principles governing the logic of the definitely operator are uncontroversial, while others are appropriate only to particular construals of it. Some relatively uncontroversial principles are the following: (T) ‘ Dj ! j (K) ‘ D(j ! c) ! (Dj ! Dc) (RN) if ‘ j then ‘ Dj. The semantic construal of the definitely operator, according to which it may be understood as akin to an operator such as ‘it is true that’, leads in addition to a principle to which epistemicists emphatically object: a rule of D-introduction. In its strong form the rule allows one to infer Dj from some premises if one can infer j from those same premises. (D-intro) if
‘ j then
‘ Dj. It is natural for a truth-value gap theorist to accept D-introduction, and in particular (since j ‘ j) to regard the inference from j to Dj as valid, given her construal of definiteness as truth. For it seems impossible for a sentence S to be true while another sentence—‘it is true that S’—that says (in effect) that 3 I’ll freely use symbols, such as : or D, as names of themselves; and also use concatenation itself, rather than the concatenation symbol _ , to stand for the concatenation function so that, for example, Dj ¼ ‘D’_ j (using name-forming quotes) ¼ ‘Dj’ (using quasi-quotes) for any formula j.

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200 / Delia Graff it’s true is not true. Standard supervaluationist semantics and conceptions of validity such as that developed by Kit Fine (1975) and endorsed more recently by Rosanna Keefe (2000)4 do indeed yield that D-introduction is validitypreserving.5 Crispin Wright (1987, 1992) has argued that acceptance of a secondorder gap principle leads to paradox. Specifically, he has argued that given D-introduction one can deduce from a second-order gap principle— for ‘definitely tall’—the following Sorites sentence: for all x, if the Soritessuccessor of x definitely isn’t definitely tall, then x definitely isn’t definitely tall as well.6 This is paradoxical: a three-foot tall man definitely isn’t definitely tall, so appeal to the deduced Sorites sentence yields that the preceding man on the series, who is just one millimetre taller, also definitely isn’t definitely tall. Repeated appeal to the Sorites sentence yields that the first man on the series, who, we may presume, is 2.5 metres tall, also definitely isn’t definitely tall. But this is false. Any 2.5-metre tall man definitely is definitely tall. With some minor variations, Wright’s deduction is as follows (line numbers in square brackets track premises): [1] (1) [1] (2) [3] (3) [4] (4) [4] (5) [1,4] (6) [1,3,4] (7) [1,3] (8) [1,3] (9) [1] (10) [1] (11)

8x(DDT(x) ! :D:DT(x’)) DDT(x) ! :D:DT(x’) D:DT(x’) DT(x) DDT(x) :D:DT(x’) D:DT(x’) ^ :D:DT(x’) :DT(x) D:DT(x) D:DT(x’) ! D:DT(x) 8x(D:DT(x’) ! D:DT(x))

Premise (1) 8-elim Premise (for conditional proof) Premise (for reductio) (4) D-intro (2,5) !-elim (3,6) ^-intro (7)[4] :-intro (8) D-intro (9)[3] !-intro (10) 8-intro

Dorothy Edgington (1993) and Richard Heck (1993) respond by claiming that one cannot apply D-introduction within sub-proofs. As Heck formulates the response, D-introduction is validity-preserving, but in its presence rules such as conditional-introduction and negation-introduction that involve 4 See also Heck (1993) and Williamson (1994, ch. 5). 5 Vann McGee and Brian McLaughlin are supervaluationists who do not accept D-introduction because they think the principle conflicts with higher-order vagueness, but for reasons different from the ones brought out here (McGee and McLaughlin 1998, forthcoming). 6 Wright formulates the deduction so that it appeals only to a weakened form of D-introduction: if D
‘ j then D
‘ Dj, where D
¼ {Dg: g 2
}.

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Gap Principles, Penumbral Consequence / 201 discharge of premises cease to be validity-preserving without restriction, with the result that premises may not be discharged via these rules once D-introduction has been applied to them or to something deduced from them. I now want to argue, revisiting this decade-old debate,7 that higher-order gap principles lead, given D-introduction, to contradiction. I will construct a new argument, based on the same idea as Wright’s, that is not subject to Edgington and Heck’s criticism. We have a Sorites series of m objects (call the first ‘1’, the second ‘2’, and so on), each R-related to its successor in the series, beginning with an object we can truly describe as tall and ending with an object we can truly describe as not tall. Then the sequence of sentences below begins with a true sentence, while each subsequent sentence follows from the one immediately preceding it either by D-introduction or by appeal to a gap principle (second conditional formulation) and modus ponens. No discharge of premises is involved. :T(m) D:T(m) :DT(m
1) D:DT(m
1) :D2 T(m
2) D:D2 T(m
2) :D3 T(m
3) .. . :Dm
1 T(1)

D-intro Gap principle for T(x) D-intro Gap principle for DT(x) D-intro Gap principle for D2 T(x) Gap principle for Dm
2 T(x)

A further argument beginning with T(1) yields Dm
1 T(1), after m
1 applications of D-introduction. Contradiction. My proof that the truth of higher-order gap principles makes it impossible (given D-introduction) to get from an object that isn’t tall, in any finite number of R-steps, to one that is tall exploits a rather obvious fact: it is not possible for any relation R to densely order only finitely many objects. We can make the point more vivid. Grasp the first member of a length-m Sorites series for ‘tall’ in your left hand; grasp the last member in your right hand. To illustrate that there’s no ‘sharp’ boundary between the tall and the not-tall, you want to move your right hand leftward to grasp a different object that is a borderline case of the predicate ‘tall’ that’s true of the object in your left hand but false of the object in your right hand.8 After one move leftward of your 7 See also Sainsbury (1991b).

8 I equate falsity with truth of the negation.

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202 / Delia Graff right hand you still have the object in your left hand that is tall, hence definitely tall, and a new object in your right hand that’s a borderline case of ‘tall’, hence not definitely tall. Now to illustrate that there is no sharp boundary between the definitely tall and the not definitely tall, you want to move your right hand leftward again, to grasp an object that’s a borderline case of the predicate ‘definitely tall’ that’s true of the object in your left hand but false of the object in your right hand. Each time you do this, you find you have an object in your left hand of which a predicate of the form ‘Dn T(x)’ is true, and an object in your right hand of which that predicate is false. The collection of m
1 gap principles appealed to in my argument entails that you can do this at least m
1 times. But you cannot do this as many as m
1 times; there were only m
2 objects between your hands at the start.9 We should get clear about what exactly has and has not been shown. It has not been shown that there are higher-order predicates of the form ‘Dn T(x)’ for which there is no true gap principle. Let me explain. Compare two Sorites series for the predicate ‘tall’, each beginning with a man whose height is 2.5 metres (definitely tall) and ending with a man whose height is 1.5 metres (definitely not tall). One series consists of 101 men, each one centimetre taller than the next; while the other series consists of 1,001, each one millimetre taller than next. All we have shown is that the truth of gap principles involving the successor relation from the first series must give out before we get to 100 iterations of the definitely operator, while the truth of gap principles involving the successor relation from the second series must give out before we get to 1,001 iterations of the definitely operator.10 So it remains an open possibility that for any predicate of the form ‘Dn T(x)’, involving any number of iterations of the definitely operator, there is some (perhaps very) finely discriminating relation involved in a true gap principle for that predicate. That that possibility remains open, however, has no bearing on the point I wish to make. The point I wish to make is that given a particular finite-length Sorites series for the predicate ‘tall’, involving a particular Sorites relation for that predicate, we are no more able to locate a point on that series marking the transition from the objects of which ‘definitely definitely . . . definitely 9 A related challenge can be posed for Timothy Williamson’s (1994) view that gaps in knowledge rather than gaps in truth value explain our inability to locate boundaries for vague predicates. See Go´mez-Torrente (1997) for presentation, and Williamson (1997) for a reply. See also Go´mez-Torrente (2002) and Graff (2002) for further developments of the challenge, and Williamson (2002) for a reply to those. 10 This has been emphasized to me by Hartry Field, Stewart Shapiro, and Timothy Williamson.

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Gap Principles, Penumbral Consequence / 203 tall’ is true to those of which it is false than we are able to locate one marking the transition from the objects of which ‘tall’ is true to those of which it is false. The truth of higher- and higher-order gap principles for the particular series—involving its particular successor relation—cannot be what explains our inability to locate such points, since (given D-introduction) not all such principles are true. We correspondingly have also not shown that there is no such thing as infinitely higher-order vagueness. I take it that on the picture we’re working with a predicate is first-order vague if, on some Sorites series for it, there is a gap between the things of which it is definitely true and the things of which it is definitely false. I take it also that a predicate is infinitely higher-order vague if it is vague to some order but not to any greatest order. We have not yet said, however, what it is for a predicate to be vague to an order greater than one. I presume that the first-order vagueness of ‘definitely tall’ would suffice for the second-order vagueness of ‘tall’, and in general that the first-order vagueness of ‘definitelyn tall’ would suffice for the (n þ 1)th-order vagueness of ‘tall’. This does not amount to a definition, however,11 since there may be other higher-order predicates, including ‘definitelyn not-tall’ and ‘borderlinen tall’, the first-order vagueness of which would also suffice for the (n þ 1)th-order vagueness of ‘tall’. But we won’t need a general definition for the purposes of the discussion here; it will be enough to have identified the sufficient conditions for nth-order vagueness already mentioned. A gap principle for ‘definitelyn tall’ is (the universal closure of): DDn T(x) ! :D:Dn T(x’). If for each such predicate there is some Sorites series—involving perhaps only very slight differences in height between adjacent elements—that furnishes us with a successor relation that renders the corresponding gap principle true, then, given the preceding considerations, that suffices for ‘tall’ to be infinitely higher-order vague. For if a gap principle for ‘definitelyn tall’ is true, relative to some given Sorites series, then nowhere on that series will we find an object which ‘definitelyn tall’ is definitely true succeeded by one of which it is definitely false. So if for every predicate of the form ‘definitelyn tall’ there is some such Sorites series, ‘tall’ is infinitely higher-order vague. One might think, in light of the foregoing discussion, that at the very least we have shown that if ‘tall’ is infinitely higher-order vague, this cannot be 11 See Williamson (1999) and the introduction to Graff and Williamson (2002, p. xxi) for a general definition of nth-order vagueness.

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204 / Delia Graff exhibited on any one finite-length Sorites series. Yet we have not shown even that much. What we have shown, using reasoning the typical truth-value gap theorist accepts, is that given a particular Sorites series for ‘tall’, not every higher-order gap principle formulated with the successor relation from that series is true. We have not, however, shown that a truth-value gap theorist must accept that any such principle is false. It is only when a gap principle for ‘definitelyn tall’ is false that we are assured of finding, on the Sorites series in question, an object of which that predicate is definitely true right next to one of which it is definitely false. If none of the higher-order gap principles are false, then no matter how many iterations of the definitely operator we ascend to, we will always find that between the objects that are definitely definitelyn tall and those that are definitely not definitelyn tall are some objects; but since not all of the higher-order gap principles are true, we will not always be able to truly describe these objects as neither definitely nor definitely not definitelyn tall. We can represent the situation in a diagram: 2.5 m.

1.5 m. T (x)

T (x)

DT (x)

DT(x)

DDT (x)

DDT (x)

DDDT(x)

DnT(x)

DDDT (x)

DnT(x)

Given D-introduction, those objects that are definitelyn tall are precisely the objects that are definitelynþ1 tall. However, the envisaged truth-value gap theorist does not always accept arguments by reductio or contraposition of implications. So in particular, those objects that are not definitelyn tall may be properly included among those objects that are not definitelynþ1 tall. (For example, the things that are definitely not tall, and also the things that are borderline tall, are among the things that are not definitely tall.) But since our series is finite, this proper inclusion cannot continue indefinitely. But still, if no higher-order gap principle is false, then for any n there will still be a gap between the objects of which ‘definitelyn tall’ is definitely true and those of which it is definitely false.

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Gap Principles, Penumbral Consequence / 205 This may count as infinitely higher-order vagueness. But what it really brings out is that there is no connection between infinitely higher-order vagueness and the presence of ‘fuzzy boundaries’, properly construed. We could have infinitely higher-order vagueness exhibited on a series of three objects: let ‘definitelyn tall’ be true of the first object, for any n; false of the last object, for any n; and let the middle object be such that for any n, we speak neither truly nor falsely when we ascribe ‘definitelyn tall’ to it. Then we have infinitely higher-order vagueness in the sense that, for any n, the segment of objects of which ‘definitelyn tall’ is definitely true is not contiguous with the segment of objects of which it is definitely false, and in that sense there is no sharp boundary (there being no boundary at all) between the two segments of objects. But there is no sense in which it would be correct to say that there is a fuzzy boundary between these two classes of objects.

2. How Classical is the Supervaluationist’s Logic? I turn now to a different set of problems, directed at supervaluationists in particular, that are raised by gap principles formulated as arguments. F(x) ^ :F(y) ;:R(x, y) Ordinary argumentation makes use of a notion of consequence with distinctively classical properties. So, one ordinary way to argue for a conditional is to assume the antecedent and argue for the consequent from that assumption. It is well known, however, that argument by conditional-introduction for example, or by contraposition or reductio ad absurdum, is not acceptable given certain treatments of vagueness involving truth-value gaps. For those who retain truth-functionality for the sentential connectives and quantifiers, no classically valid formula is deemed valid, since any such formula may have only ‘indefinite’ constituents (or instances, in the case of quantified formulas). Suppose, for example, that the sentence ‘Al is tall’ is indefinite, but that ‘Al is bald’ is false. Then given Kleene’s strong three-valued truth tables (as adopted, for example, by Soames 1999, Tappenden 1993, and Tye 1994) the classically valid conditional ‘If Al is tall then he is either bald or tall’ will itself be indefinite. But still, the antecedent of this conditional is taken to imply its consequent in the sense that it is impossible for the antecedent to be true while the conclusion is not true. So conditional-introduction (if G, j  c

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206 / Delia Graff then G  j ! c) fails. Supervaluationists, who eschew truth-functionality along with bivalence, typically claim to have the upper hand here. For them, the conditional ‘If Al is tall then he is either bald or tall’ is true (indeed, valid) since classically true for every way (‘admissible’ or not) of making the vague predicates in it completely precise. The conditional is classically true on any precisification that falsifies its antecedent; and classically true also on any precisification that verifies its antecedent, since any such precisification also verifies its consequent. Supervaluationists cannot endorse every case of argumentation by conditional-introduction, however, but it has been suggested that this restriction on our ordinary deductive practice is acceptable, because it is limited to arguments involving the definitely operator, or similar constructions. According to Fine (1975: 290), the failure of conditional-introduction ‘distinguishes the presence of D from its absence’, and according to Keefe (2000: 178) ‘the cases in which [conditional-introduction and other classical principles] fail all involve the D operator (or similar such devices)’. While in one sense these claims are incontrovertible, there is another sense in which they are false. When relations of consequence other than the rather stringent logical consequence relation are at issue, the supervaluationist must countenance failures of conditional-introduction even in the absence of a definitely operator or any similar such devices. The question whether supervaluationists really endorse classical logic has received a fair amount of attention since Williamson’s (1994, ch. 5) discussion of it. The reason some commentators have claimed that the supervaluationist’s semantics for a language with a definitely operator yields something less than classical logic is not that there turn out to be classically valid formulas or even arguments that are not supervaluationally valid, but rather that the supervaluationist’s semantics for the language with the definitely operator yields a consequence relation that isn’t closed under such classical operations as conditional-introduction, contraposition (if G, j  c, then G, :c :j), or reductio ad absurdum (if G, j  c ^ :c, then G  :j). For example, on versions of supervaluation semantics that endorse D-introduction,12 ‘Al is bald’ implies ‘Al is definitely bald’; but the conditional ‘If Al is bald then Al is definitely bald’ need not be true—it is not true when Al is a borderline case. My focus in this section is on the question whether the classical closure principles fail only in the presence of the definitely operator. 12 Fine (1975); Keefe (2000).

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Gap Principles, Penumbral Consequence / 207 Vann McGee and Brian McLaughlin (forthcoming) have recently contributed to the debate by arguing that commitment to classical logic requires only that discharge of premises be permissible in the case of arguments that are classically valid. No argument whose validity results from the special logic of the definitely operator—or their ‘determinate truth’ predicate, or any other expression not traditionally treated as a logical constant—is required to support conditional-introduction.13 I disagree with their general claim. We who accept classical logic accept forms of argumentation involving conditional-introduction and contraposition irrespective of whether the consequence relation at issue is logical consequence or just our everyday context-dependent conception of what follows from what. For example, if you accept that it in some sense follows from someone’s being a friend of yours that you have an obligation to convey your condolences to him if he has recently suffered the death of a loved one, then you must—it seems to me—accept that in the same sense it follows from your having no obligation to convey condolences to someone who has recently suffered the death of a loved one that he is not a friend of yours. Argument IA X is a friend of yours. Therefore, if X has recently suffered the death of a loved one, then you have an obligation to convey your condolences to X. Argument IB X has recently suffered the death of a loved one. You have no obligation to convey condolences to X. Therefore, X is not a friend of yours. I do not suggest that you must accept either argument; you might think that you incur an obligation to convey condolences to your bereaved friend only once you know about his loss. But that is not to object to my point, which is rather that if you accept that in the A case the conclusion follows from the premises then you must accept it in the B case as well. The converse also holds. The two arguments stand or fall together. Similarly, on the classical perspective, since Sorites paradoxes show us that there is no sense in which it follows from X’s being tall and X’s being no more 13 McGee and McLaughlin have a stake in the issue, since even though they do not endorse D-introduction, they are still committed to failures of conditional-introduction in the presence of the D operator, as pointed out by Williamson (forthcoming).

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208 / Delia Graff than a millimetre taller than Y that Y is tall as well,14 it also shows us that there is no sense in which it follows from X’s being tall and Y’s not being tall that X is more than a millimetre taller than Y. Argument IIA X is tall. X is no more than a millimetre taller than Y. Therefore, Y is tall. Argument IIB X is tall. Y is not tall. Therefore, X is more than a millimetre taller than Y. Supervaluationists, however (and other truth-value gap theorists), do think that there is at least some important sense in which in case IIB we have a genuine entailment. For they think that on a Sorites series for the predicate ‘tall’, there are borderline cases between those of which the predicate is true and those of which it is false. Argument IIB is an instance of the argumentschema formulation of a gap principle. No definitely operator is involved, but contraposition or conditional-introduction on the argument-schema formulation of the gap principle yields a Sorites paradox.

Background Since I want in this section to spell out exactly what’s meant in saying that the supervaluationist does or does not endorse classical logic, it will help for the sake of concreteness to include a brief presentation of a supervaluation semantics.15 It will be important for what follows that the language for which the semantics is given is a language whose predicates are expressions of English. A supervaluational model (‘model’ for short) is a specification space consisting of one or more specification points, each of which may be thought of as a classical model, including a domain of discourse and an assignment of 14 Richard Feldman has pointed out to me that I should qualify my claim here, since there are inductive or probabilistic notions of ‘follows from’. When I say that there is some sense in which B follows from A, I mean that there is some sense in which the truth of A guarantees the truth of B. 15 My presentation is based on Fine’s.

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Gap Principles, Penumbral Consequence / 209 extensions from that domain to the predicates in the language.16 A formula is true in a model just in case it is true at every point in the model; false in a model just in case it is false at every point in the model.17 Truth value at a point for formulas is defined with the standard classical truth clauses, with the result that at any given point in a model every formula is either true or false (not both).18 A formula is neither true nor false in a model when it is true at at least one point in the model and false at at least one other. So what happens when a definitely operator is added to the language? As before, any formula, including one containing D, is true in a (supervaluational) model just in case it is true at every point in the model. What needs to be added is a clause for truth-at-a-point for D-initial formulas. One might like to say, as Fine provisionally does, that a formula Dj is true at a point in a model just in case j is true at every point in the model. Adoption of this truth clause, however, precludes the possibility of there being borderline cases at even the second order (e.g. borderline definite cases or borderline borderline cases), since it makes it impossible for the value of a D-initial formula to vary from point to point within a model with the result that D-initial formulas are bivalent. To avoid this we first enrich the structure of the models by introducing an accessibility relation on the points in the model.19 A formula Dj is true at a point in a model just in case j is true at every point in that model accessible to it, false otherwise.20 (Truth-at-a-point remains bivalent.) One may add further constraints on the accessibility relation, depending on one’s view of the logic of the D operator. Minimally one should require that the accessibility relation be reflexive, in order to ensure the validity of Dj ! j. Nothing here will turn on any particular 16 Perhaps one would want to allow domains to vary from point to point within a space; perhaps not. Nothing here turns on that decision, but for simplicity I assume invariant domains within a space. 17 In the main text I leave implicit the relativization of truth in a model to a variable assignment. A formula with free variables is true (false) in a model on a variable assignment just in case it is true (false) at every point in the model on that variable assignment. 18 The models here are structurally simpler than those in Fine’s work, where models include an ‘extension’ relation on points and allow for there to be points at which evaluation is not bivalent. Fine’s richer semantic framework allows for semantic theories other than supervaluationism, such as ‘bastard intuitionism’. The simplifications here are appropriate for the supervaluation theory advocated in Fine’s (1975) paper. 19 Not to be confused with the ‘extension’ relation on points in Fine’s models for a D-less language. See Fine (1975: 293–4) for his accessibility-relation semantics for D. 20 Dj is true at a point on a variable assignment, just in case j is true at every accessible point on that same variable assignment.

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210 / Delia Graff properties required of the accessibility relation, though I make the minimal assumption that it is reflexive. Treating truth as the sole designated value, a relation of logical consequence is defined as follows: an argument from a premise set of formulas
to a conclusion formula j is supervaluationally valid just in case every (supervaluational) model in which all members of
are true is a model in which j is true. A formula is valid when it is the conclusion of a valid argument with an empty premise set, that is, when it is true in every model whatsoever. In what sense does the supervaluationist’s semantics for the extended language yield (or not) a classical logic? At this point I prefer to rephrase the question by asking in what sense the semantics yields a classical consequence relation. In order for a consequence relation to be a classical one, it must have certain properties. It must, for example, be reflexive and transitive. It must also contain certain arguments in its extension, for example, any argument with an instance of excluded middle as conclusion. Supervaluation semantics for a standard first-order, D-less language yields a consequence relation that coincides in extension with the classical consequence relation. The semantics for the extended language, however, yields a consequence relation possessing some of the classical properties but lacking some others. Any classically valid argument or formula (including, e.g., Dj _ :Dj) is supervaluationally valid.21 Others of the classical properties of the consequence relation are guaranteed by the given model-theoretic characterization of consequence. Examples are reflexivity and transitivity, and, more generally, reiteration (if j 2
then
 j), and generalized transitivity (if
 j and D  g for every g 2
then D  j), hence also monotonicity (if
 j and
 D then D  j) and cut (if
[ D  j and
 d for every d 2 D then
 j). But, now leading up to the main point, other classical properties are lost.22 Examples already mentioned were failure of closure under contraposition, conditional-introduction, and reductio ad absurdum. To the extent that the supervaluationist wants to endorse such standard forms of argumentation under the heading ‘classical logic’, her aim of preserving classical logic falls short of its mark. Nevertheless, if failures of the classical closure principles 21 The converse does not hold; since D is being treated as a logical constant, there are arguments (e.g. from j to Dj) and formulas (e.g. :j ! :Dj) that are supervaluationally valid yet not classically valid. 22 Williamson (1994) and Keefe (2000) discuss this point in detail. It is also discussed by Machina (1976).

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Gap Principles, Penumbral Consequence / 211 always involved the definitely operator then, as Williamson (1994: 152) puts it, ‘our deductive style might not be very much cramped’. I proceed by proving, first, that failures of conditional-introduction do all involve the definitely operator when the consequence relation involved is the supervaluationist’s strict relation of logical consequence. I then go on to argue that the result must be qualified and that in an important sense the supervaluationist is committed to more rampant failures of conditional-introduction. Suppose that a formula c is true in every supervaluational model in which j and all members of premise set
are true, i.e. that
, j SV c. Suppose also that there is a supervaluational model M in which every member of
is true, but in which j ! c is not true, i.e. that
6SV j ! c. We can prove that some formula involved in the argument contains D. In the model M there is at least one point p at which j ! c is false, hence at which j is true and c is false. Every member of
is true at p in M. Now let M’ be the model that has p as its only point. (There is only one such model given the assumed reflexivity of the accessibility relation.) Suppose that D occurs nowhere in
or in j. Then j and all members of
must have the same truth value at p in M’ that they have at p in M since the truth-at-a-point clause for D is the only one that makes reference to other points in a model. So j and all members of
are true at p in M’, so they are true in M’ since p is the only point in M’. Since
, j SV c, c is also true in M’, hence true at p in M’. Since c has a different truth value at p in M’ from the value it has at p in M, it contains D.23

More Inclusive Consequence Relations The consequence relation so far under discussion has been a relation of logical consequence characterized model-theoretically: in order for some premises to logically imply a conclusion it must be that every model in which the premises are all true is one in which the conclusion is as well. But among all the models are some seemingly very strange ones, since the language with which we are working is a so-called ‘interpreted’ one containing expressions 23 Were there other expressions in the language, such as a truth predicate, or perhaps an ‘-ish’ modifier, that had truth-at-a-point clauses that made reference to other points in a model, we could conclude only that the argument in question involved at least one such expression. This is what Keefe means by saying that the classical closure principles fail only in the presence of D ‘or similar such devices’ (Keefe 2000: 178).

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212 / Delia Graff of English. For example, there are supervaluational models that consist of exactly one point, at which ‘Tall’ and ‘Short’ have exactly the same nonempty extension: there are, for example, models in which ‘Tall(Al)’ and ‘Short(Al)’ are both true. ‘: Short(Al)’ is not a logical consequence of ‘Tall(Al)’. But arguments such as ‘Tall(Al), therefore : Short(Al)’ are ones that the supervaluationist wants in some sense to validate, since they express ‘penumbral connections’ between the two vague predicates ‘tall’ and ‘short’. Similarly, there are models in which an ordered pair in the extension of ‘Shorter’ at every point in the model has a first member in the extension of ‘Tall’ at every point in the model, but a second member in the extension of ‘Tall’ at no point in the model: there are, for example, models in which ‘(Tall(Al) ^ Shorter(Al, Joe) ) ! Tall(Joe)’ is false. The sentence is not logically true, but the supervaluationist wants in some sense to validate it. Preservation of penumbral connections is supposed to be one of the main advantages of the theory. We don’t yet have a problem for the supervaluationist, just the makings of one. The supervaluationist might try to forestall the problem by claiming that sentences expressing penumbral connections aren’t in any sense supposed to come out valid on supervaluation theory; they are merely supposed to come out true as a matter of fact. Among all the models there are is one we might call the correct (or ‘intended’, or ‘actual’) model. The correct model is the one in which sentences have the truth value that they actually have.24 (Correctness must probably be relativized to times and contexts of utterance.) Since I am a short philosopher, the sentence ‘9x (Short (x) ^ Philosopher (x) )’ is true in the correct model, though false in many others. The collection of classical models that make up the points in the correct model are those that represent the extensions our predicates would have upon admissible precisification. On no admissible precisification does a pair in the extension of ‘Shorter’ have a first member in the extension of ‘Tall’ unless its second member is as well. In the correct model, the penumbral sentence ‘8x8y( (Tall(x) ^ Shorter (x,y) ) ! Tall(y) )’ is true. This alone constitutes an advantage for supervaluations over a three-valued truth-functional theory on which, due to the presence of borderline cases, the sentence is actually neither true nor false. But still, in addition to desiring a theory on which sentences expressing penumbral connections all come out true, the super24 One way that a supervaluationist might deal with higher-order vagueness would be by saying that although it is true that there is exactly one correct model, there’s no model such that it is true that it is the correct model.

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Gap Principles, Penumbral Consequence / 213 valuationist presumably also desires a theory on which arguments expressing penumbral connections (‘Al is tall, therefore not short’) come out as good in some sense. Saying that an argument expressing penumbral connection has the property that its conclusion is true in the correct model if its premises are is not saying much. Many bad arguments have that property since any argument with a false premise has that property. Instead, the relevant notion of goodness may be characterized as a relativized consequence relation. In addition to the supervaluationist’s logical consequence relation SV , there corresponds to each class C of supervaluational models a relativized supervaluational consequence relation SVC defined as follows:
SVC j just in case j is true in every model in C in which all members of
are true. ‘: Short(Al)’ is a consequence of ‘Tall(Al)’ relative to the class of supervaluational models that verify every penumbral sentence. My point is that when we consider consequence relations that better capture what the supervaluationist regards as a good argument than the logical consequence relation does, failures of conditional-introduction become far more rampant for the supervaluationist than is typically acknowledged. It’s not that logically valid arguments are sometimes regarded as bad (except perhaps in the sense, not at issue here, of being boring or trivial), but rather that some good arguments, such as those expressing penumbral connections, are not logically valid. I do not intend to single out any one relativized consequence relation that is to be privileged as capturing, context-invariantly, a supervaluationist conception of what constitutes a good argument, or of what ‘follows from’ what. (We may, adapting an idea of Robert Stalnaker’s, think of the class of models to which we relativize our consequence relation as growing or shrinking with the context.) What I intend to show is rather that for at least one supervaluationist conception of what follows from what that can be captured as a relativized consequence relation, conditional-introduction fails in the absence of a definitely operator. Let me proceed by substantiating the point first informally. Supervaluationists, along with many other philosophers who think that bivalence should be rejected in the face of vagueness, think that on a Sorites series for a vague predicate, we never find an object of which the predicate is true adjacent in the series to an object of which the predicate is false. Presumably they do not think this to be the case merely as a matter of contingent fact but rather as matter of some kind of necessity, as the kind of thing you can figure out while sitting thinking in your office. From the supervaluationist’s perspective, this may be expressed by saying, for example, that it’s not possible for

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214 / Delia Graff it to be true that X is tall and Y isn’t unless it’s true that they’re not adjacent members of a Sorites series for ‘tall’—it follows from X’s being tall and Y’s not being tall that they differ in height by more than a millimetre. (Pick a lesser difference if you think it required.) I hesitate somewhat to describe the entailment as an analytic, conceptual, or a priori one, since it may take some experience interacting with objects and people, reading books, or using rulers to know that differing in height by no more than a millimetre from is a Sorites relation for ‘tall’. Nevertheless, just to have a suggestive term, I’ll call the entailment an a priori one: according to the supervaluationist, X’s being tall together with Y’s not being tall a priori entails that X is more than a millimetre taller than Y. But now consider the conditional ‘If X is tall and Y isn’t, then X is more than a millimetre taller than Y’. This conditional need not be true on the supervaluationist’s view; though never false, it is not true when its consequent is false, X is taller than Y, and at least one of X and Y is a borderline case. One way to see the point is to note that the conditional is classically and hence supervaluationally equivalent to the conditional ‘If X is tall and X is no more than a millimetre taller than Y, then Y is tall’. The thought that every such conditional is true is precisely what leads to the Sorites paradox. On the assumption that the actual heights of people form a sufficiently smooth curve, supervaluationists and epistemicists agree that at least one such conditional must be actually untrue.25 The relevant formal difference between supervaluation semantics and classical semantics that underlies the point is the following. Given the classical characterization of a model (a domain of discourse, an assignment of extensions to predicates, etc.) and of truth in a model, one may associate with each class C of classical models a relativized classical consequence relation CLC defined as in the supervaluational case:
CLC j just in case j is true in every model in C in which all members of
are true. The classical relation of logical consequence CL is the least inclusive relativized consequence relation, by which I mean the relation obtained by relativizing to the largest class of classical models. Not every class of classical models yields an interesting consequence relation, but since our language is an ‘interpreted’ one, many classes do yield an interesting consequence relation, such as the class of models that verify all analytic truths, or all sentences expressing metaphysical necessities or all those expressing a priori truths. In the classical case, all of the 25 Pace Kamp (1981).

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Gap Principles, Penumbral Consequence / 215 relativized consequence relations, including the uninteresting ones relativized to gerrymandered classes of models, are closed under contraposition and conditional-introduction. In the supervaluational case, however, this is not so, even if we revert to the basic semantics for the D-less language. The point can be illustrated simply by considering an uninteresting consequence relation relativized to a class C containing just one supervaluational model in which j is neither true nor false but in which c is false. In the model, j must be true at at least one point; but c is false at that point (since false in the model) so j ! c is false at that point and so not true in the model: 6SVC j ! c. But still, j SVC c: j is not true in any model in C, so vacuously, c is true in every model in C in which j is true. So SVC is not closed under conditional-introduction. Nor is it closed under contraposition: Although j SVC c, :c 6SVC :j, since :c is true in the one model in C but :j is not. In order to substantiate the main point of this section, I must describe a class of supervaluational models that yields an ‘interesting’ relativized consequence relation for which conditional-introduction fails. Building on my informal presentation of the point, let’s take the relevant class C to contain those models that faithfully represent everything the supervaluationist takes herself to know a priori; more specifically, let’s take C to contain those models that faithfully represent everything that the supervaluationist takes herself to know a priori about a Sorites series. When F is a vague predicate, and R is a Sorites relation for that predicate, no model that verifies both of F(x) and R(x, y) but falsifies F(y) is in the class C. On the simplifying yet harmless assumption that R is not vague, so that R(x, y) is false in any model in C in which it is not true, the relation SVC is not closed under conditional-introduction: F(x) ^ :F(y) SVC :R(x, y); yet 6SVC (F(x) ^ :F(y)) ! :R(x, y), since there will be models in C in which R(x, y) is true, yet which contain points at which F(x) ^ :F(y) is true. In the familiar language of precisification: when R(x, y) is true, F(x) ^ :F(y) can be true on a precisification, but not on every precisification.

Phenomenal Sorites Of special interest are instances of my main example that are connected to phenomenal versions of the Sorites paradox—cases where F is a vague observational predicate, such as ‘looks red’ or ‘tastes sweet’, and where R is

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216 / Delia Graff an associated observational sameness relation, such as the relation of looking the same as or tasting the same as. Normally, one would say that it follows from one thing’s looking red and another thing’s not looking red that the two things do not look the same; that it follows from one thing’s tasting sweet and another thing’s not tasting sweet that the two things do not taste the same. Conditional-introduction might get one into trouble here, however—that is, if one accepts something that few philosophers would deny, namely, that a thing that does look red and a thing that doesn’t look red can be members of a sequence of things each of which looks the same as the next. For then if by conditional-introduction one concludes that if X looks red and Y doesn’t then X and Y do not look the same, one can further conclude, by classical principles that the supervaluationist accepts, that if X looks red and X does look the same as Y then Y looks red as well. Repeated application of this principle along a Sorites series of the described kind results in contradiction. The last thing both does and does not look red. The argument ‘This looks red and that doesn’t; therefore this does not look the same as that’ has some features worth noting here. First, an epistemicist who accepts what I’ll call a ‘chaining principle’—that a thing that looks red can be connected by a looks-the-same-as chain with a thing that does not look red—cannot accept the argument.26 Such a person, in keeping with his general treatment of the Sorites paradox, will say that the following universal generalization is false: 8x8y ((x looks red ^ x looks the same as y) ! y looks red); and will hence say that it has a particular false instance, that for some particular values of x and y, ‘x looks red’ and ‘:y looks red’ and ‘x looks the same as y’ are all true. On this view the argument ‘This looks red and that doesn’t, therefore this does not look the same as that’ is no good. (I, being an epistemicist who does accept the argument, reject the chaining principle.27) Second, and relatedly, the supervaluationist who accepts the chaining principle is committed to accepting the argument as a special case of her view that verifiers and falsifiers do not sit next to each other on a Sorites series since they are separated by intervening borderline cases. Third, unlike non-phenomenal instances of my main example, the phenomenal instances of the argument seem to be cases of arguments expressing penumbral connections. The argument ‘This candy tastes sour and that candy 26 I thank Alexis Burgess for this forceful formulation of the problem. 27 See my ‘Phenomenal Continua and the Sorites’ (Graff 2001) for a defense. Any epistemicist who thinks there are non-epistemic senses of ‘looks’ should wholeheartedly welcome the arguments of that paper.

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Gap Principles, Penumbral Consequence / 217 doesn’t; therefore the two candies do not taste the same’ has a conclusion that follows from its premise in just the same sense that it follows from one woman’s being tall and another’s not being tall that they are not the same height. But unlike the other penumbral argument we considered—Al is tall, therefore not short—this argument will not on the supervaluationist’s view support conditional-introduction: the associated conditional ‘If this tastes sour and that doesn’t then they do not taste the same’ cannot be admitted as a sentence expressing penumbral connection, unless of course the chaining principle is, as I think, false.

Implicit Premises An important question emerges at this point. Can a relativized consequence relation always be alternatively characterized in supervaluation theory as logical consequence given some (perhaps infinite) set of implicit premises? In other words, when SVC is a relativized supervaluational consequence relation, must there be some set of formulas S, to be thought of as implicit premises, such that
SVC j if and only if S [
SV j? We know that if C is characterized as the class of models in which every member of a set S of formulas is true, then
SVC j if and only if S [
SV j. The question is whether for every relativized consequence relation SVC there is some set of formulas S meeting the stated criterion. For example, ‘8x(Tall(x) ! :Short(x) )’ is a penumbral sentence, so ‘: Short(Al)’ can be characterized as a consequence of ‘Tall(Al)’ relative to the class of models in which all penumbral sentences are true, or it can be characterized as a logical consequence, given penumbral sentences (though not all of them are needed) as implicit premises. But since we have at this point seen at least one example of an argument expressing penumbral connection for which conditionalintroduction fails, the question arises whether the supervaluationist’s penumbral consequence relation can be alternatively characterized as logical consequence plus implicit premises. It cannot be assumed on the supervaluationist picture that every relativized consequence relation can be alternatively characterized as logical consequence plus implicit premises. In fact, if we restrict ourselves to the basic supervaluational semantics for a standard first-order language without a definitely operator, we find that we have already shown that there is a relativized supervaluational consequence relation that cannot be so characterized. We

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218 / Delia Graff have shown that even in the absence of D, there is a class C of supervaluational models that yields a relativized consequence relation not close under conditional-introduction, so that for some
, j, and c:
, j SVC c but
6SVC j ! c. But we have also seen that in the absence of D, if
[ S, j SV c then
[ S SV j ! c, for all S.28 I say all this as a preface to offering a reply to the main point of this section on behalf of the supervaluationist. I argued that the relativized supervaluational consequence relation plausibly called supervaluational a priori consequence is one for which conditional-introduction fails even in the absence of the D operator. I argued also that if (contrary to my own view) some chaining principles are true, then even the supervaluationist’s penumbral consequence relation is one for which conditional-introduction fails even in the absence of the D operator. In substantiating the point I characterized a class of supervaluational models in a way that made essential use of the supervaluationist’s metalinguistic truth predicate. Noting this, the supervaluationist might like to respond to my point by arguing that the relativized a priori or penumbral consequence relations can be alternatively characterized as a relation of logical consequence plus implicit premises that do contain the D operator. Given the supervaluationist’s extended language and semantics, the class of models appealed to in substantiating my main point can be characterized as the class of models in which the universal generalization of every formula (DF(x) ^ D:F(y) ) ! :R(x, y) is true whenever F is a vague predicate and R stands for a Sorites relation for that predicate. So, the reply goes, the case I offered of a relativized consequence relation for which conditionalintroduction fails even in the absence of the D operator was really just a case of the failure of logical consequence to be closed under conditionalintroduction in the presence of the D operator, with the premises of the argument containing D serving as ‘implicit’ premises. The reply is to the point, but serves as something of a double-edged sword for the supervaluationist. If every worrisome failure of conditional-introduction and contraposition is to be accounted for as a case implicitly involving the D operator, then argumentation involving the D operator is much more common than it would at first appear. 28 In fact, even with the D operator we can show, given the compactness theorem for the classical consequence relation, that not every relativized supervaluational consequence relation can be characterized as logical consequence plus implicit premises.

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Gap Principles, Penumbral Consequence / 219

Implicit Premises and Higher-Order Vagueness But there is a sharper edge to the sword: the implicit premises now being appealed to are universal closures of gap principles, which as we saw in the first section cannot all be true. Before elaborating this point, and spelling out the dilemma it poses for the supervaluationist, we should first take stock of the dialectic so far. The supervaluationist’s relation of logical consequence, when expanded to capture the validity of arguments resulting from the logic of the definitely operator, does not satisfy certain classical closure principles such as closure under conditional-introduction. Failures of conditionalintroduction are nevertheless, it is claimed, suitably restricted to arguments involving the definitely operator.29 I showed that supervaluationists’ own view of what constitutes a lack of sharp boundary on a Sorites series commits them to failures of conditional-introduction, even in the absence of a definitely operator, once we consider consequence relations that are less restrictive than logical consequence—such as the relation I (somewhat tendentiously) called a priori consequence; and also, if chaining principles are true, the relation of penumbral consequence so dear to the supervaluationist’s heart. Examples, respectively, were the arguments ‘Al is tall and Joe isn’t, therefore they differ in height by more than a millimetre’ and ‘This looks red and that doesn’t, therefore they do not look the same’. Since these are the sorts of consequence relations often involved in ordinary argumentation, the supervaluationist who, as part of her advocacy of classical forms of argumentation, wants failures of the classical closure principles to be suitably restricted now has something to worry about. The more inclusive consequence relations were characterized model-theoretically as relativized consequence relations for it which it can be shown that conditional-introduction fails in absence of the definitely operator. On behalf of the supervaluationist, I proposed an alternative characterization of these inclusive consequence relations as logical consequence plus sets of implicit premises, and explained why, so characterized, the definitely operator proved to be involved after all—in the set of implicit premises. It emerged that the crucial implicit 29 Something stronger may be said: a valid argument will support conditional-introduction not only when it does not contain D, but also when it is classically valid. The classically valid argument from Dj to Dj _ D:j does support conditional-introduction. Something even stronger may be said, though I won’t provide a general formulation here, since the supervaluationally valid argument from Dj to j also supports conditional-introduction.

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220 / Delia Graff premises were universal closures of gap principles (conditional formulation). If the supervaluationist is to appeal to such sentences as implicit premises, then she had better think that they are true. But as we saw in the first section, she cannot think they are all true, on pain of contradiction. I’ll close by posing the following dilemma for the supervaluationist who accepts the validity of D-introduction. If she thinks that there is infinitely higher-order vagueness, in particular if she thinks that for any n, there is a gap on a Sorites series (even one of finite length) between the things of which Dn F(x) is definitely true and those of which it is definitely false, then she is committed to the validity (in some inclusive sense) of the argument from Dn F(x) and :Dn F(y) to :R(x, y), for any n. This is an argument for which conditional-introduction fails, and its validity cannot be alternatively characterized as logical consequence plus implicit premises the supervaluationist accepts as true, since we saw at the outset that for some m, the corresponding gap principle is not true. In the case where n ¼ 0, we have an argument for which contraposition and conditional-introduction fail even in the absence of a definitely operator. And since nth-order vagueness turns out not to be so intimately connected with nth-order gap principles, it is unclear why the supervaluationist should think that first-order vagueness is best expressed by a first-order gap principle. For, as we saw, nth-order vagueness can be secured by the non-falsity of an nth-order gap principle, which on the view under discussion cannot be equated with its truth. If the supervaluationist does not think that there is infinitely higher-order vagueness, in particular, if for some n, she accepts a sharp boundary (no gap) between the things of which Dn F(x) is true and those of which it is false, then it is unclear what her motivation is for rejecting such a boundary at the first level.

REFERENCES E d g i n g t o n , D. (1993), ‘Wright and Sainsbury on Higher-Order Vagueness’, Analysis, 53/4: 193–200. F i n e , K. (1975), ‘Vagueness, Truth and Logic’, Synthese, 30: 265–300. G o´ m e z - T o r r e n t e , M. (1997), ‘Two Problems for an Epistemicist View of Vagueness’, in E. Villanueva (ed), Philosophical Issues, viii: Truth (Atascadero, Calif.: Ridgeview). —— (2002), ‘Vagueness and Margin for Error Principles’, Philosophy and Phenomenological Research, 64/1: 107–25.

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Gap Principles, Penumbral Consequence / 221 G r a f f , D. (2001), ‘Phenomenal Continua and the Sorites’, Mind, 110/440: 905–35. —— (2002), ‘An Anti-epistemicist Consequence of Margin for Error Semantics for Knowledge’, Philosophy and Phenomenological Research, 64/1: 127–42. —— and T. Williamson (eds.) (2002), Vagueness (Burlington, Vt.: International Research Library of Philosophy, Ashgate). H e c k , R. G., Jr. (1993), ‘A Note on the Logic of (Higher-Order) Vagueness’, Analysis, 53/4: 201–8. K a m p , H. (1981), ‘The Paradox of the Heap’, in U. Mo¨nnich (ed), Aspects of Philosophical Logic (Dordrecht: Reidel). K e e f e , R. (2000), Theories of Vagueness (Cambridge: Cambridge University Press). M c G e e , V., and B. Mc La u g h l i n (1998), ‘Review of Timothy Williamson’s Vagueness’, Linguistics and Philosophy, 21: 221–35. —— —— (forthcoming), ‘Logical Commitment and Semantic Indeterminacy: A Reply to Williamson’, Linguistics and Philosophy. M a c h i n a , K. F. (1976), ‘Truth, Belief, and Vagueness’, Journal of Philosophical Logic, 5: 47–78. S a i n s b u r y , R. M. (1991a), Concepts without Boundaries, King’s College London, Department of Philosophy, Inaugural lecture delivered 6 Nov. 1990. —— (1991b), ‘Is there Higher-Order Vagueness?’, Philosophical Quarterly, 41/163: 167–82. S o a m e s , S. (1999), Understanding Truth (New York: Oxford University Press). T a p p e n d e n , J. (1993), ‘The Liar and Sorites Paradoxes: Toward a Unified Treatment’, Journal of Philosophy, 90/11: 551–77. T y e , M. (1994), ‘Sorites Paradoxes and the Semantics of Vagueness’, Philosophical Perspectives, 8: 189–206. W i l l i a m s o n , T. (1994), Vagueness (London: Routledge). —— (1997), ‘Imagination, Stipulation and Vagueness’, in E. Villanueva (ed), Philosophical Issues, viii: Truth (Atascadero, Calif.: Ridgeview). —— (1999), ‘On the Structure of Higher-Order Vagueness’, Mind, 108/429: 127–43. —— (2002), ‘Epistemicist Models: Comments on Go´mez-Torrente and Graff ’, Philosophy and Phenomenological Research, 64/1: 143–50. —— (forthcoming), ‘Reply to McGee and McLaughlin’, Linguistics and Philosophy. W r i g h t , C. (1975), ‘On the Coherence of Vague Predicates’, Synthese, 30: 325–65. —— (1987), ‘Further Reflections on the Sorites Paradox’, Philosophical Topics, 15: 227–90. —— (1992), ‘Is Higher-Order Vagueness Coherent?’, Analysis, 52/3: 129–39.

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II SEMANTIC PARADOXES

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10 A Definite No-No Roy A. Sorensen

From the summit of Delia Graff ’s ‘Infinitely Higher-Order Vagueness’ (Graff 2003), I spied a curiosity nestled in the valley below. I was primed by Professor Graff ’s high-altitude discussion of the relationship between definite operators and rules of inference. The puzzle I will discuss below echoes her interests; it raises, in miniature, an issue about the relationship between definiteness and proof. Since those presiding over this conference gave commentators wide latitude in their responses, I will not resist the temptation to lower the level of discussion.

1. Indefinite Denial In Greek mythology Acrisius and Proetus are twins who quarrel in the womb. Since Acrisius and Proetus were not fully formed beings, their dispute is of an indefinite nature. I name the following pair of denials in their honor: acrisius. Proetus is not definitely true.

proetus. Acrisius is not definitely true.

I have given each sentence ‘‘equal billing’’ to ensure that they are asserted in parallel. Neither is the ‘‘first’’ sentence. Consequently, there is no difference in their context or level of truth or level of determinacy. There is no principled way of assigning a different subscript to Acrisius’ use of ‘true’ than Proetus’ use of ‘true’.

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226 / Roy A. Sorensen Acrisius and Proetus are outwardly oriented. Neither sentence undermines itself. Neither sentence supports itself. Acrisius and Proetus merely deny another sentence. Acrisius and Proetus are mild caveats. You should certainly think twice before denying that Acrisius and Proetus are true. The untruth of one implies the definite truth of the other. Thus the logic of definiteness prevents us from assigning falsehood to both sentences (or an intermediate degree of truth or truth-value gaps). Happily the logic of definiteness seems to tolerate any other assignment of truth-values.

2. Symmetry and the Definite Operator However, the symmetry of the two sentences is destabilizing. We have the following feeling: As twins, Acrisius and Proetus should have matching truthvalues (or should be equally bereft of truth-values). Since we have already ruled out the possibility that both are untruths, we conclude that Acrisius and Proetus are both true. The truth of the pair is no problem in itself. But there is an ironical difficulty in us possessing such a tidy argument that Acrisius and Proetus are true. The discovery of a proof for a proposition is generally regarded as a sufficient condition for that proposition being definitely true. Since Acrisius and Proetus each deny that the other is definitely true, they would each be false if proven true. Strangely, any cogent argument to the effect that both sentences are true must backfire. Acrisius and Proetus have relatives that are triplets, quadruplets, quintuplets, and so forth. Let us reason about them abstractly: (1) (2) . . . (n
1) (n)

Statement 2 is not definitely true. Statement 3 is not definitely true.

Statement n is not definitely true. Statement 1 is not definitely true.

The symmetry principle implies that all the members of the sequence are true or none of the members are true. If none are true, then they are all definitely true (because each would be described as definitely true by some

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A Definite No-No / 227 true member of the sequence). So if none are true, then they are all true. Therefore, they are all true. But this is a proof that all are true. So all of the sentences must be definite truths. Yet each member of the sequence says of another that it is not definitely true. Contradiction. Returning to Greek mythology, Maia was the eldest of the Pleiades (the seven daughters of Atlas and Pleione). She was so shy that she lived alone in a cave. If we let n ¼ 1 in above sequence, we obtain a sentence suited to be her namesake: (Maia) Maia is not definitely true. If Maia is false, then Maia is definitely true. But definite truth implies truth. Therefore, Maia is true. This proof that Maia is true ensures that Maia is a definite truth. But Maia says Maia is not definitely true. Contradiction. Maia is more open to traditional therapies associated with the Liar paradox: the ban on self-reference, indexical theories of ‘true’, etc. Weak variations of puzzles are helpful stepping stones for diagnosis. But to test the adequacy of a solution, we should revert to Maia’s stouter cousins.

3. The No-No Acrisius and Proetus can trace their lineage to Jean Buridan’s eighth sophism. Socrates says: (8.0) What Plato is saying is false. Plato says ‘What Socrates is saying is false’. Neither Socrates nor Plato says anything further. Buridan excludes the T–F and F–T assignments because they do not respect the symmetry of the sentences. He reviews the apparent difficulty in assigning sentences the same truth-value. Nevertheless, Buridan concludes both are false. He appeals to an infallible principle: ‘‘any proposition that in conjunction with something true entails something false, is itself false’’ (Hughes 1982: 73) If 8.0 were true then the negation of Plato’s remark would be true and that truth would imply the falsehood of 8.0. Notice that Buridan’s infallible principle is not triggered for Acrisius and Proetus. If Acrisius is true then the conjunction of Acrisius and Proetus would only imply that Acrisius is an indefinite truth. In earlier ruminations (Sorensen 2001, ch. 11) I dubbed an inadvertently improved version of Buridan’s eighth sophism ‘‘the no-no paradox’’. That

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228 / Roy A. Sorensen would make Acrisius and Proetus a definite no-no paradox. I now think there is a family of no-nos paralleling the family of Liar paradoxes. There is a possible no-no (modeled on Post 1970), a knower no-no (Sorensen 2002), and a probable no-no (in which each pair denies that the other is probable). My solution to the no-frills no-no was to reject the symmetry principle. The thought was that just as the supervaluationist postulates truth-value gaps, classical logic can have truth-maker gaps—statements that have unforced truth-values. One of the sentences is true and the other is false. Since they are ungrounded, there is no way to tell which is the truth and which is the falsehood. One might extend this solution to the definite no-no. All definite truths have truth-makers but the meta-statement ‘It is definitely true that p’ may itself lack a truth-maker. What makes it true that p need not make it definitely true. Unlike the no-no, both members of the definite no-no can be true. The definite no-no uses more premises than the no-no. So one might solve the definite no-no by challenging the connection between proof and definiteness. As a first step, one might deny that proof that p implies that it is definitelyn true that p. As the definite operator is iterated, it becomes more demanding. Perhaps a proof that makes p definitely true need not make p definitely, definitely, definitely, definitely, definitely, definitely, definitely true. After all, the following pair of awkward sentences seem less fractious than Acrisius and Proetus: (M) N is not definitely, definitely, definitely, definitely, definitely, definitely, definitely true. (N) M is not definitely, definitely, definitely, definitely, definitely, definitely, definitely true. As before, we cannot consistently say the pair are untruths. Symmetry precludes one from being true while the other is false. If this is proof that M and N are true, then it may be a proof that makes M and N definitely true without making them definitely, definitely, definitely, definitely, definitely, definitely, definitely true. If proofs vary in their definitude, then perhaps there are proofs that fail at the first stage: they prove p without making p definitely true. Many sound arguments are incapable of increasing warrant for their conclusions, for instance, circular arguments and some slippery slope arguments. If ‘definite’ is sensitive to warrant (as it is under an epistemicist reading

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A Definite No-No / 229 of ‘definite’ as knowable), then an epistemically sterile proof that p would fail to make p definite. Epistemicists can admire this solution from a distance. They interpret ‘definite’ as akin to ‘knowable’. An epistemicist can take the further step of admitting that proof is a good way of making propositions knowable while denying that it is universally effective. Some proofs are too complicated or too long or otherwise inaccessible to yield knowledge of their conclusions. The logicist’s proof that a million plus a million equals 2 million is criticized in this spirit. The proof is sound but too long to give a human being extra warrant for the conclusion. Obviously, no knowledge can be gleaned from a proof that a proposition is both true but unknowable. Perhaps this is a wake-up call to oversimplistic epistemologies of proof. If this is the solution to the definite no-no paradox, then it is a solution that we can never know. My point is only to raise the solution as a possibility—an instructive possibility in the eyes of the epistemicist.

REFERENCES G r a f f , D e l i a (2003), ‘Infinitely Higher-Order Vagueness (and Other Problems for Supervaluationists)’, in JC Beall (ed.), Liars and Heaps (Oxford: Oxford University Press): 195–221. H u g h e s , G. E. (1982), John Buridan on Self-Reference (Cambridge: Cambridge University Press). P o s t , J o h n F. (1970), ‘The Possible Liar’, Nouˆs, 4: 405–9. S o r e n s e n , R o y (2001), Vagueness and Contradiction (Oxford: Clarendon Press). —— (2002), ‘Formal Problems in Epistemology’, in Paul Moser (ed.), The Handbook of Epistemology (Oxford: Oxford University Press): 539–68.

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11 Reference and Paradox Keith Simmons

Consider the phrase (B) the least integer not nameable in fewer than nineteen syllables. There are only finitely many syllables of English. And so there are only finitely many phrases of English with fewer than nineteen syllables. But there are infinitely many integers, so there are integers not nameable by any phrase of English with fewer than nineteen syllables. And among these integers there is a least, call it k. Then the phrase B denotes the integer k. But the phrase B itself has eighteen syllables. So the number not nameable in fewer than nineteen syllables is denoted by a phrase with eighteen syllables. We have a contradiction, and we are landed in paradox. This is Berry’s paradox, presented by Berry in a letter to Russell in 1904. (Russell, by the way, says that the Berry phrase B denotes 111,777.1) Berry’s paradox is one of the so-called paradoxes of definability—the others are Richard’s paradox and Ko¨nig’s paradox, both discovered in 1905, within a month of each other. It should be said that the label ‘paradoxes of definability’ is unfortunate. These paradoxes do not turn on the notion of definition in any ordinary sense of ‘definition’, and there’s nothing particularly modal going on. Rather, the paradoxes turn on the notion of reference or denotation, the relation between a referring expression and its referent. This is one of the fundamental semantic relations, along with the relation between a predicate and its extension, and a sentence and its truth value. So like Russell’s paradox 1 Berry’s paradox is reported in Russell (1908).

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Reference and Paradox / 231 about extensions and the Liar paradox about truth, the paradoxes of definability suggest that we do not have an adequate understanding of our basic semantic concepts. Now, attention has been lavished on the Liar and Russell’s paradox, while the paradoxes of definability have largely taken a back seat. So in this paper I’d like to redress the balance a little. The semantic notion of reference or denotation is no less fundamental than that of truth—indeed, arguably it’s more fundamental—and we should attend to the paradoxes of denotation just as urgently as we attend to the Liar.

1. A Simple Paradox of Denotation It will help if we work with a simple paradox of denotation, much simpler than the Berry. The story goes as follows. Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions: pi six the sum of the numbers denoted by expressions on the board in room 213. Now I am in fact in room 213, though I believe that room 213 is my colleague’s office. I set you the task of providing the denotations of these expressions. You respond as follows: But we’re in room 213! It’s clear what the first two phrases denote. But what about the third? Let’s call your third phrase P. Suppose P denotes the number k. Now, P denotes k if and only if the sum of the numbers denoted by expressions on the board in room 213 is k. So it follows that the sum of the numbers denoted by expressions on the board in room 213 is k. But then k ¼ p þ 6 þ k, which is absurd. So P is pathological— it appears to denote a number but it doesn’t, on pain of contradiction. You continue: And now, since P does not denote a number, the only expressions on the board that do denote numbers are the first two. But then the sum of the numbers denoted by expressions on the board in room 213 is p þ 6.

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232 / Keith Simmons You conclude: In the previous sentence, there is a token of the same type as P, call it P*. And P* does denote a number, namely p þ 6. There’s something remarkable about this discourse. You establish that P is pathological, but the reasoning does not stop there. You reason past pathology. By reasoning that is natural and intuitively valid, you reach a phrase, P*, composed of the very same words as P, with the very same meaning. And yet, while P fails to denote, P* succeeds. We can repeat the words of a pathological denoting phrase and successfully denote. This phenomenon calls for explanation. There is another way you could have continued. Instead of repeating P, you could revisit P: And now if P does not denote a number, then the only expressions on the board that do denote numbers are the first two. But now attend to the words on the board that make up the phrase P. They make reference to the phrases on the board that denote numbers, and the only phrases that do so are the first two. And the sum of those numbers is p þ 6. And now you conclude: So P denotes p þ 6. Again, something remarkable happens: you reason past the pathology of P, and find, by natural and intuitive reasoning, that P does denote. How can one and the same phrase be pathological and yet successfully refer? Again, an explanation is called for. (In their neglected writings on the definability paradoxes, Richard, Peano, and Poincare´ show sensitivity to the fact that we can repeat and revisit pathological phrases.2) So we want to explain these two stretches of discourse, one in which we repeat P, and the second in which we revisit P. In each case, your reasoning is natural and intuitive, and appears to be valid. We should not block it by artificial, ad hoc means. Since the reasoning is natural, I would rather regard it as data that express semantic intuitions we have about the notion of denotation or reference. My strategy will be to find a plausible analysis that preserves the validity of the reasoning, and respects the data. And, I want to 2 See Peano (1906); Poincare´ (1906, 1909); Richard (1905, 1907). For a discussion, see Simmons (1994).

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Reference and Paradox / 233 suggest, the explanation must be a pragmatic one, at least in the sense that the notion of context will play a crucial role. Given P and P*, two tokens of the same type, how can one fail to refer and the other succeed? Or given the phrase P, how can its semantic status change in the course of your reasoning? A natural response is this: because of some change in the context. So I shall be claiming that there are context-changes within our discourses. And I shall also be claiming that the term ‘denotes’ is sensitive to those changes. But let us first consider context-change and discourse analysis more generally.

2. Context-Change and Discourse Analysis We are all familiar with indexical terms, like ‘I’, ‘now’, ‘here’, and so on. It’s a familiar idea that context acts on content. But it is increasingly being recognized that this is not a one-way street. The reverse direction holds as well: content acts on context. Stalnaker writes: context constrains content in systematic ways. But also, the fact that a certain sentence is uttered, and a certain proposition expressed, may in turn constrain or alter the context. . . . There is thus a two-way interaction between contexts of utterance and contents of utterances.3

Or as Isard puts it: Utterances do more than merely display themselves before a context and then Vanish. They alter the context and become part of it.4

At a given point in a conversation or discourse, the context will in part depend on what has been said before. For example, the context may change as new information is added by the participants in the conversation. Over the last twenty years or so the kinematics of context-change has been studied by philosophers, linguists and semanticists alike—for example, by Stalnaker, Lewis, Heim, Kamp, Reinhart, Grosz and Sidner, and many others.5 According to Stalnaker, the connection between context and available information is very tight indeed. Stalnaker says that a context should be 3 Stalnaker (1999b: 66). 4 Isard (1975: 287). 5 I am grateful to Zolta´n Szabo´ for emphasizing the relevance of Stalnaker’s work to my account of the Liar. Glanzberg (2001) also stresses the relevance to the Liar of context-change and discourse analysis.

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234 / Keith Simmons identified with the body of information that is presumed to be available to the participants in the speech situation—the ‘common ground’, to use a phrase from Grice, or the shared presuppositions of the participants. As new utterances are produced, and new information is made available, the context changes. To keep track of these context-changes, then, we need a running record of the information provided in the course of the conversation. Lewis works with a related notion of the conversational score. The analogy is with a baseball score. A baseball score for Lewis is composed of a set of 7 numbers that indicate, for a given stage of the game, how many runs each team has, which half of which innings we’re in, and the number of strikes, balls, and outs. Notice that correct play depends on the score—what is correct play after two strikes differs from what is correct play after three strikes. Similarly for conversations: the correctness of utterances (their truth, or their acceptability in some other respect) depends on the conversational score. Lewis identifies a number of components of a conversational score. One is the set of shared presuppositions of the participants, the common ground. Here Lewis follows Stalnaker. As Lewis puts it: ‘‘Presuppositions can be created or destroyed in the course of a conversation.’’6 As the set of presuppositions changes, the conversational score changes. Of course, the notion of conversational score is a vivid way of capturing the notion of context. A change in the set of presuppositions is a change of context. Another component of conversational score, according to Lewis, is the standard of precision that is in force at a given stage of the discourse. Suppose I say ‘France is hexagonal’. If you have just said ‘Italy is boot-shaped’, and got away with it, then your utterance is true enough. The standards of precision are sufficiently relaxed. But if you have just denied that Italy is boot-shaped, and carefully pointed out the differences, then my utterance is far from true enough—the standards of precision are too exacting. The acceptability of what I say here depends on the conversational score, on the context, which is determined by what has been said before. In both Stalnaker and Lewis we find the idea of tracking context-change by keeping a running record of shifts in the information presumed to be available to the conversants. A number of linguists have also developed this idea. For example, in Irene Heim’s file change semantics, a ‘file’ contains all the information that has been conveyed up to that point—and the file is continu6 Lewis (1983a: 233).

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Reference and Paradox / 235 ally updated as the discourse moves on. The same basic idea informs Grosz and Sidner’s theory of discourse structure and their notion of a focus space, and also Tanya Reinhart’s analysis of sentence topics. The distinction in linguistics between given information and new information is also relevant here. For example, Chafe characterizes given information as the ‘‘knowledge which the speaker assumes to be in the consciousness of the addressee at the time of the utterance’’, and new information as ‘‘what the speaker assumes he is introducing into the addressee’s consciousness by what he says’’.7 Context-change will be marked by the introduction of new information.

3. Context-Change and the Denotation Discourse Let’s go back to the denotation discourses. Consider first the repeating discourse. It is natural to divide the discourse into four segments: first, where I produce the tokens on the board; second, where you argue to the conclusion that P is pathological and does not denote; third, where you build on that conclusion and produce P*; and fourth, where you reflect on P* and conclude that it does denote. We can identify changes in the common ground as this discourse progresses. For example, there is a change in our shared presuppositions as we move from the first segment of the discourse to the second. Your assertion that we are in room 213 changes the context: it provides new information and a change in the common ground. There is also a crucial difference between the first and second segments on the one hand, and the third and fourth on the other. The culmination of the reasoning of the second segment is the proposition that P is pathological and does not denote. This proposition is part of the common ground for the third and fourth segments. Throughout these subsequent segments, we are presupposing that P is pathological and does not denote. So in the transition from the second segment to the third, there is a shift in the context—there is a change in the body of information that is presumed to be available to us. It is this context-change that I want to focus on—the one that occurs as we move from the second segment to the third and fourth. I shall call the new context reflective with respect to P. In general, a context associated with a given stage of a discourse is reflective with respect to a given expression if at that stage it is part of the common ground that the expression is semantically pathological (and 7 Chafe (1976: 30).

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236 / Keith Simmons so does not denote—or isn’t true, or does not have an extension). So as we move from the second segment to the third and fourth, there is a change in the conversational score—a shift from an unreflective context to a reflective one. We see exactly the same shift in the case of the revisiting discourse. There too we shift to a context that is reflective with respect to P.

4. The Action of Context on Content Thus far, we have seen that content acts on context—newly available information changes the context. But there is a two-way interaction between context and content—context also acts on content. So now we want to see how the changes in context affect content. Remember the challenge posed by the repeating discourse: P and P* are composed of the very same words with the very same meaning, and yet one fails to denote while the other denotes a definite number (p þ 6). Somehow the change in context produces this phenomenon, and our task is to explain how. Now if the context acts on content, we would expect at least one expression in the discourse to be sensitive in some way to context-change. I want to claim that ‘denotes’ is that expression. (When we examine the terms that appear in the denotation discourse, it’s very hard to see a viable alternative.) So let us now see if this claim can be made out, not by any ad hoc maneuvering, but by a reasonable methodology. The methodology is this: given that the denotation discourse exhibits valid reasoning to a true conclusion, we are to find the most plausible analysis that preserves the validity of the reasoning and the truth of the conclusion. How does the term ‘denotes’ behave in our discourse? Does its extension change at any point because of a change in context? Let i be the initial context associated with the first segment, where I produce the token P. My first use of ‘denotes’ is a component of the utterance P, and so occurs in the context i. So let us represent this first use of ‘denotes’ by ‘denotesi ’. This representation does not commit us to the claim that ‘denotes’ is context-sensitive—we are only marking the fact that this use of denotes occurs in context i. Let us continue to attach the subscript ‘i ’ to each subsequent use of ‘denotes’ if no change of extension is forced upon us. If context has no effect on the extension of ‘denotes’—if ‘denotes’ is a predicate constant—then the continued appearance of the subscript i will indicate no more than this: every use of ‘denotes’ is coextensive with every other. If, on the other hand, ‘denotes’ is

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Reference and Paradox / 237 context-sensitive, then the subscript i will reappear only if, for some reason or other, subsequent uses of ‘denotes’ inherit the extension that the context i determined for my first use of ‘denotes’. So at the outset of the discourse we have: pi six the sum of the numbers denotedi by expressions on the board in room 213. Now, in order to determine the denotation of P, we have to determine what the expressions on the board denotei . So the subscript i will continue to appear in the representation of your reasoning: It’s clear what the first two phrases denotei . But what about the third? Let’s call your third phrase P. Suppose P denotesi the number k. Now, P denotesi k if and only if the sum of the numbers denotedi by expressions on the board in room 213 is k. So it follows that the sum of the numbers denotedi by expressions on the board in room 213 is k. But then k ¼ p þ 6 þ k, which is absurd. So P is pathological—it appears to denotei a number but it doesn’t, on pain of contradiction. Nothing so far forces a different extension on ‘denotes’; quite the reverse, in fact. Let us pause here. As you conduct your reasoning, something is operating in the background—what we may call a denotation schema. A denotation schema is an exact analogue of the perhaps more familiar truth schema. An instance of the truth schema is: ‘Snow is white’ is true if and only if snow is white. An instance of the denotation schema is: ‘32 ’ denotes 9 if and only if 32 ¼ 9. The instance of the schema you have used is this: P denotesi k if and only if the sum of the numbers denotedi by expressions on the board in room 213 is k. Call this the i-schema. When you assess P by the i-schema, a contradiction results. Now us return to the discourse. You continue:

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238 / Keith Simmons And now, since P does not denotei a number, the only expressions on the board that do denotei numbers are the first two. But then the sum of the numbers denotedi by expressions on the board in room 213 is p þ 6. It’s clear that the first two occurrences of ‘denotes’ receive the subscript i, since they rely on your result that P does not denotei . And for just the same reason, the occurrence of ‘denote’ in the token P* receives the subscript i as well. The occurrences of ‘denotes’ in this present stage of the discourse inherit their contextual subscripts from the previous stage. Notice that in a strict sense, P* is a repetition of P—it is composed of the very same words, with the very same meaning, and the very same extensions. Now let’s move to your final use of ‘denotes’. You continue: In the previous sentence, there is a token of the same type as P, call it P*. And P* does denote a number, namely p þ 6. Should we attach the subscript i to this final use of ‘denotes’? The answer is no. You have reached a true conclusion here through valid reasoning. According to your assessment here, P* denotes. But P* does not denotei — plug P* into the i-schema, and you’ll get a contradiction, just as you did with P. So here a shift in extension is forced upon us—P* denotesr , let us say, where ‘denotesi ’ and ‘denotesr ’ have different extensions. P* is in the extension of ‘denotesr ’, but it is not in the extension of ‘denotesi ’. What produces this shift in extension? The change in context—specifically, the shift to a context which is reflective with respect to P. When you assess P*, and declare that it denotes, you assess it in a context in which it is part of the common ground that P is pathological. The schema by which you assess P*— the r-schema—provides an assessment of P* in the light of P’s pathologicality. The relevant instance of the r-schema is this: P* denotesr k if and only if the sum of the numbers denotedi by expressions on the board in room 213 is k. The right-hand side of the biconditional is true for k ¼ p þ 6, given that P is pathological, and doesn’t denotei . And so it follows that P* denotesr p þ 6, as you say. In a nutshell, then, we explain your different assessments of P and P* this way: you assess P by the unreflective i-schema, and you assess P* by the reflective r-schema. There is no intrinsic difference between P and P*—the difference lies in the schema by which they are evaluated.

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Reference and Paradox / 239 Notice that P also denotesr p þ 6, just like P*. (Plug P into the r-schema, and that’s the result you’ll get.) So P fails to denote, and P does denote. But there is no contradiction here: P fails to denotei , but it does denoter . Compare: sometimes an utterance of ‘France is hexagonal’ is true enough, and sometimes it isn’t. It depends on the conversational score, in particular on the standards of precision that are in force. In a loosely analogous way, whether or not P denotes depends on the standard of assessment: do we apply the unreflective i-schema or the reflective r-schema? The revisiting discourse is analysed in just the same way. When we revisit P, we do so in a context that is reflective with respect to P. Having concluded that P does not denotei , and is pathological, we reassess P by the r-schema and find that P does denoter . In both discourses, the extension of ‘denotes’ undergoes a change. ‘Denotes’ is a context-sensitive term that may shift its extension according to context.

5. Singularities We’ve seen that P is excluded from the extension of ‘denotesi ’, but not from the extension of ‘denotesr ’. What else is excluded from the extension of ‘denotesi ’? And what is the relation between the extension of ‘denotesi ’ and the extension of ‘denotesr ’? A possible response here appeals to a hierarchy. The claim might be that when we move from an unreflective context to a reflective one, we move to an essentially richer language. The predicate ‘denotesi ’ is the denotation predicate of the unreflective context; the predicate ‘denotesr ’ is the more comprehensive denotation predicate of the semantically richer language of the reflective context. On such a hierarchical account, the extension of ‘denotesi ’ is properly contained in the extension of ‘denotesr ’. As far as there is an orthodoxy regarding the definability paradoxes, some kind of hierarchical account is it. (For example, van Heijenoort writes: ‘‘Today [Richard’s] paradox is generally considered solved by the distinction of language levels.’’8) But I think we should resist the hierarchy. There are, I would argue, a number of problems with the hierarchical account.9 Let me just mention here the most blatant: the hierarchical account offers too regimented an account of natural language. Surely English does not contain 8 See van Heijenoort (1967: 142).

9 For more on this, see Simmons (1993).

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240 / Keith Simmons infinitely many distinct denotation predicates, but just one. And surely the stratification of English into a hierarchy of distinct languages is highly artificial. With Tarski, we may doubt ‘‘whether the language of everyday life, after being ‘rationalized’ in this way, would still preserve its naturalness and whether it would not rather take on the characteristic features of the formalized languages’’.10 My proposal is in a strong sense anti-hierarchical. The leading idea is that in pathological denoting phrases like P, there are minimal restrictions on occurrences of ‘denotes’. At this point, a pragmatic principle of interpretation comes into play: the principle of Minimality. According to Minimality, restrictions on occurrences of ‘denotes’ are kept to a minimum: we are to restrict the application of ‘denotes’ only when there is reason to do so. Suppose you innocently say: The phrase ‘the square of 2’ denotes 4. Here, your use of ‘denotes’ is quite unproblematic. Should P be excluded from its extension? Minimality says no—because there is no need to exclude it. We have seen that in a suitably reflective context, P does denote a number (specifically, P denotesr p þ 6). Now the context of your utterance here is quite unconnected to my production of P. But Minimality says that your unconnected context should be treated as if it were reflective with respect to P. In your unconnected, neutral context of utterance, P can be counted among the phrases that denote—and so by Minimality, it is so counted. If there’s no need to exclude P, then include it. If we adopt Minimality we respect a basic intuition about predicates. Intuitively, we take a predicate to apply to everything with the property that the predicate picks out. In general, if an individual has the property picked out by the predicate w, then that individual is in the extension of w. The more restrictions we place on occurrences of ‘denotes’, the more we are at odds with this intuition. We do expect any solution to a genuine paradox to require some revision of our intuitions. But the more a solution conflicts with our intuitions, the less plausible that solution will be. For example, the hierarchical stratification of the denotation predicate involves massive restrictions on occurrences of ‘denotes’. On a standard Tarskian line, the referring expression ‘the square of 2’ is of level 0; the denoting phrase ‘‘the number denoted by ‘the square of 2’ ’’ is of level 1; and so on, through the levels. A use of ‘denotes’ in an utterance of level 1 has in its 10 Tarski (1983: 267).

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Reference and Paradox / 241 extension all referring expressions of level 0, and no others. So all sentences of level 1 and beyond are excluded from the extension of such a use of ‘denotes’. Go¨del remarks of Russell’s type theory that ‘‘each concept is significant only . . . for an infinitely small portion of all objects’’.11 A similar complaint can be made about a standard hierarchical account of denotation: an ordinary use of ‘denotes’ will apply to only a fraction of all the denoting expressions. Minimality keeps surprise to a minimum: our uses of ‘denotes’ apply to almost all denoting phrases. We are sometimes forced to restrict ‘denotes’—we must, for example, limit the extension of ‘denotesi ’ by excluding P. Still, according to Minimality, we exclude only those denoting expressions that cannot be included. So my proposal identifies what I call singularities of the concept denotes. For example, P is a singularity of ‘denotesi ’. Notice that P is a singularity only in a context-relative way—P is not a singularity of ‘denotesr ’ or your neutral use of ‘denotes’. So in my view we should not stratify the concept of denotation; rather we should identify its singularities. Go¨del once made the following tantalizing remark: It might even turn out that it is possible to assume every concept to be significant everywhere except for certain ‘singular points’ or ‘limiting points’, so that the paradoxes would appear as something analogous to dividing by zero. Such a system would be most satisfying in the following respect: our logical intuitions would then remain correct up to certain minor revisions, i.e. they could then be considered to give an essentially correct, only somewhat ‘blurred’, picture of the real state of affairs.12

I take my singularity proposal to be very much in the spirit of Go¨del’s remarks. We retain a single denotation predicate which undergoes minimal changes in its extension according to context. There is no wholesale revision of the notion of denotation; no division of ‘denotes’ into infinitely many distinct predicates; no splitting of everyday English into an infinite hierarchy of languages.

6. A Glimpse of the Formal Theory The main task of the formal theory is to identify the singularities of a given occurrence of ‘denotes’.13 And this is carried out by way of certain kinds of trees—denotation trees. We will come to those in a moment. 11 Go¨del (1944: 149). 12 Go¨del (1944: 150). 13 Once we have a formal theory that identifies singularities of a given use of ‘denotes’, we can offer suitably restricted denotation schemas; for example,

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242 / Keith Simmons But first consider P again. Suppose we represented the token P as an ordered pair , where the first element is the type of P, and the second indicates the appropriate representation of ‘denotes’ in P (namely, ‘denotesi ’). Then something is missing. This representation does not distinguish P from P*, yet the former denoting expression is pathological and the latter isn’t. There is something more to consider: the schema by which the token is given denotation conditions. In the second segment of the discourse, P is evaluated by the i-schema; in the third segment, P* is evaluated by the r-schema, a schema that is reflective with respect to P. So we capture the denotation discourse more perspicuously if we represent P by the ordered triple , where the third element indicates that the schema by which P is assessed is the i-schema, and P* by the triple , indicating that P* is assessed by the r-schema. We may of course evaluate P and P* by other schemas. But it is the evaluation of P by the i-schema—the schema associated with P’s context of utterance—that leads to the conclusion that P is pathological. And it is the evaluation of P* by the r-schema—the schema associated with P*’s context of utterance—that leads to the conclusion that P* has a determinate referent. So if we are after an analysis of the denotation discourse, the representation of P is privileged over other representations of P, and is likewise a privileged representation of P*. We will call these representations the primary representations of P and P*. Let’s now construct what we will call the primary denotation tree for P. The top node is the primary representation of P: . The third element of the triple indicates that we are after the denotationi of P. Now P makes reference to certain denoting expressions—the ones on the board. To determine a denotationi for P, we must first determine the denotation of these expressions—more specifically, we are after what these expressions denotei , since the occurrence of ‘denotes’ in P is given by ‘denotesi ’, as the second element of the primary representation indicates. We can draw a diagram to capture this: type(A)

type(B)



If ‘e’ is not a singularity of ‘denotesi ’, then ‘e’ denotesi n iff e ¼ n, where ‘e’ and ‘n’ are denoting expressions. (So in response to JC Beall’s commentary, I endorse the option presented in Sect. 6.5 of the next chapter.)

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Reference and Paradox / 243 Here A and B stand for the first two expressions on the board. Since they contain no context-sensitive terms, we represent them via their types. The representation of P at the second tier reflects the fact that we are after what P denotesi —so the third member of the triple is i. Now the primary representation of P appears again at the second tier. And to determine the denotationi of P, we must again determine the denotations of the expressions on the board. And so we are led to a third tier, and a fourth, and so on: type(A)

type(B) type(A)

type(B) type(A)

type(B)

No branches extend from A and B, since they do not make any references to other denoting phrases. But P is caught in a circle, and the tree extends indefinitely. This is the primary denotation tree for P. This tree has an infinite branch, and P repeats on this branch. The repetition indicates that P is pathological, and a singularity of ‘denotesi ’. The attempt to determine a denotationi for P breaks down. (The treatment of the Berry phrase follows the same general pattern. The primary denotation tree for the Berry phrase contains an infinite branch, indicating the pathology of the Berry phrase. The Berry phrase is identified as a singularity of the occurrence of ‘denotes’ or ‘nameable’ in the phrase itself.) While P is identified as pathological, it is a different story with P*. The primary denotation tree for P* looks like this: type(A)

type(B) type(A)

type(B) type(A)

type(B)

At the top is the primary representation of P*. The tree reflects the way in which we determine a denotationr for P*. The occurrence of ‘denotes’ in P* is

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244 / Keith Simmons given by ‘denotesi ’. So to determine what P* denotesr , we must first determine the denotationi of the phrases to which P* makes reference—and this is indicated by the second tier of the tree. Now the tree has an infinite branch too, but notice that the primary representation of P* does not repeat on this branch. The primary representation of P does repeat, but the primary representation of P* does not. In the formal account, this indicates that while P is pathological, P* is not. Intuitively, P* stands above the circle in which P is caught. Formally, we can take things a little further by pruning this tree. We prune the tree by terminating any infinite branch at the first occurrence of a nonrepeating node. Here is the pruned tree: type(A)

type(B)

The pruned tree indicates that the denotationr of P* depends only on the denotation of A and B. And that’s just what we found: the denotationr of P* is p þ 6. Here then is a glimpse of the formal treatment of the repeating discourse. The treatment of the revisiting discourse should be obvious. When you revisit P at the third stage of the discourse, you bring to bear the reflective r-schema. So at the third stage of the revisiting discourse, we should represent P by the triple , indicating that we are after the denotationr of P. This is not the primary representation of P, but what we will call a secondary representation of P, since the third element stands for a context other than P’s own context of utterance. Starting out with this secondary representation, we can construct the corresponding secondary denotation tree for P. It is easy to see that this secondary tree for P is identical to the primary tree for P*. And as before this tree may be pruned, indicating that P, like P*, denotesr p þ 6.

7. The Object Language and the Language of the Theory Here, then, is a glimpse of the formal theory. Let me now make some more general remarks about it. The theory is a theory about a natural language, English, with particular focus on the term ‘denotes’. So the object language is

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Reference and Paradox / 245 a portion of English including the term ‘denotes’. The singularity theory is a theory of a context-sensitive term, but no context-sensitive term will appear in the language of the theory. So there is a separation between the object language and the language of the theory. So we should ask: what is the relation between the two? For some theories of paradox, the relation is that of object language to metalanguage. Consider, for example, Kripke’s theory of truth (in Kripke 1975). The object language is a certain fragment of English, containing its own truth predicate. This language can express its own concept of truth, in the sense that it contains a predicate whose extension comprises exactly the true sentences of the language. But there are certain ordinary semantic concepts that the object language cannot express—for example, untrue. Now this concept can be expressed in the semantically richer language of the theory. The language of the theory is, in Tarski’s sense, a metalanguage for the object language. The object language is fully translatable into the language of the theory, and the language of the theory can say more besides. This shows the limitations of Kripke’s theory. If the theory is a theory of an object language that cannot express an ordinary semantic concept like ‘untrue’, then we do not really have a theory of truth for English. Natural languages, says Tarski, are universal: ‘‘if we can speak meaningfully about anything at all, we can also speak about it in colloquial language’’.14 In particular, says Tarski, natural languages are semantically universal, in the sense that they can express all of their own semantic concepts—not just truth, but untruth as well. The need to ascend to a hierarchy to express ordinary semantic concepts shows that the scope of Kripke’s theory is restricted. I would also argue that the same kind of ascent is forced upon other theories of truth, such as the revision theory, or those of Feferman and McGee. When the language of the theory is a metalanguage, the scope of the theory is restricted.15 In contrast, the language of the singularity theory is not a metalanguage for the intended object language—namely, the fragment of English containing ‘denotes’. The singularity theory does not attempt to provide a model of this target language. Rather, its job is to identify singularities, by describing the semantic and pragmatic behaviour of the denotation predicate. Here, the object language cannot be translated into the language of the theory: the object language is too rich for that. For one thing, ‘denotes’ is a context-sensitive 14 Tarski (1983: 164).

15 For more along these lines, see Simmons (1993, chs. 3–4).

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246 / Keith Simmons term, and the theory contains no context-sensitive term. For another, the scope of any use of ‘denotes’ is far too wide to be captured by any term of the theory. An ordinary use of ‘denotes’ applies to every denoting phrase which is not identified as a singularity. And that includes all such denoting phrases in the language of the theory itself, and indeed in all other languages, actual and possible.16 Since the scope of each use of ‘denotes’ is as close to global as it can be, it seems to me that the singularity theory goes a long way to accommodate Tarski’s intuitions about semantic universality.

8. Consequences for Deflationism I turn now to the consequences for one kind of deflationism about reference—the disquotational account. According to a disquotational account of denotation or reference, there is no more to denotation that is given by the instances of the following disquotational schema: ‘e’ denotes n if and only e ¼ n, where ‘e’ and ‘n’ are denoting expressions. This leads naturally to the following disquotational definition of denotation for a language L: x denotes n iff or or or or

x ¼ ‘e1 ’ & e1 ¼ n x ¼ ‘e2 ’ & e2 ¼ n ... x ¼ ‘ek ’ & ek ¼ n ...

where ‘e1 ’, ‘e2 ’, . . . , ‘ek ’, . . . are the denoting expressions of L. Suppose we try to maintain both disquotationalism and the singularity account. In particular, consider the disquotational definition for ‘denotesr ’. If we put x ¼ P and n ¼ p þ 6, we obtain: P denotesr p þ 6 if and only if the sum of the numbers denotedi by expressions on the board is p þ 6. 16 Notice that if we regard the language of the theory as a classical, regimented language, it will not contain its own denotation predicate—for that, we would need a richer metalanguage for the language of the theory. Thus there will be denoting phrases involving this denotation predicate that cannot be expressed in the language of the theory. But as long as these phrases are not identified as singularities, they will be in the extension of an ordinary use of ‘denotes’.

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Reference and Paradox / 247 This is a true biconditional. But what about the occurrence of ‘denotesi ’ on the right? This cannot be treated disquotationally. If we plug P into the disquotational definition of ‘denotesi ’, and suppose that P denotesi some number k, we reach the absurdity that k ¼ p þ 6 þ k. So we cannot provide a disquotational account of the occurrence of ‘denotesi ’ in P. If we accept the singularity account, we cannot accept disquotationalism. There are genuine denoting expressions containing uses of ‘denotes’ which the disquotationalist cannot handle—in particular, the use of ‘denotes’ in P cannot be treated disquotationally. And P cannot be dismissed as merely pathological—it is a genuine denoting phrase. In short, denotation cannot be disquoted away.

9. A New Paradox of Definability Finally, I’d like to turn to a new paradox of definability and see how the singularity theory handles it. It is a standard idea that the paradoxes of definability turn on self-reference—the phrase P makes reference to itself, and the Berry phrase includes itself in its scope. Every definability paradox with which I am familiar involves self-reference. But I think we can formulate paradoxes of definability without self-reference. Let E1 , E2 , ... be a list of denoting expressions—it may be finite or infinite. The list might be ‘42 ’, ‘the successor of 2’, ‘England’; or it might be ‘one’, ‘three’, ‘five’, ... ‘thirty-one’, ... . As our examples show, some of these expressions may denote positive integers (i.e. 1 or 2 or 3 or ... ). Let the expression ‘max (E1 , E2 , ...)’ denote the largest positive integer denoted by an expression on the list. If denumerably many distinct positive integers are denoted by expressions on the list (so that there is no largest positive integer denoted), ‘max (E1 , E2 , ...)’ denotes w, the first infinite ordinal; and if no expression on the list denotes a positive integer, ‘max (E1 , E2 , ...)’ denotes 0. So, for example, max(‘42 ’, ‘the successor of 2’, ‘England’) ¼ 16, max(‘London’, ‘Chapel Hill’, ‘Tel Aviv’) ¼ 0, and max(‘one’, ‘three’, ‘five’, ... ‘thirty-one’, ... ) ¼ w. Now consider the following infinite list of denoting expressions: (D1 ) 1 þ max (D2 , D3 , ..., Dn , ... ). (D2 ) 1 þ max (D3 , D4 , ..., Dn , ... ).

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248 / Keith Simmons .. . (Dn ) 1 þ max (Dnþ1 , ... , Dnþi , ...). (D .. nþ1 ) 1 þ max (Dnþ2 , ... , Dnþi , ...). . Notice that there’s no self-reference or circularity here. Each phrase makes reference only to phrases further down the list. Suppose towards a contradiction that for some arbitrary n, Dn denotes a positive integer, say p. Now Dn is given by: (Dn ) 1 þ max (Dnþ1 , ... , Dnþi , ...). Since Dn denotes p, max (Dnþ1 , ..., Dnþi , ...) ¼ p
1. So there is an expression Dk among Dnþ1 , ..., Dnþi , ... which denotes p
1.17 Dk is given by: (Dk ) 1 þ max (Dkþ1 , ..., Dkþi , ... ). Since Dk denotes p
1, max (Dkþ1 , ..., Dkþi , ... ) ¼ p
2. So there is an expression Dl among Dkþ1 , ..., Dkþi , ... which denotes p
2. And so on. Continuing in this way (for p
3 more steps), we obtain an expression Dz which denotes 1. Dz is given by: (Dz ) 1 þ max (Dzþ1 , ... , Dzþi , ... ). Since Dz denotes 1, max (Dzþ1 , ... , Dzþi , ...) ¼ 0. By the definition of ‘max’, none of Dzþ1 , ..., Dzþi , ... denote a positive integer. In particular, (i) Dzþ1 does not denote a positive integer. Now Dzþ1 is given by: (Dzþ1 ) 1 þ max (Dzþ2 , ..., Dzþi , ...). Since none of Dzþ2 , ..., Dzþi , ... denote a positive integer, max (Dzþ2 , ..., Dzþi , ... ) ¼ 0. But then Dzþ1 denotes 1 þ 0. That is, (ii) Dzþ1 denotes a positive integer, namely 1. From (i) and (ii), we obtain a contradiction. 17 There can be only one such expression. Suppose, towards a contradiction, that Dm and Dn each denote p
1, where we may assume without loss of generality that m < n. Then Dm is given by: (Dm ) 1 þ max (Dmþ1 , ...Dn , ...). Since Dm denotes p
1, max (Dmþ1 , ...Dn ,... ) ¼ p
2. But since Dn denotes p
1, max (Dmþ1 , ...Dn ,... )  p
1. So p
2  p
1, and we have a contradiction.

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Reference and Paradox / 249 By our reductio argument, we have shown that no Dn denotes a positive integer, for any n. Every Dn is pathological. So for all n, max (Dn , ..., Dnþi , ... ) ¼ 0. In particular, then, max (D2 , ..., Dn , ...) ¼ 0; max (D3 , ..., Dn , ... ) ¼ 0; and so on. But now reconsider D1 . Since 1 þ max (D2 , ..., Dn , ...) ¼ 1, D1 denotes 1. Similarly, D2 denotes 1, and in general Dn denotes 1. To sum up: no Dn denotes a positive integer, and every Dn denotes a positive integer (namely 1). We are landed in paradox.18 This paradox of definability is obviously more complicated than the simple one generated by P on the board. But the treatment of it is essentially the same. The discourse here has the same overall structure: first, denoting phrases are displayed; second, we reason to the conclusion that they are pathological and fail to denote; and third, we reason past pathology, and find that the phrases do denote. When we first assess D1 , D2 , and the rest as pathological, we assess them by an unreflective denotation schema, analogous to the i-schema in the case of P. We find that all the Dn ’s are pathological and fail to denote. And with the availability of this new information, the common ground shifts, and we move to a reflective context. We now assess D1 , D2 , ... by a new reflective schema, analogous to the r-schema. Just as we reason first that P does not denote and then conclude that it does, so we find first that D1 , and all the others, fail to denote, and then go on to conclude that they do. There is no contradiction here. Rather there is a shift in the denotation schemas by which we assess D1 , D2 , ..., triggered by the shift to a reflective context. Let I be the initial context in which the Dn ’s are first produced. And let R be the reflective context in which we assess the Dn ’s on the basis of their pathology. Then the primary tree for D1 looks like this:



This tree clearly contains infinite branches, and the formal theory will deliver the result that D1 is pathological, and a singularity of denotesI . And similarly for D2 , D3 , .... In our reasoning, we revisit D1 and reassess it via the reflective R-schema. This is captured by the following secondary tree for D1 : 18 This paradox is a companion to Yablo’s paradox about truth; see Yablo (1993).

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250 / Keith Simmons



Notice that the second tier is composed of primary representations of D2 , D3 , . . . , each of which head an unfounded primary tree. Setting aside the details of the formal theory, the idea is that this tree indicates that the denotationR of D1 is to be determined in the light of the pathology of D2 , D3 , . . . : This is just what we do at the reflective stage of our reasoning. Having determined that D2 , D3 , . . . fail to denote, we calculate that max (D2 , D3 , . . . ) ¼ 0, and hence that 1 þ max (D2 , D3 , . . . ) ¼ 1, concluding that D1 denotes 1.

10. Concluding Remarks I have focused here on denotation. But I think we can take the same kind of approach to the other fundamental semantic relations—between a predicate and its extension, and a sentence and its truth value. Parallel to our denotation discourse, there are discourses related to Russell’s paradox and the Liar, and there too, I believe, we can provide a singularity account. But that, obviously, is a topic for another day.

REFERENCES C h a f e , W. (1976), ‘Givenness, Contrastiveness, Definiteness, Subjects, Topics, and Point of View’, in C. Li (ed.), Subject and Topic (New York: Academic Press). F e f e r m a n , S. (1984), ‘Towards Useful Type-Free Theories, I’, Journal of Symbolic Logic, 49: 75–111. G l a n z b e r g , M. (2001), ‘The Liar in Context’, Philosophical Studies, 103: 217–51. G o¨ d e l , K. (1944), ‘Russell’s Mathematical Logic’, in Schilpp (1944: 123–53). G r o s z , B., and C. Sidner (1986), ‘Attention, Intention, and the Structure of Discourse’, Computational Linguistics, 12: 175–204. H e i m , I. (1988), The Semantics of Definite and Indefinite Noun Phrases (New York: Garland).

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Reference and Paradox / 251 I s a r d , S. (1975), ‘Changing the Context’, in Keenan (1975). K a m p , H. (1981), ‘A Theory of Truth and Semantic Representation’, in J. Groenendijk et al. (eds.), Truth, Interpretation and Information (Dordrecht: Foris). K e e n a n , E. (ed.) (1975), Formal Semantics of Natural Language (Cambridge: Cambridge University Press). K o¨ n i g , J. (1905), ‘On the Foundations of Set Theory and the Continuum Problem’, in van Heijenoort (1967: 145–9). K r i p k e , S. (1975), ‘Outline of a Theory of Truth’, Journal of Philosophy, 72: 690–716. L e w i s , D. (1983a), ‘Scorekeeping in a Language Game’, in Lewis (1983: 233–49); first pub. in Journal of Philosophical Logic, 8: 339–59. —— (1983b), Philosophical Papers, vol. i (Oxford: Oxford University Press). M c G e e , V. (1990), Truth, Vagueness, and Paradox (Indianapolis: Hackett). P e a n o , G. (1906), ‘Super theorema de Cantor-Bernstein et additione’, Revista de Mathematica, 8: 136–57; repr. in Opere scelte (Rome: Edizione Cremonese, 1957), vol. i. P o i n c a r e´ , H. (1906), ‘Les Mathe´matiques et la logique’, Revue de Me´taphysique et de Morale, 14: 294–317. —— (1909), ‘La Logique de l’infini’, Revue de Me´taphysique et de Morale, 17; repr. in Dernie`res Pense´es (Paris: Flammarion, 1913); Eng. trans. in Poincare´ (1963: 45–64). —— (1963), Mathematics and Science: Last Essays (New York: Dover). R e i n h a r t , T. (1981), ‘Pragmatics and Linguistics: An Analysis of Sentence Topics’, Philosophica 27/1: 53–94. R i c h a r d , Jules (1905), ‘Les principes des mathe´matiques et le proble`me des ensembles’, Revue Ge´ne´rale des Sciences Pures et Applique´es, 16: 541; also in Acta Mathematica, 30 (1906), 295–6; Eng. trans. in van Heijenoort (1967: 143–4). —— (1907), ‘Sur un paradoxe de la the´orie des ensembles et sur l’axiome Zermelo’, L’Enseignement Mathe´matique, 9: 94–8. R u s s e l l , B. (1908), ‘Mathematical Logic as Based on the Theory of Types’, American Journal of Mathematics, 30: 222–62; repr. in van Heijenoort (1967: 150–82). S c h i l p p , P. A. (ed.) (1944), The Philosophy of Bertrand Russell (LaSalle, Ill.: Open Court). S i m m o n s , K. (1993), Universality and the Liar: An Essay on Truth and the Diagonal Argument (New York: Cambridge University Press). —— (1994), ‘A Paradox of Definability: Richard’s and Poincare´’s Ways Out’, History and Philosophy of Logic, 15: 33–44. S t a l n a k e r , R. C. (1999a), Context and Content: Essays on Intentionality in Speech and in Thought (Oxford: Oxford University Press). —— (1999b), ‘Indicative Conditionals’, in Stalnaker (1999a: 63–77); first pub. in Philosophia, 5 (1975), 269–86.

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252 / Keith Simmons T a r s k i , A. (1983), ‘The Concept of Truth in Formalized Languages’, in Tarski, Logic, Semantics, Metamathematics, ed. John Corcoran, 2nd edn. (Indianapolis: Hackett). v a n H e i j e n o o r t , J e a n (ed.) (1967), From Frege to Go¨del: A Source Book in Mathematical Logic (Cambridge, Mass.: Harvard University Press). Y a b l o , S. (1993), ‘Paradox without Self-Reference’, Analysis, 53: 251–2.

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12 On the Singularity Theory of Denotation JC Beall

1. Introduction What do denotation paradoxes teach us about English? According to Keith Simmons (Chapter 11 in this volume) they teach us that ‘denotes’ is contextually sensitive, that its extension may change from context to context. As with ‘I’ or ‘here’, the meaning-constitutive principle governing ‘denotes’ is parametric; it has a parameter for context. By recognizing the parametric character of ‘denotes’ we thereby resolve the denotation paradoxes—or so Simmons argues. Parametric responses to paradox are by now common, due in no small part to Tarski [6]. Tarski did not try to solve the paradoxes; he tried to avoid inconsistency; and he did so by positing a parametric character of truth—‘is true’ is indexed to different languages. Subsequent writers [1, 2] have not so much replaced Tarski’s parameter as they have relocated it: while Tarski suggested that the parameter is a language, others have taken the parameter to be a context. Simmons’s singularity theory falls within the latter group. Positing hidden parameters, of course, is not a strategy peculiar to semantic paradox; some philosophers [5] have argued that soritical paradox (paradoxes arising from vague predicates) call for hidden parameters. Indeed, the common parametric response to vague predicates is a contextualist one: vague predicates are contextually sensitive. Having a unified solution to both the semantic and soritical paradoxes is attractive, and as such Simmons’s

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254 / JC Beall solution, if it is independently satisfactory, may well lend itself to a longsought unified approach to paradox. The question is whether Simmons’s solution is satisfactory. In this paper I raise a general problem for Simmons’s position. The problem, in effect, may be cast as a dilemma: Simmons’s solution is either inconsistent or unwarranted.

2. Simmons’s Example Simmons’s example may be formulated thus:1 p p p 6 p the sum of numbers denoted by ticked expressions in Section 2. Simmons identifies two patterns of reasoning about the ticked expressions, the repeat and revisit patterns, each having two (or so) stages.

2.1 The Repeat Pattern The repeat pattern is this: (1) ‘p’ denotes p, and ‘6’ denotes 6. What about the third ticked expression (call it ‘P’)? Suppose P denotes the number k. Now, P denotes k iff the sum of the numbers denoted by ticked expressions in Section 2 is k. So, the sum of the numbers denoted by ticked expressions in Section 2 is k. But, then, k ¼ p þ 6 þ k, which is impossible. So, P is non-denoting. (2) Given that P is non-denoting, the only denoting ticked expressions in Section 2 are ‘p’ and ‘6’, in which case the sum of numbers denoted by ticked expressions in Section 2 is p þ 6. But, now, we see that a token of P’s type—call the token ‘P*’—is a denoting expression.

2.2 The Revisit Pattern An example of the revisit pattern replaces (2) above by 1 This isn’t quite Simmons’s example; it hides what he identifies as the first change of context (where one discovers that circularity is involved). Nothing hinges on this.

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The Singularity Theory of Denotation / 255 (2’) Given that P is non-denoting, the only denoting ticked expressions in Section 2 are ‘p’ and ‘6’. As the third ticked expression makes reference only to denoting ticked expressions, we have it that it refers only to ‘p’ and ‘6’, the denotations of which sum to p þ 6, which is the denotation of P.

2.3 Analyzing the Patterns Each of the two patterns, according to Simmons, is natural and intuitively valid; indeed, Simmons takes both to be data that any theory of denotation must explain. The key feature of each pattern is twofold: We ‘‘reason past pathology’’ (as Simmons says). We conclude that two tokens of the same (expression) type differ wrt denotation: one denotes while the other doesn’t. Simmons’s driving question, then, is: Given two tokens of the same type, how can one fail to refer and the other succeed? According to Simmons, ‘‘the explanation must be a pragmatic one, at least in the sense that the notion of context will play a crucial role’’ (p. 233 above). In particular, Simmons takes ‘denotes’ to be contextually sensitive, shifting its extension from context to context.

2.4 Simmons’s Solution With respect to the ticked expressions in section 2, the crucial change of context is from (1) to (2); in (1) ‘denotes’ is evaluated according to one denotation principle while (2) requires a different denotation principle. Specifically, where i marks the ‘‘initial context’’ and r the ‘‘reflective context’’ (in the given case, these are in effect (1) and (2), respectively), ‘denotesi ’ in the ticked expression is indexed at i and, given various pragmatic principles, the expression is governed by the corresponding i-principle: (Di ) e denotesi x iff e ¼ x. When in context i we discover that P is nondenotingi we retain this information in our move to the next context—or, more accurately, this new information forces a change in context. We move to context r in which, while ‘denotes’ in the ticked expression retains its ‘‘initial’’ subscript, the ticked expression is now governed by the r-principle:

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256 / JC Beall (Dr ) e denotesr x iff e ¼ x In this context, we do not get the apparent contradiction. Instead, we conclude that while P does not denotei , it does denoter (as does the ‘‘new’’ token P*).

3. What is a Context? A crucial question for any such response is: What is a context? While he doesn’t say a lot to answer this question Simmons does endorse Stalnaker’s (similarly Lewis’s) idea that a context should be identified with the body of information that is presumed to be available to the participants in the speech situation—the ‘‘common ground’’, to use a phrase from Grice, or the shared presuppositions of the participants. As new utterances are produced, and new information is made available, the context changes. To keep track of these context-changes, then, we need a running record of the information provided in the course of the conversation. (pp. 233–4 above)

What is important for Simmons’s purposes is that a change in the ‘‘common ground’’ changes the context and vice versa.2 A shift in context occurs if the set of presuppositions (common ground) changes; and a change of context can affect content, especially when ‘‘singularities’’ are involved.

4. Common Knowledge? I am inclined to think that contexts, so understood, have to contain general principles governing our reasoning concerning certain terms—including terms like ‘denotes’. But, then, if a contextual solution is pursued (with contexts so understood), then we ought to have principles in contexts that quantify over contexts. But this leads to familiar paradox, and so Simmons (like any other contextualist, as far as I know) requires that such quantification over contexts—or, at least, the general principles governing ‘denotes’— 2 There is a nagging question here. Simmons (and others) talk about two things—the ‘‘common ground’’ (set of available presuppositions) and the ‘‘speech situation’’ to which the ‘‘common ground’’ is relative. The question is: How is a given speech situation to be individuated? I will ignore this, although I think that it is something that Simmons needs to fill out.

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The Singularity Theory of Denotation / 257 remain outside of contexts. By my lights, this makes a mystery out of contexts; they certainly aren’t the ‘‘common ground’’ they’re said to be. But I won’t dwell on this further. Instead, I will focus on an apparent dilemma confronting Simmons’s position.

5. Contextual Collapse? The problem arises by concentrating on the platitudes governing the (allegedly) contextually sensitive ‘denotes’, specifically, the various Dk principles. Simmons’s chief conclusion concerning his term P is twofold: (C1) P does not denotei p þ 6. (C2) P does denoter p þ 6. The reasoning used towards these claims, as Simmons emphasizes, essentially invokes the corresponding i- and r-denotation schemas (Section 2.4). Perhaps more accurately, the reasoning towards C1 and C2 essentially invokes the P-instances of the given schemas:3 (DPi ) P denotesi p þ 6 iff [P] ¼ p þ 6. (DPr ) P denotesr p þ 6 iff [P] ¼ p þ 6. But, now, with DPi and DPr displayed together, an obvious problem emerges. I will put the problem in terms of a dilemma.

5.1 Dilemma: Denotation Principles Either we have DPi and DPr or not. Suppose the latter. Then there is no apparent justification for either C1 or C2, in which case Simmons’s main claims are unwarranted.4 Suppose the former. Then by transitivity of ‘iff ’ we immediately get a collapse of the various contexts: (C3) P denotesi p þ 6 iff P denotesr p þ 6.

3 I use ‘P’ as Simmons does, and so use ‘[P]’ as an abbreviation for P (or, what comes to the same, a disquotation device). 4 I won’t go on to argue for this horn. Suffice to say that Simmons’s only explicit justification for C1 and C2 invokes DPi and DPr ; and I cannot think of another way of supporting C1 and C2 without the given denotation principles (or some version of them, as I’ll discuss below).

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258 / JC Beall But C3 , together with C1 and C2 , quickly yields a contradiction. The upshot is that Simmons’s position is either unwarranted or inconsistent.

6. Options As I see it, there are various options open to Simmons. I will briefly mention each option, concentrating mostly on the last (namely, restriction), as it is likely to be most attractive to Simmons’s classical bent. The key question to bear in mind is whether Simmons’s parametric approach to denotation is worth the cost of the given option.

6.1 Reject Detachment The problem (or, at least, the inconsistency) arises only if ‘iff ’ is underwritten by a detachable conditional. One option is to rely on a non-detachable conditional. There are certainly non-detachable conditionals in English;5 however, there is also an apparently detachable conditional—namely, entailment. Of course, whether entailment is expressed by any conditional in English is a controversial issue, and so the case remains open. Still, irrespective of how such debates go, it is clear that at least Simmons himself takes the various biconditionals to be underwritten by a detachable conditional; for Simmons relies on modus ponens at various steps involving the given Dk -principles.

6.2 Reject Transitivity Another closely related assumption behind the apparent problem is that ‘iff ’ in the Di is transitive. Hence, rejecting the given transitivity is an option. One worry is whether this move will immediately require non-transitive consequence; it will, especially if consequence is defined in terms of the target (non-transitive) conditional.

6.3 Contextual Identity One might posit parametric features beyond denotation; in particular, one might take identity to be parametric. For example, the problem at hand does not arise if each of the Dk is formulated 5 In fact, I think that the material ‘‘conditional’’ is one such, and McGee [4] and Lycan [3] have (allegedly) identified others.

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The Singularity Theory of Denotation / 259 e denotesk x iff e ¼k x. The idea is that ‘¼’ itself is contextually sensitive. While I think that this is interesting and may well afford solutions (or dissolutions) of other philosophical problems,6 there remains a glaring question: What does ‘‘contextual identity’’ amount to? Presumably, the only way to get identity to be contextually sensitive is to ‘‘contextualize’’ some (all?) of the elements involved in defining identity. I think that it is interesting to speculate about what a contextual approach to predication might look like; however, this is not the place. In any event, one thing is clear: that ‘‘contextualizing’’ identity may call for rethinking much of mathematics.

6.4 Contextual Singular Terms Instead of treating ‘¼’ as parametric one might avoid the noted collapse by taking every singular term to be parametric; that is, one might suggest ‘¼’ is always flanked by parametric terms. The meaning-constitutive principles governing (contextually sensitive) ‘denotes’ ought to be understood thus: P denotesi (p þ 6)i iff [Pi ] ¼ (p þ 6)i . P denotesr (p þ 6)r iff [Pr ] ¼ (p þ 6)r . So given, the denotation principles do not yield the troubling collapse (or resulting inconsistency). I am not sure whether such a theory—that posits a parametric feature of every singular term—is viable. To be sure, some singular terms (indexicals, etc.) are already parametric; however, they are also standardly taken to be ‘‘special’’ singular terms (precisely in virtue of being parametric). One immediate question arises: What of ‘‘cross-context identity’’?7 Presumably, in order to avoid the troubling collapse, the current proposal will have to rule out cross-context identity, rule out having xi ¼ xj for distinct contexts i and j. Otherwise, there is no apparent (and non-ad hoc) reason to rule out the cross-context identities at issue in the given denotation principles. The trouble is that without further explanation it remains implausible to think that each context-shift involves a shift in agents—that 6 One such problem is the statue–hunk puzzle, where the question is whether our statue is ‘‘identical’’ to the hunk of marble that constitutes it. Let ‘David’ name our statue and ‘Hunkel’ our hunk. If identity is contextual, then a plausible response (piggy-backing on contextual denotation) is that in some contexts David ¼i Hunkel while in others, not. Maybe. 7 Recall the once-hot debate about cross-world identity, arising from taking possible worlds semantics seriously (too seriously?).

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260 / JC Beall during different stages of a conversation the participants themselves are not constant throughout the stages.

6.5 Restrict the Denotation Principles The most likely reply, I think, is to impose some sort of restriction on the various denotation principles. The most natural move is to invoke Simmons’s notion of ‘‘singularities’’ and go with conditionalized versions of the denotation principles: (CDPi ) If P is not a singularity of ‘denotesi ’, then P denotesi p þ6 iff [P] ¼ p þ 6. (CDPr ) If P is not a singularity of ‘denotesr ’, then P denotesr p þ6 iff [P] ¼ p þ 6. Such restriction seems to avoid the problematic collapse (and, hence, the inconsistency horn of the dilemma); however, the restriction likewise seems to raise further problems for Simmons’s overall position—landing him squarely on the unwarranted horn discussed above.

7. A Problem with the Restriction Route The suggestion, as above, is that the fundamental meaning-constitutive principles of (our contextually sensitive) ‘denotes’ are restricted as per CDPi and CDPi . Assume that we have independent reason to accept that P is not a singularity of ‘denotesr ’.8 Then by CDPi we have it that P denotesr p þ 6 iff [P] ¼ p þ 6. Since [P] ¼ p þ 6, we may conclude C2 . No problem there; C2 is easily justified by the restricted Dr principle. The question (and the chief problem) concerns C1 . Recall that Simmons’s chief conclusion is the conjunction of C2 and C1 . But what is the import of Simmons’s chief claim C1 —that P does not denotei p þ 6? Notice that CDPi gives us no clue here—no clue as to the import (the consequences) of C1 . That is the problem. According to Simmons we ought to conclude that P does not denotei p þ 6. What are the consequences of that conclusion? Normally, 8 Presumably, this independent evidence comes from Simmons’s formal theory, which aims to specify singularities of ‘denotes’.

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The Singularity Theory of Denotation / 261 one would think that if (for some expression) e does not denote x, then e 6¼ x. But that will not follow here. There is no warrant for saying as much; the only apparent warrant for such a conclusion invokes the meaning-constitutive principle governing ‘denotesi ’, but that principle affords warrant only if we can detach its (biconditional) consequent; but we cannot do that if P is a singularity of ‘denotei ’. The upshot is that Simmons’s chief claim C1 is without content (since we don’t know the consequences of the claim) or the claim is inconsistent with the given fact [P] ¼ p þ 6.

8. Closing Remarks In this paper I have argued that Simmons’s parametric (contextual) theory of denotation is either inconsistent or unwarranted. I have also waved at different avenues that Simmons might pursue in an effort to avoid the inconsistency—for example, a ‘‘deviant’’ logic of conditionals, or a parametric identity relation, and so on. Along one (or more) of those lines, Simmons’s theory may well turn out to be a viable approach to denotation.9

REFERENCES [1] B a r w i s e , J o n , and J o h n E t c h e m e n d y (1987), The Liar: An Essay in Truth and Circularity (Oxford: Oxford University Press). [2] B u r g e , T. (1979), ‘Semantical Paradox’, Journal of Philosophy, 76: 169–98. [3] L y c a n , W i l l i a m G. (2001), Real Conditionals (Oxford: Oxford University Press). [4] M c G e e , V a n n (1985), ‘A Counterexample to Modus Ponens’, Journal of Philosophy, 82: 462–71. [5] S o a m e s , S c o t t , (1999), Understanding Truth (New York: Oxford University Press). [6] T a r s k i , A l f r e d (1956), Logic, Semantics, Metamathematics: Papers from 1923 to 1938, trans. J. H. Woodgerx (Oxford: Clarendon Press).

9 I am grateful to Graham Priest, Hartry Field, Michael Glanzberg, and Keith Simmons for discussion.

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13 The Semantic Paradoxes and the Paradoxes of Vagueness Hartry Field

Both in dealing with the semantic paradoxes and in dealing with vagueness and indeterminacy, there is some temptation to weaken classical logic: in particular, to restrict the law of excluded middle. The reasons for doing this are somewhat different in the two cases. In the case of the semantic paradoxes, a weakening of classical logic (presumably involving a restriction of excluded middle) is required if we are to preserve the naive theory of truth without inconsistency. In the case of vagueness and indeterminancy there is no worry about inconsistency; but a central intuition is that we must reject the factual status of certain sentences, and it hard to see how we can do that while claiming that the law of excluded middle applies to those sentences. So despite the different routes, we have a similar conclusion in the two cases. There is also some temptation to connect up the two cases, by viewing the semantic paradoxes as due to something akin to vagueness or indeterminacy in semantic concepts like ‘true’. The thought is that the notion of truth is introduced by a schema that might initially appear to settle its extension uniquely: the schema (T) True (hpi) if and only if p Thanks to Steve Yablo for very useful comments at the conference, and to Graham Priest for a helpful critique of my discussion of revenge problems that led to a substantial elaboration. The line on vagueness that I take here grew out of a sequence of unsatisfactory attempts to elaborate on the classical logic account offered in ch. 10 of [5]. I’m indebted to more people than I can name for their comments and criticisms of those earlier attempts, but in addition to those mentioned in note 19, I’d like to single out Joshua Schechter for detailed and helpful comments at several stages along the way.

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Semantic Paradoxes and Vagueness Paradoxes / 263 (where ‘p’ is to be replaced by a sentence and ‘hpi’ by a structural-descriptive name of that sentence). But in fact, this schema settles the extension uniquely only as applied to ‘‘grounded’’ sentences; whether a given ‘‘ungrounded’’ sentence is in the extension of ‘true’ will be either underdetermined or overdetermined (determined in contrary ways). And this looks rather like what happens in cases of vagueness and indeterminacy: our practices with a vague term like ‘bald’, or a term like ‘heaviness’ (which in the mouths of many is indeterminate between standing for mass and standing for weight), don’t appear to uniquely settle the reference or extension of the term. (There isn’t such a clear distinction between underdetermination and overdetermination in these cases: e.g. it may seem underdetermined whether Harry is bald in that Harry isn’t a paradigm case of either baldness or non-baldness, but it may seem overdetermined in that Sorites reasoning may be used to argue that he is bald and also to argue that he is not bald. Similarly, a theorist who makes no distinction between mass and weight hasn’t decided which one ‘heaviness’ stands for, so it may seem underdetermined; but he may attribute to ‘‘heaviness’’ both features true only of mass and features true only of weight, so it may seem overdetermined.) In this paper I will argue that an adequate treatment of each of the two phenomena (the semantic paradoxes and vagueness–indeterminacy) requires a nonclassical logic with certain features—features that are roughly the same for each of the two phenomena. This suggests that there might be a common logical framework, and I will propose a framework that seems adequate for treating both. The core logic for the unified framework is sketched in Section 5 (which is rather technical), though much of the discussion before and after (e.g. the discussion of rejection in Section 3 and the discussion of defectiveness in later sections) is highly relevant to the unified treatment. Sections 1, 2, 7, and 8 are primarily concerned with the semantic paradoxes, and Sections 3, 4, 6, and 9 deal primarily with vagueness and indeterminacy (especially in the case of 3 and 4), but there are close interconnections throughout.

1. The Semantic Paradoxes and Attempts to Resolve them in Classical Logic In this section and the next I will discuss the naive theory of truth and some of the semantic paradoxes which threaten to undermine it. As I’ve noted, the naive theory of truth includes all instances of the schema (T) above. Perhaps more centrally, it includes the principle that ‘True(hpi)’ and ‘p’ are always

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264 / Hartry Field intersubstitutable. (Of course I’m restricting to languages without quotation marks, intentional contexts, and so forth; also without ambiguity, and where there are no relevant shifts of context. Although it may be unnecessary, we can exclude denotationless terms as well.) In classical logic, the schema implies the intersubstitutivity and conversely; whether either direction of the implication holds in a nonclassical logic depends on the details of that logic, though I think that there is reason to prefer logics that are ‘‘classical enough’’ for both directions of the implication to hold. In the context of a very minimal syntactic theory that allows for selfreference, the naive theory of truth is inconsistent in classical logic: we can construct a sentence Q0 (‘‘the Liar’’) that is provably equivalent to :TrueðhQ0 iÞ, so in classical logic we can derive the negation of an instance of (T). It is totally unpromising to blame the problem on the syntactic theory: among other reasons, there are very similar paradoxes in the naive theory of satisfaction which don’t require syntactic premises. The real choice is, do we restrict classical logic or restrict the truth schema (and hence its classical equivalent, the intersubstitutivity of ‘True (hpi)’ with ‘p’). The attractions of keeping classical logic sacrosanct are powerful, so let’s look first at the prospects of a satisfactory weakening of the naive theory of truth within classical logic. By a satisfactory weakening, I mean one that serves the purposes that the notion of truth is supposed to serve, e.g. as a device of making and using generalizations that would be difficult or impossible to make without it. I think it is pretty clear that any weakening of the naive theory of truth will adversely affect the ability of ‘True’ to serve these purposes: the intersubstitutivity of ‘True (hpi)’ with ‘p’ is very central to the purposes the truth predicate serves. Still, if there were a sufficiently powerful but consistent classical-logic substitute for the naive theory, we might be able to learn to live with it. In fact, though, I think that restoring consistency requires massive revisions in ordinary principles about truth, revisions that would be very hard to live with. I won’t try to fully establish this here, but will make a few observations that give some evidence for it. To begin with an obvious point, the problem in classical logic isn’t simply that we can’t assert all instances of schema (T), it is that there are instances (such as the one involving Q0 ) that we can disprove; and in classical logic, that’s equivalent to proving the disjunction of (1) p ^ :True(hpi) and

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Semantic Paradoxes and Vagueness Paradoxes / 265 (2) True (hpi) ^ :p for certain specific p. Now, it would seem manifestly unsatisfactory to have a theory that proves an instance of (1): how can we assert p and then in the same breath assert that what we’ve just asserted isn’t true? But it also seems manifestly unsatisfactory to have a theory that proves an instance of (2): once we’ve asserted True(hpi), we’re surely licensed to conclude p, so going on to assert :p just seems inconsistent. We seem to have, then, that it would be manifestly absurd to have a theory that either proves an instance of (1) or proves an instance of (2). Of course, a classical theory doesn’t have to do either: it must prove the disjunction (given that it meets the minimal requirements on allowing self-reference),1 but it can remain silent on which disjunct to assert. But remaining silent doesn’t seem a satisfactory way to resolve a problem: if you have committed yourself to a disjunction of thoroughly unsatisfactory alternatives, it would seem you’re already in trouble, even if you refuse to settle on which of these unsatisfactory alternatives to embrace.2 I will not further discuss the option of biting the bullet in favor of (2), or the option of accepting the disjunction of (1) and (2) while remaining artfully silent about which disjunct to accept.3 But I’ll say a bit more about the option of biting the bullet in favor of (1). A superficially appealing way to bite the bullet for option (1) is to say that schema (T) should be weakened to the following: (T
) If True(hpi) or True(h:pi), then True(hpi) if and only if p. (Proponents of this often introduce the term ‘expresses a proposition’, and say that hpi’s expressing a proposition suffices for the consequent of (T
) to hold and that the antecedent of (T
) suffices for hpi to express a proposition.) It is easily seen that (T
) is equivalent to the simpler schema 1 As mentioned above, we wouldn’t need even these minimal requirements if we focused on satisfaction rather than truth, and used the heterologicality paradox. 2 Note that the situation is far worse than for supervaluationist accounts of vagueness. Such accounts allow commitment to disjunctions where we think it would be a mistake to commit to either disjunct. But there the only problem with choosing one disjunct over the other is that the choice seems quite arbitrary; the disjuncts are not thoroughly unacceptable, as they seem to be in the case of the paradoxes. 3 My own dissatisfaction with the ‘‘artful silence’’ option is not entirely due to the general consideration just raised, but also to the fact that a consistent view of this type must exclude so many natural principles that are crucial for the purposes that the truth predicate serves. See [11] and [18] for some important limitations on such theories.

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266 / Hartry Field (T

) If True(hpi) then p.4 Obviously these equivalent schemas can’t be anything like complete theories of truth: for they are compatible with nothing being true, or with no sentence that begins with the letter ‘B’ being true, or innumerably many similar absurdities. To get a satisfactory theory of truth that included one of these schemas, one would have to add a substantial body of partial converses of (T

); and one would presumably also want principles such as (TPMP) True (hpi) ^ True (hp  qi)  True (hqi), or better, the generalized form of this (that whenever a conditional and its antecedent are true, so is the consequent: that is, that modus ponens is truthpreserving).5 But whatever the details of the supplementation, theories based on (T
) or its equivalent (T

) are prima facie unappealing because they require a great many instances of (1). Obviously the Liar sentence Q0 is one example: since we can prove that Q0  :True(hQ0 i), (T

) yields both Q0 and :TrueðhQ0 iÞ. But in addition, Montague [19] pointed out that (T

) plus (TPMP) plus the very minimal assumption that all theorems of quantification theory are true yields a proof of the untruth of some instances of (T

): that is, there is a sentence M (a slight variant of Q0 ) such that we can prove :True(hIf True(hMi) then Mi).6 4 (T
) implies (T

): Suppose True(hpi); then by (T
), True(hpi)  p, which with True(hpi) yields p; so we have True(hpi)  p. (T

) implies (T
): This requires two instances of (T

), both (i) True(hpi)  p and (ii) True(h:pi)  :p. Suppose True(hpi); then by (i), p; so True(hpi)  p. Alternatively, suppose True(h:pi); then by (ii), :p, and by (i), :True(hpi); so again True(hpi)  p. So True(hpi) _ True(h:pi)  (True(hpi)  p). 5 Equivalently (given a minimal assumption about how elimination of double-negation leaves truth unaffected), True(hp _ qi) ^ False(hpi)  True(hqi), or the generalized form of that; where ‘False(hpi)’ means ‘True(h:pi)’. 6 Let R be the conjunction of the axioms of Robinson arithmetic (which is adequate to construct self-referential sentences). Standard techniques of self-reference allow the construction of a sentence N that is provably equivalent, in Robinson arithmetic, to True(hR  :Ni); M will be R  :N. Since this is provable in Robinson arithmetic, then R  [N  True(hR  :Ni)] is a theorem of quantification theory, hence so is its quantificational consequence [True(hR  :Ni)  (R  :N)]  (R  :N); so the claim that that is True is part of the truth theory, and that together with (TPMP) yields True[hTrue(hR  :Ni)  (R  :N)i]  True(hR  :Ni). So if our truth theory proves the negation of the consequent, it proves the negation of the antecedent, which is

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Semantic Paradoxes and Vagueness Paradoxes / 267 It seems highly unsatisfactory to put forward a theory of truth that includes (T

), and use it to conclude that some instances of (T

) (including specific instances that you can identify) aren’t true. It is sometimes thought that one can improve this situation by postulating a hierarchy of ever more inclusive truth predicates Trues , and for each one adopting (T

s), i.e. the analog of (T

) but with ‘Trues ’ in place of ‘True’. (The subscripts are notations for ordinals; the idea is that there will be truth predicates for each member of an initial segment of the ordinals, with no largest s for which there is a truth predicate. There is no notion of truths for variable s.) For any ordinal s for which we have such a predicate, we will be able to derive :Trues (hIf Trues (hMs i) then Ms i) for a certain sentence Ms that contains ‘Trues ’; but we will also be able to assert Truesþ1 (hIf Trues (hMs i) then Ms i), and this is sometimes thought to ameliorate the situation. But even if it does, the cost is high. I can’t discuss this fully, but will confine myself to a single example. Suppose I tentatively put forward a ‘‘theory of truth’’—more accurately, a theory of the various truths s—that includes all instances of (T

s) for each of the truth predicates, together with general principles such as (TPMPs ) for each of the truth predicates, and various partial consequences of each of the (T

s). Someone then tells me that my theory has an implausible consequence; I can’t quite follow all the details of his complicated reasoning, but he’s a very competent logician and the general strategy he describes for deducing the implausible consequence seems as if it should work, so I come to think he’s probably right. Since the consequence still seems implausible, it is natural to conjecture that my theory of truth is wrong—or at least, to consider the possibility that it is wrong and discuss the consequences of that. It is natural to do this even if I have no idea where it might be wrong. But I can’t conjecture this, or discuss the consequences of it, since I have no sufficiently inclusive truth predicate. (‘Wrong’ means ‘not true’.)7 the desired negation of the attribution of truth to an instance of (T

). It remains only to show that a proof theory with (T

) and arithmetic does prove :True(hR  :Ni), but that’s easy: from (T

) we get True(hR  :Ni)  (R  :N), which given arithmetic yields True(hR  :Ni)  :N; but since N is provably equivalent to True(hR  :Ni), this yields :True(hR  :Ni). 7 If the theory were finitely axiomatized I could avoid the use of a truth predicate, but it isn’t: that’s guaranteed by the need of a separate instance of (T

s) for arbitrarily high s (or rather, for arbitrarily high s such that ‘trues ’ is defined).

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268 / Hartry Field And a more specific conjecture, that my theory isn’t trues for some specific s, won’t do the trick. For one thing, I already know for each of my truth predicates trues that not all of the assertions of my theory are trues ; after all, it was because I knew that certain instances of (T

s) couldn’t be trues that I was led to introduce the notion of truthsþ1 . Might I get around that problem by finding a way of specifying for each sentence A of my theory of truth a sA such that A will be ‘‘truesA if it is true at all’’ (if you’ll pardon the use of an unsubscripted truth predicate)? I doubt that one can find a way to specify such a sA for each A: the fact that many principles of a decent truth theory contain quantifiers that range over arbitrary sentences and hence sentences that include arbitrarily high truths predicates gives serious reasons for doubt. But even if one can do that—indeed, even if one can specify a function f mapping each sentence A of the theory into the corresponding sA —it wouldn’t fully get around the problem. For it could well be that for each s, I would be confident that all members of {Aj f (A) ¼ s} are trues ; a worry that there is a s such that not all members of {Aj f (A) ¼ s} are trues does not entail a worry for any specific s. It is the more general worry, that there is a s such that not all members of {Ajf (A) ¼ s} are trues , that my story motivates; but that more general worry is unintelligible according to the hierarchical theory, for in treating quantification over the ordinal subscripts as intelligible it violates the principles of the hierarchy. There is much more that could be said about these matters, but I hope I’ve said enough to make it attractive to explore an option that weakens classical logic.

2. Semantic Paradoxes: A Nonclassical Approach The most famous nonclassical resolution of the paradoxes, due to Kripke [14], employs a logic K3 that can be read off the strong Kleene truth tables. More exactly, suppose we assign each sentence A a semantic value kAk of either 1, 0, or 12, with the assignment governed by the following rules: kA ^ Bk is min{kAk, kBk} kA _ Bk is max{kAk, kBk} k:Ak is 1  kAk kA  Bk is k:A _ Bk, hence max{1  kAk, kBk}.

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Semantic Paradoxes and Vagueness Paradoxes / 269 (Think of 1 as the ‘‘best’’ value, 0 as the ‘‘worst’’, and 12 as ‘‘intermediate’’. It may seem more philosophically natural to avoid assigning the value 12, and to instead regard certain sentences as simply having no value assigned to them; but obviously these two styles of formulation are intertranslatable, and the formulation that uses the value 12 allows for a more compact presentation at several points.) Assuming everything to have a name, as I will for simplicity,8 we also set k8xAk ¼ min{kA(x=c)k} and k9xAk ¼ max{kA(x=c)k}. Then Kripke shows that if we start with the language of arithmetic or some other language adequate to syntax, and any arithmetically standard model M for it that is evaluated by these rules, then we can extend the model by designating a subset of M as the extension of ‘True’ (leaving the ontology and the extension of the other predicates alone) in such a way that for every sentence A, kTrue(hAi)k will be the same as kAk (and where only objects that satisfy ‘Sentence’ satisfy ‘True’). More generally, if sentences B and C are alike except that some occurrences of A in one of them are replaced by True (hAi) in the other, then kBk will be the same as kCk. One way to look at this is as showing that we can keep the intersubstitutivity of True (hAi) with A in a revised logic K3 . In K3 we call an inference valid if under every assignment to atomic sentences, if the premises have semantic value 1 then so does the conclusion. And we call a statement valid if it has semantic value 1 under every assignment to atomic sentences. Note that instances of the law of excluded middle (A _ :A) come out invalid: they can have value 12. (Indeed, no statement in this language is valid, though many of the familiar classical rules are valid.) Kripke’s result shows that in the logic so obtained,9 we can consistently assume that for every sentence A and every pair of sentences B and C that are alike except that some occurrences of A in one of them are replaced by True (hAi) in the other, the inference from B to C and from C to B are valid. This is one of the two components of the naive theory of truth, and is not consistently obtainable in classical logic. It is not entirely clear that this use of nonclassical logic is what Kripke is recommending in his discussion of the strong Kleene version of his theory of truth: some of his remarks suggest it, but others suggest a classical-logic 8 Alternatively, we could extend the assignment of semantic values to pairs of formulas and functions assigning objects to variables. 9 Indeed, even in a slightly expanded logic K3þ that includes disjunction elimination as a metarule; whether the meta-rule is explicitly built into the logic makes a difference when one considers adding new validities involving new logical vocabulary.

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270 / Hartry Field theory later formalized by Feferman ([3], pp. 273–4). The classical-logic Kripke–Feferman theory postulates truth-value gaps: it says of certain sentences, such as the Liar, that they are neither true nor false. (‘False’ is taken to mean ‘has a true negation’, so the claim is that neither they nor their negations are true.) As a consistent classical theory, it gives up on the equivalence between ‘True(hpi)’ and ‘p’. (The Kripke–Feferman theory is one of the ones that commits itself to disjunct (1) in the previous section.) The Kripke theory in its nonclassical version has no commitment to truth-value gaps: indeed, since the whole point of the theory is to maintain the equivalence between ‘True(hpi)’ and ‘p’, the assertion of :[True(hpi) _ :(True(h:pi)] would be equivalent to the assertion of :[p _ :p]; that entails both p and :p in the logic, and in this logic as well as in classical that entails everything. So it is very important in the Kripke theory (on its nonclassical reading) not to commit to truth-value gaps. It will give certain sentences the value 12, but that is not to be read as ‘‘neither true nor false’’. I think the nonclassical reading of Kripke is the more interesting one, and I will confine my discussion to it. I think there are two main problems with the Kripke theory (on this reading). Perhaps the more serious of the problems is that the logic is simply too weak: as Feferman once remarked, ‘‘nothing like sustained ordinary reasoning can be carried out in [the] logic’’ ([3], p. 264). One symptom of this is that not even the law A  A is valid: since A  B is equivalent to :A _ B, this follows from the invalidity of excluded middle. And note that the intersubstitutivity of True (hAi) with A guarantees that True (hAi)  A and its converse are each equivalent to A  A; since the latter isn’t part of the logic in the Kripke theory, neither half of the biconditional True (hAi)  A is validated in Kripke models. So one consequence of the first problem for the Kripke theory is that it does not yield the full naive theory of truth. The other problem for the Kripke theory, the ‘‘revenge problem’’, has been more widely discussed, but I think much of that discussion has been vitiated by a confusion between the nonclassical version of Kripke’s theory and the classical Kripke–Feferman theory: much of it has been based on falsely supposing that the Kripke theory is committed to truth-value gaps. The only real revenge problem for the nonclassical Kripke theory has to do with the fact that the ‘‘defectiveness’’ of sentences like Q0 is inexpressible in the theory, and there is a worry that if we were to expand the theory to include a ‘‘defectiveness predicate’’ the paradoxes would return. I will be proposing a

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Semantic Paradoxes and Vagueness Paradoxes / 271 theory that has much more expressive power than the Kripke theory, and which avoids the revenge problem by having the means to express the defectiveness of paradoxical sentences like Q0 without this leading to inconsistency. Returning to the first of the two problems, a natural idea for how to avoid it is to add a new conditional ! to the Kleene logic, which does obey the law A ! A. There have been many proposals about how to do this; unfortunately, most of them do not enable one to consistently maintain the intersubstitutivity of True (hAi) with A (or even the truth schema True (hAi) $ A which that implies given the law A ! A). In fact, I know of only two workable proposals for how to do this, both by myself; and one of them ([6]) is not very attractive. (There is also a proposal in Brady [1], which is not an extension of Kleene logic but only of a weaker logic FDE, which is a basic relevance logic.) These theories not only contain A ! A, they also contain a substitutivity rule that allows the inference from A $ B to C $ D when C and D are alike except that one contains B in some places where the other contains A; thus the logic is ‘‘classical enough’’ for the two components of the classical theory of truth to be equivalent. I will say a little bit about the more attractive of the two theories ([8]). As with Kripke’s construction, we start out with a base language that doesn’t include ‘True’, or the new ‘!’, and with a classical model for this base language whose arithmetical part is standard. The semantics of the theory— which I’ll call the Restricted Semantics, since I will generalize it in Section 5— is given by a transfinite sequence of Kripke-constructions. At each stage of the transfinite sequence (‘‘maxi-stage’’), we begin with a certain assignment of values in {0, 12 , 1} to sentences whose main connective is the ‘!’. Given such an assignment of values to the conditionals, Kripke’s method of obtaining a minimal fixed point enables us (in a sequence of ‘‘mini-stages within the maxi-stage’’) to obtain a value for every sentence of the language, in such a way as to respect the Kleene valuation rules and the principle that True (hAi) always has the same value as A. It remains only to say how the assignment of values to conditionals that starts each maxi-stage is determined. At the 0th stage it’s simple: we just give each conditional value 12. At each successor stage, we let A ! B have value 1 if the value of A at the prior stage is less than or equal to the value of B; otherwise we give it the value 0. At limit stages, we see if there is a point prior to the limit such that after that point (and before the limit), the value of A is always less than or equal to that of B; if so, A ! B gets value 1 at the limit. Similarly, if there is a point prior to the limit such that

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272 / Hartry Field after that point (and before the limit), the value of A is always greater than that of B, then A ! B gets value 0 at the limit. And if neither condition obtains, A ! B gets value 12 at the limit. That completes the specification of how each maxi-stage begins; to repeat, this specification serves as the input to a Kripke construction that yields values at that stage for every sentence.10 In typical cases of sentences that are paradoxical on other theories, the values oscillate wildly from one (maxi-)stage to the next. But we can define the ‘‘ultimate value’’ of a sentence to be 1 if there is a stage past which it is always 1; 0 if there is a stage past which it is always 0; and otherwise 12. It turns out that there are ordinals D (‘‘acceptable ordinals’’) such that for any nonzero b, the value of every sentence at stage D  b is the same as its ultimate value. (This is the ‘‘Fundamental Theorem’’ of [8].) Since the Kleene valuation rules are satisfied at each stage, this shows (among other important things) that the ultimate values obey the Kleene rules for connectives other than !. As remarked, this construction validates naive truth theory, both in truth schema and intersubstitutivity form. (It validates it in a strong sense: it not only shows naive truth theory to be consistent, it shows it to be ‘‘consistent with any arithmetically standard starting model’’—conservative, in one sense of that phrase. For a fuller discussion see [8], n. 27.) One question that arises is the relation between the ! and the : The ! is not truth-functional,11 but one can construct a table of the possible ultimate values of A ! B given the ultimate values of A and of B: A!B

B¼1

B ¼ 12

B¼0

A¼1 A ¼ 12 A¼0

1 1 1

1 2,0 1, 12

1 2,0

1

1

0

It is evident from this table that A ! B is in some ways weaker and in some ways stronger than A  B. However, from the assumption that excluded middle holds for A and for B, we can derive (A  B) $ (A ! B) (and (A  B)  (A ! B)). Moreover, from the assumption that excluded middle holds for each atomic predicate in a set, we get full classical logic for all 10 Obviously there is a similarity to the revision theory of Gupta and Belnap [12]; but they use a revision rule for the truth predicate instead of for the conditional, and get a classical logic theory (one of the ones that refuses to commit between (1) and (2) ). 11 At least not in these values; but see [7] or [9] for an enriched set of semantic values in which it is truth-functional.

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Semantic Paradoxes and Vagueness Paradoxes / 273 sentences built up out of just those predicates. Thus the logic is a generalization of classical, and reduces to classical when appropriate instances of excluded middle are assumed. One way to look at the matter is that the logic without excluded middle is the basic logic, but in domains like number theory or set theory or physics where we want excluded middle, we can simply assume all the instances of it in that domain as nonlogical premises; this will make the logic of those domains effectively classical. It is only for truth and related notions that we get into obvious trouble from assuming excluded middle: there excluded middle gives inconsistency, given the naive theory of truth. I think this is a much more attractive resolution of the paradoxes than any of the classical ones. One of its most attractive features has to do with a widely held view that any resolution of the paradoxes simply breeds new paradoxes: ‘‘revenge problems’’. I claim that there are no revenge problems in this logic. More particularly, you can state in this logic the way in which certain sentences of the logic are ‘‘defective’’; because you can do so, and because there is a consistency proof of naive truth theory in the logic, the notion (or notions) of defectiveness cannot generate any new paradoxes. I will discuss this in Sections 7 and 8. I will make one remark now, which is that like the nonclassical version of the Kripke theory, this is not a theory that posits truth-value gaps. In particular, we can’t assert of the Liar sentence that it isn’t either true or false. Nor can we assert that it is either true or false. Situations like this, where we can’t assert either a claim or its negation, may seem superficially like the situation that I complained about in the case of certain classical resolutions of the paradox, where we are committed to a disjunction in which each disjunct has bad consequences, but try to avoid those bad consequence by refusing to decide which of the two disjuncts to assert. But in fact the nonclassical situation isn’t like that at all. It is true that in the nonclassical examples we would have a problem if we asserted A and we would have a problem if we asserted :A (where A is a classically paradoxical sentence). But what made that so problematic in the classical case was that there we were committed to the claim A _ :A. We’re not committed to that in the nonclassical case, so our refusal to commit to either the classically paradoxical A or to its negation is not a defect in the account. Similarly, we’re not committed to the claim that either A lacks truth value or doesn’t lack truth value, so the refusal to commit to A’s lacking truth value or commit to its having truth value is no defect.

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274 / Hartry Field

3. Vagueness and Indeterminacy Before discussing the revenge problem, let’s move away from the semantic paradoxes to other quasi-paradoxes. Many members of the right-to-life movement think that there is a precise nanosecond in which a given life begins, though we may not know when it is. Most of us think that this view is absurd, but Timothy Williamson [23] has in effect offered an interesting argument that the right-to-lifers are correct on this point. The initial argument goes as follows. Select a precise moment about a year before Jerry Falwell’s birth, and call it ‘time 0’. For any natural number N, let ‘time N’ mean ‘N nanoseconds after time 0’. By the law of excluded middle, we get each instance of the following schema: (3P) (Falwell’s life had begun by time N) _ :(Falwell’s life had begun by time N). From a finite number of these plus the fact that Falwell’s life hadn’t begun by time 0 plus the fact that it had begun by time 1018 , plus the fact that for any N and M with N < M, if Falwell’s life had begun by time N then it had begun by time M, a minimal amount of arithmetic and logic yields that (F) There is a unique N0 such that Falwell’s life had begun by time N0 and not by time N0  1. But then it seems that there is a fact of the matter as to which nanosecond his life began, namely, that between time N0  1 and time N0 (inclusive of the latter bound but not the former). That is the initial argument. And the most obvious way around it is to question the use of excluded middle. There have, of course, been attempts to get around the right-to-lifer’s conclusion without giving up classical logic: e.g. by introducing a notion of determinate truth and determinate falsehood such that sentences of form ‘Falwell’s life began in the (semi-closed) interval (N  1, N]’ are neither determinately true nor determinately false. But Williamson has given extensions of the initial argument that close off most of these attempts: the basic strategy is to argue that even if such sentences are conceded to be neither determinately true nor determinately false, in whatever sense of determinateness one favors, it’s hard to see why this should give any sense of nonfactuality to the question of when his life began, given the commitment to (F). Even if I concede that there’s no ‘‘determinate’’ truth here, in whatever sense

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Semantic Paradoxes and Vagueness Paradoxes / 275 I may give that phrase, why can’t I wonder what the unique N0 is, or wonder whether it is even or odd? Why can’t I be very worried about the possibility that the unique N0 occurred before I performed a certain act, or very much hope that N0 is odd? And even if I take the question of whether it is odd to be beyond the scope of human knowledge, why can’t I imagine an omniscient god who (by hypothesis of his omniscience) knows the answer; or a Martian who, though not knowing everything, knows this? And so forth. But if I do wonder these things or have worries or hopes like this or concede the possibility of beings with such knowledge, all pretense that I am regarding the question as nonfactual seems hollow. In the past I’ve tried to find a way around this kind of argument, in part by a nonstandard theory of propositional attitudes within classical logic, but I’ve come to see this task as pretty hopeless. It now seems to me that rejecting some of the instances (3P) of excluded middle is the only viable option (short of giving in to the right-tolifers on this issue). But will the no-excluded-middle option work any better? Let’s first get clear on an issue (which could have been raised in connection with the semantic paradoxes too) of what it is to ‘‘reject’’ certain instances of excluded middle. We don’t reject all of them, only some; what exactly is this difference in attitude we have between those that we reject and those that we don’t? First of all, ‘‘reject’’ can’t mean ‘‘deny’’, that is, ‘‘assert the negation of’’. Suppose we deny an instance of (3P), that is, assert (3N) :[(Falwell’s life had begun by time N) _ :(Falwell’s life had begun by time N)]. The expression in brackets is a disjunction, and surely on any reasonable logic a disjunction is weaker than either of its disjuncts. So denying the disjunction has got to entail denying each disjunct, and so asserting (3N) clearly commits us to asserting both of the following: (4a) :(Falwell’s life had begun by time N). (4b) ::(Falwell’s life had begun by time N). But (4b) is the negation of (4a), so (3N) has led to a classical contradiction. And as noted before, the conjunction of a sentence with its negation is also a contradiction in the Kleene logic K3 described previously, in the sense that there too it implies everything. Now, that isn’t the end of the matter: instead of using K3 we could follow Graham Priest [20] and opt for a ‘‘paraconsistent logic’’ on which classical

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276 / Hartry Field contradictions don’t entail everything, and therefore aren’t so bad as in classical logic. I wouldn’t dismiss that view out of hand. But there are problems with using it in the present context. For one thing, since the paraconsistentist accepts (4a), and (3P) is a disjunction with (4a) as one disjunct, the paraconsistentist will accept (3P) as well as (3N): (3P) follows from (4a) on any reasonable logic, including all the standard paraconsistent logics. But then we can argue from (3P) to Williamson’s conclusion that there is a unique nanosecond in which Falwell’s life began, in precisely the same way as before, so the conclusion has not been blocked. The conclusion has been denied—from (3N) we can conclude that there is not a unique nanosecond during which his life began12—but it has also been asserted. This classical inconsistency is not in itself a problem, it is just a further instance of paraconsistentist doctrine that classical inconsistency is no defect; but it is disappointing that we are left in a position of thinking that the right-to-lifers are no less correct to assert that there is a fact of the matter as to the nanosecond in which Falwell was born than we are to deny that there is a fact of the matter. So rejection must be interpreted in some other way than as denial. A common claim is that to reject A is to regard it as not true. The problem with this is that on the most straightforward reading of ‘true’—and the one I took great pains to maintain in the earlier sections on the semantic paradoxes— the claim that A is true is equivalent to A itself; so asserting that A is not true is equivalent to asserting :A, and this account of rejection reduces to the previous one. Perhaps rejection is just nonacceptance? No, that’s far too weak. Compare my attitude toward (5) Falwell’s life began in an even-numbered nanosecond with my attitude toward (6) Attila’s maternal grandmother weighed less than 125 pounds on the day she died. (5) seems intuitively ‘‘nonfactual’’, and I reject both it and its negation in the strongest terms. That is not at all the case with (6): I have no reason to doubt that this question is perfectly factual. I don’t accept (6) or its negation, for lack of evidence; but I don’t reject them either, for given the ‘‘factuality’’ of (6) and 12 Indeed, we can conclude both (i) that there are multiple nanoseconds during which his life began, rather than one, and (ii) that there is no nanosecond during which his life began.

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Semantic Paradoxes and Vagueness Paradoxes / 277 its negation I could only reject one by accepting the other. Rejection is more than mere nonacceptance.13 The same point arises for the acceptance and rejection of instances of excluded middle. The point would be easier to illustrate here with examples that have less contextual variation than does ‘life’, and where the higherorder indeterminacy is less prevalent; but let’s stick to the life case anyway. Suppose I am certain that on my concept of life, if it is determinate that a person’s conception occurred during a certain minute then it is indeterminate whether their life began during that minute, but determinate that their life didn’t begin before that minute. Then if I knew enough about Falwell to be sure that his conception occurred during some particular precisely delimited minute, and N were the nanosecond marking the end of that minute, then I would reject the corresponding instance of (3P). If however I have no very clear idea how old Falwell is, so that for all I know nanosecond N might be before his conception or after his birth, I will be uncertain about the corresponding instance of (3P): I will neither accept it nor reject it. (The same point can arise even if I do have detailed knowledge of the times of his conception and birth and the various intermediate stages: suppose that I’m undecided whether there is a God who injects vital fluid into each human body at some precise time, but think that if there is no such God then N would correspond to a borderline case of Falwell’s life having begun.) So for instances of excluded middle too, we have that rejection is stronger than mere nonacceptance.14 Should the failure of all these attempts to explain the notion of rejection required by the opponent of excluded middle lead us to suppose that there is 13 Rejecting A is also not to be identified with believing it impossible that one could have enough evidence to accept A. Why not? That depends on the notion of possibility in question. (a) On any interestingly strong notion of possibility, belief in the impossibility of such evidence does not suffice for rejection: there are intuitively factual ‘yes or no’ questions (e.g. about the precise goings-on in the interior of the sun or in a black hole or beyond the event horizon) for which there is no possible evidence, but because I take them to be factual I could only reject one answer by accepting the other. (b) On a very weak notion of possibility (e.g. bare logical possibility), we have the opposite problem: even for claims that seem ‘‘nonfactual’’, like (5), there is a bare logical possibility that there is such a thing as ‘‘living force’’ and that someone will invent a ‘‘living force detector’’ that could be used to ascertain whether the claim is true. 14 The point arises as well in connection with potentially ‘‘ungrounded’’ sentences that may not be actually ‘‘ungrounded’’. If sentence A is of form ‘‘No sentence written in location D is true’’, and I know that exactly one sentence is written in location D but am unsure whether it is ‘1 þ 1 ¼ 3’ or A itself, then I am not in a position to accept or reject either the sentence A or the sentence A _ :A.

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278 / Hartry Field no way to make sense of the no-excluded-middle position? No, for in fact there is an alternative way to explain the concept of rejection (and it doesn’t require a prior notion of indeterminacy). The key is to recognize that the refusal to accept all instances of excluded middle forces a revision in our other epistemic attitudes. A standard idealization of the epistemic attitudes of an adherent of classical logic is the Bayesian one, which (in its crudest form at least) involves attributing to each rational agent a degree of belief function that obeys the laws of classical probability; these laws entail that theorems of classical logic get degree of belief 1. Obviously this is inappropriate if rational agents needn’t accept all instances of excluded middle. But allowing degrees of belief less than 1 to some instances of excluded middle forces other violations of classical probability theory. In particular, if we keep the laws P(A _ B) þ P(A ^ B) ¼ P(A) þ P(B) and P(A ^ :A) ¼ 0, then we must accept P(A _ :A) ¼ P(A) þ P(:A). In that case, assigning degree of belief less than 1 to instances of excluded middle requires that we weaken the law (7) P(A) þ P(:A) ¼ 1 to (7w ) P(A) þ P(:A)  1. The relevance of this to acceptance and rejection is that accepting A seems intimately related to having a high degree of belief in it; say, a degree of belief at or over a certain threshold T > 12.15 So let us think of rejection as the dual notion: it is related in the same way to having a low degree of belief, one at or lower than the co-threshold 1  T. In the context of classical probability theory where (7) is assumed, this just amounts to acceptance of the negation. But with (7) replaced by (7w ), rejection in this sense is weaker than acceptance 15 We can take T to be 1, but only if we are very generous about attributing degree of belief 1. If (as I prefer) we take T to be less than 1, some would argue that the lottery paradox prevents a strict identification of acceptance with degree of belief over the threshold; I doubt that it does, but to avoid having to argue the matter I have avoided any claim of strict identification.

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Semantic Paradoxes and Vagueness Paradoxes / 279 of the negation. (It remains stronger than failure to accept: sentences believed to degrees between 1  T and T will be neither accepted nor rejected.) I take it that in a case where a sentence A is clearly indeterminate (e.g. case (5), for anyone certain that there is no such thing as ‘‘vital fluid’’), the degree of belief in A and in :A should both be 0. Some may feel it more natural to say that the degree of belief in ‘‘Falwell’s life began in an even-numbered nanosecond’’ should be not the single point 0, but the closed interval [0,1]. That view is easy to accommodate: represent the degree of belief in A not by the point P(A), but by the closed interval R(A) ¼df [P(A), 1  P(:A)].16 This is merely a matter of terminology: the functions P and R are interdefinable, and it is a matter of taste which one is taken to represent ‘‘degrees of belief ’’. (In terms of R, acceptance and rejection of A go by the lower bound of R(A).) The value of introducing probabilistic notions is that they give us a natural way to represent the gradations in attitudes that people can have about the ‘‘factuality’’ of certain questions—at least, they do when higher-order indeterminacy is not at issue. To regard the question of whether A is the case as ‘‘certainly factual’’ is for the following equivalent conditions on one’s degree of belief to obtain: P(A) þ P(:A) ¼ 1; P(A _ :A) ¼ 1; R(A) is point-valued; R(A _ :A) ¼ {1}. To regard it as ‘‘certainly nonfactual’’ is for the following equivalent conditions to hold: P(A) þ P(:A) ¼ 0; P(A _ :A) ¼ 0; R(A) ¼ [0, 1]; R(A _ :A) ¼ [0, 1]. In general, the degree to which one believes A determinate is represented (in the P-formulation) by P(A) þ P(:A); i.e. P(A _ :A); i.e. 1  w, where w is the breadth of the interval R(A); i.e. the lower bound of R(A _ :A). In the P-formulation, belief revision on empirical evidence goes just as on the 16 Note that R(A _ :A) will always be an interval with upper bound 1; its lower bound will be 1  w, where w is the width of the interval R(A).

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280 / Hartry Field classical theory, by conditionalizing; this allows the ‘‘degree of certainty of the determinacy of A’’ to go up or down with evidence (as long as it isn’t 1 or 0 to start with). The idea can be used not only for examples like the Falwell example, but for potentially paradoxical sentences as well. Consider a sentence S that says that no sentence written in a certain location is true, and suppose that we know that exactly one sentence is written in that location; our degree of belief that the sentence in that location is ‘2 þ 2 ¼ 4’ is p, our degree of belief that it is ‘2 þ 2 ¼ 5’ is q, and our degree of belief that the sentence written there is S itself is 1  p  q, which I’ll call r. I submit that our degree of belief in S should be q, our degree of belief in :S should be p, and our degree of belief in S _ :S should be p þ q, i.e. 1  r. The key point in motivating this assignment is that relative to the assumption that the sentence written there is S, then S and :S each imply the contradiction S ^ :S, and so S _ :S implies this contradiction as well; given this, it seems clear that if we were certain that the sentence written there were S, then we should have degree of belief 0 in S, in :S, and in S _ :S. As we increase our degree of certainty that the sentence written there is S, our tendency to reject the three sentences S, :S, and S _ :S should become stronger. Of course, the idea that we can attribute to an agent a determinate P-function (or R-function) is a considerable idealization. Even in the case of classical P-functions, where we don’t allow P(A) þ P(:A) to be less than 1, the issue of whether a person’s degree of belief is greater than say 0.7 often seems indeterminate. How are we to make sense of the indeterminacy here? It should be no surprise that on my view, we make sense of this by giving up the presupposition of excluded middle for certain claims of form ‘‘X’s probability function PX is such that PX (A) > 0:7’’.17 (It isn’t that we need to develop the theory of probability itself in a nonclassical language; where excluded middle is to be questioned, rather, is in the attribution of a given perfectly classical probability function to a given agent X. If you like, this gives failures of excluded middle for ‘‘claims about PX ’’, though not for claims about individual probability functions specified independently of the agent 17 A common suggestion ([17]) is that we should represent the epistemic state of an agent X not by a single probability function but by a nonempty set SX of them. That is in some ways a step in the right direction, but it too involves unwanted precision; and while that could be somewhat ameliorated by going to nonempty sets of nonempty sets of probability functions, or iterating this even further, I think that ultimately there is no satisfactory resolution short of recognizing that excluded middle fails for some attributions.

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Semantic Paradoxes and Vagueness Paradoxes / 281 X.) We can take the same position for nonclassical P-functions or R- functions too. I’m inclined to think that there is a strong connection between this indeterminacy in the degree of belief function and ‘‘higher-order indeterminacy’’: in cases where X attributes higher-order indeterminacy to A, some assertions about the value of PX (A _ :A) (or RX (A _ :A)) will be ones for which excluded middle can’t be assumed. In any case, the indeterminacy in attributions of probability doesn’t essentially change the picture offered in the preceding paragraph: to whatever extent that we can say that X’s degree of belief function attributes value 1 to A _ :A, to precisely that extent we can say that X regards A as certainly factual. So far I have not said anything about introducing a notion of determinacy into the language. I have argued that even without doing so, we can represent a ‘‘dispute about the factuality of A’’ as a disagreement in attitude: a disagreement about what sort of degrees of belief to adopt. An ‘‘advocate of the factuality of A’’ will have a cognitive state in which P(A _ :A) is high (i.e. in which R(A) is close to point-valued). An ‘‘opponent of the factuality of A’’ will have a cognitive state in which P(A _ :A) is low (i.e. R(A) occupies most of the unit interval). I think it important to see that we can do all this without bringing the notion of determinacy into the language: it makes clear that there is more substance to a dispute about factuality than a mere debate about how a term like ‘factual’ or ‘determinate’ is to be used. Still, what we have so far falls short of what we might desire, in that so far we have no means to literally assert the nonfactuality of the question of whether A: having a low degree of belief in A _ :A is a way of rejecting the factuality of A, but not of denying it. It would be very awkward if we couldn’t do better than this: debates about the factuality of questions would be crippled were we unable to treat the claim of determinacy or factuality as itself propositional. What we need, then, is an operator G, such that GA means intuitively that A is a determinate (or factual) claim, i.e. that the question of whether A is the case is a determinate (or factual) question. Actually it’s simpler to take as basic an operator D, where DA means that it is determinately the case that A. The claim GA (that it is determinate whether A) is the claim that DA _ D:A. The point of the operator is that though :(A _ :A) is a contradiction, :(DA _ D:A) is not to be contradictory. ‘‘Determinately operators’’ are more familiar in the context of attempts to treat vagueness and indeterminacy within classical logic, and their use there in representing nonfactuality is subject to a persuasive criticism. The criticism is

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282 / Hartry Field that whatever meaning one gives to the D operator, it is hard to see how :(DA _ D:A) can represent the nonfactuality of the question of whether A: for any claims to nonfactuality are undermined by the acceptance of A _ :A. But when we have given up the acceptance of A _ :A, the criticism doesn’t apply. People can have degrees of belief about determinateness, so their degree of belief function should extend to the language containing D. If we had only first-order indeterminacy to worry about, and could stick to the idealization of a determinate degree of belief function PX for our agent X, some constraints on how PX extends to the D-language would be obvious: since PX (A) þ PX (:A) represents the degree to which the agent regards A factual, which is PX (DA _ D:A), which in turn is PX (DA) þ PX (D:A), we must suppose that PX (DA) ¼ PX (A) for any A.18 That is, we must regard the lower bound of RX (DA) as the same as the lower bound of RX (A). Indeed, with higher-order indeterminacy excluded, RX (DA) should just be a point: PX (:DA) will be simply 1  PX (A). So the fact that DA is strictly stronger than A comes out in that PX (:DA) can be greater than PX (:A) but not less than it, i.e. in that the upper bound of RX (DA) can be lower than that of RX (A) but not greater than it. It is, however, important to allow for higherorder indeterminacy, and there may be some question how best to do so. A proper representation of higher-order indeterminacy presumably should allow excluded middle to fail for sentences of form DA, so we want to allow that PX (DA) þ PX (:DA) falls short of 1, i.e. that RX (DA) not be point-valued. I’m inclined to think that we ought to keep the demand that the lower bound of RX (DA) is always the same as that of RX (A); this would leave the upper bound unfixed.19 (As noted before, the situation is complicated by the fact that there is indeterminacy in the attribution of P-functions or R-functions to the agent, so we can’t assume excluded middle for all claims about PX and RX . I don’t believe that this requires modification of what I’ve said, but the matter deserves more thought than I have been able to give it.) I think that what I’ve said clarifies important aspects of the conceptual role of the determinately operator. Indeed, until recently I thought that not a 18 More generally, we could argue that for any A and B, P(DA ^ B) ¼ P(A ^ B). 19 There are intuitions that go contrary to this: sometimes we seem prepared to assert A but not to assert ‘‘It is determinately the case that A’’. I’m somewhat inclined to think that this is so only in examples where A contains terms that are context-dependent as well as indeterminate, and that it is so because we give to ‘determinately A’ a meaning like ‘under all reasonable contextual alterations of the use of these terms, A would come out true’; and this seems to me a use of ‘determinately’ different from the one primarily relevant to the theory of indeterminacy. But I confess to a lack of complete certainty on these points; for instance, another possibility

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Semantic Paradoxes and Vagueness Paradoxes / 283 whole lot more could be said to clarify that operator. I now realize that that former opinion was far too pessimistic: in fact, I’m tempted to say that the determinately operator can be defined in terms of a more basic operator, a conditional ‘!’. As we’ll see in the next section, such a conditional plays a very central role in the theory of vagueness; in Section 5 I will then make a case that the conditional required is ‘‘essentially the same as’’ the one used in connection with the semantic paradoxes. Starting in Section 6, I will explain how the conditional can be used to define a very good candidate for the determinately operator. The definition of ‘determinately’ in terms of the conditional would not make the probabilistic laws governing the determinately operator irrelevant: for they would indirectly constrain our degrees of belief in sentences involving the conditional. The conditional, however, can be given a very rich set of deductive relationships; it is these deductive relationships on it, together with the probabilistic constraints on the determinately operator defined from it, together with that definition, which would jointly clarify the conditional and the determinately operator together. The picture just sketched may eventually need to be complicated slightly: while we can certainly define a very good candidate for the intuitive determinately operator D in terms of the conditional, the defined operator D may not match the intuitive operator in every respect. But even if this turns out to be so, the intuitive operator and the defined operator will share enough properties for D to provide a very good model of D; in particular, the intuitive laws governing D, including the laws of its interaction with the conditional, will be very close to the laws provable for D. In short: I’m tempted by the view that we have a strict definition of D in terms of ‘!’, but even if not, what we do have will give rich structural connections between the two that, when combined with the probabilistic account above, do a great deal to settle the meaning of D.

4. The Conditional Again What kind of logic do we want to use for vagueness? For reasons mentioned already, it should not include the law of excluded middle. I will now assume would be to allow that in some contexts the upper bound of R(A) plays a role in governing the assertion of A. (I’d like to thank Richard Dietz, Stephen Schiffer, and Timothy Williamson for discussions of the complications of any extension of talk of probability to a language with a determinately operator, when higher-order vagueness is allowed. In particular, Dietz pointed out a substantial problem in an earlier attempt of mine, in a classical-logic context.)

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284 / Hartry Field that the logic should be like the Kleene logic K3 (or K3þ : see note 9) as regards the basic connectives :, ^ and _. This assumption is both plausible and widely accepted, and seems to have the best hope of providing a unification of the logic of vagueness with the appropriate logic for the semantic paradoxes. The standard semantics for the logic K3 (or indeed, K3þ ) is the strong Kleene valuation tables, mentioned previously, on which each sentence receives one and only one of the values 1, 0, and 12. These should not be read ‘true’, ‘false’, and ‘neither true nor false’: then assigning value 12 would be postulating a truth-value gap, which we cannot do for reasons given in Section 2. A somewhat better reading would be ‘determinately true’, ‘determinately false’, and ‘neither determinately true nor determinately false’. That isn’t quite right either: among other things, since it is assumed that each sentence has one of the three values, this reading would commit us to the claim that excluded middle holds for attributions of determinate truth and determinate falsehood,20 thereby ruling out higher-order indeterminacy. (This argument assumes that the semantics is given in a classical metalanguage; an alternative is to keep to the readings ‘determinately true’, ‘determinately false’, and ‘neither of those’, but refrain from assuming of every sentence that it has one of the three values. More on this later.) In Sections 5 and 8 I will say a bit more about these issues of how to interpret the semantics. But for now, the readings ‘determinately true’, ‘determinately false’, and ‘neither determinately true nor determinately false’ will be close enough to serve our purposes. In the nonclassical treatment of the semantic paradoxes in Section 2, we saw that there was a strong need to supplement the Kleene logic with a new conditional: a conditional A ! B not defined in the classical way, as :A _ B. For whatever the merits of that definition in a context where we have excluded middle, it fails miserably when excluded middle is abandoned: it doesn’t even obey such elementary laws as A ! A, and because of this it is very hard to reason with. Indeed, the failure of the connective :A _ B (which I’ll call the Kleene conditional, and abbreviate A  B) as a definition of A ! B is perhaps even more striking in connection with vagueness than it is in connection with the semantic paradoxes. For consider the following claims: 20 The commitment to one of the three values is DT _ DF _ (:DT ^ :DF), which is equivalent to (DT _ DF _ :DT) ^ (DT _ DF _ :DF); since DF entails :DT and DT entails :DF, this entails (DT _ :DT) ^ (DF _ :DF).

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Semantic Paradoxes and Vagueness Paradoxes / 285 (A) There are nearly 1,000 red balls in Urn 1. (B) There are nearly 1,000 black balls in Urn 1. In fact, let us suppose, there are exactly 947 red balls and exactly 953 black balls in Urn 1. Let us imagine a context where both 947 and 953 seem to be in the borderline region of ‘nearly 1,000’, so that we’d be inclined to regard A and B as each having value 12. But since the number of black balls is closer to 1,000 than is the number of red balls, we’d surely want to give ‘‘If there are nearly 1,000 red balls in Urn 1 then there are nearly 1,000 black balls in Urn 1’’ semantic value 1. But if we use the connective ‘’ to define ‘if . . . then’, we don’t get this result: ‘A  B’ will come out with value 12. So we very much need another conditional. The point is not a new one: nearly all advocates of nonclassical logics of vagueness have proposed using a conditional other than ‘’. The most popular expansion of Kleene logic is Łukasiewicz continuum-valued logic, aka ‘‘fuzzy logic’’. For the semantics of this conditional we need to further partition the values other than 0 and 1: instead of just 12, we allow arbitrary real numbers between 0 and 1. The value of A ^ B is then the minimum of the values of A and B, while the value of A _ B is the maximum; and the value of :A is 1 minus the value of A. Clearly, as far as these connectives go the valuation rules are a generalization of the Kleene rules: the Kleene rules result by restricting attention to the three values 0, 12, and 1. Moreover, taking 1 as the sole designated value, the class of valid inferences in these connectives is unaffected. The point of the new semantics (as far as logic as opposed to pragmatics goes) is that it enables us to give rules for a new connective !; the valuation rule is that the value of A ! B is 1 if the value of A is less than or equal to that of B, and otherwise is 1 minus the extent by which the value of A exceeds that of B. The Łukasiewicz semantics works reasonably well for dealing with vagueness, but I have never seen a compelling argument for it. And it does have some intuitive defects: for instance, the linear ordering of values is unintuitive, and leads to the claim that sentences such as It is either the case that if Tim is thin then John is old, or that if John is old then Tim is thin are logical truths. Moreover, using the Łukasiewicz logic would doom all hope of a combined treatment of vagueness and the semantic paradoxes which preserves the naive theory of truth, for the naive theory cannot be

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286 / Hartry Field preserved in the Łukasiewicz logic ([22], [13]). But despite these doubts about the Łukasiewicz logic, we do need a new conditional. (Whether the one proposed in connection with the semantic paradoxes would do for this purpose is something I will consider in the next section.) The issue of the conditional is also relevant to an influential argument of Kit Fine’s [10]. Fine argued that classical-logic accounts of vagueness are far superior to nonclassical-logic accounts based on the Kleene semantics, in that the Kleene-based accounts cannot handle ‘‘penumbral connections’’ between distinct vague terms. Fine’s point is this. Suppose that the claim that an object b is red and the claim that b is small both get value 12. It is unsurprising that the conjunction and disjunction of the two claims should get value 12; and that is what the Kleene tables say. But how about the claim that b is red and the claim that b is pink, when b is a borderline case of each? Fine thinks that their conjunction ought to get value 0, and that if b is clearly in the red-to-pink region their disjunction should get value 1 even though the disjuncts get the value 12. I don’t find these intuitions as compelling as Fine does, but there is a more neutral way to put his point: we ought to be able to say, somehow, that ‘red’ and ‘pink’ are contraries; and we ought to be able to say, somehow, that an object is red-to-pink, using only the terms ‘red’, ‘pink’, and logical devices. But the obvious proposals for how to do these things won’t work. For example, we can’t say that something is red-to-pink by saying that it is red or pink, since that gets value 12 when the object is on the border between red and pink; and no other logical function in Kleene logic will do any better. Put this way, Fine’s objection against the use of the unadorned Kleene logic is compelling. But once we add a new conditional to the logic, it is much less obvious that there is any problem with penumbral connections. For now we can easily explain the idea that ‘red’ and ‘pink’ are contraries: it consists in the fact that &8x[x is red ! x is not pink],21 where & indicates some kind of conceptual necessity; the claim is that what follows the & is guaranteed by a conceptual constraint on the simultaneous values that are allowed for ‘red(x)’ and ‘pink(x)’.22 And we can explain the idea of an object being in the red-to21 If ! were not contraposable, we’d have to add &8x[x is pink ! x is not red]; but both the Łukasiewicz conditional and the conditional introduced in Section 2 are contraposable (as will be the more general conditional introduced in Section 5). 22 In the case of the Łukasiewicz semantics, the appropriate constraint would be that the values assigned to ‘red(o)’ and to ‘pink(o)’ never add to more than 1. In the case of the Restricted

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Semantic Paradoxes and Vagueness Paradoxes / 287 pink region: ‘‘red-to-pink(x)’’ just means ‘‘x is not red ! x is pink’’.23 We can, if we like, even introduce a ‘‘pseudo-disjunction’’ t and ‘‘pseudo-conjunction’’ u: A t B iff (:A) ! B: A u B iff :(A ! :B): So to call something red-to-pink(x) is to say that it is red t pink, and the contrariness of the two consists in the fact that nothing can be red u pink. The logic of u and t will be slightly odd (just what it is will depend on the logic of !, obviously),24 but perhaps it is close enough to the ordinary ^ and _ to explain whatever intuitive force there is in Fine’s own way of presenting the penumbral connection problem. As we’ll see in Section 6, an appropriate conditional can also be used in clarifying the notion of determinateness, and the related notion of defectiveness. But just what sort of conditional is appropriate?

5. Generalizing the Previous Conditional Before deciding what the semantics of a conditional appropriate to vagueness and indeterminacy should be, I need to say something about the role I expect the semantics to play. There are two approaches to giving a semantic account of a language with vague terms. Semantics I suggested for the paradoxes, the appropriate constraint would have to be that the values assigned to ‘red(o)’ and to ‘pink(o)’ at any sufficiently large stage never add to more than 1. The more general semantics to be introduced in the next section will drop the applicability of stages to sentences like ‘red(o)’ and ‘pink(o)’, but will contain a more general sort of variation of extension; and the constraint will be analogous, that the values assigned to ‘red(o)’ and to ‘pink(o)’ at any ‘‘world’’ (at least, any ‘‘world’’ near ‘‘the actual one’’) never add to more than 1. 23 In Łukasiewicz semantics, an object will thus satisfy ‘red-to-pink’ iff the values assigned to ‘red(o)’ and to ‘pink(o)’ add to exactly 1; in the semantics suggested for the paradoxes or its generalization in the next section, the condition is that the values assigned to ‘red(o)’ and to ‘pink(o)’ add to exactly 1 at any sufficiently large stage, or at any world near the actual one. 24 It may be useful to display the appropriate ‘‘possible value tables’’ for t and u that we get if we use the semantics outlined in Section 2: t A¼1 A ¼ 12 A¼0

B¼1

B ¼ 12

B¼0

1 1 1

1 1, 12 1 2,0

1 2,0

1 0

u A¼1 A ¼ 12 A¼0

B¼1

B ¼ 12

B¼0

1 1, 12 0

1, 12 1 2,0 0

0 0 0

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288 / Hartry Field One is to use a perfectly precise metalanguage. This approach has one great advantage and one disadvantage. The great advantage is that we are all used to reasoning classically, and there are no real controversies about how to do it; the semantics will thus be easy to use and have a clear and unambiguous content. The disadvantage is that there is no hope of precisely capturing the content of the sentences of a vague language in a precise metalanguage: the precise metalanguage inevitably draws artificial lines. (This is why higherorder vagueness looks like such a problem when we think about it in terms of a precise metalanguage.) The only way one could conceivably give a semantic theory for a language with vague terms that accurately reflects the content of its sentences is to drop the assumption of excluded middle in the metalanguage. The metalanguage could still be like the metalanguage used before, in assigning semantic values in the set {0, 12 , 1} to sentences; but we could not assume the law of excluded middle for attributions of such values. A semantic theory that avoids excluded middle in the metalanguage may be what we should ultimately aspire to, but it does throw away the great advantage of a semantics in a classical metalanguage, and I think that for the here and now it is more useful to give our semantics in a classical metalanguage. A result of doing this is that we cannot expect our semantics to be totally faithful to the language it seeks to represent.25 What then should our standards on a semantics in a classical metalanguage be? There is a very modest conception of semantics, which Dummett ([2]) calls ‘‘semantics as a purely algebraic tool’’, where the only point of a semantics is to yield an extensionally adequate notion of logical consequence. If that were the limits on our ambition for a semantics, then it is not obvious that the Restricted Semantics offered in Section 2 in connection with the semantic paradoxes couldn’t be simply carried over to the case of vagueness and indeterminacy. In describing the construction there, I stipulated that we were to start out with a perfectly precise base language: one where each atomic predicate gets a 2-valued extension that it keeps at each stage of the 25 For independent reasons, the clarification of the meaning of logical connectives can’t proceed wholly by means of a formal semantics: any formal semantics will itself use logical connectives, often the very ones being ‘‘explained’’, and often in ways that would make the ‘‘explanations’’ grossly circular (e.g. ‘‘ ‘not A’ is true if and only if A is not true’’, ‘‘ ‘A and B’ is true if and only if A is true and B is true’’, etc.). In my view, part of what clarifies all these notions, the conditional of this section included, is its connection to degrees of belief: see the discussion at the end of Section 3.

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Semantic Paradoxes and Vagueness Paradoxes / 289 construction. I then proposed adding the term ‘True’, and I took its semantics to have two features that might be taken as typical of indeterminacy: first, ‘True’ gets only a ‘‘3-valued extension’’ (‘‘positive extension’’, ‘‘negative extension’’, and ‘‘remainder’’) at each stage, and second, this 3-valued extension varies from stage to stage. (Its varying from stage to stage is connected with the higher-order indeterminacy of ‘True’.) And so a natural thought is that we might simply allow that predicates like ‘red’ and ‘bald’ could be treated like ‘True’: we could simply assign to ‘Red’ or ‘Bald’, at each stage a, a 3-valued extension, with this extension allowed to vary from one stage to the next. There is, actually, a technical constraint we’d want to impose on the way that the 3-valued extension can vary, in order to ensure that we could still add ‘True’ to the language in accord with the naive truth theory. But the required constraint is fairly evident from the proof of the technical result mentioned in Section 2.26 With this constraint imposed, we would validate the naive theory of truth even when there are vague predicates in the language; and do so in such a way that the same logic of ! that works when we add ‘True’ to a precise language works when we add to it a language allowing vagueness. This is all fine if we accept the view that the only role of the semantics is to serve as an ‘‘algebraic tool’’ for getting a consequence relation, but I think it is reasonable to demand more. True, it is inevitable that a classical semantics for a nonclassical language won’t reflect the semantics totally faithfully, but we’d like it to represent it as faithfully as possible, and the semantics just suggested seems to me to fall short of reasonable expectations. In particular, it isn’t at all clear what the assignment of 3-valued extensions to stages ‘‘means’’ in the case of vague predicates. It does seem natural (at least when one models the semantics in a classical metalanguage, which is what’s under consideration here) to think of models that have varying 3-valued extensions. What is not so natural, though, is the well-ordering of these 3-valued extensions into stages. We certainly don’t have the clear rules constraining the transfinite sequence of 3-valued extensions of ordinary vague predicates that we have in the case of 26 A sufficient constraint on the valuation of the atomic predicates other than ‘True’, for a consistent addition of ‘True’ to be possible, is that for each n-ary predicate p and each n-tuple of objects o1 ,... , on , the sequence of members of {1, 12 , 0} assigned to p(o1 , ..., on ) satisfies the following two conditions: (i) it eventually cycles, in the sense that there are a and b, the latter nonzero, such that for any two stages g1 , g2  a, the values of p(o1 ,... , on ) at these stages is the same if g1 and g2 differ by a right-multiple of b; (ii) unless the ‘‘cycle length’’ (the smallest such b) is 1, then the value of p(o1 ,... , on ) is 12 at all sufficiently high right multiples of b. As long as this regularity condition is assumed for the base language, the Fundamental Theorem of [8] goes through with no real change. (And it tells us that the regularity condition holds for ‘True’ as well.)

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290 / Hartry Field ‘True’. We might be tempted to just require that the 3-valued extension stays the same from stage to stage; but if we do that, the model will have no representation at all of higher-order vagueness, and that doesn’t seem satisfactory. What I think is needed is a more substantial generalization of the Restricted Semantics used for the paradoxes; I’ll call it the Generalized Semantics. A natural idea for a semantics of vagueness in a classical metalanguage is to employ an infinite set W of 3-valued worlds, or just worlds for short, with one of them @ singled out as privileged. (I prefer to think of them not as alternative possibilities, but as alternative methods for assigning semantic values to actual and possible sentences, given the way the world actually is in precise respects; @ represents ‘‘the actual assignment’’.) We need to equip W with a certain structure; I propose that to each w 2 W we assign a (possibly empty) directed family -Fw of nonempty subsets of W, which I call w-neighborhoods. To say that -Fw is directed means (*) (8w 2 W)ð8U1 , U2 2 -Fw Þð9U3 2 -Fw ÞðU3  U1 \ U2 Þ. I will add a few further conditions on the -Fw as we proceed. Think of each w-neighborhood as containing the ‘‘worlds’’ that meet a certain standard of closeness to w. I’m allowing that the standards of closeness for different w-neighborhoods may be incomparable, so that two w-neighborhoods for the same w can have members other than w in common without one being a subset of the other. (If you don’t want to allow for incomparability, you could replace (*) with the simpler requirement that the members of -Fw are linearly ordered by ; this would simplify the resulting theory a bit without drastically changing its character.) It would be natural from the motivation to assume that for any w, w is a member of each U in -Fw . But for reasons that might only be relevant to the semantic paradoxes, I want to allow for the existence of abnormal worlds that do not meet this condition; all I assume in general is that @ is normal. I should also say that in my intended applications, {@} 62 -F@ (indeed, it should probably be the case that for all w, {w} 62 -Fw ); so each member of -F@ includes a point other than @. The validities that follow do not depend on this, but if {@} were a member of -F@ we would get new validities that would rule out higher-order indeterminacy and prevent the application of the theory to the semantic paradoxes. A model for a vague language L will consist of a domain V and for each w 2 W an assignment to each n-place atomic predicate of a ‘‘3-valued exten-

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Semantic Paradoxes and Vagueness Paradoxes / 291 sion’’ in V n . (That is, to each (n þ 1)-tuple whose first element is an n-place predicate and whose other elements are in V, we assign at each w 2 W a member of {0, 12 , 1}.) I will later add a constraint on the assignment of these 3-valued extensions to atomic predicates. The valuation rules for the Kleene connectives and quantifiers at a given world are just the usual Kleene rules; no reference to other worlds is required. For ‘!’ I propose the following: jA ! Bjw is 1, if ð9U 2 -Fw Þ(8u 2 U)ðjAju  jBju Þ; 0, if ð9U 2 -Fw Þ(8u 2 U)ðjAju > jBju Þ; 1 2 otherwise: (The stipulation that the members of -Fw are all nonempty is needed to keep the value of jA ! Bjw unique.) If w is abnormal, -Fw has members that don’t contain w; we could without alteration restrict the quantification in the 1 and 0 clauses to such members, given the directedness condition. I call an inference universally valid if in any model and any world w in it, whenever all the premises have value 1 at w, so does the conclusion. I call an inference strongly valid if in any model, whenever all the premises have value 1 at all normal worlds, so does the conclusion. And I call an inference valid if in any model, whenever all the premises have value 1 at @, so does the conclusion. Validity, strong validity, and universal validity for a sentence are just validity, strong validity, and universal validity for the 0-premise argument whose conclusion is that sentence. Validity is the notion that will be of direct interest, but information about the other two helps in proving results about validity; besides, it illuminates the semantics to see what is universally valid, what is merely strongly valid, and what is just valid. The above stipulations effectively generalize what was done before in the case of the semantic paradoxes, though this may not be completely obvious. There, the space W was, in effect, a closed initial segment [0, D] of the ordinals, where D was one of the ‘‘acceptable ordinals’’ proved to exist in the Fundamental Theorem of [8];27 @ was D. For any ordinal a in W, the -Fa used in specifying the semantic values of conditionals at a had as its members all (nonempty) intervals of form [b, a). So no member of W was normal, not even D; however, I used a slightly different definition of validity, so the notion of validity didn’t trivialize. How then can I say that the account here 27 To quell a possible worry, I remark that it is possible to choose the acceptable ordinal ‘‘in advance’’. (Certainly the initial ordinal for any cardinal greater than that of the power set of the domain V will be acceptable; I suspect that there are acceptable ordinals of much lower cardinality.)

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292 / Hartry Field ‘‘effectively generalizes’’ the one there? Because the Fundamental Theorem shows that if we redefine -FD to include intervals of form [b, D] (making D normal), while leaving the other -Fa as before, then the values of conditionals are unchanged; this allows us to redefine validity in accord with the general account above, and the redefined one and the original are equivalent in extension. In short: the Restricted Semantics sketched in Section 3 could be rewritten so as to be a special case of the one just outlined. Returning to the general case, let’s get some results about which sentences and inferences are valid, strongly valid, and universally valid. Even without imposing any additional constraints, we can easily prove the following: 1. The universal validities include every inference that is valid in the Kleene logic K3 . In addition, the following inferences involving the conditional are universally valid: ‘A!A ‘ ::A ! A ‘ A ! A _ B and ‘ B ! A _ B ‘ A ^ B ! A and ‘ A ^ B ! B ‘ A ^ (B _ C) ! (A ^ B) _ (A ^ C) ‘ (A ! :B) ! (B ! :A) ‘ (A ! :A) $ :(> ! A) ‘ :(A ! B) ! (B ! A) (A ! B) ^ (A ! C) ‘ A ! (B ^ C) (A ! C) ^ (B ! C) ‘ (A _ B) ! C ‘ 8xA ! At (with the usual restrictions on legitimate substitution) ‘ 8x(A _ Bx) ! A _ 8xBx, when x is not free in A. 2. In addition, the following are strongly valid: A, A ! B ‘ B ‘ :(A ! B) ! (A _ :B). 3. If A ‘ C and B ‘ C are both valid, so is A _ B ‘ C; and analogously for strong validity and universal validity. (This means that we validate K3þ , not just K3 .) The proofs of all of 1–3 are completely straightforward; indeed, the only ones that even require the directedness condition are the two universal validities with conjunctions of conditionals as premises.

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Semantic Paradoxes and Vagueness Paradoxes / 293 We can do better if we add two additional constraints on the families -Fw . First let’s define U1 is w-interior to U2 to mean: U1 , U2 2 -Fw ^ U1  U2 ^ (8x 2 U1 ) (9 X 2 -Fx )(X  U2 ). The first constraint is (i) For any normal world w, every member of -Fw has w-interior subsets. The second constraint is a generalization of directedness. If › is a cardinal number, call a family -F of subsets of W ›-directed if for any collection S of subsets of -F that has cardinality no greater than ›, there is a U 2 -F such that U is contained in each member of S. (Directedness as defined before is 2-directedness; and is equivalent to n-directedness, for any finite n  2.) Then letting kVk be the cardinality of the domain V, we stipulate (ii) For any world w, -Fw is kVk-directed. We can now prove the following additional strong validities: A ! B ‘ (C ! A) ! (C ! B) A ! B ‘ (B ! C) ! (A ! C) 8x(Ax ! Bx) ‘ 8xAx ! 8xBx 8x(:Ax ! Ax) ‘ :8xAx ! 8xAx.28 Finally, let us now add the previously promised constraint on the assignment of 3-valued extensions to atomic predicates. Actually I’ll impose the constraint originally only for what I’ll call ‘‘standard’’ predicates, by which I’ll mean, all but special predicates like ‘True’. There’s no need to be very precise here, because the idea is that for any ‘‘nonstandard’’ predicates we introduce, we’ll prove that the constraint holds for them too, though this will not be part 28 Proofs: (I) A ! B ‘ (C ! A) ! (C ! B): Suppose that w is normal and that jA ! Bjw ¼ 1. Then there is a U in -Fw throughout which jAj  jBj. By (i), there’s a U 0 that is w-interior to U. Suppose now that j(C ! A) ! (C ! B)jw < 1. Then there’s a p 2 U 0 such that jC ! Ajp > jC ! Bjp . So either jC ! Ajp ¼ 1 or jC ! Bjp ¼ 0. But both lead to contradiction. For instance, in the first case, there’s a p-region X throughout which jCj  jAj; by directedness, there’s a p-region Y  X \ U 0 ; so throughout Y, jCj  jAj  jBj, so jC ! Bjp ¼ 1, giving the desired contradiction. The second case is analogous. (II) A ! B ‘ (B ! C) ! (A ! C): Analogous to (I). (Indeed, (II) follows from (I), given the laws already established.) (III) 8x(Ax ! Bx) ‘ 8xAx ! 8xBx: Suppose that w is normal and that j8x(Ax ! Bx)jw ¼ 1. Then for all t, jAt ! Btjw ¼ 1. So for each t, there is a w-region Ut such that jAtj  jBtj throughout Ut . By (ii), there is a w-region U such that for each t, jAtj  jBtj throughout U, and hence such that j8xAxj  j8xBxj throughout U. (Recall that every element of V is assumed to have a name.) So j8xAx ! 8xBxjw ¼ 1. (IV) 8x(:Ax ! Ax) ‘ :8xAx ! 8xAx: Analogous to (III).

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294 / Hartry Field of the initial stipulation governing their 3-valued extensions. So let p be a standard n-place predicate; we assume (**) for each choice of n objects o1 , ::: , on in V, if hp, o1 , ::: , on i is assigned an integral value (0 or 1) at @, then there is an @-neighborhood throughout which hp, o1 , ::: , on i is assigned that same value. We can now prove a general lemma: that for any sentence A all of whose predicates are standard, If A takes on an integral value at @ then there is a @-neighborhood throughout which A takes on that same value. The proof is by induction on complexity; the only noteworthy cases are the conditional and the quantifier. For the conditional, the noteworthy fact is that the proof for A ! B goes not by the induction hypothesis, but rather by (i). (If jA ! Bj@ ¼ 1 then there is an @-neighborhood U throughout which jAj  jBj; letting U’ be @-interior to U, it is clear that jA ! Bjp ¼ 1 for any p in U’. Similarly for 0.) For the quantifier case, we again use (ii). Given this lemma, we get two more validities (for sentences all of whose atomic predicates are standard): B ‘ A ! B. A, :B ‘ :(A ! B). The proof of the first is totally trivial (given the lemma), and the proof of the second almost so (it uses the directedness condition). The validities here established for the Generalized Semantics include almost all the ones that I derived in [8] for the Restricted Semantics (plus two additional ones that I neglected to consider there); the only one derived there that we don’t have here is the relatively minor one :[(C ! A) ! (C ! B)] ‘ :(A ! B). Actually we would get that too if we made the additional supposition that there is an @-neighborhood in which all worlds are normal; that assumption seems fairly plausible in applications of the logic to ordinary cases of vagueness, though it does not hold in the Restricted Semantics (where the validity in question holds for a different reason). It seems then that the logic validated by the Generalized Semantics is the main core of what’s validated by the Restricted Semantics, and I propose it as the unified logic for vagueness and the paradoxes. (There

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Semantic Paradoxes and Vagueness Paradoxes / 295 may be ways to expand it slightly, e.g. to get the above-mentioned law :[(C ! A) ! (C ! B)] ‘ :(A ! B).29) One thing that needs to be shown, if this is really to be a successful unification of the logic of vagueness with the logic of the paradoxes, is that the naive theory of truth is not merely ‘‘consistent with any classical starting model (that is adequate to syntax)’’, as proved in [8], but ‘‘consistent with any generalized model for the logic of vagueness (that is adequate to syntax)’’. More precisely, let L be any language without ‘True’, but which may contain vague terms and the ‘!’; we also suppose that L contains arithmetic, so that syntax can be done within it. Let M be any generalized model of the sort just described for a language without ‘True’ (where M is built from a set W of worlds with distinguished world @, an assignment to each of an -Fw , a domain V, and an assignment of 3-valued extensions in V n to each n-place predicate that is subject to (**) ); the only restriction is that the extension of arithmetical predicates be the same at each world and take on only the values 0 and 1, and that the resulting 2-valued model be a standard model of arithmetic. The claim then is that given any such generalized model M, we can extend it to a model M
on the same structure that satisfies naive truth theory; by ‘extend it’ I mean that each predicate other than ‘True’ has the same extension in M
as in M. This can in fact be shown, by a straightforward generalization of the proof given in [8]; and (**) turns out to hold for ‘True’ as well, so that even the final two validities established above hold in the full language.30 Without 29 One natural expansion—which wouldn’t yield that law, but might yield others—is suggested by the fact that in the Restricted Semantics there is a ‘‘uniformness’’ to the structure imposed on the worlds; that is, there is a way to compare the size of a1 -neighborhoods to those of a2 -neighborhoods even when a1 6¼ a2 , namely the ordinal that must be added to the lower bound of the interval to get the upper bound. The idea seems natural in the vagueness case too. So in the general case, we might want to add to the structure an equivalence relation on the set < of pairs hw, Ui for which U 2 -Fw , with hw1 , U1 i hw2 , U2 i having the intuitive meaning that U1 is a w1 -neighborhood and U2 is a w2 -neighborhood and U1 is like U2 in size, shape, and orientation from its ‘‘base point’’ (w1 or w2 as the case may be). Axioms, besides those of being an equivalence relation and that if hw1 , U1 i hw2 , U2 i then U1 is a w1 -neighborhood and U2 is a w2 -neighborhood, should certainly include (a) and probably (b): (a) hw, U1 i hw, U2 i  U1 ¼ U2 . (b) (8w1 , w2 )[F-w1 6¼ 1 ^ -Fw2 6¼ 1  (9U1 , U2 )[hw1 , U1 i hw2 , U2 i ^ (8X  U1 ) (9Y  U2 ) (hw1 , Xi hw2 , Yi) ^ (8Y  U2 )(9X  U1 )(hw1 , Xi hw2 , Yi)]. Just what additional strength this would bring to the logic I’m not sure. 30 An alternative way to think of the procedure, which illustrates more clearly the sense in which the proposed account is a unification, is as going from the initial M to a new model M



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296 / Hartry Field going through the details, let me just say that predicates other than ‘True’ are assigned the same 3-valued extension at every stage of the construction: there is no longer a need to vary them from stage to stage to get reasonable results about higher-order vagueness, since that is already handled by the multiple ‘‘worlds’’ in the initial stage. We thus avoid the artificiality that (early in this section) we saw would be required were we to use the Restricted Semantics for vagueness.

6. Defectiveness and Determinateness Whether or not one accepts the unification just proposed between vagueness/ indeterminacy and the semantic paradoxes, we’ve seen that in each case there are strong grounds for giving up the law of excluded middle: for certain sentences A, we should not assert that A _ :A. But we should not assert that :(A _ :A) either: that lands us in contradiction. How then do we assert the ‘‘defectiveness’’ of A? Not by simply asserting that we are not in a position to assert A _ :A: that’s too weak, since our inability to assert Attila’s maternal grandfather was bald on the day he died _ : (Attila’s maternal grandfather was bald on the day he died) is due not to conviction that its disjuncts are defective but to ignorance as to whether they are defective. (He might have been clearly bald or clearly nonbald.) Nor can we assert defectiveness by asserting that no possible evidence could lead anyone to assert A _ :A; examples about the interiors of black holes in indeterministic universes, or events outside of the event horizon of any regions that can support conscious beings, can easily be constructed to show this. I’ve suggested that our treating A as certainly with the same domain V, but a new set W
of worlds. On this account, W
is W [0, D], where D is a large ordinal (a sufficiently large power of !), and the new @
is h@, Di. For hw, ai in W
, -Fhw, ai consists of precisely the subsets of form U [b, a) where in the original model U 2 -Fw and where b < a, except in the case where a is D, in which case the subsets have form U [b, a]. Predicates other than ‘True’ get the same value at any hw, ai that they got in the original model at w; the value of ‘True’ at any hw, ai is determined by a Kripke construction of the minimal fixed point given the values of the sentences with no occurrences of ‘True’ outside the scope of conditionals. The model M
(in the formulation in the main text) is simply the submodel of M

consisting of worlds of form hw, Di. This alternative requires the modest additional assumption that for each w 2 W, -Fw is kDkdirected, not merely kVk-directed. It may not really be an additional assumption, as I suspect that we can take D to have cardinality kVk; but at the moment I don’t know how to prove that.

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Semantic Paradoxes and Vagueness Paradoxes / 297 defective consists (to a first approximation anyway) in our believing A _ :A to degree 0 (or to degree [0,1], if we use R-degrees instead of P-degrees; that’s the same as R-believing A to degree [0,1]). Similarly, our treating A as probably defective consists (to a first approximation) in our believing A _ :A to a low degree. So perhaps we could assert the ‘‘defectiveness’’ of A by saying that we ought to believe A _ :A to degree less than some amount (the amount fixed by the confidence of our assertion of defectiveness).31 But it’s natural to think that we should be able to assert the defectiveness in a more direct way, without bringing ourselves or other believers into the story. Can we do so? One possibility is simply to introduce an undefined operator DA into the language, give some laws (e.g. probabilistic laws) for how it works, and define a defectiveness predicate BAD(x) by :G(True(x)), i.e. :D(True(x)) ^ :D:True(x). But one worry is that we don’t really understand such an operator. In addition, in the semantic paradox case it raises a serious issue of ‘‘Liar’s revenge’’: there is reason to worry that it might give rise to new paradoxes, e.g. for sentences that assert of themselves that they are not determinately true. It turns out that in the logic suggested in Section 2 and the generalization of it suggested in Section 5, there is a good case to be made that the only defectiveness predicates we really need are already definable in the language, using the new conditional. Obviously that would remove the worry about the intelligibility of the operator, if one grants that the conditional itself is intelligible. Moreover, it would mean that no revenge problem can arise: the construction sketched in Section 2 (and later generalized) shows that we can validate naive truth theory for that language (using a certain nonclassical logic), and that means that we can validate it even for sentences about defectiveness and truth together, so that there is no revenge problem. A first stab at defining a determinately operator D might be this: D? A ¼df > ! A (or equivalently, :A ! ?) where > is some logical truth (say B ! B, for some arbitrarily chosen B), and ? is some absurdity. In the ‘‘worlds’’ semantics of Section 5, the value of this 31 It’s worth noting that we don’t presently have a satisfactory account of the laws of ‘‘probability’’ (rational degree of belief) for the logic: we do have a satisfactory probability theory for the Kleene logic, but it is not obvious how to extend it to the logic that includes the !. (In what follows I will suggest a possible definition of D in terms of !; that together with my earlier remark on the desired probability for sentences of form DA would give a constraint on how the probability theory is extended to the language with !, but would fall far short of settling it.)

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298 / Hartry Field at a given world w is 1 if there is a neighborhood of w throughout which A has value 1; 0 if there is a neighborhood of w throughout which A has value less than 1 (it needn’t be all 0 or all 12, it could be a mixture of the two); and 12 if there is neither of these sorts of neighborhood. The only clear problem with this as a definition of determinately is that if the world w is abnormal, the neighborhood needn’t include w, so D? A could have a higher value than A at w. But the situation can be remedied by a familiar trick: define D as DA ¼df (> ! A) ^ A (or equivalently, (:A ! ?) ^ A). (I use a different typeface than for D so as not to prejudge whether D adequately defines the notion we need.) Inferentially, the proposed definition of D obeys the obvious laws. Because the semantics validates the rule (8) B ‘ A ! B, we get the validity of the D-introduction rule A ‘ DA. Modus ponens guarantees the validity of D? A ‘ A, but ‘ D? A ! A is not valid (since @ might have abnormal worlds in all its neighborhoods); but now that we’ve shifted from D? to D, we get the universal validity of ‘ DA ! A (strong D-elimination). The definition is also quite natural in connection with the Restricted Semantics used for the paradoxes in Section 2: e.g. if A has value less than 1 at a stage, DA has value 0 at the next stage. Indeed, the merits of this definition of determinateness come out especially clearly in its consequences for the semantic paradoxes, in particular as regards the ‘‘revenge problem’’. It is to this that I now turn.

7. Revenge (1) A simple illustration of how the determinately operator D works is afforded by the Liar sentence Q0 , which asserts its own untruth. It’s clear that on the semantics of Section 2, it must have value 12 at every stage; so the value of DQ0 and of D:Q0 are both 0 at every stage after stage 0, and so BAD(hQ0 i) has

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Semantic Paradoxes and Vagueness Paradoxes / 299 ultimate value 1. (‘BAD’ is defined from D in the same way that ‘BAD’ is defined from D—see the previous section.) Once we have a determinately operator D, it’s natural to consider a ‘‘weakened’’ Liar sentence Q1 , that says of itself that it is not determinately true.32 But the construction outlined in Section 2 shows that this must be consistently evaluable in the language. Indeed, it isn’t hard to see that its value is 12 at all even ordinals and 1 at all odd ordinals. DQ1 gets value 12 at all even ordinals and 0 at all odd ordinals; :DQ1 thus has the same value as Q1 , as desired. :D:Q1 gets ultimate value 1, as we might expect: so we can assert that Q1 is not determinately untrue. As for the claim that Q1 is determinately true, its ultimate value is 12, so we can’t assert DQ1 _ :DQ1 (and indeed, can reject it). So excluded middle can’t be assumed (and indeed, can be rejected) even for claims of determinateness: that is, we have a kind of second-order indeterminacy. But we can assert that Q1 isn’t determinately determinately true. So we can assert that Q1 is BAD2 , where BAD2 (x) means that :DD(True(x)) ^:DD:True(x). If we rename the earlier predicate BAD as BAD1 , then badness1 entails badness2 , but not conversely. We can now consider a ‘‘still weaker’’ Liar sentence, that says of itself that it is not determinately determinately true, and so forth. Indeed, we can iterate the determinately operator a fair way through the transfinite: for as long as our system of ordinal notations lasts.33 For each s for which there is a notation, we can then consider the s-Liar Qs , which says of itself that it is not Ds -true (Ds being the s-fold iteration of D). We will never be able to assert Ds Qs _ :Ds Qs , whatever the s, showing that indeterminacy of arbitrarily high levels34 is allowed; but we can assert that Qs is not determinately untrue, and not determinatelysþ1 true. (There will be a notation for s þ 1 whenever there is one for s.) Defining the predicates BADs in analogy to the above, we get that each is more inclusive than the previous, and for each Liar sentence, there will be a s for which the Liar sentence can be asserted to be BADs . This hierarchy of defectiveness predicates has something of the flavor of the hierarchy of truth predicates that we have in the classical case. But it is different in important ways. For one thing, it affects a much more peripheral 32 ‘Weakened’ is in quotes since though Q1 attributes a weaker property than Q0 does, it attributes it to a different sentence. 33 At limit stages we need to form infinite conjunctions; but we can do that, for limits simple enough to have a notation in some reasonable system, by using the truth predicate. 34 ‘Arbitrarily high levels’ means ‘for as large ordinals as there are notations’; for higher ordinals than that, talk of levels of indeterminacy makes no clear sense.

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300 / Hartry Field notion: the relevant notion of defectiveness (or paradoxicalness) plays only a marginal role in the lives of most of us, whereas the notion of truth is ubiquitous. I don’t think it is that hard to learn to live with the idea that we do not have a unified notion of defectiveness* that captures all ‘‘levels of defectiveness’’. A second difference is that we can reasonably hope that whatever the s, our overall theory has the virtue of nondefectivenesss ; this is in sharp contrast to the hierarchical classical truth theories, where for each s there are sentences of our overall theory that we assert while asserting not to be trues . So I don’t think that settling for the hierarchy of defectiveness predicates would be debilitating. (Indeed, it is especially undebilitating if we allow schematic reasoning about levels of determinateness: see [4], [15], and pages 141–3 of [5] for discussions of schematic reasoning in other contexts.35) So far I’ve been talking about determinately operators and defectiveness predicates that are actually definable in the language. But it might be thought that these definitions do not adequately capture the intuitive notions of determinateness and defectiveness—or to put it in a way that avoids the supposition that there are unique such notions, it might be supposed that there are intuitive notions of determinateness and defectiveness that are not adequately captured by these definitions. (I don’t myself know of any clear reason to suppose that there are intuitive notions that the definitions fail to capture, but I see no reason to be dogmatic about the issue.) So we might want to add a new primitive determinately operator, governed by axioms, from which a corresponding defectiveness predicate could be defined. By using the axioms that we know are true of the defined operator D as a guide to the choice of axioms for the primitive operator D, one can have a reasonable hope of achieving a consistent theory of determinateness slightly different from that given by D or any of its iterations. The most natural idea would be to model such a D quite closely on D. In that case, we would expect that iterations of the operator are nontrivial: more particularly, that for each value of s, the Ds -Liar can’t be asserted not to be Ds -true but can be asserted not to be Dsþ1 -true. I don’t say that it would be hopeless to add to the language a ‘‘maximum strength’’ determinately operator D
that iterates only trivially, in that D
D
is equivalent to (intersubstitutable with) D
; that would give rise to a unified 35 Allowing this is effectively the same as allowing universal quantification over the s, but not allowing such universal quantifications to be embeddable in other contexts like negations. But whereas a restriction on embedding seems grossly ad hoc, the use of schematic reasoning has a natural rationale.

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Semantic Paradoxes and Vagueness Paradoxes / 301 defectiveness predicate BAD
(x). But I have doubts as to whether we really understand this. In any case, if we do add such an operator, we must be careful that the combination of laws we postulate for it does not breed paradox, and the paradox-free ways of introducing such a D
are not altogether attractive. Option 1. Suppose we want both the inferential rules D
A  A and A  D
A (perhaps strengthening the first to  D
A ! A); if so, and we also keep the rule of disjunction-elimination as a general logical law, then we must reject excluded middle for a sentence Q
that asserts that it is not D
-true.36 But if being forced to reject excluded middle shows a kind of indeterminacy, then any determinateness operator or defectiveness predicate subject to the rules D
A  A and A  D
A thus embodies a kind of indeterminacy (in a background logic like K3þ that allows for reasoning by disjunctionelimination). And this kind of indeterminacy (if that is the right way to think of it37) isn’t expressible by :D
: to assert :D
True (hQ
i) would be tantamount to asserting Q
and thus lead to paradox. Thus D
wouldn’t seem to be the most powerful determinateness operator after all, in which case the introduction of the notion would be self-defeating. Option 2. That problem could be avoided by giving up the rule A  D
A; we can then keep  D
A ! A, disjunction-elimination, and excluded middle for sentences beginning with D
, for D
True(x) becomes very much like the unadorned truth predicate in classical theories with (T

). The problem with classical theories containing (T

) was that in them we have to assert certain claims and then deny that they are true, which seems decidedly odd. Under Option 2 we don’t have precisely that problem (indeed, whenever we can assert A, we can assert that it is true), but we have something uncomfortably close to it: we assert of specific sentences A of 36 The assumption of D
Q
leads directly to paradox, as does the assumption of :D
Q
; so by disjunction-elimination, the assumption of D
Q
_ :D
Q
leads to contradiction; but that assumption is the instance of excluded middle in question. 37 It may not be, for the italicized supposition is in fact dubious. The reason: because of higherorder indeterminacy, excluded middle must fail for claims of defectiveness, so the degree of belief that a claim is defective and the degree of belief that it is nondefective need not sum to 1. I’ve equated the degree of belief in A’s nondefectiveness with the degree of belief in A _ :A; that means that I can’t in general equate the degree of belief in A’s defectiveness with 1 minus the degree of belief in A _ :A, but can only say that the degree of belief in A’s defectiveness is no greater than 1 minus the degree of belief in A _ :A. So though we may reject Q
_ :Q
, indeed believe it to degree 0, it does not follow that the degree of belief in the defectiveness of Q
need be high.

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302 / Hartry Field our own theory that they are not D
-true, which raises the obvious question ‘‘If you think those specific A aren’t D
-true, why did you assert them?’’ For this reason, I think that Option 2 should be avoided. Option 1 may not be beyond defense. Defending it would require refusing to accept that the ‘‘D
-liar’’ Q
is either defective or nondefective; indeed, it would require having degree of belief 0 in this disjunction. So we would believe the equivalent disjunction Q
_ :Q
to degree 0, and would not be able to explain our doing so in terms of a belief that its disjuncts were defective. (Or in any sense ‘‘quasi-defective’’: for the point of D
has been stipulated to be that there is no further such notion.) The rationale for this would be that to assert Q
_ :Q
requires more than that Q
be nondefective, it requires that it be assertable that it is nondefective; so the inability to assert the nondefectiveness of Q
is enough to explain our inability to assert Q
_ :Q
without commitment as to whether Q
actually is nondefective. As I say, this may be defensible; but it is better, I think, to regard the operator D
as not fully intelligible and to simply make do with a hierarchy of determinateness operators.

8. Revenge (2) But don’t we already have a unified defectiveness predicate, namely, ‘has ultimate semantic value 12’? And a predicate corresponding to a unified determinateness operator, namely, ‘has ultimate semantic value 1’? These predicates needn’t even be added to the language: they are already there, at least if (i) the base language from which we started the construction in Section 2 included the language of set theory and (ii) the model M0 for the base language from which the construction started is definable in the base language;38 for then the set-theoretic construction of Section 2 can be turned into an explicit definition of these predicates in terms of the vocabulary of the base language. And since we have assumed excluded middle for the base language, the definitions show that excluded middle must hold for attributions of semantic value based on these definitions. Can’t we then reinstitute a paradox, based on sentences that attribute to themselves a semantic value of less than 1? 38 Condition (ii) will be met for any starting model that is at all natural (provided that condition (i) is met).

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Semantic Paradoxes and Vagueness Paradoxes / 303 No we can’t, and the construction of Section 2 shows that we can’t: our construction yields the consistency of the claim True(hkAk 6¼ 1i) $ kAk 6¼ 1, and of all other instances of (T) in the language. (Indeed, it shows that we can add all such claims to the base theory, without disrupting any given model of that theory.) But it may seem puzzling how paradox has been avoided. Why isn’t the sentence that attributes to itself a value less than 1 paradoxical? The answer to this has more to do with the limitations of explicit definition than it does with the notions of truth and determinacy. Let L be a language that includes the language of classical set theory. L may contain terms for which excluded middle is not valid, but let us assume that the set-theoretic portion of L is classical: excluded middle (and other classical axioms and rules) all hold for it. Tarski’s undefinability theorem shows that if we insist on explicitly defining a predicate, say ‘is a sentence of L that has ultimate semantic value 1’, in classical set theory, then the defined predicate can’t correspond to any normal notion of truth: it gives the intuitively wrong results even for the classical part of L. More fully, there will be a sentence A of classical set theory, not containing ‘True’, such that we can prove that either A ^ :( k A k¼ 1) or ( k A k¼ 1) ^ :A. (With minimal additional assumptions, ones that are met by our definition of semantic value, we can indeed specify a sentence B such that one disjunct fails when A is taken to be B and the other fails when A is taken to be :B.39) Thus the notion of ultimate semantic value we’ve defined in classical set theory doesn’t fully correspond to the intuitive notion of truth even for sentences that are in no way indeterminate or paradoxical. (The reason is that in order to give a definition of semantic value we have to pretend that the quantifiers of the language range only over the members of a given set, namely, the domain of the starting model, rather than over absolutely everything. What we’ve defined should really be called ‘ultimate semantic value relative to the particular starting model M0 ’.) That’s why in constructing a theory of truth we need to add ‘True’ as an undefined predicate. Since ‘having ultimate value 1 relative to M0 ’ doesn’t quite correspond to being true, even for sentences that are in no way indeterminate or paradoxical, it also doesn’t quite correspond to being determinately true; for as 39 The required assumptions are that every sentence of classical set theory gets one of the values 0 and 1, and that k :A k is 1 when k A k is 0 and 0 when k A k is 1.

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304 / Hartry Field applied to sentences of this sort, truth and determinate truth coincide. And that is why sentences that assert that their own ultimate value is less than 1 (relative to M0 ) are not genuinely paradoxical: no one who clearly thought through the limitations of Tarski’s theorem should have expected ‘having semantic value 1 relative to M0 ’ to precisely correspond to any intuitive notion of determinate truth. (In particular, it is clear from Tarski’s theorem that one of the rules ( k A kM0 ¼ 1)  A and A  ( k A kM0 ¼ 1) must fail even for sentences of set theory that don’t contain notions like truth or determinateness—indeed, both rules must fail—so the notions of truth and determinateness are in no way to blame. Without such rules, the argument for inconsistency collapses.) I’ve argued that we can’t within classical set theory define any notion of determinate truth that fully corresponds to the intuitive notion (if there is a unique intuitive notion). This is unsurprising; for even if there is a unique intuitive notion, it doesn’t obey excluded middle and so couldn’t possibly be defined within a classical language. In any case, any explicit definition of ‘has semantic value 1’ in a classical metalanguage is bound to make this notion demonstrably fail to correspond to any reasonable notion of determinateness, for the reasons just reviewed. But that raises the question: what is the point of explicitly defining semantic value within classical set theory? I addressed this question in the context of vagueness at the beginning of Section 5, but it is worth going through a parallel discussion here. One very important function of explicitly defining ‘has semantic value 1 (relative to starting model M)’ within classical set theory is that doing so enables us to increase our confidence that we have the principles of the nonclassical logic right. For we have an immense amount of experience with classical set theory, enough to make us very confident of its !-consistency, and hence of the correctness of its claims of consistency. The semantics I’ve provided for truth theory, despite its distortions, gives a proof within classical set theory of the consistency of naive truth theory in a nonclassical logic, so we know that that logic is indeed consistent. Indeed, the particular form of the proof shows that the resulting theory is far more than just consistent, it is ‘‘consistent with any arithmetically standard starting model’’; ‘‘conservative’’,

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Semantic Paradoxes and Vagueness Paradoxes / 305 in one sense of that phrase. That implies that it is !-consistent; it also implies that inconsistency can’t be generated by combining it with a consistent ‘True’-free theory, e.g. a consistent theory about the history of the Paris Commune. The explicit definition of semantic value (relative to a starting model) thus serves a very important logical purpose, even if (as we’ve seen must be the case) there is no way to turn it into an absolute definition of semantic value that completely corresponds to any intuitive notion that relates to the full universe. I don’t mean to suggest that the only function of the semantics is to give a consistency proof for naive truth theory in the logic. After all, it is certainly not the sole demand on a logic that it make naive truth theory consistent; the logic should also be intuitive, and there is no doubt that finding a notion of semantic value in terms of which the valid inferences can be characterized can play a substantial role in making the logic intuitive. If one starts the construction from a model M0 that is sufficiently natural (e.g. it is just like the real universe except for not containing sets of inaccessible rank), then the defined notion of having semantic value 1 relative to M0 is very close to what we’d want of an intuitive notion of determinate truth (e.g. in the parenthetical example, it gives intuitive results as applied to any sentence all of whose quantifiers are restricted to exclude sets of inaccessible rank); so inferences that preserve value 1 will be valid in a fairly natural sense. In my view, this fact does a great deal to help make the logic intuitive. It is hard to see how we could do much better in defining determinate truth for this logic in a classical metalanguage, given that the notion of determinate truth (if there is a unique such notion) is nonclassical. This does raise an interesting question: mightn’t we develop a nonclassical set theory or property theory without excluded middle, and within it define a semantic value relation which is appropriate to the full universe (rather than just to the domain of a classical starting model), but which is otherwise closely analogous to the definition of the model-relative notion of semantic value given earlier? And mightn’t this both validate the same logic, and be such that the defined notion ‘has real semantic value 1’ completely corresponds to ‘‘the intuitive notion of determinate truth’’? The first part of this can definitely be done: naive property theory (property theory with unrestricted comprehension) is consistent in the same extension of Kleene logic used for the semantic paradoxes: see [9]. (As I briefly discuss there, there is some difficulty in extending this is to naive set theory (i.e. adding an extensionality principle), though I don’t completely rule out that this can be done.) But as

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306 / Hartry Field for the program of using such a theory (whether the nonextensional one or the contemplated extensional one) for a semantics closely modeled on the one given here,40 I’m skeptical: the notion of having real semantic value 1 would seem to have to be a unified determinateness predicate much like the D
True(x) of the previous section; and while this might not raise difficulties about validating the full logic advocated in this paper as regards sentences not containing that predicate, it would cause difficulties about validating the application of some of the rules to sentences that do contain that predicate.41 (A similar reservation arises for the idea of simply taking a notion of having real semantic value 1 as an undefined primitive and using it in a semantics; the semantics might be OK for the language without the addition, but not for the expanded language.) How worrisome is this? Not very. As I already observed in note 25 and as countless others have observed before me, a semantic theory expressed in a given logic can explain or justify that logic only in a very minimal sense: the ‘‘explanation’’ or ‘‘justification’’ it gives of its logical principles will not only employ those very logical principles, it will usually do so in a grossly circular fashion. Logic must stand on its own; the role of a formal semantics for it (in so far as it goes beyond being a ‘‘merely algebraic tool’’ for consistency proofs), especially a nonhomophonic one (see note 40), is as a heuristic. We who use a language L governed by the logic I’m recommending get a pretty good though not perfect heuristic for that logic without expanding the language beyond L, and indeed by using merely the classical subpart of L. If we insist upon a heuristic that avoids the problem of the classical one then we probably do need to expand L a bit; perhaps this would be done by adding a relation of ‘‘real semantic value’’ that is not in L but is governed by the same logic as L. If the use of the logic of L had to depend on the semantics that might be 40 As opposed, e.g., to a ‘‘homophonic semantics’’ based on claims like True(hA ! Bi) $ [True(hAi) ! True(hBi)] or True(hDAi) $ D(True(hAi)); and also as opposed to a conceptual role semantics of some sort. 41 An alternative to modeling the notion of having semantic value 1 on a unified determinateness predicate would be to suppose it hierarchical: i.e. suppose that though the inference from ‘hAi has semantic value 1’ to ‘hhAi has semantic value 1i has semantic value 1’ is valid, still the second is stronger in that the conditional with the first as antecedent and the second as consequent is not valid. But this too would cause difficulties about validating the application of some of the rules to sentences that contain the predicate ‘has semantic value’.

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Semantic Paradoxes and Vagueness Paradoxes / 307 worrisome, but it doesn’t. Indeed, we don’t even need to regard the heuristic as completely intelligible: cf. frictionless planes. Unlike the notions of truth and determinateness, the notion of semantic value is a technical notion of formal semantics (sometimes a technical notion for giving consistency proofs, sometimes a technical notion for heuristic ‘‘explanations’’ of logical principles, sometimes both). The notion of truth, on the other hand, is certainly not a mere technical notion: as is well known ([21], ch. 1; [16]), the role of a truth predicate is to serve as a device for conjoining and disjoining sufficiently regular sets of sentences not otherwise easily conjoined and disjoined, and this role (which has great importance in everyday life) has little or nothing to do with formal semantics. To serve this role, what is needed of a notion of truth is that it adhere as closely as possible to the naive theory of truth, and the truth theory in this paper adheres to that completely. Settling for only a model-relative notion of truth would be a huge defeat, in a way that settling for only a model-relative notion of semantic value is not. The role of the notion of determinateness is also not especially technical: we would like to be able to assert the defectiveness of certain sentences in the language, such as Liar sentences and certain sentences that crucially employ vague terms, and the determinateness predicate allows us to do so. (It isn’t something we need to add on to the language; we’ve seen that we get it for free once we have a conditional suitable for reasoning in absence of excluded middle and for expressing the naive theory of truth.) We’ve also seen that a single notion of determinateness can be iterated to express the defectiveness even of sentences (e.g. extended Liar sentences) that involve the notion of determinateness. Unfortunately we can’t get a single unified notion of defectiveness, but must rest content with an increasing hierarchy; but I’ve argued that to be less problematic than a hierarchy of truth predicates would be. There’s no need to go beyond the language L except perhaps for introducing a heuristic for the logic we employ; and we don’t even need to do so for that, if we are satisfied by the heuristic given by the classical semantics.

9. Conclusion The logic I’ve proposed for dealing with the paradoxes seems quite satisfactory, but there could well be others that are equally satisfactory, or better. (I think the good ones are all likely to be ‘‘similar in spirit’’, in that they will be

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308 / Hartry Field obtained from the Kleene logic K3þ by adding a new conditional, in a way consistent with the naive theory of truth.) I’d like to believe that the good ones can all be unified with an adequate logic of vagueness, by a unified semantics something like the one suggested in Section 5 for the logic of paradoxes proposed here. For we’ve seen some rather general considerations that suggest the naturalness of unification: .

.

.

.

.

The semantic paradoxes seem to arise from the fact that the standard means for explaining ‘True’ (namely, the truth schema) fails to uniquely determine the application of the term to certain sentences (the ‘‘ungrounded’’ ones); and this seems to be just the sort of thing that gives rise to other sorts of vagueness and indeterminacy. For both the semantic paradoxes and for vagueness, it seems important to give up certain instances of the law of excluded middle. In the case of the paradoxes, this is required if we are to consistently maintain the naive theory of truth; in the case of vagueness, we must do so to resist the view that for every meaningful question there is a fact of the matter as to its answer. For both cases, giving up excluded middle doesn’t involve denying it, but rather, rejecting it, in a sense that requires explanation. (I believe that the account of rejection offered in Section 3, in terms of degrees of belief, is adequate to both cases. In the case of the semantic paradoxes that don’t turn on empirical premises, there is little need to invoke degrees of belief other than 0 and 1; which is why the discussion of degrees of belief ended up playing little role in my discussion of the semantic paradoxes.) In both cases we need a reasonable conditional, not definable within Kleene logic. This is needed to allow for natural reasoning. It is also needed for more particular purposes: in the case of the semantic paradoxes, it is needed if we are to maintain the standard truth schema; in the case of vagueness, it is needed to allow the representation of penumbral connections between vague concepts, and also needed for expressing such obvious truths as that if 947 is nearly 1,000 then so is 953. In both cases we need to be able to express not only a notion of truth (obeying the naive truth schema) but also a notion of determinate truth; so we need a determinately operator D. In both cases it is fairly natural to try to define it from the conditional. And in both cases it is natural to expect arbitrarily high orders of indeterminacy, at least in the following sense: no matter how far we iterate D—even into the transfinite, using

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Semantic Paradoxes and Vagueness Paradoxes / 309 the truth predicate—we never get excluded middle back. In other words, for any s, we will never have Ds A _ :Ds A as a general law. Do we get arbitrarily high orders of indeterminacy in the stronger sense, that for each s there are sentences A and possible circumstances under which we can assert :Dsþ1 A but cannot assert :Ds A? As we’ve seen, we get that in the case of the semantic paradoxes, so it is consistent with the generalized semantics. I doubt, though, that we want it in the case of vagueness or ordinary indeterminacy, and I doubt that any choices of the generalized models of Section 5 that are natural to employ in these cases will yield the result. It seems to me that the best way to represent higher-order vagueness in a classical model is simply to make it the case that no matter how far you reiterate the determinately operator D, you never bring back excluded middle. As I noted early in Section 5, a model for vagueness in a classical metalanguage is inevitably somewhat distorting. The most obvious distortion is that it assigns each sentence exactly one of the semantic values 1, 12, or 0. It is thus natural to think that when a sentence A gets value 12, then even when the semantics does not justify the assertion of :DA, then the semantics is really treating the sentence as indeterminate; in which case the semantics is drawing a sharp line between those sentences it treats determinate and those it doesn’t. If so, then the semantics does, in a sense, rule out higher-order vagueness, despite the fact that excluded middle can fail for sentences of form DA (and indeed, for sentences of form Ds A for arbitrarily high s). But this problem (which is an analog for vagueness of the version of the revenge problem discussed in the previous section) seems simply to be the inevitable result of using a classical metalanguage to do the semantics. As noted in both Sections 5 and 8, the only way we could hope to get a formal semantics that portrays a nonclassical language without distortion is to give that semantics in a nonclassical metalanguage. Whether even the use of a nonclassical metalanguage would enable us to develop a semantics (of a basically truth-theoretic sort) that is both more informative than a merely homophonic semantics and in no way distorting is a matter on which I take no stand. (I have expressed some skepticism as to whether such a semantics is needed: it certainly isn’t needed for an understanding of the language.) In any case, it would be hard to give the semantics in this way prior to getting a grasp on what the logic to be used in the semantic theory (and in the object

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310 / Hartry Field language) ought to be; that’s why in this paper, in exploring what the logic ought to be, I have restricted myself to the use of a classical metalanguage.

REFERENCES [1] B r a d y , R o s s T. (1989), ‘The Non-triviality of Dialectical Set Theory’, in Graham Priest, Richard Routley, and Jean Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent (Munich: Philosophia Verlag). [2] D u m m e t t , M i c h a e l (1978), ‘The Justification of Deduction’, in Dummett (ed.), Truth and Other Enigmas (Cambridge, Mass.: Harvard University Press). [3] F e f e r m a n , S o l o m o n (1984), ‘Toward Useful Type-Free Theories, I’, Journal of Symbolic Logic, 49: 75–111. [4] —— (1991), ‘Reflecting on Incompleteness’, Journal of Symbolic Logic, 56: 1–49. [5] F i e l d , H a r t r y (2001), Truth and the Absence of Fact (Oxford: Oxford University Press). [6] —— (2002), ‘Saving the Truth Schema from Paradox’, Journal of Philosophical Logic, 31: 1–27. [7] —— (2003), ‘Is the Liar Sentence both True and False?’, in JC Beall and Brad Armour-Garb (ed.), Deflationism and Paradox (Oxford: Oxford University Press). [8] —— (2003), ‘A Revenge-Immune Solution to the Semantic Paradoxes’, Journal of Philosophical Logic, 32: 139–77. [9] —— (forthcoming), ‘The Consistency of the Naive Theory of Properties’, Philosophical Quarterly, 54. [10] F i n e , K i t (1975), ‘Vagueness, Truth and Logic’, Synthese, 30: 265–300. [11] F r i e d m a n , H a r v e y , and M i c h a e l S h e a r d (1987), ‘An Axiomatic Approach to Self-Referential Truth’, Annals of Pure and Applied Logic, 33: 1–21. [12] G u p t a , A n i l , and N u e l B e l n a p (1993), The Revision Theory of Truth (Cambridge, Mass.: MIT Press). [13] H a j e k , P e t r , J e f f P a r i s , and J o h n S h e p h e r d s o n (2000), ‘The Liar Paradox and Fuzzy Logic’, Journal of Symbolic Logic, 65: 339–46. [14] K r i p k e , S a u l (1975), ‘Outline of a Theory of Truth’, Journal of Philosophy, 72: 690–716. [15] L a v i n e , S h a u g h a n (1994), Understanding the Infinite (Cambridge, Mass.: Harvard University Press). [16] L e e d s , S t e p h e n (1978), ‘Theories of Reference and Truth’, Erkenntnis, 13: 111–29.

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Semantic Paradoxes and Vagueness Paradoxes / 311 [17] L e v i , I s a a c (1974), ‘On Indeterminate Probabilities’, Journal of Philosophy, 71: 391–418. [18] M c G e e , V a n n (1985), ‘How Truthlike can a Predicate Be? A Negative Result’, Journal of Philosophical Logic, 14: 399–410. [19] M o n t a g u e , R i c h a r d (1963), ‘Syntactic Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability’, Acta Philosophica Fennica, 16: 153–67. [20] P r i e s t , G r a h a m (1987), In Contradiction (Dordrecht: Martinus Nijhoff). [21] Q u i n e , W. V. O. (1970), Philosophy of Logic (Englewood Cliffs, NJ: Prentice-Hall). [22] R e s t a l l , G r e g (1992), ‘Arithmetic and Truth in Lukasiewicz’s Infinitely Valued Logic’, Logique et Analyse, 139–40: 303–12. [23] W i l l i a m s o n , T i m o t h y (1994), Vagueness (London: Routledge).

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14 New Grounds for Naive Truth Theory S. Yablo

A theory is semantically closed if it contains, for each sentence A of the language in which it is framed, all biconditionals of the form T[A]
A.1 Tarski showed that no consistent first-order theory with a good grip on its language’s syntax could be semantically closed. This is because a theory with a good grip on its language’s syntax will (by the diagonal lemma) contain for each formula w(x) a biconditional E
w[E], hence in particular a biconditional L
:T[L]. A theory with L
:T[L] cannot in consistency also contain T[L]
L. But that is what semantic closure requires. There is no denying Tarski’s result, but one can try to steer around it. The ways of steering around it correspond to a number of things that Tarski did not show. .

He did not show that a theory with a poor grip on its language’s syntax could not be consistent and semantically closed. One way to avoid Tarski’s result is to insist that all reference to sentences be by means of quotation names.2 There is nothing to prevent a consistent theory from containing all biconditionals of the form T0 A0
A, which on the stated hypothesis is all the T-biconditionals there are. (See Gupta 1982.)

1 Every sentence is assumed to have at least one name in the language. ‘[A]’ is schematic over names of A. 2 I am oversimplifying, since paradoxes can also be fashioned out of syntactic predicates. See Gupta (1982) for details.

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New Grounds for Naive Truth Theory / 313 .

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.

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He did not show that a consistent, syntactically resourceful theory could not be weakly closed in the sense of containing, not perhaps all T-biconditionals, but at least one such biconditional for each A. A second way to avoid Tarski’s result is to construct the T-biconditional for L using a different name from the one the theory uses to affirm that L
:T[L]. (See Skyrms 1984.) Tarski did not show that a consistent, syntactically resourceful theory could not be partly closed in the sense of containing, not perhaps all T-biconditionals for every sentence A, but all for sentences A of a particularly well-behaved type. A third way to avoid Tarski’s result is to require T-biconditionals for ‘‘nice’’ sentences only, settling for approximations thereto when A is not nice. (See Feferman 1984.) He did not show that a consistent, syntactically resourceful theory could not be quasi-closed in the sense of containing all sentences of the form T[A] # A, where # expresses a type of equivalence other than the type expressed by
. A fourth way to avoid Tarski’s result is to add a nonclassical conditional ! to the language and include in your theory all instances of (T[A] ! A) ^ (A ! T[A]), that is, T[A] $ A. (See Brady 1989.) Tarski did not show that a semantically closed theory could not be consistent by the lights of a suitably chosen non-classical logic. A fifth way to avoid Tarski’s result is to concede classical inconsistency but maintain that this or that classical rule is in the present context invalid. (See Priest 1979.)

All of these strategies have been tried. But the last two have been tried the least. Field’s approach combines elements of both. He adds a non-classical connective ! while at the same time de-classicalizing the logic of the other connectives (by rejecting excluded middle). This makes for a remarkably powerful package, as you can see from the list now provided of its Top Ten Excellent Features 1. Verifies all sentences of the form T[A] $ A:3 2. Makes T[A] and A substitutable ‘‘salva valutate’’. 3. Provides an explicit model, hence . . . 4. No possibility of hidden paradoxes. 5. High degree of revenge-immunity, even though . . . 3 Here and throughout, C $ D ¼df (C ! D) ^ (D ! C).

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314 / S. Yablo 6. 7. 8. 9. 10.

Vengeance-threatening notions are expressible. ! has a natural semantics. ! has a -like logic. ! ‘‘becomes’’  in bivalent contexts. Only theory to inspire a Top Ten list.4

These features will come in for further explanation below, but let me say now what is meant by feature 2. T[A] and A are substitutable ‘‘salva valutate’’ iff for all sentential contexts w(:::), !-contexts not excluded, w(T[A]) agrees with w(A) on whatever semantic values there happen to be. Well, what semantic values do there happen to be? Field uses a three-valued scheme in which sentences are assigned either 1 (determinate truth, or something like it), 0 (determinate falsity), or 12 (indeterminate). Values are generated by a transfinite series Pa of Kripkean fixed points, each the Kripkeclosure (see below) of a ‘‘seed’’ valuation Sa that assigns values only to conditionals, defined here as statements whose main connective is !. Each Pb gives clues to the proper interpretation of !, clues that guide the construction of Sbþ1 , which forms the basis for Pbþ1 . This results in the back-and-forth process shown in Figure 14.1.5 Three things need to be explained in this figure. (a) How is Sa built up into a P ? (b) How is Sa calculated on the basis of earlier Pb s? And (c) how does the process determine semantic values for sentences? The answer to (a) comes from Kripke. The answer to (c) comes from Herzberger and Gupta. The answer to (b) is new with Field and the key to his construction. To begin with (a), we can think of valuations (e.g. Sa and Pa ) as sets of ordered pairs < A, v >, where A is a sentence and v is a truth-value, either 0 P0

S0 =

P1

S1

P2

S2

P3

S3

Pl

Sl

Pl+1

Sl+1

Sl+2

FIG. 14.1 4 OK, so I ran out of ideas. 5 Yablo (1985) also used a series of Kripkean fixed points to interpret a non-monotonic truth predicate. Mention was made at the end of extending the method to ‘‘Łukasiewicz implication’’(?), but I seem to remember having no idea what I was talking about.

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New Grounds for Naive Truth Theory / 315 or 1. A valuation V is Kleene (strictly, Kleene over M, for M a model of the base language) iff (1) (2) (3) (4) (5) (6) (7) (8)

< Fa, 1 >2 V iff M(a) 2 M(F).6 < Fa, 0 >2 V iff M(a) 62 M(F). < :A, 1 >2 V iff < A, 0 >2 V. < :A, 0 >2 V iff < A, 1 >2 V. < A v B, 1 >2 V iff < A, 1 >2 V or < B, 1 >2 V. < A v B, 0 >2 V iff < A, 0 >2 V and < B, 0 >2 V. < 8x Gx, 1 >2 V iff < Ga, 1 >2 V for each name a7 . < 8x Gx, 0 >2 V iff < Ga, 0 >2 V for some name a.

V is Kripke iff (9) < T[A], 1 >2 V iff < A, 1 >2 V. (10) < T[A], 0 >2 V iff < A, 0 >2 V. A fixed point—of Kripke’s jump operator, but let that be understood—is a valuation that is both Kleene and Kripke. Provided a valuation V is sound in the sense of satisfying the left-to-right directions of (1)–(10), it can be built up into a fixed point V* by closing under the right-to-left directions of (1)–(10). Note that Sa is automatically sound because it assigns truth-values only to conditionals, and there are no conditionals on the left sides of (1)–(10). The relation between Pa and Sa is simply that Pa ¼ Sa . Now let’s consider (b) how Sa is calculated on the basis of earlier Pb s. Given our treatment of valuations as (not necessarily single-valued) relations between sentences and truth-values, A’s semantic value in V is best understood as {vj < A, v >2 V}. Thus, limiting ourselves for now to consistent valuations, the semantic values are {1}, {0}, and {}. These values are ordered in the obvious way: {1} > {} > {0}. (Sometimes we use Field’s notation and write {1} as 1, {0} as 0, and {} as 12; then the ordering is by numerical size.) If a is a successor ordinal b þ 1, then Sa is   < A ! B, 1 > jPb (A)  Pb (B) [ < A ! B, 0 > jPb (A) > Pb (B)g. If a is a limit ordinal, then Sa is  b b < A ! B, 1 > j9g < a 8b 2 [g, a) Pb (A)  bP (B) [ < A ! B, 0 > j9g < a 8b 2 [g, a) P (A) > P (B) . 6 Assume for simplicity that all predicates are monadic. 7 Assume that every object has a name.

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316 / S. Yablo The Sa s and Pa s will as Field says ‘‘oscillate wildly’’ (Figure 14.1 is in that respect misleading), but the hope is that deserving sentences will eventually stabilize on their deserved classical value (1 or 0), while defective sentences will stabilize at 12 or never stabilize at all. Which brings us to (c), the assignment of ultimate values. kAk is limb Pb (A) if the limit exists, otherwise 1 b 1 2. That is, kAk ¼ 1 (0) if P (A) is eventually always 1 (0), and kAk ¼ 2 if 1 b P (A) is eventually always 2 or not eventually always anything. The main and most exciting claim Field makes on behalf of his semantics is that it validates the ‘‘naive theory of truth’’: (i) kT[A] $ Ak ¼ 1 unrestrictedly, and (ii) k . . . T[A] . . . k ¼ k . . . A . . . k unrestrictedly. Property (i) and most of property (ii) are immediate from the fact that each Pb is a Kripkean fixed point. For it is a feature of every fixed point that Pb (T[A]) ¼ Pb (A), and that T[A] and A are freely substitutable in otherthan-conditional contexts. Substitutivity within the scope of ! is proved by induction on the complexity of A. For instance, kB ! Ck ¼ 1 iff Pb (B) is eventually always  Pb (C) iff Pb (T[B]) is eventually always  Pb (C) iff kT[B] ! Ck ¼ 1. Field proves in fact that k . . . k is one of the Pb s and so a full-fledged Kripkean fixed point. Now to call k . . . k a Kripkean fixed point is not to say that it is a fixed point of Field’s operator, the operator taking Pb to Pbþ1 . Recall that a Kripkean fixed point needs only to satisfy (1)–(10) above. The closure S of any seed set S, obtained by slapping 1s and 0s on conditionals however one likes, does that much. To be a Fieldian fixed point, V must also satisfy (11) < A ! B, 1 >2 V iff V(A)  V(B). (12) < A ! B, 0 >2 V iff V(A) > V(B). This means that V must be S not for any old S but the particular one Sv that V itself induces: f< A ! B, 1 > jV(A)  V(B)g [ f< A ! B, 0 > jV(A) > V(B)g. But if the language contains a Curry sentence—a K identical or equivalent to T[K] ! 0 ¼ 1—then no V can do this. Either V(T[K])  V(0 ¼ 1) or V(T[K]) > V(0 ¼ 1). If the latter, then by (12), V(T[K] ! 0 ¼ 1) ¼ V(K) ¼ 0, whence (by (10) ), V(T[K]) ¼ 0  V(0 ¼ 1) after all. If the

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New Grounds for Naive Truth Theory / 317 former, then by (11), V(K) ¼ 1, whence (by (9) ) V(T[K]) ¼ 1, contradicting our assumption that V(T[K])  V(0 ¼ 1). How much should it bother us that k . . . k is not a Fieldian fixed point? What is nice about (11) and (12) is that they equip !-sentences with intuitive and comprehensible truth-conditions. By the same token, it should be no great cause for alarm if the ‘‘real’’ truth-conditions don’t take precisely the (11)–(12) form, provided that intuitive comprehensibility is not sacrificed. Indeed (11) and (12) might be considered problematic, since they prevent A ! B from assuming the value 12, thus obliterating the distinction between conditionals whose antecedents are much truer than their consequents (1 ! 0) and ones whose antecedents are only a little truer (1 ! 12 , or 12 ! 0). Better from this perspective would be an interpretation along the lines of (110 ) < A ! B, 1 >2 V iff V(A)  V(B) (120 ) < A ! B, V(A)  V(B) > iff V(A) > V(B) (the 3-valued Łukasiewicz conditional). However, (110 ) and (120 ) are no more within reach than (11) and (12). Consider K as above (T[K] ! 0 ¼ 1) and K0 which says that 0 ¼ 0 ! K. At odd stages K and K0 are 1 and 0; at even nonlimit stages they are 0 and 1. Since kKk and kK0 k are 12, kK ! K0 k should according to (110 ) be 1. But it can’t be 1, for at odd stages K has a higher value than K0 , making K ! K0 0 at even successor stages. kK ! K0 k is in fact 12. That some 12 ! 12 conditionals, like K ! K, are 1, while others, like K ! K0 are 12, shows that ! is not value-functional. Therefore nothing like (11)–(12) and (110 )–(120 ) can possibly work.8 Of course Field is under no illusions about this. He is not for a moment suggesting that ! is a standard-issue extensional connective. Still, it is natural to wonder how in that case ! is to be understood. I am not sure that a commonsensical explanation of !’s meaning is possible. What we can do is try to place it on the map between two more familiar sorts of connective. One is the extensional connective i defined by (110 ) and (120 ), the Łukasiewicz conditional. A look at the truth tables shows that ! is stronger than i, in that k A ! B k is never greater than k A i B k and occasionally less, for instance k K ! K0 k¼ 1=2 while k K i K0 k¼ 1. This raises the 8 See Field (forthcoming, a), for a type of semantic value with respect to which ! is compositional.

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318 / S. Yablo question, in what does the additional strength consist? What more is asserted by A ! B than A i B? Necessitating a claim is the obvious way to strengthen it, so one natural conjecture is that A ! B is &(A i B) for some suitable modal operator &. But ! does not appear to pack much modal punch. If it did, then one would expect A ! B not to hold unless A somehow necessitated B, and B ! A not to hold unless B necessitated A. Since ‘‘most’’ pairs of sentences necessitate in neither direction, one would expect k (A ! B) _ (B ! A) k to not often be 1. But in fact it is always 1 except under rather special conditions. For k (A ! B) _ (B ! A) k not to be 1, we need first that neither A nor B stabilizes at a classical value, and second that at least one of A, B does not stabilize at all.9 If the first condition is met but not the second, as for instance when A and B are the Truthteller and the Liar, then A ! B and B ! A are both true. This suggests that C ! D, although stronger than C i D, is not as strong as &(C iD). What does it matter, one might say, whether ! has an antecedently comprehensible meaning? A conditional that gets the job done in other respects is one we will learn how to use, and learning the use will teach us the meaning. This response is fair enough in principle. But it assumes that ! does get the job done in other respects, and that a certain semantic obscurity is just the price that has to be paid. This may be right in the end. But it seems to me that !’s performance in other respects is not beyond criticism, and that attending to the criticism makes !’s meaning less obscure rather than more. When we speak of getting the job done in other respects, we should distinguish Job 1—verifying the T-biconditionals—from Jobs 2, 3, 4, etc.— assigning appropriate values to other sentences (sentences not of the form T[A] $ A). When it comes to the evaluation of other sentences, I see three areas where ! could stand to be tweaked. The first is that ! should be less arbitrary. A ! B should not be assigned 1 when there is an equally good case for assigning it 0. The second is that ! should be more grounded. Assignments should be made on some prior and independent basis, and the dependency chains should eventually bottom out. The third is that ! should be stricter. If A contradicts B, for instance, we want not only that A ! :B is 1, but also that A ! B is not 1. 9 These conditions are necessary for a value other than 1, but not sufficient.

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New Grounds for Naive Truth Theory / 319

Arbitrariness. One objection people have raised to counting E (the Truthteller) true is that one could just as easily consider it false. Construed as true, it verifies itself, construed as false, it falsifies itself, and there is no reason to prefer one scenario over the other.10 To treat the Truthteller as true, pure and simple, strikes us as arbitrary, which in a truth-value assignment seems a Bad Thing. By the Conditional Truthteller, let’s mean a sentence F to the effect that (A ! A) ! T[F].11 F describes itself as true-if-(A ! A). Because A ! A is a tautology, this should be the same as F describing itself simply as true. The hypothesis that F is true is self-justifying, and likewise the hypothesis that F is false. Since there is no reason to prefer one outcome to the other, to consider F true, pure and simple, seems unacceptably arbitrary. However, that is the value the semantics assigns. Recall that at stage 0, all conditionals are ½, including A ! A and (A ! A) ! T[F]( ¼ F). That F’s antecedent and consequent have the same value at stage 0 makes F 1 at stage 1. That F’s consequent is 1 at stage 1 makes F 1 at stage 2, and so on indefinitely. ‘‘I am true if (A ! A)’’ has therefore an ultimate value of 1.12 Groundedness. A second reason not to call the Truthteller true is that truth attributions should be based on prior facts, facts already in place before they are made. I call this a different reason because there are cases where a nonarbitrary assignment seems objectionable simply because there is no prior and independent basis for it. Consider an infinite line-up of people each saying ‘‘I am speaking truly  the person behind me is speaking truly’’ (Sn is T[Sn ]  T½Snþ1 ). The only classically defensible assignment is 11111 . . . , since if any Sn were false it would have a false antecedent, making it true after all. Still we are hard put to regard any of these sentences as true. There is no basis for calling Sn true that does not run through assignments to later Sm s for which the same problem arises. Suppose now that the people in our infinite line decide to use arrows instead of horseshoes (Sn is T[Sn ] ! T½Snþ1 ). There is no more reason to call the sentences true than before, but this is the assignment that the semantics 10 I myself think that there is a reason and that the Truthteller is false (Yablo 1985, 1993). But I am not aware of having convinced anyone of this, so I treat the Truthteller as semantically underdetermined. A better example from my perspective would be J ¼ ‘‘S is false’’, S ¼ ‘‘J is false’’. That really is underdetermined. 11 A can be any sentence you like. 12 Another oddity is that k F k becomes 0 if in place of A ! A we put 0 ¼ 0.

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320 / S. Yablo makes. Each Sn is ½ at stage 0, so 1 at stage 1, and then we never look back. Such an outcome may not be arbitrary, but it does seem ungrounded. To suppose that Sn is true is to suppose it has a true antecedent. But then its truth is owing to the truth of its consequent T½Snþ1 , with the buck being passed on forever down the line.

Strictness. If a thing is red, then it is not orange. This seems true not only of determinately red and determinately orange objects but also objects on the red–orange border. The incompatibility of red with orange is not limited to the clear cases on either side. Similarly, if a sentence is true, then it is not false. This applies not only to determinately true and determinately false sentences but also to sentences on the border.13 The incompatibility of true with false is not limited to the clear cases on either side. Facts like these ought to be expressible in the language. (This is more or less the problem of penumbral connections.) Suppose that Jones, assured by her instructor that the Truthteller is true, asserts T0 E0 , while Smith, assured by his instructor that it is false, asserts T0 :E0 . It seems that Jones and Smith are saying incompatible things. If Smith is right in calling E false, then Jones is wrong to call it true. Similarly if Jones and Smith call each other liars: J ¼ :T[S] and S ¼ :T[ J].14 They cannot both be right. If Jones’s statement J is true, then Smith’s statement S is false, and vice versa. It would seem natural to try to register the incompatibility of T0 E0 with T0 :E0 by saying that T0 E0 ! :T0 :E0 . This sentence is certainly true for Field. But does it express the incompatibility of T0 E0 with T0 :E0 ? I would say not, since k T0 E0 ! T0 :E0 k is also 1, and, far from being incompatible with :T0 :E0 , T0 E0 , would seem to imply it. A similar situation arises if one tries to express the incompatibility of J with S by saying that J ! :S. This has ultimate value 1 as desired. However, J ! S also has ultimate value 1, even though it is precisely not the case that J is incompatible with :S( ¼ :: T[ J]).15 13 Does it apply to all sentences on the border? Some might think it false of the Liar that if it is true, then it is not false. I suspect it seems false only because of a felt tension with ‘‘If the Liar is true then it is false’’, and that we do best to regard both sentences as true. No such problems arise for the examples in the text. 14 I assume that S and J are {} (in Field’s notation ½). 15 The last three paragraphs take issue with a claim that Field has since dropped. He now says that the contrariety of red with orange consists in the fact that &8x (x is red ! x is not orange), where & is conceptual necessity (Field, Chapter 13 in this volume, and Field forthcoming, a). A couple of things still worry me, however. First, Field uses the unnecessitated conditional to express a related notion: an object x is red-to-orange iff x isn’t red ! x is orange. However if this

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New Grounds for Naive Truth Theory / 321 A thing is orange if and only if it is redder than yellow things and yellower than red ones. A sentence A is true iff A. Both claims seem correct for definite and borderline cases alike. The equivalence of orange with redder-thanyellow-and-vice-versa is not limited to the clear cases on either side, and likewise the equivalence of T[A] with A. Our theory of these matters should somehow register these equivalences. Kripke’s theory comes close to registering the T-equivalence, in that each fixed point treats the two sides alike. The trouble is that this is for Kripke an essentially metalinguistic observation. He lacks an object language connective # with the property that k B # C k¼ 1 iff Q (B) ¼ Q (C) for all fixed points Q.16 This is because his connectives are monotonic, and no monotonic connective can permit the combination of k B # C k¼ 1 with k B k¼k C k¼ 12. Now Field’s ! is not monotonic, so one might hope it would allow the equivalence of T[A] with A finally to be stated. This requires not just that k T[A] $ A k¼ 1, but also that k T[A] $ :A k not be 1 (leaving aside cases, like the Liar, where the second assignment is forced on us by the first). However, this is not what we find. k T[G] $ :G k¼ 1 for every G that stabilizes at 12, including the Truthteller and the Tautologyteller (‘‘This sentence is either true or not true’’). A related situation arises with sentences A that do not stabilize at 12. Notice first that if A0 is 0 ¼ 0 ! A, then A and A0 are intuitively speaking equivalent. One might hope then that k T[A] $ A0 k¼ 1 and k T[A] $ :A0 k¼ 0. In fact the reverse is true for certain choices of A: k T[A] $ A0 k¼ 0 and k T[A] $ :A0 k¼ 1. (This happens, for instance, if A is a Curry sentence.) To be sure, when A has ultimate value 1 or 0, biconditionals framed with ! behave as they should: k T[A] $ A k¼ 1, k T[A] $ :A k¼ 0. But that much we get already from the material biconditional
. The advantage of ! was supposed to lie with its treatment of the k A k¼ 12 case. conditional really expressed that x was red-to-orange, one would expect it not to be true that x isn’t red ! x is big, for x is not red-to-big. And it looks like it will be true if x is borderline big, or at least that is the natural assumption given the almost-extensionality of !. A different explanation of red-to-orange avoids this problem: &8y ( y is indiscernible in color from x ! ( y is not red ! y is orange)). This will be false if ‘‘big’’ is substituted since we can choose y to be small. Second, suppose with Field that P is contrary to Q iff &8A(P[A] ! :Q [A]). ‘‘True’’ and ‘‘untrue if 0 ¼ 0’’ are intuitively speaking contraries, so we should have &8A(T[A] ! : (0 ¼ 0 ! :T[A])). But 8A(T[A] ! :(0 ¼ 0 ! :T[A])) is not true on Field’s semantics, since it does not hold for the Curry sentence K. ( k T[K] ! :(0 ¼ 0 ! :T[K]) k¼ 12.) 16 Kripke (1975) does discuss the possibility of adding a modal operator.

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322 / S. Yablo A number of strategies might be tried to address these issues (I don’t call them ‘‘problems’’ since my reactions may be idiosyncratic). The ones I will look at exploit the fact that Kripke’s minimal fixed point P is, as the name implies, only one of a class of valuations all satisfying conditions (1)–(10) listed above. The idea very roughly will be to give a possible worlds semantics for !, with these valuations playing the role of worlds. I will first sketch a construction in the style of Field, building on Herzberger and Gupta; it helps with some of the examples just discussed but does not greatly clarify the truth-conditions of A ! B. This will be followed by a construction in the style of Kripke (and Yablo 1985), which addresses all of the examples and makes the truthconditions relatively straightforward. Field-style possible worlds semantics. Call a fixed point ‘‘categorical’’ if it leaves !statement unevaluated. The Field-style construction takes off from (i) the minimal fixed point P and (ii) the set Q of all categorical fixed points. Because no consistency requirement is imposed—Q 2 Q can contain neither, either, or both of and —the members of Q form a lattice under the operations Q1 _ Q2 ¼ (Q1 [ Q 2 ) Q1 ^ Q2 ¼ (Q1 \ Q 2 ) . Each Q 2 Q has a dual that turns Q’s gaps into gluts and vice versa, leaving other sentences unchanged. In particular Q contains a (unique) maximal fixed point obtained by taking the dual of P. (See Woodruff 1984.) The definition of stages is by a simultaneous induction starting from Q ¼ {Q i ji 2 I}. Each Q k is the starting point Q 0k of a sequence < Q ak >, with Q bþ1 determined by Q b ¼ {Q bi ji 2 I}. The key to the construction is k that A ! B is true in Q bþ1 iff B is valued as highly as A by all the members of k Q b . By ‘‘valued as highly as’’ I mean Q (A)  Q (B) ¼df < A, 1 > 2 Q only if < B, 1 > 2 Q , and < B, 0 > 2 Q only if < A, 0 > 2 Q :17 If Q(C) is considered {}, {1}, {0}, or {1,0} according as Q contains neither of < C, 1 > , < C, 0 >, the first only, the second only, or both, then this corresponds to the ordering that has {0} < all other values, {1} > all other values (and that’s all). Q is said to validate A ! B iff Q (A)  Q (B).

17 Q (C) ¼ Q (D) means that < C, v >2 Q iff < D, v >2 Q .

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New Grounds for Naive Truth Theory / 323 Suppose we have Q a ¼ {Q ai ji 2 I} in hand. Then Q aþ1 is the set of all Q aþ1 (i 2 I), where i Q aþ1 ¼ ({ < A ! B, 1 > j every Q ai  Q ak validates A ! B})} k [{ < A ! B, 0 > j not every Q ai  Q ak validates A ! B})  : A ! B is true in Q lk iff as g approaches l, it is eventually always the case that every Q gi  Q gk validates A ! B.18 Q lk ¼ ({ < A ! B, 1 > j9b < l 8g 2 [b, l) each Q ai  Q ak validatesA ! B} [{ < A ! B, 0 > j9b < l 8g 2 [b, l) no Q ai  Q ak validates A ! B})  :

This results in the back-and-forth process shown in Figure 14.2 Now the definition of ultimate values. For each Q k in Q , let Q 1 k be liminf b Q bk , that is b Q1 k ¼ { < C, v > j < C, v >2 Q k for all large enough b}:

If Q 0 is the minimal fixed point, then A’s ultimate value is the value it is assigned by Q 1 0 . Consider, for instance, ‘‘If J, then not S’’, where again J is :T[S] and S is :T[ J]. This receives value 1 because Q ( J)  Q (:S) ( ¼ Q (::T[J])) for all fixed points Q. ‘‘If J, then S’’ is not assigned 1 because it is not validated by consistent Qs assigning 1 to J, and such Qs can be found in every Q b . (Recall that ‘‘If J then S’’ is 1 on the original Field semantics.) I do not want to dwell too long on this system, so let me just note a few basic properties. That Q ak is always a Kripkean fixed point means that each Q a0 validates both directions of T[A] $ A, so Q 1 0 (T[A] $ A) ¼ 1. Substitutivity is harder to prove, but it holds, so we have the full naive theory of truth. ! is stricter than it was than on the original Field semantics, which helps with Q0

Q1

Q2

Q k2

Q k1

Q k0

Q 00 =P

Q 01

Ql

Q3

Q kl+1

Q kl

Q k3

Q 02

Ql+1

Q 03

Q 0l

Q 0l+1

FIG. 14.2 18 This looks a little different from the successor clause but is in keeping with Field.

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324 / S. Yablo penumbral connections (as just observed in effect with J and S). But the issues of arbitrariness and groundedness remain. It is unfortunately not the case that 1 P1 (A ! B) ¼ 1 iff 8 Q 1 k  Q 0 Q k (A)  Q k (B).19 The new truth-conditions are as obscure as the old ones. Kripke-style possible worlds semantics. This time we start not with the set R of all categorical fixed points meeting a certain condition: R 2 R must be ‘‘transparent’’ in the sense that substituting T[C] for C always preserves semantic value in R. (Note that each opaque fixed point has a least transparent extension, obtained by closing under both directions of < w(T[C]), v >2 V iff < w(C), v >2 V and (1)–(10).) < R, ^ , _ > is a lattice under the same operations as before. Each R 2 R has a dual in R that turns gaps into gluts and vice versa, leaving everything else alone. In particular R contains a maximal transparent fixed point making each conditional statement both true and false (Woodruff 1984). The idea is to develop an increasing series of fixed points Pb in tandem with a decreasing series Rb of sets of fixed points. A ! B is to be true in Paþ1 iff each R 2 Ra validates it. Raþ1 is obtained by purging Ra of any valuations that conflict with Paþ1 :Pa grows as Ra shrinks, because the smaller Ra is, the easier it becomes for all its members to validate A ! B: Ra shrinks as Pa grows, because the more opinionated a valuation is, the more it comes into conflict with other valuations. Above we said that R(A)  R(B) just in case R assigns 1 to B if to A, and 0 to A if to B. And we said that R validates A ! B iff R(A)  R(B). Now we further stipulate that R(A) > R(B) iff R(A)  R(B) and R(A) 6¼ R(B). Explicitly, R(A) > R(B) ¼df R assigns 1 to A if to B, and 0 to B if to A, and (either) 1 to A only or 0 to B only. When R(A) > R(B), we say that R invalidates A ! B. The induction goes as follows. Suppose we have Pa and Ra in hand. Paþ1 ¼ ({ < A ! B, 1 > j each R 2 Ra validates A ! B}[ { < A ! B, 0 > j each R 2 Ra invalidates A ! B}) . Valuations are incompatible when they disagree in their unique assignments, i.e. one makes a sentence uniquely true (false) that the other fails to make uniquely true (false). 19 The right-hand side holds for K, 0 ¼ 0 ! K but P1 (K ! (0 ¼ 0 ! K)) 1.

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New Grounds for Naive Truth Theory / 325 Raþ1 ¼ {R 2 Ra jR is compatible with Paþ1 }. Limit stages are similar. Suppose we have Pg and Rg in hand for all g < l. Pl ¼ ({ < A ! B, 1 > j9b < l 8g 2 [b, l) each R 2 Rg validates A ! B)}[ { < A ! B, 0 > j9b < l 8g 2 [b, l) each R 2 Rg invalidates A ! B}) .

Rl ¼ {R 2 \g
R0

P0=P

R1

P1

R3

P2

Rl

Rl+1

Pl

Pl+1

FIG. 14.3 20 Someone might worry that P1 is inconsistent, that is, assigns some sentence both 1 and 0. It is certainly not consistent if R1 is empty, for then every conditional is validated and invalidated by every possible world. If R1 has even one member, though, then conditionals have at most one truth-value, hence P1 is consistent overall. And each Ra does have at least one member, since it contains at a minimum Pa .

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326 / S. Yablo A third property clear from the construction is that P1 verifies every Tbiconditional. The proof that k T[C] $ C k¼ 1 is that both directions are validated by all accessible worlds, given that worlds are fixed points. Interchangeability of T[C] and C in all contexts is proved by induction on the complexity of C. The one non-trivial part is to show that C and T[C] are substitutable in A ! B if in A and B. Here we rely on the fact that the valuations in R1 are transparent. Suppose, for instance, that A ! (C ! D) is 1 and consider A ! (T[C] ! D): k A ! (C ! D) k¼ 1 only if for all accessible worlds R, R(A)  R(C ! D). Transparency tells us that R(C ! D) ¼ R(T[C] ! D), so for all accessible R, R(A)  R(T[C] ! D), whence k A ! (T[C] ! D) k¼ 1. Next we consider the issues of arbitrariness, ungroundedness, and ‘‘laxity’’ (insufficient strictness). Our example of arbitrariness was the conditional Truthteller F( ¼ (A ! A) ! T[F]). Field gives this a value of 1 even though 0 seems on the face of it just as justified. What value does P1 assign? First let’s show that < F, 1 > 62 P1 . R0 contains a least R assigning 1 to both A ! A and T[F]; call it R1 : R0 also contains a least R assigning 1 to A ! A and 0 to T[F]; call it R2 . Since R1 validates F and R2 invalidates it, P1 leaves (A ! A) ! T[F] unevaluated. Since P1 assigns 1 to A ! A (why?) and nothing to F, P1 is compatible with both R1 and R2 . Hence R1 and R2 survive into R1 , which by the same argument as before means that P2 leaves (A ! A) ! T[F] unevaluated. Continuing in this way we see that P1 (F) ¼ {}. Our example of ungroundedness was the infinite Curryesque sequence Ki ¼ T[Ki ] ! T[Kiþ1 ]: R0 contains for each i a smallest valuation Ri1 assigning 1 to Ki and 1 to Kiþ1 , and a smallest valuation Ri2 assigning 1 to Ki and 0 to Kiþ1 The first validates Ki and the second invalidates it, so P1 (Ki ) ¼ {} for all i. But then as above Ri1 and Ri2 survive into R1 , so P2 (Ki ) ¼ {} for all i, and so on ad infinitum. The Ki s are 1 on the Field semantics but are neither 1 nor 0 by the lights of P1 . The strictness issue was addressed already by the previous semantics, and the same considerations apply. Consider, for instance, T0 K0 ! :T0 :K0 . This has ultimate value 1 because every fixed point (including those assigning both 1 and 0 to K) validates it. T0 K0 ! T0 :K0 is not 1 because fixed points assigning just 1 to K fail to validate it, and not 0 either because fixed points assigning just 0 to K fail to invalidate it. Look back now at the Top Ten Excellent Features of Field’s system (the original system, without possible worlds). How many of these are shared by the system sketched in the last section? I have already said that the Kripke

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New Grounds for Naive Truth Theory / 327 system verifies all T-biconditionals (that’s 1), and permits the substitution ‘‘salva valutate’’ of T[A] for A (that’s 2). Certainly it provides an explicit model (that’s 3) which means no hidden paradoxes (that’s 4).21 When it comes to 7 and 8, there is a tradeoff. The Kripke system gives ! a more natural semantics—a fixed point semantics—but at the cost of a less horseshoe-like logic. For a sense of what is lost (and what is not), some key axioms, rules, and metarules of the Field system are A1 A2 A3 A4 A5 A6

A!A

::A ! A

A ! (A _ B)

A^B!A

(A ! :B) ! (B ! :A)

(A ! :A) ! :(T ! A)

R1 R2 R3 R4 R5 R6

A, A ! B B A, :B :(A ! B) A B!A (A ! B) ^ (A ! C) A ! (B ^ C) (A ! C) ^ (B ! C) (A _ B) ! C A ! B (C ! A) ! (C ! B).

A1–A4 are valid on the present semantics, but A5 and A6 hold in rule form only: A ! :B B ! :A and A ! :A :(T ! A). R1–R5 are truthpreserving but R6 fails.22 This last is important to the proof that, as in the Field system, ! ‘‘becomes’’ horseshoe in bivalent contexts (that’s feature 9).23 21 One might worry about inconsistencies, but see the previous note. 22 One key weakness of the present logic is its obliviousness in many cases to embedded conditionals (as evidenced, for instance, in the failure of A5 and A6). It might help to impose more conditions on R0 than just transparency. 23 Field’s ‘‘Theorem on ! and ’’ (Field forthcoming, a) still holds when ‘LCC is replaced by

, understood as 1-preservingness in the Kripke semantics. The modified theorem states that: (ia) (ib) (iia) (iib)

AB A!B (A _ :A) ^ (B _ :B) (A  B) ! (A ! B) (A _ :A) ^ (A ! B) A  B (A _ :A) ^ (B _ :B) (A ! B) ! (A  B).

Clearly B A ! B, and :A A ! B. _-elimination holds because k A _ B k¼ 1 only if k A k¼ 1or k B k¼ 1. By _-elimination, :A _ B A ! B, which is the same as (ia). For (ib) and (iib), if k (A _ :A) ^ (B _ :B) k¼ 1 then k A k¼ 1 or ±0, and k B k¼ 1 or 0. But then R(A) ¼k A k and R(B) ¼k B k for all R 2 R1 . It follows that k A ! B k¼k A  B k¼ 1 or 0. Either way R(A  B) ¼ R(A ! B) for all R 2 R1 , so k (A  B) ! (A ! B) k¼ 1 (proving (ib)

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328 / S. Yablo That leaves 5 and 6: high degree of revenge-immunity, and not because vengeance-threatening notions are inexpressible. One of the great attractions of Field’s theory is the astonishing mileage he gets out of a language-internal determinacy operator DA, defined as (0 ¼ 0 ! A) & A. DA is 1 at stage aþ1 iff A is true both at stage aþ1 and stage a, and 0 at stage aþ1 iff A fails either at the given stage or the one before it. DA is 1 at limit stage l iff A is 1 through the closed interval [b, l] for some b < l, and 0 at stage l if A is either 0 at l or less than 1 through the half-open interval [b, l). If L is the Liar then DL is 0; if L1 is the strengthened Liar ‘‘I am not determinately true’’ then DDL1 is 0; if L2 is the double-strength Liar ‘‘I am not determinately determinately true’’ then D3 L2 is 0; and so on until the ordinal notations run out. That none of these Liars is unevaluable, and indeed each is truly describable as defectivea (:Da A ^ :Da :A) for suitably large a, might seem to support a claim of revenge-immunity. Distinguish two questions, however. Question 1: is the language able to characterize as defective every sentence that deserves to be so characterized? Question 2: are there intelligible semantic notions such that paradox is avoided only because those notions are not expressible in the language? Field goes a long way towards addressing the first of these, but revengemongers have traditionally been more interested in second. Now it may seem that Field does address the second question, in the section of Field (forthcoming, a) called ‘Revenge (2)’. The revenge-monger (RM) asks us to imagine the chaos that would result if having an ultimate value other than 1 were expressible in the language. A Superliar L1 could then be constructed saying that k L1 k 6¼ 1. ‘‘Why imagine it?’’ Field asks. The ‘‘new’’ predicate needn’t even be added to the language: [it is] already there, at least if (i) the base language from which we started the construction in Section 2 included the language of set theory and (ii) the model M0 for the base language from which the construction . . . started is definable in the base language . . . for the construction showed how to explicitly define [it] in set-theoretic terms . . . (p. 302 above)

That Field goes so far as to lend RM his tools makes the next step all the more crushing:

and k (A ! B) ! (A  B)k ¼ 1 (proving (iib) ). For (iia), that kA _ :Ak ¼ 1 means that kAk is 1 or 0. If the latter then kA  Bk ¼ 1. If the former then 8R 2 R1 R(A) ¼ 1. Also though 8R 2 R1 R(A)  R(B) since kA ! Bk ¼ 1. So kBk ¼ 1, hence kA  B k¼ 1. QED

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New Grounds for Naive Truth Theory / 329 Can’t we then reinstitute a paradox? . . . No we can’t: our construction yields the consistency of the claim True ( ) $k A k 6¼ 1 . . . (p. 000 above)

The problem is that by Tarski’s theorem, ‘‘we can’t within classical set theory define any notion of determinate truth that fully corresponds to the intuitive notion’’ (p. 000 above), what Tarski would have called the metalinguistic notion. The paradox fails because the predicate ‘k . . . k6¼ 1’ does not mean what RM wanted it to. But of course it was clear from the start that no truth predicate definable in the language was going to express the notion of having a semantic value 6¼ 1. What then was the point of inviting RM into his workshop and offering the use of his tools? I will not speculate about motive, but the result was to get RM off the street where he might really have caused some trouble. From that external vantage point, RM would have seen the old familiar bargain at work and raised the old familiar alarm: consistency is being maintained through a sacrifice of expressive power. I am not saying I agree with RM about this, just that he should be allowed the platform from which his type has traditionally threatened revenge. Field does have a response to this: ‘‘RM, you are taking the semantics and its classical setting too seriously. The most a classical semantics can accomplish is to provide a consistency proof for the theory. The ‘real’ semantics, if there were going to be one, would be carried out in a non-classical set theory, a theory that is needed anyway to deal with the set paradoxes. Sentences like ‘k this very sentence k6¼ 1’ will not bother us any longer, when we are free of the requirement that k A k¼ 1 _ k A k6¼ 1’’. A unified treatment of the set and semantic paradoxes is the grail cup of antinomy studies. Russell thought he had it in the theory of types, but we know how that turned out. How Field’s attempt at a unified treatment will turn out, we don’t know. But the attempt bears watching.24

REFERENCES B a r w i s e , J., and J. E t c h e m e n d y (1987), The Liar: An Essay in Truth and Circularity (Oxford: Oxford University Press). B r a d y , R. (1989), ‘The Non-triviality of Dialectical Set Theory’, in Priest et al. (1989: 437–70). 24 See Field, Ch. 13 in this volume.

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330 / S. Yablo B u r g e , T. (1979), ‘Semantic Paradox’, Journal of Philosophy, 76: 169–98; repr. in Martin (1984: 83–117). F e f e r m a n , S. (1984), ‘Towards Useful Type-Free Theories, I’, Journal of Symbolic Logic, 49: 75–111; repr. in Martin (1984: 237–87). F i e l d , H. (2002), ‘Saving the Truth Schema from Paradox’, Journal of Philosophical Logic, 31: 1–27. —— (forthcoming, a), ‘A Revenge-Immune Solution to the Semantic Paradoxes’, Journal of Philosophical Logic. —— (forthcoming, b), ‘The Consistency of the Naı¨ve(?) Theory of Properties’. G u p t a , A. (1982), ‘Truth and Paradox’, Journal of Philosophical Logic, 11: 1–60; repr. in Martin (1984: 175–235). H e r z b e r g e r , H. (1982), ‘Notes on Naive Semantics’, Journal of Philosophical Logic, 11: 61–102; repr. in Martin (1984: 133–74). K r i p k e , S. (1975), ‘Outline of a Theory of Truth’, Journal of Philosophy, 72: 690–716; repr. in Martin (1984: 54–81). M a r t i n , R. (1984), Recent Essays on Truth and the Liar Paradox (Oxford: Clarendon Press). —— and P. Woodruff (1975), ‘On Representing ‘‘True-in-L’’ in L’, Philosophia, 5: 213– 17; repr. in Martin (1984: 47–51). P r i e s t , G. (1979), ‘The Logic of Paradox’, Journal of Philosophical Logic, 8: 219–41. —— R. Routley, and J. Norman (1989), Paraconsistent Logic: Essays on the Inconsistent (Munich: Philosophia Verlag). S k y r m s , B. (1984), ‘Intentional Aspects of Semantical Self-Reference’, in Martin (1984: 120–31). T a r s k i , A. (1983), ‘The Concept of Truth in Formalized Languages’, in J. H. Woodger (ed.), Logic, Semantics, Metamathematics (Indianapolis: Hackett). W o o d r u f f , P. (1984), ‘Paradox, Truth, and Logic (I)’, Journal of Philosophical Logic, 13: 213–52. Y a b l o , S. (1985), ‘Truth and Reflection’, Journal of Philosophical Logic, 14: 297–349. —— (1993), ‘Hop, Skip and Jump: The Agonistic Conception of Truth’, Philosophical Perspectives, 7: 371–96.

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15 A Completeness Theorem for Unrestricted First-Order Languages Agustı´n Rayo and Timothy Williamson

1. Preliminaries Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents, and the syntactic differences between their non-logical constituents, treated as variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals.1 Thus ‘¼’ is a logical constant because no permutation maps two Thanks to Vann McGee and Stephen Read for many helpful comments. 1 Tarski (1986) (text of a lecture delivered in 1966) suggests invariance under permutations of the universe as a criterion for a logical constant. Sher (1991) develops this approach in an extensional way (see also McGee 1996; contrast McCarty 1981). Alternatively, one might require the invariance to be necessary or a priori. One can also require logical constants to be invariant in extension across circumstances of evaluation (worlds and times), so that a predicate applicable to everything if Nelson died at Trafalgar and to nothing otherwise does not qualify as a logical constant, even though we know a priori that it is necessarily permutation-invariant in extension. Almog (1989) also characterizes logical truth by permutation invariance, although the underlying conception is quite different. Our aim here is not fine-tuning the notion of logical consequence.

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332 / Agustı´n Rayo and Timothy Williamson individuals to one or one to two; ‘2’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers, and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. For the case of first-order languages,2 interpretations are standardly cashed out in terms of what might be called model-theoretic interpretations (or MTinterpretations). An MT-interpretation for a first-order language L is an ordered pair hD, Fi. The domain D is a non-empty set, and is intended to specify the range of the variables in L. The interpretation function F is intended to specify semantic values for the variables of L (including nonlogical expressions). The semantic value of an n-place predicate-letter is a set of n-tuples of individuals in D,3 and the semantic value of a first-order variable is an individual in D. Truth on an MT-interpretation can then be characterized as follows: [MT- ¼ ] [MT-P] [MT-:] [MT- ^ ] [MT-9]

9vi ¼ vj ; is true on hD, Fi iff F(9vi ;) ¼ F(9vj ;), 9Pin (vj1 , ..., vjn ); is true on hD, Fi iff hF(9vj1 ;), ..., F(9vjn ;)i 2 F(9Pin ;), 9:c; is true on hD, Fi iff 9c; is not true on hD, Fi, 9c ^ y; is true on hD, Fi iff 9c; and 9y; are both true on hD, Fi, 99vi (c); is true on hD, Fi iff there is some d 2 D such that 9c; is true on hD, F[9vi ;=d]i,

where F[v=d] is the function that is just like F except that it assigns d to v. If G is a set of formulas, we say that G is true on hD, Fi just in case each formula in G is true on hD, Fi. Finally, an argument is said to be MT-valid

2 We take a first-order language to consist of the following symbols: the logical connectives ‘^’ and ‘:’, the quantifier-expression ‘9’, variable-symbols 9vi ; for i 2 w, the identity-symbol ‘¼’, n-place predicate-letters 9Pan ; for n 2 ! and a in some set S, and the auxiliary symbols ‘(’ and ‘)’. We do not consider function-letters, since they can be simulated by (n þ 1)-place predicates. The formulas of L are defined in the usual way. 3 To simplify our presentation, we make the assumption that, for every x, the 1-tuple of x is identical with x.

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Unrestricted First-Order Languages / 333 just in case every MT-interpretation on which the set of premises is true is also an MT-interpretation on which the conclusion is true, and a formula is said to be MT-valid just in case the argument with no premises of which it is the conclusion is MT-valid. The variables of a first-order language are sometimes intended to range over absolutely everything whatsoever (henceforth, we will use ‘everything’ and similar phrases such as ‘all individuals’ in that sense).4 For instance, the variables of the first-order language in which the theory of MT-interpretations is couched must range over everything, on pain of excluding some individuals from the semantic values of non-logical terms. But, according to standard conceptions of set-theory such as ZFU (i.e. Zermelo–Fraenkel set theory with urelements), there is no set of all individuals. And, in the absence of such a set, there is no MT-interpretation that specifies a domain consisting of everything. So, when the variables of a first-order language L are intended to range over everything, no MT-interpretation captures the intended interpretation of L. Clearly, matters cannot be improved by appealing to proper classes: no proper class can play the role of a universal domain because no proper class is a member of itself. But one might be tempted to address the problem by adopting a set theory which allows for a universal set—Quine’s New Foundations, the Church and Mitchell systems, and positive set theory all satisfy this requirement.5 Unfortunately, set theories that allow for a universal set must impose restrictions on the axiom of separation to avoid paradox. So, as long as an MT-interpretation assigns a subset of its domain as the interpretation of a monadic predicate, some intuitive interpretations for monadic predicates will not be realized by any MT-interpretation. (Throughout the rest of the paper we will be working with ZFU plus choice principles, rather than a set theory which allows for a universal set.) The problems we have discussed are instances of a more general difficulty. Regardless of the set theory one chooses to work with, trouble will arise from the fact that an MT-interpretation is an individual. For, whatever it is to G, there are legitimate assignments of semantic values to variables according to 4 A powerful defense of unrestricted quantification is set forth in Cartwright (1994), undermining some of the classic criticisms in Dummett (1981, chs. 14–16). The possibility of quantifying over everything is also defended in Boolos (1998b), McGee (2000), Williamson (1999), Rayo (2003), and Williamson (forthcoming). 5 See Quine (1937), Church (1974), Mitchell (1976), and Skala (1974). For an extended discussion of set theories with a universal set, see Forster (1995).

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334 / Agustı´n Rayo and Timothy Williamson which the predicate-letter P applies to something if and only it Gs. So, if every legitimate assignment of semantic values to variables is to be captured by some MT-interpretation, we must have the following: (1) For every individual x, MG is an MT-interpretation according to which P applies to x if and only if x Gs. But, since MT-interpretations are individuals, we may define a verb ‘R’ as follows: (2) For each individual x, x Rs if and only if x is not an MT-interpretation according to which P applies to x. Putting ‘R’ for ‘G’ in (1) and applying (2) yields: (3) For every individual x, MR is an MT-interpretation according to which P applies to x if and only if x is not an MT-interpretation according to which P applies to x. In particular, we can let x be MR itself; so (3) implies: (4) MR is an MT-interpretation according to which P applies to MR if and onlyifMR is notan MT-interpretation accordingtowhichPapplies toMR . And (4) is a contradiction. It follows that there are legitimate assignments of semantic values to variables that cannot be captured by any MT-interpretation. (It is worth noting that, although the argument is structurally similar to standard derivations of Russell’s Paradox, it does not rest on any assumptions about sets. As long as the variables in the metalanguage range over everything, all we need to get the problem going is the observation that MTinterpretations are individuals, and the claim that, whatever it is to G, there are legitimate assignments of semantic values to variables according to which the predicate-letter P applies to something if and only if it Gs.) It is best to use a semantics which is not based on MT-interpretations. Here we will work with the notion of a second-order interpretation (or SO-interpretation), set forth in Rayo and Uzquiano (1999). Informally, the idea is this: rather than taking an SO-interpretation to be an individual, like an MTinterpretation, we regard it as given by the individuals which a monadic second-order variable I is true of. The ‘domain’ of I is the collection of individuals x such that I is true of h‘8’, xi. The ‘semantic value’ which I assigns to an n-place predicate-letter 9Pi ; is the collection of n-tuples hx1 , ..., xn i such that I is true of h9Pi ;, hx1 , ... , xn ii; and the semantic value which I assigns

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Unrestricted First-Order Languages / 335 to a first-order variable 9vi ; is the unique individual x such that I is true of h9vi ;, xi (for the sake of brevity, we sometimes use ‘I(9vi ;)’ to refer to the unique x such that Ih9vi ;, xi). Such informal explanations are a kind of useful nonsense, a ladder to be thrown away once climbed, because they use the second-order (predicate) variable ‘I’ in first-order (name) positions in sentences of natural language; nevertheless, they draw attention to helpful analogies between SO-interpretations and MT-interpretations. Formally, when ‘I’ is a second-order variable, we take ‘I is an SO-interpretation’ to abbreviate the following second-order formula:6 9x(Ih‘8’, xi ) ^ 8x(FOV(x) ! 9!yIhx, yi) where ‘FOV (x)’ is interpreted as ‘x is a first-order variable’. Unlike MT-interpretations, SO-interpretations are well-suited to cover the case in which the variables of L range over everything. For, whenever Ih‘8’, xi holds for every x, the ‘domain’ of I will consist of everything. Let us now characterize the predicate ‘f is true on I’, where f is a formula of L and I is an SO-interpretation for L. It is important to note that our satisfaction predicate is a second-level predicate (i.e. a predicate taking a secondorder variable in one of its argument-places), since ‘I’ is a second-order variable.7 [SO- ¼ ] [SO-P] [SO-:] [SO- ^ ] [SO-9]

9vi ¼ vj ; is true on I iff I(9vi ;) ¼ I(9vj ;), 9Pin (vj1 , ... , vjn ); is true on I iff Ih9Pin ;, hI(9vj1 ;),... , I(9vjn ;) ii, 9:c; is true on I iff 9c; is not true on I, 9c ^ y; is true on I iff 9c; and 9y; are both true on I, 99vi (c); is true on I iff there is some d such that Ih‘8’, di and 9c; is true on I[9vi ;=d],

where I[v=d] is just like I except that I(v) ¼ d. 6 On this definition, not everything falling under ‘I’ has a role to play in the resulting semantics. If we wished to stipulate away idle clutter, we could have taken ‘I is an SOinterpretation’ to abbreviate the following instead: 9x(Ih‘8’, xi) ^ 8x(FOV(x) ! 9!yIhx, yi)^ 8x{Ix ! [9y(x ¼ h‘8’, yi) _ 9y9z9n(P r e d (y, n)^T u p l e (z, n) ^ x ¼ hy, zi)_ 9y9z(FOV(y) ^ x ¼ hy, zi)]} where the predicates are understood in the obvious way: ‘FOV (x)’ is interpreted as ‘x is a firstorder variable’, ‘Pr e d (x, n)’ is interpreted as ‘x is an n-place predicate’, and ‘Tu p l e (x, n)’ is interpreted as ‘x is an n-tuple’. 7 For more on second-level predicates see Rayo (2002). Vann McGee has pointed out that, as long as L is a first-order language, the notion of truth on an SO-interpretation for L can be

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336 / Agustı´n Rayo and Timothy Williamson If G is a set of formulas, we say that G is true on I just in case each formula in G is true on I. Finally, an argument is said to be SO-valid just in case every SOinterpretation on which the set of premises is true is also an SO-interpretation on which the conclusion is true, and a formula is said to be SO-valid just in case the argument with no premises of which it is the conclusion is SO-valid. A famous argument of Kreisel’s can be used to show that a first-order argument hG, fi is SO-valid if and only if it is MT-valid.8 [Proof sketch. Suppose explicitly characterized in a second-order language with no atomic second-level predicates (see ch. 16 in this volume). When L is a second-order language, however, the notion of truth on an SO-interpretation for L cannot be characterized in a second-order language with no atomic second-level predicates. [Proof sketch. Let L and L0 be second-order languages containing no atomic second-level predicates and let M be the intended SO-interpretation for L0 . We suppose, for reductio, that it is possible to characterize in L0 the notion of truth on an SO-interpretation for L, in other words, we suppose that there is a formula ‘Sat(x, I)’ of L0 such that the following sentence of L0 is true on M: (*) 8I8x[ [(I is an SO-interpretation for L ^ x is a formula of L) ! ([SO- ¼ ] ^ [SO-P] ^ [SO-:] ^ [SO- ^ ] ^ [SO-9 ] ^ [SO-V] ^ [SO-29])] where ‘I is an SO-interpretation for L’ and ‘x is a formula of L’ are interpreted in the obvious way, and ‘Sat(x, I)’ is substituted for ‘x is true on I’ in [SO- ¼ ], [SO-P], [SO-:], [SO-^ ], [SO-9], [SO-V], and [SO-29] ([SO-V] and [SO-29] are defined in Section 6). With no loss of generality, we may assume that L0 contains no non-logical predicate-letters which do not occur in (*), and hence that L0 contains only finitely many non-logical primitives (for simplicity, we take second-order languages to contain no function-letters). Since ‘x is a formula of L’ is true of all and only formulas of L, the domain of M must be infinite; this means that any arithmetical sentence 9f; can be expressed in L0 as the universal closure of the result of substituting the arithmetical primitives for variables of the appropriate type in 9PA ! f;, where PA is the conjunction of the second-order Dedekind–Peano Axioms. It is therefore harmless to assume that the language of arithmetic can be interpreted in the theory of M. From this it follows that L0 is able to characterize its own syntax by way of Go¨del numbering. Let P1 ,... , Pn be a complete list of the non-logical predicate-letters in L0 , and say that a correspondence function is a one–one function mapping each of the Pi (1
i
n) onto a variable of L with the same number of argumentplaces. Since L0 is able to characterize the syntax of L in addition to its own, there is a correspondence function c definable in L0 . If f is a formula of L0 , let fc be the result of substituting c(Pi ) for every occurrence of Pi in f(1
i
n), and, if necessary, relabeling variables to avoid clashes. For any formula f of L0 , c(f) is a formula of pure second-order logic, and therefore a formula of L. Moreover, it follows from the definability of c in L0 that there is a formula C(x,y) of L0 which holds of x and y just in case x is the (Go¨del number of) a formula f of L0 and y is c(f). It is also possible to characterize in L0 an SO-interpretation IM of L with the following characteristics: (a) IM (h‘8’, xi) if and only if M(h‘8’, xi) and (b) for each Pi (1
i
n), if 9Vjm ; is c(Pi ), then IM (h9Vjm ;, hx1 ,... , xm ii) if and only if M(h9Pi ;, hx1 ,... , xm ii). But ‘8y(C(x, y) ! Sat(y, IM ))’ is a truth predicate for L0 , contradicting Tarski’s Theorem.] It is worth noting that the result continues to hold when L contains unrestricted quantifiers (introduced in Sections 2 and 6). 8 See Kreisel (1967). Cartwright (1994) uses Kreisel’s argument to argue against the view that the All-in-One Principle—the principle that to quantify over certain objects is to presuppose

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Unrestricted First-Order Languages / 337 hG, fi is MT-valid. Then, by the completeness of MT-validity, f is derivable from G. But the SO-validity of derivable inferences follows from a straightforward induction on the length of proofs. So f is true on every SO-interpretation which makes G true. hG, fi is therefore SO-valid. Conversely, suppose hG, fi is not MT-valid. Then there is some MTinterpretation on which every member of G is true and f is not true. But, since every MT-interpretation has the same domain and delivers the same assignments of semantic value as some SO-interpretation, there is some SO-interpretation on which every member of G is true and f is not true. So hG, fi is not SO-valid.] Kreisel’s result guarantees that standard first-order deductive systems are sound and complete with respect to SO-interpretations. It also shows that, if the only purpose of a formal semantics is to characterize the set of standard first-order validities, then SO-interpretations are unnecessary, since they deliver the same result as MT-interpretations. But a formal semantics might have a broader objective than that of characterizing the set of standard first-order validities. For instance, one may wish to consider the result of enriching a first-order language with a quantifier ‘9AI ’, as in McGee (1992). On its intended interpretation, a sentence ‘9AI v(f(v))’ is true just in case the individuals satisfying ‘f(v)’ are too many to form a set. Accordingly, when the quantifiers range over everything, ‘9AI v(v ¼ v)’ is true, since there are too many individuals to form a set. Within an SO-semantics, we can specify the truth-conditions of ‘9AI ’ as follows: [SO-9AI ] 99AI vi (f); is true on I iff no set contains every d such that Ih‘8’, di and 9f; is true on I[9vi ;=d]. This yields the intended result. For instance, [SO-9AI ] ensures that ‘9AI v(v ¼ v)’ is true on any SO-interpretation I such that, for every x, Ih‘8’, xi. On the other hand, we run into trouble when we try to specify the truth-conditions of ‘9AI ’ within MT-semantics. For, suppose we attempt to mirror [SO-9AI ] by way of the following clause: [MT-9AI ] 99AI vi (f); is true on hD, Fi iff no set contains every d 2 D such that 9f; is true on hD, F[9vi ;=d]i. that there is one thing of which those objects are the members—derives support from MTsemantics. In particular, Cartwright notes that Kreisel’s argument undermines the thought that the MT-validity of a sentence f can only be a guarantee of f’s truth if the intended domain of f coincides with the domain of some MT-interpretation.

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338 / Agustı´n Rayo and Timothy Williamson It follows from [MT-9AI ] that no MT-interpretation can make a sentence of the form 99AI vi (f); true. So ‘;9AI v(v ¼ v)’ is an MT-validity, even though ‘9AI v(v ¼ v)’ is true on the intended interpretation.

2. Unrestricted Quantification Throughout our discussion of ‘9’ and ‘9AI ’ we have made the following standard assumption: Domain Assumption. What individuals the truth-value of a quantified sentence depends on is not a logical matter; it varies between interpretations. The presence of the Domain Assumption is evidenced by [MT-9], [SO-9], [MT-9AI ], and [SO-9AI ], which explicitly impose an (interpretation-relative) domain restriction on the individuals that the truth-value of quantified sentences depends on. Williamson (1999) argues that, even if the Domain Assumption is appropriate in the case of ‘9’ and ‘9AI ’, there is no reason to think that it is appropriate in general. Specifically, we might set forth an unrestricted quantifier ‘9U ’ such that a sentence ‘9U v(f(v))’ is true if and only if ‘f(v)’ is true of some individual, whether or not the individual is part of some domain or other. If there is such a quantifier, its application is insensitive to permutations of individuals. So, on the Bolzano–Tarski picture of logical consequence described above, it should count as a logical expression, and its semantic value should not vary between interpretations. Within an SO-semantics, we have the resources to characterize ‘9U ’. All we need to do is add the following clause to our definition of ‘truth on I’: [SO-9U ] 99U vi (f); is true on I iff some d is such that 9f; is true on I[9vi ;=d]. Extending our object-language with ‘9U ’ and our formal semantics with [SO-9U ] does not mean that we must jettison ‘9’ or ‘9AI ’. Arguments in the original language that were SO-valid before the extension will remain SOvalid, and arguments in the original language that were SO-invalid before the extension will remain SO-invalid. On the other hand, we get a number of new SO-validities from arguments in the extended language. For instance, the inference from ‘9v(f)’ to ‘9U v(f)’ is SO-valid. The sentence ‘9U v(v ¼ v)’ is

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Unrestricted First-Order Languages / 339 also SO-valid. Finally, let 99Un v(f(v)); (read ‘there are at least n vs such that f(v)’) be defined as follows: 9U1 v(f(v)) df 9U v(f(v)) 9U2 v(f(v)) df 9U v9U u(f(v) ^ f(u) ^ v 6¼ u) etc. Since the world contains infinitely many individuals (such as {}, {{}}, {{{}}},...), the sentence 99Un v(v ¼ v);, for any n, is true on every SO-interpretation. So, for any n, 99Un v(v ¼ v); is SO-valid. It is worth noting that the standard (domain-restricted) quantifier ‘9’ can be defined in terms of ‘9U ’. All we need to do is introduce a monadic predicate ‘D’, to play the role of specifying a domain and take ‘9v(f(v))’ to abbreviate ‘9U v(D(v) ^ f(v))’. By taking ‘9U v(D(v))’ as a premise, we can then recover the usual validity for ‘9’. If the language includes a predicate ‘2’ expressing settheoretic membership, then McGee’s quantifier ‘9AI ’ can also be defined in terms of ‘9U ’, since we can take ‘9AI v(f(v))’ to abbreviate ‘;9U v8U v0 (v0 2 v $ f(v0 ))’.9 When ‘9U ’ is regarded as the only primitive quantifier in the language, the notion of an SO-interpretation can be simplified. We can take ‘I is an SO-interpretation’ to abbreviate ‘8x(FOV(x) ! 9!yIhx, yi)’. It is worth noting, moreover, that MT-semantics does not provide an appropriate framework for the introduction of ‘9U ’. Although we could certainly set forth a clause analogous to [SO-9U ], [MT-9U ] 99U vi (f); is true on hD, Fi iff some d is such that 9f; is true on hD, F[9vi ;=d]i, it wouldn’t deliver the intended result. The problem is that an MT-interpretation assigns a set as the semantic value of a predicate. Since no set contains every individual, this means that, for any 9Pj ;, 99U vi (:Pj (vi )); is true on every hD, Fi. So 99U vi (:Pj (vi )); is MT-valid, even though its negation may be true on the intended interpretation of 9Pj ; (for example, as meaning self-identity). Here we will not attempt to assess the legitimacy of ‘9U ’; that project is developed in Williamson (1999). Our present task is to identify a sound and complete deductive system for first-order languages involving ‘9U ’. For simplicity, we will set ‘9’ and ‘9AI ’ aside. Thus, we let our object-language, 9 As usual, we take ‘8U u(f)’ to abbreviate ‘;9U u;(f)’.

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340 / Agustı´n Rayo and Timothy Williamson LU , be the result of substituting ‘9U ’ for ‘9’ in a standard first-order language. In addition, we let D be the result of substituting ‘9U ’ for ‘9’ in any standard first-order deductive system. Not every SO-valid sentence of LU is deducible in D. To see this note that, although 99Un v(v ¼ v); is not derivable in D for n  2, 99Un v(v ¼ v); is SO-valid for any n. It follows immediately that no deductive system weaker than D1 — the result of enriching D with an axiom 99Un v(v ¼ v); for every n—can be complete with respect to SO-validity. As it turns out, D1 itself is sound and complete with respect to SO-validity.10 Williamson (1999) outlines an argument for this result. Here we will provide a formal proof. The soundness of D1 with respect to SO-validity is immediate, since a straightforward induction on the length of proofs reveals that, if f is a sentence of LU and G is a set of sentences of LU , then f is derivable from G in D1 only if hG, fi is a valid SO-argument. To show the completeness of D1 , we prove the following: Completeness Theorem for Unrestricted First-Order Languages. Let 9f; be a sentence of LU and G a set of sentences of LU . Then hG, 9f;i is a valid SO-argument only if 9f; is derivable from G in D1 . First some preliminary remarks. We work within second-order ZFC with urelements, and assume that everything can be put in one–one correspondence with the ordinals. An immediate consequence of our assumption is that the members of any set S can be put in one–one correspondence with the ordinals less than a given ordinal a. As usual, we let jSj be the smallest such a. In the course of the proof we will make use of several standard modeltheoretic results. In order to retain their availability, we will make use of MTinterpretations, alongside SO-interpretations, with the important proviso that MT-interpretations are to treat ‘9U ’ like a standard (domain-restricted) quantifier, rather than an unrestricted one. Thus, instead of using [MT-9U ] as the truth-clause for ‘9U ’ within MT-semantics, we use the following analogue of [MT-9], [MT-9Uþ ] 99U vi (f); is true on hD, Fi iff some d 2 D is such that 9f; is true on hD, F[9vi ;=d]i. 10 Since any theory compatible with Robinson Arithmetic is recursively undecidable (see, for instance, Mendelson 1987, proposition 3.48), the recursive undecidability of the set of theorems of D1 is an immediate consequence of the observation that Robinson Arithmetic can be consistently added to the axioms of D1 .

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Unrestricted First-Order Languages / 341 On the other hand, we retain [SO-9U ] as the truth-clause for ‘9U ’ within SOsemantics. Two of the model-theoretic results we make use of are from Tarski and Vaught (1957). The first is a strengthened version of the Upward Lo¨wenheim–Skolem Theorem: Tarski–Vaught 1. If hD, Fi is an MT-interpretation for a language L, D is infinite, and k is a cardinal such that k  jDj and k  jLj, then there is an elementary extension hD , F i of hD, Fi such that jD j ¼ k. As usual, we say that an MT-interpretation hD0 , F0 i is an elementary extension of hD, Fi just in case: (i) D  D0 , and (ii) for any formula c with free variables among 9v1 ;, ... , 9vn ;, and for any a1 , ... , an 2 D, c is true on hD, F[! an ]i if and vn =! 0 0! ! ! ! only if c is true on hD , F [ vn = an ]i, where G[ vn = an ] is the function that is just like G except that it assigns a1 to 9v1 ;, a2 to 9v2 ;, ..., and an to 9vn ;. The second model-theoretic result from Tarski and Vaught (1957) is the following: Tarski–Vaught 2. Let K be a non-empty family of MT-interpretations such that, for any MT-interpretations M, M0 2 K, some MT-interpretation in K is an elementary extension of M and an elementary extension of M0 . Let DK be the union of the Db for hDb , Fb i 2 K; let FK (9vi ;) ¼ F0 (9vi ;) for some hD0 , F0 i 2 K; and let FK (9Pan ;) be the union of the Fb (9Pan ;) for hDb , Fb i 2 K. Then hDK , FK i is an MT-interpretation and, for any MT-interpretation M 2 K, hDK , FK i is an elementary extension of M. So much for preliminary remarks; we now turn to the proof. We assume that 9f; is not derivable from G in D1 , and show that G [ {9:f;} is true on some SO-interpretation. We proceed by proving each of the following three propositions in turn: Proposition 1. For each ordinal a there is an MT-interpretation hDa , Fa i such that: (a) G [ {9:f;} is true on hD0 , F0 i; (b) for any a, jDa j  @a ; and (c) for a
g, hDg , Fg i is an elementary extension of hDa , Fa i. Proposition 2. For each ordinal a there is an MT-interpretation hDa , Fa i such that: (a) for any sentence c of LU , c is true on hD0 , F0 i if and only if c is true on hD0 , F0 i; (b) for any a, jDa j  @a ; (c) for a
g, hDg , Fg i is an elementary extension of hDa , Fa i; and (d) every individual is in some Da .

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342 / Agustı´n Rayo and Timothy Williamson Proposition 3. There is an SO-interpretation I  such that G [ {9:f;} is true on I  . It is worth noting that the use of second-order resources will be confined to propositions 2 and 3.

Proof of Proposition 1 Since 9f; is not derivable from G in D1 , it follows from the standard Completeness Theorem for first-order languages that G[ {9:f;} [ N 1 is true on some MT-interpretation hD0 , F0 i, where N 1 is the set of sentences 99Un v1 (v1 ¼ v1 ); for n 2 w. Since every sentence in N 1 is true on hD0 , F0 i, D0 must be infinite. For a > 0, define hDa , Fa i as follows: (D1) Assume a ¼ b þ 1, and suppose that hDb , Fb i has been defined. By [Tarski–Vaught 1], there is an elementary extension hD, Fi of hDb , Fb i with jDj  @a . By our assumption that everything can be put in one–one correspondence with the ordinals, there is a least such elementary extension. Let that elementary extension be hDa , Fa i. (D2) Assume that a is a limit ordinal, and S suppose that hDb , Fb i has been defined for every b < a. Let Da ¼ b
It follows immediately from [Tarski–Vaught 2] that hDa , Fa i is an MTinterpretation and that, for every b < a, hDa , Fa i is an elementary extension of hDb , Fb i. Moreover, since jDb j  @b for b < a, we get the result that jDa j  @a .&

Proof of Proposition 2 Since jDa j  @a , our assumption that everything can be put in one–one correspondence with the ordinals guarantees that there is an R such that 8x(9a(x 2 Da ) ! 9!y(R (x, y))) ^ 8y9!x(9a(x 2 Da ) ^ R (x, y)). For each a, let ra be the one–one function with domain Da such that ra (x) is the unique y for which R (x, y). We may now define hDa , Fa i from hDa , Fa i as follows:

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Unrestricted First-Order Languages / 343 Da ¼ {ra (x): x 2 Da } Fa (9vi ;) ¼ ra (Fa (9vi ;)) Fa (9Pbn ;) ¼ {hra (x1 ), ..., ra (xn )i: hx1 , ..., xn i 2 Fa (9Pbn ;)}. Clause (a) of Proposition 2 can be verified by a routine induction on the complexity of formulas. Since ra is a one–one function with domain Da and since jDa j  @a , the definition of Da guarantees that clause (b) is satisfied. Clause (c) can be verified by a routine induction on the complexity of formulas. Finally, in virtue of the construction of the ra from R , the definition of the Da guarantees that clause (d) is satisfied.

Proof of Proposition 3 Let I  be any SO-interpretation with the following property: I h9Pbn ;, hx1 , ... , xn ii $ 9a(hx1 , ..., xn i 2 Fa (9Pbn ;)). We show the following: Let c be a formula of LU with free variables among v1 , ..., vn . For any ordinal a, if ha1 , ... , an i is a sequence of individuals in Da , then c is true on hDa , Fa [! v n =! vn =! an ]i if and only if c is true on I  [! an ], vn =! where I  [! an ] is just like I  except that, for k
n, ak is the unique x such ! ! 9 ; that I [ vn = an ] h vk , xi. The proof is by induction on the complexity of c. Clauses corresponding to ‘¼’, ‘:’, and ‘^’ are trivial. . Suppose c is 9Pm (vj1 , ..., vjm ); (for j1 , ..., jm
n). Then, by [MT-P], c is true b m; on hDa , Fa [! an ]i iff haDj1 , ... , ajm i 2 Fa (9PE v n =! b ). Similarly, by [SO-P], c is true on I ½! vn =! an  iff I  9PbmD;, haj1 ,... , ajm i . SoE it suffices to show that haj1 , ..., ajm i 2 Fa (9Pbm ;) iff I  9Pbm ;, haj1 , ... , ajm i . DThe definition Eof I  guarantees that haj1 , ..., ajm i 2DFa ð9Pi ;Þ only if IE 9Pbm ;, haj1 , ..., ajm i . For the converse, suppose that I  9Pbm ;, haj1 , ..., ajm i . By the definition of I  ,

.

there is a d such that haj1 ,... , ajm i 2 Fd (9Pbm ;). If a
d, then hDd , Fd i is an elementary extension of hDa , Fa i; if d < a, then hDa , Fa i is an elementary extension of hDd , Fd i. In either case aj1 , ..., ajm 2 Fa (9Pbm ;). Suppose c is 99U vi (x); (for i
n). Then, by ½MT-9Uþ , c is true on hDa , Fa ½! vn =! an i iff there is some d 2 Da such that 9x; is true on hDa , Fa ½! vn =! v n =! an ½9vi ;=di. Similarly, by ½SO-9U , c is true on I  ½! an   ! ! iff there is some d such that 9x; is true on I ½ vn = an ½9vi ;=d.

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344 / Agustı´n Rayo and Timothy Williamson an ½9vi ;=ei. Let Suppose e 2 Da is such that 9x; is true on hDa , Fa ½! v n =! ei ¼ e; and, for j
n and j 6¼ i, let ej ¼ aj . It follows that 9x; is true on hDa , Fa ½! en i. Since ek 2 Da for k
n, it follows by inductive hypothvn =! esis that 9x; is true on I  ½! en . This means that there is some d such v n =!  ! ! that 9x; is true on I ½ vn = an ½9vi ;=d. Conversely, suppose that there is an e such that 9x; is true on I  ½! an ½9vi ;=e. As before, let ei ¼ e; and, for j
n and j 6¼ i, let vn =! ej ¼ aj . It follows that 9x; is true on I ½! en . Since every individual vn =!  is in some DZ , e 2 Dd for some d  a. By inductive hypothesis, 9x; is true on hDd , Fd ½! en i. Accordingly, there is some d 2 Dd such that v n =!  9x; is true on hDd , Fd ½! an ½9vi ;=di and, by ½MT-9Uþ , 99U vi (x); is v n =! true on hDd , Fd ½! an i. But, since a
d, hDd , Fd i is an elementary vn =!   extension of hDa , Fa i. So, given that ak 2 Da (for k
n), 99U vi (x); is true on hDa , Fa ½! an i. vn =! It follows immediately that a sentence of LU is true on I  if and only if it is true on hD0 , F0 i. But, by Proposition 2, a sentence of LU is true on hD0 , F0 i if and only if it is true on hD0 , F0 i. And, by Proposition 1, G [ {;f} is true on hD0 , F0 i. So G [ {;f} is true on I  . This completes the proof. &

3. Corollaries The soundness and completeness results for unrestricted first-order languages have two immediate consequences. (As before, we use ½MT-9Uþ  rather than ½MT-9U  as the truth-clause for ‘9U ’ within MT-semantics.) Corollary 1 (Compactness). Let G be a set of sentences of LU . If, for every finite subset G of G, there is some SO-interpretation on which G is true, then there is some SO-interpretation on which G is true. The proof is immediate. Corollary 2. If hD, Fi is an MT-interpretation with D infinite, then there is some SO-interpretation I such that, for every sentence f of LU , f is true on hD, Fi if and only if f is true on I. Proof. We make use of the fact that D1 is sound with respect to the class of MT-interpretations with infinite domains. Let hD, Fi be an MTinterpretation with D infinite, and let G be the set of sentences of LU which

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Unrestricted First-Order Languages / 345 are true on hD, Fi. By the soundness of D1 with respect to the class of MTinterpretations with infinite domains, G is consistent in D1 . So, by the completeness of D1 with respect to SO-interpretations, G is true on some SO-interpretation (and no sentence outside G is, since G is negationcomplete).& Conversely, we have the following: Observation. For any SO-interpretation I, there is some MT-interpretation hD, Fi with D infinite such that, for every sentence f of LU , f is true on I if and only if f is true on hD, Fi. Proof. We make use of the fact that D1 is complete with respect to the class of MT-interpretations with infinite domains. Let I be an SO-interpretation and let G be the set of sentences of LU which are true on I. By the soundness of D1 with respect to SO-semantics, G is consistent in D1 . So, by the completeness of D1 with respect to MT-interpretations, G is true with respect to some MT-interpretation (and no sentence outside G is, since G is negationcomplete). &

4. Choice Principles Our proof of the Completeness Theorem for unrestricted first-order languages makes use of a strong choice principle. It assumes that everything can be put in one–one correspondence with the ordinals.11 This assumption is equivalent to the existence of a well-ordering of the universe together with the existence of a one–one correspondence between everything and the sets, and it follows from (but does not imply) the existence of a well-ordering of the sets together with the existence of a set containing all non-sets. As it turns out, the use of choice principles in our proof is unavoidable. The easiest way to see this is by noting that the Completeness Theorem implies, in second-order ZF, that the universe can be linearly ordered. (Proof. For an arbitrary two-place predicate R, let f be a sentence of LU stating that R is a linear ordering. Clearly, f is true on any MT-interpretation hN, Fi where F(R) is the usual ordering of the natural numbers. So, by Corollary 2, f is true on some SO-interpretation I. But then  linearly orders the universe, where x  y $ IhR, hx, yii.] A sentence stating that the universe can be linearly 11 For an interesting discussion on the question of whether everything can be put in one–one correspondence with the ordinals, see Shapiro (forthcoming).

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346 / Agustı´n Rayo and Timothy Williamson ordered is a choice principle because it is provable in second-order ZF plus the Axiom of Global Choice, but not in second-order ZF (if second-order ZF is consistent); it does not, however, imply the Axiom of Global Choice in second-order ZF (if second-order ZF is consistent).12 A result of Harvey Friedman’s shows that, for the special case where LU contains countably many non-logical primitives, the Completeness Theorem for unrestricted first-order languages is equivalent, within second-order ZF, to the claim that the universe can be linearly ordered.13 However, this result is unlikely to extend to the general case, where arbitrary non-countable sets of non-logical primitives are allowed. In addition to implying that the universe can be linearly ordered, the Completeness Theorem implies, within secondorder ZF, the Prime Ideal Theorem (which states that any Boolean Algebra has a prime ideal).14 The Prime Ideal Theorem is a choice principle because it is provable in ZFC, but not in ZF (if ZF is consistent); it does not, however, imply the Axiom of Choice in ZF (if ZF is consistent).15 Ascertaining the exact strength of our Completeness Theorem with respect to different choice principles is an interesting matter, which we do not address here. The Completeness Theorem for unrestricted first-order languages is not alone in its reliance on choice principles. A choice principle is needed to prove the Generalized Completeness Theorem for standard first-order languages (which states that, when arbitrary non-countable sets of non-logical primitives are allowed, a first-order argument hG, fi is valid only if f is derivable from G in some standard first-order deductive system). Specifically, the Generalized Completeness Theorem can be shown within ZF to be equivalent to the Prime Ideal Theorem.16 Choice principles are also needed to prove [Tarski–Vaught 1], and the Downward Lo¨wenheim–Skolem Theorem (which states that a formula of L which is true on some MT-interpretation with domain of cardinality k is also true on some MT-interpretation with domain of cardinality m, if maxð@0 , jLjÞ
m
k). They are both provably equivalent to the Axiom of Choice within ZF.17 12 See Rubin and Rubin (1985: 286). 13 See Friedman (1999, theorem D.4). 14 The proof is analogous to standard proofs that the Generalized Completeness Theorem is equivalent to the Prime Ideal Theorem within ZF. See, for instance, Jech (1973: 17). 15 See Jech (1973, §7.1). 16 The equivalence of the Prime Ideal Theorem to the Generalized Completeness Theorem is due to Henkin (1954). For a more recent exposition of the proof, see Mendelson (1987). 17 The results are due to Vaught (1956) and Tarski and Vaught (1957). For proofs, see Rubin and Rubin (1985: 163).

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Unrestricted First-Order Languages / 347 A distinctive feature of the Completeness Theorem for unrestricted firstorder languages is its reliance on a global choice principle. But non-global choice principles are largely an artefact of the use of first-order languages. When one gives purely second-order formulations of choice principles, one naturally gets the global forms.18 No choice principles are needed to prove certain special cases of the results we have considered. For instance, no choice principles are needed to prove the special case of the Generalized Completeness Theorem when the set of non-logical primitives has cardinality @a for some a (since any set of cardinality @a for some a can be well-ordered, and the well-ordering can be extended to finite sequences of its members).19 In the case of the Completeness Theorem for unrestricted first-order languages, no choice principles are required when there are only finitely many monadic predicates and no polyadic predicates other than ‘¼’, or when the language does not contain identity and the set of non-logical primitives has cardinality @a for some a. More specifically, the following propositions are provable within secondorder ZF with urelements:20 Special Case 1. Assume that LU contains only finitely many monadic predicates and no polyadic predicates other than ‘¼’. Let 9f; be a sentence of LU and G a set of sentences of LU . Then hG, 9f;i is a valid SO-argument only if 9f; is derivable from G in D1 . Proof sketch. Suppose G [ {9:f;} is consistent with respect to D1 . Then, by the standard Completeness Theorem for first-order languages, there is an MT-interpretation hD, Fi, with D infinite, on which G [ {9:f;} is true. Since there are only finitely many monadic predicates and no polyadic predicates other than ‘¼’, it is easy to verify that there is some infinite subset D of D with the following property: Let * be a one–one function from D into D such that a ¼ a if a 62 D , and a 2 D otherwise. Then, for any objects a1 , ..., an 2 D and any 9 ; 9vn ;, f is true on formula f with free variables among D hv1 , ..., !iE an i just in case f is true on D, F ! vn = an . hD, F½! vn =!

 In particular, this means that, for a, b 2 D , a 2 F 9Pa1 ; $ b 2 F 9Pa1 ; . For d 2 D , let I be any SO-interpretation such that 18 Thanks here to Robert Black. 19 For a proof, see Mendelson (1987, proposition 2.33). 20 For additional results within a metatheory lacking choice principles, see Friedman (1999).

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348 / Agustı´n Rayo and Timothy Williamson
 Ih9Pa1 ;, ai $ a 2 F
9Pa1 ; for a 2 D Ih9Pa1 ;, ai $ d 2 F 9Pa1 ; for a 62 D. An induction on the complexity of formulas shows that, for any a1 , . . . , an , if f is a formula with free variables among 9v1 ;, . . . , 9vn ;, and if  is a function from D [ {a1 , . . . , an } into D such that: (1) a ¼ a if a 2 D, and a 2 D  otherwise, and (2) ai 6¼ aj ! ai 6¼
iE i
j
n), then f is true on D aj (for h 1!  ! ! ! I ½ v = a  just in case f is true on D, F v = a . It follows that G [ {9:f;} n

n

n

n

is true on I. & Special Case 2. Assume that LU does not contain ‘¼’ and that the set of non-logical primitives in LU has cardinality @a for some a.21 Let 9f; be a sentence of LU and G a set of sentences of LU . Then hG, 9f;i is a valid SO-argument only if 9f; is derivable from G in a standard first-order system without identity. Proof sketch. Suppose G [ {9:f;} is consistent with respect to a standard firstorder system without identity. Then, by the standard Completeness Theorem for first-order languages, there is an MT-interpretation hD, Fi on which G [ {9:f;} is true. For an arbitrary d 2 D, let a ¼ a if a 2 D, and let a ¼ d otherwise. Let I be any SO-interpretation such that I h9Pan ;, ha1 , ... , an ii $ ha1 , ..., an i 2 F(9Pan ;): An induction on the complexity of formulas shows that, for any a1 , ..., an , if f is 9v1 ;, ... , 9vn ;, then f is true on I[~ a formula with free variables among vn =~ an ] just   in case f is true on D, F[~ vn =a~n ] . It follows that G [ {9:f;} is true on I.&

5. Additional Assumptions The assumption that everything can be put in one–one correspondence with the ordinals implies that the universe can be well-ordered. But it also imposes restrictions on the universe which are independent of choice principles. In particular, since the ordinals are pure sets, it implies that there can’t be more urelements than pure sets. It is important to note that this sort of restriction 21 It is worth noting that the assumption that the set of non-logical primitives in LU has cardinality @a for some a is required in the proof below to ensure that the relevant version of the Completeness Theorem for domain-relative first-order languages holds. But the assumption is not needed to extend the completeness result for domain-relative first-order languages to a completeness result for unrestricted first-order languages.

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Unrestricted First-Order Languages / 349 is inessential. The proof in Section 2 makes non-choice assumptions in order to minimize the use of second-order resources. But, by making heavier use of second-order resources, the Completeness Theorem can be proved within second ZFC plus urelements from the assumption that the universe can be well-ordered. Proof sketch. Say that < is a well-ordering of the universe, and let G be a set of sentences of LU which is consistent with respect to D1 . We show that there is an SO-interpretation on which every sentence in G is true. We begin with some notation. Let LUþ be the result of enriching LU with a constant-letter 9ca ; for every individual a. [This is possible because we may assume, with no loss of generality, that none of the primitives in LU is identical to hx, 0i for some x, and go on to identify 9ca ; with ha, 0i for every a.] In addition, let G0 be such that G0 (x) $ (x 2 G _ x ¼ 9ca 6¼ cb ;) for a 6¼ b. The consistency of G0 with respect to D1 follows from the consistency of G with respect to D1 . We wish to produce a consistent extension of G0 which contains ‘witnesses’ for all existential sentences. Since < is a well-ordering of the universe, we may say that, for any individual a, 9fa ðxia Þ; is the a-th formula of LUþ with one free variable 9xia ;. Similarly, for some subcollectionD of the constantletters in LUþ , we may say that, for any individual a, 9da ; is the a-th constant-letter in D. With no loss of generality, we may assume that, for each individual a, the a-th formula of LUþ with one free variable does not contain the a-th constantletter in D. Let G1 be the result of adding to G0 , for each individual a, the formula 99xia ðfa ðxia ÞÞ ! fa ðda Þ;. It is easy to verify that the consistency of G1 with respect to D1 follows from the consistency of G0 with respect to D1 . The next step is to produce an extension of G1 which is negation-complete in LUþ and consistent with respect to D1 . For each individual a, we define Pa as follows: . .

P0 (x) $ G1 (x), where 0 is the <-smallest individual. For a such that 0 < a, let f be the <-smallest sentence of LUþ which is <-greater than or equal to a, and let X be such that 8x(X(x) $ 9b < aðPb (x)Þ). Then Pa (x) $ (X(x) _ x ¼ f) if the result of adding f to the formulas in X is consistent with respect to D1 , and Pa (x) $ X(x) otherwise. If no sentence of LUþ is <-greater than or equal to a, then Pa (x) $ X(x).

Let G2 be such that G2 (x) $ 9aðPa (x)Þ. It is straightforward to show that G2 is an extension of G1 which is negation-complete in LUþ and which is consistent with respect to D1 .

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350 / Agustı´n Rayo and Timothy Williamson Finally, we provide an SO-interpretation on which every sentence in G2 is true. Let G3 be the result of substituting ‘9’ for ‘9U ’ throughout G2 . Expand the definition of SO-interpretations so as to allow for individual constants (in the obvious way), and let I be an SO-interpretation meeting the following conditions: . I h‘8’, xi iff there is a constant 9a; such that x ¼ 9a; and, for some constant 9c;, 9a; is the <-smallest constant such that G3 (9a ¼ c;). . if 9c; is a constant, then Ih9c;, xi iff there is a constant 9a; such that x ¼ 9a; and 9a; is the <-smallest constant such that G3 (9a ¼ c;); . if 9P; is an n-place predicate, then Ih9P ;, hx1 , . . . , xn ii iff there are constants 9a1 ;, . . . , 9an ; such that x1 ¼ 9a1 ;, . . . , xn ¼ 9an ; and G3 (9Pða1 , . . . , an Þ;). An induction on the complexity of formulas shows that a sentence f is true on I just in case G3 (f). But, since LUþ contains a constant 9ca ; for each individual a, and since G3 ð9ca 6¼ cb ;Þ whenever a 6¼ b, the individuals in the domain of I can be put in one–one correspondence with everything. This allows us to define an SO-interpretation on which every sentence in G2 is true and, hence, an SO-interpretation on which every sentence in G is true.&

6. Second-Order Languages So far our object-language has always been a first-order language, but SOinterpretations can also be used to provide a semantics for second-order languages.22 Formally, we continue to regard ‘I is an SO-interpretation’ as an abbreviation for ‘9x(Ih‘8’, xi) ^ 8x(FOV(x) ! 9!yIhx, yi)’ (or as an abbreviation for ‘8x(FOV(x) ! 9!yIhx, yi)’ if only unrestricted quantifiers are taken into account). But we add the following clauses to our characterization of ‘f is true on I’: [SO-V] 9Vin (vj1 , ... , vjn ); is true on I iff Ih9Vin ;, hI(9vj1 ;),... , I(9vjn ;)ii, [SO-29] 99Vin (c); is true on I iff 9X(8x(X(x) ! Ih‘8’, xi) ^ 9c; is true on I[9Vin ; IX]), U 9 U n [SO-29 ] 9 Vi (c); is true on I iff 9X(9c; is true on I[9Vin ;=X]): 22 See Rayo and Uzquiano (1999).

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Unrestricted First-Order Languages / 351 where I[9Vin ;=X] is just like I except that, for all x, Ih9Vin ;, xi $ X(x). SOvalidity is characterized as before. Intuitively, the ‘semantic value’ which an SO-interpretation I assigns to an n-place second-order variable 9V n ; is the collection of n-tuples hx1 , ..., xn i such that I is true of h9V n ;, hx1 , ... , xn ii. Clause [SO-29] ensures that every collection of individuals in the ‘domain’ of I is within the ‘range’ of the standard (domain-dependent) second-order quantifier ‘9’, and clause [SO-29U ] ensures that every collection of individuals is within the ‘range’ of the unrestricted second-order quantifier ‘9U ’. As in the first-order case, there are second-order sentences containing unrestricted quantifiers which are SO-valid even though their domain-relative counterparts are not. For instance, the infinity of the universe ensures that the following sentence—which states that there is a one–one function from everything onto less than everything—is SO-valid: 9U R[8U x8U y8U z(R(x, y) ^ R(x, z) ! y ¼ z)^ 8U x8U y8U z(R(x, z) ^ R(y, z) ! x ¼ y)^ 8U x9U y(R(x, y)) ^ 9U y8U x(:R(x, y))]; even though its domain-relative counterpart is not, since there are SOinterpretations with finite ‘domains’. Similarly, the existence of inaccessibly many sets ensures that the following sentence—which implies that there are inaccessibly many individuals—is SO-valid: 9U R(ZFC2) where ‘ZFC2’ is the result of substituting the unused second-order variable ‘R’ for every occurrence of ‘2’ in (an unrestricted version of) the conjunction of the axioms of second-order ZFC; even though its domain-relative counterpart is not, since there are SOinterpretations with ‘domains’ which do not contain inaccessibly many objects. Finally, consider the following two sentences, both of which are free from non-logical vocabulary: [CH] 8U X (Al e p h -1(X) $ C o n t i n u u m (X) ) [NCH] 8U X (Al e p h -1(X) ! :C o n t i n u u m (X)) where ‘Al e p h -1 (X)’ is (the unrestricted version of) a formula of pure second-order logic to the effect that there are precisely @1 -many objects falling under ‘X’, and ‘Co n t i n u u m (X)’ is (the unrestricted version of) a

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352 / Agustı´n Rayo and Timothy Williamson formula of pure second-order logic to the effect that there are precisely continuum-many objects falling under ‘X’ (see Shapiro 1991, sect. 5.1). Suppose that the continuum hypothesis is true. Then [CH] is SO-valid, and so is its domain-relative counterpart. But the existence of @1 -many individuals ensures that the negation of [NCH] is SO-valid, even though its domainrelative counterpart is not (since it is false on any SO-interpretation with a ‘domain’ consisting of less-than-@1 -many individuals). On the other hand, suppose the continuum hypothesis is false. Then [NCH] is SO-valid, and so is its domain-relative counterpart. But the existence of @1 -many individuals ensures that the negation of [CH] is SO-valid, even though its domain-relative counterpart is not (since it is false on any SO-interpretation with a ‘domain’ consisting of less-than-@1 -many individuals). So, whether or not the continuum hypothesis is true, [CH] or its negation is SO-valid, and [NCH] or its negation is SO-valid. But the same cannot be said of their domain-relative counterparts. (A similar example can be constructed for the case of the Generalized Continuum Hypothesis.) Our examples illustrate a general feature of unrestricted second-order sentences, which sets them apart from their domain-relative counterparts: every true second-order sentence containing no non-logical vocabulary or domain-relative quantifiers is SO-valid. Unfortunately, we cannot hope to obtain a completeness result for second-order languages (whether the quantifiers be unrestricted or domain-relative). It is a consequence of Go¨del’s Incompleteness Theorem that, if D is any effective second-order deductive system which is sound with respect to SO-validity, then there is a second-order sentence which is SO-valid but is not a theorem of D.23 23 For the domain-relative case, see Shapiro (1991, theorem 4.14). When the quantifiers are unrestricted, the result can be proved as follows. Let A be the conjunction of a finite, categorical axiomatization of pure second-order arithmetic, formulated in the language of pure secondorder arithmetic (for instance, the system described in Shapiro 1991, section 4.2). Let L2U be a second-order language with unrestricted quantifiers containing every non-logical primitive in A. For N an unused monadic predicate of L2U and f a sentence of the language of pure secondorder arithmetic, let fN be the sentence of L2U which results from substituting ‘8U ’ and ‘9U ’ for ‘8’ and ‘9’ in f (respectively), and relativizing the resulting quantifiers with N. Let T be the set of sentences fN of L2U such that fN is derivable in D from AN . Since D is effective, T is recursively enumerable. Say that an SO-interpretation I is arithmetical just in case AN is true on I. Since D is sound with respect to SO-validity, it follows that every sentence in T is true on every arithmetical SO-interpretation. But if f is a sentence of the language of pure second-order arithmetic and fN is true on every arithmetical SO-interpretation, then f is true. So any sentence f of the language

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Unrestricted First-Order Languages / 353

7. Higher-Order Languages We have seen that SO-interpretations can be used to provide a semantics for second-order languages. Could SO-interpretations also be used to provide a semantics for a language containing second-level predicates, such as our metalinguistic predicate ‘f is true on I’ (where ‘I’ is a second-order variable)? There is an important sense in which they cannot. Say that a semantics based on
-interpretations is strictly adequate for a language L only if every semantic value which a non-logical expression in L might take is captured by some
interpretation. Then a semantics based on SO-interpretations cannot be strictly adequate for a language containing second-level predicates. Informally, the problem is this: the semantic value of a second-level predicate might consist of any ‘supercollection’ of collections of individuals. But a (third-order) generalization of Cantor’s Theorem shows that there are ‘more’ supercollections of collections of individuals than there are collections of individuals. Since each SO-interpretation is given by the collection of individuals a second-order variable is true of, this means that there are ‘more’ semantic values a second-level predicate might take than SO-interpretations. So there are semantic values a second-level predicate might take which are not captured by any SO-interpretation. Again, this informal explanation is strictly nonsense, since ‘is a collection’ and ‘is a supercollection’ take the position of first-level predicates in sentences of natural language, even though they are intended to capture higher-order notions; nonetheless, it draws attention to a helpful analogy between first- and higher-order notions. The result can be stated formally and proved within a third-order language. In order to provide a strictly adequate semantics for languages containing second-level predicates one needs at least a third-order metalanguage enriched with a third-level predicate (i.e. a predicate taking third-order variables in some of its argument-places). And, of course, the situation generalizes. In order to provide a strictly adequate semantics for languages containing nthlevel predicates one needs at least an (n þ 1)th-order metalanguage enriched with an (n þ 1)th-level predicate. of pure second-order arithmetic such that fN is a member of T must be true. By Go¨del’s Incompleteness Theorem, the collection of true sentences of the language of pure first-order arithmetic is not recursively enumerable. Since T is recursively enumerable, it follows that there is a true sentence of first-order arithmetic c such that cN is not in T. Hence, 9AN ! cN ; is not derivable in D. But it follows from the categoricity of A that 9AN ! cN ; is true on every SOinterpretation.

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354 / Agustı´n Rayo and Timothy Williamson

8. Concluding Remarks We conclude with a historical note. The formal system which Frege set forth in the Begriffsschrift was meant to be a universal language: it was intended as a vehicle for formalizing all deductive reasoning. Accordingly, Frege took the first-order variables of his system to range over all individuals. So much is beyond dispute. However, some interpreters have recently contended that Frege’s conception of logic as a universal language prevented him from engaging in substantive metatheoretical investigation.24 The problem, they argue, is that there can be no external perspective within a universal logical system from which to assess the system itself. With this we disagree.25 The metatheoretical results in the present paper show that absolutely unrestricted quantification is not an obstacle to substantial metatheoretical investigation. Accordingly, our results show that metatheoretical investigation is possible for systems which do not allow for an ontologically external perspective. We did, of course, make use of an ideologically external perspective, since (for instance) we appealed to a higher-order metalanguage in our study of first-order object-languages. But this does not affect the claim that a universal language can be used to perform a substantial metatheoretical investigation of a fragment of itself, even when the fragment contains unrestricted quantifiers.

REFERENCES A l m o g , J. (1989), ‘Logic and the World’, in Almog et al. (1989). A l m o g , J., J. P e r r y , and J. W e t t s t e i n (1989), Themes from Kaplan (Oxford: Oxford University Press). B o o l o s , G. (1998a), Logic, Logic and Logic (Cambridge, Mass.: Harvard University Press). —— (1998b), ‘Reply to Parsons’ ‘‘Sets and Classes’’ ’, in Boolos (1998a: 30–6). C a r t w r i g h t , R. (1994), ‘Speaking of Everything’, Nouˆs, 28: 1–20. C h u r c h , A. (1974), ‘Set Theory with a Universal Set’, in L. Henkin et al. (eds.), Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics, 25

24 Relevant texts include van Heijenoort (1967), Goldfarb (1979, 1982), Dreben and van Heijenoort (1986), Ricketts (1986), and Conant (1991). 25 We are not alone. See Stanley (1996) and Tappenden (1997).

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Unrestricted First-Order Languages / 355 (Providence, RI: American Mathematical Society for the Association for Symbolic Logic), 297–308; also in International Logic Review, 8 (1977), 11–23. C o n a n t , J. (1991), ‘The Search for Logically Alien Thought: Descartes, Kant, Frege and the Tractatus’, Philosophical Topics, 20: 115–80. D r e b e n , B., and J. v a n H e i j e n o o r t (1986), ‘Introductory Note to Go¨del 1929, 1930, 1930a’, in Feferman et al. (1986). D u m m e t t , M. (1981), Frege: Philosophy of Language, 2nd edn. (Cambridge, Mass.: Harvard University Press). F e f e r m a n , S., J. D a w s o n , et al. (eds.) (1986), Kurt Go¨del: Collected Works, vol. i (New York: Oxford University Press). F o r s t e r , T. E. (1995), Set Theory with a Universal Set: Exploring an Untyped Universe, Oxford Logic Guides, 31, 2nd edn. (Oxford: Oxford Science Publications). F r i e d m a n , H. (1999), ‘A Complete Theory of Everything: Validity in the Universal Domain’, extended abstract, 16 May 1999, www.math.ohio-state.edu/~friedman/ G o l d f a r b , W. (1979), ‘Logic in the Twenties: The Nature of the Quantifier’, Journal of Symbolic Logic, 44: 351–68. —— (1982), ‘Logicism and Logical Truth’, Journal of Philosophy, 79: 692–5. H a a p a r a n t a , L., and J. H i n t i k k a (eds.) (1986), Frege Synthesized: Essays on the Philosophical and Foundational Work of Gottlob Frege (Dordrecht: Reidel). H e n k i n , L. (1954), ‘Metamathematical Theorems Equivalent to the Prime Ideal Theorems for Boolean Algebras’, Bulletin of the American Mathematical Society, 60: 388. J e c h , T. J. (1973), The Axiom of Choice, Studies in Logic and the Foundations of Mathematics, 75 (Amsterdam: North-Holland). K r e i s e l , G. (1967), ‘Informal Rigour and Completeness Proofs’, in Lakatos (1967). L a k a t o s , I. (ed.) (1967), Problems in the Philosophy of Mathematics (Amsterdam: NorthHolland). M c C a r t y , T. (1981), ‘The Idea of a Logical Constant’, Journal of Philosophy, 78: 499–523. M c G e e , V. (1992), ‘Two Problems with Tarski’s Theory of Consequence’, Proceedings of the Aristotelian Society, 92: 273–92. —— (1996), ‘Logical Operations’, Journal of Philosophical Logic, 25: 567–80. —— (2000), ‘Everything’, in Sher and Tieszen (2000). M e n d e l s o n , E. (1987), Introduction to Mathematical Logic, Cole Mathematics Series, 3rd edn. (Pacific Grove, Calif.: Wadsworth & Brooks). M i t c h e l l , E. (1976), ‘A Model of Set Theory with a Universal Set’, Ph.D. thesis, University of Wisconsin, Madison. Q u i n e , W. V. O. (1937), ‘New Foundations for Mathematical Logic’, American Mathematical Monthly, 44: 70–80; repr. in Quine (1953). —— (1953), From a Logical Point of View (Cambridge, Mass.: Harvard University Press). R a y o , A. (2002), ‘Word and Objects’, Nouˆs, 36: 436–64. —— (2003), ‘When does ‘‘Everything’’ Mean Everything?’, Analysis, 63: 100–5.

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356 / Agustı´n Rayo and Timothy Williamson R a y o , A. and G. U z q u i a n o (1999), ‘Toward a Theory of Second-Order Consequence’, Notre Dame Journal of Formal Logic, 40: 315–25. R i c k e t t s , T. E. (1986), ‘Objectivity and Objecthood: Frege’s Metaphysics of Judgement’, in Haaparanta and Hintikka (1986). R u b i n , H., and J. R u b i n (1985), Equivalents of the Axiom of Choice, II, Studies in Logic and the Foundations of Mathematics, 116 (Amsterdam: North-Holland). S h a p i r o , S. (1991), Foundations without Foundationalism: A Case for Second-Order Logic (Oxford: Clarendon Press). —— (forthcoming), ‘All Sets Great and Small’, Philosophical Perspectives. S h e r , G. (1991), The Bounds of Logic: A Generalized Viewpoint (Cambridge, Mass.: MIT Press). —— and R. T i e s z e n (eds.) (2000), Between Logic and Intuition (New York: Cambridge University Press). S k a l a , H. (1974), ‘An Alternative Way of Avoiding the Set Theoretic Paradoxes’, Zeitschrift fu¨r mathematische Logik und Grundlagen der Mathematik, 20: 233–7. S t a n l e y , J. (1996), ‘Truth and Metatheory in Frege’, Pacific Philosophical Quarterly, 77: 45–70. T a p p e n d e n , J. (1997), ‘Metatheory and Mathematical Practice’, Philosophical Topics, 25: 213–64. T a r s k i , A. (1986), ‘What are Logical Notions?’, History and Philosophy of Logic, 7: 143–54. —— and R. L. V a u g h t (1957), ‘Arithmetical Extensions of Relational Systems’, Compositio Mathematica, 13: 81–102. v a n H e i j e n o o r t , J. (1967), ‘Logic as Calculus and Logic as Language’, Synthese, 17: 324–30. V a u g h t , R. (1956), ‘On the Axiom of Choice and Some Metamathematical Theorems’, Bulletin of the American Mathematical Society, 62: 262. W i l l i a m s o n , T. (1999), ‘Existence and Contingency’, Proceedings of the Aristotelian Society, suppl. vol. 73: 181–203; repr. with printer’s errors corrected in Proceedings of the Aristotelian Society, 100 (2000), 321–43 (117–39 in unbound edn.). —— (forthcoming), ‘Everything’, Philosophical Perspectives.

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16 Universal Universal Quantification: Comments on Rayo and Williamson Vann McGee

In telling us what formulas of the lower predicate calculus are true under every interpretation in which the variables are allowed to range over everything whatever, Dr. Rayo and Professor Williamson (Chapter 15 in this volume) have given us the answer we anticipated to a question we thought we knew the answer to all along. It’s been known since the 1930s that a formula is a theorem of D1 if and only if it is true under every interpretation over some infinite domain, which happens if and only if it’s true under every interpretation over every infinite domain. So in letting us know that a formula is a theorem of D1 if and only if it’s true in every interpretation over that particular infinite domain consisting of everything whatever, Rayo and Williamson have merely reminded us of something we’ve known for seventy years. Or so it appears at first glance. The surprising thing about their completeness theorem is to realize that it isn’t something we knew all along. The key observation is that Rayo and Williamson’s theorem and the classical theorem mean different things by ‘domain’. For classical model theory, a domain is a non-empty set, and a model is a certain kind of A version of this paper was presented at the Liars and Heaps conference, organized by JC Beall and Michael Glanzberg, in Storrs, Connecticut, in October 2002. I am grateful for helpful discussion.

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358 / Vann McGee structured set. Rayo and Williamson’s concern is with interpretations in which the variables range over everything, and there isn’t any set that includes everything. An unreflective response is to say that, if the theory of sets is too impoverished to provide us the structures we desire, we should rederive the classical completeness theorem within the theory of classes or within some still more inclusive theory of collections instead. But this won’t help, since there isn’t any class or collection that includes everything, as we can see by considering Russell’s paradox about the collection of all collections that don’t contain themselves. Once we focus our attention on the fact that Rayo and Williamson really intend their models to include everything, our complacency disappears. There’s nothing remarkable about the answer they obtain to the question ‘‘What principles of inference are valid across an unrestricted domain?’’ What’s remarkable is that they are able intelligibly to ask the question at all. They work within the theory of sets with Urelemente, whose intended interpretation is one of many models with unrestricted domain, yet from within that one model they are able to inquire about all the others without the benefit of an external vantage point. There’s no place to lay down a fulcrum. Once they determine how to formulate the question, the answer they get is the expected one. It would have come as a great surprise if the principles of inference available to us when we are talking about everything should be different from those available when we restrict our attention to one or another large set, and no such surprise is forthcoming. Indeed, the soundness theorem for models of unrestricted extent can be derived directly from the soundness theorem for set-sized models, by invoking the reflection principle of Richard Montague (1961) and Azriel Le´vy (1960). If there is an unrestricted model in which f is false, the reflection principle gives us an infinite set-sized model in which f is false, and so, according to the set-theoretic version of the soundness theorem, f is not a theorem of D1 . The completeness theorem is not as simple, but it is straightforward nonetheless. Assuming the whole universe can be well-ordered, the familiar Henkin (1949) construction carries over without incident. A characteristically ingenious paper of Harvey Friedman (1999) employs an Ehrenfeucht–Mostowski (1956) construction to obtain the same result on the weaker assumption that the universe is merely ordered, not necessarily well-ordered. Friedman’s paper has an audacious title—he calls it ‘A Complete Theory of Everything’—but it’s not an audacious investigation. It’s solid and steady

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Universal Universal Quantification / 359 workmanship, laying down a secure foundation. We don’t hear any extravagant hypotheses about the ultimate nature of reality. Instead, we get calm reassurance that, whatever the ultimate constituents of reality may be, we can continue to rely on the familiar laws of logic. Though not itself audacious, the inquiry serves to advance one of the boldest scientific programs ever conceived, Aristotle’s proposal for a fully general science of ontology that comprehends everything whatever in its domain. Friedman, Rayo, and Williamson don’t advance any substantive ontological theses, but they lay the groundwork for such theses. The special utility of the first-order predicate calculus in ontological investigation was stressed, especially, by Quine (1948). One stumbles over one’s own tongue if one tries to conduct ontological inquiries in unaided English. Ontologists need to investigate questions like the following: In addition to bodies in motion that from time to time collide, are there further entities, the events that occur when two bodies bang together? Try to ask this question in plain English, and a negative answer seems absurd. To say that there was a collision between the Titanic and an iceberg is just another way of saying that the Titanic collided with an iceberg, which no one wants to deny. To give the question an intelligible sense, we employ the apparatus of the predicate calculus: Is it the case that (9x)(x is a collision ^ the Titanic is a participant in x ^ (9y)(y is a iceberg ^ y is a participant in x) )? You can tell stories that account for the facts of experience, or at least appear to, and that answer the question either way. If ontological inquiry yields an affirmative answer to our question about the existence of collisions, we can still make an utterance of the sentence ‘(9x)(x is a collision ^ Titanic is a participant in x ^ (9y)(y is a iceberg ^ y is a participant in x) )’ false while maintaining its conventional meaning by employing the quantifiers in such a way that our quantified variables only range over bodies. This makes the sentence false by evaluating it in a different context, the way we can falsify ‘It’s hot in here’ by moving to a cooler room. But ontological inquiry is a context in which we don’t restrict the range of our variables, either tacitly or out loud, so the truth or falsity within that context of the existential sentence is a substantive fact about the constituents of reality. If ontological inquiry leads us to accept ‘(9x)x is a collision’, we can nonetheless make utterances of the sentence false by restricting the domain of the variables. If ontological investigation leads us to reject the sentence, we cannot make utterances of the sentence true by expanding the range of the

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360 / Vann McGee variables, since variables can’t range over things that aren’t there. The situation is thus asymmetrical, but the asymmetry doesn’t make our inquiry easier. Although we cannot make the existential sentence true, if it isn’t true already, by tailoring the context, we can nonetheless tell a credible story according to which it is true. God, presumably, could answer the question by surveying all the things there are and seeing whether any of them are collisions, but you and I must attempt to answer ontological questions by methods that are indirect and usually inconclusive. Within the modes of reasoning available to mortals, existential quantification doesn’t act anything like disjunction. For first-order quantification to play the role we’d like it to play in ontological inquiry, we need to take it as primitive, rather than relying on translations into English. Indeed, the whole point of introducing the formalism was that the ontological implications of statements in plain English are insufficiently clear. Only after we’ve decided whether to accept ‘(9x)(x is a collision)’ will we be able to determine how best to formalize the English sentence ‘There was a collision in the North Atlantic’. First-order quantifiers have as their default value—the value they take before contextual considerations are taken into account—quantification over everything, but contextual considerations typically enforce a more restrictive value. ‘9U ’ is a variant of familiar first-order quantification that is expressly forbidden ever to move beyond its default value. Seeing the enormous utility of the methods of first-order logic in ontological inquiry, Quine (1948) goes so far as to say that a theory can make its ontological commitments explicit only if it is formulated entirely within the first-order predicate calculus. This seems excessive, particularly with regard to modal operators. Possible-world semantics has a lot of appeal, but we wouldn’t want to suppose that someone who says that Al Gore might have been president is committed to the existence of a possible world in which Gore won the election. It’s enough that, in the real world, more people might have decided to vote for Gore or those who decided to vote for Gore might have been allowed to do so less selectively. Be that as it may, the logical status of modal operators is not germane to our present concerns. A place where Quine’s strictures are crucial to our present concerns is the status of higher-order logic, since the way Rayo and Williamson solve the problem of finding a vantage point from which to examine arbitrary models of unlimited size is by employing second-order logic. Within the simple theory of types, the first-order quantifiers emphatically fail to range over

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Universal Universal Quantification / 361 everything. There are individuals, properties of individuals, relations between individuals, properties of properties of individuals, relations between properties of individuals, and so on, and the first-order variables only range over the first of these. This outcome is not an artifact of the notational system, but a characteristic feature of type theory. There are entities of some sort that the second-order variables range over. On various accounts, those entities are properties, concepts, classes, or collections, but, whatever they are, they aren’t among the ‘‘individuals’’ that the first-order variables range over. This outcome is unavoidable. The inconsistency of Frege’s Basic Law V shows that there isn’t any one–one map from concepts to individuals— ‘‘one–one’’ in the sense that concepts that aren’t coextensive are never mapped to the same individual. Since inclusion maps are one–one, it follows that the concepts aren’t included among the individuals. To get genuinely unrestricted quantification, we have to move beyond the theory of types, and that’s what the Zermelo–Fraenkel set theory enables us to do. By consolidating all the things type theory talks about into a single domain, ZFCU (Zermelo–Fraenkel set theory, including Choice, with Urelemente) permits all-inclusive quantification. It accomplishes this by adopting a bare-boned logic, treating the distinctions that type theory regarded as part of the logic as extralogical theoretical distinctions instead. From the ZFCU perspective, we can see that type theory masks its ontological commitments behind a baroque notation, and that the entities to which type theory commits itself extend far beyond what the type theorist calls ‘‘individuals’’. Now we find ourselves in a quandary. To develop the metalogic of firstorder quantification over an unrestricted domain, we need second-order logic, but second-order logic is only available if our domain is restricted. As philosophical predicaments go, this one isn’t too severe. Our ability to employ a logical system successfully doesn’t depend on possession of a metatheory. We can happily use the logical system without a completeness proof or even a soundness proof. Anyone who supposes that before we can legitimately employ a rule of inference we first have to prove its soundness neglects Quine’s (1936) observation that the inferential rules we are trying to vindicate are almost invariably the same rules we use in the metatheory to derive the soundness theorem. Even so, a metalogic would be nice to have. The breakthrough idea that permits us to devise a metalogic of unrestricted quantification was developed by George Boolos (1984, 1985), who looked at such plural constructions in English as the Geach–Kaplan sentence ‘Some critics admire only one another’. Boolos’s guiding intuition is that this

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362 / Vann McGee sentence doesn’t assert the existence of a non-empty set or collection consisting of critics who admire only other members of the collection. It asserts the existence of critics with certain attitudes, and no entities other than critics are required. Boolos fleshes out this intuition in a semantic theory that treats monadic second-order variables as ranging over individuals, just as first-order variables do. The difference between them is that first-order variables are constrained by the requirement that each variable be paired with one and only one individual, whereas second-order variables aren’t thus restricted. Boolos proves that, over set-sized domains, where we can give second-order quantification an unequivocal meaning by treating the second-order variables as ranging over the power set, his so-called ‘‘plural quantification’’, which treats second-order variables as ranging over individuals many at a time, is logically equivalent to monadic second-order quantification. However, plural quantification can be gainfully employed when the domain of our individuals is larger than any set, even when the domain is all-inclusive. Moreover, provided we can form ordered pairs of individuals, we can get the effect of full, not just monadic, second-order logic. The Boolos construction provides Rayo and Williamson a vantage point from which to examine arbitrary models over the universal domain. Actually, the machinery they use goes a bit beyond what Boolos provides. Boolos takes ordinary predicate logic and extends it by introducing a new logical operator, but he doesn’t introduce any new extralogical symbols. In particular, he doesn’t introduce any novel forms of predication. ‘‘Predicates’’ were what they’ve always been; an n-ary predicate allows n individual variables into its argument places. Rayo and Williamson are not so restrained. Truth on an SO-interpretation is inductively defined as a second-level predicate, a predicate that allows a second-order variable into one of its argument places. Thus they help themselves to logical apparatus that goes well beyond what Boolos has offered. Elsewhere (Rayo 2002), Rayo has argued eloquently (and, to my mind, persuasively) that second-level predication, as found in the formal analogue of such English locutions as ‘Some strangers carried John home from the bar’ and ‘There were toys scattered all over the living room’, which describe collective actions or collective properties, is a philosophically innocuous extension of Boolos’s plural quantification; Byeong-uk Yi (forthcoming) has advocated a similar position. Second-level predication may well be indispensable for many worthwhile purposes, but for the purpose of defining truth in an SO-interpretation, it is a

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Universal Universal Quantification / 363 needless extravagance. Let us take a variable assignment for a formula f to be a finite function that assigns a value to each individual variable that occurs free in f. We can specify what it is for a variable assignment to satisfy a formula f under a given SO-interpretation by induction on the complexity of f. There are no bound second-order variables in this inductive definition, so we can apply Frege’s (1879) technique to convert the inductive definition into an explicit second-order definition. Thus we don’t need to introduce truth in an SO-interpretation as a new second-level predicate, because we can explicitly define it using only first-level predicates. The crucial observation is that the clause for the existential quantifier doesn’t really require us to examine other SO-interpretations, only other variable assignments. Rayo and Williamson briefly consider the problem of characterizing the valid second-order inferences; that is, the valid inferences of second-order logic as it’s standardly understood, with no second-level predicates. Needless to say, no completeness theorem is forthcoming. I suspect that this time their employment of second-level predication is unavoidable. However, if we are willing to restrict our attention to inferences with finitely many premisses— bear in mind that second-order logic is incompact—we can characterize the valid inferences more simply, using Tarski’s (1936) method. The inference from f1 , f2 , ..., fn to c is valid for all SO-interpretations with unrestricted domain if and only if the universal closure (with respect to ‘8U ’) of the formula obtained from the conditional ð(f1 ^ f2 ^ ... ^ fn ) ! cÞ by replacing all the extralogical terms by variables is true. If instead we want to talk about validity over arbitrary domains, we have to restrict the quantifiers appropriately. How to characterize the valid inferences of the extended version of secondorder logic, which allows second-level predication, is an open problem. These flights into higher-order logic are taking us away from Rayo and Williamson’s principal concern, which is with rock-bottom basics. To undertake an ontological inquiry, we need to be able to reason reliably about everything at once. What we find is that, at least at the level of first-order predicate logic, the methodological principles available to us when we attempt a fully general inquiry are the same principles that have served us faithfully throughout the special sciences. With this foundation securely in place, let the ontology begin.

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364 / Vann McGee

REFERENCES B a r - H i l l e l , Y e h o s h u a , E. I. J. P o z n a n s k i , M i c h a e l O. R a b i n , and A b r a h a m R o b i n s o n (eds.) (1961), Essays on the Foundations of Mathematics (Jerusalem: Magnes Press). B o o l o s , G e o r g e (1984), ‘To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)’, Journal of Philosophy, 81: 430–49; repr. in Boolos (1998: 54–72). —— (1985), ‘Nominalistic Platonism’, Philosophical Review, 94: 327–44; repr. in Boolos (1998: 73–87). —— (1998), Logic, Logic, and Logic (Cambridge, Mass.: Harvard University Press). E h r e n f e u c h t , A n d r z e j , and A n d r z e j M o s t o w s k i (1956), ‘Models of Axiomatic Theories Admitting Isomorphisms’, Fundamenta Mathematicae, 43: 50–68. F r e g e , G o t t l o b (1879), Begriffsschrift (Halle: L. Nebert); Eng. trans. by Stefan Bauer-Mengelberg in van Heijenoort (1967: 1–82). F r i e d m a n , H a r v e y (1999), ‘A Complete Theory of Everything’, www.math. ohio_state.edu/ foundations/ manuscripts.html H e n k i n , L e o n (1949), ‘The Completeness of the First-Order Functional Calculus’, Journal of Symbolic Logic, 14: 159–66. L e´ v y , A z r i e l (1960), ‘Axiom Schemata of Strong Infinity in Axiomatic Set Theory’, Pacific Journal of Mathematics, 10: 223–38. M o n t a g u e , R i c h a r d M. (1961), ‘Fraenkel’s Addition to the Axioms of Zermelo’, in Bar-Hillel et al. (1961: 91–114). Q u i n e , W i l l a r d v a n O r m a n (1936), ‘Truth by Convention’, in Otis H. Lee (ed.), Philosophical Essays for A. N. Whitehead (New York: Longmans, Green); repr. in Quine (1966: 70–99). —— (1948), ‘On What There Is’, Review of Metaphysics, 2: 23–38; repr. in Quine (1964: 1–19). —— (1964), From a Logical Point of View (Cambridge, Mass.: Harvard University Press). —— (1966), The Ways of Paradox (New York: Random House). R a y o , A g u s t ´i n (2002), ‘Word and Objects’, Nouˆs, 36: 436–64. T a r s k i , A l f r e d (1936), ‘U¨ber den Begriff der logischen Folgerung’, Actes du Congre`s International de Philosophie Scientifique, 7: 1–11; Eng. trans. by J. H. Woodger in Tarski (1983: 409–20). —— (1983), Logic, Semantics, Metamathematics, 2nd edn. (Indianapolis: Hackett). v a n H e i j e n o o r t , J e a n (ed.) (1967), From Frege to Go¨del (Cambridge, Mass.: Harvard University Press). Y i , B y e o n g - u k (forthcoming), ‘The Language and Logic of Plurals’, Journal of Philosophical Logic.

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INDEX

Acrisus 255 Agnosticism 92, 95 ff. Aristotle 359 assertion 153 ff. Austin, J. L. 160 Barker, S. 153 Basic Law V 357n Beall, JC 4, 242n, 357n Bellow, S. 37 Belnap, N. 168, 272n Berry, G. 230 Berry’s paradox 1, 230, 231, 243, 247 Bierwich, M. 156 bivalence (principle of) 3, 92, 95 ff., 106, 151 ff. Blamey, S. 154 Bolzano, B. 331 Boolos, G. 113, 334, 361 ff., 364 borderline (cases) 40 ff., 49–51, 59–60, 63–4, 66–71, 77 ff, 80 f., 139, 196 ff, 201, 203, 206 ff., 212, 214, 216 Brady, R. 271 Braun, D. 191 Burgess, A. 216n Buridan, J. 227 Cantor’s theorem 353 Carroll, L. 39 f., see also Humpty Dumpty Cartwright, R. 336, 337n categorical perception 186 Chafe, W. 235

Chambers, T. 97n., 105 Choice principles 340, 345 ff. Chomsky, N. 157, 162 Classical logic 196, 205 f. cognition, and vagueness 27–8 Collins, J. D. 27n competence 50 f., 53, 64–8 concept, vs. predicate 26 conditionals 271 ff., 258, 283, 284–7, 291 ff., 308, 320 ff (see strict conditional) consequence relations 205–20 context 29, 32–4, 45–64, 174 ff., 256 -dependence 111 f. -sensitivity 131 f. -change 233 ff., 254n salience 174 ff ignorance of 181 ff. -ual singular terms 259 Contextualism 2, 14 f., 28–9, 118 ff., 122, 127, 253 conversational score 45–54, 57, 62, 63 f., 74–7, 234, 236, 239 completeness, see soundness continuum-valued semantics 16, 19 Curry’s paradox 1, 316 f., 321 cut-off points 10 ff., 13 ff., 16, 17, 18, 24 ff. Davidson, D. 58, 68, 164 default determinate extension, antiextension 129

366 / Index Definitiely operator 195–206, 209–11, 218, 220, 226 ff. iteration of 228 f. definiteness 40–5, 58–64, 66–8, 226 deflationism 246 f. degree of truth 19 f., 21, 226 degree of belief 278 f., 280 f., 283, 288n., 297n., 308 defectiveness 265, 270 f., 273, 287, 296 f., 299 ff., 307 denotation trees 241 ff., 249 f. designated truth value 172 determinacy (-ateness) 40–5, 58–64, 66–8, 180 ff. determinate application and undefinedness 135–47 determinately-Liar (see Liar paradox) determinately operator 274 f., 281 ff., 297–302, 304, 305, 307, 308 f., 328 Dietz, R. 283n discourse analysis 233 ff. disquotationalism 246 f. Dummett, M. 23, 57, 159 ff., 288 Dunn, J. M. 168 Ehrenfeucht, A. 358, 367 Edgington, D. 168, 200 f. Eight Sophism 227 Epistemicism 12 ff., 15, 18, 28–9, 51, 57, 61, 78, 86 ff., 90, 92, 95 ff., 107, 122, 127, 228 f. Exclusive 95 f. Non-exclusive 95 f. Euthyphro contrast 65 ff. Evans, G. 90, 114, 151 events 359 f. evidential constraint (EC) 96, 100 ff. excluded middle, law of 262, 273, 280 f., 296, 304, 305 f., 308 f. extension and antiextension 128 f., 156

Feferman, S. 245 Feldman, R. 208n, 270 Field, H. 2, 3n, 4, 202n, 312 ff. file change semantics 234 f. Fine, K. 43, 200, 206, 208n, 209n, 286 f., 181 fixed point 314 ff., 324 ff. Kripke 319 Fodor, J. A. 30n, 115n forced-march sorites 10 ff., 49, 53 f., 55 ff., 68, 70 formal semantics 331–8 role of 287 ff., 303–7, 309 ff. Foster, J. 91 Fox, D. 186 Frege, G. 354, 361, 363 f., 151 Friedman, H. 346, 358 f., 364 fuzzy boundaries 197 f., 205 fuzzy logic 11, 15 f., 18, 19, 21, 33, 44 (see also Łukasiewicz continuum-valued logic) fuzzy object 16, 17 fuzzy truth values 17 f. Garcia, J. 49, 51, 53 f., 56, 60, 64, 66, 68 Geach, P. 361 gestalt 55–7 Glanzberg, M. 3n, 4, 233n, 357n Go¨del, K. 241 Goldbach’s conjecture 101 f. Go´mez-Torrete, M. 202n gradable adjective 156 Graff, D. 3n, 4, 14, 29, 45, 61, 62, 109, 111n, 117n, 118, 119n, 120n, 127n, 202n, 203n, 216n, 225, 155 Grice, P. 234 Grosz, B. 233, 235 grounding 319 Gupta, A. 272n

Index / 367 Hart, W. D. 107n Heck, R. 114n, 124n, 200n, 200 f. Heim, I. 233, 234 Henkin, L. 358, 364 hierarchy of language levels 239 f., 240 f. Higginbotham, J. 162 higher-order languages 353 second-order 334 f., 350 ff. higher-level predicates 335, 353 higher-order vagueness 123 ff., 198 f., 200–5, 225, 281 f., 284, 288, 290, 299, 302, 308 f. Horgan, T. 10, 14, 49 Horn, L. 156 Humpty Dumpty 39, 40, 48, 58, 64, 67 f., 70 identity 10, 15 ff., 17 ff., 19 f., 25, 31, 35 substitutivity of, 16, 20, 22 transitivity of, 17, 18, 20, 21, 22, 31, 35 contextual 258 f. illocutionary act 160 implicature 186 information, given and new 235 information structure 153 intension (partial) 177 f. Intuitionism 95 intuitionist logic 22 f. Isard, S. 233 judgement dependence 41 f., 65 ff., 71 Kamp, H. 214n, 233 Kaplan, D. 153, 361 Keefe, R. 3, 10, 30n, 200, 206, 211n Kennedy, C. 180 Kenstowicz, M. 186 Kleene logic 268 f., 284, 292, 315 Klein, E. 180 Knower paradox 228 Ko¨nig’s paradox 1, 230

Kreisel, G. 336, 337n Kripke, S. 112, 154, 245 Kripke theory of truth 268–71 Kripke–Feferman theory 270 Larson, R. 185 law of excluded middle (see excluded middle) law of non-contradiction (see noncontradiction) LePore, E. 30n, 115n Le´vy, A. 358, 364 Lewis, D. 45–9, 52, 58, 74, 76, 110n, 233 f. Liar paradox 231, 250 possible 228 determinately- 328 meta- 328 f. local validity 17, 19 ff., 22, 35 logical consequence 331 f. Łukasiewicz continuum-valued logic 285 f. Maia 227 many-valued logic 172 f. margin of error principle 13 McDowell, J. 191 McGee, V. 3n, 4, 5n, 30n, 40 ff., 58–60, 181, 200n, 207, 245, 335, 336n, 337 McLaughlin, B. 30n, 40 ff., 48 ff., 181, 200n, 207 metalanguage (metatheory) 19 ff., 21 ff., 244 ff. truth in 19 ff. validity in 19, 21 ff. indeterminacy in 284, 288, 305 f., 309 f. Misconceived Conditional, the 97, 98, 99 model theory 331–8 modus ponens 17, 21, 22

368 / Index Montague, R. 358, 364 Montague sentence 266 Moore’s paradox 89 Mostowski, A. 358, 364

proof 226 ff. propositions 153 ff., 265 Putnam, H. 96n., 97n., 104, 105 Quine, W. V. O. 359 ff., 364, 168

negation (internal) 172 f. non-classical logic 5, 313, 327 non-contradiction, law of 9 non-monotonicity 315 f. no-no paradox 227 ff. No Sharp Boundaries paradox, the 98 normativity 63 f., 68 open-texture 43 ff., 49 f., 59, 64 f., 66 f., 77, 80 f. object, fuzzy 34 f. ontology 359 f., 363 pairwise indistinguishability 116–21 paraconsistent logic 276 paradoxes of self-reference 1, 5, 9, 253 paradox of denotation 1, 231 ff., 253 ff. paradoxes of definability 1, 230, 239, 247, 253 ff. Simmons’ infinitary 247 ff partiality and weak partiality (of predicates) 128 f., 144–7, 156 ff. Peano, G. 231 penumbral connection 212 f., 216 ff., 286 f., 308, 320 perlocutionary effects 189 Plato 227 plural predication 362 plural quantification 361 f. Poincare, H. 231 possible worlds semantics 322 ff. Post, J. 228 f. precisification 30 presupposition 179 Priest, G. 3, 16, 17, 20, 24–37, 49, 276 Proetus 225

Raffman, D. 50, 54 ff., 60 ff., 64–71, 73 f., 77, 80, 82, 109, 116n, 117n, 118n, 119n, 180 Rawls, J. 161 Rayo, A. 1n, 3n, 4, 5n, 334, 357–64 Read, S. 96n., 105 reflection principle 358 Reinhart, T. 233, 235 rejection 275–80, 308 revenge problem 270 f., 273, 297–307, 328 f. Richard, J. 231 Richard’s paradox 1, 230, 239 rigidification 118n Rooth, M. 186 Russell, B. 230, 329 Russell’s paradox 1n, 3n, 230 f., 250, 329, 354, 358 Russell’s theory of types 241, 360 f. Sainsbury, R. M. 43, 44n, 197, 201n Second-order logic 334 f., 350 ff., 360 ff. Segal, G. 185 semantics 64–8 -pragmatics interface 185 f. semantic nihilism 12 semantic view of vagueness 85 ff. and higher-order vagueness 89 sentence topics 235 set paradoxes 1n, 329 Shapiro, S. 3, 73–83, 118n, 119n, 202n Sidner, C. 233, 235 similarity constraint 14 f. Simmons, K. 239n, 245n, 253–61 (passim)

Index / 369 singularities 239 ff. Slater, M. 24n Smith, P. 10 Soames, S. 2n, 4, 119n, 151 ff., 205 Socrates 227 Sorensen, R. 3n, 4, 12, 13, 29, 228 f. Sorites paradox 1 f., 5, 9 ff., 42–71, 74 ff., 79 f., 81 ff., 114–21, 132–5, 149 f., 197, 200 ff., 207 f., 214 ff., 253 sorites series 195 ff., 201 ff., 214 ff., 219 f. soundness and completeness 21 ff. Stalnaker, R. 176 f., 213, 233, 234 Stanley, J. 118n Strawson, P. F. 151 Strengthened Liar (see revenge problem) strict conditional 320 ff. substitutivity 313 ff. superdetermination and superundefinedness 147–50 Supervaluationism (supervaluation) 43, 44, 47, 49, 63, 87 ff., 98, 100n, 114 f., 121–7, 196, 200 ff., 207 f., 214 ff. supervaluationist semantics 265, 274 f. symmetry principle 226 f. Szabo, Z. 233n Tappenden, J. 30n, 111n, 156 205 Tarski, A. 240, 245 f., 253, 312 f., 331, 341, 363 f. Tarski undefinability theorem 303 ff., 329 Tautology-teller 321 Third Possibility 92 ff., 98n tolerance (of vague predicates) 14 f., 41–5, 50–4, 71, 74 f., 82 quasi-tolerance 117, 120 f., 125 of small changes 117n truth 40, 43 f., 47 f., 58 f., 63 f. T-rules 112 f. T-scheme 122 f.

naı¨ve 316 ff. T-biconditionals 312 ff. -assertion (purpose of) 164 truth-maker gap 228 Truth-teller 319 truth-value gaps 151 ff., 195 f, 205 f, 208, 212, 226, 270, 273, 284 Tye, M. 205 types, theory of (see Russell’s theory of types) Unger, P. 48 universal set 333 universality of natural languages 245 semantic universality 245 f. unrestricted quantification 333 f., 338 ff., 350–4 Urquhart, A. 154 Uzquiano, G. 334 vagueness in rebus 3, 86, 88 ff., 95 van Deemter, K. 180 van der Sandt, R. 179 van Heijenoort, J. 239 Varzi, A. 3, 9, 27n Vaught, R. 341 Weakly vague predicates 147 Wedgwood, R. 164 Wiggins, D. 164 Williamson, T. 1n, 3n, 4, 5n, 10, 12, 13, 25, 29, 41, 57, 78, 79, 86, 88, 91n., 104n., 105, 107 f., 112n, 117, 122, 135n, 155, 164, 180, 200n, 202n, 203n, 206, 210 f., 274, 338 ff., 357–64 Wright, C. 3, 14, 41, 43, 50, 64 f., 70, 96n., 101n., 103n., 105, 106–27 (passim), 164, 197, 199 ff. Yablo, S. 3n, 4, 249n Yi, B. 362, 364