The Riddle of Vagueness: Selected Essays 1975-2020 0199277338, 9780199277339

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The Riddle of Vagueness

OUP CORRECTED AUTOPAGE PROOFS – FINAL, 16/06/21, SPi

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The Riddle of Vagueness Selected Essays 1975–2020 C R I SP I N W R IG H T With an Introduction by

R IC HA R D K I M B E R LY H E C K

1

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1 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom 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. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Crispin Wright 2021 Introduction © Richard Kimberly Heck 2021 The moral rights of the authors have been asserted First Edition published in 2021 Impression: 1 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, by licence 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 work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2020947868 ISBN 978–0–19–927733–9 DOI: 10.1093/oso/9780199277339.001.0001 Printed and bound in the UK by TJ Books Limited Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Contents Preface and Acknowledgements vii Origins of the Essays xv

Introduction by Richard Kimberly Heck 1 1. On the Coherence of Vague Predicates

41

2. Language-­Mastery and the Sorites Paradox

79

3. Hairier than Putnam Thought with Stephen Read 103 4. Further Reflections on the Sorites Paradox

107

5. Is Higher-­Order Vagueness Coherent?

167

6. The Epistemic Conception of Vagueness

181

7. On Being in a Quandary: Relativism, Vagueness, Logical Revisionism

209

8. Rosenkranz on Quandary, Vagueness, and Intuitionism

261

9. Vagueness: A Fifth Column Approach

271

10. Vagueness-­Related Partial Belief and the Constitution of Borderline Cases

293

11. ‘Wang’s Paradox’

303

12. The Illusion of Higher-­Order Vagueness

335

13. On the Characterization of Borderline Cases

367

14. Intuitionism and the Sorites Paradox

393

Appendix to Chapter 14

423

References General Index Index of Names

427 435 445

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Preface and Acknowledgements I started grappling with the philosophical challenges presented by vagueness in the early 1970s. At that time, I think it fair to say, almost nothing of real significance had been written on the topic since the contributions of Eubulides of Megara.1 In the modern era, in particular, philosophers of language from Frege on had been for the most part content to theorize in ways that marginalized vagueness, or to focus on idealized languages in which there was none. No one writing before 1970 seemed fully to have taken the measure of the awkwardness of the Sorites paradox,2 or the depth of its roots, as usually ­formulated, in our intuitive thinking about what kind of ability mastery of a language is. My curiosity about the topic was originally piqued by conversations with my friend the mathematician Aidan Sudbury and with Michael Dummett, then my colleague at All Souls, who around that time was working on the stunning lecture that he later published as ‘Wang’s Paradox’ (Dummett 1975). My own interest initially stemmed from concerns in the philosophy of mathematics: I was drawn to the thought that the apparent open-­endedness of the extension of a vague predicate might provide a fruitful model for the manner in which a finitist should think about the putatively infinite extension of ­nat­ural number, and that a correct logic of vagueness might accordingly be appropriate for a finitist number theory. My subsequent paper ‘Strict Finitism’ (Wright 1982) was the upshot of my reflections in that direction. But while thinking about finitism I became preoccupied with the Sorites paradox itself. Dummett’s paper argued, inter alia, that vague expressions do indeed affect natural language with inconsistency—that is, that the paradox shows that our use of vague expressions is governed by rules that are actually inconsistent. That struck me then as an incredible conclusion,3 but one that was nevertheless forced by a certain conception of the significance of the kind of theory of meaning for a natural language to which philosophers of the time aspired, at least in Oxford in the mid-­1970s, in the throes of the then reverberating 1  An interesting study of the early history of the Sorites is Moline (1969). 2  An exception is Black (1937). 3  Others, of course, have endorsed Dummett’s response, notably Matti Eklund. A useful conspectus of his views is Eklund (2019).

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viii  Preface and Acknowledgements ‘Davidsonic boom’. This conception is what I punningly dubbed the ‘governing view’—crudely, that understanding a natural language is, through and through, a rule-­governed competence. The idea, crudely, was that we are able to parse a novel sentence by (in some sense) working out the conjoint implications of the rules of syntax relevant to its mode of construction and the semantical rules governing its occurrent primitive expressions. My first two chapters4 elaborate and critique that thought, and indeed my efforts to refine it resurface in several places in this volume. But at that time I attempted no specific resolution of the paradox other than, in this way, to try to undercut one kind of motivation for (one form of) its major premise. It was more than a decade before I felt that I had anything further to say on the issues. By then Hilary Putnam (1983) had suggested that a resort to in­tu­ition­ist rather than classical logic might contribute to a solution. After some skirmishing (Wright (1987a, this volume, Chapter 4), I became convinced that there might be something to this. The other development in my thinking at that time was the realization that we need to distinguish a ­variety of Sorites paradoxes, differing in the form taken by their major premises, the various lines of motivation for those premises, and even—in recognition of the so-­called Forced March Sorites—in whether they involve explicit inference from premises at all (Wright  1987a, this volume, Chapter 4). I was, however, still thinking of vagueness as essentially a phenomenon of semantics—as some kind of deficiency, or partiality, of content, or lack of instruction from s­ emantic rules—and it was only after trying to come to terms with Timothy Williamson’s brilliant book (1994, critiqued in this volume, Chapter  6) in the mid-­1990s that a different way of thinking about the matter began to dawn on me. In essentials, Williamson’s ‘Epistemicism’ grafts together two thoughts: a classical, bivalent metaphysics of indicative content coupled with a view of the vagueness of a predicate as essentially an epiphenomenon of our difficulty in judging its application in the area close to the sharp ‘cut-­off ’ required by the first thought. It now occurred to me that dispensing with the first thought while developing the second (shorn, therefore, of the presupposition of sharp cut-­offs) might provide the motivation for a thoroughgoing intuitionistic treatment of the 4  ‘On the Coherence of Vague Predicates’ (this volume, Chapter 1) was eventually published in the same volume of Synthese as ‘Wang’s Paradox’. The volume also included Kit Fine’s ‘Vagueness, Truth and Logic’ (Fine 1975), and proved to be key in launching the intense discussion of vagueness and the Sorites, now into its fifth decade, that has followed since. ‘Language-­Mastery and the Sorites Paradox’ (1976, this volume, Chapter 2) was published in Gareth Evans’s and John McDowell’s influential edited anthology Truth and Meaning, exploring the issues raised by Davidson’s proposal for meaning-­theory.

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Preface and Acknowledgements  ix topic—a treatment broadly modelled on the Mathematical Intuitionists’ treatment of classical number theory and analysis—providing both for a satisfying deconstruction of the plausibility of the major premises in Sorites paradoxes, and for a well-­motivated framework in which those premises can be denied without the essentially and regrettably superstitious resort to sharp cut-­offs. That remains my view. But it takes a bit of working out. My first explicit foray in the intuitionistic direction was ‘On Being in a Quandary’ (Wright 2001b, this volume, Chapter 7; also published in Philosopher’s Annual, 24). The paper was initially rejected by Mind with a comment by the referee that it ‘contained no discernible line of argument’. I requested the editor at the time, Mark Sainsbury, to determine whether that should be the final view of the journal. Mark solicited other opinions and, gratifyingly, subsequently saw fit to publish. The in­tu­ition­istic project underwent further motivation and development in a paper I wrote for the memorable Liars and Heaps conference organized by Jc Beall and Michael Glanzberg at the University of Connecticut in 2002 (Wright 2003c, this volume, Chapter 9). A further opportunity for a more complete statement of it was provided by the invitation to contribute to the Library of Living Philosophers volume for Michael Dummett that was published in 2007 (Wright 2007, this volume, Chapter 11). Michael’s graceful but incredulous reaction to my proposal in his Reply5— echoed, at least in point of incredulity, by Ian Rumfitt6—spurred me into thinking further about the question, what kind of semantics might be appropriate for a language containing vague expressions and the basic logical resources involved in the derivation of Sorites paradoxes, if the needed in­tu­ ition­istic/logical distinctions, especially the potential contrast between the

5 ‘I am left, then, with admiration for the beautiful solution of the Sorites paradox advocated by Crispin Wright, clouded by a persistent doubt whether it is correct . . . I do not say that Wright’s proposed solution of the Sorites is wrong; I say only that we need a more far-­going explanation than Wright has given us of why intuitionistic logic is the right logic for statements containing vague expressions before we can acknowledge it as correct. It is not enough to show that the Sorites paradox can be evaded by the use of intuitionistic logic: what is needed is a theory of meaning, or at least a semantics, for sentences containing vague expressions that shows why intuitionistic logic is appropriate for them rather than any other logic . . . If Crispin Wright is to persuade us that he has the true solution to the Sorites paradox, he must give a more convincing justification of the use of intuitionistic logic for statements containing vague expressions: a justification namely, that does not appeal only to the ability of that logic to resist the Sorites slide into contradiction. We need a justification that would satisfy someone who was puzzled about vagueness but had never heard of the Sorites: a justification that would sketch a convincing semantics for sentences involving vague expressions’ (Dummett 2007, pp. 453–4). 6  Rumfitt does indeed proceed to offer an intricate semantics for vagueness that has some prospect for validating intuitionist logic (Rumfitt 2015, pp. 227ff). I have expressed reservations about its fitness for the philosophical purpose elsewhere (Wright 2020, pp. 378ff).

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x  Preface and Acknowledgements acceptability conditions of some kinds of sentence and those of their double negations, are to be semantically grounded? My recent contribution to the volume on the Sorites edited by Elia Zardini and Sergi Oms (this volume, Chapter 14) takes that question head on and also takes the opportunity to try to provide a more rounded and complete overview of the problems and treatment of vagueness from an intuitionistic point of view than was accomplished either in the Quandary paper, the article for Liars and Heaps, or the contribution to the Dummett festschrift. Maybe a preface to a philosophical book may allowably make a purely philosophical point. If so, perhaps here is a place to emphasize that I do not myself resonate with the Dummett–Rumfitt thought that it is only after the provision of a satisfactory such semantics that the intuitionistic proposal will be ready for the philosophical market. The proposal, after all, recommends a revision of logic. So someone who thinks that it requires validation by a background semantics has to suppose also that logical principles generally stand on firm ground only when sustained by an appropriate semantic story about the logical operators involved. That thought is, in my opinion, by no means mandatory. Notwithstanding my own sortie into the knowledge-­theoretic semantics outlined in Chapter 14, I want to sound a note of reservation about the need for any such underpinning for proposed logical restrictions specifically in response to paradox. There are delicate questions in the vicinity here concerning what should count as a solution to a paradox—questions concerning how much, and what kind of, explaining of what is going wrong, one is required to accomplish. It is useful to be mindful here of Stephen Schiffer’s distinction (1996, 2003) between ‘happy-­face’ and ‘unhappy-­face’ cases. In ‘happy-­face’ cases, a paradox is successfully diagnosed as owing to a determinate mistake, or oversight, which is identifiable as such by standards of our practices that are in place before the paradox is considered. (The reader may re-­attend here to the concluding sentence of the quote from Dummett in n. 5.) Regrettably, happy-­face paradoxes have proved historically relatively rare. With a paradox of the latter, ‘unhappy-­face’ kind by contrast, there may be little to offer by way of diagnosis and explanation other than to say that the paradox is spawned by concepts and conceptual practices that are in some way inherently incoherent, or otherwise objectionable, and that have become entrenched, and that the only solution is to modify them in ways which, perhaps because of their entrenchment, may have no independently arguable sanction.

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Preface and Acknowledgements  xi So in the present case. As far as the Sorites is concerned, I myself am compelled by the following train of thought: • First, that we know the major premise of a Sorites is false because it is inconsistent, by absolutely elementary reasoning, with truths (the relevant polar verdicts); • Second, that, in the usual run of examples, we clearly do not know that the hypothesis of a sharp cut-­off has a witness in the relevant series of cases; and hence • Third, that there, therefore, has to be something wrong with the ­reasoning—classical reasoning—that, granted plausible forms of closure of knowledge across entailment, forces us to deny the latter ignorance if we think we have the former knowledge. And this conclusion must stick before we identify a mishap—indeed, even if we cannot readily do so— in the double-­negation elimination step that concludes in the postulation of a sharp cut-­off. In a recent seminar I attended in New York, a well-­respected colleague was heard to say that what the Sorites teaches us is that we ‘just have to get used to’ the idea that there really are sharp boundaries in all vagueness-­related Sorites series.7 I am vividly aware that a whole generation of (mostly Oxford-­trained) professionals have indeed habituated themselves to that idea (or anyway profess that they have.) But I venture to suggest that, for most, the three-­step train of thought articulated above will present an immensely more powerful, commonsensical appeal. If the reader concurs, they will see that the idea we need to get used to is not that of a crystalline world of unknowable sharp cut-­ offs that spares us Sorites paradoxes, but rather the idea of situations where classical logic lets us down.8 Why should we need a semantic theory before we can accept the charge that there is here a gap that classical reasoning evidently illicitly crosses? Someone may of course, like Dummett, still insist that, if classical reasoning really is here unjustified, it must be recognizably so in the light of a proper account of how we already implicitly (when fully lucid and reflective)

7  Actually, some of those who take this line on the ‘regular’ Sorites propose that certain forms of modal Sorites—for example, what has come to be known as Chisholm’s paradox—require a different response. This is not the place to pursue the putative distinction. 8  A common rejoinder is that arguments for the revision of classical logic may rationally be discounted, since it has, after all, for approaching a century and a half, performed sterling service for us. No doubt it has—in mathematics and the exact sciences. But not at all as a logic of vagueness.

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xii  Preface and Acknowledgements understand the logical operators involved. Semantic theory is therefore needed to give a correct account of that alleged prior implicit non-­classical understanding. This insistence can be appropriate, however, only if we take the view that the Sorites-­paradoxical reasoning, extended to the conclusion that there has to be a sharp cut-­off, involves a mistake that is in principle recognizable as such by the lights of the understanding of the key notions involved that we already have. We have to be, in other words, in the territory of a possible ‘happy-­face’ solution. And, if we are confident that that is so, there will now be a constraint on any semantic theory to be offered that it present a plausible account both of the antecedent understanding of ordinary thinkers and of why they are here inclined to misperceive its requirements. Good luck with that project. Revisionary proposals have historically sometimes been mo­tiv­ated by a sense that certain logical principles involve distortion of our understanding of the operators involved—‘relevance’ critiques of classical logic are one example. But the revisionism of the intuitionist is not of this character. The intuitionist’s revisionism issues from a reformist stance. The semantic project is not to recover an account of extant distinctions which, if someone is seduced by the extended paradoxical reasoning, they overlook but, in the wake of already well-­motivated revisions of classical logic, to propose a framework in which those revisions have an independent theoretical setting, so that we can restore a sense of knowing what we are doing in inferential practice and of how the suspect transitions may be conceived to fail. It is in this spirit that I offered the knowledge-­ theoretic clauses proposed in Chapter 14. However, if it is only for this purpose that it is useful, then semantic theory is precisely not needed to justify the relevant logical revisions, to persuade us of ‘the true solution to the Sorites paradox’. Instead, like Hegel’s Owl of Minerva, it spreads its wings of insight only with the coming of the dusk, when the day’s (revisionary) work is already done. In arriving at and developing over the years the views offered in this volume, I have benefited from the published contributions of two colleagues in particular. First, the stability of the intuitionistic proposal, as argued for in the Quandary chapter, involving, as it does, maintaining a kind of agnosticism about sharp cut-­offs simultaneously with the thesis of Evidential Constraint concerning a large class of vague predications, was challenged early on by Sven Rosenkranz. My original attempt to defuse his objections is contained in  Chapter  8.9 Second, Stephen Schiffer’s writings have been especially 9  Rosenkranz pursues his criticisms in Rosenkranz (2009).

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Preface and Acknowledgements  xiii influential in persuading me of the importance of what I term the Characterization Problem: the challenge of saying what exactly a borderline case is. Too much of the literature has neglected this or proceeded on ­unexamined assumptions about the answer. But a satisfactory account of what the vagueness of a soritical predicate consists in is the essential first step both to understanding the place and significance of vagueness in natural language and to the dissolution of the Sorites. Schiffer’s own answer, latterly abandoned, is that the borderline cases of a vague predicate are those which distinctively excite a certain kind of partial belief in competent judges—and that vagueness is thus, in a certain sense, a psychological phenomenon. The relevant special notion of partial belief is, I believe, very difficult to substantiate in detail—some of the wrinkles are explored in Chapters 10 and 13—but, in thinking of borderline case as a status grounded in features of our judgemental psychology, rather than in the semantics of the relevant predicate, while simultaneously rejecting Bivalence, Schiffer made a key move in common with the intuitionistic proposal. One other issue is prominent in the chapters that follow. Anyone thinking about vagueness needs to address the putative phenomenon of higher-­order vagueness: the apparent fact that the distinction between the clear cases of a predicate and its borderline cases itself seems to have no sharp boundary, and that the point must reiterate in vertiginous fashion. The apparent fact ­troubled me for a long time before I hit on the argument of Chapter  5 that higher-­ order vagueness is itself distinctively soritical. A different argument to the conclusion that the very notion of higher-­order vagueness is intrinsically incoherent is given in Chapter 9. But a third argument, gratifyingly endorsed by Riki Heck in their introductory chapter and drawing directly on my preferred approach to the Characterization Problem, has to wait until Chapter 12. If it is correct, there is simply no such thing as higher-­order vagueness, as usually conceived, and it is accordingly no constraint on a satisfactory account of vagueness that it accommodate, still less explain it. Forty-­five years thinking about these matters has built up large enough debts to completely swamp my own investments. Besides the conversations at the beginning with Aidan Sudbury and Michael Dummett, my work in this area has probably benefited more extensively than even I realize from inputs from so many sources over the years. But the following deserve special mention. In the late 1990s, when I was fortunate enough to be awarded a Leverhulme Personal Research Professorship for, inter alia, a project on Vagueness, I enjoyed countless valuable discussions with my former St Andrews research students Patrick Greenough and Sven Rosenkranz, who were then working

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xiv  Preface and Acknowledgements on doctoral theses concerning, respectively, vagueness and agnosticism. Later, I learned a huge amount from the regular seminar sessions with the participants in the 2003–6 AHRC-­ supported Arché Vagueness project: Agustin Rayo, Stewart Shapiro, Richard Dietz, Sebastiano Morruzzi, Elizabeth Barnes, and Elia Zardini. Elia, in particular, has, then and since, given me invaluable detailed written feedback on early drafts of several of the papers reprinted here. I also have profited greatly from interactions with my NYU colleagues Hartry Field, Kit Fine, and especially Stephen Schiffer, with whom I taught an exceptionally interesting graduate seminar on the topic in 2007. In more recent times I have enjoyed very helpful conversations with Susanne Bobzien and Ian Rumfitt, whose excellent, recently published co-­ authored paper (Bobzien and Rumfitt 2020) has important points of affinity with the views developed here. My thanks to Dirk Kindermann, and Yu Guo for help at different times with the Bibliography, and to Yu again and Sergi Oms for extensive and meticulous work in preparing corrected copy for the Press. Special thanks to Peter Momtchiloff for his usual tact and patience while I repeatedly put off the work necessary to compile the final manuscript. Finally, I am beyond grateful to Riki Heck for their patient, searching, and perceptive critical reconstruction of my journey. Early in their essay, they advise readers not to attempt the chapters that follow before they have assimilated Dummett’s ‘Wang’s Paradox’. I strongly endorse the same advice about Riki’s Introduction. Crispin Wright Kemback, Fife June 2020

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Origins of the Essays The Introduction by Richard Kimberly Heck was specially written for this volume. Chapter 1 ‘On the Coherence of Vague Predicates’, was first published in Synthese, 30 (1975), 325–65. It is reprinted here by kind permission of Springer Nature. Chapter 2 ‘Language-­Mastery and the Sorites Paradox’ was first published in G.  Evans and J.  McDowell (eds.), Truth and Meaning (Oxford University Press, 1976), 223–47. Chapter 3 ‘Hairier than Putnam Thought’ (co-­authored with Stephen Read) was first published in Analysis, 45 (1985) (Oxford University Press), 56–8. Chapter 4 ‘Further Reflections on the Sorites Paradox’ was first published in Philosophical Topics, 15/1 (Spring 1987), 227–90. © 1987 The Board of Trustees of the University of Arkansas. It is reprinted here with the kind permission of the University of Arkansas Press, www.uapress.com. Chapter 5 ‘Is Higher-­ Order Vagueness Coherent’ was first published in Analysis, 52 (1992) (Oxford University Press), 129–39. Chapter 6 ‘The Epistemic Conception of Vagueness’ was first published in Southern Journal of Philosophy, 33 (1995), special number on Vagueness, 133–59. It is reprinted here by kind permission of John Wiley and Sons Inc. Chapter 7 ‘On Being in a Quandary: Relativism, Vagueness, Logical Revisionism’ was first published in Mind, 110 (2001) (Oxford University Press), 45–98. Chapter 8 ‘Rosenkranz on Quandary, Vagueness, and Intuitionism’ was first published in Mind, 112 (2003) (Oxford University Press), 465–74. Chapter 9 ‘Vagueness: A Fifth Column Approach’ was first published in J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox (Oxford University Press, 2004), 84–105. Chapter 10 ‘Vagueness-­ Related Partial Belief and the Constitution of Borderline Cases’ was first published in a Book Symposium on Stephen Schiffer’s The Things We Mean in Philosophy and Phenomenological Research, 73/1 (Wiley, 2007), 225–32. It is reprinted here by kind permission of John Wiley and Sons Inc.

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xvi Origins of the Essays Chapter 11 ‘ “Wang’s Paradox” ’ was first published in The Library of Living Philosophers, volume XXI, The Philosophy of Michael Dummett, edited by Randall E. Auxier and Lewis Edwin Hahn (Open Court, 2007), 415–45. It is reprinted here by kind permission of the Open Court Publishing Company. Chapter 12 ‘The Illusion of Higher-­Order Vagueness’ was first published in Richard Dietz and Sebastiano Morruzzi (eds.), Cuts and Clouds (Oxford University Press, 2010), 523–49. Chapter 13 ‘On the Characterization of Borderline Cases’ was first published in Gary Ostertag (ed.), Meanings and Other Things: Themes from the Work of Stephen Schiffer (Oxford University Press, 2016), 190–210. Chapter 14 ‘Intuitionism and the Sorites Paradox’ was first published in Sergi Oms and Elia Zardini (eds.), The Sorites Paradox (Cambridge University Press, 2019), 95–117. © Cambridge University Press 2019, reproduced with permission.

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Introduction Richard Kimberly Heck

The present volume collects most of Crispin Wright’s published papers on vagueness.1 These papers represent the fruits of a career-­long investigation of the Sorites paradox and its significance for our understanding of language and cognition. Anyone at all familiar with the literature on vagueness will already know the early papers, as many of them have long featured as standard fare in any course on the subject. The more recent material, too—meaning the papers published in the twenty-­first century—will be known to aficionados. But having recently read through the corpus myself, it seems safe to say that reading these papers together allows for an understanding of the development of Wright’s thought, and of the connections between the papers, that repays the effort required many times over. The relations between the earlier papers and the later ones are especially intriguing. What I want to do in this introduction is to provide a sense of what I have learned from my own recent rereading, and of the issues that seem to me most to need attention. I will begin, however, by providing a high-­level guide to the papers themselves: to their content and their context. Afterwards, I will take up five themes that seem to me of particular interest.

An Overview By his own account,2 Wright’s interest in vagueness—like that of many other philosophers at the time—was inspired by Sir Michael Dummett’s now classic 1  First, let me thank Crispin for asking me to write this introduction. It is an honour to do so. Though I have been reading and thinking about his papers on vagueness for almost as long as I have been doing philosophy—‘Language-­Mastery’ was an early favourite—the reflections to follow grew most immediately out of a graduate seminar on vagueness that I gave at Brown University in the autumn of 2008. (This is what accounts for the fact that references to more recent literature are mostly lacking.) Thanks to all the members of that seminar for their contributions, but especially to my colleagues David Christensen and Josh Schechter, whose participation was purely optional and most welcome. 2  See the introductory remarks to ‘ “Wang’s Paradox” ’ (this volume, Chapter 11).

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Richard Kimberly Heck. DOI: 10.1093/oso/9780199277339.003.0001

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2  The Riddle of Vagueness paper ‘Wang’s Paradox’ (Dummett 1978: 248–68).3 Perhaps the most im­port­ant contribution of Dummett’s paper, as Wright again notes, was to make it plain, first, what meagre logical resources are required to generate the Sorites paradox and, second, how utterly plausible the major premise of the paradox is, at least in central cases. This is especially so in the observational case, which Dummett clearly sees as critical: if two patches of colour are visually indiscriminable by me, how could it possibly be that one of them looked red to me but the other did not? But, if visually indiscriminable patches must both look to be red if either does, then quite simple reasoning—reasoning that does not, in particular, need to appeal to induction—will lead quickly to absurd results. Dummett’s own dramatic conclusion was more or less Frege’s: that observational predicates in particular, and vague predicates more generally, are logically incoherent, so that no proper semantics for them is possible. One reaction to Dummett’s argument is to attempt to provide the semantics Dummett had claimed could not be given.4 Wright’s reaction was different. It was to see Dummett’s argument as revealing an incoherence in certain very general assumptions about the nature of language use. Given those premises, Wright’s thought was, Dummett’s argument was correct, and what had to go were therefore the very general assumptions with which the argument began.5 It is the burden of ‘On the Coherence of Vague Predicates’ (this ­volume, Chapter 1)6 to make that argument. The general assumptions in question are ones any philosopher working in Oxford in the mid-­1970s might well have conceived as orthodoxy. Hence, Wright styles them the ‘governing view’. The first thought behind the ­governing view is that language use is, by and large, an activity governed by rules, rules that competent speakers really do follow, not so much in the sense that they consciously appeal to those rules—it is, familiarly, often a difficult matter to say what the rules are—but rather in the sense that these rules are norms that govern our linguistic behaviour, which is partly to be explained in terms of our allegiance to them. The second thought then follows more or less 3  Anyone who is considering reading Wright’s papers but who has not already read Dummett’s is hereby strongly advised to read Dummett first. 4 This is the reaction of supervaluationists (Fine  1975) and degree theorists (Goguen  1969; Peacocke 1981). Contextualism and its variants (Raffman 1996; Soames 1999; Fara 2000; Shapiro 2006: ch. 7) represent another class of responses. Wright pays very little attention to the latter. As it happens, I also tend to think that such views miss the point. See n. 29 for my reasons. 5  An argument with much the same structure is given by Jamie Tappenden (1995), though his target is Timothy Williamson’s argument (1994) for the epistemic conception of vagueness. Tappenden’s conclusions are very different from Wright’s. 6 And of the shortened version, ‘Language-­ Mastery and the Sorites Paradox’ (this volume, Chapter 2).

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Introduction by Richard Kimberly Heck  3 naturally. If these rules really are ones we follow, if our behaviour is supposed to be guided by these norms, then their content ought in some sense to be available to reflection.7 It should be clear enough that Dummett himself is committed to the ­governing view, and it may well be true that it plays an essential role in the argument of ‘Wang’s Paradox’. What is less clear is whether the governing view really can enforce the key premise of the Sorites paradox, as both Dummett and Wright require. We will worry about this below. Not long after the two papers so far mentioned were published, however, Wright himself became somewhat dissatisfied with his treatment of vagueness. The paper ‘Further Reflections on the Sorites Paradox’ (this volume, Chapter 4) is his effort to do better. That is a long paper, and there is a lot in it. Much of it consists of critical reaction to a paper by Christopher Peacocke (1981) that had sought to defend the governing view against Wright’s ­criticisms by developing a degree-­theoretic account of vagueness. Wright answers some of Peacocke’s arguments against his own proposals, and he adds new criticisms of degree-­theoretic semantics. Many of these criticisms still seem to me quite damaging. But, all these many years on, the engagement with Peacocke is a sideshow, and what is most important in the paper lies elsewhere. Perhaps the most important contribution is Wright’s analysis of what he calls the ‘Tachometer paradox’, which I will discuss in some detail below. Almost as important, however, is Wright’s isolation of what he calls the ‘No Sharp Boundaries paradox’. As usually formulated, the key premise of the Sorites paradox is, for ex­ample, that, if a patch is red, then any patch pairwise indistinguishable from it (in colour, of course) must also be red. The key premise is thus a universally quantified conditional: (1)  ∀x(Rx ∧ x ∼ y → Ry )

7  This second condition is what characterizes ‘implicit’ knowledge, in the sense in which that notion figures in Dummett’s writings on the philosophy of language between roughly 1974 and 1985. One can know something implicitly and not be able, at that time, to articulate it; but one is supposed, in principle, to be able to articulate it. Note how this contrasts with what has come to be called tacit knowledge. One’s knowledge of the rules of syntax—of universal grammar, for example—is in no sense supposed to be available to reflection. It is nowadays a common alternative to regard our ‘knowledge’ of whatever rules might be involved in concept-­use as merely tacit. It would be worth investigating the significance of this point for present concerns. Some of what follows contributes to such a project, though it hardly completes it.

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4  The Riddle of Vagueness where ∼ expresses indistinguishability.8 The intuitive justification for this premise is supposed to be that redness is tolerant of small changes, and that is very much how the Sorites paradox is usually presented: surely removing one grain, plucking one hair, and so on, cannot ‘make a difference’. The No Sharp Boundaries paradox, on the other hand, proceeds from the assumption that there cannot be two indistinguishable patches one of which is red while the other is not. This premise, then, is the negation of an existential: (2)  ¬∃x(Rx ∧ x ∼ y ∧ ¬Ry ) And the justification for this claim is supposed to be that the very vagueness of ‘red’ requires that there not be a sharp boundary between the red and the not-­red. For this reason, Wright suggests, the No Sharp Boundaries paradox threatens to disclose an incoherence in the very notion of vagueness (this volume, Chapter 4, 143–145). Of course, (1) and (2) are classically equivalent. But they are not intuitionistically equivalent, and this observation was part of what fuelled a proposal, due to Hilary Putnam (1983a), that intuitionistic logic might help here. This suggestion is criticized in ‘Hairier than Putnam Thought’ (this volume, Chapter 3), for which Stephen Read joined Wright. That is actually where the No Sharp Boundaries paradox originally appears. It is worth pausing to appreciate, before we get too far along to notice such things, just how different these two premises feel, even if they are classically equivalent. Perhaps what is most important here is that (2) is weaker than (1) in the absence of excluded middle. For how might one argue for (1) given (2)? Well, suppose Rx and x ∼ y. By (2), it cannot be that ¬Ry . So, given excluded middle (or double-­negation elimination), Ry. But, without excluded middle, we cannot draw that conclusion. And it is a common move in discussions of vagueness to reject the law of excluded middle—or, at least, the law of ­Bivalence—for borderline cases.9

8  Not many years later, Timothy Williamson (1990) would launch a wide-­ranging investigation of the notion of indiscriminability and, shortly thereafter, revolutionize the study of vagueness in ways we will consider below (Williamson 1994). 9  Let me quickly pet a peeve. It is often said that intuitionists cannot deny the law of Bivalence. This  is simply false. What is true is that an intuitionist cannot deny any particular instance of the law of Bivalence. That law, understood as ∀x(T (x ) ∨ ¬T (x )) —that is, as the universal quantification of  an  instance of excluded middle—can consistently be denied. And that is so even though ¬∃x¬(T (x ) ∨ ¬T (x )) is a logical truth. This is a consequence of the fact that ¬∀x(Fx ) and ¬∃x(¬Fx ) are intuitionistically consistent.

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Introduction by Richard Kimberly Heck  5 We will return to this issue below. For now, let me mention a third contribution made in ‘Further Reflections’: the paradox of higher-­order vagueness. It is, as was just said, natural to respond to the Sorites by saying that the paradox somehow assumes Bivalence—for example, that every statement of the form ‘Patch 𝑥 is red’ is either true or false. Rather, the thought is, there are some patches to which redness can neither truly nor falsely be ascribed; it is objectively indeterminate whether they are red. The now standard objection is that we are no more able to identify the boundary between the red and the indeterminate than we are to identify the boundary between the red and the not-­red. So these intermediate boundaries are themselves vague, and that in turn gives rise to what is nowadays called ‘higher-­order vagueness’. One sometimes has the sense that higher-­order vagueness is ultimately what makes the problem of vagueness so utterly intractable. What Wright suggests, first in Section 6 of ‘Further Reflections’ and then again in ‘Is Higher-­Order Vagueness Coherent?’ (this volume, Chapter 5), is that higher-­order vagueness itself suffers from a special kind of incoherence. We will discuss this in some detail in Section 6. About the same time Wright published ‘Is Higher-­ Order Vagueness Coherent?’ Timothy Williamson published his first paper defending an epi­ stem­ic account of vagueness: Williamson (1992b) argued that there really is a last hair before baldness, a last red patch in the series; it is just that we do not know, and perhaps cannot know, which one it is. This view had previously been held,10 but it had never been defended with such resourcefulness. It suffices to say that things have not been the same since. Wright’s initial reaction  appears in ‘The Epistemic Conception of Vagueness’ (this volume, Chapter 6).11 Wright’s central criticism is the same as that of many others: the epistemic conception has no plausible account of what determines the sharp cut-­offs of whose existence it assures us, and yet whose location it insists we cannot know.12 But Wright makes other criticisms of the epistemic view, too, and these will repay close study. We will discuss one below. What is most intriguing about this paper, however, is that Wright does not disagree with Williamson’s claim that vagueness is an epistemic phenomenon. On the contrary, Wright concedes that the notion of a borderline case is to be

10  Williamson (1994) discusses the history. 11  For Williamson’s response, see his paper ‘Wright on the Epistemic Conception of Vagueness’ (Williamson 1996b). 12  My own version of this worry appears in ‘Semantic Accounts of Vagueness’ (Heck 2003: sect. 1), which was itself a commentary on what became Wright’s paper ‘Vagueness: A Fifth Column Approach’ (this volume, Chapter 9).

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6  The Riddle of Vagueness explained in broadly cognitive terms. What distinguishes Wright’s view is his insistence that vagueness is not just an epistemic matter but also a semantic one. Anyone familiar with Dummett’s writings on anti-­ realism should see quickly both where this is headed and where it comes from. Anti-­realism, as Dummett understands it, is founded on two thoughts:13 First, the semantic properties of our words must supervene on the way we use them; second, our use of at least certain parts of our language is inadequate to secure classical truth conditions for our utterances. The familiar worry about epistemicism is strikingly similar: our use of such predicates as ‘red’, ‘heap’, and the like cannot fix determinate classical extensions for them. To be sure, and importantly, one does not have to agree with Dummett’s arguments for anti-­realism about mathematics (say) to share this worry about vague predicates. But the similarities are nonetheless suggestive, and that makes a broadly intuitionistic approach to vagueness worth exploring.14 It is with the exploration of such an approach that Wright’s twenty-­first-­ century writings on vagueness have largely been concerned. There are hints of it in ‘The Epistemic Conception of Vagueness’, but it first emerges with clarity in ‘On Being in a Quandary’ (this volume, Chapter  7). That paper is not exclusively concerned with vagueness. Indeed, in some ways, it is more concerned to elaborate and defend aspects of the approach to questions about realism that are explored in Wright’s book Truth and Objectivity (1992b)15 and to reinforce a certain form of argument for rejection of the law of excluded middle. And that, to my mind, is precisely what makes Wright’s latest approach to problems about vagueness so interesting: he has always seen issues about vagueness as intertwined with some of the deepest issues in philosophy. The question Wright makes central in his most recent work is how so-­ called borderline cases are to be characterized. Until recently, it was widely taken for granted that borderline cases of a vague predicate are ones in which 13  For Dummett’s view, see, e.g., The Logical Basis of Metaphysics (Dummett 1991). The introductory essay to Wright’s earlier collection, Realism, Meaning, and Truth (Wright 1993), remains an excellent survey. 14  For someone attracted to this sort of argument against epistemicism, these reflections should raise the question where exactly they want to get off Dummett’s train, and why. Why, that is to say, should these sorts of reflections undermine epistemicism but not undermine mathematical realism? One can understand Williamson as pushing precisely this line of argument, but from the other side: At least some of the objections to epistemicism threaten to commit one to a much broader anti-­ realism (about mathematics, the past, etc). 15  Indeed, ‘On Being in a Quandary’ has previously been reprinted in Wright’s book Saving the Differences: Essays on Themes from Truth and Objectivity (Wright 2003b).

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Introduction by Richard Kimberly Heck  7 the predicate can neither truly nor falsely be ascribed. But Wright now agrees with Williamson (1992b: sect. I), as he did not in ‘The Epistemic Conception’, that such a description is flatly inconsistent: in the presence of the so-­called Tarski biconditionals (or T-­sentences), such a description leads to contra­dic­ tion, and Wright, like Williamson, regards the Tarski biconditionals as non-­ negotiable. Unlike Williamson, however, Wright does not conclude that Bivalence must therefore hold. Rather, he concludes, again following the intuitionists, that borderline cases cannot be ones in which Bivalence fails, in the sense that they are cases in which a particular statement is neither true nor false. Bivalence fails only in the sense that we cannot endorse it:16 we cannot say, affirmatively, that the statement in question is either true or false, but nor can we preclude its being either true or false. We must remain agnostic about the matter. We must, in particular, remain open-­minded about the question whether, say, a particular patch midway between red and orange might be red. We might have no idea how the question might be decided. We might even, Wright suggests, have no idea whether it is even metaphys­ ically possible for the question to be decided. Still, Wright’s claim is, we c­ annot, on pain of contradiction, say that the question can have no answer. If that seems puzzling—if one finds oneself wondering how borderline cases so much as could be resolved—well, let me simply assure the reader that they are not alone. So, that is the broad sweep of Wright’s discussion. Let us now discuss some of these issues in more detail. We will start by exploring some themes from the early papers.

Tolerance as Putatively a priori In his own later work, Wright often characterizes the conclusion of the early papers, ‘On the Coherence of Vague Predicates’ and ‘Language-­Mastery’, as being that our use of vague predicates is not governed by ‘rules’. In fact, however, the ‘governing view’ he wants to challenge has two components: first, that the use of language is rule governed and, second, that, in uncovering  the rules in question, ‘we may legitimately approach our use of language from within, that is, reflectively as self-­conscious masters of it, rather than ­ externally, equipped only with behavioural notions’ (this volume, 16  Cannot endorse it for any particular case. As I mentioned earlier, in n. 8, there is no general reason to suppose that we cannot deny Bivalence, as a general principle.

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8  The Riddle of Vagueness Chapter 1, 43). I think it is clear, on a careful reading, that the arguments of the early papers are directed not so much at the first thesis as at the second one. The claim is that, if the rules governing our use of vague predicates have  to be discernible ‘from within’, then those rules will, among other things, commit us to the major premise of the Sorites and so to the incoherence of vague predicates. In particular, it will prove to follow from (if not just  to be among) the rules that govern our use of vague predicates that their application is, as Wright puts it, tolerant of small changes to the objects to which they apply. But lots of small changes add up to big ones, and so ­paradox ensues. One point that deserves emphasis is that, on Wright’s treatment, as on Dummett’s, the tolerance of vague predicates is supposed to be a putatively conceptual truth, since, as was said, tolerance is supposed to emerge from the rules that govern our use of vague predicates. And if one reflects upon the sort of paradox most commonly discussed under the heading ‘Sorites’, this should seem obviously correct: in so far as one is attracted to the claim that, say, anything that is pairwise indistinguishable from a red thing is red, it is on broadly a priori grounds. Indeed, it is hard to see how Dummett’s conclusion that observational predicates as such are incoherent could be reached except via some general, a priori argument from observationality to tolerance, and any such argument is going to make tolerance a conceptual matter. This point, it seems to me, has been lost in some work that exploits Sorites-­ style reasoning.17 There is a tendency nowadays to regard almost all ­expressions of natural language as vague. Now, perhaps they are all in some sense imprecise. But it is far from obvious that expressions such as ‘dog’, or even ‘chair’, are vague in the way that predicates like ‘red’ are vague. I occasionally encounter a general argument to the contrary, one that purports to demonstrate the Sorites susceptibility of (almost?) every property. It goes like this: anything that is a chair (dog, flea) will still be a chair (dog, flea) if one removes just one atom from it; quod erat demonstrandum.18 Or again: anything that is a molecule of water will still be a molecule of water if one moves its constituent atoms some incredibly minute distance further apart from one another than they originally were. But, for one thing, neither of these claims is clearly true. As far as the chair is concerned, removing the crucial atom that 17  It is emphasized, however, by Elia Zardini (2008). 18  An argument of this general sort is used by Theodore Sider (2001) to establish unrestricted mereological composition, and it may be that he is responsible for the popularity of this argument. But it is not clear to me whether Sider’s own use of the argument is vulnerable to the criticisms I am making here.

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Introduction by Richard Kimberly Heck  9 was holding the back onto the seat will not leave one with a chair.19 And, as far as water molecules are concerned, I would have thought that there would come a point at which one had moved the atoms far enough apart to break the atomic bonds holding the molecule together, at which point the atoms would proceed to go their separate ways. But, for present purposes, it does not matter at all whether I am right about this. My point is simply that, even if they are true, neither of these ‘Sorites premises’ has any chance whatsoever of being a conceptual truth. And it is in this respect that the vagueness of ‘red’ is very different from the vagueness of ‘dog’, if, indeed, ‘dog’ is vague (a question about which I hereby register serious doubt). Whether ‘chair’ is vague in the privileged sense, I am not sure. Maybe a series based upon a slow change of shape, or of ability to support weight, would give rise to a Sorites premise whose appeal was relatively a priori. But, if so, then that will be because having a distinctive shape and being able to support weight are, as Frege would have put it, ‘marks’ of the concept of a chair. In any event, it is important to recognize that the target of Wright’s argument, in the early papers, is really much broader than a certain conception (then locally popular) of how the philosophy of language ought to be done. To see this, we need to ask how precisely the governing view is supposed to conceive the way that ‘self-­conscious reflection’ is supposed to inform the theoretical account of our use of language. Is it supposed to be distinctive of the governing view that it permits self-­conscious reflection as an analytical tool? To put it differently: is it enough to make one an adherent of the ­governing view that one is prepared to reflect self-­consciously on what justifies the application of a particular predicate as one attempts to uncover the rules that control our use of it? Or does the governing view instead assume that the character of these rules must be wholly available to self-­conscious reflection? Much of what Wright has to say suggests the former interpretation, but only the latter will serve his purposes. Justifying this claim would require more careful, and more lengthy, exposition of Wright’s discussion than I can undertake here. So let me just say that I believe careful study will support this reading.20 If so, however, then the governing view might well be redescribed simply as one that is committed to the possibility of a priori conceptual 19  I owe this quip to Josh Schechter. 20  Consider this remark: ‘[C]onclusions [about the content of semantic rules] are, apparently, not to be drawn by reference to the character of our response to a predication of “looks red” when, by ordinary criteria, all the provisos are fulfilled. And in order for this exclusion to be legitimate, the dictates of the semantic rules for “looks red” have to be constituted independently of such responses’ (this volume, Chapter 4, 159; emphasis in original). Note that Wright speaks here of the governing view as excluding certain sorts of data that would not be available to self-­conscious reflection.

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10  The Riddle of Vagueness a­ na­lysis: the character of our concepts, quite generally, can in principle be understood from within, by the deployment of the typical tools of armchair philosophical reflection. But then Wright’s argument against the governing view becomes one against a priori conceptual analysis: in the case of vague predicates, a priori conceptual analysis leads inexorably to the nihilist conclusion that vague predicates are incoherent. Wright’s argument that the governing view is committed to tolerance is therefore of quite breathtaking scope. I will not try to evaluate that argument here—though, for what it is worth, I am inclined to think it may well have significant implications for contemporary discussions of the place of conceptual analysis (that is, ‘intuitions about cases’) in philosophy. This observation about the conceptual basis of Sorites premises explains, by the way, why Sorites premises take the form they do, at least in central cases. Consider the following principle: (3) If a thing is red, then anything not very different in colour from it must also be red. The phrase ‘not very different in colour’ could be given various in­ter­pret­ ations. Perhaps it means ‘is just discriminable’, or ‘connectable by a sequence of three pairwise indiscriminable patches’. It does not matter. The point is that (3) has nothing like the intuitive force that the usual Sorites premise has. Note carefully: I am not saying that (3) has no force at all; I am just saying that it does not strike me, anyway, as having anything like the irresistibility that the usual version has. This is particularly true in the phenomenal case: (4) If a thing looks red, then anything that looks not very different in c­ olour from it must also look red. I think this has essentially no force. As we will discuss in the next section, what makes the usual version of (4) seem so utterly undeniable is the thought that, if two patches are visually indiscriminable, then they look the same. And that, of course, is not something we discovered empirically. If it strikes us as true at all, it does so on the basis of reflection. One might worry that no such point applies to heaps (so that the point just made does not generalize). That is, it might seem implausible that the one-­ grain version of the Sorites premise should be a conceptual truth but that the two-­grain version should not be. But what is the intuitive status of a Soritical principle concerning the removal of two grains? Surely it must matter how big the grains are. What if the heap were made of flour or, better yet, of some

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Introduction by Richard Kimberly Heck  11 purely homogeneous substance? How much would you then be able to remove without threatening the heap’s heaphood? The answer is implicit in a remark of Wright’s: ‘Our conception of the conditions which justify calling something a heap of sand is such that the justice of the description will be unaffected by any change which cannot be detected by casual observation’ (this volume, Chapter  1, 51; emphasis in original). So we can remove only as much as will not affect how the heap appears to casual observation. If so, then what is really driving the paradox of the heap is a Sorites premise of this sort: (5) If a thing is a heap, then anything that cannot be distinguished from it by casual observation must also be a heap. What makes the usual premise about grains plausible is that we suppose, rightly or wrongly, that removing one grain will not affect how the heap appears. So, if all of that is correct, then the Sorites for ‘heap’ is just as observational as the one for ‘red’, and the points made in the last paragraph do indeed apply, mutatis mutandis.21 This point turns out to matter more than one might suppose. In ‘The Epistemic Conception of Vagueness’, Wright argues that Williamson cannot explain the scope of our ignorance concerning where the cut-­off lies in a Sorites series for ‘red’ (this volume, Chapter 6, sect. 7). Williamson’s explanation (1994: 227 ff.) proceeds in terms of what he calls ‘margin for error’ principles. In the case of ‘red’, such a principle would nat­ural­ly take the form: (6) If one knows that a thing is (not) red, then anything pairwise indiscriminable from it must (not) be red, whether one knows it or not. It takes only a little effort to discover that, while this principle excludes our knowing that x is red and that x′ is not-­red, and also our knowing that x is red and that x ″ is not-­red,22 it allows that we should know that x is red and that x′″ is not-­red. But, Wright observes, the borderline region seems much broader than three patches. Williamson (1996b: 40–1) replies that how wide the borderline region is will depend upon how wide the margin of error for such judgements is, at least implying that he did not mean to commit himself to the claim that the margin for error principle must be stated in terms of 21  The same goes for ‘bald’. If it does not go for ‘small number’, maybe that is because Wang’s paradox was not so plausible to begin with. 22  Because then x′ will be indiscriminable from x, hence red, but also from x ″ , hence not-­red. (By x′ , I mean the next patch in the series after x.)

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12  The Riddle of Vagueness pairwise indiscriminability.23 But it seems to me that Williamson is committed to that claim, at least if margin for error principles are going to do the ex­plana­ tory work he wants them to do—namely: to explain why Sorites premises strike us as plausible, indeed, as compelling. Wright elsewhere complains that epistemicism lacks any account of the plausibility of the Sorites premise (this volume, Chapter  9, 274). But this is uncharitable: Williamson does have an account on offer. The explanation is supposed to be that we mistake the true margin for error principle—that, if we know that x is red, then x′ must be red—for the false Sorites premise that, if x is red, then x′ too must be red.24 But now suppose that the margin for error is not pairwise indiscriminability but instead: being not very different in colour. If so, then the correct margin for error principle is: (7) If one knows that a thing is red, then anything not very different in ­colour from it must also be red, whether one knows it or not. And then (7) will ‘explain’ the intuitive plausibility of (3), just as (6) was supposed to explain the intuitive plausibility of the usual Sorites premise. As we have seen, however, (3) simply does not have the intuitive plausibility of the usual Sorites premise, so something somewhere is wrong. In sum, then, if Williamson sticks with (6), he cannot explain the width of the borderline region. So he has to go for something else, like (7). But then (7) explains too much, and Wright’s complaint, that epistemicism has no account of the special plausibility of the Sorites premise, in its usual form, is vindicated.

The Tachometer Paradox Surely the most intractable form of the Sorites paradox is the phenomenal one, the one that concerns predicates like ‘looks red’. It is certainly this form of the Sorites that always bothers my students the most, by which I mean: it is the one that makes them really, fundamentally puzzled. Suppose you are looking at a computer screen, and the screen is completely covered by two large patches of colour, one red, one blue. Over time, the colours change. At a certain point, the patch on the left is so close in colour to the patch on the 23  There is anyway something fishy about this: the borderline region turns out to be precisely thrice the width of the margin for error? That just seems too neat and tidy. 24  Or, perhaps, we wrongly derive the latter from the former and the so-­called KK principle that, if one knows that p, one must also (be in a position to) know that one knows that p.

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Introduction by Richard Kimberly Heck  13 right that no line appears between them.25 Then it seems consistent with what you are seeing that there is a single, double-­sized patch of colour before you. (As, indeed, there may be, for all I have said.) If so, however, then it is very hard to see how, if the left-­hand side of the screen looks to be red, the right-­ hand side of the screen could fail to look red. After all, one wants to say, they look exactly the same! But that way there be monsters, since we have just committed ourselves to the claim that, if two patches of colour are pairwise indistinguishable, then, if one looks red, so does the other, and we can, of course, link an obviously red patch to an obviously blue one by a sequence of pairwise indistinguishable patches. The right response to this, I suggest, is to deny that, if two patches of colour are pairwise indistinguishable, then, if one looks to be red, then so must the other. Indeed, I suggest, essentially with Wright, that this claim is flatly refuted by the Sorites itself. Probably we do not even need the Sorites. I believe I have seen a direct counterexample: three patches a, b, and c, of which the first and last two are pairwise indiscriminable, where a looks to be red and c does not.26 One might think that this was impossible, since surely a must look the same colour as b, which must look the same color as c. But there is no reason to suppose that looks the same colour is transitive. It cannot be that a is the same colour as b and b is the same colour as c, but a is not the same colour as c. But I see no obstacle to a similar phenomenon in the case of appearances: it can simultaneously be that a looks the same colour as b, that b looks the same colour as c, and yet that a does not look the same colour as c.27 Nonetheless, such remarks do remarkably little to remove one’s sense of unease. And things get worse if one reflects on the fact that, if only for the noise present in our perceptual systems, it would seem as if there simply must be some sufficiently small degree of change, along any dimension you like, that we are not capable of registering. That is not, of course, to say that our perceptual systems cannot respond differently to stimuli that differ only by such a tiny amount. But our perceptual systems will respond differently at different 25  I am not at all sure, but it seems important to put it as I have in the text, rather than as: it appears that there is no line between them. So what I mean is: it is not the case that it appears that there is a line between the two patches. 26  The counterexample was one created on my computer. It is not that hard to do. Just start with a patch that is near the limit of what looks red. 27  This observation is relevant to Delia Graff Fara’s discussion in ‘Phenomenal Continua and the Sorites’ (Fara 2001). Her treatment of the phenomenal Sorites depends crucially upon the claim that, though a and b might be phenomenally indistinguishable when viewed together, and b and c might be phenomenally indistinguishable when they are viewed together, these two statements cannot both be true together, whence there is no reason to deny that phenomenal indistinguishability is transitive (Fara 2001: 934). My difficulty with her argument is that she seems to be assuming that we cannot have three patches that synchronically illustrate the non-­transitivity of phenomenal indistinguishability. But we can, and it is easy to construct such an example on the computer.

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14  The Riddle of Vagueness times to precisely the same stimulus—if only, again, because of the effects of noise—so the mere possibility of differential response does not seem to help. Wright’s discussion of what he calls the Tachometer paradox (this volume, Chapter  4, 119–123) constitutes, it seems to me, a completely satisfying response to this way of motivating the phenomenal Sorites. The key thought is simply that, in appealing to the presence of noise, we are misconstruing what is involved in a system’s having a threshold of sensitivity. To say that a system has a certain threshold of sensitivity is, as I just said, not to say that it will never respond differently to stimuli differing only by a very small amount but only that it will not reliably respond differently to such stimuli; it will reliably respond differently only when the stimuli differ by more than the threshold. The name Tachometer paradox comes from the example Wright uses to make the point: a digital tachometer. If the tachometer reads in tens of RPMs, say, then one RPM is below its threshold of sensitivity: it does not report differences of just one revolution per minute. But a difference of one RPM can cause a change in its output; it will do so, we might imagine, right around 965 RPM or so. Wright himself does not spend much time with the Tachometer paradox. In the context of ‘Further Reflections’, its main role is to underwrite a reply to an objection of Peacocke’s. But the more I think about it, the more important the Tachometer paradox seems to me. It is exceedingly tempting to think that, if one judges that patch a is red, and if patch b is pairwise indistinguishable from a, then it would be irrational not also to judge that b is red. After all, the thought would be, one has exactly the same evidence that b is red as one has that a is red—namely, how the patches look. In fact, however, that is too strong: we have taken no account of background knowledge and the like; one might have reason to think something was funny about patch b, or that one’s perceptual systems were not functioning normally, or what have you. But is not one rationally committed at least to judging that b looks red? Well, I am not so sure. What the Tachometer paradox teaches, in the first instance, is that one can respond differently to the two patches—regard the one but not the other as looking red—even if they are indiscriminable in the present sense, and even if they differ to a degree below our visual threshold of sensitivity.28 Still, one might nonetheless want to insist that it cannot be rational to respond differently to indiscriminable patches. Maybe we sometimes do respond differently, in a purely descriptive sense, but it does not follow that 28  Wright suggests in ‘Intuitionism and the Sorites Paradox’ (this volume, Chapter 9, section V) that similar ideas can be used to respond to Williamson’s famous anti-­ luminosity arguments (Williamson 1996a).

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Introduction by Richard Kimberly Heck  15 we may respond differently, in a prescriptive sense. And is not the prescriptive question the one that really matters? Well, that is our next topic. As we will see, the prescriptive question is not the only one that matters.

Rules and Rationality on the Borderline I emphasized in Section  2 that Wright’s concern in the early papers is not, strictly speaking, to argue that our use of vague predicates is not governed by rules, but rather that the content of those rules cannot be regarded as wholly available to a priori philosophical reflection. Nonetheless, it is clear that Wright really does want to argue that our use of vague predicates is, in some sense, not governed by rules. And, while this suggestion is conspicuously absent from most of his twenty-­first-­century papers on vagueness, it surfaces again in ‘ “Wang’s Paradox” ’ (this volume, Chapter 11). As Wright himself recognizes, it has never been very clear exactly what he means to deny when he denies that our use of vague predicates is governed by rules. It does not help that he also wants to concede, and indeed to insist, that, in some platitudinous sense, our use of such predicates is governed by rules. It has to be, since the use of language, if it is to be use of language, must be ­subject to norm.29 Wright’s suggestion is then that what makes the Sorites paradox so intractable is the fact that we tend to assume that the use of language is governed by rule in a non-platitudinous sense. But what is ­ that sense? The suggestion in ‘ “Wang’s Paradox” ’ is that the culprit is something Wright calls the ‘modus ponens model of rule-­following’. On the modus ponens model, a rule for the use of a predicate takes the form of ‘a conditional statement whose antecedent formulates the initial conditions of the operation of the rule, and whose consequent then articulates the mandate, permission, or prohibition that the rule involves’ (this volume, Chapter 11, 319). So what we are looking for is a rule of the form: If something is X, then it should be classified as “red”. As Wright says, it is hard to see what X might be other than red. If so, however, then 29  This is more a point about cognition than one about language. See below.

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16  The Riddle of Vagueness the price of continuing adherence to the modus ponens model . . . is that we are forced to think of grasping the concept of what it is to be . . . red as underlying and informing competent practice with the predicate ‘red’ [and so as] independent of the linguistic practice. (this volume, Chapter 11, 320; emphasis in original)

Wright goes on to argue that it is difficult to see how a non-­linguistic creature could possess our concept of red; in particular, how such a creature could possess a concept that matched the vagueness of ours. I do not find this approach promising. The reason, in brief, is that I myself tend to regard it as axiomatic that vagueness has nothing particular to do with language.30 Wright himself often suggests that vagueness is an intrinsic feature of observational concepts. But one would suppose that, if non-­linguistic creatures can have any concepts at all, they can have observational concepts. Indeed, Wright almost says so himself, saying that he does ‘not deny that [a] chimpanzee might behave in ways which went some way toward giving sense to the idea that the concepts he was working with were vague . . .’ (this volume, Chapter 11, 322). But then, he denies that the chimp’s concepts might really be vague. No argument for this conclusion is offered, however, other than, as already mentioned, that the chimp’s concept could not be our concept, which is clearly insufficient even if correct. Moreover, so far as I can see, the modus ponens model simply does not imply that our grasp of the concept red must be independent of our use of the word ‘red’. It could be both that the rule governing our use of the word ‘red’ was ‘Apply the word “red” to all and only the red things’ and that the content of that concept was somehow bound up with our use of the word ‘red’. But this issue is very difficult, and discussing it would take us far from our current topic.31 None of this matters very much here, however, because little of what Wright has to say about whether our use of vague predicates is governed by rules actually depends upon the claim that those concepts are essentially linguistic. 30  From which it follows, by the way, that contextualist accounts are off-­base. And here, I really do mean contextualist accounts, ones that seek to explain the phenomenon of vagueness in terms of variation in what proposition is expressed by the utterance of a sentence containing a vague predicate. (Contextualism about vagueness is not just the view that ‘red’ and the like are context sensitive. That could be true in ways that have nothing to do with vagueness.) This sort of view has been defended by Scott Soames (1999); it is explicitly not Fara’s view (2000). Unfortunately, however, many who call themselves ‘contextualists’ are very unclear whether they are contextualists in this sense. In her reply to Jason Stanley’s criticisms (2003) of contextualism, for example, Diana Raffman (2005) emphatically denies that she is a contextualist in this sense, but then says several things that seem to commit her to that very view. See n. 34 for further remarks. 31 For some preliminary remarks, see the closing section of my paper ‘Reason and Language’ (Heck 2006).

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Introduction by Richard Kimberly Heck  17 What matters to Wright is that ascriptions of colour (to continue, for the moment, with this example) should in some important sense be immediate. That is supposed to be what relieves us of having to make the unappealing choice between semantic incompleteness, Dummettian nihilism, and some form of epistemicism. The trilemma arises if, but only if, it is a good question concerning the rules which govern our competence with ‘red’: what do they really have to say when our colour classifications fall into the complex patterns which show that we are in the borderline area?—what is their message, what would we do if we were to do exactly as they require and nothing else? That the three answers canvassed seem, among them, to exhaust all of the alternatives, yet each to be objectionable, might have made one suspect that there is something wrong with the question. Now we see how that suspicion might be substantiated. There is no requirement imposed by the rules—not if we understand such a requirement as something whose character may be belied by our practice in the borderline area and into which there is scope for independent inquiry. There is no such requirement because to suppose otherwise is implicitly to commit oneself to the modus ponens model of rule-­following for the classification in question, and hence . . . to open an—at least locally—unsustainable gap between conceptual competence and the linguistic capacities that manifest it. (this volume, Chapter 11, 323–324; emphasis in original)

The word ‘linguistic’, as it occurs in the last sentence quoted, is a gratuitous addition.32 What is doing the work here is the claim that our ascriptions of colour should be regarded as having a certain sort of primacy, and it does not matter, so far as that point is concerned, whether those ascriptions are linguistic or purely mental. Indeed, rejection of the modus ponens model seems all the more necessary if we focus upon ascription of the concept red rather than application of the adjective ‘red’. In the former case, a rule that accorded with the modus ponens model would have to take the form: If something is X, then it should be classified as red.

32  The capacities might just as well be purely cognitive ones, of the sort to which a fan of conceptual role semantics might appeal.

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18  The Riddle of Vagueness And now it is really hard to see what X might be. The only remotely plausible proposal would be: If something looks red, then it should be classified as red. But there are well-­known difficulties here. For one thing, there is the familiar problem of background knowledge: not everything that looks red should be classified as red. The real issue, however, is deeper. It is not just that the concept looks red is arguably posterior to the concept red, in which case it seems wrong to suppose that our grasp of red might somehow depend upon our grasp of looks red. The fundamental problem is that judgements of colour are not based upon judgements of appearance. What is true, rather, is that one’s judgements of colour are based upon how things appear to one—that is, upon appearances themselves—and that is quite a different matter.33 But, however that may be, it should be clear that we cannot easily rescue the modus ponens model by denying that vague concepts are essentially available only to languaged creatures. To see what is really bothering Wright, I suggest we turn to ‘Further Reflections’. There, Wright is concerned with something he calls the ‘proviso biconditional’: Necessarily: if the subject and the circumstances are as required by [various sorts of] provisos, then ‘ x looks red’ is true if and only if the subject assents to ‘ x looks red’.  (This volume, Chapter 4, 158)

The provisos address the problem of background knowledge and require, familiarly, that the subject understand ‘looks red’, that his visual system not be malfunctioning, and the like. Wright observes that there are two ways of ­taking this biconditional. On the first, whether a thing looks red is regarded as constitutively independent of how we respond to it (how we are inclined to classify it), and the proviso biconditional says that, at least in appropriate circumstances, we can infallibly track this independent state of affairs. On the second, by contrast, ‘for x to look red just is for subjects to be willing to assent to that judgement when the provisos are met’ (this volume, Chapter 4, 159).

33 I emphasize this point in my paper ‘Non-­Conceptual Content and the “Space of Reasons” ’ (Heck 2000), though the point is not mine alone. In some ways, the idea goes back to Peacocke (1983). We will return to this sort of issue below.

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Introduction by Richard Kimberly Heck  19 Here again, the point that really matters to Wright is that our responses should have a certain sort of primacy. Someone who succumbs to the arresting, apparently simple thought that ‘looks red’ must be applicable to both, if to either, of any pair of indiscriminable items, is likely to be taking it one of two ways. Taken one way, it is the central move in the Tachometer paradox: how can a signalling device, even a properly functioning signalling device, discriminate among stimuli whose difference is smaller than its sensitivity threshold? We now know, I hope, how to respond to that question. But the second and, as I think, more nat­ ural way of taking the thought conceals, in effect, a presupposition of the first perspective on the biconditional. You have to forget that we do not, or would not, so apply the predicate [‘looks red’] in every case and fall in, instead, with the idea that something can be discerned about its proper conditions of application just34 by intuitive reflection on the kind of content which it overtly seems to have. (This volume, Chapter 4, 160; emphasis in original)

Before I address the ‘second way’ of taking the soritical thought, I want to emphasize again how important it is that Wright has shown us how to respond to the first. It is very easy indeed to talk oneself into the view that there just has to be some degree of change so small that it cannot change whether a thing looks red. And if, indeed, that is true, then it entails something about the ‘proper conditions of application’—namely, that one ought to keep judging that the thing looks red, since that is, indeed, how it looks. As Wright says, however, we do indeed know how to respond to that suggestion. Our present concern, though, is to understand Wright’s denial that our use of vague predicates is governed by rule. What Wright is suggesting we must abandon is not so much the idea that there are rules governing our use of ‘looks red’, but rather the presumption that those rules must have priority over our actual responses, so that it might turn out the rules have something to say about borderline cases even if we do not. And here again, while everything is put in broadly linguistic terms—in terms of sentences and our assent to them—there is nothing fundamentally linguistic about the point Wright is making. The denial that ascriptions of colour are governed by rules amounts, then, on this interpretation, to the thesis that such ascriptions have a certain sort of 34  This is further evidence for the claim that the governing view is constitutively committed to the possibility of a priori conceptual analysis.

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20  The Riddle of Vagueness primacy in the determination of the extension of our colour concepts. We might put it this way: the only norms that govern our use of colour concepts are internal to our use of them, where the term ‘internal’ is meant to ring Putnamian bells. I see no reason to suppose that there is any ‘natural’ property that our concept red picks out: the region of colour space that falls within the extension of red is not, in any objective sense, privileged, and it is not privileged in any subjective sense, either (as it would be if, say, our perceptual systems were critically attuned to a particular range of colours). If that is not already obvious, reflect upon the fact that ‘red’—and now I am talking about the adjective—is, prima facie, context sensitive,35 so that, in fact, there is no single concept red that the word ‘red’ always expresses. But, if there is no nat­ ural property that our (uses of) ‘red’ pick out, then there is nothing to determine the extension of our concept red other than the ascriptions we make. It is therefore unsurprising that we might be left without any binding instruction in the borderline region. So, if what is ultimately at stake in Wright’s discussions of the rules underlying our use of vague predicates is a point about the primacy of our ascriptions, then it is not really a point about the presence or absence of rules, after all. It is not even really a point about the ‘immediacy’ of our judgements. And this, I suggest, is all to the good, because ascriptions of colour—let alone judgements of appearance—are not, in these respects, obviously representative of the class of vague predicates. Consider, for example, ‘bald’. It may not be obvious what should go in for X in: If something is X, then it should be classified as bald. but the kind of reason we had, in the case of red, to suppose that nothing informative could be said is just lacking here. Even in the case of red, one might suggest that there is a rule of sorts to  be  had.36 Surely when Wright says that ‘[t]here is no scope for an . . .  35  I say ‘prima facie’ only because some philosophers are inclined to deny that any but the most obvious expressions—‘I’, ‘you’, ‘this’, and the like—are context-­sensitive. See, e.g., Cappelen and Lepore (2005). For present purposes, it suffices to note that the case for the context-­sensitivity of ‘red’ is every bit as good as the case for the context-­sensitivity of many other expressions, e.g., quantifiers. And even Cappelen and Lepore would hold that the propositions that are actually communicated by utterances of sentences containing ‘red’ do not uniformly concern ‘the’ concept red but will concern different concepts on different occasions. Let me emphasize again that claiming that ‘red’ is context-­sensitive is not, of course, tantamount to endorsing contextualism about vagueness. Contextualism about vagueness is not the view that vague expressions are context-­dependent but that the characteristic mani­fest­ ations of vagueness are to be explained in terms of context-­dependence. Fara (2008) is admirably clear on this point. 36  There are a few more remarks along these lines in my paper ‘Semantic Accounts of Vagueness’ (Heck 2003: §2).

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Introduction by Richard Kimberly Heck  21 information-­theoretic standpoint whose goal is to outline rules which we implicitly follow in our use of vague expressions . . .’ ( this volume, Chapter 11, 324–325), he does not mean to pre-­empt ongoing empirical investigation into how we make the colour ascriptions we do. Such an ­empirical investigation might well reveal a rule characterizing our ascriptions of colour. One might respond that such a rule would not be one we follow but rather one with which our judgements merely accord. But even that is not clear. It is, of course, an option to say that our colour judgements (and other perceptual judgements) are not made on any rational basis but are merely caused by our perceptual experience. But one might have reason to be dissatisfied with that view.37 Colour judgements, as I said earlier, are based upon how things appear to us, and how things appear is thus-­and-­so, to borrow a phrase. That is to say, in speaking of how things appear, I mean to be speaking of how things are represented in perception, which means that I am speaking of states with content. If so, then the rule that governs our ascriptions of colour will have to make reference to the contents of the perceptual states that ground those ascriptions. And if, as I myself hold, the contents of perceptual states are non-­conceptual, then any such rule will resist verbal formulation and will be un­avail­able to a priori reflection, simply because verbal formulations are necessarily conceptual and because a priori reflection is itself a conceptual exercise.38 The point about immediacy does matter, however, in one case, which just happens to be the really fundamental case: that of phenomenal predicates, like ‘looks red’. As I mentioned at the end of Section 3, Wright’s analysis of the Tachometer paradox frees us from the question how we even can respond differently to indiscriminable patches: how we even can judge that the one looks red but not judge that the other does.39 But it leaves us with the question how it can be rational to do so. And here, I suggest, is where the appeal to the immediacy of such judgements does its work. The thought behind the 37  It is unclear to me exactly how Wright feels about this broadly Davidsonian view. His insistence that ‘not everything judged rationally is judged for reasons’ (this volume, Chapter 11, 326) suggests some sympathy with it. And, in his review of John McDowell’s Mind and World, he expresses some doubt about McDowell’s argument against it (Wright 1996). But still, I do not know where he ul­tim­ ate­ly stands on the matter, and one would hope that his treatment of vagueness did not depend upon the resolution of such contentious issues in epistemology. 38  Of course, this assumes that belief and the like are conceptual, which might be, and has been, denied (Stalnaker 1998). 39  As so often, the case of agnosticism is the one that matters. We need not suppose that the subject will judge that the one looks red and judge that the other does not look red. It is enough that she responds differently: that she hesitates or refuses to commit herself. This, unfortunately, causes problems for my proposal, in ‘Semantic Accounts of Vagueness’, that a principle I call ‘quasi-­tolerance’ might suffice both to explain the plausibility of the Sorites and to disarm it (Heck  2003: 117). The puzzle is to see how any difference in response can be rational.

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22  The Riddle of Vagueness question is, presumably, something like this. Suppose we take an unfortunate subject Lily on a tour of our colour patches. Eventually, Lily will reach a point where, having said that one patch looks red, she will nonetheless decline to say of the other that it does. So now we ask her: how can you not say that this one too looks red? What difference between the patches can you identify that might justify your different responses? Of course, there is none. But she could, I think, say something like this:40 I know that, in a sense, the two patches look the same. I put them together, and I can see no line between them. But nonetheless, when you ask me whether they look red, I find myself wanting to say that this one does and yet find myself hesitating about that one, even though I am happy to grant that I cannot point to any other difference between them. That is just how they seem to me.

And, if Lily is philosophically inclined, she will perhaps add that she has reached bedrock, that her spade is turned (Wittgenstein 1968: §217). Is that irrational? It seems unlikely, to be sure, that this situation is terribly stable. If our subject stares a while longer at the patches, maybe she will change her mind,41 and if we repeat the trial, we should not expect her to fall off the wagon at the same point every time. But is that enough to make her response irrational? Does it even make her response less rational than it ought to be? That follows, or even can follow, only if there is some standard, prior to and in some sense independent of our considered judgements in such cases, to which those judgements can be held responsible. And what Wright is ur­ging is precisely that there is no such standard: The reply should be that competence with such predicates is nothing to do with the capacity to fit one’s usage to the dictates of rules . . . What it is correct to say using such a predicate is a function only of what we are actually inclined to say when there is no reason to doubt that the provisos are met [i.e., that our perceptual systems are functioning normally, etc.]. (This volume, Chapter 4, 160)

40  Whether this is in fact what people say is of no particular interest at the moment, which is not to say that it is of no interest what people in fact say in such situations. 41  Can one even be sure that the patches, or one’s perception of them, are not subtly changing in ways one cannot visually discriminate?

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Introduction by Richard Kimberly Heck  23 So the point about the priority of our responses, as I am reading it, has both a meta-­semantic aspect and an epistemological one: it is both a point about what fixes the extensions of vague predicates and a point about what it is for judgements involving vague concepts to be rational. As I have said, I think this latter point probably applies less broadly than Wright implies—I have in mind the case of ‘bald’ again—but that is much less important than whether it applies here: to the case of phenomenal predicates. If it does apply here, then I think Wright has a compelling resolution of the phenomenal Sorites. And that can arguably be parlayed into a compelling resolution of the next most intractable form of the Sorites—namely, that concerning plausibly observational concepts, like ‘red’. What makes the Sorites with ‘red’ so powerful? I suggest that it is the combination of two thoughts: that a thing’s being red is a matter of how it looks and that phenomenally indistinguishable patches look relevantly the same; in particular, if this patch looks red, so does that one. But, if Wright has shown us how to avoid a commitment to the soritical principle for looks red, then he has ipso facto shown us how to avoid commitment to the soritical principle for red, as well. If we do not even have to say that indiscriminable patches must both look red if either does, why should we have to say that they must both be red, if either is? There are issues here, to be sure. One problem is that it is hard to know how Wright’s point might be affected by questions about the epistemology and aetiology of perceptual judgements. If, as I have suggested above, perceptual judgements are rationally based upon perceptual states, states that have non-­ conceptual content, then one would suppose that phenomenal judgements too will be rationally based upon those same states, and in that case they will not, in the most obvious sense, be immediate. That said, I am not sure the sort of mediation we would then have is what matters here. I have long had a dim sense that the really important part of Wright’s point concerns not so much the limits of rationality but the limits of conceptuality.42 But let me not try to pursue that issue here. 42  ‘Language-­Mastery and the Sorites Paradox’ was Wright’s contribution to the now classic volume Truth and Meaning: Essays in Semantics (Evans and McDowell 1976). It is an interesting exercise to read it together with John McDowell’s contribution to that same volume, ‘Truth Conditions, Bivalence, and Verificationism’ (McDowell 1976). Wright’s language—particularly, his contrast between the governing view, which ‘approach[es] our use of language from within’ (this volume, Chapter 2, 81) and his own, ‘more purely behaviouristic’ (this volume, Chapter 2, 101) alternative—is strikingly similar to language McDowell uses in criticizing what he sees as Dummett’s overly behaviouristic approach to the theory of meaning. In particular, McDowell wants to argue that our use of language is all but unintelligible unless we approach it from within, whereas Wright wants to argue that, if we approach our use of language from within, we can only regard it as incoherent. I do not want to suggest that Wright’s real target is McDowell. But it might as well have been, since McDowell wholeheartedly embraces precisely those elements of Dummett’s view that Wright finds troubling. Dummett is thus

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24  The Riddle of Vagueness A second problem is that Wright’s resolution of the phenomenal Sorites seems to depend crucially upon the thought that, in the borderline region, we might just be unsure whether a particular patch looks red. Now, granted, it is one thing to suggest that our subject might coherently judge one patch to look red and simultaneously judge that the next one does not look red, which seems insane, and entirely another to suggest that she might just not judge the next one to look red.43 But can one really be unsure whether a thing looks red? There is at least a long tradition of supposing that whether a thing looks red is, in Williamson’s sense, ‘luminous’ (Williamson 2000a: ch. 4): if a thing looks red, then a competent judge will be in a position to know that it looks red, in which case, contraposing, we should conclude—and, indeed, our subject herself will be able to conclude, if she has done enough philosophy—that, if she cannot be sure whether the patch looks red, then it does not look red, after all. So Wright’s treatment of the phenomenal Sorites would seem to involve rejecting the luminosity of ‘looks red’. That, in itself, is not a problem, just a consequence. But it is in some tension with Wright’s frequent endorsement in such contexts of what he calls ‘Evidential Constraint’ (EC): the principle, as applied to a predicate F, that, if a thing is F, then it is feasibly knowable that it is F.44 EC for ‘looks red’ is, to be sure, logically weaker than luminosity, but, if it is not currently knowable by our subject whether a patch looks red to her, it is hard to see how it could be known by anyone else whether that patch looks red to her, either. We will return to this issue.

The Intuitionistic Approach to Vagueness The Sorites paradox is unlike other prominent paradoxes. In the case of the truth-­ theoretic paradoxes, for example, we are not even sure what is caught in the middle, with Wright and McDowell pulling him in opposite directions. And he stays firmly there, moving away from the behaviouristic view in his later writings, beginning with ‘What Do I Know when I Know a Language?’ (Dummett 1996), but never adopting the ‘modest’ view for which McDowell (1998) had argued (Dummett 1987). 43  Compare the notion of ‘quasi-­tolerance’ I discuss elsewhere (Heck 2003: 117). 44 Evidential Constraint was introduced (and figures prominently) in Truth and Objectivity (Wright 1992b). It is first discussed in connection with vagueness in ‘On Being in a Quandary’. The seeming tension between Evidential Constraint and Wright’s views on vagueness is discussed by Patrick Greenough (2009). For what it is worth, I am not so sure about the ‘feasibly’ part. If an analogy with intuitionism is intended, then one would have thought ‘feasibly’ inappropriate. The principle should rather take the form: if a thing is F, then it can in principle be known to be F. This difference is, of course, precisely what is at issue in ‘Wang’s Paradox’ (Dummett 1978: 248–68) and also in Wright’s early papers on vagueness, not to mention ‘Strict Finitism’ (Wright 1993: 107–75). So I am sure Wright must have some reason to formulate the principle as he does. I just do not know what it might be.

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Introduction by Richard Kimberly Heck  25 re­spon­sible for the paradox. Is it the Tarski biconditional for the liar sentence? Is it the reasoning from the biconditional to a contradiction? If so, which bit of that reasoning? In the Sorites, by contrast, almost everyone agrees that the culprit is the Sorites premise: ∀x(Fx → Fx′) .45 But it is not widely agreed that the Sorites straightforwardly shows that the Sorites premise is false, whereas the set-­theoretic paradoxes are almost universally taken simply to show that Basic Law V, naive abstraction, or what have you is false,46 the question then being what we should say instead. Indeed, in the case of the Sorites, there is even a prior question—namely, how it is even compatible with the vagueness of F that the Sorites premise should not be true. As Timothy Chambers (1998) pointed out, the difficulty is all the more pressing if we take the crucial premise to be ¬∃x(Fx ∧ ¬Fx′) , since its truth can easily seem, as Wright himself had more or less said in ‘Further Reflections’, simply to constitute the vagueness of F. If so, then the very vagueness of F seems to enforce the no sharp boundaries premise, and so the first question the Sorites poses is how we can rationally refuse to accept that premise. It is this question that we have been discussing to this point. What is striking is how little our discussion so far bears upon the other question I just identified: whether the Sorites premise is false.47 And it bears even less on the related question how we should understand the semantics of vague predicates—for example, what we should say about the semantic values of sentences concerning borderline cases. Those problems, it seems to me, remain every bit as intractable as they have always been, despite the fact that Wright has given us, or so I have just suggested, a solution to the Sorites paradox, in the sense that he has explained why we do not have to accept the Sorites premise. What I want to do in this section, then, is to say a few words about Wright’s approach to questions about the logic and semantics of vagueness, an approach that is developed in a series of papers beginning with ‘On Being in a Quandary’ (this volume, Chapter 7). I will have rather less to say about these papers than about the earlier ones, not because I find them less interesting but, rather, because I have not been thinking about them for twenty-­plus years and still find myself with more questions than answers. So I will try to 45  This being philosophy, there are of course exceptions. Elia Zardini (2008) has suggested that we should reject the transitivity of implication. And some people, like Dominic Hyde (1997), are happy to accept the contradiction and find nothing wrong. 46  Here again, there are of course exceptions, including Alan Weir (1998) and Graham Priest (2006). 47  Wright himself gives these problems little attention in the papers from the 1970s and 1980s, being much more concerned with the first problem. Which, of course, is why we have been discussing it.

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26  The Riddle of Vagueness provide a guide to the terrain and highlight some of the issues that most puzzle me. Let us begin with the status of the Sorites premise, and let us concentrate upon the no sharp boundaries form: ¬∃x(Fx ∧ ¬Fx′) , which we shall henceforth call NSB. Using only logical laws that ought not to be denied,48 we can derive a contradiction from NSB and premises that only a nihilist would deny, so NSB cannot be true (unless some contradictions are true, which none is). There are then two options: we can regard NSB as false, or we can regard it as indeterminate. The latter option is, in principle, available, and it would be embraced by degree-­theorists, for example. But Wright regards the required abandonment of reductio ad absurdum as too high a cost to pay and so proposes to regard NSB as false.49 This is of a piece with Wright’s rejection of what he calls ‘Third Possibility’, of which more below. The difficulty with this view is that it seems to commit us to regarding NSB’s negation, ¬¬∃x(Fx ∧ ¬Fx′) , as true, which classically commits us to regarding ∃x(Fx ∧ ¬Fx′) as true, and that seems simply to say that F has a sharp boundary in the series in question, which therefore seems to commit us to epistemicism (or to something no less objectionable). One way out is supervaluationism, which regards ∃x(Fx ∧ ¬Fx′) as true but as not having a true instance. But Wright has objections to supervaluationism that he takes to be conclusive.50 His alternative proposal is that we should reject the inference from ¬¬∃x(Fx ∧ ¬Fx′) to ∃x(Fx ∧ ¬Fx′) , and his model, of course, is in­tu­ ition­ism in the philosophy of mathematics. Motivating this view is one of the central tasks of ‘On Being in a Quandary’, and it is taken up again in ‘Vagueness: A Fifth Column Approach’ (this volume, Chapter 9). The basic line of argument is this. Suppose ∃x(Fx ∧ ¬Fx′) holds, and that it does not hold in the way the supervaluationist thinks it holds. Then there is some true instance Fa ∧ ¬Fa′. Given the structure of our Sorites series, then, everything preceding a is F and everything following a′ is not-F, whence every­ thing in our series is either F or not-F. Which is to say that Bivalence holds, at least as far as this particular series is concerned. So reason to doubt Bivalence is reason to doubt that ∃x(Fx ∧ ¬Fx′) holds, even in the presence of reason to endorse ¬¬∃x(Fx ∧ ¬Fx′) .

48  These include the transitivity of implication, which Zardini (2008) proposes to deny, and the inference ¬( p ∧ q), p  ¬q , which Kit Fine (2017) has explored denying. 49  This issue is discussed by Stephen Schiffer (2020). 50  These are especially prominent in ‘Vagueness: A Fifth Column Approach’. As I have discussed them elsewhere (Heck 2003: 121–7), I will not repeat that discussion here. Suffice it to say that I do not find Wright’s argument conclusive.

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Introduction by Richard Kimberly Heck  27 One might want to do a modus tollens where Wright does a modus ponens and say instead that we have ample reason to endorse Bivalence and so ought simply to accept ∃x(Fx ∧ ¬Fx′) . It is a testament to Williamson’s influence how many people would be tempted by precisely such a response. But it is important to remember here that philosophers have, for a very long time, been inclined to reject Bivalence where the statements in question are vague, and there is, so far as I can see, very little mileage in Williamson’s frequent insistence that, since Bivalence has served us so well in the past, we would be fools to abandon it now (Williamson 1992b: 162). That looks like nothing so much as a fallacy of hasty generalization. Bivalence has indeed served us well. But perhaps that is because the development of modern logic proceeded with almost complete disregard for vagueness for almost the whole of its first century (meaning: 1879–1975). Indeed, as Wright emphasizes, a commitment to Bivalence is a commitment to something’s determining a precise boundary for, say, ‘tall’ on every occasion of its use. If so, however, Wright has a question to ask. [C]an anyone, even the most rampant epistemicist, put her hand on her heart and say 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.51 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.  (This volume, Chapter 9, 291)

My guess is that committed epistemicists will indeed take the oath, but whether they are right to do so is another matter. So suppose we reject Bivalence for vague statements, as philosophers have been doing since Frege. Typically, such philosophers have wanted to say that some vague statements—in particular, statements concerning borderline cases—are neither true nor false. As said earlier, however, Wright now thinks that this view is incoherent, since, given the Tarski biconditionals, it leads to contradiction. For what it is worth, I do not find this line of thought conclusive, since I do not regard the Tarski biconditionals—or, perhaps more to the point, the intersubstitutivity of A and A—as non-­negotiable.52 But 51 This is a point Williamson makes in several places. Wright cites Williamson’s reply to ‘The Epistemic Conception of Vagueness’ (Williamson 1996b: 36). 52  My reasons are not unlike Dummett’s (1978: 4–5). Anyone who is prepared to acknowledge truth-­ value gaps must be prepared to recognize predicates that validate the so-­ called T-­rules— A  T ( A) , and the like—but that will not validate the Tarksi biconditionals. So it looks as

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28  The Riddle of Vagueness there is a more interesting line of thought, anyway—namely, that endorsement of what Wright calls Third Possibility—the view that borderline statements ‘have some kind of third status, inconsistent with each of the poles’ (this volume, Chapter 9, 278)—is inconsistent with the phenomenology of borderline cases.53 The idea is very simple. When one is confronted by a borderline case, Wright insists, one does not have the sense that one ought not to make any judgement. Rather, one feels pulled this way and that, ‘It is and it isn’t’ being as common a reaction as ‘It’s not really either’. And if someone else says, ‘Well, I think it’s red’, one does not always feel as if she has no right to such an opinion. But if Third Possibility held, and if we knew that the case was borderline, then we would know that it was not true that the thing in question was red, and the opinion in question would be improper. Of course, it might be excused by ignorance, but that is a different matter. More generally, Wright wants to suggest that the characterization of ­borderline cases ought, fundamentally, to be epistemic. As he puts it in ‘Vagueness: A Fifth Column Approach’: The central manifestation of borderline cases is not a convergence on such unwillingness [to make a judgement], but . . . in weakness of confidence in such verdicts as are offered, in their instability, and in the unwillingness of some to endorse any verdict. (This volume, Chapter 9, 279)

So Wright is insisting here on a thesis he calls ‘Permissibility’: [W]ith the kind of vague concepts with which we are concerned, a verdict about a borderline case is always permissible; it is 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 . . . were . . . 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. (This volume, Chapter 9, 280–281; emphasis in original)

if an insistence on the biconditional begs the question, and the same goes for intersubstitutivity. There is a nice discussion of this issue by Manuel García-­Carpintero (2007). 53  A different sort of argument for the same conclusion, but one with much wider applicability, is presented by Michael Glanzberg in his paper ‘Against Truth-­Value Gaps’ (Glanzberg 2003).

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Introduction by Richard Kimberly Heck  29 It is important to understand that this description is meant, in the first instance, to be interpersonal—though Wright also wants to insist that similar remarks could also be made about ‘the judgements of a single competent subject about a single case made on a number of separate occasions’ (this volume, Chapter  9, 280). Thus, Wright says a bit later that regarding something as a  borderline case ‘is, first and foremost, a failure to come to a view’, which ‘is in general quite consistent with there being a true view; and with someone who holds it doing so knowledgably’ (this volume, Chapter  9, 281). That someone will, of course, have either to be someone else or to be oneself at a different time. But this is not all there is to Wright’s conception of borderline cases. The real core of his conception is the thought, first voiced in ‘On Being in a Quandary’ and explored further in ‘On the Characterization of Borderline Cases’ (this volume, Chapter  13), that borderline cases are a type of ‘quandary’, that being a term of art. One is, in Wright’s proprietary sense, in a quandary with respect to some proposition p if one is, as we might put it, utterly ignorant as regards p: one does not know whether p; one does not know if there is any way to decide whether p; and one does not even know if it is metaphysically possible for someone to know whether p.54 The fundamental question is how knowledge concerning borderline cases might be possible, even though we are now in a quandary with respect to them.55 There are powerful intuitions that such knowledge just is not to be had. Williamson, for example, takes it to be a requirement on his own view that it should explain why we cannot know where the boundary lies, even though there is one. Indeed, the claim that we cannot know where the boundary lies is the central premise in Sven Rosenkranz’s ‘Wright on Vagueness and Agnosticism’ (2003), and assuring the consistency of his view with that claim is the primary goal of Wright’s reply, ‘Rosenkranz on Quandary, Vagueness, and Intuitionism’ (this volume, Chapter  8).56 That there is no immediate conflict is due to the fact that it might be possible for us to know that Fa and possible for us to know that ¬Fa′ without its being possible for us to know that Fa ∧ ¬Fa′. Wright needs the former to be true, not the latter, and it might even be that knowing that Fa would preclude

54  As Patrick Greenough (2009) emphasizes, this should really be put in terms of what one is in a position to know, but I will ignore this subtlety here. 55  There is some discussion of this range of issues in ‘Intuitionism and the Sorites Paradox’ (this volume, Chapter 14). Wright now seems to regard this kind of argument for an intuitionistic approach to vagueness as, at least, having some real problems. 56  Rosenkranz (2008) has since continued the discussion.

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30  The Riddle of Vagueness our  knowing that ¬Fa′, perhaps even for Williamson’s reasons. Still, one might want to know how exactly we are to conceive of the possibility of our knowing of some borderline patch that it is, as it happens, red. The sense I often get from Wright’s discussions is that he means to be allowing that, even if I am in a quandary regarding whether our borderline patch Jack is red, someone else might still know that he is, or at least have a warranted belief that he is. But this is not easy to swallow. It helps a bit to realize how weak Wright’s claim here is: it is not that I must positively regard knowledge that Jack is red as possible; it is only that I must not regard such knowledge as definitely unavailable. But it is still very hard to see how anyone might know whether Jack is red. Here I am, looking at Jack in a strong white light, with well-­functioning eyes, the excellence of my epistemic position being well known to me. If I am in no position to know whether Jack is red, how could anyone else be? How could their epistemic position be any better than mine?57 I do not myself know of any good answer to this question. But there is a different way of understanding Wright’s position. It is ­consistent to regard it as presently impossible for anyone to know whether Jack is red while also leaving it open that there might yet be advances in our understanding that would make the question decidable. Perhaps a comparison will help. Consider Goldbach’s Conjecture: every even number greater than two is the sum of two primes. At present, we are in a quandary with respect to it: we do not know whether it is true; we do not know whether it is (even meta­phys­ic­al­ly) possible for its truth to be decided by us.58 But there are two quite different ways in which its truth might yet be decided. One would be for the Conjecture to be proven or refuted by means of some currently accepted form of mathematical argument, say, by means of an argument formalizable in ZFC. Another would be for there to be some advance in our understanding of mathematics that led us to acknowledge new axioms, or new forms of argument, maybe even forms of argument we cannot now understand, and these would then suffice to decide the Conjecture. Perhaps, for example, some genius of the future will discover new sorts of set-­theoretic principles very different in kind from the ‘large cardinal’ axioms that are now the focus of so much research. And, indeed, a willingness to countenance such a development as a possibility is very much of a piece with intuitionistic thinking. When 57  This is what Greenough (2009: 4) calls the ‘Unknowability Problem’. It is, as he notes, similar in spirit to the problems raised by Rosenkranz. 58  The thought is that it might just happen that every even number is the sum of two primes, but each one (so to speak) for a different reason, so that there would be no finitely statable proof of that fact.

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Introduction by Richard Kimberly Heck  31 intuitionists speak of ‘proof ’ and ‘provability’, they do not mean proof in some specified formal system but prove in an informal (indeed, intuitive) sense.59 Could something like that be true of colour? Can we imagine making ­discoveries about colour that would put us in a position to decide whether Jack is red? For Wright’s purposes, nothing so strong is required. All Wright needs is that we cannot now foreclose the possibility of our making such discoveries. And, given how weak that claim is, it is hard to see how one can really quarrel with it. Note, however, that going this way involves abandoning the thesis of Permissibility, at least in one of the ways Wright sometimes seems to interpret it—namely, as allowing that others’ judgements about a borderline case might now be knowledgeable. But that strong construal of Permissibility is itself in some tension with another way Wright often speaks—for example, ‘it is always all right to have a (suitably qualified) ­opinion’ (this volume, Chapter 9, 280; emphasis removed). Suitably qualified opinions can, of course, be warranted or not, and so one might want to read this as saying that, for all one knows, a suitably qualified opinion regarding a borderline case can be appropriately held. But that is far weaker than allowing that a ‘verdict’ on such a case might now be ­knowledgeable. And it seems to account just as well for the phenomenology of borderline cases, as Wright describes it. But there are, of course, still problems. One concerns the objectivity of ­colour judgements.60 Suppose, as suggested, that some ‘improvement’ in our understanding of colour will one day put us in a position to decide whether Jack is red. Say we will decide that Jack is, in fact, red. The question is whether this is the only possible ‘improvement’. Is it now possible for us to make some other ‘improvement’ in our understanding of colour that would lead us to decide that Jack is not-­red? If so, then it does not seem as if there could be anything now that makes it the case that Jack is red. If coming to know that Jack was red was like proving something in PA, that would be one thing: there is a clear sense in which such a proof is already provided for and, similarly, in which a proof of the negation is already foreclosed. But we are supposing that coming to know that Jack is red is not like that. And now the analogy with mathematics does not help, for the same question arises in that case. Sure, we

59 That, it is often said, is why Brouwer found Gödel’s incompleteness theorem unsurprising: Brouwer seems to have regarded it as almost obvious that no formal system can encompass everything we are prepared to count as a proof. (Anyone who has studied Brouwer’s theory of the creative subject will certainly be prepared to agree that no formal system is likely to encompass everything he was prepared to count as a proof.) 60  I owe this problem to David Christensen.

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32  The Riddle of Vagueness might come to accept new axioms sufficient to decide Goldbach’s Conjecture. But must we suppose that those axioms were implicit in our practice all along? Might we instead have come to accept new axioms sufficient to decide it the other way?61 The analogies with relativism explored in ‘On Being in a Quandary’ suggest that Wright himself might not be terribly unhappy with the thought that this sort of objectivity is a casualty of the Sorites. But, if so, then there is still a tension in his position, since it would seem as if, right now, borderline cases do indeed occupy a middle ground between truth and falsity.62 A second problem concerns the case on which we were focused above: phenomenal predicates. It is one thing to imagine some improvement in our understanding of colour that would allow us to decide whether Jack is red. But what kind of epistemic miracle would it take to put us in a position to decide whether Jack looks red—to me, right now? How could anyone possibly be in a better position to answer that question than I presently am?63 Maybe more and better information about redness itself would help, but even that is hard to see. So maybe, at least in this case, we should reconsider whether it is really so bad to deny that knowledgeable resolutions of such cases are, even in principle, possible. We should reconsider, that is, whether we might accept what Wright calls ‘Verdict Exclusion’, the principle that knowledgeable verdicts about borderline cases are not, in general, possible—and so trade the position he calls ‘Agnosticism’ for the one he calls ‘Pessimism’ (this volume, Chapter 9, 279, 282). Wright’s only argument against Pessimism is that it is incompatible with Evidential Constraint, which he regards as plausible (this volume, Chapter 9, 282–283). And, in the case of ‘looks red’, maybe it is. In that case, EC tells us that, if Jack looks red, then it is knowable that Jack looks red, and that, if he does not, then it is knowable that he does not. But Verdict Exclusion would tell us that it is not in principle knowable whether Jack is red. So Jack is not-­ red, and he is not not-­red, either. That is a contradiction, and so something has to go. Wright jettisons Verdict Exclusion. But EC, in the form in which Wright uses it in this 61  Actually, in this case, no. If the Conjecture is false, its falsity can already be proven in Robinson arithmetic. This is because the negation of the Conjecture is (equivalent to) a Σ1 sentence, and every true Σ1 sentence is provable in Robinson arithmetic. But there are other examples (e.g., the twin prime conjecture). 62  Of course, as the analogy with Goldbach’s Conjecture shows again, it would be a middle ground of a sort already familiar from intuitionism, so perhaps Wright would be happy with that, too. 63  Would the oft-­used analogy with wine tasting help here? The refinement of taste that comes with learning how to taste wine can help one to realize, say, that something tastes briny that one did not previously realize tasted briny. But then it is not obvious that one’s sense of taste itself is left unchanged by the refinements.

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Introduction by Richard Kimberly Heck  33 argument, seems to me too strong, anyway. What we want here is the idea that the truth of claims of a certain kind requires that it be possible to know those claims. But, even if one accepts such a thesis, one might still think it odd methodology to investigate whether, say, Fred is bald by investigating whether it is possible to know whether he is bald. The point applies equally in the case of intuitionistic mathematics. We want to allow that, if it is in fact true that every even number is the sum of two primes, then it can be proven that every even number is the sum of two primes. But we might yet want to deny that one could find out that every even number is not the sum of two primes by doing epistemology.64 And, indeed, despite the fact that the informal gloss often given on intuitionistic negation would permit such an inference, the more precise account one usually sees identifies ¬A with A →⊥. So ¬A is true if there is a very particular sort of obstacle to a proof of A—namely, the existence of a procedure that would transform a proof of A into a proof of a contra­dic­tion. The empirical case is, familiarly, harder, because it is hard to know how an anti-­realist should explain negation, but much the same point ought to apply: it is one thing to say that, if Fred is bald, we can in principle know that he is, and another to allow that one could find out that Fred is not bald (not by examining Fred but) by doing epistemology.65 We might just jettison EC.66 As mentioned at the end of the previous section, EC for phenomenal predicates appears to be in some tension with Wright’s resolution of the phenomenal Sorites. And, indeed, the sort of issue we have been discussing poses serious problems for EC. Suppose that there are borderline cases of ‘looks red’. These, according to Wright’s account of what borderline cases are, will be cases of quandary: cases in which we do not know if the thing looks red (or not), or even whether it can be known to look red (or not). But that appears to be flatly inconsistent with EC: anything that cannot feasibly be known to look red does not look red, and so on and so forth.67 In his most recent work on this problem, ‘Intuitionism and the Sorites Paradox’ (2019; this volume, Chapter  14), Wright explores a different 64  Exactly this problem is discussed in Wright’s ‘Anti-­Realism, Timeless Truth, and Nineteen EightyFour’ (Wright 1993: 176–203), though the issue there is anti-­realism about the past. We are also, of course, in the vicinity of Fitch’s paradox. 65  For recent discussion of this line of argument, see ‘Intuitionism and the Sorites Paradox’ (this volume, Chapter 14). 66  Another suggestion—actually, a familiar sort of suggestion, by now—is to reconstrue EC as a rule. So we will have, not: if A, then it is in principle knowable that A; but rather: from A, infer that it is in principle knowable that A. No contradiction will then be forthcoming from Verdict Exclusion, if we can motivate the appropriate restriction on conditional proof (or proof by cases). 67  But see n. 43.

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34  The Riddle of Vagueness approach, one that is compatible with agnosticism about both Evidential Constraint and Verdict Exclusion. The idea is to develop a semantics for vagueness modelled on the Brouwer–Heyting–Kolmogorov interpretation of intuitionistic logic, which proceeds in terms of proof—or, in Wright’s more general case, knowledge conditions. A recent paper by Suzanne Bobzien and Ian Rumfitt (2020) nicely complements Wright’s approach. They argue in favour of a certain modal logic, S4M,68 as the logic of ‘It is clearly the case that . . .’ (or, I assume, ‘It is definitely the case that . . .’). They then argue that we should think of intuitionistic logic as arising from a translation—the McKinsey–Tarski translation—of this modal logic into the language of first-­ order logic. Both of these strike me as fruitful avenues for further exploration. But, as new as this material is, it is perhaps best if I not try to comment on it here.69

Higher-­Order Vagueness The final theme I want to discuss is that of higher-­order vagueness. As mentioned earlier, Wright has long suspected that there is something problematic about the very notion of higher-­order vagueness. The issue first surfaces in ‘Further Reflections’, where Wright presents an argument that purports to show that higher-­order vagueness is incoherent, because the claim that some predicate F is higher-­ order vague leads via otherwise ­unobjectionable assumptions directly to contradiction.70 This ‘paradox of higher-­order vagueness’ attracted a fair bit of attention, first from Mark Sainsbury (1991). This led to a reiteration of the paradox in ‘Is Higher-­Order Vagueness Coherent?’ (this volume, Chapter  5), which sparked yet further replies by Dorothy Edgington (1993) and myself (Heck 1993); these in turn 68  This is the familiar logic S4 plus the principle: ¬  (¬  A ∧ ¬  ¬A) . 69  One point I will make is that, if what is argued in the next section is correct, then Bobzien and Rumfitt need not worry about making their account compatible with higher-­order vagueness. It also seems worth noting that, though Bobzien and Rumfitt purposely avoid discussion of the semantics appropriate to their approach, the usual sorts of models for S4M can be thought of as Kripke models for intuitionistic logic, with the ‘worlds’ representing states of information. Since truth, on this account, is clear truth, the approach thus has something in common (formally speaking) with supervaluationism. 70  Wright’s attitude towards the argument has changed over time. Originally, he seems to have regarded it as a paradox akin to the Liar or the Sorites, one whose conclusion—that ‘higher-­order vagueness is per se paradoxical’ (this volume, Chapter 5, 176)—is obviously false and whose point is thus to expose a contradiction lurking within otherwise plausible assumptions. Nowadays, he seems to take the paradox more at face value and, indeed, to regard higher-­order vagueness as seriously problematic.

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Introduction by Richard Kimberly Heck  35 gave rise to a new version of the paradox due to Delia Graff Fara (2003), one which avoids use of the inference I had questioned,71 and now to an even stronger form due to Kit Fine (2008). I will not discuss the formal issues here. What I do want to discuss is Wright’s attempt to dispose of higher-­order vagueness, in ‘The Illusion of Higher-­ Order Vagueness’ (this volume, Chapter 12). As Wright notes, much of the literature on vagueness takes the existence of higher-­order vagueness to be something like a datum. What is meant by ‘higher-­order vagueness’ is not always clear, however. Wright identifies two different things it has been taken to mean.72 These are somewhat different in spirit, though arguably equivalent,73 emphasizing, in different ways, the difficulty of drawing any boundary between the clear cases and the borderline cases. Any such boundary, it seems, will just be infected by the same pathogens that were responsible for the vagueness of the original predicate. One way to express this is to say that being a borderline case of F is itself vague— that is, that there are cases on the borderline between the Fs and the borderline cases of F. That is the first of the conceptions of higher-­order vagueness Wright distinguishes. And, if we now say that being definitely F is just being F and not being borderline F, then it would appear that this can also be put, more or less equivalently, as: being definitely F is itself vague.74 That is the second of the conceptions of higher-­order vagueness Wright distinguishes. If they are not quite equivalent, that is probably because the latter formulation leads to a more complex, but likely more adequate, conception of the structure of the various borderline regions that are now in play (Williamson 1999). It is worth emphasizing a point just made in passing. The notion of def­in­ ite­ness is sometimes derided as a philosopher’s invention—and some of Wright’s remarks suggest that he harbours such an attitude himself.75 But it is no more a philosopher’s invention than is the notion of a borderline case. The idea of a definite case is just the idea of a case that is not a borderline case: df

Def ( A) ≡ A ∧ ¬BL( A) , where ‘BL ( A)’ means: A is borderline. Of course, this 71  In fact, the model I gave to show the consistency of higher-­order vagueness does not actually do what I claim. That is the most immediate lesson of Fara’s paper. 72  Actually, there is a third: the alleged vagueness of ‘vague’. But Wright correctly (in my view) sets that aside. 73  Wright rehearses arguments that would establish the equivalence, given classical logic. Whether these arguments depend essentially upon classical logic is unclear to me. It is also unclear to me how the arguments might be affected by the definition of ‘Def(A)’ in terms of ‘BL(A)’ to be given below, the point being that constructive reasoning is extremely sensitive to details of formulation. 74  The same goes, of course, for ¬F . 75  I, on the other hand, have never expressed such a view in print. Much thanks to Josh Schechter for keeping it that way.

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36  The Riddle of Vagueness means that the notion of a definite case is no clearer than the notion of a borderline case, and Wright is absolutely correct to insist that the notion of a borderline case stands in need of serious clarification. He is also correct to point out that the notion of a definite case, so described, is prima facie puzzling: how could there be a case that was also a borderline case? If it is a case, then is not it for that very reason not a borderline case? Wright himself, however, needs to resist this sort of reasoning, since otherwise we can adapt a line of argument we explored above: if being F entails not being borderline, then being borderline entails not being F; mutatis mutandis for being not-F; and contradiction ensues. So any reasonable account of vagueness must, it seems to me, make room for the fact that Def(A) is, as Wright puts it, logically stronger than A itself—that is, that A must be logic­ al­ly consistent with BL(A). Exactly how this should be done is a delicate question, and different views of what vagueness is can be expected to entail different answers to the question why A is consistent with BL(A). But my point is simply that the problem is one many views have. The familiar ‘line of argument’ just mentioned is one that would be resisted by someone who held a form of the Third Possibility view, one that insisted that borderline cases can be neither truly nor falsely described as F (e.g. Schiffer 2020). But Wright, as we have seen, takes himself to have good ­reasons to reject Third Possibility views, and those reasons are independent of worries about higher-­order vagueness. This point is important because the standard objection to Third Possibility views is that they fail to provide any account of higher-­order vagueness. This is most obvious as regards simple three-­valued views, but it is equally true of supervaluational views and degree-­theoretic views. In the degree-­theoretic case, the worry is that, despite all the degree-­ theoretic machinery, there is yet a sharp line between the things to which F applies to degree 1.0 and the things to which it does not. Similarly, in the supervaluational case, there just has to be a sharp line between the things to which F super-­truly applies and the things to which it does not. In both cases, the existence of such a precise boundary is obscured, since the existence and location of the boundary cannot be expressed in the object-­language. But  that, arguably, is only because the object-­languages are expressively impoverished. As a result, both degree-­theoretic and supervaluational views face a problem akin to the ‘revenge problems’ that tend to afflict solutions to the liar.76 76  The simplest form of the revenge problem is the one that arises for any view that says that ‘I am false’ is neither true nor false: just consider ‘I am not true’.

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Introduction by Richard Kimberly Heck  37 It is the burden of Section 6 of ‘The Illusion of Higher Order Vagueness’ (this volume, Chapter 12) to argue for a kind of generalization of the foregoing reflections. The idea is to define a new operator of absoluteness in terms of definiteness and then to show that the Sorites paradox resurfaces once this new operator is in place, given fairly minimal assumptions about how ‘Def(A)’ itself behaves. The argument Wright gives is, as he acknowledges, similar in spirit to an argument due to Williamson (1994: 160–1). Williamson defines a new operator ‘Def *(A)’ in such a way that Def *(A) is equivalent to the infinite conjunction Def(A) ∧ Def(Def(A)) ∧ . . . .77 Williamson then argues that Def *(A) satisfies principles sufficient to reinstate the Sorites, in particular, that it yields Def *(A) implies Def *(Def *(A)). My own view about Williamson’s argument, however, has long been that a supervaluationist might have reason to deny that Def *(A) makes proper sense. The infinitary character of the ­reasoning, in particular, seems suspect. The argument Wright considers inherits some of my suspicions about Williamson’s. There is, however, a better version of the argument to be had. I have not worked out all the details, but it seems clear that the argument can be reconstructed in terms of a truth predicate and some appropriate assumptions about how it behaves.78 In particular, absoluteness can be defined as follows: df

Ab( A) ≡ ∀n(Tr (Def n ( A) )) Thus: Absolutely A just in case every sentence of the form ‘Definitely . . .  def­i n­ite­ly A’ is true. It would be nice to know exactly what assumptions about the truth predicate are needed if we are to show that Ab( A) → Ab( Ab( A)), which is the crucial ingredient in the restoration of the Sorites, but it seems unlikely that the necessary assumptions will prove to be ones a fan of higher-­ order vagueness might antecedently have found suspect.79 If not, however, then the view that vague predicates exhibit indefinitely higher-­order vagueness is in serious trouble.

77  As Williamson notes, this has a natural semantic characterization, as well. 78  The idea that the truth predicate can be used to express infinite conjunctions is usually credited to Quine. It figures heavily in recent discussions of deflationary theories of truth. Hartry Field (2001) goes so far as to claim that only a deflationary truth predicate can be used to express infinite conjunctions, but I think this is clearly mistaken (Heck 2004). 79  They are likely to include such things as: Ab(Tr ( A)) ≡ Tr ( Ab( A)) . The semantics for Ab(A) that Williamson develops—accessibility in its case is just the transitive closure of accessibility for Def(A)—makes this pretty secure, whether truth is understood as super-­ truth or as disquotational truth.

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38  The Riddle of Vagueness If that is correct, then there are two ways to respond. The first, and most obvious, is simply to accept that there are unknown (perhaps unknowable) boundaries somewhere in the series of borderlines, perhaps between the def­ in­ite­ly red and the borderline red. But then it is natural to suppose that, if we are going to have to swallow unknown boundaries, we might as well swallow them at the outset and accept that, despite appearances, there is a sharp boundary between the red and the not-­red—that is, accept Bivalence for vague predicates. So argue Williamson (1994: 5.6) and Fara (2003), each in their own way. Wright wants no part of that option, however, and so attempts to develop an alternative. The key thought is that, in a certain sense, we have no such concept as being borderline red. Wright is, of course, clear that every­thing hinges upon the sense in which this is so. Surely we do have a concept of a borderline case, and many of Wright’s attempts to motivate his denial that we have a ‘settled’ concept of the borderline red are a bit gestural for my taste. Nonetheless, I think Wright is onto something im­port­ ant. What is really moving him is the thought that borderline red is not ‘a  competitor’ with red, in the way that blue and yellow are (this volume, Chapter  12, 363). The concept of the borderline is a ‘sociological’ one, ­characterized in terms of a region in which we simply find ourselves unable to come to any confident judgement (this volume, Chapter 12, 364). As Wright puts it in ‘Intuitionism and the Sorites Paradox’: For F to be vague is for it to have borderline cases, but its possession of borderline cases is . . . a matter of our propensity to certain dysfunctional patterns of classification outside the polar regions. F ’s being vague is thus a fact about us, not about the patterns that may or may not be exhibited by the Fs and the non-Fs in a sorites series. (This volume, Chapter 14, 416; emphasis added)

I think this is absolutely right. Indeed, I think the point has been lying there unnoticed, implicit in much of the literature, for some time. According to certain sorts of supervaluationists, for example, to say that a is borderline red is to say that the semantic and non-­semantic facts simply provide no answer to the question whether a is red (McGee and McLaughlin 1995). According to epistemicists, to say that a is borderline red is to say that we do not, and cannot, know whether it is red. In neither case is borderline red itself a colour concept. It is rather a meta-­linguistic or

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Introduction by Richard Kimberly Heck  39 epistemic concept. And something similar is true on many other views of vagueness.80 So Wright’s point, as said, is that borderline red is not a colour concept. But why should that be important? It is important because this observation undermines the considerations that seem to lead so inexorably to higher-­ order vagueness. The thought, recall, is that we can no more locate the boundary between the borderline cases and the definite ones than we can locate the boundary between the red and the not-­red. If so, then it appears that both boundaries are vague, and off we go. Wright, however, wants to insist that, if the concept borderline red is not even a colour concept, then there is no reason to suppose that there even is a boundary between the red and the borderline red. To think there had to be one would be to suppose that what is borderline red ipso facto is not red, and that is what Wright means to deny when he says that borderline red is not ‘a competitor’ with red in the way that orange is. Here is another way to put the same point. It is, on reflection, entirely unclear why we should regard the boundary between the red and the borderline red as vague. True, we are unable to find that boundary. But why suppose that the boundary is vague?81 Presumably, the thought is that the Sorites premise for borderline red (8) If a thing is borderline red, then anything pairwise indiscriminable from it must also be borderline red. ought to be just as appealing as the Sorites premise for red was. But we can now see, I think, that this is just a mistake. Recall, first, that the apparent ‘tolerance’ of a vague concept to small changes, which is what underwrites the original Sorites premise, is supposed to be an a priori consequence of the nature of the concept at issue. What makes the Sorites premise for red seem so irresistible are the facts (a) that we suppose, reasonably enough, that whether a thing is red depends solely upon how it (normally) appears colour-­ wise and (b) that pairwise indiscriminable patches look relevantly the same. We now know how to avoid commitment to the latter. But the present point is that the Sorites premise for borderline red cannot be motivated in anything like these same terms. In particular, (8) will seem irresistible only if something akin to (a) holds for borderline red, which it will only if borderline red is a colour concept. Which it is not. If ‘borderline red’ means something like 80  Degree-­theoretic views may be an exception. 81  I have made this kind of complaint before (Heck 2003: 124). What follows, however, is new, and was directly inspired by my reading of ‘Illusion’ (this volume, Chapter 12).

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40  The Riddle of Vagueness ‘such that we can come neither to a stable judgement that it is red nor to a stable judgement that it is not’, then it just is a fact that there are patches we can stably judge are red which are pairwise indiscriminable from ones we cannot stably judge are red, in which case (8) ought uncontroversially to be false. I do not say that (8) has no appeal whatsoever. But I do not think its appeal can survive the realization that borderline red is not a colour concept but a sociological, epistemic, or meta-­linguistic one. If it seems still to have some appeal, then that is likely because one has not yet assimilated the lessons of the Tachometer paradox. But it is just a mistake to think that, if we cannot reliably discriminate two patches, then people cannot in fact react to them differently. Not only can they, they do and indeed must, lest it follow that ­people react the same way to all colour patches. And now, if being borderline is a sociological matter, one concerned with how we do in fact react, then, again, (8) has little to be said for it.

Conclusion In sum, then, I myself take the papers collected here to establish a number of important conclusions about vagueness. First, I have suggested that a compelling diagnosis and resolution of the Sorites paradox can be extracted from them. The resolution rests, most fundamentally, on Wright’s treatment of the phenomenal form of the Sorites. This, in turn, rests upon Wright’s discussion of the Tachometer paradox, which we discussed in Section 3, and the reflections on what rationality requires on the forced march, discussed in Section 4. Second, I take Wright to have shown, on his third attempt, that higher-­order vagueness really is an illusion. That conclusion was reached in the previous section, but the point emphasized in Section 2, that the Sorites premises are supposed to be a priori conceptual truths, played a critical role there. These are significant contributions, but, as I mentioned earlier, it is striking how much these achievements leave yet undone. Despite Wright’s work towards an intuitionistic theory of vagueness, which we discussed in Section 5, we still do not have an adequate account of the logic and semantics of vague predicates. But there is lemonade in that lemon. What it implies, I think, is that the general shape of Wright’s resolution of the Sorites, and of his dismissal of higher-­order vagueness, does not depend upon specific views on the logic and semantics of vagueness. And that kind of independence is all to the good.

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1 On the Coherence of Vague Predicates I Frege came to believe that a language containing vague predicates was essentially defective—that it was philosophically intolerable that predicates should occur for which it was not always determinate whether or not they could truly be ascribed to an object. Expressions of this conception are scattered throughout his writings. When, more seldom, he argues for it, it is on the ground that logical transformations may fail when applied to sentences containing expressions whose range of application has been only partially defined. It is not just a matter of the Law of Excluded Middle. Let F be a predicate defined only among, and universally applying to, individuals which are G. Then anything G is F. But the contrapositive fails: we cannot say that anything not-F is not-G, since the concept of having or lacking F has been fixed only for things which are G.1 Frege does not seem to have spared a thought for the idea that vague terms might require a special logic. The vagueness of ordinary language is seen rather as a flaw both needful and capable of remedy. This conception was endorsed by Russell2 in his widely despised introduction to the Tractatus. Ordinary language is always more or less vague; a logically perfect language, however, is not vague at all, and ordinary language is deemed to approach fulfilment of its function—that of having meaning—only in proportion as it approaches logical perfection—that is, in inverse proportion to its vagueness. Of course, we have long since abandoned the Frege–Russell view of the matter. We no longer see the vagueness of ordinary language as a defect. But we retain a second-­order wraith of the Frege–Russell view in the notion that, even if the senses of many expressions in natural language are not exact, there This discussion was largely motivated by two papers: Aidan Sudbury’s unpublished, ‘Are Imprecise Terms Essential?’, and Michael Dummett’s, ‘Wang’s Paradox’ (Dummett 1975). The chapter benefited from extensive discussion with both Sudbury and Dummett. 1  Cf. Frege (1903) Grundgesetze, vol. II, §65. 2  Elsewhere Russell takes the vagueness of ordinary language more seriously; e.g. in his (1923), the notion that vagueness is a flaw is tempered by pessimism about its remediability.

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0002

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42  The Riddle of Vagueness is a precise semantical description for a given natural language—that is, a theoretical model of the information assimilated in learning a first language or, equivalently, of the conceptual equipment in whose possession mastery of the language may be held to consist. Even if ‘bald’, say, is imprecise, this does not require any inexactitude in an account of its sense. We suppose our use of language to be fundamentally regular; we picture the learning of language as the acquisition of grasp of a set of rules for the combination and application of expressions. The task of a philosophical theory of meaning is naturally interpreted as that of giving a systematic account of the contribution that constituents of a complex expression make to its content; we are concerned, that is to say, when we attempt such a theory, with the nature of the path from familiarity with the senses of the sub-­sentential components of a new sentence to recognition of the sense of the whole. Such a theory will normally only concern itself with describing the contribution of a constituent qua expression of a certain logical type; it is in this connection that problems to do, for example, with the nature of the distinction between proper names and other singular terms, or that between singular terms generally and predicative expressions, or with the question whether the notion of reference may properly be extended to apply to predicative expressions, derive their interest. The completion of such a theory would thus only be a preliminary to a full semantic description of a natural language; for we think not just of the type of contribution but of the specific contribution which a constituent makes as determined by semantic rules. It is no obstacle to such a conception that we cannot in general informatively state such rules—that is, provide a statement which could be used to explain the sense of an expression to someone previously unfamiliar with it. Consider, for example, the following schematic rule for a one-­place predicate, F: F may truly be applied to an individual, a, if and only if a satisfies the ­condition of being φ . How should we specify φ if F is ‘red’? Plainly the only such condition which actually captures our understanding of ‘red’, rather than, in a merely extensional way, its conditions of application, is . . . the condition of being red. So we  may not in general expect instances of the schematic rule to be of any ­explanatory use if they are stated in a given language for a predicate of the same language; as a corollary, it will not in general be possible to appeal to such a rule to settle a dispute about the applicability of an expression. It remains open to us, nevertheless, to regard such a rule as exactly encapsulating (part

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On the Coherence of Vague Predicates  43 of) what is understood by someone who understands, for example, ‘red’, for it states conditions recognition of whose actualization is sufficient to justify them in describing an object as ‘red’; it is merely that such a capacity of recognition cannot be bestowed by stating the rule. Our picture, then, is that to use language correctly is essentially nothing other than to use it in conformity with a set of instructions, of semantic rules. Of course our handling of language is in general quite automatic, but so is a  chess player’s recognition of the moves allowable for a piece in a certain position; it remains true that an account of their knowledge is to be given by reference to the rules of chess. The question now arises, what means are legitimate in the attempt to discover features of the substantial rules for expressions in our language, the rules which determine specifically the senses of such expressions? The view of the matter with which we are centrally concerned in this chapter is that we may legitimately approach our use of language from within—that is, reflectively as self-­conscious masters of it, rather than externally, equipped only with behavioural notions. We may appeal to our conception of what justifies the application of a particular expression; we may appeal to our conception of what we should count as an adequate explanation of the sense of a particular expression; to the limitations imposed by our senses and memories on the kind of instruction which we can actually implement; and to the kind of consequence which we attach to the application of a given predicate, to what we conceive as the point of the classification which the predicate effects. The notion that forms the primary concern of this chapter—henceforward referred to as the governing view—is that we can derive from such considerations a reflective awareness of how expressions in our language are understood, and so of the nature of the rules which determine their correct use. The governing view, then, is a conjunction of two theses: that our use of language is properly seen, like a game, as an activity in which the allowability of a move is determined by rule, and that properties of the rules may be discovered by means of the sorts of consideration just described. The governing view does not involve a psychologistic conception of understanding, according to which understanding would be regarded as an essentially mental state of which the correct employment of an expression was a mere behavioural symptom; but it provides means in excess of the behaviourist minimum, = the description of when an expression is actually used, for an attempt to investigate the nature of the semantic rules operative with respect to a given class of expressions. It may be that, as so far characterized, the view will seem ­platitudinous; the purpose of this chapter is to question its coherence.

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44  The Riddle of Vagueness A striking feature of the Philosophical Investigations is the hostility displayed by the author of the notion of a language-­game to the idea that it is explanatory of our use of language to appeal to the concept of rule-­following. It is not, of course, that there cannot be a resoluble dispute about the correct use of a particular expression. Rather Wittgenstein seems to argue for an indeterminacy in the identity of the rules which someone supposedly follows. Here it is ­irrelevant whether we can supply, for example, what seems an informative completion of the above schematic rule for a particular predicate, F. The ­question is, what content is there to the claim that such a rule faithfully ­incorporates someone’s understanding of F? For they and we may sincerely agree on a particular formulation and then, sincerely and irresolubly, apply F in mutually inconsistent ways. It is useless to protest that all that follows is that they use F in accordance with a different rule. We agreed on how the rule was to be characterized; now, it seems, we reserve the right to offer some other characterization of what governs their use of the expression. (Though there is no necessity that any such characterization should occur to us.) From their point of view, however, the initial characterization remains perfectly adequate; it is of our use of F that a recharacterization of the determining rule is required. But, now, what objectivity is there in the idea of the correct characterization of their or our respective rules? And, if there is none, how can we penetrate, as it were, to the real nature of the rule which we follow, if we wish to explain our use of F? Thus the terminology of semantic rules, adequate perhaps for certain purposes as a picture, fails to express a domain of objective fact from which our use of language may be seen to flow. This line of argument poses a difficulty for the first thesis of the governing view. It is mentioned as a familiar type of criticism of the notion of a semantic rule, and by way of contrast with the character of the difficulty for the governing view to be posed in this chapter. Here it is contended rather that the second thesis, concerning the means whereby features may be discovered of the semantic rules which we actually follow, constrains us to recognize semantic incoherence in our understanding of a whole class of predicates—elements whose full exploitation would force the application of these expressions to situations where we should otherwise regard them as not applying. The second thesis requires the recognition of rules which, when considered in conjunction with certain general features of the situations among which their associated expressions are to be applied, issue in contradictory instructions. Nevertheless we succeed in using these expressions informatively, and to use language informatively depends on using it, in large measure, consistently. It follows that our use of these expressions cannot correctly be pictured purely

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On the Coherence of Vague Predicates  45 as the implementation of the rules which the second thesis yields for them— these rules cannot be obeyed by consistent behaviour. The governing view is, therefore, incoherent; for, if its second thesis is true, the semantic rules which are operative with respect to certain predicates are capable by consistent beings only of selective implementation and thus, contrary to the first thesis, are not constitutive of what we count as the correct use of these expressions. Predicates of the relevant kind are all examples of a certain sort of vagueness: not exactly borderline-­case vagueness, if that is understood, as Frege sometimes describes it, as the existence of situations to which it is indeterminate whether or not a predicate applies, but something which, under the guise of a favourite metaphor, he constantly runs together with the possession of borderline cases—namely, the idea of lacking ‘sharp boundaries’, of dividing logical space, as a blurred shadow divides the background on which it is reflected. The conflation is plausible because the image equally exemplifies the idea of the borderline case, a region falling neither in light nor in shadow. But there seems no reason why having borderline cases should imply blurred boundaries. Borrowing another of Frege’s analogies, we may assimilate a predicate to a function taking objects as arguments and yielding the True or the False as ­values; in these terms a predicate with borderline cases is simply a partial such function, and that is consistent with the obtaining of a perfectly sharp distinction between cases for which it is defined and cases for which it is not. Borderline-­case vagueness simpliciter presents no difficulty for the governing view; it is merely that we are presented with situations to which no response is determined by the semantic rules of our language as the correct one. On the other hand, if the second thesis of the governing view is correct, then predicates with ‘blurred boundaries’ are, in typical cases, to be regarded as semantically incoherent. This incoherence resides in their vagueness as such. It is plausible to suppose that the vagueness of many expressions is not, as Frege and Russell thought, merely a reflection of our intellectual laziness. Rather, the utility and point of the classifications expressed by many vague predicates would be frustrated if we supplied them with sharp boundaries. (If it is an empirical truth that stress diseases are more widespread in highly concentrated populations, it is doubtful whether it would survive an exact numerical definition of ‘highly concentrated’.) The sorts of consideration admitted by the second thesis will transpire, in the succeeding sections of the chapter, to yield support for the idea that such predicates are essentially vague. The thesis equips us to argue that lack of sharp boundaries is not just a surface phenomenon reflecting a hiatus in some underlying set of semantic rules. Lack of sharp boundaries is not the

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46  The Riddle of Vagueness reflection of an omission; it is a product of the kind of task to which an expression is put, the kind of consequences which we attach to its application, or, more deeply, the continuity of a world which we wish to describe in purely observational terms. Lack of sharp boundaries is semantically a deep phenomenon. It is not generally a matter simply of lacking an instruction where to draw the line; rather the instructions we already have determine that the line is not to be drawn. This conclusion might seem a welcome contribution to our understanding of the nature of vagueness, were it not that it is supplied in the form not merely that no sharp distinction may be drawn between cases where such a predicate applies and cases where it does not, but that no such distinction may be drawn between cases where it is definitely correct to apply the predicate and cases of any other sort. Thus it is that adherents of the governing view simply have no coherent approach to the Frege–Russell view of vagueness. Their second thesis requires them to reject the suggestion that vagueness is a superficial, eliminable aspect of natural language with no real impact upon its informative use. But it does so by means of considerations which require them to regard many vague predicates as semantically incoherent, so that, unless the Frege–Russell view is right, they cannot maintain their first thesis with respect to such expressions. Only if their vagueness is an incidental feature can they maintain that the essential semantics of such expressions conform to their first thesis. The programme for the remaining part of the chapter is as follows. In the next section, three examples will be presented of predicates to which the second thesis attributes semantic incoherence; specifically, it sustains for each of them the reasoning of the Sorites paradox. The character of the incoherence will be generalized, and arguments afforded by the second thesis for such  an account of the semantics of these expressions will be developed. In  Section  III a fourth and deeper-­rooted example will be presented. In Section  IV we shall consider an obvious strategy for solving paradoxes in the Sorites group which, if allowed, would undercut the considerations of Sections II and III. This strategy will be rejected. In Section V it will be argued that one class of predicate to which the second thesis attributes semantic incoherence is, in a certain sense, ineliminable. It will be shown that a seemingly promising adaptation of certain of the ideas in Goodman (1951) fail to provide an adequate refashioning of the semantics of these predicates; and that a simpler suggestion, while indeed liberating them from semantic incoherence, does so at the cost of generating other predicates with the same feature. In the final section our conclusions will be drawn together.

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On the Coherence of Vague Predicates  47 If the governing view is unacceptable, that is something which it is as well to know. The interest of the chapter, however—if any—derives equally from that of the issues which we shall have occasion to discuss passim: the nature of vagueness and the correct logic for vague expressions; ostensive definition and observational language; the empirical sources of the concept of continuity, and the notion of order within phenomenal continua like the spectrum of colours.

II Example 1 Our first example is the Megarian paradox itself.3 If we begin with a pile of salt large enough to be fairly described as a heap, the subtraction of a single grain of salt cannot make a relevant difference; if n + 1 grains of salt may constitute a heap, so may n such grains. Of course, the plausibility of this ­supposition derives from the species of sense with which ‘heap’ has been endowed; we have not fixed exact boundaries for the concept of a heap, either in terms of the precise number of grains contained or, indeed, in terms of any other precise measure. To allow that at some stage the subtraction of a single grain might transform a heap into a non-­heap would be exactly to anticipate the determination of such a specific boundary. However, in the present semantic situation of ‘heap’, it would be merely an error to insist that, for some particular value of n, n + 1 grains of salt would amount to a heap while n grains would not; for that is simply not the sense of ‘heap’. If it were agreed in some particular case that n + 1 grains did amount to a heap, no one could produce a telling reason for withholding the predicate from the same pile minus one grain; except perhaps avoidance of the incoherence implicit in the situation. ‘Heap’, then, would appear to be semantically incoherent, for its sense is such that it essentially lacks exact extensional boundaries; and its lack of such boundaries demands particularly that a transition from n + 1 grains to n grains can never be recognized to transform a case where ‘heap’ applies to a case where it does not. So we are seemingly equipped to force the application of ‘heap’ through successively smaller aggregations of salt grains, terminating 3  Traditionally, attributed along with its variant, the ‘Bald Man’ to Eubulides. (See, e.g., Diogenes Laertius, Lives, ii. 108.)

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48  The Riddle of Vagueness in items which amount not to heaps but merely, say, to pinches. Here we gravitate towards the idea that lack of exact boundaries is, as such, an essentially incoherent semantic feature. This idea, however, will need qualification.

Example 2 Predicates of degree of human maturity—‘infant’, ‘child’, ‘adolescent’, ‘adult’— display the same peculiarity. They are mutually inconsistent yet lack sharp boundaries with respect to their neighbours in the scale of human maturation. More exactly, if we take some sufficiently small interval of time and suppose that someone matures in a typical fashion, then at no stage will they effect within such an interval of time a transition from one stage of maturity to the next. To illustrate the point with an example from Esenin-­Volpin (1961): take as the relevant interval the span of time from one heartbeat to its successor; then the concept of childhood—the sense of ‘child’—is such that one does not, within a single heartbeat, pass from childhood to adolescence. To be sure, one is not a child forever; but at least childhood does not evaporate between one pulse and the next. If one’s nth heartbeat takes place in childhood, then so does the n + 1th.4 ‘Infant’, ‘child’, ‘adolescent’, ‘adult’, are thus all semantically incoherent expressions; for the sense of each of these predicates is such that, in a typical process of growing up, their correct application will always survive the transition from one heartbeat to its successor or to its predecessor. So again, by appropriately many steps of modus ponens, we may force the application of each of these predicates to cases we should otherwise regard as falling within the domain of a competitor.

Example 3 Consider a series of homogeneously coloured patches, ranging from a first, red patch to a final, orange one, such that each patch is just discriminable in 4  Esenin-­Volpin’s interest in the example derives from the possibility, which it suggests, of satisfying orthodox arithmetical postulates for the successor function in an open-­ended but finite domain. The existence of such domains depends upon the admissibility of the concepts of which they are the apparent extensions; that is, it depends upon the correctness of the view that we may have a coherent understanding of predicates whose semantics are strictly incoherent. For, unless a predicate of the genre in question really does have an incoherent semantics, it will not yield a structure in which the relevant postulates are satisfied; but, unless such predicates are acceptable, we shall not be much interested in the pure mathematics of their extensions. There will be, unfortunately, no further discussion in this chapter of the significance of these examples for the concept of infinity.

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On the Coherence of Vague Predicates  49 colour from those immediately adjacent to it, and is more similar to its immediate neighbours than to any other patches in the series. Marginal, uni-­ directional changes of shade are thus involved in every transition from a patch to its successor. Now, it is notable that the sense of colour predicates is such that their application always survives a very small change in shade. Given that one is content to call something ‘red’, one will still be so content if its ­colour changes by some just discriminable amount. There is a notion of a degree of change in respect of colour too small to amount to a change of colour. Only if a substantial difference intervenes between two patches shall we consider ourselves justified in ascribing to them incompatible colour predicates. Obviously this is an incoherent notion. In particular, in view of the proximity in shade of neighbouring patches in our series, it provides an easy proof that all the patches are red (or orange, or doubtfully either). Moreover, any two colours can be linked by such a series of samples; so any colour predicate can likewise be exported into the domain of application of one of its rivals. Colour predicates as a class are semantically incoherent. In these examples we encounter the feature of a certain tolerance in the concepts respectively involved, a notion of a degree of change too small to make any difference, as it were. There are degrees of change in point of size, maturity, and colour which are insufficient to alter the justice with which some specific predicate of size, maturity, or colour is applied. This is quite palpably an incoherent feature, since, granted that any case to which such a predicate applies may be linked by a series of ‘sufficiently small’ changes with a case where it does not, it is inconsistent with the exclusivity of the predicate. More exactly, let φ be a concept related to a predicate, F, in the following way: that any case to which F applies may be transformed into a case where it does not apply simply by sufficient change in respect of φ; colour, for example, is such a concept for ‘red’, size for ‘heap’, degree of maturity for ‘child’, number of hairs for ‘bald’, degree of complexity for ‘memorable’ as applied to patterns, and so on. Then F is tolerant with respect to φ if there is also some positive degree of change in respect of φ insufficient ever to affect the justice with which F is applied to a particular case. These wholly intuitive ideas are sufficiently clear for present purposes. At this point it has to be conceded that the manner in which these examples have been presented has been wholly tendentious. It is not to be doubted that the predicates in question do lack sharp boundaries; and the antiquity of the paradox bears witness to how easy it is to interpret this as involving the possession by these predicates of a principle of reapplication through marginal change. But is this a correct interpretation? If ‘heap’, for example, lacks sharp

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50  The Riddle of Vagueness boundaries, then certainly we are not equipped to single out any particular transition from n to n – 1 grains of salt as being the decisive step in changing a heap into a non-­heap; no one such step is decisive. But that is not to say that such a step always preserves application of the predicate. Should we not instead picture the situation as comparable to that in which neighbouring states fail to agree upon a common frontier? Their failure to reach agreement does not vindicate the notion that, for example, a single pace in the direction of the other country always keeps one in the original country. For they have at least agreed that there is to be a border, that some such step is to be decisive; what they have not agreed is where. The analogue of this conception for the predicates which interest us is exactly that their vagueness is purely a reflection of our intellectual laziness. We have, as it were, decided that a disjunction is to be true—at some stage, n grains will be a heap whereas n – 1 grains will not—without following up with a decision about which disjunct is true. The notion that these predicates are tolerant confuses a lack of instruction to count it the case that a proposition is false with the presence of an instruction to count it true. This conflation would be permissible only if our set of semantic rules were in a certain sense complete—that is, if it contained instructions for every conceivable situation. But for there to be vague expressions in our language is precisely for this not to be so. If we avail ourselves of the types of consideration afforded by the governing view, we shall reject this suggestion as a deep misapprehension of the nature of this species of vagueness. The lack of sharp boundaries possessed by these examples is correctly interpreted as tolerance, provided that we may so ­discover elements of their senses. The point is not that there is any decisive formal obstacle to providing general stipulations by means of which we might identify in such examples a last point at which the predicate in question could definitely correctly be applied. We could set upper and lower limits for ‘heap’ in terms of number of grains.5 We could do the same for ‘child’, and so on, by setting precise age ­limits to the successive phases of life—perhaps in terms of numbers of heartbeats. For colour predicates it would prima facie be less easy to provide such a refinement, since the notion of being of the same shade encounters difficulties which those of being of the same age, or containing the same number of

5  We spare the reader the irrelevant complications involved in how such a stipulation would apply to substances of varying granular coarseness, viscosity, etc.

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On the Coherence of Vague Predicates  51 grains, do not. But suppose it is possible. Then what in the semantics of these examples is already inconsistent with our so refining their senses? ‘Heap’ is essentially a coarse predicate, whose application is a matter of rough and ready judgement. We should have no use for a precisely demarcated analogue in the contexts in which the word is typically used. It would, for example, be absurd to force the question of the execution of the command, ‘Pour out a heap of sand here’, to turn on a count of the grains. Our conception of the conditions which justify calling something a heap of sand is such that the justice of the description will be unaffected by any change which cannot be detected by casual observation. A different argument is available for colour predicates. We learn our basic6 colour vocabulary ostensively—that is, by exposure to samples illustrative of its application. Evidently it is a precondition of our capacity to do so that we can reasonably accurately remember how things look. We can imagine people who can recognize which simultaneously presented objects match in colour, and so are able to use a colour chart, but who cannot in general remember shades of colour sufficiently well to be able to employ without a chart a colour vocabulary anything like as rich as ours. Such people, for example, having been shown something yellow, might later be quite unable to judge whether what they were earlier shown would match the orange sample now before them. Thus, for such a community, an ostensive definition of ‘yellow’ would not be feasible, and, in order to make the distinctions of our basic colour vocabulary, they would need to employ charts. Of course, we also use charts for some purposes; but never to make distinctions of the magnitude of those, say, between the colours of the rainbow. Any object to which one of these predicates definitely correctly applies may be recognized as such just on the basis of our ostensive training. But, if this is so, it has to be a feature of the senses thereby bestowed upon these predicates that changes too slight for us to remember—that is, a change such that exposure to an object both before the change is undergone and afterwards leaves one uncertain whether the object has changed, because one cannot remember sufficiently accurately how it was before—never transform a case to which such a predicate applies into one where such is not definitely correctly the right description. The character of the basic colour training which we receive, and which we hand on to our children, presupposes the total memorability of the

6  Intuitively, a colour predicate is basic in a given language if it does not express a shade of a more general colour for which means of expression exists in the same language. Obviously, as it stands, the notion is fragile.

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52  The Riddle of Vagueness distinctions expressed by our basic colour predicates; only if single, unmemor­ able changes of shade never affect the justice of a particular, basic colour description can the senses of these predicates be explained entirely by ­methods reliant upon our capacity to remember how things look. With respect to ‘child’, and so on, the governing view allows us a third type of consideration. Broadly, we do not make distinctions in phases of maturity in the fashion of a naturalist, just to record a variety with which we are confronted; rather, these classifications are of substantial social importance in terms of what we may appropriately expect from, and of, persons who exemplify them. Thus infants have rights but not duties, whereas of a child outside infancy we demand at least a rudimentary moral sense; we explain the anti-­social behaviour of some adolescents in terms of their being adolescents; and we make moral and other demands of character on adults which we would not impose on the immature. It is plausible that the predicates of Example 2 could not endure this treatment, were they not tolerant with respect to marginal changes in respect of degree of maturity. It would be irrational and unfair to base substantial distinctions of right and duty on marginal—or even non-­existent7—such differences; if we are forced to do so—for example, with electoral qualifications—it is with a sense of artificiality and absurdity. There is another point: without tolerance these predicates could no longer sustain the explanatory role which they now have for us. Only if a substantial change is involved in the transition from childhood to adolescence can we appeal to this transition to explain substantial alterations in patterns of behaviour; if some adolescents differ barely, if at all, from some children, so that no significant change need be undergone in making the transition, to have done so can explain nothing. That predicates of degree of maturity ­possess tolerance is a direct consequence of their social role; very small differences cannot be permitted to generate doubt about their application without correspondingly coming to be associated with a burden of moral and ­explanatory distinctions which they are too slight to convey. Our embarrassment about where to ‘draw the line’ with these examples is thus a reflection not of any hiatus in our semantic ‘programme’ but of the ­tolerance of the predicates in question. If casual observation alone is to determine whether some predicate applies, then items not distinguishable by casual

7 As they would sometimes be, if the distinctions were made in terms of precise numbers of heartbeats.

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On the Coherence of Vague Predicates  53 observation must receive the same verdict.8 So single changes too slight to be detected by casual observation cannot be permitted to generate doubt about the application of such a predicate. Likewise, if the conditions under which a predicate applies are to be generally memorable, it cannot be unseated by ­single changes too slight to be remembered. Finally, very slight changes cannot be permitted to generate doubt about the application of predicates of maturity without contravening their moral and explanatory role. The utility of ‘heap’, the memorability of the conditions under which something is red, the point of ‘child’ thus appear to impose upon the semantics of these predicates tolerance with respect to marginal change in their various  relevant respects. Not that, on the governing view, those consider­ ations provide a whole account of their vagueness. For example, the considerations applied to ‘heap’ presumably apply to ‘red’ and, in some measure, to ‘child’ also. However, it is clear that to allow just the foregoing sketchy considerations is to concede both that the vagueness of our examples is a phenomenon of semantic depth—that is, it is sacrificed at much more than the cost of the intellectual labour of the stipulation—and that it is a structurally incoherent feature. Two things follow. First, there is no special logic for predicates of this sort, crystallizing what is distinctive in their semantics in contrast with those of exact predicates; for what so distinguishes them is their inconsistency. A  ‘logic’ of this species of vagueness is chimerical. Second, the manner in which we typically use these expressions needs some other model than the simple following of rules, if these are to incorporate all the features of their senses which we should wish to recognize. It is perhaps more nearly comparable to the behaviour of a public-­spirited citizen in relation to fiscal law; for, as is familiar, the overall effectiveness of a system of taxation may well depend both upon the presence of loopholes and on people’s forbearance to exploit them.

III Anyone who holds the second thesis of the governing view should recognize a distinct and more profound source of tolerance in adjectives of colour than the inability of our memories to match the sharpness of discrimination possessed by our senses. Colour predicates, it is plausible to suppose, are in the 8  There is no presupposition here that a definitely correct verdict can always be reached; but, if it cannot, that in turn must be the situation with respect to each item in question.

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54  The Riddle of Vagueness following sense purely observational: if one can tell at all what colour something is, one can tell just by looking at it. The look of an object decides its ­colour, as the feel of an object decides its texture or the sound of a note its pitch. The information of one or more senses is decisive of the applicability of an observational concept; so a distinction exemplified in a pair of sensorily equivalent items cannot be expressed by means solely of predicates of observation, for any observational expression applying to either item must apply to both. What is about to be illustrated with respect to colour predicates is, under appropriate assumptions, a feature of any expression whose sense is observational in the fashion just sketched. Since colour predicates are observational, any pair of objects indistinguishable in point of colour must satisfy the condition that any basic colour predicate applicable to either is applicable to both. It is, however, familiar that we may construct a series of suitable, homogeneously coloured patches in such a way as to give the impression of a smooth transition from red to orange, where each patch is indiscriminable in colour from those immediately next to it; it is the non-­transitivity of indiscriminability which generates this possibility. So, since precise matching is to be sufficient for sameness of colour, we can force the application of ‘red’ to all the patches in the series, some of which are not red but orange. That is: since ‘red’ is observational, its sense must be such that from the premises, that x is red and that x looks just like y, it follows that y is red, no matter what objects x and y may be. This rule enables us to conclude that each successive patch in our series is red, given only the true premise that the first patch is red. The purpose of Example 3 was to illustrate, from the standpoint of the second thesis, a tolerance of predicates of colour with respect to marginal changes of shade. If we retain such an account for this new example—Example 4—we shall be forced to regard identity of shade as a non-­observational notion. We shall be admitting that changes in shade take place between adjacent patches where none seems to have taken place, where the most minute mutual comparison reveals no difference. But it is clear that we shall be driven to some such admission even if, rather than construe it as non-­ observational, we abandon the notion of identity of shade altogether. For some sort of non-­observational changes are taking place in this example, however we choose to describe them. Example 4, if allowed, reveals colour predicates as tolerant with respect to changes which cannot be directly discerned in objects which undergo them; an object may suffer such a change without it being possible to discover that it  has done so save by comparing it with something else. This feature

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On the Coherence of Vague Predicates  55 differentiates this example from the others. Moreover, we do not have ready to hand a concept in terms of which we can describe what these changes essentially are. Not that we could not offer an account in terms, for example, of the physics of light; but it is contingent that the changes in question are associated with any particular physical changes which we might independently discover. On the other hand, it is seemingly not a full account of the matter just to say that a patch now matches something which it did not match before, while not itself seeming to have changed in the meantime; this is unsatisfactory as a full account, because we are inclined to say that, when objects come into a relation which they did not share before, one or both of them must have changed in independently specifiable respects; whereas, in this case, neither patch may seem to have changed. We are lacking, for example, a notion standing to the concept of shade as that of real position stands to phenomenal position. Had we such a concept, we could account for the changes in question in real terms. This lack would be a prima facie obstacle to stipulating away examples of this fourth type; for the stipulations envisaged in the other three cases each made use of a concept—number of grains, number of heartbeats, phenomenal shade—in terms of which we could describe the small changes involved. Later we shall consider how this obstacle might be overcome. For the present it is more significant that the difficulty arises at all. Had we a notion of shade comparable to that of real position, we might provide such a stipulation; but, had we such a notion of shade, we should already have surrendered its observationality, and to provide the stipulation would be to surrender that of colour as well. To stipulate away the tolerance of a predicate is to provide a general explanation of where, in a series of the relevant sort, it may be applied correctly for the last time. But, if we did so with respect to ‘red’, ‘orange’, and the type of series illustrated in Example 4, we should have to forgo our entire present conception—namely, the look of a thing—of what justifies the application of these predicates. To say that a predicate is observational requires that the conditions under which it may be applied to an object must be determinable simply by observation of it; whereas we are contemplating a situation where ‘red’ may definitely correctly be applied to only one of a pair of cases which are observationally exactly similar—whose looks match exactly. Clearly, then, any observational predicate must display tolerance in a series in which it is initially but not everywhere exemplified and whose every member cannot be observationally distinguished (save numerically) from those immediately adjacent to it. Naturally, no series can satisfy these conditions unless the relevant relation of indistinguishability behaves ­ non-­transitively.

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56  The Riddle of Vagueness These considerations are broadly analogous to what was said of the Heap: if we so fix the sense of a predicate that whether it applies has to do with nothing other than how an object seems when casually observed, then changes other than can be determined by casual observation of it cannot transform a case to  which the predicate applies into one where it does not, or to which its application is somehow doubtful. The point remains good if we omit the word ‘casual’. But Example 4 is seemingly deeper reaching, at any rate for someone who, in the spirit of Frege, required that language should be purified of vague expressions. The cost of eliminating predicates of casual observation is convenience; but the cost of eliminating the use of expressions tolerant in the manner of Example 4 would be the abandonment of predicates of strictly observational sense. Might there not then be a higher price to pay—namely, the jeopardizing of contact between language and empirical reality? We shall not pursue this thought immediately. First, we need a deeper perspective for Example 4. We require an explanation of the observationality of certain predicates—or, what in this context comes to the same thing, some reinforcement of the supposition that their semantics are purely observational—and we require to know under what circumstances we may expect our sensory discriminations to be non-­transitive. The intersection of these explanations will yield a general indication of the range of the type of case which Example 4 illustrates. That we do intuitively regard the semantics of adjectives of colour as purely observational is beyond doubt; and simply illustrated by the fact that we should regard it as a criterion of lack of understanding of such an adjective if someone was doubtful whether both of a pair of objects which they could not tell apart should receive the same description in terms of it. We regard it as a criterion of understanding such a predicate that someone, presented under suitable conditions with an object to which it applies, can tell that it does so just on the basis of the object’s appearance. But can an explanation be provided of why there should be predicates with such a semantics? It might be supposed that any ostensively definable predicate must be observational. If an expression can be ostensively defined, it must be possible to draw to someone’s attention those features in their experience which warrant its application; and, if this is possible, there can be no question of the expression applying to some but not others among situations which they cannot distinguish experientially. It would be a poor joke on the recipient of an ostensive definition if the defined expression applied selectively among ­situations indistinguishable from one which was originally displayed to them as a paradigm.

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On the Coherence of Vague Predicates  57 Unless it is disallowed that aspects of the semantics of an expression can be discovered by appeal to such considerations, we are bound to regard this suggestion as basically correct. Accordingly, we can no longer combine the conventional idea of the place which ostensively defined predicates occupy in our ‘conceptual scheme’—the base of the epistemic pyramid—with the view that strict semantic coherence is a necessary condition of intelligibility. Rather, we have to recognize that such predicates are endowed by their very mode of introduction with a kind of original sin—a species-­ liability to tolerance. We are not yet, however, in a position to draw this conclusion. Certainly an ostensive definition must be regarded as issuing a licence, so to speak, to apply an expression to any situation relevantly matching that which the definition uses; an ostensive definition must do this, or it is of no use. But that is not to issue a licence to apply the expression to a situation which does not match the original situation, but merely matches an intermediate situation, indistinguishable from both. If we think of an ostensive definition as a command: ‘Apply F to situations like this’, the command obviously does not apply to situations which are not ‘like this’ but merely like something like this. This is so, whether the required resemblance is conceived as indistinguishability or as some less exact likeness. What, then, is the connection between an expression’s being observational— its applying to both, if to either, of any pair of observationally indistinguishable situations—and its being ostensively definable? It is as follows. The picture  of acquiring concepts by experience of various cases where they do apply and various cases where they do not—a picture which surely has some part to play in a philosophically adequate conception of the learning of a first language—cannot be wholly adequate for concepts which differentiate among situations which look, feel, taste, sound, and smell exactly alike. So, if that picture is wholly adequate for any concepts, they must be concepts whose range of application does not include situations which experience cannot distinguish from situations which may not definitely correctly be regarded as falling within that range. To master the sense of a predicate is, at least, to learn to differentiate cases to which it is right to apply it from cases of any other sort. If such mastery can be bestowed ostensively, a comparison of two such cases must always reveal a difference which sense experience can detect. The notion, then, that the whole range of application of a predicate can be made intelligible by ostensive means presupposes that it is never the case that only one of a pair of objects, which the senses cannot tell apart, is characterized by it.

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

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On the Coherence of Vague Predicates  59 position and sufficient conditions for their being truly described as being so. A change of position need not be noticeable; a change of colour must be. Naturally we may be driven to abandon this very aspect of the semantics of colour predicates as a result of the tension between observationality and non-­ transitive indiscriminability; but, until we do so, it cannot be an explanation of the non-­transitive indiscriminability of suitable colour samples that differences of colour exist too fine for our gross perceptions. This, however, is still not quite the point. For plainly, as we noted, the non-­ transitivity of matching requires that not every feature of colour patches can be a directly observational one; colour patches evidently allow of changes, whether these changes are described as changes in colour or not, of a kind which cannot be directly discerned. So we have no alternative but to admit a gap among such items between seeming not to have changed in any respect, and actually not having done so. We have to admit the described objectivity, if not in the notion of change in colour, at least in the notion of change in some respect. What is really wrong with the explanation is not the presupposition of objectivity, but the circumstance that the sole ground for affirming that there is a distinction between seeming and being here is that indiscriminability is behaving non-­transitively; whereas we are trying independently to ­circumscribe the circumstances under which we may expect such behaviour. So we require an alternative account. Suppose that we are to construct a series of colour patches, ranging from red through to orange, among which indiscriminability is to behave transitively. We are given a supply of appropriate patches from which to make selections, an initial red patch C1, and the instruction that each successive patch must either match its predecessor or be more like it than is any other patch not matching it which we later use. Under these conditions, it is plain that we cannot generate any change in colour by selecting successive matching patches; since indiscriminability is to be transitive, it will follow if each Ci in the first n selections matches its predecessor, that Cn matches C1. The only way to generate a change in colour will be to select a non-­matching patch. When the series is complete, how will it look in comparison with the series of Example 4? It is clear that we shall have lost what is distinctive of that series: the appearance of continuous change from red to orange. In the new series, the shades are exemplified in discrete bands (containing perhaps no more than one patch), and all the changes take place abruptly in a transition from a patch to its successor. So it appears that, were our judgements of ­indis­criminability to be universally transitive among samples of homogeneous colour, no field of colour patches could be ordered in the distinctive fashion

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60  The Riddle of Vagueness now possible; that is, so as to give the impression of a perfectly smooth change of hue. If matching generally behaved transitively among shades, no series of colour patches could give the impression of continuous transformation of colour; contrapositively, then, for matching to function non-­transitively among a finite set of colour patches, it is sufficient that they may be arranged so as to strike us as forming a phenomenal continuum. This reasoning may obviously be generalized. Any finite series of objects, none of which involves any apparent change in respect of φ, may give an overall impression of continuous change in respect of φ only if indiscriminability functions non-­transitively among its members. Evidently, though, not all processes of seemingly-­continuous change come, as it were, ready made out of finitely many stages. Seemingly-­continuous processes in time do not generally do so; and nor do certain purely spatial seemingly-­continuous changes—for example, the convergence of a pair of near-­ parallel lines. The question now arises whether there is not a more ­general connection between seemingly-­continuous change and non-­transitive indiscriminability than that illustrated in the somewhat artificial case of the colour patches. Let us consider the case of processes of change in time. Let D be such a process, and let a stage of D be the state of D at a particular point in time, an instantaneous exposure, as it were, of the process at that point. D is to be monotonic in the following sense: if ϕ is the respect in which changes in D take place, no distinct stages, x, y, z in D, are to be such that z is in respect of ϕ more like x than y is when x is earlier than y and y is earlier than z. It is analytic that any process of change is monotonic up to some stage. We claim that, on a very natural presupposition, D will be seemingly-­continuous only if there is some finite selection of stages of D among which indiscriminability behaves non-­transitively. Suppose, on the contrary, that, while D itself is seemingly-­continuous, there is no way of selecting stages from D, however close together, so that indiscriminability behaves non-­transitively among them. Consider a maximal set, S, of stages of D yielded by the following rule of selection: x is a member of S  if and only if x is discriminable in respect of φ from every y previously selected as a member. (Let the first selection be regarded as satisfying this condition vacuously.) Since S is maximal, every stage in D must be indiscriminable in respect of φ from some member of S. Suppose now that S is finite and consider the series , of all its members in order of temporal succession. Let Di, Di+1, be a pair of stages adjacent in this series.

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On the Coherence of Vague Predicates  61 Plainly any stage occurring in D later than Di, but earlier than Di+1, must be indiscriminable in respect of φ from either Di or Di+1 but not both; it cannot match both, since matching is hypothesized to be behaving transitively; it cannot match neither, or it must match some other stage in , so violating the hypothesis that D is non-­recurrent. Clearly if such an in-­between stage matches, say, Di, then all stages lying temporally between it and Di in D must likewise match Di, or D will not in this region be monotonic; mutatis mutandis if it matches Di+1. So the region of D between Di and Di+1 must divide into two contiguous segments, every stage in one of which will match Di while every stage in the other matches Di+1. Evidently, then, D cannot present an impression of continuity of change as it moves through this region, contrary to hypothesis. To obstruct this reasoning, it will not be sufficient to hypothesize that S  is infinite; we need specifically that is densely ordered by temporal succession, that between any pair of stages of D discriminable in  respect of φ lies a stage discriminable from them both and from any stage outside the region of D which they flank. We have to suppose that we  have in this sense infinite powers of discrimination in D, that we can always directly discern some distinction more minute than any discerned so far. The ‘very natural presupposition’ earlier referred to is that this is not so. To summarize our conclusions here: if D is a non-­recurrent temporal process of change such that indiscriminability behaves transitively among every selection of stages from it, and if we can directly discern only a finite variety of stages of D—at least in some of its regions—then D must contain seemingly absolutely abrupt changes. Hence, if D is everywhere to give an impression of  continuous change, indiscriminability cannot behave transitively among every selection of stages from it; specifically, if Di and Di+1 are adjacent in derived as above, at least one stage occurring between Di and Di+1 in D must match both. These considerations are, of course, incomplete. We have not considered purely spatial processes of change, though one might venture to expect that exactly analogous reasoning would apply; and we have not considered whether there is not a sufficiency condition for seeming continuity of change in terms of non-­transitive indiscriminability. But enough has been done to give this phenomenon a certain dignity. It is not something confined to the psychological laboratory, comparable say, to people’s inability to judge the relative lengths of:

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62  The Riddle of Vagueness and

—something of which someone might reasonably require experimental confirmation that they too were subject to it. Granting the presuppositions of the above reasoning, we have rather to regard the non-­transitivity of the relation ‘is not discriminably different from’ as a reflection of a pervasive structural feature of our sense experience—the continuity of phenomenal change. The general lesson then of Example 4 is this. If we attempt to mark off regions of a seemingly-­continuous process of change in terms of predicates which are purely observational—predicates of which it is understood that ostensive definition gives their whole meaning—these expressions are bound to display tolerance in a suitable series of stages selected from the process. In his book on the Philosophy of Mathematics, Stephan Körner several times characterizes perceptual concepts as essentially inexact. This section can be regarded as an attempt to bring this insight into sharp relief. Absolutely any ostensively defined predicate may come to display tolerance, for absolutely anything which it characterizes might undergo seemingly-­continuous change to a point where it could be so characterized no longer. Example 4 is but a tremor, signalling a basic fault, as it were, buried deep in the relation between the nature of our experience and those parts of language by means of which we attempt to give the most direct, non-­theoretical expression to it. This conclusion rests upon two premises which might be held open to question: that it is right to regard the senses of colour predicates, and so on, as purely observational; and that this is a very fundamental fact about their senses, whose sacrifice would be possible only at great cost. For the first, no further argument will be provided in this chapter. The considerations adduced earlier in the section are surely decisive, provided it is allowed that they are relevant—provided the second thesis of the governing view has not been rejected. For the second, however, no argument has so far been presented; we merely voiced concern that ‘contact’ between language and the empirical world might be attenuated if the use of purely observational predicates was abandoned. No special considerations have been advanced in relation to Example 4 to correspond to the points about convenience, memorability, and social role in relation to the earlier examples. The matter must await the next but one section. First we must consider a general objection to our treatment of all four examples.

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On the Coherence of Vague Predicates  63

IV To concede that the vagueness of our examples is correctly interpreted as tolerance is to concede that there can be no consistent, non-­classical logic for such predicates.9 But it is natural to suggest that the arguments for this interpretation may have overlooked an essential feature of this sort of predicate: that they typically express distinctions of degree. There are degrees of redness, of childishness, and, if a smaller heap is regarded as less of a heap, of heaphood also. When is the distinction between being F and not being F one of degree? Typically, when the comparatives ‘is less/more F than’ are in use and when iteration of one of these relations may transform something F into something not F, or vice versa. Moreover, the semantic relations between the comparatives and the simple descriptions ‘is F ’ and ‘is not F ’ are such that, if a is less/more F than b, then the degree of justice with which a can be described simply as F is correlatively smaller or larger than that with which b can be so described. That is, a twofold classification of possible states of affairs into those which would justify the judgement ‘a is F ’, and those which would not, misses what is distinctive about a predicate whose application is a matter of degree. For that to be so is exactly for there to be degrees of such justice. It is thus plausible to suppose that a logic for distinctions of this sort cannot be based upon simple Bivalence. With such predicates, there are, as it were, degrees of truth whose collective structure is that of the set of degrees of being F. We shall not speculate what this structure might be, but it does not seem too fantastic to suppose that an indefinitely large or even densely ordered series might be involved, so issuing in infinitely many truth values. In this sense it is arguable that our examples do require a special—that is, non-­classical—logic. The view that classical logic is inadequate for distinctions of degree is not contested in this chapter. What is contested is the idea that the seeming tolerance of the examples is generated by overlooking that the predicates in question express distinctions of degree. This impression views the paradoxical reasoning as essentially depending upon the constraints of Bivalence—thus no attention is paid to the point that distinctions of degree are involved. Consider a pair of objects, one of which, a, we are happy to describe as F, while b is slightly less F than a. How is b to be described? If our admissible descriptions are restricted to ‘F ’ and ‘not-F ’, if we have to say one 9  At least, any consistent such logic will have to reject either modus ponens or universal instantiation.

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64  The Riddle of Vagueness or the other, then presumably we shall describe b as F. For, if something is more like something F than something not-F, to describe it as F is the less misleading of the two alternatives. But the justification with which ‘F ’ is applied in successive such cases successively decreases. We have no principle of the form: if a is F and b differs sufficiently marginally from a, then b is F; with distinction of degree, there are no ‘small changes insufficient to affect the ­justice with which a predicate applies’; they are, on the contrary, small changes in the degree of justice with which the predicate may be applied. Of course we do have the principle: if the judgement that a is F is justified to some large degree and b is marginally less F than a, then the description of b as F will be better justified than its description as not-F. But that is not a paradoxical principle. Anyone who thinks they at last feel the cool wind of sanity fanning their brow would do well to be clear why we do not still have this principle: if b is marginally less F than a, then, if the less misleading description of a is ‘F ’, the less misleading description of b is ‘F ’. Yet, if this principle is false, there must, in any Sorites-­type series, be a last case of which we are prepared to say that, if we had to describe it either as F or as not-F, the better description would be ‘F ’. Why, then, is it usually embarrassing to be asked to identify such a case without any sense of arbitrariness? Let us say that ‘a is F ’ has a positive value just in case ‘F’ is a less misleading description of a than ‘not-F ’. Then our embarrassment is exactly to identify a last object to which the application of F would receive a positive value. But then the suspicion arises that tolerance is with us still; only it is not now the truth of the application of F that would survive small changes but, so to speak, its positivity. Is this suspicion justified? One thing is clearly correct about the assumption of Bivalence: faced with a situation and a predicate, we have only two choices— to apply or to withhold. There is not a series of distinct linguistic acts in which we can reflect every degree of justification with which a predicate may be applied. The crucial notion to be mastered for practical purposes is thus that of a situation to which the application of F is on balance justified. Without mastery of this notion, no amount of information about the structure of variations in the degree with which F applies entails how the predicate is to be used. Now, of this notion, may it not still be a feature that it always survives sufficiently small changes—that if a and b are dissimilar only to some very small extent, then, if describing a as F is on balance justified, so is thus describing b? It is clear that all the considerations adduced in the previous sections now apply. The introduction of a complex structure of degrees of justification has

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On the Coherence of Vague Predicates  65 got us no farther; for among these we have still to distinguish those with which for practical purposes the application of the predicate is to be associated; otherwise we have not in repudiating Bivalence done anything to replace the old connection between justified assertion and truth. The distinction in question need not of course be exact (so one’s embarrassment at having to identify a last case to which the application of F is preferable to that of its negation is understandable). On the contrary, in the kind of case which we have been considering, the concept will be tolerant. To rehearse the reasons: if we are to be able to remember how to apply F, then differences too slight to be remembered cannot transform a situation to which its application is on balance justified into one which is not so; if we are to be able to apply F just on the basis of casual observation, the same applies to differences too subtle to be detected by casual observation; if the distinction between cases to which the application of F is on balance justified and others is to be made just on the basis of how things look, or sound, and so on, then any pair of indistinguishable situations must receive the same verdict; finally, if F is associated with moral or explanatory distinctions which we are unwilling to tie to very small changes, we shall likewise be unwilling to allow such changes to generate doubt about the status of a situation previously regarded as on balance justifying description as F. Of course, a quite uncritical use is here being made of the notion of a situation to which the application of F is ‘on balance’ justified. But this is legitimate. As remarked, there must be some such notion if a many-­valued logic for distinctions of degree is to have any practical linguistic application.

V We turn to the question whether we could not, at not too heavy a cost, ­eliminate the tolerance of observational predicates. What is necessary, it seems, is rejection of ostensively defined predicates; hence the initial doubt whether such a purified language could engage with the observational world at all. But, on reflection, it is clear that the dislocation of language and the world of appearance does not have to be as radical as that. When three situations collectively provide a counterexample to the transitivity of indiscriminability, there is nothing occult, as it were, in the circumstance that they do so. It is an observationally detectable difference between indiscriminable situations that one is distinguishable from a third situation from which the other is not; the  relation ‘a matches b matches c does not match a’ is an observational

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66  The Riddle of Vagueness relation—that is, one whose application to a trio of objects can be determined just by looking at them, listening to them, and so on. Observational concepts evidently require narrower criteria of reapplication than indistinguishability, if they are to be purified of tolerance. But we should not jump to the conclusion that to provide such criteria will require surrender of observationality altogether, for the phenomenon which is causing the trouble is itself observational. Indeed, the only kind of observationally detectable ­difference which there can be between indiscriminable items is that one should be distinguishable from some third item from which the other is not. So, if the class of expressions in question is to remain in contact with observation, we have to look for some form of stipulation which exploits the non-­ transitivity of indistinguishability to provide a basis for describing indiscriminable situations differently. No other explanation can correspond to a distinction which sense experience can determine to obtain, a distinction which we can simply be shown. One implication of this suggestion is already apparent. All along our discussion has centred on the use of certain predicates. Now there is a tradition, dating at least from Frege, in accordance with which a predicate of individuals is essentially an open sentence of only one individual argument place. The status of a predicate in this sense is indifferent to the occurrence of individual constants and bound variables within the expression, and so to the nature and scope of the procedure required to determine whether the predicate applies. But there is also the narrower traditional conception in whose terms predicates essentially express properties, whose application to an individual is a question of scope no broader than the boundaries, so to speak, of the individual concerned; it is this which distinguishes properties from relations. Such was the conception of quality to which Locke and Berkeley appealed. Of course, it may be appropriate to decide whether a property applies by comparison with cases where it does; but that only serves to emphasize that its application cannot always be a matter of comparison. What is the point of this distinction here? Simply that it is not coherent to demand more than indistinguishability as sufficient for reapplication of an observational quality. The question whether such a quality is shared by a pair of indistinguishable items cannot essentially turn on what relations they bear to other things; it has to be possible to decide whether an object has the quality by observation of it alone. So the envisaged kind of modification to the senses of observational predicates requires that we abandon their use as expressions of observational qualities. Indeed, the interest of the proposal depends upon our abandoning the notion of an observational quality

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On the Coherence of Vague Predicates  67 altogether; for, if there are such qualities, there can be no objection to introducing predicates to express them. This is not to say, of course, that language must cease to contain expressions whose syntax is that of a simple predicate—that is, expressions containing no singular term or quantifier and having but a single individual argument place. But, if the conditions of application of such an expression can be determined by observation, they will not be determinable by observation of a single ­individual. In a certain sense, the world of observation is to be a world of relations. The semantics of any observational predicate will be implicitly relational. The sense of this last claim becomes clearer if we now take note of a striking aspect of the philosophical psychology of non-­transitive matching. Summarily, it does not seem to be possible to conduct experiments with non-­transitively matching triads in memory. For, suppose that a predicate, F, is defined ostensively by reference to some individual, a, which, it is noted at the time, perfectly matches another individual, c; it is understood that F is not to be applied to individuals which match a unless they also match c. Later the trainee comes across b, which, so far as they can determine, matches a perfectly; the question is, does b match c? It is evident that the issue is only resoluble by direct comparison, and especially that it cannot be settled by memory, however accurate. For the most perfect memory of c can give no further information than that it looked just like a; which, when non-­transitive matching is a possibility, is simply insufficient to determine whether it would match b. This, it should be emphasized, in contrast with our conclusions concerning Example 3, is not a limitation imposed by the feebleness of our memories; it is a limitation of principle. So, if we are to be able to exercise expressions whose application to matching individuals depends upon their behaviour in relation to a third, possibly differentiating individual, it is clear that we have to be able to ensure the availability of the third individual. Expressions of this species will be practicably applicable only in relation to a system of paradigms. Thus we can see, even in advance of attempting a specific stipulation to remove the tolerance of ‘red’ as displayed in Example 4, that the kind of semantic construction it will have to  be is going to tie the application of expressions of colour to the use of a colour chart. Let us then consider, as a test case, how we might go about the construction of such a chart. What we require of the chart is that it should enable us to identify a last red patch in any series of the type of Example 4. How is this to be achieved? In The Structure of Appearance, Goodman (1951, ch. IX, sect. 3)

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68  The Riddle of Vagueness notes the difficulty that the non-­transitivity of matching provides for his concept of a quale: . . . this is somewhat paradoxical; for since qualia are phenomenal individ­ uals, we can hardly say that apparently identical qualia can be objectively distinct.

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

(The strategy is not original to Goodman. It was also adopted by Russell in Inquiry into Meaning and Truth and indeed there described as ‘familiar’ (Russell 1940, ch. VI).) Let us say that patches a and b are of the same Goodman-­Shade just in case they satisfy such a condition; that is, any c matches a if and only if it matches b. Then it would appear that an ideal colour chart—a Goodman Chart—would contain samples of all Goodman-­Shades, together with instructions about how each is to be described. Evidently, however, we must abandon this ideal straight away. We might chart a region of colour comprehensively in the sense of being confident that any sample of colour which we should intuitively regard as falling in that region would match something on the chart; but we could never have reason to be sure that every Goodman-­ Shade falling within that region was represented. For there is no way of foreclosing the possibility of finding a sample which is Goodman-­ distinguishable from every sample on the chart from which it is not simply distinguishable; that is, it matches something which such a chart sample does not, or vice versa. It is, indeed, a peculiarity of the notion of a Goodman-­Shade that, on a certain natural presupposition, a comprehensive Goodman Chart could not be achieved in principle. Imagine a band of colour varying uniformly in the following sense: that there is a constant distance, d inches, say, such that we can always distinguish the colours on the band more than d″ apart but can never distinguish those less than d″ apart. It follows that the colours at any distinct points on the band are Goodman-­distinct, since associated with any pair of distinct points is a point less than d″ from one and more than d″ from

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On the Coherence of Vague Predicates  69 the other. On such a band, the claim of identity of Goodman-­Shade at any distinct points is defeasible. There seems, a priori, no reason why there should not be a band of colour varying uniformly in this sense. But then there can no more be a specific, finite, comprehensive totality of samples of Goodman-­Shades than there can be such a totality of points in the linear continuum. The assumption that colour may vary uniformly thus generates a somewhat startling consequence for the notion of a Goodman-­Shade: single Goodman-­Shades do not require that regions in which they are exemplified should be extended. This is, perhaps, paradoxical only if, like Goodman, we suppose that these notions are still in any meaningful sense phenomenal. Besides, the kind of comprehensiveness which we were contemplating is not necessary anyway. What is necessary is the construction upon the set of Goodman-­Shades of an effective total order; such an order should, moreover, coincide with that generated by the intuitive relation of likeness among colour samples in any case where the latter issues a clear verdict. Given a decidable such order, there is no obstruction to stipulating intelligibly a determinate last red shade; and then we may in any series like that of Example 4 identify a last sample preceding or coinciding with this shade. Let us, then, consider what may be achieved in the direction of explaining such an effective order. Plainly, we have to be able to arrange a series of samples in such a way that we can tell of any new sample whether it falls between any pair of samples already located; such recognition will, of course, be crucial in cases where one of the samples is the last red shade. Conversely, if we are given the two outermost flanking samples, it will be sufficient for the task to have an effective notion of betweenness. When should a shade, b, come between a and c in an intuitively correct arrangement of colours? The notion we really need is that of b coming nearer to a than c does; for b comes between a and c just in case b is nearer to a than c is and nearer to c than a is. Plainly, a colour patch b should come nearer to a than c does if and only if it is more like a than c is. Let us consider then the construction of a Goodman Chart in terms of this basic principle; that sample b is to be placed nearer sample a than sample c is just in case it is more like a than c is. The principle has one very clear-­cut application: among a trio of non-­ transitively matching samples, the sample which matches the other two must be located between them. And surely there is this much to be said for it: if on a Heaven-­sent, correctly constructed Goodman Chart of sufficient degree of refinement of distinction we find a sample b located between a and c, that it is

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70  The Riddle of Vagueness so should be explicable by (iterated) application of the principle; if a comes to the left of b, then there should be something to the left of a which a matches and b does not, and so on. It is thus plausible to claim that the principle at least gives necessary and sufficient conditions for correctness of mutual location among samples in an existing Goodman Chart. This is not, of course, to say that it can serve as a principle of construction in an effective sense. We are, moreover, making a very substantial assumption. Suppose that a matches b matches c, while a and c do not match; what, then, if some d matches a and c but not b? The basic principle will require both that b and d  be placed between a and c and that a and c be placed between b and d, ­contrary to the supposition of total order. It is, thus, a necessary condition of the possibility of a total order among the relevant Goodman-­Shades in terms associating ‘is nearer to’ with ‘is more like to’ that this cyclic matching ­situation should not be a possibility. It will be assumed, since our main conclusions will be unaffected if it does not, that this necessary condition obtains. Suppose then that we have been given a red and an orange flanking sample, and have advanced, using the basic principle and the kind of ability on which Hume remarked, to a Goodman Chart in which every sample matches just those immediately adjacent to it and no others. Consider a section of the chart: . . . a;  b;  c;  d;  e;  f;  g;  h . . . Our task is to locate an x which matches both d and e; does it go between them? Evidently in two cases it does: the basic principle puts x between d and e if it matches both or neither of c and f. But suppose one of the other two cases obtains—for example, that x matches c but not f. Still, we may get a decision if x matches b, for then it must lie between c and d. But suppose x does not match b; suppose, in fact, that x and d match and are distinguishable from all the same samples located so far. Then all the basic principle tells us is that x must lie somewhere in the region between c and e. We have no way of locating x in relation to d on present information; apparently we need more samples on the chart. A sample between b and c, or between e and f, matching x or d but not both would do the trick. Yet, we have just noted that the question whether something goes between samples adjacent at this degree of construction is effectively decidable only in two cases of four; so we have no guarantee of being able to locate such additional samples; indeed, the spectre is introduced of reciprocally undecidable cases, where the location of x is decidable only by deciding that of y and conversely.

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On the Coherence of Vague Predicates  71 This spectre at least can be banished. The discovery of a y matching x but not d or vice versa would actually guarantee a solution to the question of their relative locations in the region between c and e.10 Still we cannot convincingly claim to have an effective principle of construction, since we have no effective means of finding such a y. There is a yet more radical deficiency. If x and d are as a matter of fact of the same Goodman-­Shade, then not only have we no effective way of determining that they are; we have no way of recognizing the fact at all. This is a fundamental point. Goodman, we noted earlier, congratulated himself on being able to explain a distinction between matching and identity among qualia without going beyond the realm of appearance. Certainly, the notion of Goodman-­identity is explained by means of a phenomenal relation and a 10  Proof: Let y match d but not x, and e but not c. Then, letting ‘M’ = ‘matches’, ‘M ’ = ‘does not match’, ‘a b c’ = ‘b is between a and c’: (i) y M d, d M c, y M c;  ∴ y d c; hence y is right of d. (ii) y M d, d M x, y M x;  ∴ y d x;  hence by (i), x is left of d. The case where y matches c but not e is quite similar. Suppose, on the other hand, that y differentiates x and d the other way about, i.e., that y matches x but not d. Let y again match e but not c. (The converse case is again essentially similar.) Then: (i) y M e, e M d, y M d;  ∴ y e d; hence y is right of d. (ii) y M x, x M d, y M d;  ∴ y x d;  hence by (i), x is right of d. So if a differentiating y matches only one of c and e, the question of the location of x in relation to d is soluble. What, though, if y matches both or neither of c and e? Somewhat surprisingly the basic principle, conceived as determining a total order, proves strong enough to rule out both cases. Again, let y match d but not x: Suppose y M c, y M e; then: (i) y M e, e M x, y M x;  ∴ y e x (ii) y M c, c M x, y M x;  ∴ y c x. Hence both c and e lie between y and x; so x cannot lie between c and e, contrary to the hypothesis that, save for y, x matches all and only the patches matched by d (whence c x e). Suppose alternatively y M c, y M e; then: (i) y M d, d M e, y M e;  ∴ y d e. (ii) y M d, d M c, y M c;  ∴ y d c. Hence y is required, absurdly, to lie both right and left of d. Now let y match x but not d: Suppose y M c, y M e; then: (i) y M c, c M d, y M d;  ∴ y c d. (ii) y M e, e M d, y M d;  ∴ y e d. Hence, absurdly, y is required to lie both left of c and right of e. Suppose y M c, y M e; then (i) y M x, x M c, y M c;  ∴ y x c. (ii) y M x, x M e, y M e;  ∴ y x e. Here (i) and (ii) are conjointly satisfiable only if both y and x lie to the left of the region between c and e; but x must lie within that region since, prior to the discovery of y, it matched and was distinguishable from the same samples as d.

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72  The Riddle of Vagueness quantification over ostensibly phenomenal individuals. Yet, evidently, it is only in a formal sense that we have not gone beyond appearance in this account; for, as Russell noticed, the resulting relation is not one which may appear to obtain. The concept of a Goodman-­Shade is not merely not an observational notion; it is a transcendental notion. Nothing counts as discovery of identity of Goodman-­Shade. This casts serious doubt on the suitability of the notion to serve in the construction of a semantic rule. For, surely, sense can be given to an expression only by reference to conditions whose satisfaction we can determine at least in principle. After a stipulation of the kind considered, the ability to recognize whether something was red would sometimes require the ability to recognize identity of Goodman-­Shade—namely, when what we have is in fact a sample of the last red shade. But this is an ability which we do not have. There is no effectively decidable order among Goodman-­Shades based on likeness. A fortiori, there is no effective stipulation of the sort considered: a stipulation based on the notion of a Goodman-­Shade and providing a sharp, decidable red/orange distinction in any series of the type of Example 4. The concept of the last red shade is underdetermined, for nothing amounts to ­recognition of which shade it is. Indeed, Goodman’s whole strategy for surmounting the difficulty of non-­transitive matching amounts to nothing other than the introduction of a spuriously phenomenal identity to which nothing in our experience can correspond. Not that Goodman introduced the notion with a view to providing the kind of stipulation which we are looking for; but we must look elsewhere. Could we not instead simply devise ad hoc paradigms? Consider a colour chart complete in the sense distinguished earlier; that is, we are confident that anything we should wish to regard as falling in the red/orange region will match something on the chart. It is plausible to suppose that likeness provides an effective order among any finite set of colour samples, so suppose the chart samples to be so arranged. Then we can generate a decidable red/orange distinction as follows. Select some patch towards the middle; then any colour patch matching something on the chart either matches the selected patch or it does not; if it does, it is red; if it does not but matches a sample to the left of the selected patch, it is red; otherwise it is orange. Naturally we could not guarantee that duplicates of a chart would always deliver the same verdict. Charts could look absolutely similar, and even satisfy the condition that the nth sample on either matched and was distinguishable from exactly the same samples on the other chart as its own nth sample, yet deliver discrepant results. But they would not often do so. Besides, the

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On the Coherence of Vague Predicates  73 situation is nothing new. Rulers, for example, sometimes give different results. A final criterion for one system is deposited in Paris; and we could do the same with a colour chart. Nevertheless the generalization of this proposal seems quite ludicrous in practical terms. We are confronted with the imaginary spectacle of a people quite lost without their individual wheelbarrow loads of charts, tape recordings, smell- and taste-­samples, and assorted sample surfaces. But this does the proposal an injustice. There will be no need for all this portable semantic hardware. To pursue the analogy with the use of rulers: it is true that, without a ruler, we can only guess at lengths; but after the introduction of an ad hoc paradigm for colours, the use of colour predicates will presumably be analogous not to that of expressions like ‘two feet long’, but rather to that of expressions like ‘less than two feet long’—that is, expressions of a range of lengths. Of such expressions, the criterion of application is still measurement; but, unless the case is peripheral, we can tell without measuring what the outcome of measurement would be. Training in the use of paradigms might be essential if one is to grasp the sense of such expressions; but, once grasped, most cases of practical application could be decided without the use of paradigms; for most practical purposes the wheelbarrow could be left behind. It appears, then, that, were we to adopt such stipulations as a general strategy, it would not have to affect our use of observational language very much at all. At present we can tell of anything red that it is so just by looking at it. This would still usually be true after the proposed stipulation; and, if the new distinction was suitably located, cases where it was not true could generally coincide with borderline cases of the old red/orange distinction. The use of predicates so refined could thus greatly resemble their present use; the distinctions which they expressed would be empirically decidable; and there would be one crucial disanalogy—they would be tolerance-­free. It is, indeed, apparent that exactly parallel considerations may be brought to bear upon our earlier treatment of Examples 1 and 3. Even after a precise redef­ inition of ‘heap’, we would be able to learn to tell in most cases just by casual observation what verdict the new criterion would give if applied; it would seldom be necessary actually to count the grains. And the distinction between red and orange, supposing an exact distinction were drawn by means of a chart, would be unmemorable only within that small range of shades which could not by unaided memory be distinguished from the last red sample. It would thus appear that the cost of eliminating tolerance in cases of these two types need not after all be high, since we could expect to be able to tell in general just by looking at, and so on, an item on which side of the dividing line it would fall.

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74  The Riddle of Vagueness If there need after all be no substantial sacrifice in endowing formerly observational predicates with exact boundaries, what has become of the alleged profound tension between phenomenal continuity and language designed to express how things seem to us? The answer is that we have simply swept it under the carpet. The possibility of our dispensing with paradigms for most practical purposes depends upon our capacity, for example, to distinguish between cases where we could tell whether or not ‘red’ applied just by looking and cases where we could not, where we should have recourse to a chart. If we are able to make such a distinction, what objection can there be to introducing a predicate to express it? But then, it seems, the semantics of this predicate will have to be observational. For on what other basis should we decide whether something looks as though comparison with a chart would determine it to be red than how it looks? Of any pair of colour patches which look exactly alike, if either looks as though the chart would deliver the verdict ‘red’, both must. So the new predicate, introduced to reflect our capacity to make this distinction, will be applicable to both members of any pair of matching colour samples if to either. Of course, there is no reason to have any such predicate; but equally there is no reason not to. If we were sometimes able to tell without using a chart whether something is red, it would surely be possible to make intelligible to us a predicate designed to apply in just such circumstances. A language all of whose observational concepts were based on paradigms would avoid containing tolerant predicates only by not containing means of expression of all the observational distinctions which we are in fact able to make. Naturally it would make no difference to this point if we insisted on actually using the paradigms in every case, and hang the inconvenience, as it were. We could dispense with them almost always, so we should just be insisting on a charade. Would it not, moreover, be quite absurd to propose that the tolerance of such new predicates—‘looks as though it would lie to the left of the last red shade’, ‘looks as though it contains fewer than ten thousand grains’, and so on—might in turn be stipulated away? Their meaning will not permit it; it cannot be allowed of things which look exactly alike that one may look as though it satisfies some condition which the other looks as though it does not—unless how a thing looks may not be determined by looking! But the earlier treatments of Examples 1 and 3 involved overestimation of our interest in preserving the tolerance of the predicates concerned only if we possess a coherent understanding of these new predicates; if Examples 1 and 3 do not, after all, pose a substantial problem for the governing view, it is because of our capacity to handle expressions falling within the scope of Example 4.

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On the Coherence of Vague Predicates  75

VI Conclusion Let us, then, review the character of the difficulty which Example 4 would appear to pose for the governing view. It is a fundamental fact about us that we are able to learn to classify items according to their appearance, that we are able consistently, as it seems to us, to apply or to withhold descriptions just on the basis of how things strike the senses. In a discrete phenomenal world there would be no special difficulty— no difficulty, that is, not inherent in the idea of a semantic rule as such—in viewing our use of such expressions as essentially nothing but the following of rules of which it was a consequence that indiscriminable phenomena should receive the same description. But, if mutually exclusive use is made of a pair of such predicates, and if cases to which one applies permit of continuous transformation into cases where the other applies, it cannot be correct to represent the use made of either predicate just as the doing of what is required by a set of rules with such a consequence. Yet we are constrained—if the relevance is allowed of considerations to do with what we should regard as adequate ­explanation of such expressions, or with certain criteria which we should accept of misunderstanding such an expression—to attribute to the rules governing these predicates precisely such an implication; and all the phenomena which we confront in our world impress us as capable of continuous variation. In the Introduction to this chapter, the difficulty was presented, starkly, as that of the inadequacy of any inconsistent set of rules to explain a consistent pattern of behaviour. This needs a little refinement. It is, to begin with, unclear how far our use of, for example, the vocabulary of colours is consistent. The descriptions given of awkward cases may vary from occasion to occasion. Besides that, the notion of using a predicate consistently would appear to require some objective criteria for variation in relevant respects among items to be described in terms of it; but what is distinctive about observational predicates is exactly the lack of such criteria. So it would be unwise to lean too heavily, as though it were a matter of hard fact, upon the consistency of our employment of colour predicates. What, however, may be depended upon is that our use of these predicates is largely successful; the expectations which we form on the basis of others’ ascriptions of colour are not usually disappointed. Agreement is generally possible about how colours are to be described; and this, of course, is equivalent to saying that others seem to use colour predicates in a largely consistent way.

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76  The Riddle of Vagueness It is of this fact which the governing view can provide no account. A semantic rule is supposed to contribute towards determining what is an acceptable use of its associated expression. The picture invoked by the first thesis of the governing view is that there is, for any particular expression in the language, a set of such rules completely determinant of when the expression is used correctly; such a set thus provides a model of the information of which a master of the use of the expression may be deemed to be in possession. Clearly, however, the feasibility of such a picture requires that the rules associated with an expression, about whose use we generally agree, be consistent. For, if they issue conflicting verdicts upon the correctness of a particular application of the expression, it cannot be explained just by appeal to the rules why we agree that the application is, for example, correct. The problem presented for the first thesis by the occurrence of tolerant predicates, or of any other kind of semantically incoherent expression, is not that, in a clear-­cut way, nothing can be done to implement an inconsistent set of instructions. It is true that, strictly, anything that is done will conflict with a part of them. But we can imagine a game whose rules conflict but which is nevertheless regularly and enjoyably played to a conclusion by members of some community because, for perhaps quite fortuitous reasons, whenever an occasion arises to appeal to the rules, the players concur about which element in the rules is to be appealed to, so that an impasse never comes about. We need not enquire whether they have noticed the inconsistency in the rules. The point of the analogy is that in practice they always agree whether a move is admissible, as we generally agree whether something is red. The analogue of the first thesis in relation to this example is the notion that the rules completely determine when a particular move is admissible. But, while it may be true that the authority of the rules can be cited for any of the moves the community actually makes, it is plain that the rules alone do not provide a satisfactory account of the practice of the game. For someone could master the rules yet still not be able to join in the game, because they was unable to guess what sort of eclectic application of them an opponent was likely to make in relation to any given move. An outsider attempting to grasp our use of a tolerant predicate would presumably not encounter exactly this difficulty; it would be clear that we were not prepared to allow remote consequences of its tolerance, inferred by means of reasoning of the Sorites type. But the difficulty of principle for the first ­thesis is the same. The rules of the game do not provide an account of how the game is played, for it is possible that someone might grasp them yet be unable to participate. The semantic rules for an expression are supposed to provide an

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On the Coherence of Vague Predicates  77 account of its correct use; they cannot do so if someone whose use of it differed radically from ours could still be thought of as in possession of exactly the same brief—as they can be if it consists in an inconsistent set of instructions. The comparison of language with a game is in many ways an extremely ­natural one. What better explanation could there be of our ability to agree in our use of language than if, as in a game, we are playing by the same rules? So we are attracted towards the assimilation of our situation to that of people to whom the practice of a game has been handed down via many generations but of which the theory has been lost. Our task is to infer the theory. The burden of this chapter, however, has been that this image is in conflict with the principles of investigation into the semantics of an expression which we find it natural to allow. These principles yield an account of the senses of certain vague predicates which pushes our agreement in their application beyond explanation by appeal to what the rules for their use require. And it is doubtful whether an intuitively satisfactory conception can be achieved of what an investigation into the semantics of an expression might be which did not admit these principles— unless the notion is abandoned that such an investigation is something which only a master of the investigated language is optimally placed to carry out. Such a conclusion would seem to force on us a more purely behaviouristic concept of how a theory of language use should be accomplished, and a corresponding shift in the concept of a semantic rule. There would no longer be any room for the idea that such rules might be discoverable by means of such a sort that our use of the associated expression(s) could prove to be in conflict with or otherwise inexplicable by reference to them; there would no longer be any room for the idea that by reflection on the kind of training which we have received in the use of some expression, the criteria which we would employ to judge that someone misunderstands it, their concept of the purpose or interest of the classification which it effects, or their awareness of their own intellectual and perceptual limitations, speakers of a language have access to sound conclusions about their understanding of an expression which a mere observer of their use of it has not. This would be one response to the difficulties generated by the richer, more natural methodology of the governing view. It can be expected to encounter severe difficulties of its own: in particular, it is unclear that an adequate characterization can be achieved from so restricted a standpoint of in what the vagueness of the sorts of example with which we have been concerned consists. But, if the notion of a semantic rule is not to be abandoned altogether, some more restricted account of the epistemology of semantic rules is required than that afforded by the governing view.

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2 Language-­Mastery and the Sorites Paradox* I Throughout Frege’s writings are scattered expressions of the conception that the vagueness of ordinary language, and especially the occurrence of predi­ cates for which it is not always determinate whether or not they may truly be applied to an object, is a defect. His reason for such a view seems to have been that orthodox logical principles fail when applied to sentences containing expressions whose range of application has been defined only partially.1 Thus Frege seems not to have considered, or not to have thought worth consider­ ing, the possibility that vague terms might require a special logic. Vagueness is rather something which can and should be expurgated from language, if it is to be suitable for ‘scientific purposes’. The same conception is to be found in Russell’s introduction to the Tractatus. Ordinary language is always more or less vague, but a logically perfect language would not be vague at all; so the degree of vagueness of a natural language is a direct measure of its distance from being everything which it ‘logically’ ought to be. Of course, we have since learned a greater respect for language as we find it; we no longer regard the vagueness of ordinary language as a defect. But a higher-­order analogue of the Frege–Russell view continues to figure in our thinking about language: even if many predicates in natural language are vague, there can still be a precise semantics for such expressions and indeed for the whole language—that is, a theoretical model of the information as­simi­lated in learning it as a first language or, equivalently, of the conceptual apparatus possession of which constitutes mastery of the language. There need be no imprecision, it seems, in such a model; at any rate, none occa­ sioned purely by the vagueness of the expressions of mastery of whose senses it is to provide an account. *  This discussion is a synopsis of, or, better, a series of excerpts from my ‘On the Coherence of Vague Predicates’ (Wright 1975, this volume, Chapter 1). 1  Excluded Middle is the obvious example. But, as Frege points out, contraposition also fails (Frege 1903, p. 65).

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0003

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80  The Riddle of Vagueness We tend to picture our use of language as something essentially regular. We tend to think of language-­learning as ingestion of a set of rules for the com­ bin­ation and application of expressions. Thus the task of a philosophical the­ ory of meaning, in one natural sense of that phrase, would be to give a systematic account of the contribution made by the constituents of a se­man­ tic­al­ly complex expression to its overall sense; and the theory would be con­ cerned especially with the epistemology of the transition from understanding of subsentential components of a new sentence to recognition of the sense of the whole. Such a philosophical theory will normally only be concerned with types of contribution made by constituent expressions; in just this connec­ tion arise the familiar questions concerning the nature of the distinction between proper names and other singular terms, between singular terms generally and predicative expressions, whether the notion of reference may illuminatingly be extended to predicative expressions, and so on. So the completion of such a theory would only be a preliminary to what we think of as a full semantic description of a natural language; for it is not just the type of contribution but the specific contribution which a constituent expression makes to complex expressions containing it which we think of as determined by rule. It is worth emphasizing that no obstacle to such a conception is posed by the fact that we cannot in general state such rules in such a way as to explain the sense of an expression to someone previously unfamiliar with it. Consider a schematic rule for a one-­place predicate, F: F may truly be applied to an individual, a, if and only if a satisfies the ­condition of being φ. How should we specify φ if F is ‘red’? Clearly the only completion of the rule which is actually constitutive of our understanding of ‘red’, rather than a mere extensional parallel, is to take φ as the condition of being red. In general we cannot expect instances of such a schematic rule to be of explanatory use if they are stated in a given language for a predicate of the same language; in consequence, it will not generally be possible to appeal to such rules to settle questions about the applicability of an expression. Nevertheless, we may still legitimately regard such a rule as an exact expression of (part of) what is understood by someone who understands, for example, ‘red’, for it states con­ ditions recognition of which is sufficient to justify them in describing an object as ‘red’; the statement of the rule is uninformative only in the sense that such a capacity of recognition cannot be imparted just by stating it.

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Language-Mastery and the Sorites Paradox  81 So our picture is that correct use of language is essentially nothing other than use of it which conforms with a set of instructions, a set of semantic rules, which we have learned. Of course we handle language in general in a quite automatic way. But a chess player’s recognition of the moves allowed for a piece in a certain position can be similarly automatic; it remains true that an account of their knowledge is to be given by reference to the rules of chess. If language-­mastery is thought of in such terms, the question arises, what means are allowable in the attempt to discover general features of the substantial rules for expressions in our language, the rules which determine specific senses? The view of the matter on which this chapter centres is that here we may legitimately approach our use of language from within—that is, re­flect­ ive­ly as self-­conscious masters of it—rather than externally, equipped only with behavioural notions. Thus it is legitimate to appeal to our conception of what justifies the application of a particular expression; to our conception of what we should count as an adequate explanation of the sense of a particular expression; to the limitations imposed by our senses and memories on the kind of instruction which we can actually carry out in practice; and to the kind of consequence which we associate with the application of a particular predicate, to what we think of as the point or interest of the distinction which the predicate implements. The primary concern of this chapter is with the idea, henceforward referred to as the governing view, that from such consid­ erations can be derived a reflective awareness of how we understand expres­ sions in our language, and so of the nature of the rules which determine their correct use. The governing view, then, is a conjunction of two claims: that our use of language is rightly seen, like a game, as a practice in which the admis­ sibility of a move is determined by rule, and that general properties of the rules may be discovered by means of the sorts of consideration just described. What I am going to argue is that these theses are mutually incoherent. The difficulty has to do with the fact that the second thesis of the governing view, concerning the means whereby general features may be discovered of the semantic rules which we actually follow, forces us to recognize semantic incoherence in our understanding of a whole class of predicates—elements whose full exploitation would force the application of these expressions to situations in which we should otherwise regard them as not applying. The second thesis requires us to recognize rules which, when considered in con­ junction with certain general features of the situations among which their associated expressions are to be applied, issue in contradictory instructions. Nevertheless we succeed in using these expressions informatively; and it seems that to use language informatively depends on using it, in large

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82  The Riddle of Vagueness measure, consistently. It follows that our use of these expressions cannot cor­ rectly be pictured purely as the implementation of rules of the character which the second thesis yields for them; these rules cannot be implemented by any consistent pattern of behaviour. The governing view is, therefore, inco­ herent; for, if its second thesis is true, the semantic rules governing certain predicates are capable by consistent beings only of selective implementation and thus, contrary to the first thesis, are not constitutive of what we count as the correct use of these expressions. The predicates in question are all vague; but their vagueness is not just a matter of the existence of situations to which it is indeterminate whether or not they apply. Rather it is something which Frege, under the guise of a favourite metaphor, constantly runs together with possession of borderline cases—namely, the idea of lacking ‘sharp boundaries’, of dividing logical space, as a blurred shadow divides the background on which it is reflected. The conflation is natural, because the figure equally exemplifies the idea of the borderline case, a region falling neither in light nor in shadow. But there is no clear reason why possession of borderline cases should entail possession of blurred boundaries. If, following Frege, we assimilate a predicate to a function taking objects as arguments and yielding a truth value as value, then a predi­ cate with borderline cases may be seen simply as a partial such function— which is consistent with the existence of a perfectly sharp distinction between cases for which it is defined and cases for which it is not. Borderline-­case vagueness of this straightforward kind presents no difficulty for the governing view; it is merely that there are situations to which no response in terms of a certain range of predicates is determined by their associated semantic rules as correct. In contrast, if the second thesis of the governing view is correct, then predicates with ‘blurred boundaries’ are, in typical cases, rightly regarded as semantically incoherent. This incoherence is implicit in the very nature of their vagueness. Vagueness is hardly ever, as Frege and Russell thought, merely a reflection of our not having bothered to make a predicate precise. Rather, the utility and point of the classifications expressed by many vague predicates would be frustrated if they were supplied with sharp boundaries. The sorts of argument allowed by the second thesis of the governing view will transpire to yield support for the idea that such predicates are essentially vague. The thesis equips us to argue that lack of sharp boundaries is not in general merely a superficial phe­nom­ enon, a reflection of a mere hiatus in some underlying set of semantic rules. In almost all the examples one comes across, lack of sharp boundaries is not the consequence of an omission, but, for example, a product of the kind of

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Language-Mastery and the Sorites Paradox  83 task to which an expression is put, the kind of consequences which we attach to its application or, more deeply, the continuity of a world which we wish to describe in purely observational terms. Lack of sharp boundaries is a phe­ nomenon of semantic depth. It is not usually a matter simply of our lacking an instruction where to ‘draw the line’; rather the instructions we already have determine that the line is not to be drawn. This conclusion might seem a welcome contribution to our understanding of the nature of vagueness, even from the standpoint of the governing view, were it not that it comes out in the form that no sharp distinction may be drawn between cases where it is definitely correct to apply such a predicate and cases of any other sort. But that is obviously a paradoxical concept. Thus it is that someone who espouses the governing view simply has no coherent approach to the Frege–Russell view of vagueness. Their second thesis fur­ nishes them with conclusive reasons to reject the suggestion that vagueness is a superficial, eliminable aspect of natural language with no real impact upon its informative use. But it does so in such a way that they are constrained to regard many vague predicates as semantically incoherent—specifically, as prone to the reasoning of the Sorites paradox—so that, unless the Frege– Russell view is right, they cannot maintain their first thesis with respect to such expressions. Only if their vagueness were an incidental feature could they maintain that the essential semantics of such expressions conformed to their first thesis.

II Let us then consider some examples of the Sorites paradox in order to be clear how the governing view cuts off traditional lines of solution, indeed, all lines of solution. To begin with the classical case: if a pile of salt is large enough to be fairly described as a heap, the subtraction of a single grain of salt cannot make a relevant difference; if n + 1 grains of salt constitute a heap, so do n grains. Thus, one grain, and, indeed, zero grains, constitute a heap. To block the paradox, it seems we have to be able to insist that, for some particular value of n, n + 1 grains of salt would amount to a heap while n grains would not. But that is simply not the sense of ‘heap’. Exact boundaries for the con­ cept of a heap, either in terms of the precise number of grains contained or, indeed, in terms of any other precise measure, simply have not been fixed. But without such boundaries, a transition from n + 1 grains to n grains can never be recognized as transforming a case where ‘heap’ applies into a case where it

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84  The Riddle of Vagueness does not. Here we gravitate towards the idea that lack of exact boundaries is, as such, an essentially incoherent semantic feature. A second example is given by Essenin-­Volpin (1961, p. 203). Consider the typical span of time between one human heartbeat and its successor. Then the concept of childhood—the sense of ‘child’—is such that one does not, within a single heartbeat, pass from childhood to adolescence. Not that we are chil­ dren for ever; but at least childhood does not evaporate between one pulse and the next. Similarly, for the transition from infancy to childhood, and from adolescence to adulthood. ‘Infant’, ‘child’, ‘adolescent’, ‘adult’ are thus all semantically incoherent expressions; for the sense of each of these predicates is such that, in a typical process of growing up, their correct application will always survive the transition from one heartbeat to its successor or to its pre­ decessor. So, by appropriately many steps of modus ponens, we may force the application of each of these predicates to cases we should otherwise regard as falling within the domain of one of the others. As a third example, consider a series of homogeneously coloured patches, ranging from a first, red patch to a final, orange one, such that each patch is just discriminable in colour from those immediately next to it, and is more similar in respect of colour to its immediate neighbours than to any other patches in the series. That is, marginal changes of shade are involved in every transition from a patch to its successor, and each such transition carries us further from red and closer to orange. Now, the sense of colour predicates is such that their application always survives a very small change in shade. If one is content to call something ‘red’, one will still be so content if its colour changes by some just discriminable amount. There is a notion of a degree of change in respect of colour too small to amount to a change of colour. Only if a substantial difference comes between two patches of colour shall we con­ sider ourselves justified in ascribing to them incompatible colour predicates. This, obviously, is to attribute semantic incoherence to colour predicates. We have an easy proof that all the patches in the example are red, or that they are all orange, or that they are all doubtfully either. Moreover, any two colours can be linked by such a series of samples; so any colour predicate can be exported into the domain of application of one of its rivals. What is involved in treating these examples as genuinely paradoxical is a certain tolerance in the concepts which they respectively involve, a notion of a degree of change too small to make any difference, as it were. The paradoxical interpretations postulate degrees of change in point of size, maturity, and ­colour which are insufficient to alter the justice with which some specific predicate of size, maturity, or colour is applied. This is quite palpably an

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Language-Mastery and the Sorites Paradox  85 incoherent feature, since, granted that any case to which such a predicate applies may be linked by a series of ‘sufficiently small’ changes with a case where it does not, it is inconsistent with there being any cases to which the predicate does not apply. More exactly, suppose φ to be a concept related to a predicate, F, as follows: that any object which F characterizes may be changed into one which it does not simply by sufficient change in respect of φ. Colour, for example, is such a concept for ‘red’, size for ‘heap’, degree of ma­tur­ity for ‘child’, number of hairs for ‘bald’, and so on. Then F is tolerant with respect to φ if there is also some positive degree of change in respect of φ insufficient ever to affect the justice with which F applies to a particular case. In essentials, then, the Sorites paradox interprets certain vague predicates as tolerant. But this might seem a tendentious interpretation. Not that there is any doubt that the predicates in question do lack sharp boundaries; and the antiquity of the paradox bears witness to how easy it is to interpret this as involving the possession by these predicates of a principle of reapplication through marginal change. But is this a correct interpretation? Because ‘heap’ lacks sharp boundaries, it is plain that we are not entitled to single out any particular transition from n to n–1 grains of salt as being the decisive step in changing a heap into a non-­heap; no one such step is decisive. That, however, is not to say that such a step always preserves application of the predicate. Would it not be better to assimilate the situation to that in which bordering states fail to agree upon a common frontier? Their failure to reach agreement does not vindicate the notion that, for example, a single pace in the direction of the other country always keeps one in the original country. For they have at least agreed that there is to be a border, that some such step is to be a decisive one; what they have not agreed is where. If we regard the predicates in the example in the terms of this model, we shall conclude that their vagueness is purely a reflection of our intellectual laziness. We have, as it were, decided that a disjunction is to be true—at some stage, n grains will be a heap where n – 1 grains will not—without following up with a decision about which dis­ junct is true. On this view, the notion that these predicates are tolerant con­ fuses a lack of instruction to count it the case that a proposition is false with the presence of an instruction to count it true. This conflation would be per­ missible only if the semantic rules for our language were in a certain sense complete—that is, if we possessed instructions for every conceivable situ­ ation. But for there to be vague expressions in our language is, on this view, precisely for this not to be so. Someone who holds the governing view is bound to reject this suggestion as a deep misapprehension of the nature of the vagueness of these predicates.

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86  The Riddle of Vagueness The lack of sharp boundaries possessed by these examples is correctly inter­ preted as tolerance, provided that we may discover elements of their senses in accordance with the second thesis. It would be inconsistent with elements already present in the semantics of these predicates so to refine their senses that the Sorites reasoning was blocked. How is this? ‘Heap’ is essentially a coarse predicate whose application is a matter of rough-­and-­ready judgement. We should have no use for a precisely demar­ cated analogue in contexts in which the word is typically used. It would, for example, be ridiculous to force the question of obedience to the command ‘pour out a heap of sand here’ to turn on a count of the grains. Our concep­ tion of the conditions which justify calling something a heap is such that the appropriateness of the description will be unaffected by any change which cannot be detected by casual observation. A different argument is available for supposing colour predicates tolerant with respect to marginal changes in shade. We learn and teach our basic ­colour vocabulary ostensively. Evidently it is a precondition of the feasibility of so doing that we can reasonably accurately remember how things look. Imagine someone who can recognize whether simultaneously presented objects match in colour, so that they are able to use a colour chart, but they cannot in general remember shades of colour sufficiently well to be able to handle without a chart colour predicates for which we are able to dispense with charts. Such people might, for example, be quite unable to judge whether something yellow, which they were shown earlier, would match the orange object now before them. Thus, for such people, an ostensive definition of ‘yel­ low’ would be useless; in order to apply ‘yellow’ as we apply it, they would have to employ a chart. We, in contrast, are able to dispense with charts for the purpose of making distinctions of colour of the degree of refinement of ‘yellow’. Any object to which a colour predicate of this degree of refinement definitely correctly applies may be recognized as such just on the basis of our ostensive training. Plainly, then, it has to be a feature of the senses thereby bestowed upon these predicates that changes too slight for us to remember— that is, a change such that exposure to an object both before the change is undergone and afterwards leaves us uncertain whether the object has changed, because we cannot remember sufficiently accurately how it was before—never transform a case to which such a predicate applies into one where such is not definitely the right description. The character of our basic colour training presupposes the total memorability of the distinctions expressed by our basic colour predicates; only if single, unmemorable changes of shade never affect the justice of a particular basic description can the senses

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Language-Mastery and the Sorites Paradox  87 of these predicates be explained entirely by methods reliant upon our capacity to remember how things look. For the tolerance of ‘child’, and so on, the governing view affords a third type of argument. The distinctions expressed by these predicates are of sub­ stantial social importance in terms of what we may appropriately expect from, and of, persons who exemplify them. Infants, for example, have rights but not duties, whereas of a child outside infancy we demand at least a rudimentary moral sense; we explain the anti-­social behaviour of some adolescents in terms of their being adolescents; and we make moral and other demands of character on adults which we would not impose on the immature. Plausibly, these predicates could not endure such treatment, were they not tolerant with respect to marginal changes in degree of maturity—certainly with respect to the changes involved in the transition from one heartbeat to the next. It is ceteris paribus irrational and unfair to base substantial distinc­ tions of right and duty on marginal differences; if we are forced to do so, for example, with electoral qualifications, it is with a sense of injustice. Moreover, it is only if a substantial change is involved in the transition from childhood to adolescence that we can appeal to this transition to explain substantial alterations in patterns of behaviour. That predicates of degree of maturity should possess tolerance is a direct consequence of their social role; very small differences cannot be permitted to generate doubt about their application without correspondingly coming to be associated with a burden of moral and explanatory distinctions which they are too slight to carry. On the second thesis of the governing view, then, our embarrassment about where to ‘draw the line’ with these examples is to be viewed as a consequence not of any hiatus in our semantic programme but of the tolerance of the predi­cates in question. If casual observation alone is to determine whether a predicate applies, then items not distinguished by casual observation must receive the same verdict. So single changes too slight to be detected by casual observation cannot be permitted to generate doubt about the application of such a predicate. Similarly, if the conditions under which a predicate applies are to be generally memorable, it cannot be unseated by single changes too slight to be remembered. Finally, very slight changes cannot be permitted to generate doubt about the application of predicates of maturity without contra­ ven­ing their moral and explanatory role. The utility of ‘heap’, the mem­or­abil­ ity of the conditions under which something is ‘red’, the point of ‘child’ impose upon the semantics of these predicates tolerance with respect to marginal change in the various relevant respects.

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88  The Riddle of Vagueness To allow these considerations is to concede that the vagueness of these examples is a phenomenon of semantic depth—that it is sacrificed at much more than the cost of the intellectual labour of the stipulation—and that it is a structurally incoherent feature. Two things follow. First, there is no special logic for predicates of this sort, crystallizing what is distinctive in their semantics in contrast with those of exact predicates; for what is so distinctive is their inconsistency. Second, the fashion in which we typically use these expressions needs some other model than the simple implementation of rules, if these rules are to incorporate all the features of their senses which we should wish to recognize on the basis of the second thesis.

III There is a fourth and more profound, way in which tolerance, according to the second thesis, would seem to arise. Colour predicates will again serve as an illustration. Plausibly, these predicates are in the following sense purely observational: if it is possible to tell at all what colour something is, it can be told just by looking. The look of an object decides its colour, as the feel of an object decides its texture, or the sound of a note its pitch. The information of one or more senses is decisive of the applicability of an observational predi­ cate; so a distinction exemplified by a pair of sensorily equivalent items can­ not be expressed solely by means of such predicates. What is about to be illustrated is a feature of any predicate whose sense is purely observational in the fashion just adumbrated. If colour predicates are observational, any pair of patches indistinguishable in colour must satisfy the condition that any colour predicate applicable to either is applicable to both. Suppose, then, that we build up the series of ­colour patches of the third example, interposing new patches to the point where every patch in the resultant series is indiscriminable in colour from those immediately adjacent to it. The possibility of doing so, of course, depends upon the non-­transitivity of our colour discriminations. The obser­ vationality of ‘red’ requires it to be tolerant with respect to the kind of change involved in passing from any patch in this series to an immediate neighbour, so we have a Sorites paradox. If ‘red’ is observational, its sense must be such that, from the premises, that x is red and that x looks just like y, it follows that y is red, no matter what objects x and y may be. Thus we are equipped to conclude that each successive patch in the series is red, given only the true premise that the first patch is red.

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Language-Mastery and the Sorites Paradox  89 The memorability, then, of the conditions of application of ‘red’ requires that it be tolerant with respect to changes which, under favourable circum­ stances, we can actually directly discern. Now, however, it appears that, even if our memories were to be as finely discriminating as our senses, colour predicates and others would still possess tolerance; only the changes which their application tolerated would not be changes which we could directly dis­ cern in objects which underwent them. This tolerance has nothing to do with the limitations of our memories; it is a consequence of the observationality of these predicates. These considerations are broadly analogous to what was said of ‘heap’: if we so fix the sense of a predicate that whether it applies has to do with nothing other than how an object seems when casually observed, then changes other than such as can be determined by casual observation cannot transform a case to which the predicate applies into one to which there is some question whether it applies. The point remains good if we omit the word ‘casual’. But this fourth example is, prima facie, deeper reaching, at any rate for someone who, like Frege, believes that language should be purified of vague expres­ sions. The cost of eliminating predicates of casual observation would be no more than convenience; to require, however, that language should contain no expression tolerant in the manner of the fourth example would be to require that it contained no expressions of strictly observational sense. If we stipu­ lated away the tolerance of colour predicates, we should have to forgo our whole present idea of what justifies the application of these predicates— namely, the look of a thing. In general, there would be no predicate whose application to an object could be decided just on the basis of how it looked, felt, sounded, and so on. Might there not then be a higher price to pay— namely, the jeopardizing of contact between language and empirical reality? We shall return to the last thought. First, we require to see how the govern­ ing view sustains the idea that there is a large class of predicates whose senses are purely observational. If we are to understand the scope of the fourth example, we also require to know under what circumstances we may expect our sensory discriminations to be non-­transitive. That we do intuitively regard the semantics of colour predicates as purely observational is beyond doubt; and simply illustrated by the fact that we should regard it as a criterion of lack of understanding of such a predicate if someone was doubtful whether both of a pair of objects which they could not tell apart should receive the same description in terms of it. We regard it as a criterion of understanding such a predicate that someone, presented under suitable conditions with an object to which it applies, can tell that it does so

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90  The Riddle of Vagueness just on the basis of the object’s appearance. Certainly, then, our ordinary ­conception of how to tell that a particular colour predicate applies, of what justifies its application, would involve that these predicates are purely observational. In addition, it is plausible to suppose that any ostensively definable predi­ cate must be observational. If an expression can be ostensively defined, it must be possible to draw to someone’s attention those features in their experi­ ence which warrant its application; and, if this is possible, there can be no question of the expression applying to some but not others among situations which they cannot distinguish in experience. It would be a poor joke on the recipient of an ostensive definition if the defined expression applied se­lect­ ive­ly among situations indistinguishable from one which was originally dis­ played to them as a paradigm. In general, the connection between an expression’s being observational— its applying to both, if to either, of any pair of observationally in­dis­tin­guish­ able situations—and its being ostensively definable is as follows. The picture of acquiring concepts by experience of cases where they do apply and cases where they do not—a picture which surely has some part to play in a philo­ sophically adequate conception of the learning of a first language—cannot be wholly adequate for concepts which differentiate among situations which look, feel, taste, sound, and smell exactly alike. So, if that picture is wholly adequate for any concepts, they must be concepts whose range of application does not include situations which experience cannot distinguish from situ­ ations which may not definitely correctly be regarded as falling within that range. To master the sense of a predicate is, at least, to learn to differentiate cases to which it is right to apply it from cases of any other sort. If such mas­ tery can be bestowed ostensively, a comparison of two such cases must always reveal a difference which sense experience can detect. The notion that the whole range of application of a predicate can be made intelligible by ostensive means presupposes that it is never the case that only one of a pair of objects, which the senses cannot tell apart, is characterized by it. This is a clear, absolutely general connection. If there is in the conditions of the correct application of a predicate nothing which is incapable of ostensive communication, then the predicate must apply to both, if to either, of any pair of indistinguishable objects. But it seems manifest that adjectives of colour, and many others, do precisely not involve any such further condition of ­correct application; on the contrary, ostensive training would appear fully determinant of their meaning—or, if it is not, it is the only training which we get. The governing view thus vindicates the observationality of colour predicates

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Language-Mastery and the Sorites Paradox  91 twice over: as a consequence both of our general conception of what justifies their application, and of the character of the training in their use which we receive. The other question was to do with the scope of the phenomenon of non-­ transitive indiscriminability. Suppose that we are to construct a series of ­colour patches, ranging from red through to orange, among which indiscrim­ inability is to behave transitively. We are given a supply of appropriate patches from which to make selections, an initial red patch, and the instruction that each successive patch must either match its predecessor or be more like it than is any other patch not matching it which we later use. Under these con­ ditions it is plain that we cannot generate any change in colour by selecting successive matching patches; if indiscriminability is to be transitive, then if each patch in the first n selections matches its predecessor, the nth selection must match the first patch. The only way to generate a change in colour will be to select a non-­matching patch. When the series is complete, how will it look in comparison with the series of the fourth example? It is clear that we shall have lost what was distinctive of that series: the appearance of continuous change from red to orange. In the new series, the shades are exemplified in discrete bands, containing perhaps no more than one patch, and all the changes take place abruptly in a transi­ tion from a patch to its successor. It thus appears that, were our judgements of indiscriminability to be universally transitive among samples of homo­ge­ neous colour, no field of colour patches could be ordered in the distinctive fashion now possible: that is, so as to give the impression of a perfectly smooth change of colour. If matching generally behaved transitively among shades, no series of colour patches could give the impression of continuous trans­ form­ation of colour; by contraposition, then, for matching to function non-­ transitively among a finite set of colour patches, it is sufficient that they may be arranged so as to strike us as forming a phenomenal continuum. This reason­ing may obviously be generalized. Any finite series of objects, none of which involves any apparent change in respect of φ, may give an overall impression of continuous change in respect of φ only if indiscriminability functions non-­transitively among its members. The reasoning may in fact be generalized further. It can be shown (cf. Wright 1975, this volume, Chapter 1) that the non-­transitivity of our discrim­ inations may be seen as a consequence of the continuity of change, viewed as a pervasive structural feature of our sense experience. The general lesson of the fourth example is thus as follows. If we attempt to mark off regions of a seemingly-­continuous process of change in terms of predicates which are

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92  The Riddle of Vagueness purely observational—predicates of which it is understood that ostensive ­definition gives their whole meaning—these expressions are bound to display tolerance in a suitable series of stages selected from the process. An analogue of the fourth example may thus in principle be constructed for any ostensively defined predicate; for absolutely anything which it characterizes might undergo seemingly-­continuous change to a point where it could be so charac­ terized no longer. The fourth example indicates a basic fault, as it were, lying deep in the relation between the nature of our experience and those parts of language by means of which we attempt to give the most direct, non-­ theoretical expression to it. This conclusion rests upon two assumptions: that it is right to regard the senses of colour predicates, and so on, as purely observational; and that this is a very fundamental fact about their senses, whose sacrifice would be possible only at great cost. The governing view, as we have seen, yields the first assump­ tion. For the second, however, no argument has so far been presented; I  merely voiced concern that ‘contact’ between language and the empirical world might be attenuated if the use of purely observational predicates was abandoned. Before this concern is evaluated, and the general implications assessed of stipulating away the tolerance, and so the observationality, of the relevant predicates, we must consider a general objection to the way in which all four examples have been treated.

IV If it is conceded that the vagueness of these examples is correctly interpreted as tolerance, then plainly no consistent logic does justice to the semantics of such predicates. It is natural to suggest, however, that the argument for this interpretation may have overlooked an essential feature of this sort of predi­ cate: that they typically express distinctions of degree. There are degrees of redness, of childishness, and, if a smaller heap is regarded as less of a heap, of heaphood. What is it for the distinction between being F and not being F to be one of degree? Typically, it is required that the comparatives ‘is less/more F than’ are in use and that iteration of one of these relations may transform something F  into something not-F, or vice versa. In addition, the semantic relations between the comparatives and the simple descriptions, ‘is F’ and ‘is not-F’, are such that, if a is less/more F than b, then the degree of justice with which a can be described simply as F is correlatively smaller or larger than that with

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Language-Mastery and the Sorites Paradox  93 which b can be so described. That is, a twofold classification of possible states of affairs into those which would justify the judgement ‘a is F’ and those which would not misses what is distinctive about the predicate whose appli­ cation is a matter of degree. For that to be so is exactly for there to be degrees of such justice. It is thus plausible to suppose that a logic for distinctions of this sort cannot be based upon simple Bivalence. With such predicates there are, as it were, degrees of truth, whose collective structure is that of the set of degrees of being F. In this sense it is arguable that the examples do require a special, non-­classical logic. But how did the earlier arguments for the tolerance of these predicates overlook that they expressed distinctions of degree? The suggestion is that the paradoxical reasoning essentially depends upon the constraints of Bivalence. Consider a pair of objects one of which, a, we are  happy to describe as F, while b is slightly less F than a. How is b to be described? If our admissible descriptions are restricted to ‘F’ and ‘not-F’, if we have to say one or the other, then presumably we shall describe b as F. For, if something is more like something F than something not-F, to describe it as F is the less misleading of the two alternatives. But the justification with which ‘F’ is applied in successive such cases successively decreases. We have no prin­ ciple of the form: if a is F, and b differs sufficiently marginally from a, then b is F; with distinctions of degree, there are no ‘small changes insufficient to affect the justice with which a predicate applies’; they are, on the contrary, small changes in the degree of justice with which the predicate may be applied. Of course, we do have the principle: if the judgement that a is F is justified to some large degree, and b is marginally less F than a, then the description of b as F will be better justified than its description as not-F. But that is not a para­ doxical principle. Anyone who thinks they here feel the cool wind of sanity fanning their brow would do well to be clear why we do not still have this principle: if b is marginally less F than a, then, if the less misleading description of a is ‘F’, the less misleading description of b is ‘F’. Yet, if this principle is false, there must in any Sorites-­type series be a last case of which we are prepared to say that, if we had to describe it either as F or as not-F, the better description would be ‘F’. Why, then, is it usually embarrassing to be asked to identify such a case without any sense of arbitrariness? Let us assign to ‘a is F’ a designated value just in case ‘F’ is a less misleading description of a than ‘not-F.’ Then our embarrassment is exactly to identify a last object to which the application of ‘F’ would receive a designated value. But now the suspicion arises that tolerance is with us still; only it is no longer

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94  The Riddle of Vagueness the truth of the application of ‘F’ that would survive small changes but its designatedness. Is this suspicion justified? One thing is clearly correct about the assump­ tion of Bivalence: faced with a situation and a predicate, we have only two choices—to apply or to withhold. There is not a series of distinct linguistic acts in which we can reflect every degree of justification with which a predi­ cate may be applied. The crucial practical notion to be mastered for a predi­ cate associated with the distinction of degree is thus that of a situation to which the application of the predicate is on balance justified. Without mas­ tery of this notion, no amount of information about the structure of vari­ ations in the degree with which ‘F’ applies entails how the predicate is to be used. Now of this notion may it not still be a feature that it always survives sufficiently small changes?—that, if a and b are dissimilar only to some very small extent, then, if describing a as F is on balance justified, so is thus describing b? It is clear that all the previous considerations will apply, and that the intro­ duction of a complex structure of degrees of justification will get us no fur­ ther. For among these degrees we have still to distinguish those with which for practical purposes the application of the predicate is to be associated; other­wise we have not in repudiating Bivalence done anything to replace the old connection between justified assertion and truth. But plainly, once we attempt to make such a distinction, the arguments afforded by the governing view sweep aside this proposed solution to the Sorites paradox as an irrele­ vance. To rehearse the reasons: if we are to be able to remember how to apply ‘F’, then differences too slight to be remembered cannot transform a situation to which its application is on balance justified into one which is not so; if we are to be able to apply ‘F’ just on the basis of casual observation, the same applies to differences too subtle to be detected by casual observation; if the distinction between cases to which the application of ‘F’ is on balance justified and others is to be made just on the basis of how things look, or sound, and so on, then any pair of indistinguishable situations must receive the same ver­ dict; finally, if ‘F’ is associated with moral or explanatory distinctions that we are unwilling to tie to very small changes, we shall likewise be unwilling to allow such changes to generate doubt about the status of a situation previ­ ously regarded as on balance justifying description as F. Of course, the use here being made of the notion of a situation to which the application of ‘F’ is ‘on balance’ justified is quite uncritical. But this is legitimate. For, as remarked, there must be some such notion if a many-­valued logic for distinctions of degree is to have any practical linguistic application.

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Language-Mastery and the Sorites Paradox  95

V Let us turn then to the question whether we could not eliminate, at not too heavy a cost, the tolerance of observational predicates. The resulting predi­ cates would no longer be strictly observational; hence the initial doubt whether such a purified language could engage with the observational world at all. On reflection, though, it is clear that the dislocation of language and the world of appearance generated by such a purification would not have to be as radical as that. When three situations collectively provide a counterexample to the transitivity of indiscriminability, there is nothing occult in the circum­ stance that they do so. It is an observationally detectable difference between indiscriminable situations that one is distinguishable from a third situation from which the other is not; the relation ‘a matches b matches c does not match a’ is an observational relation—that is, one whose application to a trio of objects can be determined just by looking at them, listening to them and so on. Observational concepts evidently require narrower criteria of reapplication than indistinguishability, if they are to be purified of tolerance. But we should not jump to the conclusion that to provide such criteria will require surrender of observationality altogether, for the phenomenon which is causing the trouble is itself observational. Indeed, the only kind of observationally ­ ­detectable difference which there can be between indiscriminable items is that one should be distinguishable from some third item from which the other is not. So, if the class of expressions in question is to remain in contact with observation, we have to look for some form of stipulation which exploits the non-­transitivity of indistinguishability to provide a basis for describing indiscriminable situations differently. No other explanation can correspond to a distinction which sense experience can determine to obtain, a distinction which we can simply be shown. After such a stipulation, the question whether a pair of indiscriminable ­colour patches should receive the same colour description may turn on their respective relations of indiscriminability/discriminability with respect to some third patch. But now we have to take note of a striking aspect of the philosophical psychology of non-­transitive matching: it does not seem to be possible to conduct experiments with non-­transitively matching triads in memory. For suppose that a predicate, F, is defined ostensively by reference to some individual, a, which, it is noted at the time, perfectly matches another individual, c; it is understood that F is not to be applied to individuals which match a unless they also match c. Later the trainee comes across b, which, so

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96  The Riddle of Vagueness far as they can determine, matches a perfectly; the question is, does b match c? It is evident that the issue is only resoluble by direct comparison, and espe­ cially that it cannot be settled by memory, however accurate. For the most perfect memory of c can give no further information than that it looked just like a; which, when non-­transitive matching is a possibility, is simply insuffi­ cient to determine whether it would match b. This, it must be emphasized, in contrast with our conclusions concerning the third example, is not a limita­ tion imposed by the feebleness of our memories; it is a limitation of principle. It thus appears that, if we are to be able to exercise expressions whose appli­ cation to matching individuals depends upon their behaviour in relation to a third, possibly differentiating, individual, then we have to be able to ensure the availability of the third individual. Expressions of this species will be prac­ ticably applicable only in relation to a system of paradigms. So we can see, even in advance of attempting a specific stipulation to remove the tolerance of ‘red’ as displayed in the fourth example, that the kind of semantic construc­ tion it will have to be is going to tie the application of expressions of colour to the use of a colour chart. Let us then consider, as a test case, how we might go about the construction of such a chart. What we require of the chart is that it should enable us to identify a last red patch in any series of the type of the fourth example. There is one obvious way to achieve this—namely, to devise a single ad hoc para­ digm. It is plausible to suppose that we could complete a colour chart for the red/orange region at least in the sense that anything which we should wish to regard as falling within that region would match something on the chart. Consider, then, an arrangement of colour patches which form in this sense a complete colour chart for the red/orange region and which are simply ordered by similarity—that is, every patch on the chart more closely resembles its immediate neighbours than any other patches on the chart. Then a sharp red/ orange distinction can be generated as follows. Select some patch towards the middle of the chart; then any colour patch matching something on the chart either matches the selected patch or it does not; if it does, it is red; if it does not, but matches a sample to the left of the selected patch, it is again red; other­wise, it is orange. Naturally it could not be guaranteed that duplicates of this chart would always deliver the same verdict. Charts could look absolutely similar, and even satisfy the condition that the nth sample on either matched and was dis­ tinguishable from exactly the same samples on the other chart as its own nth sample, yet deliver discrepant results. But they would not often do so. Besides, the situation is not novel. Rulers, for example, sometimes give different

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Language-Mastery and the Sorites Paradox  97 results. A final criterion for one system is deposited in Paris; and we could do the same with a colour chart. Generalized, this proposal might seem quite ludicrous in practical terms. We are confronted with the spectacle of a people quite lost without their indi­ vidual wheelbarrow loads of charts, tape recordings, smell- and taste-­samples, and assorted sample surfaces. But this caricatures the proposal. There would be no need for all this portable semantic hardware. This is clear if we pursue the analogy with the use of rulers: it is true that, without a ruler, we can only guess at length; but after the introduction of an ad hoc paradigm for colours, the use of colour predicates will presumably be analogous not to that of expressions like ‘two feet long’, but rather to that of expressions like ‘less than two feet long’—that is, expressions of a range of lengths. Of such expressions the criterion of application is still measurement; but, unless the example is a peripheral one, we can tell without measuring what the outcome of measure­ ment would be. Training in the use of paradigms might be essential if one is to grasp the sense of such expressions; but, once grasped, most cases of prac­ tical application could be decided without the use of paradigms—for most practical purposes, the wheelbarrow could be left behind. It would appear, then, that, if we adopted such stipulations as a general strategy, it would not have to affect our use of observational language very much at all. At present we can tell of anything red that it is so just by looking at it. This would still usually be true after the proposed stipulation; and, if the new distinction was suitably located, cases where it was not true would in general coincide with borderline cases of the old red/orange distinction. The use of predicates so refined could thus greatly resemble their present use; the  distinctions which they expressed would be empirically decidable; and there would be one crucial disanalogy—they would be tolerance-­free. It is apparent that exactly parallel considerations may be brought to bear upon the earlier treatment of the first and third examples. Even after a precise redefinition of ‘heap’, we would be able to learn to tell in most cases just by casual observation what verdict the new criterion would give if applied; it would seldom be necessary actually to count the grains. And the distinction between red and orange, supposing an exact distinction were drawn by means of a chart, would be unmemorable only within that small range of shades which could not by unaided memory be distinguished from the last red sam­ ple. It would thus appear that the cost of eliminating tolerance in cases of these two types need not after all be high, since we could expect to be able to tell in general just by looking at, and so on, an item on which side of the dividing line it would fall.

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98  The Riddle of Vagueness If there need, after all, be no substantial sacrifice in endowing formerly observational predicates with exact boundaries, what has become of the alleged profound tension between phenomenal continuity and language designed to express how things seem to us? The answer, of course, is that it has simply been swept under the carpet. The possibility of our dispensing with paradigms for most practical purposes depends upon our capacity, for example, to distinguish between cases where we could tell whether or not ‘red’ applied just by looking and cases where we could not, where we should have recourse to a chart. But, if we are able to make such a distinction, there can be no objection to introducing a predicate to express it. And then, it seems, the semantics of this predicate will have to be observational. On what other basis should we decide whether something looks as though comparison with a chart would determine it to be red than how it looks? Of any pair of colour patches which look exactly alike, if either looks as though the chart would deliver the verdict ‘red’, both must. So the new predicate, introduced to reflect our capacity to make this distinction, will be applicable to both mem­ bers of any pair of matching colour samples, if to either. It is not that there is any compelling reason to have such a predicate; only that there is no reason not to. If we were sometimes able to tell without using a chart whether something was red, it would surely be possible to make intel­ ligible to us a predicate designed to apply in just such circumstances. So it transpires that a language all of whose observational concepts were based on paradigms would avoid containing tolerant predicates only by not containing means of expression of all the observational distinctions which we are in fact able to make. The dispensability of the wheelbarrow requires the exercise of observational concepts. It would, of course, be absurd to propose that the tolerance of such new predicates—‘looks as though it would lie to the left of the last red shade’, ‘looks as though it contains fewer than ten thousand grains’, and so on—might in turn be stipulated away. Their meaning will not permit it; it cannot be allowed of things which look exactly alike that one may look as though it satisfies some condition which the other looks as though it does not, unless how a thing looks may not be determined by looking. The earlier discussion of the first and third examples involved an overestimation of our interest in pre­ serving the tolerance of the predicates involved only if we possess a coherent understanding of these new predicates; if the first and third examples do not, after all, pose a substantial problem for the governing view, it is because of our capacity to handle expressions falling within the scope of the fourth example.

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Language-Mastery and the Sorites Paradox  99

VI Let us, then, finally review the character of the difficulty for the governing view which originates in the fourth example. It is a fundamental fact about us that we can learn to classify items accord­ ing to their appearance, that we are able, consistently as it seems to us, to apply or to withhold descriptions just on the basis of how things strike the senses. In a discrete phenomenal world, there would be no special difficulty— no difficulty not inherent in the idea of a semantic rule as such—in viewing our use of such expressions as essentially nothing but the following of rules of which it was a consequence that indiscriminable phenomena should receive the same description. But, if mutually exclusive use is made of a pair of such predicates, and if cases to which one applies permit of continuous trans­form­ ation into cases where the other applies, it cannot be correct to represent the use made of either predicate just as the doing of what is required by a set of rules with such a consequence. Yet we are forced—if the relevance is allowed of considerations to do with what we should regard as adequate explanation of such expressions, or with certain criteria which we should accept of ­misunderstanding such an expression—to attribute to the rules governing these predicates precisely such an implication; and all the phenomena which we confront in our world are in principle capable of seemingly-­continuous variation. It will not quite do, though, to present the difficulty as that of the in­ad­ equacy of any inconsistent set of rules to explain a consistent pattern of behaviour. To begin with, it is unclear how far our use of, for example, the vocabulary of colours is consistent. The descriptions given of awkward cases may vary from occasion to occasion. Besides that, the notion of using a predi­ cate consistently would appear to require some objective criteria for variation in relevant respects among items to be described in terms of it; but what is distinctive about observational predicates is exactly the lack of such criteria. So we may not lean too heavily, as though it were a matter of hard fact, upon the consistency of our employment of colour predicates. The point rather has to do with the fact that our use of these predicates is largely successful; the expectations which we form on the basis of others’ ascriptions of colour are not usually disappointed. Agreement is generally possible about how colours are to be described, which is equivalent to saying that others seem to use ­colour predicates in a largely consistent way. It is this fact of which the governing view can provide no account. A seman­ tic rule is supposed to contribute towards determining what is an admissible

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

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Language-Mastery and the Sorites Paradox  101 exactly the same brief—as they can be, if it consists in an inconsistent set of instructions. What, then, have we learned? The comparison of language with a game is an extremely natural one. What better explanation could there be of our abil­ ity to agree in our use of language than if, as in a game, we are playing by the same rules? We are thus attracted towards the assimilation of our situation to that of people to whom the practice of a highly ramified, complex game has been handed down via many generations, but of which the theory has been lost. Our central task, as philosophers of language, is to work towards the recovery of such a theory: a theory which will explain the mechanism of our recognition of the senses of new complex expressions by displaying them as functions of the senses of their constituents and their mode of combination, which will explicate our apprehension of valid inferences—which, in short, will explicate the overall character of our mastery of the language game. What we have learned is that we probably cannot combine this conception of what a theory of meaning should accomplish with the notion that the investigation is something which, as masters of the language in question, we are better placed to carry out than an observer of our practice. We have to avoid appeal, at any rate, to a range of considerations which it is our antecedent prejudice to con­ sider must be relevant: considerations to do with what we should deem a proper explanation of the sense of an expression, the criteria which we should employ for determining that someone misunderstands it, what we use the expression for, that is, what issues turn on its application, the limitations imposed by our senses and memories on the information which we can absorb from our linguistic training, and our general conception of what justi­ fies the application of the expression. And what privilege do we enjoy in the quest for a theory of meaning which an observer of our usage does not, if all these traditionally accepted guidelines for sense are dismissed? But we have seen that we must dismiss them if we want a coherent account of the senses of vague expressions. The methodological approach to these expressions, at any rate, must be more purely behaviouristic and anti-­reflective, if a general the­ ory of meaning is to be possible at all.

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3 Hairier than Putnam Thought with Stephen Read

From one point of view, it may seem that Sorites paradoxes are a consequence of the very nature of vagueness. If ‘bald’, for example, lacks sharp boundaries, that can only mean, it appears, that there is no determinate number of hairs, n, such that any person with n hairs on their head is bald while some person with n + 1 hairs on their head is not. Accordingly it seems we must deny (1)  (∃n)(Bn & ¬Bn + 1) (where ‘Bn’ is short for ‘Any person with n hairs on their head is bald’). Classically this denial is equivalent to assertion of (2)  (∀n)(Bn → Bn + 1), which generates a Sorites paradox. Conjoined, for example, with the undeniable (3)  B0, (2) will lead, either by successive applications of universal instantiation and detachment, or by induction, to (4)  B50,000 which is clearly false. If we essay to treat this result as a reductio of (2), we wind up with (5)  ¬(∀n)(Bn → Bn + 1), which is classically equivalent to (1). Hence, it appears, if any predicate is vague, it must be Sorites-­generating (or tolerant); and the option of treating Sorites paradoxes as a reductio of their major premises is available only at the The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0004

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104  The Riddle of Vagueness cost of denying the vagueness of the predicates involved—which is hardly less paradoxical. In ‘Vagueness and Alternative Logic’, Hilary Putnam (1983, pp. 285–­6) puts forward a suggestion for a formal treatment of the logic of vagueness. His leading thought, although not presented in the context of exactly the foregoing perspective on the problem, is in effect that the equivalence of (5) with (1) is not intuitionistically valid. May not a suitable treatment for vague predicates be available by treating them in the manner in which intuitionistic logic treats predicates in number theory which are not effectively decidable? The prospect is thereby opened of treating the paradox as indeed a reductio of (2), so accepting (5), without commitment to (1) with its paradoxical imputation of precision to ‘Bx’. Putnam admits that, at the time of writing, he had not thought this idea through. What will already be apparent to the alert reader is that, in order to disclose serious difficulties for the proposal, Putnam would not have had to think very far. For, if the denial of (1) is indeed a satisfactory expression of the vagueness of ‘Bx’, intuitionistic logic provides all the materials needed for the paradox. This may be seen in either of two ways. The denial of (1) is intuitionistically equivalent both to (6)  (∀n)(Bn → ¬¬Bn + 1) and to (7)  (∀n)(¬Bn + 1 → ¬Bn) Now, appropriately many steps of contraposition (which is of course intuitionistically valid) will obtain from (6) any principle of the form (8)  (∀n)(¬k Bn → ¬k +2 Bn + 1) (where ‘¬k Bn’ is to indicate the result of prefixing ‘Bn’ by some even number, k, of occurrences of ‘¬’). By appropriate successive appeal, then, to the members of the series of such principles, there will be no difficulty (in principle) in advancing from (3) and (6) as initial premises to, say (9)  ¬100,000 B50,000. But the admitted falsity of (4) yields

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Hairier than Putnam Thought  105 (10)  ¬B50,000, which in turn entails (11)  ¬¬99,998 B50,000, of which (9) is the negation. Alternatively, and more simply, (7) will generate a Sorites paradox back-­to-­front, as it were, starting from (10) and leading to the denial of (3). So the paradox is with us yet. Although Putnam’s proposal is unsatisfactory, its failure is instructive: if the denial of principles like (1) is considered to follow from the vagueness of  the relevant expressions, then no logic which, like intuitionistic logic, endorses reductio ad absurdum and the classical handling of the quantifiers and conjunction can prevent the derivation of Sorites paradoxes. This is evident from the starkest form of the paradox, a back-­to-­front version, each phase of which proceeds as follows: i ii iii ii, iii ii, iii i, ii

(i) (ii) (iii) (iv) (v) (vi)

¬(∃n)(Bn & ¬Bn + 1) ¬Bk + l Bk Bk & ¬Bk + l (∃n)(Bn & ¬Bn + 1) ¬Bk

Assumption Assumption Assumption (ii), (iii), &I (iv), EI (iii), (i)/(v), RAA

Certain Relevant Logics do indeed, of course, modify the appropriate classical rules.1 But intuitionistic logic does not. Moreover, it is difficult to see how an intuitionistic semantics for negation, appropriately generalized to handle non-­mathematical contexts, could fail to endorse the negation of (1). For, intuitionistically, one is entitled to assert the negation of a statement just in case one has verified that no constructive proof of that statement is in ­principle possible; and to recognize the vagueness of ‘Bx’ is surely exactly to  recognize that all possibility of a constructive verification of (1) can be discounted.

1  The central tenet of relevant logic is that the mere lack of a counterexample does not guarantee an entailment. Hence, (i) will not, without qualification, entitle us to pass from (ii) to (vi). Note that this move would be an instance of modus ponendo tollens, classically and relevantly equivalent to modus tollendo ponens—i.e. Disjunctive Syllogism, which is, familiarly, relevantly invalid. What in detail a ‘relevantist’ will demur at in the reasoning is another matter.

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106  The Riddle of Vagueness Suppose it is agreed that it cannot be a satisfactory response to the paradox to admit the inconsistency of vague expressions. What we have seen is that the denial of (1) cannot be, for anyone with classical or intuitionistic sympathies, a satisfactory way of expressing the vagueness of ‘Bx’; and, as noted, the assertion of (1) merely flies in the face of its vagueness. It follows that a satisfactory account of the matter must render it coherent to hold that (1) may correctly be neither asserted nor denied; although we may take the view that its negation, and (2), may, and indeed must, be denied, since refuted by the paradox. Consequently the unrestricted validity of double negation elim­in­ ation for statements containing vague expressions is indeed in doubt, and Putnam’s proposal has point to that extent. But only to that extent. For, as noted, the motivation for intuitionistic-­type restrictions in the logic of vague expressions cannot closely resemble orthodox intuitionistic semantics if we are to avoid denial of (1); and, further, until a way has been found of properly acknowledging vagueness while avoiding such a denial, recourse to intuitionistic logic is no help.2

2  We are indebted to Peter Clark and John Slaney for their contributions to the discussion which suggested this chapter.

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4 Further Reflections on the Sorites Paradox In two papers published just over a decade ago (Wright 1975, this volume, Chapter 1; 1976, this volume, Chapter 2), I argued that the major premises— usually universally quantified conditionals1—which feature in Sorites para­ doxes owe their plausibility to an assumption which it then seemed appropriate, from the perspective of a philosopher of language in Oxford, to dub ‘the governing view’. The governing view was a combination of two claims. The first was that a master of a language is someone who has, at some deep level, internalized a definite set of semantic and syntactic rules, defini­ tive of the language. The second was something of a pot pourri of specific ideas about how such rules might be brought into the light or more fully char­ acterized and understood. It included the admissibility of, for instance, con­ sid­er­ations concerning speakers’ known limitations (of memory or perceptual acuity, for example), standardly accepted criteria of (mis-)understanding, salient features of standard linguistic training, and the purposes and role which a particular distinction might have. The argument was that, once the relevance of such considerations was granted, the major premises for Sorites paradoxes could be seen as guaranteed by aspects of the semantic rules gov­ erning the relevant expressions (together with such general facts as the non-­ transitivity of indiscriminability). Since the other premises involved in such paradoxes, and the principles of inference which they utilize, seem incontest­ able, the conclusion suggested was that the governing view was responsible for them. That (at least some) Sorites paradoxes are genuine—do no more than reflect features of the meanings of the affected expressions and aspects of the world, and so commit no identifiable fallacy nor involve any identifiable error—is a view which has been adopted by a number of writers, most not­ ably  by Dummett (1975). My suggestion, in contrast, was that this is an ­illusion, generated by the governing view; specifically, that its second, epis­ temo­ logic­ al ingredient is unacceptable, and that its first ingredient—the

1  Though other forms of major premise will serve. Overlooking this has caused problems for some writers—see the remarks about Peacocke and Putnam later in this chapter.

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0005

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108  The Riddle of Vagueness overall conception of language mastery which it embodies—needs, at the least, severe qualification. Despite their antiquity, the Sorites paradoxes had received comparatively little attention in the literature prior to 1975. Since then, there has been some­ thing of an explosion. One purpose of the present chapter is to try to respond to various criticisms of my earlier efforts which have appeared, notably in  Christopher Peacocke’s very interesting article ‘Are Vague Predicates Incoherent?’ (Peacocke 1981). I now wish to restrict the scope of my original proposal. Major premises for Sorites paradoxes can actually be motivated in a number of quite different ways, which it is essential to distinguish. Only for a limited class of cases does it now seem to me that my earlier diagnosis is apt. The widespread conviction that the paradox afflicts all vague expressions (and hence virtually all our language) is best represented as the product of a ­different and, I believe, confused line of thought. And we have to reckon also with a third line of thought, leading to what I have called in this chapter the Tachometer paradox. Moreover, this threefold division is probably not exhaustive. Even in the restricted class of cases in which I still wish to press my original proposal, I am no longer content with the kind of formulation which I offered. Where earlier I spoke, vaguely, of the inappositeness of the comparison of language with a game, and of the need for a more purely ‘behaviouristic’ con­ ception of language-­mastery, I now wish to offer something different. It is less that I think these formulations wrong than that they appeal to contrasts which may seem unclear and which are anyway very difficult to sharpen. A first statement of the particular moral which these formulations tried to capture would better proceed, it seems to me now, in somewhat different terms. I have tried, so far as is possible, to write the present chapter in a fashion which will not presuppose that a reader is familiar with my earlier two papers. But since my principal objective has been to improve on them, it is un­avoid­ ably, to that extent, a sequel.

I The Governing View and the Sorites Paradox There is an idea about the kind of ability which mastery of a language (ba­sic­ ally) is which would figure in virtually anyone’s first philosophical thoughts on the topic. Suppose we had to teach somebody to play chess without any

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Further Reflections on the Sorites Paradox  109 kind of verbal or written exchange. Could it be done? There seems no reason why not; no reason, that is, why an intelligent and receptive subject could not acquire a knowledge of the moves and of checkmate and stalemate, and so on, just by immersion in the practice of the game, suitably reinforced by nods and shakes of the head or punishment and reward. Such a subject might become an average or better than average chess-­player without ever learning any of the vocabulary which we use in description of the game. However, once there was unmistakable evidence of mastery, we would not hesitate to attribute to the trainee the knowledge which we express by a statement of the rules and object of the game and also, if the trainee were an accomplished player, the knowledge which we can express by a description of certain strategies and of  their rationale. We would think of the trainee as having this knowledge although unable—for the time being anyway—to talk chess theory at all. Such knowledge would be viewed as implicit; but its content would coincide with that of the sort of explicit articulate knowledge which is normally acquired when someone learns to play chess. This way of describing the example seems natural, almost inevitable, because the ability in question is par excellence a rational one, involving intel­ ligence, insight, and purpose. If recourse to rationalistic explanation—ex­plan­ ation involving attribution of beliefs, goals, and intentions—were for some reason excluded, we simply should not know how to begin to describe the trainee’s accomplishment. But beliefs, for example, are contentful states; and we should be equally at a loss if none of the beliefs allowably attributable to the trainee could have a content which we would specify using the standard vocabulary of chess. So, provided the eventual performance were good enough, the idea that the trainee possessed implicit knowledge of the rules of chess would be just as attractive if they mastered chess before acquiring any linguistic competence; or, indeed, if the trainee were a deaf mute whose ­prospects of ever understanding a language were slight. The first philosophical thought about language-­mastery to which I allude is, in effect, a large-­scale analogue of the foregoing. Language-­mastery, like mastery of chess, is apt to seem an exemplar of rational ability, differing only in that the set of rules—syntactic, semantic, and pragmatic—whose know­ ledge it involves are inordinately more complex than the rules of chess, and the purposes to which it is adapted far more various. Yet we perforce acquire language-­mastery in much the manner of the fictional chess trainee, at least when learning a first language: we are immersed in the practice of those who can already ‘play’, and explicit verbal instruction can be received only when enough of the language has already been grasped to enable an understanding

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110  The Riddle of Vagueness of the terms in which it is couched. No doubt a good deal of the instruction is explicit—in typical school curricula, a great deal is given in the way of exact definitions of terms, and precise characterizations of what is good grammar. But it remains true that the greater part of anyone’s competence in their native language is not and could not be acquired that way. In short: it comes naturally to think of someone with a good level of mas­ tery of their native language as knowing a great deal more than they were ever explicitly taught; and it follows that there can be at best a contingent connec­ tion between the possession of this knowledge and the ability to articulate it. If the chess trainee’s performance when playing the game is completely con­ vincing, we shall not be any less inclined to credit them with knowledge of the rules of chess should it prove, when they acquire explicit chess vocabulary, that they find it difficult to articulate them for themselves. So too with language-­mastery. The idea of implicit knowledge thus has just the same attractiveness, just the same seemingly inevitable part to play in the ex­plan­ ation of linguistic ability, that it should have in accounting for the per­form­ ance of our fictional chess trainee. The conclusion of this way of thinking about the matter, then, is that, as someone proceeds towards mastery of a first language, it must be true at cer­ tain stages, and may be true at any later stage, that some or all of the know­ ledge which they have acquired is implicit. But that, it seems, need pose no barrier to the possibility of an explicit description, either in a different lan­ guage or in a larger fragment of the same language, of what is thereby im­pli­ cit­ly known. On the contrary, since this ‘implicit knowledge’ has been characterized from the outset as a contentful state, as in effect propositional knowledge, it must be possible in principle to specify the contents known by language learners, even if they themselves cannot yet do so or will never be able to do so. It would follow that there must be a formulable theory which stands to the competence of speakers of English rather as a complete codifica­ tion of the rules of chess stands to the competence of someone whom we credit with a complete implicit knowledge of those rules but who is able to give, for whatever reason, at most a partial characterization of them. But there is a difference. Chess is a complex enough game, but it would nevertheless be surprising if, once introduced to the names of the pieces and other terms of art such as ‘castling’ and ‘checkmate’, a subject whose knowledge of chess had originally been implicit did not prove to be able, on relatively brief reflection, to produce a passable statement of the rules of chess. In sharp contrast, it is obvious even on superficial reflection that the task of producing an explicit statement of the rules governing just the fragment of English concerned with

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Further Reflections on the Sorites Paradox  111 literal statement-­making would be one of the greatest difficulty, even if we restricted our concern to legality, as it were, and prescinded from all con­sid­ er­ation of what constituted ‘good play’. We are as if in the situation of people who have all been taught, by directly practical methods and with a minimum of formal explanation, to play a game hugely more complex than chess, but now find that all the instructors—those who might have produced a written codification of the rules of the game if asked—have disappeared. We therefore confront, if we choose to rise to the challenge, an intriguing project: work out what the rules are. The result, if the task could be accomplished for an entire natural language, would be something that nobody has ever beheld: a com­ plete explicit formulation of the knowledge which any master of the language knows but knows (mostly) only implicitly. These ideas represent one possible route into sympathy with the modem philosophical project of a ‘theory of meaning’. (They may, indeed, represent the only such route, though that is a nice question.2) I am not recommending them; I hope only to have made it intelligible, if there is anyone to whom they have never occurred, how they might nevertheless constitute an attractive vision. Their significance in the present context is that the Sorites paradox, or so I have argued (Wright 1975, this volume, Chapter  1; 1976, this volume, Chapter 2), casts serious doubt on their acceptability in general. That is one claim which I wish to review in this chapter. The claim is important; for the most fundamental task in the philosophy of language is the achievement, in the most general terms, of an understanding of the nature of language-­ mastery. It is here, if I am right, that the paradox has most to teach us. If the tension is to be perceived, however, we have to ride the analogy a lit­ tle bit harder. Consider the task of trying to draw up the rules of a game which we have all been able to play since childhood but of which there is no extant codification. How should we set about it? No doubt we should first need to invent some vocabulary to describe the materials and situations which are specific to the game; but then what? Recognition of one’s own hitherto unfor­ mulated intentions is a peculiar business in any case; but the natural approach would be for a number of competent players to draw up severally as complete a description as they could by reflection, and then to compare notes. But when, as is likely, differences of opinion are revealed, how should they be resolved? The extent of the consensus on a particular point would be im­port­ ant, of course. But if somebody, for example, listed a rule which no one else had, that would not necessarily be decisive against that particular rule. If, 2  For some reflections on it, see Wright (1986d); reprinted as Wright (1986c, ch. 6).

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112  The Riddle of Vagueness for example, there was universal consensus that behaviour in the course of the game which was in fact a breach of the putative rule would be regarded as outré and unacceptable, we would probably take that as strong prima facie evidence that the rule should be included and that the majority who did not list it had overlooked something. If, on the other hand, the putative rule seemed too complex, or incorporated distinctions too subtle to be remem­ bered with any degree of surety, or relied on our ability to make contrasts which we felt that the direct, ‘immersive’ training which we had received could not have possibly got across—if any consideration of that sort applied, there would be a strong case for discounting the rule. Perhaps the most intriguing thing about Sorites paradoxes is that an im­port­ant class of them seems to arise very directly from the application of this sort of thinking to the rules of which implicit knowledge is supposed, on the other side of the analogy, to constitute our language-­mastery. Once the content of those rules is supposed to be constrained by considerations like the  limitations of the methods whereby we were trained in them, and our ­limitations, for example, of memory, and the criteria for correct understand­ ing of them which we now find intuitively satisfying, it is relatively straight­ forward to argue that they imply the unrestricted acceptability of the major premises—standardly, universally quantified conditionals—involved in Sorites paradoxes. The arguments to this effect developed in Wright (1975, this volume, Chapter 1; 1976, this volume, Chapter 2) were various, and I shall give only the most general indication of them now. One thought was that it could not be part of understanding an expression to be able to make unmemorable dis­ tinctions if the learning and use of the expression standardly involves no reli­ ance on external aids of any kind. Another was that distinctions too fine to be detected by casual observation could not be incorporated into the conditions of application of a predicate like ‘bald’ or ‘heap’ whose utility depends on being applicable on the basis of casual observation. A third was the thought that the conditions of application of an expression which, like ‘adult’, is associ­ ated with substantial moral and social significance cannot incorporate dis­ tinctions too refined to sustain that significance. A fourth was that the conditions of application of any ostensively teachable expression cannot incorporate distinctions—in particular the kind of distinction that may obtain between observationally indiscriminable items—not amenable to ostensive display. A fifth was that no distinction can mark a watershed in the conditions of application of an expression if treating it as such would stand­ ardly be taken to display a misunderstanding of that expression.

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Further Reflections on the Sorites Paradox  113 Typically, the major premises in plausible Sorites paradoxes can be sup­ ported by considerations of more than one of these kinds. The paradox for ‘red’, for example, may draw on any, or all, of the first, second, fourth, and fifth. A paradox for ‘heartbeat of childhood’ may draw on either the second or third. And so on. The exact manner in which the major premises are sup­ posed to follow from such considerations will bear rather more careful scru­ tiny than I gave it before and we shall return to the question in Section VII. But what seems beyond dispute is that, if we accept that linguistic competence is constituted by sensitivity to the dictates of internalized rules, it is overwhelm­ ingly natural to suppose that features of these rules may be discerned by reflection on our practical limitations, the ways the application of an expres­ sion would be defended if challenged, the interest which we attach to the clas­ sification effected by it, the way it is standardly taught, and the criteria for someone’s misunderstanding it. How we should respond to the situation depends, of course, on whether the specific arguments are truly watertight. But there is little at least in the lit­ erature with which I am familiar which tries to meet the problem head-­on, to show that it is simply mistaken to think that relevant major premises can be validated in this way. For the present, let us continue to suppose that they can. On that assumption, the position which we arrive at is as follows. Two sep­ arable claims have been made (together, they constitute the ‘governing view’). There is the general claim that our language-­mastery is to be seen as the prod­ uct of (mostly) implicit knowledge of rules; and there is the more specific claim that the enterprise of attempting to arrive at an explicit statement of these rules—or to enlarge on features of them which need not be reflected in such a statement, if we are doing semantics in the ‘homophonic’ mould—is rightly seen as constrained by considerations of the seemingly platitudinous sort outlined. If both these claims are accepted, we are faced with the realiza­ tion that Sorites paradoxes (or at least some of them) merely serve to unravel certain features of the semantics of the expressions with which they deal, which must accordingly be viewed as at least de facto incoherent; that is, the rules for the use of those expressions, taken in conjunction with undisputed features of the world, enable flawless cases to be made for simultaneously withholding them from and applying them to certain situations. There are then only three possible modes of response. The first is accept­ ance of both claims, with their consequence that our use of a large class of expressions is governed by incoherent rules. Second, we might essay to retain the first claim while jettisoning the second: the result would be to hold that while the implicit-­knowledge conception of our language-­mastery is

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114  The Riddle of Vagueness fundamentally correct, the platitudinous-­seeming constraints on the explica­ tory enterprise are not. Third, there is the apparently radical step of rejecting the first claim. How deep such a rejection would have to go would be a matter for determination, but, at the very least, it would be necessary to construct some alternative conception of our mastery of the use of Sorites-­susceptible expressions, to supplant the idea that it is comprised by implicit knowledge of the requirements of rules. It is important to be clear that there cannot be any further alternative. (There is, of course, no possibility of retaining the second claim at the expense of the first.) Or rather: the only alternative is to disclose error in the sup­pos­ition that the two claims, in conjunction with further undis­ puted premises (for example, that indiscriminability is a non-­transitive relation), inescapably generate Sorites paradoxes. A fully satisfactory treat­ ment of the problem must therefore accomplish one of two things: either it must demonstrate that the link between the two ingredient claims of the governing view and the major premises of Sorites paradoxes can be ­broken—in which case it will need, in addition, to explain how exactly those premises fail of truth—or it must explain why one of the three responses outlined is superior to the others and intuitively satisfying when properly fleshed out.3

II Accepting the Paradox Towards the end of ‘Wang’s Paradox’, Dummett summarizes some conclu­ sions as follows: (1) Where non-­discriminable difference is non-­transitive, observational predicates are necessarily vague. (2) Moreover, in this case, the use of such predicates is in­ trin­ sic­ al­ ly inconsistent. 3  In comparison with what is needed, it is not unduly harsh to say that much of the literature on this topic which has mushroomed over the last decade or so has amounted to little more than tinker­ ing. Writers have been content to devise more or less ad hoc semantics in whose terms the major premises fail of strict truth without doing anything to disclose why their plausibility is specious, still less confront the intuitively powerful arguments on their behalf. It should go without saying that the solution of a paradox requires more than designing wallpaper.

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Further Reflections on the Sorites Paradox  115 (3) Wang’s paradox merely reflects this inconsistency. What is in error is not the principles of reasoning involved, nor, as on our earlier diagno­ sis, the induction step. The induction step is correct, according to the rules of use governing vague predicates such as ‘small’: but these rules are themselves inconsistent, and hence the paradox . . . (Dummett 1975, pp. 319–20)

This is a clear enough example of the first kind of response. Why not this response? Why should it not be the case that the ‘language-­game’ is governed by (de facto) inconsistent rules? Clearly, appeal to our knowledge of such rules couldn’t provide a completely satisfactory explanation of our actual lin­ guistic practice—some supplementary account would be wanted of why we do not fall into confusion. But the idea of making some sort of successful enterprise out of an incoherently codified practice is not in itself incoherent.4 So what, if any, is the real objection to it here? There are, it seems to me, three related objections. First it is open to ques­ tion whether the idea of inconsistency can actually get any grip unless we are concerned with rules which have been explicitly codified. Suppose that we play an inconsistently formulated game, but never notice the inconsistency, sometimes going by one rule in a particular kind of situation and sometimes going by another which conflicts with it. If a third party, trying to find out the exact character of the rules of the game, was restricted to observation of play and never allowed to see a written version of the rules, what reason could they possibly have to suspect that the game was inconsistent rather than that at least two different courses of action were permissible in the situation in ques­ tion? The analogy is not, indeed, accurate, since it is not as if we sometimes go along with the results of Sorites reasoning but sometimes prefer the response which such reasoning can be made to contradict. But just for that reason a radical interpreter of our use of colour vocabulary, for instance, would have even less reason to suspect an inconsistency than the observer of the game. If we say that there is an inconsistency nevertheless, with what right do we claim an insight into the character of the rules governing this vocabulary from which an observer of the way we use it is barred? The answer will be, pre­sum­ ably, that the methodology implicit in the second claim of the governing view already transcends the resources available to radical interpretation if the latter must prescind from consideration of our intellectual and sensory limitations, the kind of training which we receive, and so on. But then is not deliverance 4  A point Wittgenstein (1978) stresses repeatedly in the observations on consistency.

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116  The Riddle of Vagueness of so peculiar a conclusion some cause for disquiet with the richer methodology? Second, what is actually responsible, on this view, for the large degree of coherence and communicative success which our use of colour vocabulary enjoys? Indeed, what is the justification for continuing to think of the use of such expressions as governed by rule? Knowledge of appropriate rules was supposed to constitute linguistic competence. But it cannot do so if competent usage essentially has a coherence which, in Dummett’s view, the rules lack. Dummett’s response needs supplementing with an explanation of our com­ municative success with such vocabulary in which the idea of knowledge of inconsistent rules has an ineliminable part to play. For either such know­ledge is still to be a basic ingredient in competence, or we should drop the idea. But it is quite unclear, to me at least, how such an explanation should proceed. That brings us to the third, and, I think, a decisive objection to Dummett’s response. I do not see how we can rest content with the idea that certain implicitly known semantic rules are incoherent when nobody’s reaction, on being presented with the purported demonstration of the inconsistency, that is, the paradox—even if they can find no fault with it—is to lose confidence in the unique propriety of the response—for example, ‘That’s orange’—which the demonstration seems to confound. Think of your reaction when, having received as explanation of the notion of class only the usual informal patter plus the axioms of naive set theory, you first confronted Russell’s Paradox. If, which is unlikely, you held any intuitive conviction about whether Russell’s class was a member of itself, you will have been forced to recognize an exact parity in the opposing case; and the effect should have been to cause you to realize that there is just no view to take on the matter before some refinement of the notion of class has been made. But that is exactly not our response to the Sorites paradox for ‘red’. Our conviction of the correctness of the non-­ inferential ingredient in the contradiction is left totally undisturbed. So, far from bringing us to recognize that, pending some refinement in the meaning of ‘red’, there is just no such thing as justifiably describing something as ‘red’ or not, our conviction is that no one ought to be disturbed by the paradox— and this conviction is not based on certainty that we shall be able to disclose some simple fallacy. If the rules for the use of ‘red’ really do sanction the para­ dox, why do we have absolutely no sense of disturbance, no sense that a real case has been made for the inferential ingredient at all? Are we so abjectly irrational that we cannot recognize our confusion even when it is completely explicit? A different account is called for.

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Further Reflections on the Sorites Paradox  117

III Resolving the Paradox by Jettison of the First Claim of the Governing View In Wright (1975; 1976) I came down fairly strongly in favour of the third approach: that of rejecting the claim that understanding the relevant expres­ sions should be seen as consisting in the implicit knowledge of semantic rules. This response need not involve total banishment of the concept of semantic rule from any satisfactory account of what it is to understand such an expres­ sion. Indeed, it would be a serious objection if banishment were involved, since some sort of notion of rule would appear to be prerequisite for the very idea that such expressions have meaning, which we wish, presumably, to retain. So I am now inclined to acknowledge little real contrast between the third approach and the second—that of retaining the first part of the govern­ ing view while rejecting the patchwork epistemology of semantic rules mooted in its second part. What I had in mind in my earlier proposal was that we should, in any case, repudiate the idea that knowledge of the character of the rules governing Sorites-­affected expressions is to be construed as prop­os­ ition­al knowledge, acquired by immersion, and largely implicit. That is the idea which sets up the project of rendering the purported content of this knowledge explicit, and which makes the second ingredient seem so natural and harmless. If we insist that, in these cases at least, there are no ‘contents’ implicitly known—so, in only that sense, no rules—there is no space for an argument that the rules are incoherent. However, if it continues to be appro­ priate to think of correct use of the relevant expressions as subject to rule, in whatever appropriate sense, then it ought also to be harmless to continue to think of competent speakers as knowing the rules in question. Something like the letter of the first part of the governing view may accordingly yet be sus­ tained. But, if we follow my former proposal, the knowledge which competent speakers have of the rules relating to the relevant expressions will not be implicit, propositional knowledge; it will be knowledge which—like the knowledge how which constitutes practical skills such as ice-­skating and bal­ ancing—calls for no content as its object. This proposed response to the Sorites may still seem frustratingly in­def­in­ ite. If it can be worked into a satisfactory treatment of the problem, two mat­ ters, in particular, need extensive clarification. We need an account of the restricted class of cases, if any, in which it remains appropriate to think of

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118  The Riddle of Vagueness understanding as constituted by implicit knowledge of the contents of rules. And we need a much fuller account of what positive conception of under­ standing should supplant the implicit knowledge conception in the problem­ atic cases. If understanding ‘red’, for instance, is to be thought of on the model of a practical skill, comparable to the ability to hit a good crosscourt backhand or ride a bicycle, we need both a sharp theoretical account of the distinction between those abilities in whose explanation it is legitimate and needful to invoke the idea of the agent’s implicit propositional knowledge and the abil­ ities, like those two examples, where we feel it is not; and also, ideally, inde­ pendent reason to regard our understanding of Sorites-­affected expressions as constituted by abilities of the latter sort. Pending clarification of these matters, my earlier proposal offers, at most, a strategy for resolving the Sorites paradox, and a pointer only to the general tenor of the lesson which it has to teach us. Christopher Peacocke (1981) is unsanguine about the strategy, believing that it can be shown in advance that certain versions of the paradox will sur­ vive even if we abandon the conception that understanding is informed by (implicit) propositional knowledge of semantic rules. Peacocke invites con­ sid­er­ation of a predicate, C, which is to apply to an object just in case the community will agree in calling that object ‘red’. Suppose that some small ­difference, d, in the wavelength of light is not visually discriminable by any member of the community. And let light of wavelength k be definitely red. Then, according to Peacocke, we still have this paradox: If a reflects light of wavelength k, then C(a). If an object differs in the wavelength of light when it reflects by just d from something that is C, it too is C. ∴ All visible objects (reflecting pure light) are C.

Hence, ‘the paradox seems to arise even if we do not suppose that the use of these expressions is governed by rules’ (Peacocke, 1981, p. 122). This may seem a rather puzzling thought. The example ought to be one, it seems, where a paradox arises for a predicate understanding which does not involve propositional knowledge of a rule. So how can a predicate expressly defined for the purpose, understanding which is thereby explicitly associated with knowing the content of a semantic rule, fit the bill? Well, there are two different points to whose service Peacocke’s example might be put. First, if C really is Sorites-­susceptible, there seems, for just the reason given—namely, that it is introduced by explicit statement of a rule— simply to be no space for the response recommended in my earlier papers.

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Further Reflections on the Sorites Paradox  119 How can we possibly avoid bringing in the idea of propositional knowledge of the content of a rule if pressed to give an account of what it is to understand C? What Peacocke should claim, it seems, is not that the paradox arises in certain cases even after we have made the recommended response but that it arises in cases where the recommended response is not a possible option. But second, and more interesting, there is, in any case, reason to think that C is Sorites-­ susceptible only if there is reason to think that communal consensus with ‘red’ will be stable under indiscriminably small variations in light wavelength. Now, of course, one reason for that thought would be if one supposed that ‘red’ was governed by incoherent rules and that the community would stick by them. But, if that were the only reason, dissolving the ‘red’ paradox in the way I re­com­mend­ed would dissolve the C-paradox as well. So that cannot be Peacocke’s reason. Rather, his thought is best interpreted as follows. Think, if you will, of competence with ‘red’ as a practical skill, uninformed by prop­os­ ition­al knowledge. The operation of the skill involves differential sensitivity to varying visual stimuli, but, if—as is so—the stimuli permit variation too slight to be detected by our visual apparatus, the skill cannot involve dis­crim­in­ation exercised over differences of that order of magnitude. So, if there is communal consensus that a patch of colour is red, and if all that the participants in the consensus are responding to is the visual stimulus, it may seem quite unintel­ ligible how the response can vary if the situation is changed only by altering the stimulus by an amount insufficient to be picked up by our v­ isual systems. That this way of looking at the matter really does prescind from the prop­ os­ition­al knowledge conception is testified to by the fact that it now seems incidental to the example that it concerns subjects who are capable of grasp­ ing contents at all. Think instead of a digital tachometer on a car. All such a device goes on, all it is designed to respond to, are variations in electronic impulses. There will be limitations to its sensitivity—so sufficiently slight vari­ ations in the incoming impulse will not, presumably, provoke any variation in the reading. And now it is neither more nor less plausible than before to con­ clude that, provided we are careful at each stage not to vary the impulse too greatly, the reading will never vary over a series of steps, no matter how many. It would, of course, be quite unphilosophical to take comfort in the thought that actual tachometers do not behave this way—it is only because they do not so behave that we have the appearance of paradox. So Peacocke should be granted that some apparent paradoxes of the Sorites family are not amenable to the response I suggested. That, of course, completely disposes of the value of that suggestion only if we assume that the entire family must admit of a uniform solution. What, I suggest, Peacocke’s example teaches us is that that

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120  The Riddle of Vagueness is not so. In fact, when Sorites paradoxes are individuated by reference to the type of ground adduced for their major premises, they fall, I believe, into at least three separate groups. One group involves support from the governing view; the C-paradox and the Tachometer paradox exemplify another group; and we shall be concerned with the third in Section VI. (The groups are not, of course, individuated by the expressions which they concern: there is no reason a priori why one and the same expression should not be argued to be Sorites-­susceptible by reasoning of each of the three kinds.) If it would be unphilosophical to take comfort in the fact, it remains that actual functioning tachometers cannot be stabilized through any number of unidirectional but sufficiently marginal changes in the incoming impulse: sooner or later they jump—and a good one will do so often enough, no matter how marginal the changes, to continue to serve as a practically reliable instru­ ment. Therefore, the premises of the Tachometer paradox are not true of actual tachometers; and the paradox must be resolved by explaining how exactly that is so. The obvious and presumably correct thought is nothing very exciting: the major premise, to the effect that, if the instrument gives reading R in response to impulse i, it will also give R to any i′ differing from i by no more than some specified amount, will turn out to be a mis­in­ter­pret­ation of what is entailed by possession of a sensitivity threshold. Quite what the cor­ rect account will be will depend on the design of the particular instrument concerned. But one would expect it to be fairly typical that, within the con­ tinuous interval, < i1 ,…, in > , there will be a finite subset of points, {b1,…,bk}, such that, in response to an impulse of any of these values, bj or lying within some margin, d, of bj, the instrument will always respond with the reading fj; while responses to other impulses with values in the interval, < i1 ,…, in > will depend additionally upon the prior state of the instrument.5 One conse­ quence, of course, is that such an instrument will, on different occasions, give different readings in response to the same impulse, depending on how that impulse is ‘approached’; but perhaps that is just what such instruments do. The salient point is that it has to be consistent with the claims that a tachometer has a sensitivity threshold, and that its response is entirely to 5 Pictorially: variable reading

fk : impulse value

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Further Reflections on the Sorites Paradox  121 stimuli of a certain sort, to suppose that the major premise in the paradox is not everywhere true. Presumably, then, since communal consensus in the use of ‘red’ does not extend all the way down to orange objects, the major premise in the C-paradox will be false on similar grounds. And the explanation of its falsity will be consistent with supposing that we all possess visual sensitivity thresholds and respond only to visual stimuli in our use of ‘red’. Peacocke is aware of the possibility of this kind of response to the C-paradox, and is dissatisfied with it. He writes: any model for an application of vague observational predicates must provide analogues of three things involved in such application: there must be states which are the analogues of having experiences, there must be something analogous to the . . . non-­transitivity of non-­discriminable difference, and there must be some analogue of the application of an observational predi­ cate upon a particular occasion. (Peacocke 1981, p. 123)

Peacocke believes that it is impossible to provide for all three conditions in the sort of model illustrated by the tachometer. Suppose we assimilate the state of the instrument, induced by the reception of a particular impulse, to the having of an experience; and the reading which it issues to the application of the predicate. What about the non-­transitivity of indiscriminability? In the case illustrated, for instance, we can, by choosing a pair of values which are respectively within and just outside the d-margin of some bj, elicit—at least sometimes—differential responses from the instrument no matter how small the difference between the two chosen values may be. How then can it be claimed that sufficiently small differences are indiscriminable for the instru­ ment, so that—since larger ones are not—the relation ‘. . . is not discriminable by the instrument from . . .’ behaves non-­transitively? But Peacocke is wrong about this. Or rather: he is right to think it essential that some analogue of non-­transitive indiscriminability should be a feature of the model but wrong to suppose that one cannot be provided. The impres­ sion to the contrary requires that the instrument’s sometimes issuing differ­ ent responses to a pair of stimuli should be regarded as sufficient for its being able to discriminate between them. And the error in that supposition is easily demonstrated if, reverting once again to a case in which human beings are the ‘instruments’, we consider Michael Dummett’s example (1975, p. 316) of the slowly moving pointer. You observe a pointer which is initially at rest but then begins to move, too slowly, however, for you to see that it is moving. You are asked to give a signal—to raise your right hand, say—as soon as it seems

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122  The Riddle of Vagueness to you to occupy a different position from the initial position; and—let us add to Dummett’s example—to raise both hands as soon as it seems to you to have moved into another position again. Suppose that you raise one hand after four seconds, and both after eight. Are we to conclude that the position which the pointer seemed to you to occupy after the third second looked different from the position which it seemed to you to occupy after the fourth? The answer, of course, is that you cannot detect any difference between those positions—what you meant to indicate when you first raised one hand was only that the position now seemed to you to be different from the start­ ing point. As Dummett brings out, there are ways of describing the phenomenology of the situation which are threatened with incoherence—and perhaps none which is not. But it is legitimate to suppose, for the purposes of the analogy with the tachometer, that your permissible ‘signals’ are restricted to the rais­ ing and lowering of hands. Suppose, then, that the actual positions of the pointer after each second are determined by some appropriately finely cali­ brated instrument. The situation will be that you sometimes will and some­ times will not respond to the sorts of changes involved from one second to the next by varying your signal; but that, asked about any single such change, you will report the positions involved seem to you to be the same. More: if we imagine that, once you have raised both hands, the movement of the pointer is reversed, we would expect to find that, when it regained the position ori­ gin­al­ly occupied after three seconds, you would still have one hand raised. If we—as is permissible—suppose further that your signals for the positions occupied initially, after four seconds, and after eight seconds, are by and large respectively uniform, then your performance is now in all relevant respects assimilable to that of the hypothetical tachometer. And, crucially, the fact that you always (more or less) have a hand raised when the four-­second position is presented, but only sometimes have a hand raised when the three-­second position is presented is no indication that those positions seem different to you. On the contrary, if they did seem different, you would always give a different signal. So far, then, from its being a sufficient reason for regarding the tachometer as able to discriminate between a pair of impulses that it some­ times issues different readings in response to them, the fact that it does not always do so should be regarded as decisive for regarding the relation in which they stand to it as an analogue of indiscriminability. And the modelling of non-­transitivity is then provided by the reflection that, among three dis­ tinct impulses, the tachometer may only sometimes respond differentially with respect to the first and second, and only sometimes respond differentially with

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Further Reflections on the Sorites Paradox  123 respect to the second and third, but always respond differentially with respect to the first and third.6 I conclude that this class of Sorites paradoxes need not detain us. Their major premises are false, and false in a way which classical two-­valued seman­ tics is quite adequate to describe. The details of why they are false will, of course, depend upon the character of the particular ‘instruments’ concerned; and I am not seriously suggesting that the tachometer provides a satisfactory model of all aspects of human competence with colour vocabulary. My point is only that, once the first claim of the governing view is abandoned, the re­sidual case which Peacocke thinks he sees for supposing C to be Sorites-­ susceptible is perfectly matched by the Tachometer paradox; and the model solution sketched for the latter depends on no feature of the tachometer with­ out an analogue in the ‘instrument’ constituted by a human being responding to colour. Accordingly, while different things might need to be said about why precisely the major premise in the C-paradox is false, we can at least under­ stand why we shall not be committed to its truth simply by supposing that our descriptions of colour are nothing but responses to visual stimuli among which our discriminations are not everywhere transitive.

IV Tolerance and Observationality Peacocke (1981, p. 124) concludes his objection to the tachometer analogy with the complaint that, if it were apt, there would be no difficulty in the idea of an observational predicate . . . def­ in­ite­ly applying to one but not to the other of two objects which produce experiences which are not discriminably different in quality. But in fact we can make no sense of this idea. 6  It may occur to the reader that, as I have presented the example, the raising and lowering of hands will involve judgements in which an observational relation—‘seems to be differently positioned from’—rather than an observational predicate is an ingredient. I do not think this qualifies the force of the example, however. First, the holding aloft of zero, one, or two hands—while undoubtedly ‘observa­ tional’ responses in the sense which currently occupies us—would have to be encashed in terms some­ what like ‘looks to be in the original position’, ‘looks to be in the four-­second position’, and ‘looks to be in the eight-­second position’, respectively. It is not obvious that these are not properly regarded as observational predicates. Second, the point that the example is being used to make—that a tendency sometimes to respond differentially to a pair of stimuli need imply no capacity to tell them apart— could be illustrated in lots of other ways in any case.

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124  The Riddle of Vagueness Undoubtedly, we can make no sense of it if observational predicates are ­governed by the rule that they apply to both, if to either, of any pair of observationally indistinguishable items. But that, I want to suggest, is just the misunderstanding of what an observational predicate is which the governing view encourages. The question is therefore, what should we replace it with? Of course, the question whether any vocabulary is worth distinguishing as ‘observational’ has been intensively debated by twentieth-­century philosophy of science; and the received wisdom has been in favour of a negative answer. Still, it is natural to feel, there is a class of predicates—those expressing c­ olours, musical pitch, or perhaps any Lockean secondary quality—for which Peacocke’s thought has prima facie force. Whether we style these predi­ cates as observational or not, how exactly are they equipped to effect the sort of distinction, among qualitatively indistinguishable items, whose intelligibil­ ity Peacocke is questioning? We shall get an inkling about how an answer to this question might go by reflecting on what may seem a fairly obvious lacuna in the response to the Sorites which I have been canvassing. The response claims, in essentials, that the motivation for the major premises in (a large class of) Sorites paradoxes is misguided; that the pressure to assent to the major premises is generated only by the mistaken assumptions of the govern­ ing view. But can it suffice for a solution just to undercut the motivation for the major premise in, for example, the Sorites paradox for ‘red’. Do we not need, in addition, to disclose some specific fault with the major premise? If there is really no paradox, then that quantified conditional cannot be true; and we cannot rest content until we have explained exactly why not, and, if neces­ sary, have devised a semantic apparatus in terms of which the manner of its untruth can be properly described. We are now in a position to see how at least the outline of such an account might run. Let us essay to see ourselves, in accordance with rejection (or, at least, severe qualification) of the first claim of the governing view, on the model of signalling instruments for qualities like red—albeit voluntary signal­ lers. Then there ought to be the same kind of connection between an item’s being red and the description to which we are prepared to assent, when appro­ priately placed and functioning normally in normal circumstances, as obtains between, say, an engine’s operating at 4,500 rpm and the reading issued by a number of correctly connected tachometers which are functioning properly and are free from external interference. This connection is actually such as to generate a truth condition: for a particular reading to be correct is for it to be the case that the mean value of the readings issued by sufficiently many appro­ priate instruments, which are appropriately situated, functioning properly, and

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Further Reflections on the Sorites Paradox  125 free from external interference, will coincide with that reading. So, too, for red, or any quality of the kind which Peacocke’s quoted thought seems plausible. For an object to be (definitely) red is for it to be the case that the opinion of each of a sufficient number of competent and attentive subjects who are appropriately situated to command a clear perception of the object, function­ ing normally, and free from interfering background beliefs—for instance, some doubt about their situational competence—would be that it was red. What follows? Well, one thing we know—and, having disposed of the Tachometer paradox, can bring to bear with a clear conscience—is that if a Sorites series of indistinguishable colour patches is run past a group of such subjects under the specified conditions, and they are agreed about the redness of the initial elements, there will be, as the series moves towards orange, a specific first patch over which the consensus fractures. The occurrence of the fracture need not raise any doubt about the continuing competence of the subjects or about the satisfaction of the specified conditions at that point; nor need any of the subjects be able to re-­identify that patch if it is subsequently re-­presented with its immediate predecessor in a different setting. So the upshot is simple. If observational qualities, whatever should be their proper characterization, essentially sustain the sort of equation which the tachometer analogy suggests—if their application is a function of what competent ob­ser­ vers, under suitable conditions, and so on, will assent to—then a predication of red, or any observational quality, fails of definite truth as soon as the right hand side of the equation fails of truth. We have therefore to acknowledge, surprising as it may seem, that a Sorites series of indistinguishable colour patches can contain a last patch which is definitely red: it will be a patch about whose redness there is a consensus meeting the described conditions, and its immediate successor will be a patch about which the consensus breaks down in a way that still observes those conditions. It may be difficult or impossible to identify such a patch in a particular case, either because it is difficult to verify that consensus has broken down in a way which still observes the conditions, or because it is known that one or more of the conditions is unsatisfied. But it remains that there can be such a patch; that it can be identified under favourable circumstances; and that, even in cir­ cumstances which are not favourable, when the consensus has broken down for irrelevant reasons, it is legitimate to suppose that there is such a patch which could have been identified had the conditions been different. No doubt, it would usually be of absolutely no interest to effect such an identification. And it should be noted that there is, a priori, no reason to suppose that ‘the last definitely red patch’ would turn out to have a stable reference; if it did not,

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126  The Riddle of Vagueness that would disclose an element of context relativity in the concept of red which we normally do not suspect. Provided, then, that observational qualities indeed sustain the relevant sort of equation, we can, more than undercut the motivation for believing them to be true, explain how specifically the major premises in Sorites paradoxes for observational predicates fail of truth. (The question remains how such ex­plan­ ations could proceed for other kinds of Sorites—I shall return to it below.) We can also make sense of the idea of an observational predicate definitely apply­ ing only to one of a pair of phenomenally indiscriminable objects, which Peacocke claimed was unintelligible. What is unintelligible is how someone might tell just by inspection of a pair of indiscriminable items that they in fact mark the watershed, in a particular context, between what is definitely red and what is not. But Peacocke’s claim follows from that only if we suppose that any observational predicate worth the title must be such that this distinc­ tion can be drawn purely observationally. And that is the mistake: the raising of an arm, in the example of the slowly moving pointer, is a gesture made purely in response to how things look—it is explicitly instructed that this is how it should be made—and so may be regarded as analogous to the applica­ tion of an expression which is observational in the sense which Peacocke intends. The fact remains that, if we suppose that having a single arm in the air is definitely an appropriate response only in situations when—supposing the satisfaction of all the relevant conditions—the subject always has a single arm raised, the observationality of the response is consistent with the exist­ ence of a sharp boundary to the cases where it is definitely appropriate. On the account of observationality which figures in Dummett (1975) and in my earlier papers, a predicate is observational if whether it applies to an object can invariably be determined just by (unaided) observation of that object. It seems to follow immediately that any such predicate must be ap­plic­ able to both, if to either, of any pair of objects which (unaided) observation cannot tell apart; so that the account renders all observational predicates tol­ erant with respect to indiscriminable change and thus takes us straight over the precipice. But there is an unclarity here: is it intended that the distinction between definite cases of application and all others be observationally decidable,7 or merely that between definite cases of application and those 7  Note that observational decidability, in the intended sense, does not preclude the intrusion of considerations from scientific theory into our conception of when ‘unaided observation’ is to be able to deliver the correct verdict. So ‘red’ can count as observationally decidable consistently with the occur­ rence of red illusions associated, e.g., with red shift and the Doppler effect on light traversing galactic distances. Observational decidability, like all concepts of possibility, is a concept of what is possible

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Further Reflections on the Sorites Paradox  127 where the predicate definitely fails to apply? A little reflection suggests that what we really want is the second, weaker, version. The observationality of ‘red’, for instance, should entail only that something is red only if any suitable subject may, in appropriate circumstances, observe it to be so; and that some­ thing is not red only if any such subject may, likewise, in appropriate circum­ stances, observe it not to be red. A predicate’s being observational in this sense need not entail that a distinction can always be drawn, just by observation of the objects concerned, between cases to which it definitely applies and cases of any other sort. Indeed, there is something absurd about the idea that that distinction might be observationally decidable, since borderline cases of ‘red’, for instance, are exactly cases about whose classification observation, even in  appropriate circumstances, leaves at least some suitable subjects essen­ tially unsure. Let me pause to review the two principal suggestions of this section so far. The first is that it is not, contrary to Peacocke’s claim, unintelligible how any­ thing worth regarding as an observational predicate might definitely apply only to one of a pair of observationally indistinguishable items; on the con­ trary, the possibility is implicit in the consideration (to which we led by con­ sid­er­ation of the analogy with the tachometer but which stands independently) that it is a necessary condition for an observational predicate to apply to a particular item that any suitable subject who observes the item under appro­ priate circumstances will judge that it does so. The second is that it is an error to think that Peacocke’s idea follows if we agree that anything worth regarding as an observational predicate must be such that, whether or not it applies to an object can be determined, in appropriate circumstances, and so on, just by observation of it. What that commits us to is only the second, weaker prin­ ciple sketched in the preceding paragraph. To be sure, the governing view arguably provides grounds for thinking that ‘red’ is observational in a sense which subserves the stronger, paradoxical principle. We shall review those grounds in Section VII. There is much more that needs to be said about the notion of an observa­ tional predicate. Here, though, I can only briefly canvass two further sugges­ tions. First, on the relation between the two thoughts just summarized. There is no objection, one would imagine, to interpreting the reference to ‘appropriate circumstances’ in the weaker principle along exactly the lines featured in the biconditional for qualities like red to which consideration of the tachometer in appropriate circumstances; but our idea of which circumstances are appropriate may well be the­or­ et­ic­al­ly informed.

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128  The Riddle of Vagueness analogy led us. ‘Appropriate circumstances’ are those where the subject is appropriately placed to observe the object concerned; ‘suitable subjects’ are competent, attentive, perceptually normal subjects who are free of interfering background beliefs. So the weaker principle, so interpreted, has just the effect in its own right which the biconditional had: reason to doubt that competent, and so on, subjects would, in appropriate circumstances, judge an object to be red—reason afforded, perhaps, by such subjects actually failing to do so—is reason to doubt that the object should be classified as red. Thus, the weaker principle, so far from motivating acceptance of the major premises for Sorites paradoxes involving observational predicates, actually subserves the form of rejection of such premises suggested by the analogy with the tachometer. Note, therefore, the perspective in which the stronger principle is now placed. The stronger principle differs from the weaker only in its second clause. But the effect of the weaker principle just described is wholly owing to its first clause. So the stronger principle likewise has that effect: to endorse the principle is to be committed to treating a lack of consensus among compe­ tent, and so on, subjects in appropriate circumstances as a sufficient reason for doubting that something is definitely red, even if the object in question is indistinguishable from one about which a consensus obtains. If, therefore, as it seems, the stronger principle also entails that ‘definitely red’ tolerates indis­ criminable difference, the conclusion is forced that the ingredient clauses of the stronger principle are actually incoherent: collectively they entail that a difference must always be apparent to observation between anything which is definitely red and anything of any other sort; but the first clause also entails that different responses by competent, and so on, subjects may suffice for such a difference, even if the items in question are observationally in­dis­tin­guish­ able. This is a quite different conclusion from anything that might be inferred from the role of the stronger principle in the generation of one kind of Sorites paradox for ‘red’. For one thing, that paradox depends, in addition, on assumption of the non-­transitivity of indiscriminability; for another, the kind of semantic incoherence which the paradox has been thought—by Dummett, for instance—to disclose, must not be confused with the kind of incoherence just bruited. It is not inconsistent to suppose that there might be semantically incoherent predicates; but no predicate can be observational in a sense which subserves the stronger principle—at least if the foregoing is correct—since to be so would be to meet an inconsistent specification.8 8  Peacocke’s discussion takes a rather strange turn just after the passage quoted on p. 123, on which he proceeds to base a refinement of the idea of an observational predicate (see Peacocke  1981,

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Further Reflections on the Sorites Paradox  129 Finally, let me sketch a possible connection with a proposal for the charac­ terization of observational expressions which I have entertained elsewhere (Wright 1986b, especially pp. 276–80). Observational expressions are those which connect most intimately and immediately with our experience. The account of observationality given by Dummett (and Peacocke’s purported refinement9) both locate this intimate, immediate connection in the conditions of application of the expression. Observational expressions are to apply, or not, in virtue of how things appear. But an alternative would be to seek an account by exploration, rather, of the conditions of understanding. To be sure, items qualified by an observational predicate ought, under appropriate circum­ stances, to present a distinctive appearance; and understanding the predi­cate ought to involve attaching a proper significance to the appearance, or range of appearances, associated in this way with it. But there are some expressions—a good example is a natural kind term like ‘banana’—which we would not wish to regard as observational because, for example, they carry implications about constitution and origin, even though the things they qualify have distinctive enough appearances. And it is striking that somebody’s use of an expression

pp. 127–8). According to the refinement, observational predicates are distinguished by their satisfac­ tion of the condition that, with respect to any pair of indiscriminable items x and y, it cannot be the case that such a predicate definitely applies to x but does not definitely apply to y; formally,

Ixy → ¬(Def (Fx ) & ¬Def (Fy ))

How exactly does this proposal prevent observational predicates from being genetically Sorites-­ susceptible? Peacocke’s answer, in a familiar tradition, is that such predicates are predicates of degree. His own positive response to the paradox, to be reviewed in the sequel, is that the consequents of the conditionals which are instances of the major premise will typically be slightly less true, accordingly, than the antecedents. He is thus able to maintain that the consequent of the above conditional may have a lower degree of truth than its antecedent consistently with the conditional’s retaining its dis­ tinctive acceptability for observational expressions; it will be so acceptable just in case, whenever the antecedent is true, the consequent is almost true. Now, the consequent will, presumably, be almost true just in case the conjunction Def (Fx) & ¬Def (Fy) is almost false; which, intuitively, should be the situ­ ation when one, or both of the conjuncts are almost false, although neither is actually false. Hence, if Def (Fx) is true, ¬Def (Fy) will be almost false, so Def (Fy) will be almost true. But that, on Peacocke’s view, will be the most that can be inferred from the acceptability of the original conditional, the indis­ criminability of x and y, and the truth of Def(Fx). So the paradox is obstructed. (The reader will be able to see how similar remarks would apply to the attempt to work the paradox from right to left, as it were, taking as premise ¬Def (Fy).) What immediately strikes one, however, is that, once Peacocke has a conditional which allows of almost-­truth, there is no need for this reformulated account of observationality. For the mere almost-­ truth of the conditional

Ixy → (Def (Fx ) → Def (Fy ))

would not subserve the Sorites paradox either. The way in which Peacocke makes his preferred account of observationality avoid paradox is equally open to the original, seemingly inevitably para­ doxical version. Still, that is just a quibble. The substantive point remains that, if we wish to avoid traf­ ficking in degrees of truth, we owe a different kind of account of what observationality is—if not the sort outlined in the text, than another. 9  See n. 8.

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130  The Riddle of Vagueness could be quite properly responsive to the kind of appearances associated with it and yet they might not understand it precisely because ignorant of such additional implications. A natural suggestion, then, is that observational expressions are marked off from this wider class precisely by the circumstance that, while anyone who understands such an expression must display an appro­ priate sensitivity to a certain distinctive range of appearances with which its application is standardly associated; there must in add­ition be no space for the possibility of their doing so yet failing to understand the expression. ‘Gene’, ‘torque converter’, and ‘pulsar’ fail to qualify as observational because there is no particular kind of appearance possessed by the things to which they apply; ‘banana’ and ‘human being’ fail to qualify because someone who had no grasp of the kinds of thing which bananas and human beings respectively are might still be quite clear about the way they tend to look, and might show as much in their employment of ‘banana’ and ‘human being’. This indicates, at most, merely the rough shape which a satisfactory account of observationality might assume.10 What is salient is that any expression which conforms to it is presumably going to sustain something very close to the clause which is common to the weaker and stronger principles which we have been discussing. If ‘φ’ is such an expression, for instance, then it follows from the sketched account that anything which it qualifies will present, under appropriate circumstances, a distinctive kind of appearance to which anyone suitable who understands ‘φ’ will be appropriately responsive; that is, will be prepared to judge that the item is so qualified provided other appropriate conditions are met. Natural kind concepts, like ‘banana’ and ‘human being’, will likewise sustain such clauses, provided it really is essential that their instances have a certain sort of appearance. (If it is not, then they should be classified with ‘pulsar’ and ‘torque converter’ after all.) The question arises, therefore, whether the add­ition of a stipulation that such sensitivity is to suffice for understanding ‘φ’ though not ‘banana’,11 somehow imports commitment to the tolerance of ‘φ’ with respect to indiscriminable change. It is clear that the stipulation pre-­empts the 10  This proposal is actually very close to the account of observationality offered by Peacocke him­ self (1983, ch. 4). Peacocke is there concerned not with the Sorites but with the prevailing scepticism among philosophers of science concerning the very idea of an observational concept/statement. I doubt if I should have conceived the present suggestion without the benefit of conversations and cor­ respondence with Peacocke prior to the publication of his 1983 work. But the reader should be wary of assuming that Peacocke would endorse my suggestion; it contains, in particular, no explicit ­analogue of the play with experience which Peacocke’s treatment involves. 11  The proposal that if someone’s use of ‘red’ manifests an appropriate sensitivity to colour appear­ ances, that suffices for their understanding ‘red’, leaves us free, of course, to ponder just what kind of sensitivity is appropriate. In particular, we are free to insist that the sensitivity be informed by the conception that red is a property of the object of predication, and so to require, for example, that the subject acknowledge certain a priori principles which evince this conception—for instance, that red

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Further Reflections on the Sorites Paradox  131 kind of explanation which can be given, in the case of natural kind predicates, of how they may distinguish between observationally indiscriminable items. An artefact—a ‘fool’s banana’, for instance—may be in­dis­tin­guish­able from an instance of natural kind; understanding ‘banana’ involves grasping this possi­ bility. Needless to say, pre-­empting one kind of explanation need not be to foreclose on all, but the question is still outstanding: how might the cor­res­ pond­ing explanation go in the case of predicates which conform to the pro­ posed account? How could the ‘appropriate kind of responsiveness’ operate selectively among indiscriminabilia? If I am right that the kind of account of observationality mooted will entail that observational expressions comply with the first clause of the stronger (and weaker) principle, then what is asked for is, in effect, an explanation of why the mooted account does not entail their com­ pliance with the second clause of the stronger principle—the clause whose inclusion makes it impossibly strong. I shall revert to the matter in Section VII.

V Peacocke’s Positive Proposals: Degree-­Theoretic Approaches Vague expressions tend to be associated with a comparative: red things can be more or less red. It is natural to suppose that, if something can be more or less φ, then that gives sense to the idea that the simple positive predication ‘. . . is φ’ can be more or less true of it. Peacocke’s idea is that, once we take degrees of truth seriously, we can diagnose the Sorites paradox as arising from the attri­ bution of contradictory properties to the conditional in the major premise. Specifically, the paradox will result from assuming simultaneously that modus ponens inferences are valid without restriction for the conditional and that it suffices for the truth of a conditional that its antecedent and consequent enjoy degrees of truth which, if not identical, are at least extremely close. In order to apply this proposal to the case of colour predicates, and other observational expressions, we shall require, of course, that the degrees of truth of ‘a is red’ and ‘b is red’, for instance, may be different even though a and b are indis­ criminable in point of colour. Peacocke is thus committed to utilizing a notion akin to Goodman’s idea of a quale.12

things retain their colour when unperceived. Analogues of all such principles may be expected to hold for natural kind terms also, so there is no threat here of compromise of the contrast drawn in the text. 12  Locus classicus: Goodman (1951a, ch. IX, sect. 2).

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132  The Riddle of Vagueness Appealing as it may seem, Peacocke’s treatment is open to a number of objections which are, I think, collectively decisive against any similar degree-­theoretic account. More specifically, it is only the third and fifth of the six objections which follow which, so far as I can see, relate to features of Peacocke’s paper which connect inessentially with his overall approach. (i) Can a degree-­theoretic account explain the plausibility of the major premises? There is no difficulty, of course, with the usual, quantified condi­ tional form of premise. The explanation will claim that each instance Fa → Fa′, of (∀x)(Fx → Fx′) is almost true: that its consequent enjoys a degree of truth ever so nearly but not quite as great as that of its antecedent. And this claim will then be followed, presumably, by a stipulation that the degree of truth of any universally quantified statement is the minimum of the degrees of truth enjoyed by its instances. There could be some debate about this stipulation for the universal ­quantifier—why should it not be, for example, that (∀x)(Fx → Fx′) takes on the sum of the errors of its instances instead? But let us now forgo that debate. More important is the reflection that the major premise does not need to be conditional at all. In the case of the Sorites series of indiscriminable colour patches, for instance, we could just as well take it in the form (∀x ) ¬(red(x )& ¬red(x′)) All the ways of making the conditional form of major premise seem intui­ tively plausible would be applicable to this conjunctive form.13 Now, we have already seen, in effect, how, if we assume that each instance of the conjunctive major premise is merely almost true, it follows quite naturally that successive inferences from red(x) to red(x′), or from ¬red(x′) to ¬red(x) will involve marginally diminishing degrees of truth.14 But Peacocke needs to explain, in addition, with what right such a conjunctive major premise may be regarded as almost true; otherwise he cannot explain its plausibility, or duly ac­know­ ledge the force of the arguments which seem to sustain it. However, the attempt to extend his account in the most natural way fails to produce the right results. Plainly we need degree-­theoretic matrices for negation and con­ junction of such a sort that ¬(P & ¬Q) turns out to be almost true whenever the truth value of Q is almost as high as that of P. Assuming that the matrix for negation yields that ¬A is almost true just in case A is almost false, it 13  And it would be unwise to rule out the possibility of disjunctive formulations as well, although (∀x)(¬red(x) ∨ red(x′))—the simplest disjunctive analogue of the conditional and conjunctive ­formulations—would intuitively offend against the fact that ‘red’ possesses borderline cases. 14  See n. 8.

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Further Reflections on the Sorites Paradox  133 follows that P  & ¬Q has to be almost false in that same case. Classically, a conjunction is false just in case at least one conjunct is false; so a conjunction which marginally fails to meet the minimum standard for falsity ought to be one in which at least one of the conjuncts is almost false while the degree of falsity of the other is no greater. The result—if I may be forgiven for spelling it out in detail—is that P & ¬Q cannot be almost false if P has a middling sort of degree of truth and Q, as hypothesized, a marginally smaller degree. For, in that case, P is not almost false and neither, by the sort of matrix for negation envisaged, is ¬Q. Accordingly, although Peacocke displays the materials suf­ ficient to explain how, if it is agreed that the instances of a quantified conjunc­ tive premise are all almost true, reiterated applications of modus ponendo tollens will lead to successive conclusions of a deteriorating degree of truth, he has provided absolutely no explanation of why the close proximity in degree of truth of successive Fx and Fx′ should make for the almost-­truth of a quan­ tified conjunctive major premise, even granted the stipulation for the univer­ sal quantifier noted above. And it is unclear what direction a degree-­theoretic account might take which would fare better on the point. (ii) Is the notion of difference of degree which the account demands fully intelligible? Since it is essential to Peacocke’s purposes that the degrees of redness of a pair of indiscriminable colour patches can be different, an account of identity and distinction among degrees is called for which sustains that possibility and, indeed, enables us to regard the degree of redness as diminishing step by step as we move along the relevant sort of Sorites series. Peacocke’s offering (1983, p. 126), inspired by Goodman, is (a) x and y are red to the same degree if and only if any colour matching the colour of x matches the colour of y and vice versa; and (b) The degree to which x is red is greater than the degree to which y is red if and only if some colour matching the colour of x is redder than the ­colour of y. One evident problem with clause (b) is that it will deliver what is wanted only if it is true to say of some pairs of colour patches which are just discriminable, that is, form the first and third members of a non-­transitively matching trio, that one is redder than the other. I do not know why Peacocke thinks that this is so. It seems certain that, with shades of colour as close as that, no one would

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134  The Riddle of Vagueness feel able to say that either in particular was redder than the other when presented with them in isolation; and presentation in the context of a Sorites series in which one appeared later than the other would be of no help unless one was assured that the series was moving uniformly away from red (which, in view of the variability of colour in several parameters—saturation, intensity, hue need not be so.) But that assurance is precisely the assurance that, of each suc­ cessive just-­discriminable pair of patches, the first is redder than the second— and it is the content of that claim in such cases that is at issue. Peacocke offers nothing by way of an account of the comparative ‘is redder than’ utilized in clause (b). He is, however, making a substantial assumption about it. Let us say a relation R is counter-­transitive with respect to a domain of entities just in case, whenever Rxy holds good for x and y in that domain, x also bears the ancestral of the contrary of R to y, the intermediate links in the ances­ tral chain being supplied from the domain in question. Thus, with respect to a suitable domain of colour patches, the relation of discriminability is counter-­ transitive, since any pair of discriminable colour patches can be linked by a chain of pairwise indiscriminable ones. Peacocke’s assumption, then, is that the relation ‘is redder than’ is not counter-­transitive with respect to any domain of colour patches, any pair of which are at least just-­discriminable. I cannot see that our intuitive conception of the relation val­id­ates this assumption; it seems evident that we make no use of the comparative except in cases involving differ­ ences much greater than just-­discriminability. In any case, Peacocke needs a demonstration that that is not so, which would presumably have to draw on the analysis of this and similar comparatives which he does not give. Unless the lacuna can be filled, the degree-­theoretic treatment simply cannot proceed as far as Sorites paradoxes involving indiscriminability are concerned. (iii) Can the notion of a colour which features in Peacocke’s clauses (a) and (b) be satisfactorily accounted for? It seems doubtful whether the idea can rea­ sonably be taken as primitive in the present context. But we strike trouble pretty speedily as soon as we attempt to explain what sort of things the colours are which constitute the range of the quantifiers in Peacocke’s two clauses. A natural proposal would be to treat colours as Goodmanian qualia so that Colour c1, is the same as c2 just in case any colour matching c1 matches c2 and vice versa. But this has the drawback that it quantifies over colours in attempting to explain the criterion of identity distinctive of the sort. That, admittedly, need not vitiate an explanation whose purpose is simply to characterize a feature of a concept rather than to introduce it. Indeed, it is arguable that even certain

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Further Reflections on the Sorites Paradox  135 introductory explanations can afford this kind of apparent circularity (see Wright 1983, ch. 4, pp. 180–4, n. 8). But it is doubtful whether anything mitigates the charge of circularity in the present case. We do need an introductory explanation of qualia in general and of colours, so understood, in particular. And introductory explanations which involve quantification over instances of the concept we are trying to explain can get away with that only when it is possible for a trainee, so far innocent of the concept, to understand statements involving such quantifications. If a case can be made for saying that that is so, it will be because the quantification is involved in the formulation of condi­ tions which can be recognized to be true or false of a domain of objects quite independently of full knowledge of the sortal distinctions exemplified in that domain. (A trivial example would be (∀x)(x = x).) The proposed account of colour identity, and indeed Goodman’s account of identity for qualia gener­ ally, do not come into that category. The obvious response is to take the matching relation as obtaining between coloured objects rather than colours themselves and modify the account to read something like The colour of x is the same as the colour of y just in case it is not possible that there should be a z which matches one but not the other, where x, y, and z are all variables for individuals. But, of course, it is always possible that there should be such a z—if x and y are allowed to shift their colour! If we then stipulate that the colours are to remain the same across the relevant range of possible circumstances, we once again use the notion we are trying to explain. And, if we leave the modal operator out, the precise colour of an object will turn out to be relative to what other objects happen to exist: objects which are currently the same in colour may suddenly become distinct in virtue of the creation of some third object which sets up non-­transitive matching, and may once again come to coincide in colour if that object is destroyed—all this without themselves undergoing any change.15

15  Peacocke is fully aware of the difficulties which beset attempts to elucidate the kind of notion of identity of colour which he wants, but he conceals their force from himself with a rather curious thought, namely: ‘Since these difficulties equally effect sharp observational notions . . . [they] . . . cannot be the source of the Sorites Paradox . . .’ (Peacocke 1981, p. 126). I am not sure what the sharp observa­ tional notions are which he has in mind. But the excuse would be appropriate, anyway, only if it was necessary to have recourse to Goodmanian ideas in order to develop the Sorites paradox in the first place. To someone who then thought that the solution to the paradox might be to make trouble for such ideas, Peacocke’s remarks could possibly be appropriate. But the fact is that the ideas in question are not involved in the paradox in that sort of way at all: what they are involved in is Peacocke’s attempted solution, and any difficulties which they encounter are difficulties for it.

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136  The Riddle of Vagueness (iv) Can degree-­theoretic approaches fulfil their intuitive promise? Let us be clearer what makes such approaches seem promising. The motivation is not merely the thought that if the relevant expressions admit degrees of ap­ plic­ abil­ ity, the major premises can fail of strict truth. It goes deeper. Consider the series of conditionals that make up the intervening links in a Sorites. The dilemma, basically, is that, if we say that they are all true, we are bereft of the means to explain how the transition from, for example, redness to non-­redness is effected; and, if we say that one of them is false—or, anyway, other than true—we secure an explanation only at the cost of imputing a determinate sharp boundary when none seems to exist. The preceding section tried to explain how it might be that our scruples about the second horn of this dilemma are ill-­founded in an important class of cases. But, whether in that way or another, the apparent problem is to explain how the change from redness to non-­redness is effected across these conditionals. Redness disap­ pears as we work our way along the series—if there is not a first point at which the patch has ceased to be definitely red, how is the trick pulled? What we would most like to be able to say is exactly that the change takes place at no particular point. And, if that idea can be saved from incoherence at all, some­ thing close to Peacocke’s suggestion seems to be the only way of doing so: each of the conditionals contributes towards the change, but contributes too small an amount to make it worth saying, in any practical context, that its consequent is less acceptable than its antecedent. It then needs to be recog­ nized merely that this notion of the acceptability of a conditional will not sustain indefinite reiteration of modus ponens, and the paradox is solved. The real attraction, then, of the degree-­theoretic approach is that it promises simultaneously to explain how the transition from Fs to non-Fs is effected in a Sorites series and so to block the paradox, to explain the plausibility of the prem­ ises which generate it, yet to avoid postulation of any determinate boundary of the sort that is apt to strike us as fictitious. So it is important to realize that, when matters are inspected closely, this promise cannot be redeemed by Peacocke’s account (nor, so far as I can see, by any degree-­theoretic approach) irrespective of whatever other objections there may be to it. A major difficulty emerges when we reflect that there is intuitively no pinnacle, as we might put it, to the scale of degrees to which something can be red or exemplify any other vague concept of degree. There is, for example, no more-­or-­less-­red among crimson, scarlet, and vermilion.16 In general, any vague concept F 16  No doubt many people’s grasp of these distinctions is somewhat shaky; the words derive from the various animal, vegetable, and mineral origins of the strikingly red dyes to which they were once

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Further Reflections on the Sorites Paradox  137 admits of quite a wide variety of discernible cases all of which are definitely and absolutely F. Thus, although scarlet is more like orange than vermilion is, it is not less red: both are paradigms of what is absolutely red. The result is that, if we start the red-­orange Sorites series of matching colour patches with a vermilion one, we have to say that the conditionals which correspond to the early transitions in the series involve no diminution of degree of redness at all. And now we can scarcely forbear to ask how the transition is effected from conditionals of this kind to conditionals where such a dim­in­ution is involved. If it takes place at some specific nth conditional, that is as much as to say that there is a first patch whose degree of redness is not absolute; and, if we are willing to admit that, it is not clear why we should scruple over the idea that there is a first patch which is not definitely red—and so rescind the con­sid­er­ ation which gave the degree-­theoretic approach its special appeal. Again, what we would like to be able to say is that the transition from the one kind of conditional to the other takes place at no particular point but still takes place. The obvious ploy would be to attempt to activate the degree-­ theoretic account at a higher level, considering, for example, the series of con­ ditionals of the form: If Cn is absolutely acceptable, then Cn+1 is absolutely acceptable, where Ck is the kth conditional called on in the Sorites paradox for ‘red’, and such a conditional is absolutely acceptable just in case the degree of truth of its consequent is no smaller than the degree of truth of its antecedent. But the trouble with this is twofold. First, the notion of a degree of absolute acceptability, in contrast with a degree of redness, seems to make no sense—it is like Orwell’s joke about political equality. Second, even if it can be made to make sense, the problem recurs; for sufficiently small values of n, such conditionals will themselves be absolutely acceptable, so, if those taking larger values are not, when and how is the transi­ tion between the two kinds of cases effected? I suggest the conclusion not merely that Peacocke’s account fails to solve the original dilemma in the attractive-­ seeming way promised, but that there is no solution to the dilemma of that sort. (v) C is not a predicate of degree. Peacocke’s predicate, C, you will recall, applies to an object just in case the community will agree in calling it red. Consensus, of course, admits of degrees; but, as the notion figures in Peacocke’s definition of C, it is intended to be complete and absolute. But it was, in Peacocke’s view, the Sorites paradox for C which showed the error of

applied. ‘Crimson’, e.g., was the name of a dye derived from the pregnant female Kermes beetle, which in turn took its name from the Kermes oak, native to southern Europe and North Africa, on which it feeds.

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138  The Riddle of Vagueness my diagnosis (Wright 1975; 1976) and called for a different account. Yet Peacocke grants that C is not a predicate of degree, and indeed allows that it is precise, determinately either applying or failing to apply to any particular object. Ought he not, then, to admit that his own proposal can cope no better with the C-paradox than he believes mine did? His response is perplexing. He writes: [the degree-­theoretic approach] covers the metalinguistic paradox in­dir­ ect­ly. Sometimes we have to state a theory about the extension of a sharp predicate by using a vague predicate. In particular is this true of the sharp predicate C and the vague predicate ‘red’. We have to say that: Any object which is red the community will agree in calling ‘red’; Any object which is not red the community will not agree in calling ‘red’. I suggest that the only reason that we feel tempted to accept the major prem­ ise of the metalinguistic paradox is that we employ reasoning using condi­ tions with properties (1) and (2) [namely, unrestricted iterativity of modus ponens, and ‘if Fa, then Fb’ true whenever F is observational and a and b are indistinguishable] and which contain ‘red’, and then go on to apply these two general principles to draw conclusions which contain the predi­cate C. (Peacocke 1981, p. 128)

Otherwise, Peacocke claims, there is no reason to believe the major premise of the C-paradox. Rather, since C is sharp, some instance of this premise must be false if the principles displayed earlier [in the immediately preceding quotation] are true. Since C itself is not an  observational predicate, there is no pressure against this conclusion . . . (Peacocke 1981, p. 129)

What is happening here? The thought, I take it, is that in the presence of the two cited principles determining the extension of C, we can negotiate the transition, for arbitrary indistinguishable a and b, from Ca to Cb provided we take it that the indistinguishability of a and b verifies ‘if a is red, b is red’ and this conditional sustains the modus ponens inference. The point is immediate if we assume that Peacocke intends the two principles cited to have the effect that ‘red’ and C have the same extension—but that he cites two principles, rather than just the biconditional of the first, suggests that he may intend, rather, a relation like that between the extension of ‘red’ and the extension of

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Further Reflections on the Sorites Paradox  139 an admissible sharpening of it (in the sense of Fine  1975).17 Even so, the sharpness of C, and the consequent validity of double negation elimination, affords us Cx → ¬¬ red(x) by contraposition on the second principle, and ¬¬ red(x) → Cx by double contraposition on the first. Since we have ¬¬ red(a) → ¬¬ red(b) by double contraposition on the conditional for ‘red’, the reasoning thus goes through. Peacocke’s claim, then, is presumably that, in this precise sense, the major premise for the C-paradox stands or falls with that for the ‘red’ paradox. But, of course, if that is true, the C-paradox cannot be a special problem for any­ one! In particular, it cannot be a special problem for the view that the major premise for ‘red’ derives a specious plausibility from the governing view. By attempting to treat the C-paradox in this way, Peacocke undercuts his case for thinking that a different approach to the Sorites was called for in the first place. The wood gets further tangled when we ask whether Peacocke himself accepts the two principles (hedged about, as he presumably intends them to be understood, by appropriate constraints on the normality and competence of the members of the community and the appropriateness of the conditions of observation). The argument of Section  IV would have it that, so hedged

17  Fine (1975, especially pp. 271 ff.). Cf. Dummett (1975, pp. 310–11). A sharpening of a vague predicate is a stipulation which renders it (more) precise, and is admissible just in case it respects uncontroversial cases, i.e., does not reclassify any definite Fs or definite non-Fs. Then if S is such a sharpening of ‘red’, and ‘reds’ expresses the result, it seems we can affirm both If a is red, a is reds, and If a is not red, a is not reds, although the converse of either may fail since both red items and non-­red items may be borderline cases of ‘red’. Likewise things that are not C may—since even an ideal community may fail to agree about the redness of borderline cases—nevertheless not be non-­red, so the converse of Peacocke’s second principle fails in similar manner. Not so the converse of his first principle, presumably, but that is not really a point of disanalogy: C corresponds to the, as it were, left-­most admissible sharpening of ‘red’, which classifies all but the definitely red things as falling under the contrary of the sharp predicate. I have unfortunately no space in this chapter to discuss the bearing on the Sorites paradox of Fine’s approach to vagueness. The basic idea, as is familiar, is that sentences containing vague expressions are true only if they remain so under every admissible sharpening of those expressions. Hence the neg­ ations of major premises for Sorites paradoxes will generally turn out true—without implication of a ‘fictitious boundary’—because, for instance, no matter how we sharpen ‘red’, there will be in any appropriate series—if the sharpening is both admissible and complete—a determinate last item to which the sharpened predicate applies immediately followed by a first item to which it does not. I regard the approach as unsatisfactory principally because there is no evidence that our use of sen­ tences containing vague expressions is sensitive to such truth conditions, and plenty of evidence that it is not—for instance, a widespread squeamishness about Excluded Middle; and also because the notion of an admissible sharpening manifestly inherits Sorites-­susceptibility from any predicate which is so susceptible—if ‘red’ tolerates indiscriminable change, for instance, so does ‘would be classified as ‘red’ under any acceptable sharpening of that expression’—and so cannot contribute towards a satisfactory solution of the paradox until independently sanitized.

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140  The Riddle of Vagueness about, their acceptability is of a piece with the most promising account of the sense in which ‘red’ is observational.18 But Peacocke cannot accept them, at least as stated. To do so would be inconsistent with his whole approach. Consider the situation at the cut-­off point for C which Peacocke allows to exist. Let a be the last C element, and b the first non-C one. Then, contrapos­ ing on the first principle, b is not red. Whereas, contraposing on the second and eliminating the double negation on ¬¬ Ca we have that a is not-­not-­red. So there is after all a sharp boundary which we could describe in terms of ‘red’: if we run the paradox from right to left, as it were, and take ‘not-­red’ as its subject, then there is a determinate last case to which the predicate applies and a determinate first case to which it does not. But ‘not-­red’ is, in this con­ text, as much an observational predicate as ‘red’ is; and the entire motif of Peacocke’s approach was to find a way of avoiding the paradox while ac­know­ ledg­ing what he considered essential, that observational expressions lack sharp boundaries. Needless to say, if Peacocke were to respond to this by rejecting the two principles, then he would either have to admit that the C-paradox is no paradox—if it has to depend on the two principles19—or 18  The reflections of Section IV (above) would actually support a stronger second principle than Peacocke’s stated version, viz. Any object which is not red the community will agree in not calling ‘red’. But there are ambiguities here—see n. 19. 19  It is worth considering—to anticipate somewhat the business of the next main section—how the connection demonstrated between the Sorites paradoxes for ‘red’ and C would fare if ‘definitely’ was appropriately interpolated into the argument. The first of Peacocke’s principles then becomes Any object which is definitely red the community will agree in calling ‘red’. But what is the second? Is it Any object which is definitely not red the community will not agree in calling ‘red’? Or the stronger Any object which is not definitely red the community will not agree in calling ‘red’? It is the former which is supported by the reflections of Section IV (via something with a stronger consequent—see n. 18) but the latter which, as the reader may verify, is required to link the major premises for C and ‘(definitely) red’ in the manner illustrated. And, given the stronger version of the second principle, it is still true that, if C has a sharp cut-­off in a Sorites series for ‘red’, then, where k is the last C patch, it will follow that it is not-­not-­definitely-­red; whereas its successor, qua not-C, will— by Peacocke’s first principle—be not-­definitely-­red. So the dilemma remains: either Peacocke cannot link the ‘red’ and C-paradoxes in the way he—ill-­advisedly—wants, or the relevant kind of series must always contain a determinate sharp boundary describable in terms of ‘red’. To revert just for a moment to the ‘supervaluational’ approach of Fine and others, briefly considered in n. 17: clearly—if a definiteness operator is to occur in its antecedent—the principle for ‘reds’ cor­res­pond­ ing to Peacocke’s second principle has to be If a is definitely not red, a is not reds, rather than If a is not definitely red, a is not reds. So there is no prospect of a similar argument that ‘reds’ has to be Sorites-­susceptible if ‘red’ is. But that does nothing to mitigate the corresponding charge about ‘would be classified as ‘red’ under any

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Further Reflections on the Sorites Paradox  141 allow that the degree-­theoretic approach provides no means for coping with it—if it does not. Actually, I think that Peacocke’s diagnosis of the plausibility of the major premise in the C-paradox is far-fetched and that its appeal derives rather from the train of thought outlined in presenting the Tachometer para­ dox above. But that is a minor criticism at this stage. (vi) Can degree-­theoretic approaches engage Sorites paradoxes which involve phenomenal predicates? I argued in Wright (1975, pp. 348–51, this volume, Chapter 1) and (1976, pp. 237–40, this volume, Chapter 2) that there are certain examples of the Sorites paradox for which no degree-­theoretic approach can be adequate. To grant, for instance, that ‘x is red’ admits of indefinitely many degrees of truth is of no immediate help when it comes to addressing the paradox that arises for the assertibility of ‘x is red’. Let us pur­ sue the matter a little. In Section  IV I tried to give cause to doubt that a proper account of the observationality of ‘red’ would sustain the principle, which leads directly to paradox, that, of any pair of indistinguishable items, both must be red if either is. It is notable, in addition to the considerations advanced earlier, that any account with that consequence is going to have the effect that there is no dif­ ference between looking and being red; at least, it is going to have that effect unless it is believed that whatever factors can bring it about that the look and the reality come apart could not apply selectively to one of a pair of items which are indiscriminable in a single sensory presentation. But there is no reason to endorse such a belief. It certainly isn’t a priori, for instance, that a pair of objects only one of which is red cannot both assume indistinguishably red appearances when bathed in red light of the right intensity and hue. But, if ‘red’ is not obser­ vational in any such sense, that does nothing to mitigate the threat of a paradox for ‘looks red’. It is not entirely clear whether a proponent of the degree-­theoretic approach may legitimately assume that, whenever ‘is φ’ admits of degrees of truth, so will ‘looks φ’. Certainly, ‘looks φ-er than’ will then exist; but is that the right kind of comparative—if a looks φ-er than b, is ‘looks φ’ truer of a than of b? I don’t know, but let’s suppose it is. Even so, there is surely no sense in the idea that, when a and b are indistinguishable, one might nevertheless look redder

acceptable sharpening of that predicate’. (Note, by the way, that the two ‘definitely’-free conditionals for ‘reds’ cited in n. 17 are both supervaluationally unacceptable, since there may well be ways of sharpening ‘red’ so as to make it false that, e.g., If a is red, a is reds, namely, sharpenings that place the red/not-­red cut-­off further across towards the non-­reds than the cut-­off for reds. Thus, from the supervaluational point of view, what those two conditionals try to say can only be said coherently by recourse to the definiteness operator.)

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142  The Riddle of Vagueness than the other in the context in which they are indistinguishable. If ‘looks red’ is a predicate of degree, it is still utterly unclear how it might be true of indistin­ guishable objects to differing degrees. What settles the degree to which some­ thing looks red—if there is indeed any question of ‘degree’—is the way it looks; so how could things that look absolutely similar look red to different degrees? These simple reflections seem to dispose of all prospect of a degree-­theoretic treatment of the Sorites paradoxes for the family of predicates which ‘looks red’ exemplifies. There is no discussion of the matter in Peacocke’s paper. But there is a footnote at least on the paradox of warranted assertibility to which we are now rapidly led. For the thought is a compelling one that the conditions of warranted assertion of a statement of the form ‘x is red’ will coincide, ceteris paribus, with the truth conditions of ‘x looks red’. More specifically, if you have no reason to suspect the conditions of observation, or your own powers of observation, or your competence with ‘red’, then you are warranted in asserting that something is red if it looks red; under the same provisos, it follows—granted the reflections of the preceding paragraph—that the degree of warranted assertibility cannot be different between two statements of that form concerning indistinguishable items. Accordingly, every φ apt for the formulation of statements whose ­assertion conditions can be characterized using an appropriate ‘looks φ’—or ‘sounds φ’, and so on—is apt for the generation of Sorites paradoxes to which the degree-­theoretic approach would seem to have no response. The objection presented in Wright (1975; 1976) was formulated slightly dif­ ferently. I argued that the degree-­theoretic approach could provide no satis­ factory treatment of predicates of the type ‘x is such that its description as ‘red’ is on balance justified’. Peacocke’s response (1981, p. 140, n. 11)—hardly cal­ culated to inspire confidence after our reflections in subsection (v)—is that he would treat this predicate as he treated C if it is not a predicate of degree, otherwise as he treats ‘red’ itself. Plainly it is no more a predicate of degree than is ‘is heavy enough to tip the scales’. So Peacocke’s response would be, in effect, to propose an extension-­determining theory via two principles differ­ ing from those discussed on above only in that their consequents would be formulated in terms of this predicate rather than C, and then to suggest that there is no reason to accept the major premise of the relevant paradox except the defective one that goes via our acceptance of the major premise of the paradox for ‘red’. But, even if two such principles would have sufficient plausi­ bility to subserve the suggestion, it should be clear that it is wide of the mark anyway: the reason for thinking that the major premise holds for ‘x is such that its description as ‘red’ is on balance justified’ is exactly the role of looking red in the conditions of warranted assertion of ‘x is red’, and this reason is entirely independent of the status of the major premise in the ‘red’ paradox.

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Further Reflections on the Sorites Paradox  143 The stark problem posed by Sorites paradoxes involving predicates like ‘looks red’ is that the claim that ‘. . . is indiscriminable from . . .’ is a congruence for them seems utterly convincing. The function of such predicates is purely to record how things appear—so what space can they make for any kind of selective response among things whose appearances are absolutely similar? I believe that this kind of thought is convincing if, but only if, we model our understanding of such predicates in accordance with the first part of the gov­ erning view. Rules for the use of such predicates—predicates geared just for the description of how things appear—would indeed have to assign exactly similar appearances exactly similar worth. But if, in contrast, competence with such predicates consists not in the internalization of rules but in habituation in certain patterns of response, then the thought that the responses must be the same to a pair of indiscriminable stimuli is only another version of the reasoning behind the Tachometer paradox, and may be treated accordingly. Since the apparent Sorites-­ susceptibility of ‘it is warrantedly assertible that . . . is red’ depends entirely upon the possibility of characterizing the assertibility conditions of ‘x is red’ in terms of ‘looks red’, the same treatment will also serve for the Sorites paradoxes of assertibility. It is still on our agenda to try to get much clearer about what this proposal comes to. But the foregoing reflections ought to be persuasive that a degree-­ theoretic approach, at any rate, is no alternative.

VI Higher-­Order Vagueness and the No Sharp Boundaries Paradox It was remarked earlier that, in addition to the lines of reasoning which the governing view encourages, major premises for Sorites paradoxes may be motivated in (at least) two further, quite different ways. One such is the train of thought that generates the Tachometer and C-paradoxes. But it is another which seems to make the connection between vagueness and Sorites-­ susceptibility most explicit and most intimate. Surely, this train of thought proceeds, the very vagueness of φ must entail that, in a series of appropriately gradually changing objects, φ at one end but not at the other, there will be no nth element which is φ while the n + 1st is not; for, if there were, the cut-­off between φ and not-φ would be sharp, contrary to hypothesis. Accordingly, the vagueness of φ over such a series must always be reflected in a truth of the form: (i)

¬(∃x )(ϕx & ¬ϕx′)

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144  The Riddle of Vagueness That, of course, is a classical equivalent of the universally quantified condi­ tional which is the major premise in standard formulations of the Sorites para­ dox—a thought which has prompted Putnam20 to suggest that a shift to intuitionistic logic might be of value in the treatment of the paradox. So indeed it might. But it will not be enough; for intuitionistic logic will yield a paradox direct from the negative existential premise (as will any logic with the standard ∃- and &-Introduction rules + reductio ad absurdum.)21 This form of the para­ dox—the No Sharp Boundaries paradox—thus appears to constitute a proof that (one kind of) vagueness is eo ipso a form of semantic incoherence. Only ‘appears’ though. What the No Sharp Boundaries paradox brings out is that, when dealing with vague expressions, it is essential to have the expres­ sive resources afforded by an operator expressing definiteness or determinacy. I have heard it argued that the introduction of such an operator can serve no point, since there is no apparent way whereby a statement could be true with­ out being definitely so. That is undeniable, but it is only to say that—in terms of Dummett’s distinction (1981, pp. 446–7)—the content senses of ‘P’ and ‘Definitely P’ coincide; whereas the important thing, for our purposes, is that their ingredient senses differ, the vital difference concerning the behaviour of the two statement forms when embedded in negation. Equipped with an appropriate such operator, we can see that a proper expression of the vague­ ness of φ with respect to the relevant sort of series of objects is not provided by the above negative existential but rather requires a statement to the effect that no definitely φ element is immediately succeeded by one which is def­in­ ite­ly not φ; formally (ii)

¬(∃x )(Def (ϕx ) & Def (¬ϕx′)).

20  Putnam (1983, especially pp. 284–6). Intuitionistically, ¬ (∃x)(φx & ¬φx′) is equivalent to (∀x) (φx → ¬¬φx′), but not to (∀x)(φx → φx′) unless φ is effectively decidable; for non-­effectively-­ decidable φ, the last may consistently be denied without commitment to (∃x)(φx & ¬φx′). 21  See Read and Wright (1985; this volume, Chapter 3). Putnam has since protested (1985, p. 203) that he never intended to recommend denial of (∃x)(φx & ¬φx′); his point was only that, intuitionisti­ cally, we could accept ¬(∀x)(φx → φx′)—as the paradox seems to force us to—without commitment to the fictitious boundary imported by asserting (∃x)(φx & ¬φx′). But Read and I said nothing to the contrary. Our point was that bringing the distinctions of intuitionist logic to bear on vague state­ ments—for which, Putnam acknowledges, the semantic motivation must be quite unlike that appealed to in the mathematical case—will not stop the Sorites paradox. We have, independently, to find some­ thing wrong with the impression that ¬(∃x)(φx & ¬φx′) is a satisfactory expression of the vagueness of φ in the relevant series of objects. In Read and Wright (1985; this volume, Chapter 3) we remarked that there would still be a place for at least one intuitionistic ploy—rejection of the inference from ¬¬(∃x)(φx & ¬φx′) to (∃x)(φx & ¬φx′). Putnam acknowledges this, but in fact the argument to ‘unmotivate’ ¬(∃x)(φx & ¬φx′) about to be developed in the text shows that even this is wrong: ¬(∃x)(φx & ¬φx′) is as infelicitous an expres­ sion of the proposition of which the paradox is a legitimate disproof by reductio as ¬(∃x)(φx & ¬φx′) is of φ’s vagueness in the relevant series.

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Further Reflections on the Sorites Paradox  145 This principle generates no paradox. The worst we can get from it, with or without classical logic, is the means for proving, for successive x ′, that (iii) Def (¬ϕx′) → ¬Def (ϕx ). And nothing untoward follows from that. A believer in higher-­order vagueness22 may want to reply that this merely postpones the difficulty. If, for example, the distinction between things which are φ and borderline cases of φ is itself vague, then assent to (iv) ¬(∃x )(Def (ϕx ) & ¬Def (ϕx′)) would seem to be compelled even if assent to (i) is not. Someone who sympa­ thized with the drift of Section IV will want to dispute that predicates like ‘red’ do indeed possess higher-­order vagueness—the truth is merely that the dis­ tinction between the last definitely φ item and the first case where some measure of indefiniteness enters is not to be placed just by looking. For pre­ sent purposes, though, let us suppose that the phenomenon is real. Then, once again, the materials for paradox seem to be at hand, each ingredient move taking the form, for instance, of a transition from ¬Def (ϕk′) to ¬Def (ϕk ) . I believe that this thought, that higher-­order vagueness is per se a source of paradox, may quite possibly be correct. But some complication is needed. For the following is the immediate reply. Of any pair of concepts which share a blurred boundary, we shall want to affirm (v)

¬(∃x )(Def (ϕx ) & Def (ψx′)),

when x ranges over the elements of an appropriate series in which the blurred boundary between φ and ψ is crossed. The original problem occurred when, with ¬φ in place of ψ, we overlooked the necessary role of the definiteness operator. And now we are guilty of the same oversight again; it is merely that this time ψ has been replaced by ¬Def (ϕ) . As soon as the inclusion of the definiteness operator is insisted on, all that emerges is 22  A nice example of succumbing to the allure of the idea is provided by Dummett (1978, p. 182): the vagueness of a vague predicate is ineradicable. Thus ‘hill’ is a vague predicate, in that there is no definite line between hills and mountains. But we could not eliminate this ­vagueness by introducing a new predicate, say ‘eminence’, to apply to those things which are neither definitely hills nor definitely mountains, since there would still remain things which were neither definitely hills nor definitely eminences, and so ad infinitum [sic]. A sophisticated discussion of the topic is Fine (1975, sect. 5).

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146  The Riddle of Vagueness (vi) ¬(∃x )[Def (Def (ϕx )) & Def (¬Def (ϕx′))], which yields nothing more than the harmless (vii)  Def (¬Def (ϕk′)) → ¬Def (Def (ϕk )). Evidently the strategy will generalize; we need never, it seems, be at a loss for a way of formulating φ’s possession of vagueness, of whatever order, in a way that avoids paradox. But this is too quick. We are able to be confident that the sort of formula­ tion illustrated avoids paradox only because we have so far no semantics for the definiteness operator, and are treating it as logically inert. Without con­ sidering what form a semantics for it might take, the crucial question is whether it would be correct to require validation for this principle: Γ P (DEF) ; provided Γ consists of propositions, all of which are Γ  Def (P ) prefixed by ‘Def ’ For, in the presence of DEF, and assuming that the corrected formulation, (vi) above, of what it is for the borderline between φ and its first-­order borderline cases to be itself blurred, is itself definitely correct, the harmless (vii) gives way to (viii)

Def (¬Def (ϕk′)) → Def (¬Def (ϕk ))

a generalization of which will enable us to prove that φ has no definite instances if it has definite borderline cases of the first order.23 DEF has this effect because it sanctions the inference from Def (ϕk ) to Def (Def (ϕk )) . More generally, assuming that what is definitely true is true, DEF yields the biconditional theorem. (IT)

 Def ( A) ↔ Def (Def ( A))

If, as was supposed, A and Def (A) may have differing ingredient senses, this may seem obviously unacceptable. But matters are not so straightforward. The question is whether the difference is one to which the context, ‘Def ( )’, is, like ‘Not( )’, itself sensitive. Suppose, for instance, we had a semantics which accounted Def (A) false when A was anything other than definitely true, and Not-A as borderline when A was borderline. Not-Def (A) would then, pre­sum­ ably, be true when A was borderline—diverging, as it intuitively should, from 23  A derivation is given on p. 148 below.

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Further Reflections on the Sorites Paradox  147 Not-A; but Def (A) and Def (Def (A)) would both be false. An approach having these features would not, I suppose, be a non-­starter—the idea has some appeal that if, A is on any sort of borderline, the claim that it is anything else is false. But, unless we found more to say, there would be no evident objec­ tion to DEF. There is more to say, of course. To take higher-­order vagueness seriously is just to allow that cases may arise where it is indeterminate whether a state­ ment is true or borderline. To say that its definitization was false in such a case would be, in effect, to rule that the original was borderline—to ignore its leanings, as it were, towards truth. So the sort of semantics adumbrated which promises to validate DEF is anyway guilty of failing to take higher-­order vagueness seriously: to repeat, taking higher-­ order vagueness seriously involves allowing that Def (A) may itself, on occasion, be borderline. So to allow, however, will make no difference as far as IT is concerned unless, when Def (A) is borderline, Def (Def (A)) may either be false or, at any rate, of an inferior degree of truth to that of Def (A). But the idea that Def (Def (A)) may be false when Def (A) is borderline can hardly be separated from the corresponding claim about Def (A) and A respectively—the claim just accused of failing to take higher-­order vagueness seriously. So a defender of higher-­order vagueness should prefer the second type of proposal: when a statement is borderline, so should its definitization be, but not in such a way as to sustain IT, left to right. How might this proposal be elaborated? Let us pretend for a moment that we really do understand the idea of an indefinite hierarchy of orders of ­vagueness, along the following lines: Level – ω . . . Level – 2 Level – 1 Level 0 Level 1 Level 2 . . . . Level ω

:       : : : : :         :

statements which are false       statements which are neither definitely false nor definitely at level – 1 statements which are neither definitely false nor definitely at level 0 statements which are neither definitely true nor definitely false statements which are neither definitely true nor definitely at level 0 statements which are neither definitely true nor definitely at level 1         statements which are true

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148  The Riddle of Vagueness where it is understood that each statement of finite positive level is closer to truth than any statement which is definitely of level 0; and each statement of finite nega­ tive level is closer to falsity. Then a relatively straightforward proposal would be that, for any statement of level k, its definitization is of level k–1, if k is finite or 0, but otherwise is also of level k. So, if A is definitely true, or false, so is Def (A); but if A is on some sort of borderline, Def (A) lies on the immediately inferior border­ line. And A ↔ B will hold only if A and B are of the same level. But the problem with this is evident enough. Any statement, A, of finite level k, >0, is part-­characterized as one which is not definitely true. So the statement of which that part-­characterization is the negation ought to be false—and that state­ ment ought to be Def (A). It is accordingly impossible to marry the sense given to ‘Def’ by the proposal with our intuitive understanding of ‘definitely’. The point is a general one. We cannot intelligibly characterize higher orders of vagueness in terms which invoke statements’ failure to be definitely true, yet simultaneously require definitization to generate falsity only when applied to false statements. We could, of course, drop the latter requirement without reverting to the original idea that Def (A) always polarizes to truth or falsity. A (somewhat messy) compromise would be that Def (A) continues to drop a level on A just in case A occupies some finite negative level, but polarizes to falsity if A occupies any finite positive level or 0. But, as will be apparent if we  now construct an explicit derivation, this compromise does nothing to obstruct the relevant version of the No Sharp Boundaries paradox: 1 2 3 3 2,3 1 1,2 1,2 1

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Def (¬(∃x)[Def (Def (φx)) & Def (¬Def (φx′))]); Ass. Def (¬Def (φx′)); Ass. Def (φx′)); Ass. Def (Def (φx)); IT. (∃x)[(Def (Def (φx)) & Def (¬Def (φx′))]: 2,4, ∃-intro. ¬(∃x)[Def (Def (φx)) & Def (¬Def (φx′)); 1, Def-elim. ¬Def (φx); 3,5,6, RAA. Def (¬Def (φx)); 7, DEF. Def (¬Def (φx′)) → Def (¬Def (φx)); 2,8 CP

Clearly, IT and DEF do not survive the mooted compromise without restrictions: Def (A) will fail of equivalence to Def (Def (A)) for any A of finite negative level; and Def (A) will be of level lower than A for A of any finite level, negative or positive. But remember that each φx or φx′ with which we are concerned in this version of the paradox is of level 0 or greater—we start with an x′ which is a definite borderline case of φ and ‘work left’, so to speak.

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Further Reflections on the Sorites Paradox  149 So the compromise poses no obstacle to the application of IT at line (4): Def (φx) and Def (Def (φx)) will both be false for every germane φx. Similarly, the c­ om­prom­ise offers no objection to the application of DEF at line (8): if φx is of level 0 or greater finite level, ¬Def (φx) and Def (¬Def (φx)) will both be true, and, if φx is of level ω (in which case why doesn’t it mark a sharp bound­ ary?), ¬Def (φx) and Def (¬Def (φx)) will both be false. And, to stress: every φx at which we arrive by ‘working left’ is in one of those two cases. I am under no illusion that these sketchy remarks constitute a treatment of the topic. But they do suggest that the No Sharp Boundaries paradox may have something to teach us, not about vagueness generally but about the idea of an indefinite hierarchy of orders of vagueness. The idea, which has smitten some writers,24 that lack of sharp boundaries is per se paradoxical is merely retribu­ tion for working with too crude a formulation of what lack of sharp bound­ar­ ies is. It is essential to have the expressive resources of a definiteness operator. But the case for thinking that higher-­order vagueness—always a difficult and vertiginous-­seeming idea—may be per se paradoxical still needs an answer. There is a further, very important point to which the introduction of a def­ in­ite­ness operator into our formulations gives rise. Earlier, in Section IV, the complaint was briefly considered that it could not suffice to resolve a Sorites paradox simply to undermine any motivation for believing the major prem­ ise. For there is a paradox in any case unless the major premise, mo­tiv­ated or not, is untrue—the paradox is, indeed, a reductio of the major premise—and an account is therefore owing of how, specifically, it is untrue. Such an account is provided, for expressions like ‘red’, by the considerations—if sound—which were advanced in Section  IV to support the idea that such expressions can have determinate thresholds of application in the appropriate kind of series. But, correct or not, would nothing less than such an account, or, for example, an account along Peacocke’s lines, suffice to discharge the obligation? The doubt whether such an account is necessary arises because—obviously enough—the presence of vague expressions may divest complex sentences of determinate truth value no less than simple predications. So it is plausible to insist that we have other options open besides affirmation or denial of the major premise in a Sorites—the premise may itself lack any determinate sta­ tus. Once that is recognized, the point about the insufficiency of responses which do no more than undercut the motivation for the major premises is likely to seem less compelling. Perhaps we will have said enough about the 24  The most notorious case, I suppose, is Peter Unger. A typical example of his use of the Sorites is Unger (1979a).

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150  The Riddle of Vagueness ‘manner of untruth’ of the major premises when we have shown that they may reasonably be regarded as lacking any determinate truth status. And, showing that there is no good reason to regard them as true would accomplish just that—provided that there is also no good reason to regard them as false. Well, the evident problem with this is that a Sorites, as remarked, itself appears to constitute a reductio of its major premise, and so to exclude the response that the premise may be viewed as indeterminate. As a result, the challenge to explain how specifically the major premise is false—does one of the ingredient condi­ tionals, for example, cross a threshold, or are they all marginally false, or what?— is apt to seem compelling. Perhaps it could be resisted—it is a suggestive thought, for instance, that a conjunction of inde­ter­min­ate conjuncts may have de­ter­min­ ate­ly false consequences. But with definiteness operators in our weaponry, we can now see, once again, that the challenge need not arise. What, we should claim, a Sorites establishes as false is the claim that the major premise is definitely true. The attempt to express the major premise without recourse to the definiteness operator is a source of needless difficulty no less than the corresponding attempt with the expression of lack of sharp boundaries. The paradox establishes only the negation of the definitization of the major premise—a conclusion quite consist­ ent with its indeterminacy. In contrast, the challenge presupposes, illegitimately, that what is established is the definitization of the negation.25 The suggestion demands, of course, sufficient of a semantical account for the definiteness operator to sustain the intuitively plausible idea that what is really up for reductio is the definitization of the major premise—that it is somehow improper to put it forward undefinitized. But it does not seem far­ fetched to suppose that such an account can be provided. If it can, then undercutting the motivation for the major premises can be enough, provided it is acceptable methodology to regard compound statements involving vague expressions as indeterminate in truth status if no evidence is available to the contrary. If so, one corollary is that the way adopted with observational expressions in Section IV involved biting off more than it was strictly neces­ sary to chew. (That is not to say that the conclusions there drawn are not good in any case.) Another, more general, is that the problem of explaining how the transition from F-ness to non-F-ness is effected in a Sorites series can be dis­ missed as spurious. It may be that there is, after all, some kind of sharp bound­ ary, and it may be, in particular cases, that diminutions of degree are uniformly involved. But we are under no obligation to provide an account—there may be nothing determinately correct to say about the matter. 25  Refer to n. 21 above, concluding remarks.

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Further Reflections on the Sorites Paradox  151

VII The Governing View and the Major Premises I now revert to the various types of ground, briefly canvassed in Section  I, which the governing view provides for the major premises in Sorites para­ doxes. Such grounds involve reference to a number of factors including (i) standard criteria for understanding and misunderstanding an expression, (ii) the way the use of the expression is standardly explained, (iii) what is known about relevant cognitive limitations of ours—of visual acuity, or ­memory, for instance—, (iv) the way the applicability of the expression can standardly be determined—by casual observation, for instance—, and (v) our understanding of the point or significance—the practical consequences—of its applying. These are all considerations of a kind which, once it is accepted that our use of an expression is governed by implicitly known semantic rules, must at least be relevant to determining the character of the content which those rules have. But it is a different question how convincing in detail are the applications which were made of them in Wright (1975; 1976). There is, to begin with, some doubt about the use there is made of the first. It is one thing to claim that standard criteria for misunderstanding ‘red’ should have implications for the character of the rules which govern the use of that predicate; quite another to argue, as I did, that a subject who was pre­ pared to describe one but not the other of a pair of indiscriminable colour patches as ‘red’ would invariably give cause to think that they misunderstood the predicate (Wright 1976, p. 234, this volume, Chapter  2). Certainly, that would very often be so; it would, I suggest, always be so if the pair was pre­ sented to the subject in isolation and in a context in which they had no reason to suspect any abnormality of lighting, and so on. But it is not obviously so precisely in the context that interests us—namely, that of a Sorites series: here we should want to recognize the right of the subject to ‘switch off ’ at some point—and, since that is our response to the case, an argument for saying that their doing so would nevertheless be in contravention of the rules for ‘red’ would better draw on considerations of a different sort. There are also reservations about the way the line of thought, which appealed to the moral and explanatory significance of terms such as ‘adult’ and ‘child’, was supposed to support the major premise for the ‘heartbeats of childhood’ paradox (Wright 1976, pp. 231–2, this volume, Chapter 2). It is, of  course, true that giving these concepts mutually sharp boundaries would  bring it about that important differences concerning rights and

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152  The Riddle of Vagueness re­spon­si­bil­ities would come to be associated, in certain cases, with no sub­ stantial change in the subject, and useful lay-­ psychological and lay-­ sociological generalizations would be placed in jeopardy. But the fact is that, in order for a certain distinction—say that between ‘adolescent’ and ‘adult’—to serve such purposes and carry such consequences, it is not necessary that absolutely every instance of the distinction be able to do so. If we troubled to have an absolutely sharp distinction between adolescence and adulthood, it would still be true that most adolescents would differ sufficiently from most adults to justify the sort of differential treatment and expectations visited upon them. So the absence of a sharp boundary cannot be imposed on us by the wish to have these concepts associated with their standard moral and explanatory significance. Only if the association had to be unfailing would there seem to be such an im­pos­ition. But it would be quite good enough if it were merely usual; or so it could be plausibly argued. At this point, however, it becomes urgent to enquire how exactly the kinds of consideration canvassed are supposed to yield the major premises in Sorites paradoxes. The point just considered, for example, would seem to have been whether mutual precision in the ‘child’, ‘adolescent’, ‘adult’ family of predicates would be consistent with certain important features of their use. But surely, even if we decided that it would not, the most that could follow—on the standpoint of the governing view—would be that the semantic rules for these expressions prescribed that there should be no sharp cut-­offs; and that conclu­ sion would foster paradox only if lack of sharp boundaries was thought to be tantamount to Sorites-­susceptibility—a thought which, unqualified, is just an endorsement of the No Sharp Boundaries paradox, to which an outline solu­ tion was offered in the previous section. Presenting considerations which, under the aegis of the governing view, deepen our understanding of the vagueness of certain expressions is absolutely not to the point unless it is sup­ posed (that one who endorses the governing view must hold) that to be vague is to be Sorites-­prone. I was, I think, alive to this consideration in my previous discussions (Wright 1975, pp. 334–5, this volume, Chapter 1; 1976, pp. 229–30, this vol­ ume, Chapter 2). But how exactly do the arguments advanced negotiate the pitfall? The second, third, and fourth types of consideration—those concern­ ing standard explanation, cognitive limitations of users, and methods for assessing application of an expression—can each, it seems, be made to furnish a case for the quantified conditional

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Further Reflections on the Sorites Paradox  153

(∀x) (it is not the case that the rules for ‘red’ prescribe its application to x′ → it is not the case that the rules for ‘red’ prescribe its application to x). where x, as usual, ranges over the members of an appropriate series of colour patches and x′ is the successor of x therein. We proceed by putting forward for reductio the suggestion that there is a last case in the series to which the rules prescribe application of ‘red’. That is distinct, of course, from the suggestion that a case to which the rules prescribe application of ‘red’ is immediately fol­ lowed by one to which they prescribe that ‘red’ does not apply. The argument, whether we bring the second, third, or fourth kind of consideration to bear, is very straightforward. Mastery of a set of rules involves knowledge not just of what they require, permit, and prohibit, but also of their limits: if the rules for ‘red’ do not prescribe application of the predicate to certain kinds of case to which they also do not prescribe its non-­application, then one who has mas­ tered those rules ought to know this and have the capacity, other things being equal, to recognize such a case if presented. So much will be part of under­ standing ‘red’. Suppose, then, that all the distinctions which such an under­ standing empowers one to draw can be drawn by someone with normal limitations of memory without reliance on external aids; that nearby survey­ able objects which are red can, in normal circumstances, be recognized to be so just by casual observation; and that a full understanding of ‘red’ can be bestowed by ostensive training. Each of these suppositions is plausible and, apparently, inconsistent with the hypothesis. If the hypothesis were true, a k to which the rules prescribed application of ‘red’ would be followed by a k′ to which they did not. Someone who fully understood ‘red’ ought, therefore, to be able to recognize the pair as such. But doing so would require reliance upon a distinction—that exemplified by k and its successor—which was unmemorably fine, undetectable by casual observation, and—in the case where adjacent patches are indiscriminable—incapable of ostensive display. So, on the stated suppositions, the hypothesis is false for arbitrary choice of k′; and the quantified conditional is thereby established. One response to these considerations would be to wonder whether appro­ priate interpolation within the argument of occurrences of the definiteness operator might make some material difference. Someone who fully under­ stands the rules for the use of ‘red’ ought indeed to be empowered to recog­ nize any k and k′ such as the rules definitely prescribe applications of ‘red’ to k and definitely do not prescribe its application to k′. But that is to say that understanding bestows grasp of the distinction between things which are

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154  The Riddle of Vagueness definitely red and things which are definitely something else—either def­in­ ite­ly not-­red or definitely borderline. Can anything weaker be so much as coherently formulated? In Wright (1975, p. 342, this volume, Chapter  1) I spoke of the distinction between things which are definitely red and things of ‘any other sort’. But it seems reasonable to insist that any distinction to which understanding demands sensitivity must be one between items of determinate status—otherwise it cannot be determinate that there is any relevant distinc­ tion between them, and of no response in particular can it therefore be cor­ rect to say that it is what understanding requires. Well and good. What follows is that all we may legitimately put up for reductio is the supposition of a sharp threshold between the definite reds and the definite borderline cases. And no problem will be presented if this is expressed by straightforward definitization of what we had above: ‘The rules prescribe applications of ‘red’ to k’ and ‘The rules do not prescribe application of ‘red’ to k′ ’ respectively. For the worst that will then be forthcoming is (Definitely: it is not the case that: the rules prescribe application of ‘red’ to k′) → (It is not the case that: definitely: the rules prescribe application of ‘red’ to k.) However, the supposition of sharp threshold would be as well captured by conjoint assumption of Definitely: it is not the case that: definitely: the rules prescribe applica­ tion of ‘red’ to k′; and Definitely: the rules prescribe application of ‘red’ to k. And a reductio of this assumption does afford a conditional of paradoxical potential—one of uniform antecedent and consequent—provided, as we know, the definiteness operator may be manipulated in accordance with DEF. The position, then, is that, in the presence of DEF and assuming the (def­in­ ite) correctness of the suppositions about memorability, casual observability, and ostensive teaching which were made, there is a good case for thinking that ‘red’ and its ilk are subject to the higher-­order version of the No Sharp Boundaries paradox. That is not, indeed, the conclusion drawn in Wright (1975, pp. 333–4, this volume, Chapter 1; 1976, p. 229, this volume, Chapter 2),

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Further Reflections on the Sorites Paradox  155 that the governing view constrains us to regard such predicates as tolerant with respect to indiscriminable (or unmemorable) degrees of change. But it is no more comfortable a conclusion for one who wants to hold the governing view.26 What room is there for manoeuvre? DEF can be challenged, of course, though the prospects do not seem encouraging. A more plausible response, at least at first sight, is to claim that the background suppositions are at fault. The claim, for instance, that the whole meaning of ‘red’ may be taught by ostensive means is open to the same reproach as that earlier directed against Dummett’s account of observationality: if it were right, there would be no space for a distinction between looking red and really being so. For that dis­ tinction cannot, presumably, be ostensively explained—(or, if it can, it is no longer evident why ostensive teaching could not encompass an explanation of how only one of a pair of indiscriminable items could be red). The same con­ sid­er­ation, indeed, seems to disarm the other suppositions as well. If we allow that a k which is definitely red can be indiscriminable, or only just barely dis­ criminable from a k′ which is definitely borderline, the memorability is thereby jeopardized of the distinction between things which are definitely red and things which are not only if that distinction has to be drawn by reference solely to appearances. But to recognize the contrast between looking red and being red—and a corresponding contrast, presumably, between looking a borderline case of red and being one—is to recognize that appearance need not be the whole basis of that distinction. Likewise, it is possible to tell that something is red by casually observing it only insofar as its appearance is, in the circumstances, a good guide to its colour; for the appearance is all that, to a casual observer, is apparent. There is, therefore, no requirement that every 26  It is not clear, however, that the argument about ‘child’, ‘adult’, etc., based on considerations about moral and explanatory role, can be similarly reconstructed. Even if—contrary to the suggestion in the text above—every instance of the adolescent/adult distinction has to involve substantial difference if the moral and explanatory role of these expressions is not to be compromised, the corresponding claim about the adolescent/definite-­borderline-­case-­of-­adolescence distinction is rather less plausible. The trouble is that whereas each of If x is an adolescent, there are certain moral demands which it would not be reasonable to make on x but which it would be reasonable to make, ceteris paribus, on an adult; and If x is an adult, there are no moral demands which it would not be reasonable to make on x but which it would be reasonable to make, ceteris paribus, on an adult; is unexceptionable, no such conditional beginning If x is definitely on the adult/adolescent borderline, . . . is clearly correct. And some such conditional is going to be needed in order to reconstruct the argu­ ment, since it will have to be argued that a sharp distinction between adolescence and definite-­ borderline status would be associated with disproportionate moral/explanatory significance. But I leave the reader to ponder the matter.

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156  The Riddle of Vagueness distinction in real colour status be one to which casual observation can be sensitive. This response does not, however, take us very far. Even if ‘red’ could be saved by such appeals to the ‘is red’/‘looks red’ contrast, no parallel hope seems to be proffered for the other contrasted term. Surely the distinction between what definitely looks red and what does not is one which it must be possible to make salient by ostensive means;27 and surely it is a distinction which the competent can draw just by casual observation and without reli­ ance upon external aids. These suppositions remain vividly plausible for ‘looks red’ even if they do not hold for ‘red’, and the problem remains, even if restricted in scope. Moreover, it is not even clear that the scope of the prob­ lem has been restricted. For to be red is to look red in circumstances of obser­ vation which leave nothing to be desired—however exactly that condition should be explicated. So, if there is still a Sorites paradox to solve for ‘looks red’, then, relative to the assumption that such are the prevailing circum­ stances of observation, there is still a Sorites paradox for ‘red’. Looking red suffices for being red when other things are equal, and, as noted towards the conclusion of Section V, for being justifiably assertible to be red if there is no reason to doubt that other things are equal. It is as well, of course, to pay proper heed to the distinction between looking and being red; but doing so gets us no further forward in the present context. Whether it is correct to suppose that ‘looks red’ is ostensively definable or not,28 it is certain that competence with that predicate is practised by subjects who have only quite ordinary powers of observation and rely on nothing but casual exercise of those powers, eschewing in particular any use of the kind of external aids—colour charts, or whatever—which could compensate for limi­ tations of memory. So the escape route has to involve finding a way of avoid­ ing drawing paradoxical conclusions from this undeniable fact. But how—if we say that competence consists in knowing certain rules and their limits? For it then seems inescapable that both every distinction prescribed by the rules, and the distinction between cases where the rules have something to say and 27  That is not the same as saying that ‘looks red’ is ostensively definable. The claim in the text would still hold if ‘looks more than six feet tall’ were substituted for ‘looks red’. So the question of ostensive definability turns on whether ‘looks red’ is genuinely semantically structured, with ‘red’ a significant component in that structure. A bad reason for an affirmative answer would be based on confusing ‘looks red’ with ‘looks as if it is red’ or ‘looks the way red things look’, since in special circumstances it may be necessary, in order to look the way red things look, to look, e.g., brown (if, say, bathed in green light). To be sure, it is a priori that looking red is looking the way red things look in good lighting, etc. But that does not entail that understanding ‘looks red’ involves grasp of this a priori truth. For a sophisticated discussion of such matters, see Peacocke (1984). 28  See n. 27.

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Further Reflections on the Sorites Paradox  157 cases where they do not, can be based only on contrasts which may be detected by casual exercise of ordinary powers of observation, without reli­ ance on external aids; and, hence, that no such distinction can be exemplified by items which conjointly exhibit no such contrast. So the conclusion still looks good: competence with expressions of the class which ‘looks red’ typifies cannot consist in knowledge of the requirements of certain rules and of their limits. There is still the possibility of making trouble for DEF, without which the particular version of the Sorites which threatens is not, apparently, obtainable. But, once we concentrate on ‘looks red’ and its ilk, the route to paradox out­ lined is anyway needlessly indirect. The best argument afforded by the gov­ erning view for a major premise for ‘looks red’ is the one bruited towards the end of Section V. If items are visually indiscriminable, they look the same. So visual indiscriminability is necessarily a congruence relation for any predicate whose rules of application take account only of how things look. Our concep­ tion of the role and purpose of ‘looks red’, ‘looks orange’, and so on is, unques­ tionably, that we use them purely to record public appearances. So if, as the governing view permits, we may legitimately base upon this conception con­ clusions which concern the character of the rules governing such predicates’ application, those rules must have the feature, it seems, of relating only to appearance, of prescribing application of such predicates only and purely on the basis of appearance. But then they must prescribe application of such a predicate to both members, if either, of any pair of items whose appearances are the same. Now there is terribly little room for manoeuvre. It would suffice if we could somehow drive indistinguishability and sameness of appearance apart—if, while granting that the rules for ‘looks red’ relate only to appearance, we could provide some principled basis for denying that indistinguishability suf­ fices for sameness of appearance. But I do not see any hopeful direction for such an attempt to take. I mention it only by way of noting one formal alter­ native to the response which I want to recommend. This, as the reader will anticipate, is to drop the idea that the harmless truism that we use predicates like ‘looks red’ to record how things appear to us has any bearing on the char­ acter of the putative semantic rules which govern their use. Since—so it seems to me—the truism could hardly fail to have the very direct bearing illustrated so long as there are governing semantic rules for such expressions at all, the recommendation is that we drop that assumption, and adopt, as the matter was expressed in Wright (1976), a ‘more purely behaviouristic’ conception of what competence with such expressions involves.

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158  The Riddle of Vagueness But what does that mean? A way to provide at least the beginnings of a proper account of the contrast is to enquire what connection there is between the obtaining of an instance of the type of state of affairs which confers truth on a token of the sentence ‘x looks red’, and our willingness to assent to that token. It should not be controversial, I think, that each is necessary and suffi­ cient for the other so long as certain provisos are met which are distinctive of this class of expression; that is, for no other class of expressions do exactly these provisos subserve such a biconditional dependence. The provisos are that the judging subject understands ‘looks red’, that their perceptual faculties are functioning normally, that they are otherwise in good cognitive order, that x is presented to them in clear view, and that they are attentive to x.29 Subject to these provisos, assent and truth necessarily coincide. We have, that is, Necessarily: if the subject and the circumstances are as required by the pro­ visos, then ‘x looks red’ is true if and only if the subject assents to ‘x looks red’. Now, two quite different broad perspectives on this principle—the proviso-­ biconditional—are possible. One—required by the governing view—will hold that the circumstance that ‘looks red’ applies to x is settled, independently of any subject’s response to x, by the semantic rules which govern the use of ‘looks red’ in English and by how, objectively, x appears. The role of the pro­ visos, on this view, is to ensure the subject’s ability to ‘track’, or detect, this independent state of affairs: thus, their understanding of ‘looks red’ and their being in ‘good cognitive order’ ensure their sensitivity to the requirements of the relevant semantic rules; and their normal perceptual function and atten­ tiveness to x, and the clear presentation of x to them, ensure their sensitivity to x’s objective appearance. On this view, then, appeal to the idea of how a subject will or would respond to the utterance when the provisos are satisfied should play no essential part in an account of its truth conditions. Such an account need consider nothing but the semantics of the utterance and the characteristics of x. What the provisos do is to foreclose on every possible explanation of fracture between the fact of the matter and a subject’s response. The reason why this perspective is required by the governing view is not that the latter platonizes semantic rules, as it were—writes us out of the story 29  The corresponding provisos for ‘x is red’ would include not merely that x be presented in clear view but that the circumstances of presentation be normal, or appropriate.

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Further Reflections on the Sorites Paradox  159 altogether—but rather that we are allowed to feature only in a limited way. To be sure, it is by reference to our cognitive limitations, what we can casually observe, our conception of the function of a certain class of predicates, and so on, that conclusions are to be drawn about the content of semantic rules. But conclusions are, apparently, not to be drawn by reference to the character of our response to a predication of ‘looks red’ when, by ordinary criteria, all the provisos are fulfilled. And, in order for this exclusion to be legitimate, the dictates of the semantic rules for ‘looks red’ have to be thought of as consti­ tuted independently of such responses. The point is, indeed, quite general and simple. In order for there to be a Sorites paradox of this kind, our actual clas­ sificatory responses have to be out of accord with the requirements of the rele­vant semantic rules. So to think you have such a paradox, you have to be working with an epistemology of semantic rules which allows firm conclu­ sions to be drawn about their character independently—or anyway, in a man­ ner sufficiently inattentive to—the shape which those responses are disposed to assume. Such a view need nevertheless pose no barrier to recognizing the proviso-­ biconditional. Whatever epistemology is utilized, it will contain the resources for a declaration that, where responses are not in accord with the rules as it construes them, subjects are, for instance, muddled or incon­ stant in their apprehension of the requirements of those rules and so, to that extent, out of good cognitive order. That is the first broad perspective. But the alternative perspective turns it all around. Now the proviso-­biconditional is seen, instead, as itself supplying the canonical form of a statement of the truth conditions of ‘x looks red’. The provisos are no longer seen as serving to describe the conditions under which a subject succeeds in tracking an independent fact; rather, for x to look red just is for subjects to be willing to assent to that judgement when the provisos are met. Whatever else we want to say about the meaning of such expressions, and about the epistemology of their meaning, it is all answerable to this point. Hence, there is no possibility of such an epistemology teaching us that the provisos are in fact not satisfied in circumstances where, by normal criteria, we should have been satisfied that they are. It is the other way about: if we are tempted to opinions about the meanings of such expressions which force us to draw such a conclusion, it is those opinions which are at fault. There is no ulterior fact which meeting the provisos ensures that we can detect, so no possibility—by reference to independent criteria for the existence or non-­ existence of such a fact—of surprising conclusions about when the provisos really are not met, notwithstanding the satisfaction of ordinary criteria for saying that they are.

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160  The Riddle of Vagueness What does this distinction do for us?30 Well, suppose a group of subjects are agreed that ‘looks red’ applies to the first colour patch in a Sorites series in circumstances when the provisos are met. As the series is run from apparently red to apparently orange patches, a point will be reached where, despite its indiscriminability from the immediately preceding patch, a consensus in sub­ jects’ responses breaks down for the first time. That is undeniable. It is also undeniable that, when that happens, no single subject in the group need, by  ordinary criteria, have fallen out of accord with any of the provisos. (Borderline cases are exactly cases about which competent subjects are allowed to differ.) The second perspective on the proviso-­biconditional gives us the right to treat such a circumstance as raising a doubt about the predic­ ability of ‘looks red’.31 The first perspective, by contrast, leads to the cancella­ tion of that right. Someone who succumbs to the arresting, apparently simple thought that ‘looks red’ must be applicable to both, if to either, of any pair of indiscrimina­ ble items is likely to be taking it in one of two ways. Taken one way, it is the central move in the Tachometer paradox: how can a signalling device, even a properly functioning signalling device, discriminate among stimuli whose difference is smaller than its sensitivity threshold? We now know, I hope, how to respond to that question. But the second and, I think, more natural way of taking the thought conceals, in effect, a presupposition of the first perspective on the proviso-­biconditional. You have to forget that we do not, or would not, so apply the predicate in every case and fall in, instead, with the idea that something can be discerned about its proper conditions of application just by intuitive reflection upon the kind of content which it overtly seems to have. If there were facts about the proper application of such predicates which were constituted independently of our best responses—the responses we would have when, by ordinary criteria, all the provisos were met—what else could they be but the offspring of rules which correlated their proper use with appearances? And how, when following such rules, could it ever be justifiable, in consequence, to assign to identical appearances distinct responses? The reply should be that competence with such predicates is nothing to do with the capacity to fit one’s usage to the dictates of rules of that kind. Indeed, it is not a matter of compliance with rules at all, if that is taken to imply the pro­ priety of a ‘detective’ direction of interpretation of the proviso-­biconditional. What it is correct to say using such a predicate is a function only of what we 30  This very important form of distinction was utilized by Mark Johnston in classes on ethics in Princeton in the spring of 1986. It is further discussed in Wright (1988); and put to work in Wright (1987b). 31 Because A → (B ↔ C) entails B → (A → C).

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Further Reflections on the Sorites Paradox  161 are actually inclined to say when there is no reason to doubt that the provisos are met. The distinction between these two perspectives on the proviso-­ biconditional would undoubtedly benefit from further work. But it is, I hope, tolerably clear. It unravels the most important strand in what I was attempting to say in Wright (1975; 1976). The ‘behaviouristic’ conception of meaning advocated towards the conclusion of the latter is just the non-­detective read­ ing of the proviso-­biconditional which lets us take ‘best behaviour’ seriously. It is, indeed, the belief in the conception of meaning which sustains a de­tect­ ive reading of proviso-­biconditionals that constitutes the real essence of the governing view, of which its involvement with the idea of implicit knowledge of semantic rules is merely a corollary. For, as noted, the separation, effected by the detective reading, between, on the one hand, what constitutes the cor­ rectness of a particular response with ‘looks red’, for instance, and, on the other, the fact that the response observes the provisos, enjoins viewing the provisos as collectively ensuring a subject’s ability to keep cognitive track of the independent, constitutive fact. So there has to be substantial knowledge of the ingredients of that fact, including the requirements of the relevant seman­ tic rules. Since the subject’s response will usually come to them involuntarily and unreflectively, it would appear that the relevant cognitive processes must usually be implicit. One does not need to be wary of the notion of implicit knowledge in general to reject this way of involving it in the theory of linguis­ tic understanding. The rules to which competent use of ‘looks red’ con­ forms—if indeed we wish to continue to think in terms of semantic rules for such expressions at all—are not rules knowledge of whose content, at some deep level, makes competence possible; rather their content is given by the course assumed by competent use. It remains to respond to a question outstanding from the end of Section  IV. An observational expression, it was there suggested, should be characterized as, inter alia, one which a subject could not fail to understand if their use of it displayed an appropriate sensitivity to a certain distinctive range of appearances. The question was whether the presence of this condition in an account of observationality could be seen to spare observational expressions the tolerance which Dummett’s account, for instance, would lumber them with. ‘How could the appropriate kind of sensitivity operate selectively among indiscriminabilia.’32 If what I have been saying has any force, it should now seem as if this question has rather disappeared. ‘Looks red’ ought certainly to qualify as observational on this count. But the kind of sensitivity to 32  See above, p. 131.

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162  The Riddle of Vagueness appearance which someone who understands ‘looks red’ (and is normally sighted) must have just does operate selectively among items which, in respect of apparent colour, cannot be told apart. If, for instance, we present a subject with each of the elements of a Sorites series of indistinguishable colour patches but out of sequence, such selectivity will have to be manifest in the collation of their responses to the question ‘Does this look red?’, so long as we are to be satisfied that the provisos have been met in every case. The sugges­ tion that there is some kind of tension between such selectivity of response and its being the sort of response appropriate in the case of an observational predicate depends on the thought that it cannot then be purely in response to appearance—either it is unprincipled or else it is a principled response to more than the appearances. But ‘unprincipled’ here just means: not guided by rules correlating responses with appearances. So we should embrace the first alternative: such responses are indeed unprincipled, and no less appropriate or less purely ‘to’ appearances on that account.

VIII Conclusion We have covered a lot of ground, and a substantial proportion of our ­findings—most especially about degree-­theoretic responses to the Sorites— have been negative. But a number of positive points have emerged, which, if I am right about them, are worthy of emphasis. First, it is a mistake to think of the Sorites as a single paradox, which admits of a large variety of illustrations. Rather, we have here a family of different paradoxes, formally similar but ­distinct when identified by reference to the type of ground which supports their major premises. By this criterion, the Tachometer paradox, the No Sharp Boundaries paradox, and the various Sorites paradoxes which are the progeny of the governing view, all need to be distinguished and treated differently. A  further important group are constituted by the Sorites paradoxes, which affect predicates of practical intellectual possibility—like ‘intelligible’ when applied to expressions, ‘memorable’ when applied to patterns, ‘surveyable’ when applied to proofs, and so on. The proper response to paradoxes in this last group is a matter of great consequence for finitism in the philosophy of ­math­em­at­ics, but I have no space to engage the issues here.33 33  Some preliminary discussion may be found in Wright (1982, pp. 262–9). The paper is reprinted in Wright (1993, ch. 4).

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Further Reflections on the Sorites Paradox  163 Second, the distinctions among the Sorites family correspond to a variety of lessons which the broad lines of solution to these paradoxes contain for us. The Tachometer paradox is perhaps the least interesting in this respect. The positive lesson from it is only the very local point that what it is for an instru­ ment or human subject to have a threshold of sensitivity is something which needs careful description if incoherence is to be avoided. That is a point which it is as well to know, but it hardly trembles the foundations of the subject. Still, as recently noted, the paradox represents one way of being seduced by the thought that phenomenal predicates like ‘looks red’ apply to both, if to either, of any pair of indiscriminable items, so it needs a proper resolution. And pur­ suit of the signalling instrument analogy which arose in the course of our discussion of the paradox led us indirectly towards what, I would contend, is the germ of a good account of the distinguishing features of observational vocabulary. The essence of the No Sharp Boundaries paradox, on the other hand, is the thought that vagueness is per se Sorites-­generating, and hence that the paradox afflicts almost the entirety of our language. This thought is defective, or so I argued, but points up the need for definiteness operators in any coherent philosophical treatment of vagueness, and for a proper account, in turn, of the semantics and logical behaviour of such operators. Even so, the question was left unresolved whether paradox might not be inherent in higher-­order vagueness at least. Since, as noted, at least two important writers on vagueness have taken the view that higher-­order vagueness is essential to a large class of vague expressions, this is an issue on which further work is much needed. Third, we have found ourselves taking a somewhat unexpected route to one of the central concerns of Wittgenstein’s later philosophy of language, math­ em­at­ics, and mind: the issues to do with what I have elsewhere called the objectivity of meaning (Wright 1993, intro., pp. 3–8 and 26–9; 1986b) which seem to me to be the principal focus of his discussions of rule-­following in the Investigations and Remarks on the Foundations of Mathematics.34 One who 34  One is more wary of quoting Wittgenstein in support of a particular interpretation of his writ­ ings than virtually any other philosopher. But a sceptical reader should be reminded of these passages from Philosophical Investigations: 188. Here I should first of all like to say: your idea was that that act of meaning the order had in its own way already traversed all those steps: that when you meant it your mind as it were flew ahead and took all the steps before you physically arrived at this or that one. Thus you were inclined to use such expressions as: ‘The steps are really already taken, even before I take them in writing or orally or in thought.’ And it seemed as if they were in some unique way pre-­determined, anticipated—as only the act of meaning can anticipate reality. 218. Whence comes the idea that the beginning of a series is a visible section of rails invisibly laid to infinity? Well, we might imagine rails instead of a rule. And infinitely long rails correspond to the unlimited application of a rule.

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164  The Riddle of Vagueness accepts the objectivity of meaning thinks of understanding an expression in quasi-­contractual terms: there is, for instance, when one confronts the ques­ tion whether or not to classify a particular object in a particular way, a response to which one is obligated if one is to keep faith with the meaning of the relevant expressions as they were taught to one (and pay due heed to the relevant worldly facts).35 Less metaphorically, finite, over-­and-­done-­with epi­ sodes of explanation and other forms of linguistic behaviour may succeed in fully determining the meaning of a so far unmade utterance in a fashion inde­ pendent of any reaction which we will or would have to it: there is only the question of whether our reaction, if one is forthcoming, is in line with the meaning so independently constituted. Wittgenstein’s discussion has tempted many commentators to conclude that he is advocating scepticism about the 219. ‘All the steps are really already taken’ means: I no longer have any choice. The rule, once stamped with a particular meaning, traces the lines along which it is to be followed through the whole of space.—But if something of this sort really were the case, how would it help? No; my description only made sense if it was to be understood symbolically. —I should have said: This is how it strikes me. When I obey a rule, I do not choose. I obey the rule blindly. Likewise consider these passages from the Remarks on the Foundations of Mathematics: IV. 47. Might I not say: if you do a multiplication, in any case you do not find the math­ em­at­ic­al fact, but you do find the mathematical proposition? For what you find is the non-­ mathematical fact, and in this way the mathematical proposition. For a mathematical proposition is the determination of a concept following upon a discovery. You find a new physiognomy. Now you can, e.g., memorize or copy it. A new form has been found, constructed. But it is used to give a new concept together with the old one. The concept is altered so that this had to be the result. I find, not the result, but that I reach it. And it is not this route’s beginning here and ending here that is an empirical fact, but my having gone this road, or some road to this end. 48. But might it not be said that the rules lead this way, even if no one went it? For that is what one would like to say—and here we see the mathematical machine, which, driven by the rules themselves, obeys only mathematical laws and not physical ones. I want to say: the working of the mathematical machine is only the picture of the working of a machine. The rule does not do work, for whatever happens according to the rule is an interpretation of the rule. VII. 42. When I said that the propositions of mathematics determine concepts, that is vague; for ‘2 + 2 = 4’ forms a concept in a different sense from ‘p ⊃ p’, ‘(x) . fx ⊃ fa’ or Dedekind’s Theorem. The point is, there is a family of cases. The concept of the rule for the formation of an infinite decimal is—of course—not a specifically mathematical one. It is a concept connected with a rigidly determined activity in human life. The concept of this rule is not more mathematical than that of: following the rule. Or again: this matter is not less sharply defined than the concept of such a rule itself.—For the expression of the rule and its sense is only a part of the language-­game: following the rule. One has the same right to speak of such rules in general, as of the activities of follow­ ing them. Of course, we say: ‘all this is involved in the concept itself ’, of the rule for example—but what that means is that we incline to these determinations of the concept. For what have we in our heads, which of itself contains all these determinations? 35  The point of Wittgenstein’s concentration upon simple arithmetical series is that it enables him to drop this part of the proviso.

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Further Reflections on the Sorites Paradox  165 very notions of meaning and of correct or incorrect linguistic practice.36 But  that is as much a misreading as the opposite tendency which finds in Wittgenstein’s discussion only a corrective to a crudely phenomenalistic con­ ception of understanding, intention, and other cognate notions.37 What is at issue is the conception of our relationship with the meaning of a novel utter­ ance as being purely cognitive, of its meaning as something which, with any necessary help from the world, can determine the utterance’s truth value quite independently of any contributive reaction from us. There is and can be no such independence. We are ceaselessly actively involved in the determination of meaning. This is not to say that a reference to ‘us’ or to the ‘community’ is somehow implicit in a full account of the truth conditions of any sentence— still less that truth is what we take to be the truth, or anything of that sort.38 Wittgenstein, like anyone else, was aware that whole communities can be in error. The point is rather that failure to keep track of independently consti­ tuted meanings is not intelligible as a separate source of such error, alongside factual ignorance, misperception, illusion, prejudice, forgetfulness, and the like. The platitude, that the truth value of an utterance is a function only of its mean­ ing and of the context and prevailing states of affairs, is not in question either. What is in question is the idea that the meaning of the utterance is something which is as much constituted independently of our response to it as are the germane features of the context and prevailing states of affairs, and no less something which, in appraising the utterance’s truth value, we have to cognize. As Wittgenstein saw, the conception of meaning as objective in this way is prerequisite for the sort of platonism in mathematics which views math­em­at­ ic­al activity as the exploration of our conceptual constructs. It is also pre­ requis­ite for the thought—essential to the ‘private linguist’—that, just by a personal ostentive definition, or by otherwise going through the motions of forming an intention to use an expression in a certain way, it is possible to generate facts about the correct use of that expression on subsequent occa­ sions which are independent of our reaction on reaching them, and supply, at least in prin­ciple, the standard which those reactions have to meet.39 It is the objectivity of meaning which is at the heart of the governing view and which 36  See most notably Kripke (1982). 37  A criticism which could be levelled at the otherwise useful contributions of Budd (1984), and C. McGinn (1984). 38  The matter is so widely misunderstood that it is still worth emphasizing Wittgenstein’s explicit denials (1968, §241; 1978, VII, §40, and elsewhere). 39  That is not to say that ‘private language’ falls with objective meaning. It has still to be shown that there is no other way of establishing content for the ‘seems right’/‘is right’ contrast which the private language has to have. For an attempt to show just that (and to explain why the contrast is necessary), see Wright (1986a).

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166  The Riddle of Vagueness sustains the detective reading of the sort of proviso-­biconditional discussed in the previous section. The principal lesson of that discussion is thus that, more than being deeply problematical for the reasons which Wittgenstein’s deliberations bring out, the objectivity of meaning is actually a source of paradox. The third species of Sorites paradoxes—those issuing from the governing view—thus bring us up against absolutely basic questions in the philosophy of language. But how general is the challenge to the objectivity of meaning which they uncover? It is not confined to ‘looks red’ and its ilk since, as noted above, being red is looking red when other things are equal. So both secondary-­quality predicates and, when they are distinct, what we might term their phenomenalizations—the corresponding ‘looks’, ‘sounds’, and so on predicates—come within range. Also in range are all predicates of the sort ‘is justifiably characterizable as . . .’, for whose correct application looking, or sounding, and so-­and-­so suffices, ceteris paribus. This is a wide class but hardly a comprehensive one. While it is possible to argue for a global rejec­ tion of the objectivity of meaning on the basis of its failure for this class (see Wright 1986b, pp. 292–4), I do not know of any such argument which does not draw on considerations quite different from those with which we have been occupied here. In my earlier papers I rather tended to write as though the governing view paradoxes show us a route to rejection of the objectivity of meaning which is quite different from anything to be found in Wittgenstein’s writings but of no less generality. But that now seems to be an overstatement. The best paradoxes of the kind—par excellence that for ‘looks red’ on which we have concentrated—depend on quite special considerations, and no global extension seems to be in prospect.40

40  Versions of parts of this material were presented at colloquia at MIT, Johns Hopkins University, and the University of Michigan at Ann Arbor in Spring 1986. I am grateful to those who participated for a number of useful comments and questions.

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5 Is Higher-Order Vagueness Coherent? I It is widely assumed not merely that the Sorites afflicts vague expressions only, but that it is a paradox of vagueness—that vagueness is what gives rise to it. Since almost all expressions in typical natural languages are vague, that belief brings one uncomfortably close to the thought, advocated by such philo­sophers as Peter Unger (see, for example, Unger 1979b), that natural languages, and the conceptual systems which they embody, are typically incoherent. But I think the thought that vagueness, per se, generates Sorites-susceptibility is a muddled thought.1 When spelled out it goes, presumably, something like this. If F is vague, its very vagueness must entail that in a series of appropriately gradually chan­ ging objects, F at one end but not at the other, there will be no nth element which is F while the n + 1st is not; for, if there were, the cut-off between F and not-F would be sharp, contrary to hypothesis. Accordingly, the vagueness of F over such a series must always be reflected in a truth of the form: (i)  ¬(∃x )(Fx & ¬Fx′), (where x′ is the immediate successor of x ). That, of course, is a classical equivalent of the universally quantified conditional which is the major premise in standard formulations of the Sorites—a thought which prompted Hilary Putnam to suggest that a shift to intuitionist logic might be of value in the treatment of vagueness. So indeed it might, in other connections. But it will not help here; for intuitionistic logic will yield a paradox from (i) (as will any logic with the standard ∃- and &-Introduction rules + reductio ad absurdum).2 This form of the paradox—the No Sharp 1  And that the most convincing and troublesome versions of the Sorites draw on other features of the (vague) expressions they concern. See Wright (1987a, sects. I and VII, this volume, Chapter 4). 2  Of course, what intuitionistic logic does make possible is a response to the paradox which treats it as a reductio of the major premise, hence as a proof that ¬¬(∃x)(Fx & ¬Fx′), without sustaining the double negation elimination step which forces that conclusion into a statement the precision of F. But the basic difficulty remains: whether or not the double negation is sustained, to treat the paradox as a reductio is to deny a premise which seems to say merely that F is vague. Any genuine solution to the

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0006

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168  The Riddle of Vagueness Boundaries paradox—thus appears to constitute a proof that (one kind of) vagueness is eo ipso a form of semantic incoherence. Only ‘appears’ though. What, it seems to me, the No Sharp Boundaries paradox brings out is that, when dealing with vague expressions, it is essential to have the expressive resources afforded by an operator expressing def­in­ite­ness or determinacy. Someone might think that the introduction of such an op­er­ ator can serve no point, since there is no apparent way whereby a statement could be true without being definitely so. That is undeniable, but it is only to  say that—in terms of a distinction of Michael Dummett (1973, pp. 446–7)—the content senses of ‘P’ and ‘Definitely P’ coincide; whereas the important thing, for our purposes, is that their ingredient senses—their contribution to contexts embedding them—differ, the vital difference concerning the behaviour of the two statement forms when embedded in negation. Equipped with an appropriate such operator, we can see that a proper expression of the vagueness of F with respect to the relevant sort of series of objects is not provided by (i) but rather requires a statement to the effect that no def­in­ite­ly F element is immediately succeeded by one which is definitely not F; that is (ii)  ¬(∃x )(Def (Fx )& Def (¬Fx′)). And this principle generates no paradox. The worst we can get from it, with or without classical logic, is the means for proving, for successive x′, that (iii)  Def (¬Fx′) → ¬Def (Fx ). Nothing untoward follows from that.

II If, however, we take seriously the idea of higher-order vagueness, then a case can be made that this merely postpones the difficulty. For, if the distinction between things which are F and borderline cases of F is itself vague, then assent to (iv)  ¬(∃x )(Def (Fx ) & ¬Def (Fx′)) paradox has therefore to explain how that appearance is illusory—how the major premise fails as a schematic description of vagueness. There is no way around the obligation and no reason to think, once it is met, that any further purpose will be served by imposing intuitionistic restrictions in this context.

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Is Higher-Order Vagueness Coherent?  169 would seem to be compelled even if assent to (i) is not. So once again the materials for paradox seem to be at hand, each ingredient move taking the form of a transition from ¬Def (Fx′) to ¬Def (Fx). But the following is the obvious reply. Of any pair of concepts, F and H, which share a blurred boundary, we shall want to affirm (v)  ¬(∃x )(Def (Fx )& Def (Hx′)), when x ranges over the elements of an appropriate series in which the blurred boundary between F and H is crossed. The original problem occurred in (i) when, with ¬F in place of H, we overlooked the need to prefix the predicates with a definiteness operator. And now we are guilty of the same oversight again in (iv); it is merely that this time H has been replaced by ¬Def (F). As soon as the inclusion of the definiteness operator is insisted on, all that emerges is (vi)  ¬(∃x )[Def (Def (Fx )) & Def (¬Def (Fx′))], which yields nothing more than the harmless (vii)  Def (¬Def (Fx′)) → ¬Def (Def (Fx )). Evidently the trick will generalize; so we need never, it seems, be at a loss for a way of formulating F’s possession of vagueness, of whatever order, in a way that avoids paradox.

III But this is too quick. It is possible to be confident that the sort of formulation illustrated by (vii) avoids paradox only because we have so far no semantics for the definiteness operator, and are treating it as logically inert. Without considering in detail what form a semantics for it might take, a crucial question is whether it would be correct to require validation for this principle:   

(DEF)

A1 … An  P A1 … An  Def (P )

provided {A1 . . . An} consists of propositions all of which are ‘definitized’. For, in the presence of DEF, and assuming that the corrected formulation, (vi) above, of what it is for the borderline between F and its first-order

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170  The Riddle of Vagueness borderline cases to be itself blurred, is itself definitely correct, the harmless (vii) gives way to (viii)  Def (¬Def (Fx′)) → Def (¬Def (Fx )), whose generalization will enable us to prove that F has no definite instances if it has definite borderline cases of the first order.3 By contrast, (iii) gives way, by parallel reasoning, only to the innocuous (ix)  Def (¬Fx′) → Def (¬Def (Fx )). The trouble is thus distinctively at higher order. DEF says, in effect, that the truth of each of a set of fully definitized pro­posi­ tions ensures that every consequence of that set is likewise definitely true. This may get some spurious plausibility from conflation with the distinct and indisputable principle that whatever is a consequence of a set of propositions each of which is definitely true is itself definitely true. But DEF is plausible in any case. In effect it comes to the claim that when a proposition of the form: it is definitely the case that P, is true, it cannot be less than definitely true. If DEF is valid, and (vi) is a satisfactory characterization, then higher-order vagueness— always a difficult and vertiginous-seeing idea—would seemingly be an intrinsically paradox-generating phenomenon (ergo, presumably, a delusory one).

IV Interesting recent work of Mark Sainsbury’s raises a number of points bearing on this paradox.4 I shall comment on three aspects of his discussion. 3 Proof: 1 (1) Def¬(∃x)[Def (Def (Fx)) & Def (¬Def (Fx′))] Ass. 2 (2) Def (¬Def (Fx′)) Ass. 3 (3) Def (Fx) Ass. 3 (4) Def (Def (Fx)) 3,DEF. 2,3 (5) (∃x)[Def (Def (Fx)) & Def (¬Def (Fx′))] 2,4,∃-intro. 1 (6) ¬(∃x)[Def (Def (Fx)) & Def (¬Def (Fx′))] 1,Def-elim. 1,2 (7) ¬Def (Fx) 3,5,6,RAA. 1,2 (8) Def (¬Def (Fx)) 7,DEF. 1 (9) Def (¬Def (Fx′)) → (Def (¬Def (Fx)) 2,8,CP. 4  Mark Sainsbury (1991). Sainsbury’s paper is a reaction to Wright (1987a, this volume, Chapter 4), in which the No Sharp Boundaries paradox for higher-order vagueness was first, to the best of my knowledge, presented. See also Sainsbury (1990).

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Is Higher-Order Vagueness Coherent?  171 First, Sainsbury objects that in (vi), which he takes as its classical equivalent (vi)c  Def (Def (Fx )) → ¬Def (¬Def (Fx′)), I picked a needlessly vulnerable characteristic sentence for higher-order vagueness. The motivation for (vi), recall, was that, if x′ is a borderline case of F, it will at least be true that it is not definitely F; and that, if it is a definite borderline case, then the same will be definitely true. Thus (vi), or, if you will, (vi)c, says that no (definitely) definite F thing is succeeded by a definite borderline case—that the distinction between the Fs and the definite borderline cases is not one with an abrupt threshold, not a sharp one. Isn’t that just what second-order vagueness ought to be?5 Sainsbury (1991, p. 176) believes that there are other equally well-motivated candidates for the characteristic sentence, from which one cannot by a proof of the same general structure as Wright’s, derive anything paradoxical . . . So their entitlement to represent vagueness needs to be undermined before any conclusion antithetical to vagueness can be drawn from Wright’s proof.

What candidates? Well, it is natural, when vagueness is at issue, to want to work with some notion, however intuitive, of truth-value gaps, or of truth values other than truth and falsity. Either will set up the possibility of a distinction within the notion of negation. One notion, the proper negation of A, may be defined as true if A is false, and false if A is true. The broad negation of A, by contrast, while false if A is true, will be true in any other case—true just in case A is other than true. In these terms, it is natural to characterize a borderline case of F as something such that neither the claim that it is F nor the proper negation of that claim is true. Writing ‘Neg A’ for the proper neg­ation of A and ‘Not A’ for the broad, x is thus a borderline case of F if    Not Fx & Not Neg Fx.

5  There is a slight infelicity here, in so far as ‘¬Def (Fx)’ is not actually definitive of x’s being a borderline case of F, but will also be true if x is a negative case. But nothing important hangs on this. The thought in the text is restored—if good at all—by restricting the range of ‘x’ to positive and borderline cases of F. Alternatively, the reader may prefer to treat ‘¬Def (Fx)’ as characterizing the agglomerate of borderline and negative cases together: (vi) and (vi)c then plausibly capture what it is for the distinction between F’s and this agglomerate to be vague—which is just what it is for F to be second-order vague.

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172  The Riddle of Vagueness This gives us something else to play the role of ‘¬Def (Fx′)’ in (vi)c. Reflecting that no intuition is offended by replacing the occurrence of ‘Def (Def (Fx))’ by one of ‘Def (Fx)’, we arrive at Sainsbury’s first alternative: (vi)s  Def (Fx ) → Not Def (Not Fx′ & Not Neg Fx′). Alternatively, suppose we stipulate, as is quite natural, that the claim that A is definitely true, while true when A is true, ranks as false in all cases save where A is true. So ‘Def (Fx′)’ is false when x′ is borderline for F. Then   

Neg Def (Fx′)



will be true of such an x′. The thought that no x′, next to a definite F, is a borderline case of F, can then be expressed by (vi)ss  Def (Fx ) → Not Neg Def (Fx′). 6 Sainsbury’s claim is that neither (vi)s nor (vi)ss generates a paradox in the fashion of (vi) and (vi)c; but that their credentials as characteristic sentences for higher-order vagueness are no less plausible. But cannot we get paradox with (vi)s and (vi)ss? It is a consequence of the stipulation by which the latter was motivated that no provision is made for the untruth of Def (A) save by its being false. So there is no contrast between Neg Def (Fx′) and Not Def (Fx′), and (vi)ss is equivalent to   

Def (Fx ) → Not Not Def (Fx′).



Since reductio ad absurdum is presumably valid in the form:

  

A1 … An  B & Not B A1 … An −1  Not An



that is going to ensure the availability of   

Not Def (Fx′) → Not Def (Fx ),



6  Sainsbury does not quite motivate (vi)ss this way. I have simplified his discussion slightly, which had a generalized version of (vi)s characterizing vagueness of order n +1 as Def n (Fx ) → Not Def n (Not Fx′ & Not Neg Fx′), where ‘Def n’ expressed n iterations of ‘Def ’. As he remarks, this works out only for n > 0—hence his shift to (vi)ss. Note that the latter shares the harmless infelicity of (vi) and (vi)c—see n. 5.

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Is Higher-Order Vagueness Coherent?  173 and hence reintroduce paradox.7 The situation with (vi)s is slightly more complicated. Presumably we will want to sustain (x)  Def (Fx ) → Not Neg Fx′, which says, merely, that each definite F is immediately succeeded by something which may not truly be denied to be F and ought surely to hold in any series of the sort which concerns us. But, if (x) is a definite truth, then, in the presence of (DEF), we may strengthen it to (xi)  Def (Fx ) → Def (Not Neg Fx′), which, in company with (vi)s, ensures that (xii)  Def (Fx ) → Not Def (Not Fx′) holds.8 Now (xii) must not be confused with (iii), whose contrapositive it resembles. For (xii), unlike (iii), is formulated in terms of broad negation, and it is very unclear, in consequence, whether, like (iii), it may be regarded as harmless. To see the difficulty, reflect that, whatever we stipulate about the semantic value of Def (A) when A is borderline, there is a powerful intuition that its value ought to coincide with that of A when A is polar—is true or false. But broad negation was characterized as false when A is true and true in any other case, including any form of indeterminacy. So it is quite unclear how the broad negation of A can be anything but polar. Hence the semantic value of Def (Not Fx′) cannot diverge from that of Not Fx′, so the occurrence of ‘Def’ in the consequence of (xii) is idle, and there is no evident way of resisting its conversion to (xiii)  Def (Fx ) → Not Not Fx′. Since there is no evident objection to double negation elimination for ‘Not’,9 that yields 7  Sainsbury in effect remarks on this. 8  The principle here appealed to is A1 … An  Not Def ( A & B); A1 … An  Def (B) A1 … An  Not Def ( A) 9 Suppose A is something other than true. Then Not A is true; so, by the matrix for ‘Not’, Not Not A is false—so of truth value equal or inferior to that of A. Hence

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174  The Riddle of Vagueness (xiv)  Def (Fx ) → Fx′. And, while (xiv) won’t formally yield a Sorites, all that then stands in the way of the paradox is a failure to exploit the powerful intuition canvassed above, that the semantic value of Def (A) should coincide with that of A whenever A is polar, and hence that Def (Fx′) should be true when Fx′ is—that is, by (xiv), whenever Def (Fx) is. It is far from clear, then, that either of Sainsbury’s proposed alternatives, (vi)s and (vi)ss, contributes to any case at all that the apparent link between higher-order vagueness and paradox is merely an artefact of reliance on (vi) or (vi)c as characteristic sentences. Moreover, there is a serious doubt whether anything better could be forthcoming in the framework he favours. I suggested above that, when dealing with vague expressions, it is essential to have the expressive resources afforded by Def. One moral of the foregoing, I suspect, is that it is essential to lack the expressive resources of the sort of broad negation operator produced by Sainsbury—an operator which always generates a polar sentence, no matter what the status of the sentence it operates on. That is the feature which gives rise to the eliminability of double broad n ­ egations.10 But the basic natural deduction proof of the double neg­ation of the law of excluded middle uses only rules which seem perfectly unexceptionable where broad negation is concerned.11 So there is no obstacle to the law itself in the form A or Not A.



And its arrival marks the demise of any hope of a satisfactory characterization of higher-order vagueness. With the law in place, we are powerless to refuse the claim that, for each Def (Fx′), either it or its broad negation will hold. The cost of blocking the Sorites which would result if we were to accept each

Not Not A → A will be acceptable on the standard sort of many-valued semantics for ‘→’. (Double Negation Introduction, by contrast, will fail for broad negation, as the reader will swiftly verify.) 10  See n. 9. 11 Viz. reductio ad absurdum, in the form remarked on above, and vel-introduction. Thus 1 2 2 2 1  

(1) (2) (3) (4) (5) (6)

Not (P ∨ Not P) P P ∨ Not P Not P P ∨ Not P Not Not (P ∨ Not P)

Ass Ass 2,vel-intro. 2,3,1,RAA 4,vel-intro. 1,5,1,RAA

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Is Higher-Order Vagueness Coherent?  175 Def (Fx′) is thus that we must at some point broadly deny one—tertium non datur. And, as soon as we do, we have established a sharp boundary to def­in­ ite F-ness, when the whole point was to provide an apparatus to describe what is involved in there being none. Prescinding from the detail of Sainsbury’s proposals, there is, in any case, a more basic question about his strategy. Recall that the purpose in introducing (vi)s and (vi)ss was to provide at least prima facie coherent characteristic sentences for higher-order vague predicates whose entitlement to represent vagueness needs to be undermined before any conclusion antithetical to vagueness can be drawn from Wright’s proof.

Surely this thought has matters precisely backwards. It is rather the entitlement of (vi), or (vi)c to represent vagueness which needs to be undermined before any comforting conclusion about the status of vagueness could be drawn from the availability—if any are available—of paradox-free char­ac­ter­ iza­tions. If a notion has an intuitively acceptable characterization which generates paradox, it is not progress towards a resolution of matters merely to devise other seemingly acceptable characterizations which, so far as one can see, avoid paradox. So long as nothing is done to disarm the intuitive credentials of the villain, they—the credentials—merely transfer into grounds for thinking that the apparently innocent characterizations either fail to do just­ ice to the intended notion or are not really innocent.

V Sainsbury’s second main reservation about the paradox concerns the role of the inference rule, DEF. He writes (Sainsbury 1991, pp. 175–6): . . . one cannot feel happy with the introduction of the undefined ‘Def ’ followed immediately by an assumption about its logic which leads to paradox. It would seem a clear possibility that there should be a conception of ‘Def ’ upon which it demands progressively higher standards. Such a conception would fail to validate . . . the definitization rule, DEF, and would need to be argued against if higher order vagueness is to be shown paradoxical by the argument.

These are sensible reactions, but they betray some misunderstanding of my original discussion. One thing I regard as definite progress, in an area where it

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176  The Riddle of Vagueness is exceedingly hard to make any, is the modest insight that the No Sharp Boundaries paradox may be defused by appropriate use of an operator of def­ in­ite­ness. What I sought to show was that that point, which ought to extend to a coherent characterization of vagueness of higher order if that notion is coherent at all, will not so extend unless DEF fails in some relevant way. The alternatives are thus to find fault with (vi), to disclose a relevant failing in DEF or face the consequence that higher-order vagueness is per se paradoxical. Sainsbury writes as if I had claimed to establish the third disjunct. But my aim was the disjunction. Nevertheless, he seems to underestimate the problem of disclosing a relevant failing in DEF. It is perfectly true that if, as Sainsbury puts it, ‘Def ’ demands ‘progressively higher standards’—if, in other words, in order for Def (A) to reach a certain level of acceptability, A must in general surpass it— then DEF will be invalid for a range of cases in which not all of the premises in a sequent to which it is to be applied are polar. To take the simplest case: if, when Def (A) is non-polar, Def (Def (A)) drops lower in degree of acceptability, then—since valid inference ought to be degree-of-acceptability preserving—the sequent    Def ( A)  Def (Def ( A)) will be invalid. But to advance that reflection is not yet to find fault with the kind of application of DEF essentially involved in the proof of the paradox. Consider the application at line 8.12 The objection is in effect that DEF may fail for cases in which the conclusion of its premise-sequent was not polar, since the definitization of that conclusion might drop crucially further in truth value, as it were. But such a case can arise only if (some of) the assumptions of the premise-sequent are already themselves non-polar—otherwise the premise-sequent would not be a valid entailment in the first place. And in that case the objection is beside the point. For assumption 1, we are taking it, is true—so much is the force of any assumption; and we presumably so select our starting point for the relevant Sorites that assumption 2 is likewise true. But if 1 and 2 are true claims, then so is line (vii). Similarly, at line (iv), DEF is used to obtain a conclusion from an assumption—3—presumed true. So, if either of the uses made of DEF in the proof is to fail, definitization must be capable of producing a drop in truth value even when applied to true premises. Sainsbury does nothing to motivate that idea, and it is not clear how it 12  See n. 3.

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Is Higher-Order Vagueness Coherent?  177 might be motivated. For it is just the denial of the ‘powerful intuition’ bruited earlier in discussion of (vi)s.

VI But Sainsbury’s most far-reaching and distinctive contention (1991, p. 170) is that it is in any case a mistake to try to capture vagueness, of whatever ‘order’, by means of a characteristic sentence— a sentence schema, containing a schematic predicate position, such that the sentence resulting by replacing the schematic element by a predicate is true iff that substitute is a vague predicate.

The impression to the contrary derives, in Sainsbury’s view, from the lingering influence of what he styles the ‘classical conception’ of vagueness: the idea that what defines a vague predicate is that it effects a tripartite div­ ision—into positive, negative, and borderline cases, respectively—where a precise predicate determines a merely bipartite one. The difficulty is then to say something coherent about how the tripartite division can itself be blurred at the edges—so that a merely tripartite distinction is seemingly not enough. My offering of (vi)  ¬(∃x )[Def (Def (Fx )) & Def (¬Def (Fx′))] was exactly an attempt to produce the execrated sort of characteristic sentence: it tries to say what it is for the distinction between the definite Fs and the definite borderline cases to be itself vague—for the transition between the two kinds of case not to occur at an abrupt threshold. According to Sainsbury, this attempt is misguided in principle. Rather we should recognize (Sainsbury 1991, pp. 179–80) that [t]he right way to characterize the vagueness of a predicate is by the fact that it classifies without drawing boundaries: it is boundaryless. A boundaryless predicate allows for borderline cases, but this is not its defining feature. A boundaryless predicate draws no boundary between its positive and negative cases, between its positive cases and its borderline cases, between its positive cases and those which are borderline cases of borderline cases. The phenomena which, from a classical viewpoint, lead to notions of ‘higher

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178  The Riddle of Vagueness order vagueness’ are accounted for by boundarylessness . . . To convince you that boundaryless classification is possible, I would ask you to think of the colour spectrum. It contains bands but no boundaries. The different colours stand out clearly, as distinct and exclusive, yet close inspection shows that there is no boundary between them. The spectrum provides a paradigm of classification, yet it is boundaryless.

He continues (Sainsbury 1991, p. 182): We must shift away from the classical perspective. We are carried away by images which make us find boundarylessness problematic. We think of a system of classification as like a grid, a system of pigeon-holes, a way of drawing a line, dividing a field. In this way of thinking, Frege’s idea that a boundaryless concept is no concept at all seems irresistible. But we should shift images. Classification is better likened to providing magnetic poles around which some objects cluster more or less closely and from which ­others are more or less repelled; some fall between a number of poles, drawn by more than one but especially close to none.

I have quoted Sainsbury at length here in order to emphasize that the criticisms I have been making focus on aspects of his discussion which are in any case rather distant from his main concern. This is not the occasion to attempt to appraise the conception of vagueness which the passage conveys, but I certainly do not venture to deny that there is something correct in the adjustment to much contemporary thought about vagueness which Sainsbury is trying to teach. I merely offer one deflationary thought. It is one question whether the idea of boundarylessness offers a useful fresh perspective on the nature of vagueness. It is a different question whether the adjustment points to an understanding of vagueness which is paradox-free. Even if the ‘classical conception’ mislocates what defines vagueness, it is quite another matter whether it altogether misdescribes it. Sainsbury himself acknowledges that ‘a boundaryless predicate allows for borderline cases’—so vagueness is associated with a tripartite division, or taxonomy of relevant cases, even if this is ‘not its defining feature’. And now, why should it matter whether a feature is a defining feature or not, provided it is a feature? How could the classical conception have been led to avoidable paradox by correct characterization of non-defining features, even if it mistakenly took them to  be defining ones? Above all, how is the conception of vagueness as boundarylessness fundamentally at odds with the characteristic sentence

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Is Higher-Order Vagueness Coherent?  179 approach—why should there not be ‘a sentence schema, containing a schematic predicate position, such that the sentence resulting by replacing the schematic element by a predicate is true iff that substitute is a boundaryless predicate’? And why in particular is the approach illustrated by (vi) not suit­ able to generate such a sentence-schema? Perhaps these questions somehow miss the whole point about boundaryless predicates. If so, it will be interesting to learn how. I tentatively conclude that the case—which must, of course, be flawed!—for thinking that higher-order vagueness is per se paradoxical is so far unanswered.

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6 The Epistemic Conception of Vagueness I According to the usual way of thinking, the indeterminacy on the margins of a vague concept is real. Vague concepts carve out their extensions rather as a blurred shadow carves out a region of the background on which it is cast. Thus, between the extension of such a concept and that of its complement, lies a blurry penumbral region—the domain of the borderline case. Vague concepts define the world rather as an imperfectly focused slide defines an image. Perfect precision, by contrast, is perfect focus. This harmless-­seeming imagery, however, seems prone rapidly to de­sta­bil­ ize. Imagine that the blurred shadow and its background are quite large—apt for measurement in terms of tens of metres, say; and imagine a straight line run horizontally across them and calibrated in terms of centimetre points. Then it seems irresistible to say that no two adjacent such points are one in light and the other in shadow. But, if shadow is conceived, as in the analogy, as if it were the conceptual complement of light, then that is to say that no point is such that, while it is in light, an immediate neighbour is not. And at least in classical logic, that is equivalent to the claim that, if a point is in light, so is its immediate neighbour—which is all that is needed for a Sorites paradox. Perhaps the single largest programme in contemporary work on vagueness consists in attempts, one way or another, to save something akin to the Fregean image of vagueness from paradox. One would think it must be ­possible—after all, blurred shadows and penumbral regions really do occur, so must admit, it would seem, of some form of distinctive yet coherent characterization. Any such approach may be viewed as indeterminist: it ­ accepts the reality, however exactly it should be described, of the conceptual analogue of the penumbral region. According to proponents of the epistemic conception, by contrast, the Fregean imagery actually utterly mistakes the character of (what we take to be) vague concepts. There is no genuine ­indeterminacy, no region of borderline cases between the red and the non-­red,

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0007

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182  The Riddle of Vagueness the bald and the non-­bald, the small and the large. In truth, these distinctions are—and must be—completely sharp. Vagueness is rather a matter of ignorance. The various forms of reaction within a speech community that we are accustomed to take to be a reflection of indeterminacy—hesitancy when all relevant information is in, or irreducible conflicts, or perhaps a consensus that any clear verdict would be somehow improper—these various phenomena come about not because there is any genuine indeterminacy, or a lack of a ‘fact of the matter’, but because speakers do not know any better, do not know where the sharp boundaries lie. Somewhere in the progression along any Sorites series, however marginal the differences between its adjacent elements, will come a sharp cut-­off point; a last case where the target concept applies, immediately succeeded by a case where it does not. It is merely that we do not—and cannot—know where this happens. This bizarre-­seeming view has one clear attraction and one clear merit. The attraction is that it allows an acknowledgement of the linguistic phenomena which we associate with vagueness to sit perfectly comfortably alongside classical logic and semantics, and thus to provide the simplest possible dis­sol­ ution of the Sorites paradox. The merit is that of bringing out that the ordinary idea of genuine semantic indeterminacy is not itself a datum, but a proto-­ theory of data—the linguistic phenomena noted—which do not themselves constitute or clarify what vagueness, as conceived indeterministically, should be taken to be. The—at least initial—availability of the epistemic view brings it home that this is material for theoretical, philosophical description, and that much discussion has tended to bypass this essential task.1 Even so, the epistemic proposal is apt to seem utterly unmotivated—would hardly seem better than a superstition were it not for the recent efforts of philo­sophers such as Timothy Williamson and Roy Sorensen2 to take on at least some of the explanatory obligations which it must incur. Two such ­obligations are paramount. First, and most obvious, some sort of case must be made to make it credible that the sharp cut-­offs that the epistemicist postulates

1  Proponents of the enduringly fashionable supervaluational approach to vagueness, for example, typically simply help themselves to the idea that there are cases to which acceptable valuations may assign differing truth values—no explanation is attempted of what distinguishes such cases from those—the definite cases—on which all acceptable valuations must agree. 2  Sorensen (1988, pp. 199–253). Williamson (1992b, pp. 145–62; 1994). The epistemic view also receives sympathetic treatment at the hands of Cargile (1969; see also 1979, sect. 36) and Campbell (1974). For further references, see Williamson (1994, p. 300, n. 1).

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The Epistemic Conception of Vagueness  183 really do exist. Second, and less obvious but equally pressing, an account is owing of why we cannot locate these cut-­offs—why, indeed, we lack any clear conception of what it would be to locate them. I think it fair to say that, given the technically cumbersome and variously unsatisfactory treatments of vagueness currently on offer from indeterminists, strong responses to each of these obligations would suffice to put Epistemicism in a very competitive position. That would be a remarkable turnabout for the modern debate, which has been conducted almost exclusively within an indeterministic setting. In what follows I shall accordingly review the best responses known to me on each count, all deriving from the work of Sorensen and Williamson. Then I shall outline what seems to me the most basic type of reason why the philosophical treatment of vagueness must dismiss these defences of Epistemicism, resourceful as they may be, as merely tours de force, and continue to struggle towards a coherent indeterminist conception of the matter.

II For the Existence of Sharp Cut-­Offs: (1) Williamson on Bivalence Suppose it accepted that the Disquotational Scheme,    (DS) ‘P ’ is true ↔ P , holds for every contentful indicative substitution for ‘P’. And suppose the indeterminist characterizes (determinate) borderline cases for a given such sentence as failures of Bivalence—as cases where neither that sentence nor its negation express truths. A demonstratively presented borderline case of red, for instance, will be a case where neither ‘That is red’ nor ‘That is not red’ is true. Since DS entails that the denial that ‘P’ is true is equivalent to the af­fi rm­ ation of ‘Not P’, and that the latter is equivalent to the predication of truth on ‘Not P’, we are immediately ensnared in contradiction. The epistemicist’s preferred conclusion will be that, since the denial of Bivalence is incoherent, so is the notion of a borderline case. Since vague concepts are characterized by the generation of borderline cases, it follows that no concepts are genuinely vague. QED.

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184  The Riddle of Vagueness This simple train of thought does a lot of work in Williamson’s exposition in particular.3 If it is right, then there is no coherent option but to recognize that there have to be sharp cut-­offs in Sorites series.4 But, of course, a number of assumptions are in play, namely: (a)   that the Disquotational Scheme is acceptable for vague substituends; (b)  that any biconditional entails the biconditional which results from negating both its halves; (c)   that borderline cases involve truth-­value gaps. Each of these could conceivably be denied. Which would it be best for the indeterminist to deny? My own view, to which I shall return shortly, is that (c)—the ‘gappy’ conception of the borderline case—is a mistake in any case. But there is room for manoeuvre, even if (c) is accepted.5 There are some fairly immediate moves to rehearse. To begin with, it may seem doubtful whether there could be any coherent unqualified rejection of the DS for vague sentences or any others. A sentence may be said to be true just in case the proposition it expresses is true. But the analogue of the DS for propositions, sometimes known as the Equivalence Scheme for truth,    (EQ) It is true that P ↔ P , may seem uncontestable. If so, then a biconditional has to hold between any assertoric sentence and the sentence which says of that sentence that it is true. And that claim is just what the DS schematizes. If that is accepted, the question becomes accordingly: what is the gist of that biconditional? Plainly, someone who accepts that a truth-­apt sentence, S, may lack a truth value cannot allow that S and the claim that it is true always 3  See Williamson (1992b, pp. 145–50; 1994, pp. 187 ff.). Essentially the same line is run in sect. 26 of Paul Horwich (1998, pp. 80 ff.); but Horwich seems to fight shy of an outright endorsement of Epistemicism. 4  Note the (suspicious) strength of the argument: it implies that there cannot be genuine in­de­ter­ min­acy even as a result of explicit intention. What if we deliberately only partially determine the application conditions of a concept? Williamson (1994, pp. 213–14) envisages a word ‘dommal’, whose use we determine by stipulating that it is to apply to all dogs and to no non-­mammal. Suppose we intentionally refrain from saying anything else. Is a cat a dommal? The natural view, that the matter has been left indeterminate, is squeezed out by Williamson’s argument—a consequence with which he is content. But suppose a range of candidates for a job, and a preliminary shortlisting meeting at which we decide that Jones will be interviewed and that no one without a Ph.D. will be interviewed, but then adjourn without any other decision. Can one agree with Williamson’s reasoning that cats are not dommals without being willing to conclude that Dr Smith will not be shortlisted? I will return to the matter of vagueness by intention and the problems, if any, which it raises for Epistemicism in Section VIII. 5  The discussion to follow may be compared with the concluding section of Peter Simons’s response to Williamson (1992b, pp. 173–7).

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The Epistemic Conception of Vagueness  185 enjoy the same truth status—are always true, false, or ‘gappy’ together. For, when S is gappy, ‘S is true’ ought, intuitively, to be false (since S determinately has some status other than truth). The most such a theorist can accept is only that, whenever either is true, so is the other. Let it suffice for the validity of the weak biconditional of A with B that necessarily, whenever A is true, so is B, and vice versa, the validity of their strong biconditional demanding, in add­ ition, coincidence in their truth status in all cases other than truth. Then a believer in determinate truth-­value gaps can accept the validity of the DS— but only as schematizing a range of weak biconditionals. The upshot would be a qualified rejection of assumption (b). For weak biconditionals are not unqualifiedly contrapositive, if I may so put it. More accurately, suppose we distinguish corresponding weak and strong negations: a strong negation of A is true in the one case when A is false (it does not matter for present purposes what value it takes when A is neither true nor false). A’s weak negation, by contrast, is true when A has any truth status other than truth. Then the principle:   

A↔B Not A ↔ Not B

holds good only if both ingredient connectives are weak or if the biconditional is strong. By contrast, if the biconditional is weak, but negation strong, then the validity of A ↔ B will be consistent with, say, A’s being false in a case where B is gappy; when only the strong negation of A will be true, so that the lower biconditional, even weakly interpreted, will fail. Since, so we are supposing, our theorist accepts the DS only as a weak biconditional, it follows that Williamson’s purported reductio will have to be run in terms of weak negation. But when so run, it terminates in no inconsistency—to predicate truth of ‘Not P’ is not, when negation is weak, to say something inconsistent with ‘P’s being gappy. Williamson’s argument should accordingly be reckoned ineffectual unless it can be shown that an endorsement merely of the weakly biconditional DS is less than is demanded: that the reasons for endorsing the DS at all enjoin viewing its biconditional as strong. Is that so? The biconditional would have to be regarded as strong if the sentences it connects—instances of ‘P’ and ‘P is true’—were equivalent in content: expressive of the same proposition, or whatever. That would entail that their introduction into a single form of context would always preserve content, and it would hence be unintelligible how their negations could differ in

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186  The Riddle of Vagueness truth status. Manifestly, however, such sentences do not coincide in content. For one thing, they deploy quite different conceptual resources—it is poss­ ible fully to understand a sentence of the form ‘P is true’ without understanding the mentioned ‘P’. For another, their cross-­substitution in certain types of context may generate changes in truth conditions: compare, for instance, If ‘red’ meant green, ‘grass is red’ would be true

and If ‘red’ meant green, grass would be red.6

This is not decisive, however. For Williamson’s thought could just as well proceed not on the basis of the DS but via the Equivalence Scheme itself: It is true that ‘P’ if and only if ‘P’,

which will generate contradiction in just the same way once harnessed to the supposition that a given proposition lacks a truth value, so that both It is not true that P

and It is not true that Not P

are enjoined. And there are not the same obstacles as with the DS to the idea that instances of the two halves of the Equivalence Scheme do express the

6  This, of course, is to parse the consequents of these conditionals as, respectively: . . . it would be the case that [‘grass is red’ is true] and it would be the case that [grass is red]. But that seems alright. Alternatively, consider any example of this shape. Suppose the actual F would not be the F, and would not be G, under C-conditions. Then the counterfactual, Were it the case that C, the F would not be G will have a reading under which it expresses a truth; but that is manifestly no g­ uarantee that Were it the case that C, ‘the F is not G’ would be true.

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The Epistemic Conception of Vagueness  187 same content; surely the thought that P just is the thought that it is true that P. To this the defender of the truth-­value gap conception of the borderline case must reply, I think, that those simply are not the same thought—precisely because when the proposition that P is gappy, the thought that it is true that P will not be a gappy thought but a false one. The ‘gappy indeterminist’ must stick to it that even the Equivalence Scheme is valid only as a weak biconditional. It remains to be disclosed if any instability lurks in that proposal. It is not clear that there is any.

III Do Borderline Cases Involve Truth-­Value Gaps? The foregoing line of response to Williamson’s argument is premised on an acceptance, with Williamson, of assumption (c). The response is of interest mainly because, if coherent, it shows how we might live with that assumption without falling rapidly into the incoherence Williamson expects. But actually there is nothing mandatory about (c)—indeed it arguably quite mistakes the kind of indeterminacy involved in the usual run of borderline cases. Reflect that we do not, in general, expect agreement among otherwise competent judges either about which the borderline cases of a vague predicate are, or in their verdicts on what others are agreed are borderline cases. On the contrary, the normal idea of a borderline case is one on which competent judges may unite in hesitation but about which they may also permissibly differ. If Jones is on the borderline of baldness, that will, of course, make it allowable if we each judge that he is borderline, but also allowable—at least in very many cases—if you regard him as bald and if I do not. It is crucial to recognize that this phenomenon—of permissible ­disagreement at the margins—is of the very essence of vagueness, and that to leave it out of account is merely to miss the subject matter. I am not denying that there are often definite borderline cases of a given distinction. But, wherever a stable consensus can be elicited that something is on the borderline between two concepts, that is merely an indication that we could, if we wished, employ a concept intermediate between them and hence that they are not really complementary. Even where such a consensus operates, however, the vagueness of the concepts concerned will surface in the permissibility of

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188  The Riddle of Vagueness competent judges’ coming to divergent verdicts on their margins. If this were not so—if divergence were never permissible—then some particular verdict would have to be mandated in each case, and the boundaries between the concepts concerned would have to be sharp. This point accords ill with a conception of indeterminacy as failure of truth value for each of two contradictory claims. For so to construe the borderline case is to conceive of it as having a status inconsistent with the truth of each, and hence is a commitment to regarding either polar verdict as mistaken. We therefore do better, I suggest, to try to conceive being borderline as a status consistent with both the polar verdicts: for an item to be a borderline case on the red–orange border is for it to have a status consistent both with being red and with being orange (so not red), precisely because it is for that item to have a status under which it has not been determined whether it is red or not. If it has not been settled whether or not x is F, that cannot amount to x’s having a status inconsistent both with being F and with being not-F; if it were, then matters would have been settled after all—x would be neither. Vague predicates, non-­epistemically viewed, are predicates associated with indeterminacy. But indeterminacy should be conceived as a matter of things having been left open—which requires consistency with each of the relevant polar verdicts in a particular case. It is because the truth-­value gap conception, by contrast, requires borderline cases to occupy a status inconsistent with each of the polar verdicts that it is apparently open to the threat that Williamson tried to develop. If I am right, that is already a misconception, compounded by Williamson, of the way a non-­epistemic conception of vagueness should proceed.

IV For the Existence of Sharp Cut-­Offs: (2) Sorensen on Limited Sensitivity Sorensen’s argument likewise purports to show that there is no stable option but to credit what we take to be vague concepts with sharp boundaries, and that any non-­epistemic conception of vagueness is consequently incoherent (see Sorensen 1988, pp. 246–52). According to the epistemic conception, any Sorites series for a vague predicate F will contain a determinate last F and a determinate first non-F. This will be true no matter how similar in relevant respects the adjacent elements of the series are. So vague predicates, as

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The Epistemic Conception of Vagueness  189 conceived by the epistemic conception, are, in a clear intuitive sense, of unlimited sensitivity: that is, there is no degree of change, however small, in those relevant respects which is always insufficient to change the status of an item in point of F-ness. So far so good. It may now seem—and so Sorensen takes it—that one who rejects the epistemic conception has to be one who accepts that vague predi­cates are of correspondingly limited sensitivity: more precisely, that, for each such predicate, there will be some degree of change, u, in some relevant parameter(s) such that no possible pair of items, one a positive, the other a negative instance of the predicate, differ only to degree u or less.7 Sorensen’s argument is now directly to the conclusion that this concept of limited sensitivity, and hence any concept of vagueness which incorporates it, is actually incoherent. The argument proceeds by a meta-­Sorites. Let F be any vague predicate, say ‘short’, and u a degree of change in some relevant respect to which F is insensitive. As like as not, F will be sensitive nevertheless to changes of the order of, say, thousands of u. Sorensen (1988, p. 249) accordingly constructs this illustrative argument: (1) A Sorites argument concerning ‘short man’ has a false induction step if the step’s increment equals or exceeds ten thousand millimetres. (2) If a Sorites argument concerning ‘short man’ has a false induction step if the step’s increment is n millimetres, it also has a false induction step if the step’s increment is n–1. ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­_________________________________________________________ (3) All Sorites arguments concerning ‘short man’ having induction steps with increments convertible to millimetres have false induction steps.

The conclusion is tantamount to the unlimited sensitivity of ‘short’. But Premise 1 is undeniable. So the opponent of Epistemicism must either reject Premise 2 or dispute the validity of the argument. Sorensen is surely right that the latter course is no real option. But why would it be awkward for a believer in limited sensitivity to reject Premise 2? Because, according to Sorensen, to suppose Premise 2 false is to suppose an exact threshold—though presumably an unknowable one—to the degree of limited sensitivity of ‘short’. That, however, would be to go epistemicist at

7  Limited sensitivity is thus to be equated with tolerance in the sense introduced in Wright (1975, this volume, Chapter 1).

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190  The Riddle of Vagueness second order, and so to give the game away, since there is no conceivable prin­ cipled motivation for the indeterminist’s making such an exception. There is some room for manoeuvre against this. Someone might want to query, for instance, Sorensen’s implicit equation of rejection of Premise 2 with its (classical) denial. But his argument is easily strengthened so as to pre-­empt such skirmishes, and to provide a nastier pay-­off for the indeterminist than to force him into ad hoc-ery. Consider this principle: (4) If some Sorites argument for F that works with a series each pair of ­adjacent elements of which differ by exactly n u contains a major premise to which there is a counterexample in the series, then some Sorites argument for F that works with a series each adjacent elements of which differ by exactly n–1 u will contain a major premise to which there is a counterexample in its series.

This entails that there are no predicates of limited sensitivity, for it entails that there is no lower limit to the sensitivity of any predicate which has both positive and negative instances. But Premise 4 is entailed by the supposition that F is of limited sensitivity.8 So, if F is of limited (but non-­null) sensitivity, then it follows that there are no such predicates. So there are no such predicates. All predicates with both positive and negative instances are of unlimited sensitivity, and hence perfectly precise! It merits remark that this argument depends on no distinctively classical principles. It is intuitionistically valid, and indeed will run in any logic which sustains conditional proof, reductio ad absurdum, and the usual quantifier rules. It is another question what precisely it shows. One thing Sorensen would presumably want to claim for it is that—like the version he himself presents— it pre-­empts the incoherentist response to the Sorites offered by, among 8  Proof. Suppose F is insensitive to differences of one u, and assume the antecedent of Premise 4, i.e., suppose a Sorites argument for F that works with a series, S, each adjacent two elements of which differ by exactly n u, and let k and k′ be an adjacent pair of elements in S which constitute a coun­ter­ exam­ple to the relevant major premise. And now consider any item k* which differs by exactly n–1 u from k and by exactly one u from k′ (and in no other respect from either). Construct a series, S*, whose nth element is k, whose next element is k*, and whose every non-­initial element differs from its immediate neighbours by exactly n–1 u. We have Fk. Suppose Fk*. Then there are a pair of items, k* and k′, differing but by a single u, such that Fk* and not-Fk′, contrary to the hypothesis that F is insensitive to such differences. So not-Fk*, and k and k*, must accordingly be a counterexample to the major premise for an S*-based Sorites. Hence, there is a Sorites argument for F that works with a series each adjacent elements of which differ by exactly n–1 u, and which contains a major premise to which there is a counterexample in its series. So the consequent of Premise 4 holds on the supposition of its antecedent; so Premise 4 is true.

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The Epistemic Conception of Vagueness  191 ­ thers, Dummett (1975) and Unger (1979b), according to which vague predi­ o cates really are Sorites-­prone, sustaining appropriate major premises by virtue of genuine features of their sense.9 This response demands the limited sensitivity (tolerance) of vague expressions. Limited sensitivity is what it has just been proved no predicate has. That the argument militates against incoherentism seems, however, a misconception: if anything, just the opposite seems to be true. What is shown by the argument is that no predicate can be of limited but non-­null sensitivity. The problem is to combine the hypothesis of limited sensitivity with the claim that larger changes transform F’s into non-F’s and vice versa. However, the incoherentist should not want to make the latter claim: their point is precisely that there are no determinate F’s and non-F’s. If anyone is put in trouble by the argument, then it is the commonsensical indeterminist who essays to  accept both the coherence of vague expressions—their possession of at least some determinate positive and negative instances—and their limited sensitivity. That said, it must immediately be granted that an argument that reduced the options to Epistemicism and incoherentism would be of the greatest interest. On reflection, however, this weaker contention, too, has certainly not been demonstrated. To credit the argument, in either version, with that significance depends upon the so far unjustified assumption that to deny that a vague predicate is possessed of the sharp boundaries attributed to it by the epistemicist must be to attribute limited sensitivity (tolerance) to it instead: the assumption, indeed, that vagueness just is limited sensitivity. That identification must and will simply be denied by any indeterminist—be they supervaluationists, degree-­theorists, fuzzy logicians, or whatever—who believes they have the resources to take principled exception to the major premises of Sorites paradoxes.

V Definiteness It is worth reflecting that drawing a principled distinction between vagueness and tolerance need not depend upon any heavyweight—for example, ­supervaluational or degree-­theoretic—semantic apparatus. The intuitive 9  See Sorensen’s remarks (1988, p. 250) against Unger.

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192  The Riddle of Vagueness tradition of finding use for an operator of definiteness in the description of  phenomena to do with vagueness easily makes the necessary initial ­distinctions. Williamson assumed a characterization of the borderline cases of F as items of which neither ‘x is F’ nor ‘x is not F’ holds good. Adjust that to: items of which neither ‘x is definitely F’ nor ‘x is definitely not F’ holds good, and reads—in accordance with my earlier suggestion that borderline-­case status is a matter of genuine indeterminacy, rather than occupancy of an estate inconsistent with both poles—the claim that x is not definitely F in such a way that it does not entail that x is not F. Then we can now express F’s lack of sharp boundaries in an appropriate Sorites series as consisting in the truth of the claim:    ¬ (∃x )(Def (Fx ) & Def (¬Fx′)), a principle which, unlike the negation of the epistemicist’s ascription of a sharp boundary:    ¬(∃x )(Fx & ¬Fx′), is not at the service of a Sorites paradox.10 The epistemicist can hardly deny that it is permissible to introduce such an operator. For an operator of knowability will, in their view, have both stipulated features: the borderline cases of F will indeed be cases such that they are ­neither knowably F nor knowably not-F, and there will indeed be no ­entailment from ‘x is not knowably F’ to ‘x is not-F’. What they doubtless will, when challenged, deny is that such an operator has any coherent interpretation suited to the needs of indeterminism. Sorensen, for his part, offers no explicit argument for that claim. Williamson, however, does express scepticism on the point. He writes: what more could it take for an utterance to be definitely true than just for it to be true? Given that it cannot be neither true nor false, how could it fail to be definitely true other than by failing to be true? Such questions are equally pressing with ‘false’ in place of ‘true’. Again, ‘TW is thin’ is no doubt ­def­in­ite­ly true if and only if TW is definitely thin, but what is the difference between 10 I am simplifying. There are issues here about the coherence of higher-­order vagueness— spe­cif­ic­al­ly, a case to be made that, if the definiteness operator has certain seemingly not-­unmotivated features, it may yet be possible to generate a Sorites whose major premise is to the effect that F is higher-­order vague. See Wright (1992a, this volume, Chapter 5); and, for criticism, Edgington (1993) and Heck (1993).

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The Epistemic Conception of Vagueness  193 being thin and being definitely thin? Is it like the difference between being thin and being very thin? Can ‘definitely’ be explained in other terms, or are we supposed to grasp it as primitive? (Williamson 1994, pp. 194–5)

The first two questions come to: why does not the schema (DEF→):    P → Def (P ) hold generally good? It does not seem intelligible that there should be any way for an utterance to be true save by being definitely true—at any rate, there is no species of indefinite truth. But, if that is decisive, then—since the ­converse principle is uncontroversial—statements and their ‘definitizations’ will be generally equivalent, and the introduction of an operator of ­definiteness will be pointless. This is quite a subtle matter and needs treatment in detail. But let me air some reflections on it and suggest a direction. First, to scotch one natural friendly thought. It might be supposed that what the indeterminist requires is not to explain how (DEF→) may fail by dint of some statement’s being true while its definitization is not—maybe there can indeed be no explaining that—but to explain how such a pair can contribute differently to the truth conditions of contexts containing them. That need not be the same thing, as we know from consideration of the weak biconditional earlier. But this is a blind alley. Remember that the situation under review is that of an indeterminist who proposes no truck with truth-­value gaps or additional values. We therefore have just the classical truth values in play. And in that case, it seems that if ‘P’ and ‘Def (P)’ are indeed weakly equivalent—if each is always true if the other is—they must extensionally coincide in all circumstances, there being only one other truth status— namely, falsity—to occupy. So no extensional contexts differing only in that one embeds ‘P’ where the other has ‘Def (P)’ can vary in truth status. Since negation is extensional, it appears that Williamson’s point has to stick. Mustn’t the indeterminist either concede that ‘Not P’ and ‘Not Def (P)’ always have the same truth status—so that the definiteness operator is pointless—or resign himself to working with more than the classical truth values—with gaps, or additional values? Well, let’s review the data of the problem. The indeterminist wants ‘Not P’ and ‘Not Def (P)’ to differ in their content in such a way that (i) there is no entailment from the latter to the former—but it does nevertheless seem very intuitive that (ii) ‘P’ and ‘Def (P)’ should be, if not weakly equivalent, then at

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194  The Riddle of Vagueness least equivalent in some sense. (iii) There are not to be truth-­value gaps, or additional truth values. (iv) The truth conditions of negations are to be a function of those of the statements negated. How can these four constraints be met simultaneously? We can learn from a simple case where they are. Consider any statement ‘P’, and the corresponding statement that ‘P’ is warranted. Such a pair will have the same acceptability conditions, yet there will in general be no entailment from the first to the second; and indeed their negations, consistently with the truth functionality of negation and the currency of just the two classical truth values, have different conditions of acceptability. The explanation is simple. Any pair of statements which necessarily coincide in cognitive status will provide an example of co-­satisfaction of the four constraints provided their truth conditions diverge. And the moral, I suggest, is that the indeterminist should look for a broadly cognitive interpretation of the definiteness operator. Such an interpretation may put us in position to affirm, what at first seems absurd, that ‘P’ and ‘Def (P)’ do indeed have different truth conditions and that (DEF→) may possess a true antecedent but false consequent. Here is a first shot, intended just as an illustration of the territory in which the search for an account might begin. The governing idea is to construe ‘def­ in­ite­ly’ along the lines of operators like David Wiggins’s ‘There is nothing else to think but that’ (Wiggings 1990, pp. 61–86), or—more graphically—‘No one in their right mind could doubt that’. In the typical run of cases, the clear cases of a vague predicate will be decidable—items which are definitely, or definitely not, red, or heaps, or bald will be effectively recognizable as such. Suppose such a clear, decidable case of a predicate F—‘red’, say—and suppose someone who believes of it that it is not-F, not on the basis of testimony or inference, or groundlessly, but by an, as it transpires, abortive attempt at the relevant basic mode of decision (in this case, looking and seeing). There must be an explanation of the unfortunate upshot: maybe their vision is defective, or the light is bad, or their view is obscured; maybe they misunderstand the predicate. Whatever the correct explanation, it will advert to factors such that to know that they were in operation would be to have a reason to mistrust their opinion even without knowing what it was or knowing anything about the colour of the object. Call an opinion cognitively misbegotten if a factor of this kind—a factor whose operation could be used to explain the subject’s falling into error, and to know of which would be to have a reason to mistrust the subject’s opinion—contributes to its generation. Then the suggestion is that, in the sense that concerns us, for ‘P’ to be definitely true is for any appropriately generated opinion that ‘Not P’ to be cognitively misbegotten.

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The Epistemic Conception of Vagueness  195 Now, on this proposal, the thought that borderline cases may be marked by the permissibility of conflicting opinions will amount to the idea that ­associated with a typical vague predicate will be a range of cases for which neither of a pair of conflicting opinions need be cognitively misbegotten. About a borderline case of ‘red’, you and I may hold respectively that it is, and  that it is not, red without any factor operating in the generation of either opinion such that someone who knew of it would have a reason to mistrust that opinion. Earlier I argued that the phenomenon of permissible ­disagreement at the margins is of the very essence of vagueness. In terms of  the present proposal, we should refashion that suggestion: the basic ­phenomenon of vagueness is one of the possibility of faultlessly generated— cognitively un-­misbegotten—conflict.11 Note that the epistemicist need not disagree with this, as far as it goes. For the epistemicist, the borderline cases of ‘red’ are all determinately but in­scrut­ably red; naturally, therefore, should opinions conflict about such a case, no factors need be at work of the kind which would explain error, or conflict, in the decidable—definite—cases. Those are factors which would explain misfirings of the ordinary basic way of determining colour. Since that way of determining colour cannot engage inscrutable cases anyway, there is no a priori need to ascribe conflicting opinions about such cases of misfirings in its operation. Interestingly, then, on the proposed conception of ­definiteness, the epistemicist differs from the indeterminist not by rejecting the latter’s conception of what a borderline case is but by adding something to it: the addition, namely, of the hypothesis of universal determinacy in truth value: the Principle of Bivalence. Call an opinion based neither on testimony nor on inference, nor ­groundlessly held, a primary opinion. Then the suggested interpretation of ‘Definitely P’ amounts, to recapitulate, to this: any primary opinion that ‘Not P’ is cognitively misbegotten. How do matters stand with (DEF→), once the operator of definiteness is so interpreted? Evidently, the principle must fail if it is indeed a possibility for a pair of subjects to conflict in primary opinions 11  It is a further question whether such a conflict, once in the open, can be faultlessly sustained. Of course, it might be wrong publicly to profess either of the conflicting opinions. But would it be permissible to persist in holding it in the light of another’s apparently faultlessly generated ­ ­disagreement? Well, evidence of irreducible conflict is not always—not, for instance, in the moral sphere—received as a defeater of what one regards as admissible opinion. I know now that many ­opinions which I regard as unacceptable could be debated only to a stalemate by ordinary moral ­reasoning. But I shall not pursue the issue here. What the reader should note is that the proposal in the text only concerns faultless generation: one who construes the permissibility of conflicting ­opinions about borderline cases in this way has so far no commitment to the idea that such opinions remain permissible as conflict comes to light.

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196  The Riddle of Vagueness concerning ‘P’ without either component being cognitively misbegotten. For in that case we have—just as intended—both ‘Not definitely P’ and ‘Not ­definitely not-P’ holding true. If (DEF→) were good, we should therefore ­possess the materials, by a double contraposition, for a contradiction. The situation should seem quite straightforward from the epistemicist’s point of view. If the case is one on the borderline where ‘P’ nevertheless un­detect­ably holds true, ‘Def (P)’ may nevertheless be false, since a primary opinion that ‘Not P’ need not, in the precise relevant sense, be cognitively misbegotten. The indeterminist, by contrast, cannot give quite that ex­plan­ation of the failure of (DEF→), since—unwilling to accept Bivalence—they lack the materials for an insistence that there are such undetectable truths. Nevertheless, it is their intended conception that there is real in­de­ter­min­acy— that to be a borderline case is not to occupy a status inconsistent with either ‘P’ or its negation being true. So it has to be, if not required, then at least ­consistent to suppose either of the parties in an appropriately but faultlessly generated dispute over ‘P’ to be right—and hence, since both ‘Not def­in­ite­ly P’ and ‘Not definitely not P’ will hold in such a case, (DEF→) cannot be valid. As both theorists, however, can readily acknowledge, it remains that, for any vague statement ‘P’, admitting of primary opinions in the sense outlined, any warrant for ‘P’ will be a warrant to suppose that a primary opinion that ‘Not P’ will be somehow cognitively misbegotten. So it can be agreed on all hands that there be no identifying a counterexample to (DEF→); in any ­circumstances in which ‘P’ may rightly be accepted, the acceptability of ‘Def (P)’ is assured. There is accordingly a clear form of equivalence— co-­warrantability—between ‘P’ and ‘Def (P)’: all four constraints on the problem noted above are respected.12 These remarks are, to stress, only offered as preliminary pointers, albeit in an, as it seems to me, rather intuitive and obvious direction. Much would need to be clarified by an adequate treatment, including in particular the question of how this kind of proposal might be developed to accommodate  higher-­order vagueness—the vagueness of statements prefixed by the 12  An additional constraint, of course, is that the converse of (DEF→) should hold—that ‘Def ’ should be factive. But this should not give rise to difficulty. Here is one supportive line of thought, conditional on the hypothesis that the subject matter of ‘P’ and its kin is within primary reach, as it were; i.e., is open to competent, non-­inferential, testimony-­independent assessment—as are colour, heaphood, baldness, and height. Suppose ‘Def (P)’. Then any primary opinion that ‘Not P’ is cognitively misbegotten. But the same cannot, consistently with the hypothesis, be true of ‘P’. So it has to be possible faultlessly to generate a primary opinion that ‘P’. Hence—since, by hypothesis, the same is not true of ‘Not P’—P cannot be a borderline case. But nor can P be false—or, by the hypothesis, it would not be true that any primary opinion to that effect is cognitively misbegotten. So ‘P’ must be true. QED.

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The Epistemic Conception of Vagueness  197 def­in­ite­ness operator. Perhaps the most basic problem for the indeterminist is to characterize what vagueness consists in—to say what a borderline case is. It is also one of the least investigated. The epistemic conception should not be allowed to draw strength from this neglect. There is no cause to despair that the situation can be remedied.

VI For the Existence of Sharp Cut-­Offs: (3) Sorensen’s Clones Sorensen (1994) has one other interesting and independent line of argument for the epistemic conception. I shall prescind from some of the detail of his presentation, considering what I take—I hope correctly—to be the nerve. For all the vague properties which interest us we can envisage a process of gradual change whereby an item slowly acquires, or sheds such a property— gradually turns red, or goes bald, or becomes tall. Imagine a pair of qualitatively absolutely similar items—Sorensen’s clones—which undergo ­ such a process: say the process of growing tall, having started short. Suppose they grow at exactly the same rate. But suppose Clone A starts growing before Clone B. Now, the following principle may seem quite compelling: (C) If an item undergoes some finite process of change, then had it started earlier and changed at just the same rate, it would have finished sooner.

An immediate corollary would be that, if a pair of items undergo isomorphic processes from absolutely similar starting points, and if the processes proceed at exactly the same rates, then, if one starts first, it finishes first. It follows that Clone A finishes the process of growing tall before Clone B. But to say that Clone A finishes first is to say that at some time, t, it has become tall—so is then tall—while Clone B has not yet become tall. Let HA be A’s height at t and HB that of B. Then ‘tall’ has a threshold in the interval between HA and HB. But this interval could be as small as you like—for we could have B start to grow as soon after A as you like. It follows that ‘tall’ has a precise threshold. QED. There is nothing the matter with this imaginative and resourceful argument except its basic premise, (C). Say that a process is limited by predicates, F and G, just in case its inception, and conclusion, are, respectively, marked by the acquisition of F and G. Thus, the process of evaporation of water from

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198  The Riddle of Vagueness a bowl, for instance, is limited by the predicates of the bowl: ‘contains less water than it did’—the process has started, ceteris paribus, just as soon as the bowl contains less water than it did—and ‘contains no water’—it is over as soon as the bowl contains no water. Likewise, the processes of the clones growing tall are limited by the predicates ‘has grown taller’ at their inception—A’s process, for instance, has started just as soon as it is true to say of A that it is taller than it was—and ‘is tall’ at their conclusion: each process is over just as soon as it is true of the clone that it is tall (though it may, of course, continue to grow). Then the point is merely that the principle (C) stands in need of qualification: that it is good only if the process in question is either limited by the application of a precise predicate at its conclusion, or— more generally—by a predi­cate no more vague than that which limits its inception. What is it for a predicate to be more vague than another? The following limited case will do for our purposes. Say that F has a certain span of insensitivity, s, as measured in terms of some precise parameter, just in case no pair of items differing within that span and in no other respect can be such that, while one is definitely F, the other is definitely not F. Then G is more vague than F if each is sensitive to changes in the parameter in question but there is some s which is a span of insensitivity for G but not for F. Consider principle (C) as applied to any process limited at its inception by a predicate F and at its cessation by a vaguer G. The counterfactual ‘. . . had it started earlier . . .’ will invite consideration of a range of cases which will include some in which F first applies to the subject item at times at which it did not actually apply but which lie within the span of insensitivity, s, of G. But clearly of no such hypothetical process will it be true to say that it would have finished sooner than the actual process—since that would require that G apply to the item at a point when its condition would differ by less than s from its condition at various stages in the actual process at which it was definitely not G, contrary to the hypothesis that s is a span of insensitivity for G. It remains to reflect that the processes undergone by Sorensen’s growing clones are limited by predicates—‘has grown taller’ and ‘is tall’— which are in exactly this case; indeed, ‘has grown taller than’ is arguably a precise predicate. Of course, this way with the argument needs the tolerance-­free construal of the idea of imprecision, and the associated recourse to an operator of def­in­ite­ ness proposed in response to Sorensen’s limited sensitivity argument above. It stands or falls with that proposal.

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The Epistemic Conception of Vagueness  199

VII Why Can’t we Know where the Sharp Cut-­Offs Lie? At this point, no unanswered argument to mandate the epistemicist’s belief in universal sharp boundaries remains in play. In fact, though, Williamson (1992b, p. 162) for one takes the stance that Epistemicism would be justified purely by the operational advantages of its conservatism: Classical logic and semantics are vastly superior to the alternatives in simplicity, power, past success, and integration with theories in other domains. In these circumstances it would be sensible to adopt the epistemic view in order to retain classical logic and semantics even if it were subject to philosophical criticisms in which we could locate no fallacy.

and even presumably if the view was otherwise entirely unmotivated! Given this hard-­nosed line, it comes as something of a surprise to find Williamson ready to acknowledge that Epistemicism has any explanatory obligations. But acknowledge it he does: For most vague terms, there is knowledge to be explained as well as ig­nor­ ance. Although we cannot know whether the term applies in a borderline case, we can know whether it applies in many cases that are not borderline. The epistemic view may reasonably be expected to explain why the methods successfully used to acquire knowledge in the latter cases fail in the former. (Williamson 1994, p. 216)

This is the second main explanatory obligation distinguished earlier: to explain why—granted the sharp boundaries in which Epistemicism believes, and notwithstanding the shortcomings in the arguments so far reviewed for that belief—we seem barred from knowledge of where exactly they fall. Williamson is alone among the epistemicists whose writings are known to me in taking this challenge very seriously—to his credit.13 His answer, very 13  The obligation is surely quite general: anyone who takes it that the truth values of a certain range of statements are (potentially) unknowable ought to explain why. Platonism about number theory, for instance, and realism about the past have such obligations—albeit obligations which, it seems, they are easily able to meet. The platonist can build straightforwardly on the contrast between the finitude of our abilities and the potentially infinite range of epistemically independent consequences of statements involving quantification over all the natural numbers, and the realist about the past needs to point only to the contingency of the existence of effects distinctive of any particular event. Neither

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200  The Riddle of Vagueness roughly, is that any broadly reliabilist conception of knowledge will require that knowledge always be cushioned by a margin of error. In order to know where the sharp boundary falls in a spectrum of patches ranging smoothly and barely discriminably from red to orange, I would have to know of some patch j that it is red while at the same time knowing that j′, its neighbour, is orange. But I cannot know that an item, x, is red unless my impression that x is red is a reliable indicator that it is so—which will be true only if x is flanked on both sides by patches that are red, and which therefore—since knowledge is factive—cannot be known to be orange. In short: I could not reliably practice, just on the basis of ordinary vision and memory, a perfectly sharp red/ orange distinction. Hence, if there is such a distinction, I cannot know since I could not be reliable about precisely where it falls. This is an elegant response to the problem. Is it sufficient? Let us put on one side any reservation about the reliabilist conception of knowledge which drives Williamson’s suggestion. The more immediate doubt is whether it in any case has the wherewithal to explain everything that needs explaining. Consider any Sorites series for a predicate F, and let j be the last F element in the series, and j′ the first non-F element. What Williamson has explained, if anything, is why we cannot know the conjunction, ‘Fj & ¬Fj′’; that is, he has explained why we cannot know where the borderline is. But it also needs explaining why we cannot know where, in a large class of cases, the borderline is not. There are, on the epistemicist account, many conjunctions associated with the series in question whose truth value we cannot know: each k lying close to but not immediately at the borderline will be associated, on the epistemicist view, with a determinately true conjunction, ‘Fk & Fk′’, or ‘¬Fk & ¬Fk′’, whose effect is that the borderline lies elsewhere but which we cannot know. What explanation of that is in the offing? Clearly, the reasoning outlined for ‘Fj & ¬Fj′’ has no direct application. That reasoning works by appealing to the idea of a margin of error to justify the principle that ‘K(Fx) → Fx′’, and to the factivity of knowledge to conclude therefrom that never ‘K(Fx & ¬Fx′)’. There is no way of reasoning similarly to the conclusion that truths of the form ‘Fx & Fx′’, or ‘¬Fx & ¬Fx′’, cannot be known for any x lying in the appropriate range.

type of account would provide any justification for the idea that the relevant kinds of statement must be determinately true or false, of course—that is a different battle the realist must fight another day. The question now is not to justify a belief in the relevant range of determinate, truth- or falsity-­ conferring states of affairs but to explain why, granted that belief, such states of affairs may conspire to elude detection.

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The Epistemic Conception of Vagueness  201 What is true, of course, is that knowledge, per impossibile, of the true ‘Fj & ¬Fj′’ would thereby place a subject in position to know all the ­problematical instances of ‘Fx & Fx′’, or ‘¬Fx & ¬Fx′’. But Williamson has provided an explanation of the impossibility of knowledge of the latter only if knowledge of the true ‘Fj & ¬Fj′’ would not merely suffice but is necessary for that knowledge—only if there can be no way to know any problematical instance of ‘Fx & Fx′’, or ‘¬Fx & ¬Fx′’, save by knowing the true ‘Fj & ¬Fj′’. Without a vindication of that principle, his explanation is at best incomplete. Let us put the point somewhat differently. Suppose it is in fact impossible to know any more than we do about the property which, on the epistemicist’s view, is denoted by F. What Williamson’s explanation requires, if it is to extend to explaining not merely the unknowability of the true ‘Fj & ¬Fj′’ but that of each true problematical conjunction, is the truth of the following subjunctive conditional: were it possible to know more about that property—enough to know of certain of the problematical places where the cut-­off does not come that it does not come in those places—then one would know enough to locate the cut-­off. If this conditional fails, then it ought to be quite coherent to ­recognize for Williamson’s reasons that there can be no knowing where the cut-­off comes but to wonder whether some improvement in our information might not resolve some presently imponderable claims about where it does not come. And in that case, if the latter are indeed essentially imponderable, Williamson has not yet explained why. There are other possible misgivings. It is a datum for explanation not merely that we cannot know—do not know what it would be like to know— where the cut-­off comes, but that we have no conception of what it would be like to be able to justify any particular belief about where it comes. So Williamson’s explanation cannot be fully satisfactory if it depends on features of the concept of knowledge which are not shared by justified belief and other relevant forms of epistemic relationship. But on the face of it the explanation does so depend: it depends on the factivity of knowledge. Clearly no non-­factive attitude, ‘R’, can be subject to a margin of error in the sense Williamson defines for knowledge: it cannot be that whenever ‘R(P)’, then ‘P’ holds in all sufficiently similar cases, since ‘R(P)’ will not ensure the truth of ‘P’ in the first place. So, as Williamson is of course well aware, something additional needs to be said. He proposes a variety of ­suggestions (see Williamson 1992b, sect. 6; 1994, ch. 8, sect. 7); but I should have thought that, just so long as ‘R’ is a notion with some significant internalist component, the not-­very-­sophisticated reflection should suffice that, in the ordinary run of cases, our standards of justification find no significant

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202  The Riddle of Vagueness distinction between the adjacent elements in Sorites series. For instance, since the basic mode of colour appraisal is to look and see, it is ceteris paribus absurd to suppose that we might justify incompatible colour judgements about items that look exactly alike. That is a weaker principle than the tolerance of ‘may justifiably be regarded as red’ with respect to indiscriminable difference;14 but any solution to the associated Sorites paradox which disarms that tolerance principle should surely conserve the weaker one. However, all this skirmishing may distract from what is most fundamentally unsatisfying about Williamson’s proposal. Revert to the case of know­ ledge, and consider, say, a series of canes running from something below 5' 6'' to something in excess of 6' 6'' in length, each differing by 1/16'' from those adjacent to it. Now, if I am restricted to a merely visual assessment, any knowledge I can achieve of the lengths of the canes will certainly be subject to a margin of error in excess of 1/16''. So we can explain à la Williamson why I will not be able to know the cut-­off point for the predicate ‘is less than 6' long’ by means of such assessment. But, of course, it has not thereby been explained why—and indeed it is not true that—I cannot know the cut-­off point at all. For in this case I can merely measure up. I know exactly what property is denoted by ‘is less than 6' long’, and can recognize the appropriateness of other methods of assessment besides the visual, par excellence measurement, and apply them.15 That is: the most that Williamson has done is to outline a form of explanation why there is no knowing where the cut-­off for a vague predicate comes provided we are restricted to knowledge acquired by the means of assessment on which we actually rely in the application of that predi­cate. Nothing has been done to explain why knowledge or justified belief about the exact location of the cut-­off is impossible tout court. It might at this point be questioned whether that absolute impossibility has to be reckoned among the explananda. Are not the only clear data first, that we have no way of knowing where the cut-­off lies if restricted to usual methods and, second, that we have no inkling of other appropriate methods? But the question deserves to be pressed: Is there, in the epistemicist’s view, or is there not, an absolute impossibility here? If there is, it has not been explained by Williamson. Further explanation is still wanted of why it is impossible to determine the appropriateness of other methods of assessment 14  The weak principle implies merely that, if x and y look exactly alike, and if x may justifiably be regarded as red, then y may not justifiably be regarded as not red. 15  This objection is quite compatible with the idea that measurement-­based knowledge is also subject to a margin of error principle. I am assuming merely that the margin of error is smaller than 1/16''.

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The Epistemic Conception of Vagueness  203 and bring them to bear. Such an explanation might proceed, of course, by appeal to the impossibility of our knowing precisely what property is denoted by a vague predicate.16 But then an explanation will be owing of that. If, on the other hand, there is no absolute impossibility, then it has to be in principle possible to know more about the property denoted by a vague predicate ­sufficient to identify additional pertinent methods of assessment; and an ex­plan­ation will now be owing of how this additional knowledge might be accomplished. Either way, Williamson’s play with margins of error falls short of what, in recognizing that there is an explanatory obligation here at all, he implicitly undertook to provide.

VIII Epistemicism, Reference, and Deliberate Approximation What kind of property in general does the epistemicist take vague expressions to stand for? Precise ones, of course. But what type of precise properties, and why? There is usually no difficulty in finding a precise comparative relation to order a Sorites series. Such a comparative will be associated with a range of (relatively) precise properties, potentially at the service of the description of a sharp cut-­off. Thus, ‘is taller than’ can order a Sorites series for ‘is tall/small’ and is so associated with properties like being exactly 5' 11" tall. Is it the epistemic view that, for example, ‘tall’ denotes such a familiar precise property or not? If so, then—if we cannot know where the sharp boundaries lie—it must be unknowable which one; if not, then what would appear to be unknowable is what kind of property ‘tall’ denotes: some precise property which cuts off sharply in any suitable Sorites series, but is not a metric property of height. Surely, though, the second option is a non-­starter. ‘Tall’ has to stand for a property with the features that anyone taller than someone who has the ­property likewise has it, and that anyone not taller than someone who lacks it, likewise lacks it. It is hard to believe that there are any precise properties which have these congruences on ‘is taller than’ other than the familiar ­metric ones. If that is right, then the epistemicist position in general must be that, while we can readily form a conception of the kind of property referred to by a 16  We may, I suppose, discount the possibility of the property’s being identifiable but recognizably undecidable over the relevant range of cases.

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204  The Riddle of Vagueness vague expression, we do not know which such property is actually its ­reference. What is owing, therefore, is an account of what makes it the case that, for example, ‘tall’ refers—on a particular occasion of use, if you like: an assumption of context-­ dependence makes no difference to the essential point—to a particular metric property, F, rather than to any other as close to F as you please. Epistemicists say nothing, or almost nothing about this. Understandably: what could there possibly be to say? The root and most fundamental criticism of Epistemicism is that it has no account to offer of what constitutes reference. And this lack is not merely what we all lack—a philosophically watertight such account. Epistemicism cannot even gesture at an account of how the transcendent referential relations in which it believes might be constituted. What is it about our attitudes, or intentions, or practices with ‘tall’, or about the causation of our uses of the word, which might suffice to settle that—unknowably—it is exactly at 5' 10 13/31'', say, that its lower threshold is reached? We are ineluct­ ably ignorant, it seems, not merely of the alleged thresholds of vague expressions, but of any idea of the nature of the facts which constitute their reference to the relevant precise properties. I will come back to this. Is there, in any case, any scope for Epistemicism in every case where or­din­ ary thought would impute some degree of indeterminacy? Counterexamples might be thought to be provided by the particles which we seemingly use to  introduce vagueness deliberately: particles such as: ‘approximately’, ‘roughly’, ‘about’, ‘almost’, ‘not quite’, and so on. Usually such particles are redundant. In most ordinary contexts—buying canes in the ­garden centre, for instance—a more relaxed interpretation of the truth conditions of what people say will prevail than is literally warranted. I say I want six 7' canes; but  it will be good enough if the canes are approximately 7 feet in length. In most ordinary contexts, the approximate truth of what is strictly a precise statement will be good enough; and, since it will be ­understood that this is so, there will be no need to avoid a precise formulation. The contexts where particles of approximation have work to do are ones in which a precise interpretation will otherwise prevail. For instance, in a context in which its removability through a 3' 1''-wide door frame is about to be settled with tape measure, you had better express your merely visual ­assessment of the width of a sofa by saying, for example, that it is about 3' wide. In such a context the particle will have the effect of relaxing, to some unspecified extent, what will otherwise be taken as a relatively precise set of truth conditions. In general, the role of such particles seems unquestionably to be to introduce some conveniently indeterminate degree of flexibility. But for the epistemicist, there

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The Epistemic Conception of Vagueness  205 can simply be no such thing as relaxing the truth conditions of a precise ­statement in such a way—for there is no such thing as introducing a genuine indeterminacy. So what is the epistemic view of such particles—what do we use them to do? The epistemicist has no option but to say that when a particle of ­approximation is added one precise statement is supplanted by another. For instance, if I claim that Jones is roughly 6' tall, the epistemicist must so ­construe the truth conditions of what I say that, for some fixed k and j, the coincide with those of the statement: Jones is more than 5 k and less than 6 j tall. Of course, I do not and cannot know what these values are. And it is utterly obscure, for all any epistemicist has said, what mechanism determines them. But that is not now my point. The question is rather how, on the ­epistemic view, the relaxation—which it is the whole point of the added ­particle to effect—can be seen as being effected. The epistemicist will presumably have to elaborate their view along these lines. What a speaker tries to do by the relaxation is to allow that their statement can be accounted true in each of a range of cases when Jones’s height differs from—is greater or smaller than—6. Consider every case in which a speaker would reckon their relaxed statement true. The greatest and smallest of these will be flanked by a range of cases about which they will probably hesitate whether or not their statement should still be regarded as true; and these in turn will be flanked by cases which they would regard as beyond what is allowed for even by the relaxed statement. So, all the epistemicist has to do, it seems, is to ensure that the relevant values for k and j above effect cuts within the area of their hesitation. Provided the mechanism—whatever it is—which determines these values places them in this region—and provided speaker and audience tacitly know this—there will be something like the intuitively intended relaxation. It will be as if they had said: Jones’ height lies within a precisely bounded region of whose endpoints I can tell you only that they are respectively somewhere between (say) 5' 10'' and 5' 11 1/2'', and 6' 1/2'' and 6' 2'' respectively.

It must be essentially along these lines that the epistemicist should propose to handle such particles. For it is utterly obscure how motivation might be supplied for the concession that particles of approximation generate genuine semantic indeterminacy, which would not compromise the epistemic view across the much wider, ordinary range of cases, where an intuitive vagueness has nothing to do with the presence of such particles.

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206  The Riddle of Vagueness The question arises, however, what relation such an account of the content of a relaxed statement is supposed to bear to the semantic intentions of its speaker—the intention, that is, that the approximate truth of the precise statement be good enough. Obviously, it would be no good for the epistemicist to concede that the content of this intention is indeterminate—that it is, just as common sense suggests, associated with only vaguely demarcated regions of fulfilment and non-­fulfilment. For, if that were granted, what objection could there be to the idea that the semantic role of ‘roughly’ and its kin is simply to carry over the indeterminacy of such an intention into a linguistic content? The view has to be that, when I intend that you should stand roughly here, for instance, the demarcation of the range of cases in which you would comply from that in which you would not is already perfectly precise, even before it comes to speech. I cannot but have a wholly precise intention; the attempt to relax precision results only in the supplanting of an intention whose exact conditions of fulfilment I know by another of whose exact conditions of fulfilment, broader but no less precisely demarcated than those of the former, I can have no idea. Maybe this upshot—that I have intentions and other attitudes whose exact conditions of fulfilment I do not and cannot know—may not seem too bizarre to those weaned on semantic externalism. I may intend to drink a glass of water, for instance, without knowing exactly what water is, or to avoid arth­ritis without knowing exactly what arthritis is. Sure, in both cases, I can also knowingly give one perfectly adequate formulation of my intention; but then the epistemicist will presumably rejoin that my preferred formulation—that you should stand roughly here—is likewise a knowing, perfectly adequate formulation of a condition of fulfilment whose precise boundaries I do not know. But there are evident differences. If the concept of water is correctly taken to be that of a natural kind, it is courtesy of the intentions of those who deploy it that it is so. We intend, that is, that there should be conceptual space for an account of what water essentially is—an account of whose details we may well be in ignorance—and we have a prior conception, however inchoate, of how such an account might be achieved. By contrast, no ordinary subject who knowingly possesses the sort of approximated intention we are concerned with will conceive of themselves as intending to introduce a cor­res­ pond­ing conceptual space—space for an account, for example, of where exactly ‘roughly here’ begins and leaves off—nor pretend to any conception whatever of what it would be to even look for one. There is simply nothing to inform such a search—beyond the manifestly inadequate constraint that the

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The Epistemic Conception of Vagueness  207 bound­ar­ies not be so located as to infringe clear verdicts—and absolutely no intuitive preparation for the idea that there is anything to search for. Ordinary forms of content externalism always proceed on the basis of an intentional abrogation of semantic responsibility. We, so to speak, knowingly delegate the determination of the content and boundaries of certain concepts to external factors. But the kind of determination postulated by the epistemicist is nothing we bargain for in this kind of way and would have to proceed by mechanisms of which we have no conception. Epistemicism simply helps itself to a rampant form of content externalism. It should acquire not one iota of plausibility from association with the moderate, local, carefully argued, externalist claims familiar in modern philosophy of language and mind. Of course, the dialectical balance would be different if the arguments to enforce global sharp boundaries had worked. Then, we should be compelled to work for a conception of reference to redeem Epistemicism’s borrowing. But those arguments do not work, and there is every cause for scepticism whether any could. In this setting, the epistemicist proposal comes across not as a serious philosophy of vagueness but as an invitation to a kind of semantic mysticism.17

17  Thanks to Bob Hale, Mark Sainsbury, John Skorupski, Roy Sorensen, Charles Travis, and Tim Williamson.

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7 On Being in a Quandary Relativism, Vagueness, Logical Revisionism

In this chapter, I shall propose a unified treatment of three prima facie unrelated problems. Two are very well known. One is the challenge of providing an account of vagueness which avoids the Sorites paradox. This has been discussed almost to tedium, but with the achievement, it is fair to say, of increasing variety rather than convergence in the proffered solutions.1 Another is the  problem of formulating a coherent relativism (in the sense germane to matters of taste, value, and so on). This is also well known. However, it has had rather less intense recent attention; part of my project in what follows (Section  I) will be to recommend a view about what the real difficulty is. But  the third problem—an awkward-looking wrinkle in the standard kind of  case for revision of classical logic first propounded by the Intuitionists and  generalized in the work of Michael Dummett—has, I think, not been widely perceived at all, either by revisionists2 or by their conservative opponents.3 The link connecting the problems, according to the diagnosis here entertained, runs via the notion of indeterminacy. Specifically: I will propose and commend a—broadly epistemic—conception of what (at least in a very wide class of cases) indeterminacy is which not merely explains how vagueness does not ground the truth of the major premises in Sorites paradoxes but also assists with the question what form an interesting relativism (whether global, or restricted to a local subject matter) may best assume, and helps to bring out what the basic intuitionistic—‘anti-realist’—misgiving about classical logic really is. Though differing in at least one—very significant—respect from the conception of indeterminacy defended in the writings of Timothy Williamson, Roy Sorensen, and other supporters of the so-called Epistemic Conception of 1  I am afraid that the direction of the present treatment will be to add to the variety. My hope is that it will draw additional credibility from its association with resources to treat the other problems. 2  Henceforward, I restrict ‘revisionism’ and its cognates to the specific form of logical revisionism canvassed by the Intuitionists and their ‘anti-realist’ descendants. 3  An insightful exception to this general myopia is Salerno (2000).

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0008

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210  The Riddle of Vagueness vagueness,4 I doubt if it would have occurred to me to explore the ensuing proposal without their precedent. Indeed, a second important sub-project of the discussion to follow is indirectly to make a case that the Epistemicists have hold of an insight which may be detached from the extreme and, for many, bizarre-seeming metaphysical realism which—with their own encouragement—is usually regarded as of the essence of their view. Our work will be in eight sections. Sections I–III will lay out the problems in the order indicated in the subtitle; Sections IV–VI will then take them in the reverse order and develop the advertised uniform treatment. Section VII will comment on the relation of the proposal to the Epistemic Conception of vagueness. Section VIII is a concluding summary.

I Relativism I.1. Let me begin with a reminder of the crude but intuitive distinction from which the relativistic impulse springs. Any of the following claims would be likely to find both supporters and dissenters: That snails are delicious That cockroaches are disgusting That marital infidelity is alright provided nobody gets hurt That a Pacific sunset trumps any Impressionist canvas and perhaps That Philosophy is pointless if it is not widely intelligible That the belief that there is life elsewhere in the universe is justified That death is nothing to fear. Disputes about such claims may or may not involve quite strongly held convictions and attitudes. Sometimes they may be tractable disputes: there may be some other matter about which one of the disputing parties is mistaken or

4  See Sorensen (1988, pp. 199–253); Williamson (1992b, 1994). The epistemic view receives earlier sympathetic treatment in Cargile (1969; 1979, sect. 36) and Campbell (1974) and is briefly endorsed in Horwich (1998, p. 81). For further references, see Williamson (1994, p. 300, n. 1).

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On Being in a Quandary  211 ignorant, where such a mistake or ignorance can perhaps be easily remedied, with the result of a change of heart about the original claim; or there may be a type of experience of which one of the disputing parties is innocent, and such that the effect of initiation into that experience is, once again, a change of view. But there seems no reason why that should have to be the way of it. Such a dispute might persist, even though there seemed to be nothing else relevant to it about which either party was ignorant or mistaken, nor any range of ­rele­vant experience which either was missing. The details of how that might happen—how the dispute might be intransigent—vary with the examples. But, in a wide class of cases, it would likely be a matter of one disputant ­placing a value on something with which the other could not be brought to sympathize; or with them being prone to an emotional or other affect which the other did not share; or with basic differences of propensity to belief, ­perhaps associated with the kinds of personal probability thresholds which show up in such phenomena as variations in agents’ degrees of risk aversion. Intuitively, claims of the above kinds—potentially giving rise to what we may call disputes of inclination—contrast with claims like these: That the snails eaten in France are not found in Scotland That cockroaches feed only on decomposing organic matter That extramarital affairs sometimes support a marriage That sunset tonight will be at 7.31 p.m. That there are fewer professional analytical philosophers than there were That there are living organisms elsewhere in the solar system That infant mortality was significantly higher in Victorian times than in Roman. Any of these might in easily imaginable circumstances come into dispute, and in some cases at least we can imagine such disputes too being hard to resolve. Relevant data might be hard to come by in some cases, and there are also material vaguenesses involved in most of the examples, on which a difference of opinion might turn. Then there is the possibility of prejudice, ignorance, mistake, delusion, and so on, which in certain circumstances—perhaps ­far-fetched—it might be difficult to correct. However, what does not seem readily foreseeable is that we might reach a point when we would feel the disputants should just ‘agree to differ’, as it were, without imputation of fault on either side. Opinions about such matters are not to be exculpated, to use a currently modish term, by factors of personal inclination, but have to answer to—it is almost irresistible to say—the facts.

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212  The Riddle of Vagueness This crude but intuitive distinction—disputes of inclination versus disputes of fact—immediately gives rise to a problem. Both types of dispute are focused on straightforward-seeming, indicative contents. But all such contents are naturally treated as truth-evaluable, and truth, one naturally thinks, is a matter of fit with the facts. So the very form of disputes of inclination seems ­tailor- made to encourage the idea that they are disputes of fact after all: disputes in which, ceteris paribus, someone is out of touch with how matters really stand. The problem is therefore: how to characterize disputes of in­clin­ ation in such a way as to conserve the species, to disclose some point to the layphilosophical intuition that there are such things at all, genuinely contrasting with—what one’s characterization had better correlatively explain—disputes about the facts. I.2. So far as I can see, there are exactly four broadly distinguishable types of possible response: (i)  Rampant Realism denies that the illustrated distinction has anything to do with non-factuality. For rampant realism, the surface form of disputes of inclination has precisely the significance just adumbrated: such disputes do centre on truth-evaluable contents, and truth is indeed a matter of fit with the facts. So, even in a radically intransigent dispute of inclination, there will, ceteris paribus, be a fact of the matter which one of the parties will be getting wrong. It may be that we have not the slightest idea how a particular such dispute might in principle be settled, and that, if charged to explain it, we would hesitate to assign any role to ignorance, or prejudice, or mistake, or vagueness. These facts, however, so far from encouraging relativism, are best attributed to the imperfection of our grasp of the type of subject matter which the dispute really concerns. I mean this option to be parallel in important respects to the Epistemic Conception of vagueness. The Epistemicist5 holds that vague expressions like ‘red’, ‘bald’ and ‘thin’ actually denote properties of perfectly definite extension. But we do not (or, in some versions, cannot) know which properties these are—our concepts of them, fixed by our manifest understanding of the rele­ vant expressions, fail fully to disclose their nature. There is thus a quite straightforward sense in which when I say that something is, for instance, red,

5 I shall capitalize—‘Epistemicist’, ‘Epistemicism’, etc.—whenever referring to views which, like those of Sorensen and Williamson, combine a conception of vagueness as, broadly, a matter of ig­nor­ ance with the retention of classical logic and its associated bivalent metaphysics.

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On Being in a Quandary  213 I (necessarily) imperfectly understand what I have said.6 Clearly there is space for a similar view about the subject matter of a dispute of inclination. It can happen that we express a concept by ‘delicious’ which presents a property whose nature it fails (fully) to disclose. This property may or may not apply to culinary snails. There is no way of knowing who is right in the dispute, but somebody will be. At any rate, the issue is no less factual than that of whether culinary snails are indigenous to Scotland. I do not propose to discuss the rampant realist proposal in any detail here. No doubt a fuller discussion of it would recapitulate many of the moves and counter-moves made in the recent debates about the Epistemic Conception of vagueness. Still, there are some interesting, foreseeably additional issues. Here are three: First, is there any principled ground whereby a theorist might propose an Epistemicist treatment of vagueness but refuse to go rampant realist over what we are loosely characterizing as matters of inclination? Second, can a rampant realist treatment of matters of inclination match the conservatism of the Epistemic Conception of vagueness? The Epistemicist 6  Experience shows that Epistemicists incline to protest at this. Suppose ‘tall’, say, as a predicate of human beings, applies to an individual just if they are precisely 5′ 11″ tall or more—that 5′ 11″ tall or more is the property actually denoted by the vague ‘tall’, so used. Then why, in saying that an individual is tall, should I be regarded as understanding what I have said to any lesser an extent than when, in circumstances where I do not know the identity of the culprit, I say that whoever broke the clock had better own up? Why should ignorance of what, in fact, I am talking about be described as an imperfection of understanding? Although it is not my purpose here to develop criticisms of the Epistemic Conception, I’ll take a moment to try to justify the charge. The foregoing protest assumes that the epistemicist is entitled to regard us as knowing what type of sharply bounded property an understood vague expression denotes, and as ignorant only of which property of that type its use ascribes. I know of no justification for that assumption. What type of sharply bounded property does ‘red’ denote? Something physical? Or a manifest but sharply bounded segment of the ‘colour wheel’? Or something else again? On what basis might one decide? And if the understanding of some common-or-garden vague expressions gives rise to no favoured intuitive type of candidate for their putative definitely bounded denotations, why should we favour the obvious candidates in cases—like ‘tall’—where there are such? Intuitively, to understand a simple, subject–predicate sentence, say, is to know what object is being talked about and what property is being ascribed to it. To be sure, the purport of that slogan should not be taken to require that one invariably has an identifying knowledge of the former: I can fully understand an utterance of ‘Smith’s murderer is insane’ without knowing who the murderer is. But it is different with predication. Here what is demanded of one who understands is, at least in the overwhelming majority of cases, that they know—in a sense parallel to the possession of identifying know­ledge of the referent of a singular term—what property the use of a particular predicate ascribes. Since the overwhelming majority of natural language predicates are vague, that is what the Epistemicist denies us. It would be no good for them to reply: ‘But you do know what property ‘red’ denotes—it is the property of being red!’ On the Epistemic account, I know neither which property that is, nor what type of property it is, nor even—in contrast to, say, my understanding of ‘. . . has Alex’s favourite property’ where, while ignorant in both those ways, I at least know what a property has to do in order to fit the bill—what would make it true that a particular property was indeed ascribed by the normal predicative use of ‘red’. It is the last point that justifies the remark in the text; if you were comparably ignorant in all three respects about the content of a definite description—thus ignorant, in particular, of what condition its bearer, if any, would have to meet—it would be absolutely proper to describe you as failing fully to understand it.

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214  The Riddle of Vagueness does not, properly understood, deny there is any such thing as vagueness; rather, they attempt a distinctive account, in epistemic terms, of what vagueness consists in. A similar account would be desirable, if the approach is to be extended to matters of inclination, of what it is that really distinguishes them from those matters which the opposing, mistaken view takes to be the only genuinely factual ones. A satisfying account must somehow save the crude and intuitive distinction, rather than merely obliterate it. Third, the question arises whether rampant realism can be reconciled with the good-standing of our ordinary practice of the discourses in question. If irremediable ignorance—for instance, a gulf between our concept of the property denoted by ‘delicious’ and the nature of that property—is at work in disputes of inclination, one might wonder with what right we take it that there is no serious doubt in cases where there is consensus that the property applies. Of course, the same issue arises for epistemic treatments of vagueness: if we do not know enough about the sharply bounded property we denote by ‘red’ to be sure where its boundaries lie, what reason have we to think we have not already crossed those boundaries in cases where we are agreed that something is red? However, the problem may be a little more awkward for an epistemic treatment of matters of inclination. For the Epistemicist can presumably rejoin that, however the reference of ‘red’ is fixed, a good account will constrain the word to refer to a property which does at least apply to the paradigms, on which we concur. The possible awkwardness for the extended, rampant realist view is that there are not, in the same way, paradigms for many of the examples of matters of inclination. That is: there are shades of colour that must be classified as red on pain of perceptual or conceptual incompetence, but there are no tastes that must similarly be classified as delicious. If matters of inclination—for instance, of gastronomic taste—even where not contested in fact, are as a class essentially contestable, at least in principle, without incompetence, then in contrast with the situation of ‘red’ and vague expressions generally, there would seem to be no clear candidates for the partial extensions that a competitive account of the reference of the distinctive vocabulary—‘delicious’, ‘over-salted’, and so on—might plausibly be required to conserve. (ii)  The second possible response to the problem of characterizing disputes of inclination is that of Indexical Relativism. On this view, truth-conditional contents are indeed involved in ‘disputes’ of inclination, but actually there are no real disputes involved. Rather, the seemingly conflicting views involve implicit reference to differing standards of assessment, or other contextual parameters, in a way that allows both disputants to be speaking the literal

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On Being in a Quandary  215 truth. Snails are delicious for you—for someone with your gastronomic susceptibilities and propensities—but they are not delicious for me—for one whose culinary taste is as mine is. Hurt-free infidelities can be acceptable to you—perhaps, to anyone inclined to judge the moral worth of an action by its pleasurable or painful effects alone—but they are not acceptable to me—to one inclined to value openness and integrity in close personal relationships for its own sake, irrespective of any independently beneficial or harmful consequences. This, very familiar, kind of relativistic move is still supported in recent phil­ oso­phy—for instance by Gilbert Harman on morals.7 Its obvious drawback is that it seems destined to misrepresent the manner in which, at least as or­din­ ar­ily understood, the contents in question embed under operations such as the conditional and negation. If it were right, there would be an analogy between disputes of inclination and the ‘dispute’ between one who says ‘I am tired’ and their companion who replies, ‘Well, I am not’ (when what is at issue is one more museum visit). There are the materials here, perhaps, for a (further) disagreement, but no disagreement has yet been expressed. But ordinary understanding already hears a disagreement between one who asserts that hurt-free infidelity is acceptable and one who asserts that it is not. And it finds a distinction between the denial that hurt-free infidelity is acceptable and the denial that it is generally acceptable by the standards employed by someone who has just asserted that it is acceptable. Yet, for the indexical relativist, the latter should be the proper form of explicit denial of the former. In the same way, the ordinary understanding finds a distinction between the usual understanding of the conditional—that, if hurt-free infidelity is acceptable, so are hurt-free broken promises—and the same sentence taken on the understanding that both antecedent and consequent are to be assessed relative to some one particular framework of standards (that of an actual assertor of the sentence, a framework which might or might not treat infidelity and promisebreaking in different ways). Of course there is room for skirmishing here, some of it no doubt quite intricate. But it is not clear that we should expect that indexical relativism can save enough of the standard practice of discourses within which disputes of inclination may arise to avoid the charge that it has simply missed their ­subject matter.

7  Harman has been, of course, a long-standing champion of the idea. The most recent extended defence of his views is in Harman and Thomson (1996, part one). For many-handed discussion, see Harman et al. (1998).

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216  The Riddle of Vagueness (iii)  The third possible response to the problem of characterizing disputes of inclination is that of Expressivism: the denial that the discourses in question genuinely deal in truth-conditional contents at all. Of course, on this view, there are, again, no real disputes of inclination at all—merely differences of attitude, feeling, and reaction. There has been a significant amount of recent discussion of this kind of approach, stimulated by the sophisticated versions of it proposed by writers such as Simon Blackburn and Allan Gibbard.8 But it confronts a very general dilemma. What is to be the expressivist account of the propositional—seemingly truth-conditional—surface of the relevant discourses? The clean response is to argue that it is misleading— that what is conveyed by discourse about the delicious, the morally ac­cept­ able, or whatever this kind of view is being proposed about, can and may be better expressed by a regimented discourse in which the impression that truth-conditional contents are being considered, and denied, or hypothesized, or believed, and so on is analysed away. However, it seems fair to say that no one knows how to accomplish this relatively technical project, with grave difficulties in particular attending any attempt to reconstruct the normal ap­par­ atus of moral argument in such a way as to dispel all appearance that it moves among truth-evaluable moral contents.9 The alternative is to allow that the propositional surface of moral discourse, to stay with that case, can actually comfortably consist with there being no genuinely truth-conditional contents at issue, no genuine moral beliefs, no genuine moral arguments construed as movements from possible beliefs to possible beliefs, and so on. But now the danger is that the position merely becomes a terminological variant for the fourth response, about to be described, with terms such as ‘true’ and ‘belief ’ subjected to a (pointless) high redefinition by expressivism, but with no ­substantial difference otherwise. (iv)  Of the options so far reviewed, the first allows that a dispute of in­clin­ ation is a real dispute, but at the cost of conceding that one of the disputants will be undetectably wrong about a subject matter of which both have an essentially imperfect conception, while the other two options deny, in their respective ways, that there is any genuine dispute at all. The remaining option—I will call it True Relativism—must, it would seem, be the attempt to maintain that, while such disputes may indeed concern a common truthevaluable claim, and thus may be genuine—may involve incompatible views 8  Blackburn (1984, ch. 6, ‘Evaluations, Projections and Quasi-Realism’),still remains the best introduction to his view, but the most recent official incarnation is Blackburn (1998); Allan Gibbard’s ideas are developed systematically in his magisterial work (Gibbard 1990). 9  For exposition and development of some of the basic difficulties, see Hale (1986; 1993; and 2002).

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On Being in a Quandary  217 about it—there need be nothing about which either disputant is mistaken, nor any imperfection in their grasp of what it is that is in dispute. Opinions held in disputes of inclination may, in particular cases, be flawed in various ways. But in the best case, the true relativist thought will be, such a dispute may oppose two opinions with which there is no fault to be found, even in principle, save by invocation of the idea that there is an ulterior, undecidable fact of the matter about which someone is mistaken. That hypothesis, dis­ tinct­ive of the first option, is exactly what true relativism rejects: for true relativism, genuinely conflicting opinions about a truth-evaluable claim may each be unimprovable and may involve no misrepresentation of any further fact. I.3. In the light of the shortcomings, briefly noted, of the three available alternatives—and because it has, I think, some claim to be closest to the commonsense view of the status of disputes of inclination—it is of central importance to determine whether the materials can be made out for a stable and coherent true relativism. In Truth and Objectivity (Wright 1992b) I proposed—without, I think, ever using the word ‘relativism’—a framework one intended effect of which was to be just that. The key was the contrast between areas of discourse which, as it is there expressed, would be merely minimally truth-apt, and areas of discourse where, in addition, differences of opinion would be subject to the constraint of cognitive command. The idea that there are merely minimally truth-apt discourses comprises two contentions, about truth and aptitude for truth respectively. The rele­ vant—minimalist—view about truth, in briefest summary, is that all it takes in order for a predicate to qualify as a truth predicate is its satisfaction of each of a basic set of platitudes about truth: for instance, that to assert is to present as true, that statements which are apt for truth have negations which are ­likewise, that truth is one thing, justification another, and so on.10 The view 10  A fuller list might include: The transparency of truth—that to assert is to present as true and, more generally, that any attitude to a proposition is an attitude to its truth—that to believe, doubt or fear, for example, that P is to believe, doubt or fear that P is true. (Transparency) The opacity of truth—incorporating a variety of weaker and stronger principles: that a thinker may be so situated that a particular truth is beyond their ken, that some truths may never be known, that some truths may be unknowable in principle, etc. (Opacity) The conservation of truth aptitude under embedding: aptitude for truth is preserved under a  variety of operations—in particular, truth-apt propositions have negations, conjunctions, disjunctions, etc. which are likewise truth-apt. (Embedding) The Correspondence Platitude—for a proposition to be true is for it to correspond to reality, ac­cur­ate­ly reflect how matters stand, ‘tell it like it is’, etc. (Correspondence) The contrast of truth with justification—a proposition may be true without being justified, and vice versa. (Contrast)

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218  The Riddle of Vagueness about truth aptitude, likewise in briefest summary, itself comprises two contentions: that any discourse dealing in assertoric contents will permit the definition upon its sentences of a predicate which qualifies as a truth predicate in the light of the minimalist proposal about truth;

and that a discourse should be reckoned to deal with suitable such contents just in case its ingredient sentences are subject to certain minimal constraints of syntax—embeddability within negation, the conditional, contexts of propos­ ition­al attitude, and so on—and discipline: their use must be governed by commonly acknowledged standards of warrant.

A properly detailed working-out of these ideas11 would foreseeably have the effect that almost all the areas of discourse which someone intuitively sympathetic to the ‘crude but intuitive’ distinction might want to view as hostage to potential disputes of inclination will turn out to deal in contents which, when the disciplinary standards proper to the discourse are satisfied, a supporter is going to be entitled to claim to be true. That however—the proposal is—ought to be consistent with the discourse in question failing to meet certain further conditions necessary to justify the idea that, in the case of such a dispute, there will be a further fact in virtue of which one of the disputants is in error. What kind of condition? The leading idea of someone—the factualist—who believes that a given discourse deals in matters of fact—unless they think that their truths lie beyond our ken—is that soberly and responsibly to practise that discourse is to enter into a kind of representational mode of cognitive function, comparable in relevant respects to taking a photograph or making a wax impression of a key. The factualist conceives that certain matters stand thus

The timelessness of truth—if a proposition is ever true, then it always is, so that whatever may, at any particular time, be truly asserted may—perhaps by appropriate transformations of mood, or tense—be truly asserted at any time. (Timelessness) That truth is absolute—there is, strictly, no such thing as a proposition’s being more or less true; propositions are completely true if true at all. (Absoluteness) The list might be enlarged, and some of these principles may anyway seem controversial. Moreover it can be argued that the Equivalence Schema underlies not merely the first of the platitudes listed— Transparency—but the Correspondence and Contrast Platitudes as well. For elaboration of this claim, see Wright (1992b, pp. 24–7). For further discussion of the minimalist conception, and adjacent issues, see Wright (1998; 2001a). 11  A partial development of them is offered in Wright (1992b, chs. 1–2).

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On Being in a Quandary  219 and so independently of us and our practice—matters comparable to the photo­graphed scene and the contours of the key. We then engage in the appropriate investigative activity—putting ourselves at the mercy of the standards of belief-formation and appraisal appropriate to the discourse in question (compare taking the photograph or impressing the key on the wax)—and the result is to leave an imprint on our minds which, in the best case, appropriately matches the independently standing fact. This kind of thinking, while doubtless pretty vague and metaphorical, does have certain quite definite obligations. If we take photographs of one and the same scene which somehow turn out to represent it in incompatible ways, there has to have been some kind of shortcoming in the function of one (or both) of the cameras, or in the way it was used. If the wax impressions we take of a single key turn out to be of such a shape that no one key can fit them both, then again there has to have been some fault in the way one of us went about it, or in the materials used. The tariff for taking the idea of representation in the serious way the factualist wants to is that, when subjects’ ‘representations’ prove to conflict, then there has to have been something amiss with the way they were arrived at or with their vehicle—the wax, the camera, or the thinker. That is the key thought behind the idea of cognitive command. The final formulation offered in Truth and Objectivity was that a discourse exerts cognitive command just in case it meets this condition: It is a priori that differences of opinion formulated within (that) discourse, unless excusable as a result of vagueness in a disputed statement, or in the standards of acceptability, or variation in personal evidence thresholds, so to speak, will involve something which may properly be regarded as a cognitive shortcoming. (Wright 1992b, p. 144)

To stress: the constraint is motivated, in the fashion just sketched, by the thought that it, or something like it, is a commitment of anyone who thinks that the responsible formation of opinions expressible within the discourse is an exercise in the representation of self-standing facts. Conversely: any suggestion that conflicts in such opinions can be cognitively blameless, yet no vagueness be involved of any of the three kinds provided for in the formulation, is a suggestion that the factualist—seriously representational—view of the discourse in question is in error. Broadly, then, the implicit suggestion of Truth and Objectivity was that true relativism about a particular discourse may be formulated as the view that, while qualifying as minimally truth-apt, it fails to exhibit cognitive command.

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220  The Riddle of Vagueness I.4. However, there is an awkwardness to be confronted by any proposal of this general kind. The key to true relativism, as we have it so far, is somehow to make out that a discourse deals in contents which are simultaneously truth-apt yet such that, when they fall into dispute, there need in principle be nothing wrong with—nothing to choose between—the disputed opinions. But, in granting that the contents in question are minimally truth-apt, the relativist allows, presumably, that they are subject to ordinary propositional– logical reasoning. So, where P is any matter of inclination which comes into dispute between a thinker A, who accepts it, and a thinker B, who does not, what is wrong with the following Simple Deduction? 1 2 3 4 2,4 2,3 1,2,3 1,2

(1) A accepts P (2) B accepts not-P (3) A’s and B’s disagreement involves no cognitive shortcoming (4) P (5) B is guilty of a mistake, hence of cognitive shortcoming (6) Not-P (7) A is guilty of a mistake, hence of cognitive shortcoming (8) Not -[3]

— — — — — — — —

Assumption Assumption Assumption Assumption 2,4 4,5,3 RAA 4 3,3,7 RAA

The Simple Deduction seems to show that, whenever there is a difference of opinion on any—even a merely minimally—truth-apt claim, there is—quite trivially—a cognitive shortcoming, something to choose between the views. And, since this has been proved a priori, cognitive command holds for all truth-apt discourses. So the alleged gap between minimal truth aptitude and cognitive command, fundamental to the programme of Truth and Objectivity, disappears. Obviously there has to be something off-colour about this argument. So much is immediately clear from the reflection that the disagreement it concerns could have been about some borderline case of a vague predicate: nothing that happens in the Simple Deduction is sensitive to the attempt made in the formulation of cognitive command to exempt disagreements which are owing to vagueness (one way or another). Yet the Deduction would have it that even these too must involve cognitive shortcoming. And the notion of shortcoming involved is merely that of bare error—mismatch between belief and truth value. So, if the argument shows anything, it would appear to show a priori that any difference of opinion about a borderline case of a vague predicate will also involve a mismatch between belief (or unbelief) and actual truth value. It would, therefore, seem that there has to be a truth value in all such cases, even if we have not the slightest idea how it might be determined. We appear to have been saddled with the Epistemic Conception! I believe that

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On Being in a Quandary  221 means, with all due deference to the proponents of that view, that the Simple Deduction proves too much.12 So where does it go wrong? It may be felt that the trouble lies with an overly limited conception of ‘cognitive shortcoming’. The considerations used to motivate the cognitive command constraint—the comparison with the idea of representation at work in the examples of the photograph or the waximpression—license something richer: a notion of cognitive shortcoming that corresponds to failure or limitation of process, mechanism, or materials, and not merely a mismatch between the product and its object. The two cameras that produce divergent—conflicting—representations of the same scene must, one or both, have functioned less than perfectly, not merely in the sense that one (or both) gives out an inaccurate snapshot but in the sense that there must be some independent defect, or limitation, in the process whereby the snapshot was produced. So too, it may be suggested, with cognitive command: the motivated requirement is that differences of opinion in regions of genuinely representational discourse should involve imperfections of pedigree: shortcomings in the manner in which one or more of the opinions involved were arrived at, of a kind that might be appreciated as such in­de­ pend­ent­ly of any imperfection in the result. Once shortcoming in that richer sense is required, it can no longer be sufficient for its occurrence merely that a pair of parties disagree—it needs to be ensured in addition that their dis­ agree­ment betrays something amiss in the way their respective views were arrived at, some independently appreciable failure in the representational mechanisms. That, it may be felt, is what the cognitive command constraint should be understood as really driving at. 12  It may be rejoined (and was, by Mark Sainsbury, in correspondence) that we could accept the Simple Deduction without commitment to the stark Bivalence espoused by the Epistemic Conception if we are prepared to allow that A’s and B’s respective opinions may indeed both reflect cognitive shortcoming where P’s truth status is borderline—on the ground that, in such circumstances, both ought to be agnostic about P. The point is fair, as far as it goes, against the gist of the preceding paragraph in the text. However, I believe—and this will be a central plank of the discussion to follow—that it is a profound mistake to regard positive or negative verdicts about borderline cases as eo ipso de­fect­ ive. If that were right, a borderline case of P should simply rank as a special kind of case in which— because things are other than P says—its negation ought to hold. In any case, the Simple Deduction will run no less effectively if what B accepts is not ‘not-P’ when understood narrowly, as holding only in some types of case where P fails to hold, but rather as holding in all kinds of case where things are not as described by P—all kinds of ways in which P can fail of truth, including being borderline (if, contra my remark above, that is how being borderline is conceived.) So, even if Bivalence is rejected, the Simple Deduction still seems to commit us to the more general principle Dummett once called Determinacy: that P always has a determinate truth status—of which Truth and Falsity may be only two among more than two possibilities—and that at least one of any pair of conflicting opinions about P must involve a mistake about this status, whatever it is. That is still absolutely in keeping with the realist spirit of the Epistemic Conception, to which it still appears—at least in spirit—the Simple Deduction commits us if unchallenged.

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222  The Riddle of Vagueness Such an emended understanding of cognitive shortcoming is indeed in keeping with the general motivation of the constraint. But it does not get to the root of our present difficulties. For one thing, the Simple Deduction would still run if we dropped all reference to cognitive shortcoming—thereby ­finessing the issue of how that notion should be understood—and replaced line 3 with: (3*) A’s and B’s disagreement involves no mistake. The resulting reasoning shows—if anything—that any pair of conflicting claims involve a mistake. If it is sound, there just is not any fourth, that is, true-relativistic response to the original problem. To suppose that P is merely minimally truth-apt in the sense of allowing of hypothesis, significant neg­ ation, and embedding within propositional attitudes is already, apparently, a commitment to rampant realism. Surely that cannot be right. But the modified Deduction, with (3*) replacing (3), shows that refining the idea of cognitive shortcoming in the manner just indicated has nothing to contribute to the task of explaining why not. Perhaps more important, however, is the fact that we can run an argument to much the same effect as the (unamended) Simple Deduction even when ‘cognitive shortcoming’ is explicitly understood in the more demanding sense latterly proposed.13 One reason why rampant realism is unattractive is because by insisting on a fact of the matter to determine the rights and wrongs of any dispute of inclination, no matter how intransigent, it is forced to introduce the idea of a truth-making state of affairs of which we have a necessarily imperfect concept,14 and whose obtaining, or not, thus necessarily transcends our powers of competent assessment. This is unattractive in direct proportion to the attraction of the idea that, in discourses of the relevant kind, we are dealing with matters which essentially cannot outrun our appreciation: that there is no way in which something can be delicious, or disgusting, or funny, or obscene, and so on, without being appreciable as such by an appropriately situated human subject because these matters are, in some very general way, constitutively dependent upon us. What we—most of us—find it natural to think is that disputes of inclination typically arise in cases where were there a ‘fact of the matter’, it would have to be possible—because of this constitutive dependence—for the protagonists to know of it. Indeed, the ordinary idea that such disputes need concern no fact of the matter is just a modus tollens on that conditional: 13  This point was first made in Shapiro and Taschek (1996). 14  See n. 6.

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On Being in a Quandary  223 were there a fact of the matter, the disputants should be able to achieve consensus about it; but it seems manifest in the character of their disagreement that they cannot; so there isn’t any fact of the matter. So for all—or at least for a wide class of cases—of claims, P, apt to figure in a d ­ ispute of inclination, it will seem acceptable—and the recoil from rampant realism will provide additional pressure—to hold to the following principle of evidential constraint (EC): P → it is feasible to know that P 15



and to hold, moreover, that the acceptability of this principle is a priori, ­dictated by our concept of the subject matter involved.16, 17 Consider, then, the following EC-Deduction: 1

(1)

2 (2) 2 (3) 1,2 (4) 1,2 1 1 1

(5) (6) (7) (8)

1  

(9) (10)

A believes P, B believes not-P, and neither has any cognitive shortcoming. P It is feasible to know that P B believes the negation of something feasibly knowable. B has a cognitive shortcoming Not-P It is feasible to know that not-P A believes the negation of something feasibly knowable. A has a cognitive shortcoming Not-[1]

— Assumption — Assumption — 2, EC — 1,3 — — — —

4 2,1,5 RAA 6,EC 1,7

— 8 — 1,1,9 RAA

15  One substitution instance, of course, is: Not-P → it is feasible to know that not-P. 16  To forestall confusion, let me quickly address the quite natural thought that, where EC applies, cognitive command should be assured—since any difference of opinion will concern a knowable ­matter—and hence that any reason to doubt cognitive command for a given discourse should raise a doubt about EC too. This, if correct, would certainly augur badly for any attempt to locate disputes of inclination within discourses where cognitive command failed but EC held! But it is not correct. What the holding of EC for a discourse ensures is, just as stated, that each of the conditionals P → it is feasible to know that P Not-P → it is feasible to know that not-P, is good for each proposition P expressible in that discourse. That would ensure that any difference of opinion about P would concern a knowable matter, and hence involve cognitive shortcoming, only if in any such dispute it would have to be determinate that one of P or not-P would hold. But of course it is of the essence of (true) relativism to reject precisely that—(and to do so for reasons unconnected with any vagueness in the proposition that P.) 17  The modality involved in feasible knowledge is to be understood, of course, as constrained by the distribution of truth values in the actual world. The proposition that, as I write this, I am in Australia is one which it is merely (logically or conceptually) possible to know—the possible world in question is one in which the proposition in question is true, and someone is appropriately placed to recognize its

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224  The Riddle of Vagueness This time ‘cognitive shortcoming’, it is perhaps superfluous to remark, must involve less than ideal procedure, and not just error in the end product, since it involves mistakes about feasibly knowable matters. So: it seems that 1 and EC are inconsistent—that is, evidential constraint is incompatible with the possibility of cognitively blameless disagreement. If the EC-Deduction is sound, then it seems that, wherever EC is a priori, cognitive command is met. And it is plausible that EC will be a priori at least for large classes of the types of claim—par excellence simple predications of concepts like delicious—where relativism is intuitively at its most attractive, and where a gap between minimal truth aptitude and cognitive command is accordingly called for if we are to sustain the Truth and Objectivity proposal about how relativism should best be understood.18 I.5. What other objection might be made to either Deduction? Notice that there is no assumption of Bivalence in either argument; both can be run in an intuitionistic logic. But one might wonder about the role of reductio in the two proofs. For instance, at line 6 in the Simple Deduction, the assumption of P having run into trouble, RAA allows us to infer that its negation holds. Yet surely, in any context where we are trying seriously to make sense of the idea that there may be ‘no fact of the matter’, we must look askance at any rule of inference which lets us advance to the negation of a proposition just on the ground that its assumption has run into trouble. More specifically: in any circumstances where it is a possibility that a proposition’s failing to hold may be a reflection merely of there being no ‘fact of the matter’, its so failing has surely to be distinguished from its negation’s holding. Natural though the thought is, it is not clear that there is much mileage in it. Let’s make it a bit more specific.19 The idea is best treated, we may take it, as involving restriction of the right-to-left direction of the Negation Equivalence,

T¬P ↔ ¬TP ,

b   eing so. By contrast, the range of what is feasible for us to know goes no further than what is  actually the case: we are talking about those propositions whose actual truth could be recognized by the implementation of some humanly feasible process. (Of course there are further parameters: recognizable when? where?, under what if any sort of idealization of our actual powers?, etc. But these are not rele­ vant to present concerns.) 18  To stress: it is not merely Truth and Objectivity’s implicit proposal about relativism that is put in jeopardy by the EC-Deduction. According to the project of that book, cognitive command is a significant watershed but is assured for all discourses where epistemic constraint fails and realism, in Dummett’s sense, is the appropriate view. Thus, if the EC-Deduction were to succeed, cognitive command would hold universally and thus fail to mark a realism-relevant crux at all. 19  I draw here on a suggestion of Patrick Greenough.

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On Being in a Quandary  225 expressing the commutativity of the operators, ‘it is true that’ and ‘it is not the case that’. In circumstances where there is no fact of the matter whether or not P, it will be the case both that ¬TP and that ¬T¬P. The proper conclusion, on the assumptions in question, of the reductio at line 6 of the Simple Deduction is thus not that the negation of P holds, but merely that it is not the case that P is true. And from this, since it is consistent with there being ‘no fact of the matter’ whether or not P, we may not infer (at line 7) that A is guilty of any mistake in accepting P. Or so, anyway, the idea has to be. Rejecting the Negation Equivalence has repercussions, of course, for the Equivalence Schema itself:

TP ↔ P

since one would have to reject the ingredient conditional:

P → TP20

That flies in the face of what would seem to be an absolutely basic and constitutive property of the notion of truth, that P and TP are, as it were, attitudinally equivalent: that any attitude to the proposition that P—belief, hope, doubt, desire, fear, and so on—is equivalent to the same attitude to its truth. For, if that is accepted, and if it is granted that any reservation about a conditional has to involve the taking of some kind of differential attitudes to its antecedent and consequent, then there simply can be no coherent reservation about P → TP. A more direct way of making essentially the same point is this. At line 6 of each Deduction, even with RAA modified as proposed, we are entitled to infer that it is not the case that P is true. By hypothesis, however, A accepts P. Therefore, unless that somehow does fall short of an acceptance that P is true, A is guilty of a mistake in any case. But how could someone accept P without commitment to its truth? Indeed, there is actually a residual difficulty with this whole tendency, independent of issues to do with the attitudinal transparency of truth. Simply conceived, the mooted response to the two Deductions is trying to make out/ exploit the idea that A and B may each be neither right nor wrong because there is ‘no fact of the matter’, where this conceived as a third possibility, contrasting with either A or B being right. That idea may well demand some 20  There will be no cause to question the converse conditional, which is needed for the derivation of the uncontroversial T¬P → ¬TP.

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226  The Riddle of Vagueness restriction on the form of reductio utilized in the two Deductions. But the problem they are bringing to light will persist even after the restriction. For the simple fact now seems to be that A is taking matters to be one way, and B is taking them to be another, when in truth they are neither—when, precisely, a third possibility obtains. In that case there is indeed nothing to choose between A’s and B’s respective views, but only because they are both equally off-beam. We achieve the parity between their views essential to any satisfactory working-out of a true relativism only by placing them in parity of ­disesteem. This general point—broadly, the intuitive inadequacy of ‘third ­possibility’ approaches to the construal of indeterminacy—will recur in the sequel. So, that is the first of the three problems to which I want to work towards a unified approach: it is the problem of stabilizing the contrast between min­ imal truth aptitude and cognitive command or, more generally, the problem of showing how there can indeed be a coherent true relativism—a coherent response of the fourth kind to the challenge of providing a proper account of the character of disputes of inclination. 

II The Sorites II.1. Even after all the attention meted out to it, the simplicity of the Sorites paradox can still seem quite breathtaking. Take any example of the standard sort of series. Let F be the predicate in question. Let x′ be the immediate successor in the series of any of its elements, x. The first element in the series— call it ‘o’—will be F and the last—‘k’—will be non-F. And, of course, F will be vague. If it were precise, there would be a determinate cut-off point—a last F-element in the series, immediately succeeded by a first non-F one. It would be true that (∃x)(Fx & ¬Fx′). So, since F is vague, that claim is false. And its being false would seem to entail that every F-element is succeeded by another F-element: that (∀x)(Fx → Fx′). But that is trivially inconsistent with the data that Fo and that not-Fk. What is startling is that it is, seemingly, child’s play to replicate this structure with respect to almost every predicate that we understand; and that the motivation for the troublesome major premise—

(∀x ) (Fx → Fx′)

—seems to flow directly just from the very datum that F is vague—that is,  from the denial that it is precise. Again: if (∃x)(Fx&¬Fx′) just

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On Being in a Quandary  227 says—falsely—that F is precise in the relevant series, then surely it’s (classical) contradictory, (∀x)(Fx → Fx′), just says—truly—that F is vague. But it was given that Fo and that not-Fk. Seemingly incontrovertible premises emerge— extremely simply, if a little long-windedly—as incompatible. Vague predicates, in their very nature, seemingly have all-inclusive extensions. II.2. Hilary Putnam once suggested that an intuitionistic approach might assist (see Putnam  1983, pp. 271–86). How exactly? Not, anyway, by so restricting the underlying logic that the paradox cannot be derived.21 It is true that it takes classical logic to motivate the major premise, (∀x)(Fx → Fx′), on the basis of denial of the unpalatable existential, (∃x)(Fx & ¬Fx′). But the paradox could as well proceed directly from that denial: ¬(∃x )(Fx & ¬Fx′)



in intuitionistic logic. To be sure, we cannot then reason intuitionistically from Fo to Fk. (To do so would require double negation elimination steps.) But we can still run the Sorites reasoning backwards, from not-Fk to not-Fo, using just n applications of an appropriate sub-routine of conjunction introduction, existential introduction, and RAA. So what profit in Intuitionism here? Putnam’s thought is best taken to have been that there is no option but to regard the major premises, ¬(∃x )(Fx & ¬Fx′)

or

(∀x )(Fx → Fx′),



as reduced to absurdity by the paradox, and that we are therefore constrained to accept their respective negations, ¬¬(∃ x )(Fx & ¬Fx′)

and

¬ (∀x )(Fx → Fx′),

21  I do not suggest that Putnam was under any illusion about this.

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228  The Riddle of Vagueness as demonstrated. The advantage secured by an intuitionistic framework is then that, lacking double negation elimination—and also the classical rule, ¬(∀x)(…x…) ⇒ (∃x) ¬(…x…), in consequence—we are not thereby constrained to accept the unpalatable existential:

(∃x )(Fx & ¬Fx′) .

So we can treat the Sorites reasoning as a straightforward reductio of its major premise without thereby seemingly being forced into denying the very datum of the problem—namely, that F is vague. The trouble is that this suggestion, so far, deals with only half the problem. Avoiding the unpalatable existential is a good thing, no doubt. Yet equally we have to explain what is wrong with its denial. And does not recognition of the vagueness of F in the relevant series precisely enforce that denial? Does not the vagueness of F just consist in the fact that no particular claim of the form, (Fa & ¬Fa′), is true? And is not the problem compounded by the fact that the usual style of antirealist/intuitionist semantics will require us to regard recognition that nothing could justify such a claim as itself a conclusive reason for denying each particular instance of it for the series in question? It is true that intuitionistic resources would avoid the need to treat the Sorites reasoning as a proof of the unpalatable existential claim. But that thought goes no way to explaining how to resist its negation, which seems to be both an apt characteristic expression of F’s vagueness and mandated by intuitionist style-semantics in any case. And to stress: the negation leads straight to the paradox, whether our logic is classical or intuitionist (cf. Read and Wright 1985, this volume, Chapter 3). This brings out sharply what I regard as the most natural perspective on what a solution to the Sorites has to accomplish. Since the reasoning is a reductio of the major premise, we have to recognize that ¬¬(∃x)(Fx & ¬Fx′) is true. So we need to understand (i) how the falsity of ¬(∃x)(Fx & ¬Fx′) can be consistent with the vagueness of F; and (ii) how and why it can be a principled response to refuse to let ¬¬(∃x)(Fx & ¬Fx′) constitute a commitment to the unpalatable existential, and hence—apparently—to the precision of F.22

22  This perspective is not mandatory, of course. In particular, it will not appeal to any dyed-in-thewool classicists. Supervaluationist and Epistemicist approaches try, in their different ways, to allow us the unpalatable existential while mitigating its unpalatability. But those are not the approaches we ­follow here.

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On Being in a Quandary  229

III Revisionism III.1. It is generally though not universally assumed among interested philo­sophers that anti-realism in something close to Dummett’s sense—the adoption of an evidentially constrained notion of truth as central in the theory of meaning—should lead to revisions in classical logic. But why? Truth plays a role in the standard semantical justification for classical logic. Persuasion that truth is essentially—or locally—evidentially constrained might thus lead to (local) dissatisfaction with classical semantics—and hence with the standard justification for classical logic. But why should that enjoin dissatisfaction with the logic itself? There would seem to be an assumption at work that classical logic needs its classical justification. But maybe it might be justified in some other way. Or maybe it needs no semantical justification at all.23 Is there a revisionary argument that finesses this apparent lacuna? Here is one such proposal—I will call it the Basic Revisionary Argument— advanced by myself (see Wright  1992b, ch. 2, pp. 37–44.). Assume the discourse in question is one for which we have no guarantee of decidabilty: we do not know that it is feasible, for each of its statements P, to come to know P or to come to know not-P. Thus this principle holds

(NKD) ¬K (∀P )(FeasK(P ) ∨ FeasK(¬P )).

Then given that we also accept

(EC) P → FeasK(P )

—any truth of the discourse in question may feasibly be known—we get into difficulty if we also allow as valid

(LEM) P ∨ ¬P .

For LEM and EC sustain simple reasoning to the conclusion that any P is such that either it or its negation may feasibly be known.24 If we know that both 23  I pursued these doubts about Dummett’s revisionary line of thought in (Wright 1993, pp. 433–57). 24  For a reason to emerge in n. 25, we should formulate the reasoning like this: LEM EC   EC EC   EC LEM, EC  

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)  

P ∨ ¬P P → FeasK(P) FeasK(P) → (FeasK(P) ∨ FeasK(¬P)) P → (FeasK(P) ∨ FeasK(¬P)) ¬P → FeasK(¬P) FeasK(¬P) → (FeasK(P) ∨ FeasK(¬P)) ¬P → (FeasK(P) ∨ FeasK(¬P)) FeasK(P) ∨ FeasK(¬P)  

      (ii), (iii)     (vi), (vii) (i), (iv), (vii) disjunction elimination

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230  The Riddle of Vagueness LEM and EC are good, this reasoning presumably allows us to know that, for each P, (FeasK(P) ∨ FeasK(¬P)). But that knowledge is inconsistent with NKD. Thus it cannot stably be supposed that each of EC, LEM, and NKD is known. Anti-realism supposes that EC is known a priori, and NKD seems incontrovertible—(for does it not merely acknowledge that, relative to extant means of decision, not all statements are decidable?). So the anti-realist must suppose that LEM is not known—agnosticism about it is mandated so long as we know that we do not know that it is feasible to decide any significant statement. Since logic has no business containing first principles that are uncertain, classical logic is unacceptable in our present state of information. Of course, there are three possible responses to the situation: to deny, with the anti-realist, that LEM is known; to deny, with the realist, that EC is known; or to accept the reasoning as a proof that NKD is after all wrong. The last might be reasonable if one had provided consistent and simultaneous mo­tiv­ ation for LEM and EC. But it is not a reasonable reaction when the grounds— if any—offered for LEM presuppose an evidentially unconstrained notion of truth, (or at least have not been seen to be compatible with evidential constraint). Note that, provided disjunction sustains reasoning by cases, it is LEM—the logical law—that is the proper target of the argument, not just the semantic Principle of Bivalence. (And reasoning by cases would be sustained in the rele­vant case if, for example, the semantics was standard-supervaluational, rather than Bivalence-based.)25 So this really is an argument for suspension of classical logic, not just classical semantics. 25  Actually, as Tim Williamson has reminded me, this point depends on how reasoning by cases (disjunction elimination) is formulated. If ‘P ∨ Q’ is super-true and so is each of the conditionals, ‘P → R’ and ‘Q → R’, then so is R. That is ungainsayable, and enough to sustain the proof in n. 30 and the letter of the argument of the text, that one who believes that EC and NKD are each known should be agnostic about LEM. However, reasoning by cases fails from the supervaluational perspective if the required auxiliary lemmas take the form

P  R, Q  R,

rather than the form

 P → R,  Q → R.

A counterexample would be the invalidity of the inference from

‘P ∨ ¬P ’ to ‘Definitely P ∨ Definitely ¬P ’,

notwithstanding the validity of each of and

P  Definitely P , ¬P  Definitely ¬P .

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On Being in a Quandary  231 Note too that the argument is for not endorsing LEM in our present state of information. It is not an argument that the Law allows counterexamples—that it is false. That view is indeed inconsistent with the most elementary properties of negation and disjunction, which entail the double negation of any instance of LEM.26 III.2. But there is a problem—the advertised ‘awkward wrinkle’—with the Basic Revisionary Argument. It is: what justifies NKD? It may seem just obvious that we do not know that it is feasible to decide any significant question (what about vagueness, backwards light cones, Quantum Mechanics, Goldbach, the Continuum Hypothesis, and so on?) But, for the anti-realist, though not for the realist, this modesty needs to be able to stand alongside our putative knowledge of EC. And there is a doubt about the stability of that combination. To see the worry, ask: what does it take in general to justify the claim that a certain statement is not known? The following seems a natural principle of agnosticism: (AG) P should be regarded as unknown just in case there is some possibility Q such that if it obtained, it would ensure not-P, and such that we are ­(warranted in thinking that we are) in no position to exclude Q.27

If AG is good, then justification of NKD will call for a Q such that, were Q to obtain, it would ensure that

¬(∀P )(FeasK(P ) ∨ FeasK(¬P )).

(From the supervaluational perspective, we lose the inference from

P  Definitely P



 P → Definitely P ,

to

so that the premises for the form of disjunction elimination that is supervaluationally sound are ­unavailable in the particular instance. Cf. Williamson (1994, p. 152); also Fine (1975, p. 290). 26  Here is the simplest proof. Suppose ¬(P ∨ ¬P). And now additionally suppose P. Then P ∨ ¬P by disjunction introduction—contradiction. So ¬P, by reductio. But then P ∨ ¬P again by disjunction introduction. So ¬¬(P ∨ ¬P). This proof is, of course, intuitionistically valid. 27  If ‘are in no position to exclude’ means: do not know that not, then of course this principle uses the notion it constrains—but that is not to say that it is not a correct constraint. Admirers of ‘relevant alternatives’ approaches to knowledge may demur at the generality of (AG) as formulated; but it will make no difference to the point to follow if ‘there is some possibility Q’ is replaced by ‘there is some epistemically relevant possibility Q’, or indeed any other restriction.

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232  The Riddle of Vagueness And now the problem is simply that it would then follow that there is some statement such that neither it nor its negation is feasibly knowable—which in turn, in the presence of EC, entails a contradiction. So, given EC, there can be no such appropriate Q.28 So, given EC and AG, there can be no justifying NKD. Thus the intuitive justification for NKD is, seemingly, not available to the anti-realist. III.3. There is a response to the problem which I believe we should reject. What NKD says is that it is not known that all statements are such that either they or their negations may feasibly be known. So an AG-informed justification of NKD will indeed call for a Q such that, if Q holds, not all statements, P, are such that FeasK(P) ∨ FeasK(¬P). But the advertised contra­dic­tion is in effect derived from the supposition that some particular P is such that ¬(FeasK(P) ∨ FeasK(¬P)). So, to refer that contradiction back to the above, we need the step from ¬(∀P)(…P…) to (∃P) ¬(…P…)—a step which is, of course, not generally intuitionistically valid. In other words: provided the background logic is intuitionistic, no difficulty has yet been disclosed for the idea that there are grounds for NKD which are consistent with AG. The trouble with this, of course, is that we precisely may not take it that the background logic is (already) intuitionistic; rather the context is one in which we are seeking to capture an argument to the effect that it ought to be (at least to the extent that LEM is not unrestrictedly acceptable). Obviously, we cannot just help ourselves to distinctively intuitionistic restrictions in the attempt to stabilize the argument if the argument is exactly intended to motivate such restrictions. A better response will have to improve on the principle AG. Specifically it will need to argue that it is not in general necessary, in order for a claim of ignorance whether P to be justified, that we (recognize that we) are in no position to exclude circumstances Q under which not-P would be true—that, at least in certain cases, it is possible to be in position to exclude any such Q while still not knowing or being warranted in accepting P. And, of course, it is actually obvious that the intuitionist/anti-realist needs such an improved account in any case. For, while the right-hand-side of AG is presumably uncontentious as a sufficient condition for ignorance, it cannot possibly give an acceptable necessary condition in any context in which it is contemplated that the double negation of P may not suffice for P. In any such 28  Epistemically relevant or otherwise.

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On Being in a Quandary  233 case, we may indeed be in a position to rule out any Q sufficing for not-P, yet still not in a position to affirm P. What the anti-realist needs, then, is a conception of another sufficient condition for ignorance which a thinker can meet even when in position to exclude the negation of a target proposition. And that there is this type of sufficient condition needs to be appreciable independently and—since we are seeking this in order to refurbish an argument for revising classical logic—in advance of an endorsement of any broadly intuitionistic understanding of the logical constants. Is there any such alternative principle of ignorance? Our third problem29 is the challenge to make out that there is and thereby to stabilize the Basic Revisionary Argument.

IV Revisionism Saved IV.1. We shall work on the problems in reverse order. To begin with, then, how might AG fail—how might someone reasonably be regarded as ignorant of the truth of a proposition who rightly considered that they were in a pos­ ition to exclude (any proposition entailing) its negation? A suggestive thought is that a relevant shortcoming of AG is immediate if we reflect upon examples of indeterminacy.30 Suppose we take the simplest possible view of indeterminacy—what I will call the third possibility view: that indeterminacy consists/results in some kind of status other than truth and falsity—a lack of truth value, perhaps, or the possession of some other truth value. Then it is obvious—at least on one construal of negation, when not-P is true just when P is false—how being in position to exclude the negation of a statement need not suffice for knowledge of that statement. For, excluding the negation would leave open two possibilities: that P is true and that it is in­de­ ter­min­ate—that it lacks, or has a third, truth value. Hence, if that were the way to conceive of indeterminacy, we should want to replace AG with, as a first stab, something like:

29  Of course, friends of classical logic are not likely to perceive this as a problem. 30  Under this heading I mean at this point to include both linguistic vagueness—the phenomenon, whether semantic, or epistemic, or however it should be understood, which is associated with the Sorites paradox—and also indeterminacy in re, as might be exhibited by quantum phenomena, for instance, or the future behaviour of any genuinely indeterministic physical system.

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234  The Riddle of Vagueness (AG*) P should be regarded as unknown just in case either there is some possibility Q such that if it obtained, it would ensure not-P, and such that we are (warranted in thinking that we are) in no position to exclude Q or P is recognized, in context, to be indeterminate.

This (in one form or another very widespread31) conception of indeterminacy is, however, in my view, un premier pas fatal. It is quite unsatisfactory in general to represent indeterminacy as any kind of determinate truth status—any kind of middle situation, contrasting with both the poles (truth and falsity)— since one cannot thereby do justice to the absolutely basic datum that in general borderline cases come across as hard cases: as cases where we are baffled to choose between conflicting verdicts about which polar verdict applies, rather than as cases which we recognize as enjoying a status inconsistent with both. Sure, sometimes people may non-interactively agree—that is, agree without any sociological evidence about other verdicts—that a shade of ­colour, say, is indeterminate (though I do not think it is clear what is the content of such an agreement); but more often—and more basically—the in­de­ter­ min­acy will be initially manifest not in (relatively confident) verdicts of indeterminacy but in (hesitant) differences of opinion (either between subjects at a given time or within a single subject’s opinions at different times) about a polar verdict, which we have no idea how to settle—and which, therefore, we do not recognize as wrong. In any case, even if indeterminacy is taken to be third-possibility in­de­ter­ min­acy, AG* is indistinguishable from AG in the present dialectical setting. The standard anti-realist/intuitionist semantics for negation will have it that P’s negation is warranted/known just when the claim is warranted/known that no warrant for/knowledge of P can be achieved.32 It follows that, for the intuitionist/anti-realist, recognizable third-possibility indeterminacy would be a situation where the negation of the relevant statement should be regarded as holding and is hence no ground for agnosticism about anything. (Unrecognizable third-possibility indeterminacy, for its part, would be a sol­ ecism in any case, in the presence of EC for the discourse in question.) 31  It is a common assumption, for instance, both of any supervaluational theorist of vagueness who accepts it as part of the necessary background for a supervaluational treatment that vague statements give rise to a class of cases in which we may stipulate that they are true, or that they are false, without (implicit) reclassification of any case in which they would actually be true, or false; and of defenders of degree-theoretic approaches (in accepting that there are statements which are neither wholly true nor wholly false.) 32  This account of negation is actually enforced by EC and the Disquotational Scheme—see Wright (1992b, ch. 2).

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On Being in a Quandary  235 IV.2. A better conception of indeterminacy will allow that it is not in general a determinate situation and that indeterminacy about which statement, P or its negation, is true, is not to be conceived as a situation in which neither is. The latter consideration actually enjoins the former. For, to comply with the latter, indeterminacy has to be compatible both with P and with its negation being true and clearly no determinate truth status can be so compatible: if it is a truth-conferrer for either, it is inconsistent with the other; if it is a truthconferrer for neither, then neither is true, and contradictions result (at least in the presence of the Disquotational Scheme). To reject the third possibility view is thus to reject the idea that in viewing the question, whether P, as in­de­ ter­min­ate, one takes a view with any direct bearing on the question of the truth value of P. I know no way of making that idea intelligible except by ­construing indeterminacy as some kind of epistemic status. To accept this view—I shall call it the Quandary view—is, emphatically, less than to subscribe to the Epistemic Conception of vagueness, according to which vague expressions do actually possess sharp, albeit unknowable, limits of ­extension. But it is to agree with it this far: that the root characterization of in­de­ ter­min­acy will be by reference to ignorance—to the idea, as a starting character­ ization, of cases where we do not know, do not know how we might come to know, and can produce no reason for thinking that there is any way of coming to know what to say or think, or who has the better of a difference of opinion.33 The ­crucial question how a Quandary view of indeterminacy can avoid becoming a version of the Epistemic Conception will exercise us in due course. How does AG look in the context of the Quandary view? Consider for P a  borderline-case predication of ‘red’. The materials about it which the Quandary view, as so far characterized, gives us are that we do not know, do not know how we might come to know, and can produce no reason for thinking that there is any way of coming to know whether the item in question is correctly described as ‘red’. Now, if what we are seeking to understand—in our attempt to improve on AG—is how someone could remain ignorant of 33  All three clauses are active in the characterization. If a subject does not know the answer to a question nor have any conception of how it might be decided, they are not thereby automatically bereft of any ground for thinking it decidable. One such ground might be to advert to experts presumed to be in position to resolve such a question. Another might be some general reason to think that such questions were decidable, even while lacking any specific idea of how. Neither will be available in the range of cases on which we shall shortly focus—simple predications of colour of surfaces open to view in good conditions. A difference of descriptive inclination in such a case among otherwise competent and properly functioning subjects is not open to adjudication by experts, nor do we have any general reason to think that the issue must be adjudicable in principle, in a way beyond our present ken. To be sure, we are forced to say so if we cling to the Law of Excluded Middle while retaining the belief that these predications are subject to EC. But then—again—we owe a ground for LEM consistent with that belief. I shall add a fourth clause in due course.

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236  The Riddle of Vagueness the correctness of this predication who already knew that no Q inconsistent with it was true, then clearly the Quandary view helps not at all. For, if I knew that no Q entailing not-P was true, that would surely be to resolve the in­de­ter­ min­acy, since it would rule out the case of not-P and—on the Quandary view, though not the Third Possibility view—no other case than P is then provided for. If all I am given is, not some additional possibility besides P and not-P but merely that I do not know, and do not know how to know, and can produce no reason for thinking that there is any way of coming to know which of them obtains, then there seems to be no obstacle to the thought that to learn that one does not obtain would be to learn that the other does. IV.3. The situation interestingly changes, however, when we consider not simple indeterminate predications like ‘x is red’ but compounds of such in­de­ter­ min­ate components, as conceived under the Quandary view. In particular the Basic Revisionary Argument, that LEM is not known to hold in general, arguably becomes quite compelling when applied to instances of that principle whose disjuncts are simple ascriptions of colour to surfaces in plain view. It is a feature of the ordinary concept of colour that colours are transparent under suitable conditions of observation: that, if a surface is red, it—or a physical duplicate34—will appear as such when observed under suitable conditions; mutatis mutandis if it is not red. Colour properties have essentially to do with how things visually appear, and their instantiations, when they are instantiated, may always in principle be detected by our finding that they do indeed present appropriate visual appearances. So, according to our ordinary thinking about colour—though not of course that of defenders of the Epistemic Conception—EC is inescapable in this setting: when x is any coloured surface in plain view under what are known to be good conditions, each of the conditionals: if x is red, that may be known, and if it is not the case that x is red, that may be known, is known. 34  The complication is to accommodate ‘altering’—the phenomenon whereby implementing the very conditions which would normally best serve the observation of something’s colour might, in special cases, actually change it. Rapid-action Chameleons would be an example.

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On Being in a Quandary  237 Now EC for redness, so formulated, would of course be inconsistent with recognizable third possibility indeterminacy: with our recognition, of a particular such x, that it could not be known to be red and could not be known not to be red. But it is perfectly consistent with our recognition merely that, among some such possible predications, there will be a range where we do not know, and do not know how we might come to know, and can produce no reason for thinking that there is any way of coming to know whether the objects in question are red or not—it is only knowledge that we cannot know that is foreclosed. (If we could know that we could not know, then we would know that someone who took a view, however tentative—say that x was red— was wrong to do so. But we do not know that they are wrong to do so—the indeterminacy precisely leaves it open.) The key question is therefore the status of NKD as applied to these predications: the thesis that the disjunction, that it is feasible to know P or feasible to know not-P, is not known to hold for all P in the range in question. Sure, in the presence of EC, it cannot be that— so we cannot know that—neither disjunct is good in a particular case; that is the point just re-emphasized. But we surely do know of suitable particular instances—particular sample surfaces, in good view—that we do not know, and do not know how we might know, and can produce no reason for thinking that there is any way of coming to know what it is correct to say of their colour or who has the better of a dispute. And it may therefore seem plain that, the contradictoriness of its negation notwithstanding, we are thus in no position to affirm of such an instance, x, that the disjunction, that either it is feasible to know that x is red or it is feasible to know that it is not the case that x is red, may be known. Since LEM—in the presence of EC—entails that disjunction, it follows—granted that there is a compelling case for EC over the relevant subject matter—that we should not regard LEM as known. IV.4. But there is a lacuna in this reasoning. An awkward customer may choose to query the passage from the compound ignorance described by the three conditions on Quandary to the conclusion that we do not know the target disjunction, that either it is feasible to know that x is red or it is feasible to know that it is not the case that x is red. Suppose I do not know, and do not know how I might know, and can produce no reason for thinking that there is any way of coming to know that P; likewise for not-P. Then I might— loosely—describe myself as not knowing, and not knowing how I might know, and able to produce no reason for thinking that there is any way of coming to know what it is correct to think about P or who has the better of a dispute about it. Still, might I not have all those three levels of ignorance and

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238  The Riddle of Vagueness still know that it is the case either that P is knowable or that its negation is? For not knowing what it is correct to think about P might naturally be taken as consisting in the conjunction: not knowing that it is correct to think P and not knowing that it is correct to think not-P; likewise not knowing who has the better of a difference of opinion about P might be taken as the conjunction: not knowing that the proponent of P has the better of it and not knowing that the proponent of not-P has the better of it. And all that, of course, would still be consistent with knowing that there is a correct verdict—that someone has the better of the dispute. The objection, then, is that it does not strictly follow from the too-informal characterization offered of Quandary that if ‘x is red’ presents a quandary, then we have no warrant for the disjunction, FeasK(x is red) ∨ FeasK(it is not the case that x is red). All that follows, the awkward customer is pointing out, is that we are, as it were, thrice unwarranted in holding either disjunct. To say that someone does not know whether A or B is ambiguous. Weakly interpreted, it implies, in a context in which it is assumed that A or B is true, that the subject does not know which. Strongly interpreted, it implies that the subject does not know that the disjunction holds. The objection is that we have illicitly mixed this distinction: that to suggest that to treat borderline cases of colour predicates as quandaries enjoins a reservation about the displayed disjunction is to confuse it. It is uncontentious that such examples may be quandaries if that is taken merely to involve ignorance construed as an analogue of the weak interpretation of ignorance whether A or B. But to run the Basic Revisionary Argument, a case needs to be made that borderline cases of colour predicates present quandaries in a sense involving ignorance under the strong in­ter­pret­ ation. What is that case? A first rejoinder would be to challenge the objector to say, in the examples that concern us, what if any ground we possess for the claim that our ig­nor­ ance goes no further than the weak interpretation—what residual ground, that is, when x is a borderline case of ‘red’, do we have for thinking that the disjunction, FeasK(x is red) ∨ FeasK(it is not the case that x is red) is warranted? It will not do, to stress, to cite its derivation from EC and clas­ sic­al logic—not before a motivation for classical logic is disclosed consistent with EC. Yet no other answer comes to mind.

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On Being in a Quandary  239 However a second, decisive consideration is to hand if I am right in thinking that the kind of quandary presented in borderline cases has so far been under-described. As stressed, it is crucial to the conception of indeterminacy being proposed that someone who takes a (presumably tentative) view for or against the characterizability of such a case as ‘red’ is not known to be wrong. But that is consistent with allowing that it is also not known whether know­ ledge, one way or the other, about the redness of the particular case is even metaphysically possible—whether there is metaphysical space, so to speak, for such an opinion to constitute knowledge. I suggest that we should ac­know­ ledge that borderline cases do present such a fourth level of ignorance: that, when a difference of opinion about a borderline case occurs, one who feels that they have no basis to take sides should not stop short of acknowledging that they have no basis to think that anything amounting to knowledge about the case is metaphysically provided for. And, if that is right, then there cannot be any residual ground for regarding the above disjunction as warranted. The strong interpretation of our ignorance whether it is feasible to know that x is red or feasible to know that it is not the case that x is red, is enforced. IV.5. Let us take stock. Our project was to try to understand how it might be justifiable to refuse to endorse a claim in a context in which we could nevertheless exclude the truth of its negation. For the case of simple predications of colour on surfaces open to view in good conditions, the situation is seemingly this: (i) that what I termed the transparency of colour enjoins acceptance of EC, in the form of the two ingredient conditionals given above; (ii) that we know that there is a range of such predications where we do not know nor have any idea how we might come to know whether or not they are correct, and moreover where we can produce no independent reason for thinking that there must be a way of knowing, or even reason to think that knowledge is metaphysically possible. Nevertheless (iii) we have a perfectly general disproof of the negation of LEM.35 If we now essay to view the latter as a proof of LEM, something will have to give: either we must reject the idea that even simple colour predications obey 35  See n. 26.

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240  The Riddle of Vagueness EC—specifically its two ingredient conditionals—and so reject the transparency of colour, or we must repudiate (ii), treating the putative proof of LEM precisely as a ground for the claim that there must be a way of adjudicating all borderline colour predications. But, again, it just seems plain that the proof does not show that; what it shows is merely that denial of the law cannot consistently be accommodated alongside the ordinary rules for disjunction and reductio ad absurdum. The move to ‘So one of the disjuncts must be knowably true’ should seem like a complete non sequitur. If that is right, then one who accepts both the transparency of colour and that borderline cases present quandaries as most recently characterized must consider that there is no warrant for LEM as applied to colour predications generally—even though the negation of any instance of it may be disproved— and hence that double negation elimination is likewise without warrant. Thus there has to be a solution to the problem the Intuitionist has with the Basic Revisionary Argument if it is ever right to accept EC for a given class of vague judgements and simultaneously allow that some of them present quandaries. And the solution must consist in the disclosure of a better principle of ig­nor­ ance than AG. IV.6. Does the example of colour guide us towards a formulation of such a principle? According to AG, it is necessary and sufficient for a thinker’s ig­nor­ ance of P that there be some circumstances Q such that, if Q obtained, not-P would be true and such that the thinker has no warrant to exclude Q. The improved principle the anti-realist needs will allow this to be a sufficient condition, but will disallow it as necessary. Here is a first approximation. Consider any compound statement, A, whose truth requires that (some of) its con­stitu­ ents have a specific distribution of truth values or one of a range of such specific distributions. And let the constituents in question be subject to EC. Then (AG+)A is known only if there is an assurance that a suitably matching distribution of evidence for (or against) its (relevant) constituents may feasibly be acquired.36, 37 36  As it stands this—more specifically, its contrapositive—provides a second sufficient condition for ignorance, restricted to the kind of compound statement it mentions. That is all that is necessary to explain how someone can be properly regarded as ignorant of a statement who, by being in position to discount any Q inconsistent with that statement, fails to meet the other sufficient condition of ig­nor­ ance offered by AG. 37  This is, to stress, only a first approximation to a full account of the principle required. Quantified statements, for instance, do not literally have constituents in the sense appealed to by the formulation— though it should be straightforward enough to extend the formulation to cover them. More needs to be said, too, about how the principle should apply to compounds in which negation is the principal operator. But the provisional formulation will serve the immediate purpose.

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On Being in a Quandary  241 A purported warrant for a compound statement meeting the two stated conditions thus has to ground the belief that some appropriate pattern of evidence may be disclosed for its constituents. In particular, nothing is a basis for knowledge of a disjunction which does not ensure that at least one of the disjuncts passes the evidential constraint in its own right. More generally, when the truth of any class of statements is evidentially constrained, know­ledge of statements compounded out of them has to be conservative with respect to the feasibility of appropriate patterns of knowledge of their con­stitu­ents. One may thus quite properly profess ignorance of such a compound statement in any case where one has no reason to offer why an appropriate pattern of knowledge for its constituents should be thought achievable. The great insight of the Mathematical Intuitionists—and the core of their revisionism—was that a thinker may simultaneously both lack any such reason and yet be in a position to refute the negation of such a compound using only the most minimal and uncontroversial principles governing truth and validity. The proof of the double negation of LEM sketched in n. 26, for instance, turns only on the standard rules for disjunction, reductio ad absurdum, in the form that no statements collectively entailing contradictory statements can all be true, and the principle (enjoined, remember, by the Equivalence Schema) that the negation of a statement is true just in case that statement is not. These principles are themselves quite neutral on the question of evidential constraint but are arguably constitutive of the content of the connectives—disjunction and negation—featuring in LEM. The assurance they provide of the validity of its double negation is thus ungainsayable. But, when the truth of the ingredient statements is taken to involve evidential constraint, then that assurance does not in general amount to a reason to think that the appropriate kind of evidence for one disjunct or the other must in principle be available in any particular case. The assurance falls short in quandary cases—like borderline cases of simple colour predications—where we do not know what to say, do not know how we might find out, and can produce no reason for thinking that there is a way of finding out or even that finding out is metaphysically possible. Quandaries are not, of course, restricted to cases of vagueness as usually understood. They are also presented, for instance, by certain unresolved but—so one would think—perfectly precise mathematical statements for which we possess no effective means of decision. So add the thought—whatever its motivation—that mathematical truth demands proof, and there is then exactly the same kind of case for the suspension of classical logic in such

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242  The Riddle of Vagueness areas of mathematics.38 That is what the Intuitionists are famous for. But, if the account I have outlined is sound, then—whether or not there are compelling reasons derived within the philosophy of meaning for regarding EC as globally true—there will always be a case for suspension of classical logic wherever locally forceful grounds for EC combine with the possibility of quandary.39

V An Intuitionistic Solution to the Sorites V.1. Our problem was to make out how the Sorites reasoning could justly be treated as a reductio of its major premise without our incurring an obligation to accept the unpalatable existential, and further—when the existential is unpalatable precisely because it seems to express the precision of the relevant predicate in the Sorites series—to explain how the major premise might properly be viewed as a misdescription of what it is for that predicate to be vague. The essence of the solution that now suggests itself is that the vagueness of F

38  Note that this way of making a case for basic intuitionistic revisions needs neither any suspect reliance on AG nor appeal to specific non-truth-based proposals—in terms of assertibility conditions, or conditions of proof—about the semantics of the logical constants. The key is the combination of epistemic constraint and the occurrence of quandary cases. Any semantical proposals offered can sound exactly the same as those of the classicist. 39 An interesting supplementary question is now whether a revisionary argument might go through without actual endorsement of EC, just on the basis of agnosticism about it in the sense of reserving the possibility that it might be right. The line of thought would be this. Suppose we are satisfied that the outlined revisionary argument would work if we knew EC, but are so far open-minded— unpersuaded, for instance, that the usual anti-realist arguments for EC are compelling, but sufficiently moved to doubt that we know that truth is in general subject to no epistemic constraint. Suppose we are also satisfied that NKD, as a purely general thesis, is true: we have at present no grounds for thinking that we can in principle decide any issue. The key question is then this: can we envisage—is it rational to leave epistemic space for—a type of argument (which a global proponent of the revisionary argument thinks we already have) for EC which would ground its acceptance but would not improve matters as far as NKD is concerned? If the possibility of such an argument is open, then it must be that our (presumably a priori) grounds for LEM are already inconclusive—for what is open is precisely that we advance to a state of information in which EC is justified and yet in which NKD remains true. But in that case we should recognize that LEM already lacks the kind of support that a fundamental logical principle should have—for that should be support which would be robust in any envisageable future state of information. That seems intriguing. It would mean that revisionary anti-realism might be based not on a positive endorsement of EC but merely on suspicion of the realist’s non-epistemic conception of truth. Would this provide a way of finessing Fitch’s Paradox?—the well-known argument (Fitch  1963) that, in the presence of EC, it is contradictory to suppose that some truths are never known? No: if nothing else was said, the paradox would stand as a reason for doubting that it is rational to reserve epistemic space for a convincing global argument for EC.

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On Being in a Quandary  243 should be held to consist not in the falsity of the unpalatable existential claim but precisely in its association with quandary in the sense latterly introduced. To expand. Assume that F is like ‘red’ in that, though vague, predications of it are subject to EC. Then any truth of the form Fa & ¬Fa′ would have, presumably, to be recognizably true. The unpalatable existential, (∃x)(Fx & ¬Fx′), has only finitely many instances in the relevant type of (Sorites-)context. So its truth too would have to be recognizable. And to recognize its truth would be to find an appropriate Fa and ¬Fa′ each of which was recognizably true. We know that there is no coherently denying that there is any such instance, since that denial is inconsistent, by elementary reasoning, with the data, F(o) and ¬F(n). But we also know that we cannot find a confirming instance so long as we just consider cases where we are confident re­spect­ ive­ly that Fa, or that ¬Fa′. Thus, if there is a confirming pair, Fa and ¬Fa′, it must accordingly be found among the borderline cases. If these are rightly characterized as presenting quandary—that is, if we do not know whether to endorse them, do not know how we might find out, and can produce no reason for thinking that there is, or even could be, a way of finding out—then the status of (∃x)(Fx & ¬Fx′) is likewise a quandary, notwithstanding the proof of its double negation. And the plausibility of its (single) negation, notwithstanding the paradox it generates, is owing to our misrepresentation of this quandary: we are prone to deny the truth of the unpalatable existential when we should content ourselves with the observation that all its instances in the series in question are either false or quandary-presenting—an observation that merits denial of no more than its (current or foreseeable) assertibility. V.2. Again, it is crucial to this way with the problem that the quandary posed by borderline cases be exactly as characterized and in particular that it falls short of the certitude that there can be no deciding them. There can be no intuitionistic treatment of the Sorites unless we hold back from that concession. The indeterminacy associated with vague predicates has to fall short of anything that fits us with knowledge that one who takes a determinate—positive or negative—view of such an example, however tentative, makes a mistake. For, once we allow ourselves to cross that boundary—to rule out all possibility of finding a confirming instance of the unpalatable existential— EC, where we have it,40 will enforce its denial and the paradox will ensue.

40  Is EC always plausible for basic Sorites-prone predicates? It does seem to be a feature of all the usual examples. See Section VIII below.

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244  The Riddle of Vagueness This limitation—that we lack the certitude that there can be no finding a validating instance of the unpalatable existential—may seem very difficult to swallow. Let it be that atomic predications of vague expressions present quandaries in just the sense characterized; in particular, that we do not know that there is no knowing that such a predication is true, or that it is false. Still, that both P and Q present quandaries is not in general a reason for regarding their conjunction as beyond all knowledge: if Q is not-P, for instance, we can know—one would think—that the conjunction is false even though each conjunct is a quandary. It may seem evident that instances of the unpalatable existential are in like case: that even if Fx and ¬Fx′ are quandaries, we do still know that there is no knowing that both are true. In general, quandary components are sure to generate quandary compounds only if verdicts on those components are mutually unconstrained; but the whole point about Sorites series is that adjacent terms lie close enough together to ensure that differential verdicts cannot be justified—ergo cannot be known. Plausible as this train of thought may seem, it must be resisted—at least by a defender of EC for the range of predications in question. For suppose we knew that any adjacent terms in a Sorites series lie close enough together to ensure that differential verdicts about them cannot both be known. Then we would know that

FeasK(Fx ) → ¬FeasK(¬Fx′).

By EC, we have both

Fx → FeasK(Fx )

and

¬ Fx′ → FeasK(¬Fx′).

So, putting the three conditionals together,

Fx → ¬¬Fx′.

Hence, contraposing and collapsing the triple negation,41

¬Fx′ → ¬Fx.

So, if we think we know that any adjacent terms in a Sorites series lie close enough together to ensure that differential verdicts about them cannot both 41  The equivalence of triple to single negation is, of course, uncontroversial.

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On Being in a Quandary  245 be known, we have to acknowledge that each non-F item in the series is preceded by another. Thus we saddle ourselves with a Sorites paradox again.42 Let me again stress the two morals:

42  The general thrust of our discussion involves—as one would naturally expect of an advertised intuitionistic treatment—a heavy investment in EC. As I have said, I believe the principle is plausible for the kinds of statement that feature in the classic examples of the Sorites paradox—though the relationship between vagueness and evidential constraint is a crucial and relatively unexplored issue (see remarks at the end of the chapter). But I should stress that I regard the conception of borderline cases which I am proposing, of which it is an essential feature that we do not know that there is no knowledgeable verdict to be returned about a borderline case, as plausible in dependently of the ­incoherence of its denial when EC is accepted. Let me quickly rehearse a further corroborative consideration. According to the opposing view—the verdict exclusion view—a borderline case is something about which we know that a knowledgeable positive or negative verdict is ruled out. The verdict exclusion view would be imposed by the third possibility view, but, whatever its provenance, it faces great difficulty in accommodating the intuitions that ground the idea of higher-order vagueness. For consider: if a (first-order) borderline case of P is something about which one can know that one ought to take an agnostic stance—a situation where one ought not to believe P and ought not to believe not-P—then (one kind of) a higher-order borderline case is presumably a situation where one can know that one ought not to believe P and ought not to believe that P is (first-order) borderline. Since on the view proposed P’s being first-order borderline is a situation where one ought not to believe P and ought not to believe not-P, it follows that, confronted with a higher-order borderline case, one can know that: (i)  one ought not to believe P;  and (ii)  one ought not to believe that one ought not to believe P and ought not to believe not-P. However in moving in the direction of (putative) borderline cases of P and the first-order P/not-P borderline, we have moved towards P, as it were, and away from not-P. Since—according to the verdict exclusion view—the first-order borderline cases were already cases where it could be known that (iii)  one ought not to believe not-P, it should follow that the relevant kind of higher-order borderline cases are likewise cases where (iii) may be known. So one gets into a position where one may knowledgeably endorse both (i) and (iii) yet simultaneously know—by dint of knowing (ii)—that one ought not to endorse their conjunction—a Moorean paradox (at best).  In sum: the idea that agnosticism is always mandated in borderline cases cannot make coherent sense of higher-order vagueness. The distinction between cases where a positive or negative view is mandated and cases where agnosticism is mandated cannot itself allow of borderline cases, on the verdict exclusion view. That is very implausible, and provides a powerful reason to be suspicious of the verdict-exclusion view. This conclusion would be blocked, of course, if the verdict exclusion view were qualified: if it were conceded that agnosticism is only mandated for some borderline cases and that for others, perhaps less ‘centrally’ borderline, something like the permissibility-conception which I have been recommending—that in such cases those who incline to return positive or negative verdicts are not known to be incorrect but are, as it were, ‘entitled to their view’—is the stronger account. Arguably, though, such a compromise would give the game away. For if the permissibility-conception is correct at least for cases towards the borderline between definite cases of P and—the alleged—definite cases on the borderline between P its negation, the question must immediately arise what good objection there could be to allowing the negation of P to cover the latter, agnosticism-mandating cases. None if they are conceived as by the third possibility view—for then they are exactly cases where P is other than— so not—true. But, after that adjustment, the only remaining borderline cases would be just those where conflicting opinions were permissible, and the permissibility-conception would therefore seem to have the better case to capture the basic phenomenon.

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246  The Riddle of Vagueness (i) EC plus knowledge of the irresolubility of borderline cases is a cocktail for disaster. Any compelling local motivation for EC with respect to a vague discourse enforces an acknowledgement that our ignorance with respect to the proper classification of borderline cases can extend no further than Quandary, as characterized, allows. We—innocent witnesses, as it were, to a difference of opinion—do not know what to say about such a case, do not know how to know, cannot produce any reason for thinking that there is any way of knowing nor even that there could be. But we do not know that there is none. (ii) EC plus knowledge of the undifferentiability of adjacents in a Sorites series—the unknowability of the truth of contrasting verdicts about them—is similarly explosive. So, we must take it that, where the statements in question are quandaries, we do not know that verdicts of the respective forms Fa and not-Fa′ can never knowingly be returned. That allows each conjunction of such quandaries, Fa & ¬Fa′, to be itself a quandary; whence we may infer that the unpalatable existential is also a quandary, by the reasoning outlined in 5.I.43 V.3. What are we now in position to say about the following conditional:

(∃x )(Fx & ¬Fx′) → ‘F ’is not vague,

rightly focused on by Timothy Chambers in recent criticism of Putnam (Chambers 1998)? If it is allowed to stand as correct, then—contraposing— any vague expression will be characterized by the negation of the antecedent and the all too familiar aporia will ensue. What fault does the broadly intuitionistic approach I have been canvassing have to find with it? Well, there is no fault to be found with it as a conditional of assertibility: to be in position to assert the antecedent with respect to the elements of a Sorites series must be to be in position to regard ‘F’ as sharply defined over the series. So an intuitionist who insists on the familiar kind of assertibility-conditional semantics for the conditional, whereby ‘P → Q’ is assertible just if it is assertible that any warrant for asserting P would be (effectively transformable into) 43  Timothy Williamson’s (see Williamson 1996b) otherwise cogent recent criticisms of Putnam— specifically, his reductio of the combination of Putnam’s proposal about vagueness and the ideal-justification conception of truth which Putnam favoured at the time—precisely assume that our knowledge of the status of borderline cases extends far enough to let us know that there can be no justified differentiation of adjacents, even under epistemically ideal circumstances. But we have seen, in effect, that Putnam should refuse to grant that assumption. A would-be intuitionistic treatment of vagueness must respect the two morals just summarized.

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On Being in a Quandary  247 a warrant for asserting Q, will be put in difficulty by Chambers’s simple point. However, that style of semantics is arguably objectionable in any case, ob­lit­er­ at­ing as it does the distinction in content between the conditionals, If P, then Q and If P is assertible, then Q.44 What is wrong with the Chambers conditional from our present perspective is rather that, if its antecedent—the unpalatable existential—is rightly regarded as presenting a quandary in cases where F is vague in the series in question, then it is not something whose truth we are in a position to exclude. So, for all we know, the antecedent of the Chambers conditional may be true while its consequent is false; for F is vague by hypothesis. So there is—as there needs to be—principled cause to regard the conditional as unacceptable.45

44  This assumes that ‘P’ and ‘P is assertible’ are always co-warranted. 45  A skirmish about this is possible. If the unpalatable existential is justly regarded as a presenting a quandary, then we shouldn’t rule out the possibility of coming to know that (∃x)(Fx & ¬Fx′) is true. But, if we did know it, we should presumably not then know that the relevant predicate, F, is vague— for we would know that it was sharply bounded in the series in question. So it seems we can rule out (*) (∃x )(Fx & ¬Fx′) & ‘F ’ is vague as a feasible item of knowledge. And now, if (*) is subject to EC, it follows that it is false and hence— again, an intuitionistically valid step—that the Chambers conditional holds after all. (I am grateful to Timothy Williamson for this observation.) On the other hand, if (*) is not subject to EC; then the question is why not—what principled reason can be given for the exception when so much of our discussion has moved under the assumption that many contexts involving vague expressions are so? The answer is that (*) cannot be subject to EC—at least in the simple conditional form in which we have been considering that principle—for just the reason that Fitch’s well-known counterexamples cannot be. These counterexamples are all contingent conjunctions where knowledge of one conjunct is inconsistent with knowledge of the other. The simplest case is: P and it is not known that P. Knowledge of the second conjunct would require—by the factivity of knowledge—that the first conjunct was not known; but, if the conjunction could be known, so could each conjunct simultaneously. Hence EC must fail if the Fitch schema has true instances. It now suffices to reflect that, on the conception of vague expressions as giving rise to quandary, (*) is merely a more complex Fitch case. For to know that ‘F’ is vague is to know that predications of it give rise to quandaries in a series of the appropriate kind and hence—by the reasoning sketched in the second paragraph of V.1—that the unpalatable existential is itself a quandary and hence is not known. Of course, this comparison would not be soothing for someone sympathetic to the sketched intuitionistic response to the Sorites who was also a proponent of EC globally. But there is no evident reason why the viability of the intuitionistic response to the Sorites should depend upon the global proposition. For one for whom the case for EC always depends on the nature of the local subject matter, there should be no discomfort in recognizing that ‘blind-spot’ truths—truths about truths of which we are, de facto or essentially, ignorant— will provide a region of counterexamples to EC.

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248  The Riddle of Vagueness This is not inconsistent with allowing that the unpalatable existential does indeed characterize what it is for F to be precise relative to the series of objects in question. But, if that is insisted upon, then we learn that it was a mistake to view vagueness as entailing a lack of precision. Rather, the vagueness of a predicate involves the combined circumstances that atomic predications of it are prone to present quandary and that we are unwarranted in  regarding Bivalence/Excluded Middle as valid for such predications. Vagueness so conceived is an epistemic notion; precision, if enjoined by the truth of the unpalatable existential, is a matter of ontology—of actual sharpness of extension. I will return to the issue of the characterization of vagueness later. V.4. Earlier we set two constraints on a treatment of the Sorites: it was to be explained (i) how the falsity of ¬(∃x)(Fx & ¬Fx′) can be consistent with the vagueness of F; and (ii) how and why it can be a principled response to refuse to let ¬¬(∃x)(Fx & ¬Fx′) constitute a commitment to the unpalatable existential, and hence— apparently—to the precision of F.

The answers of the present approach, in summary, are these. The major premise for the Sorites may unproblematically be denied, without betrayal of the vagueness of F, if F’s vagueness is, in the way adumbrated, an epistemic ­property—if it consists in the provision of quandary by some of the atomic predications of F on objects in the series in question. And such a denial need be no commitment to the unpalatable existential—or other classical equivalents of that denial which seem tantamount to the affirmation of precision—if the ­latter are also quandaries and are thus properly regarded as objects of agnosticism. Rather, the classical-logical moves which would impose such commitments are to be rejected precisely because they allow transitions from known premises to quandary conclusions.46 46  The reader should note that no ground has been given for reservations about double negation elimination as applied to atomic predications, even in quandary-presenting cases. For—in contrast to the situation of the double negation of the unpalatable existential—no purely logical case will be available to enforce acceptance of ¬¬Fa in a case where Fa presents a quandary. However, an acceptance of DNE for vague atomic predications will, not, of course, enforce an acceptance of the Law of Excluded Middle for them. (Recall that the proof of the equivalence of DNE and LEM requires that the former hold for compound statements, in particular for LEM itself.)

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On Being in a Quandary  249

VI Relativism Stabilized VI.1. Our problem was to block both the apparent lesson of the Simple Deduction, that any dispute about a truth-apt content involves a mistake, and that of the EC-Deduction, that any dispute about an evidentially constrained truth-apt content involves a substantive cognitive shortcoming—so that, at least with subject matters constrained by EC, the intended gap between min­ imal truth aptitude and cognitive command collapses. It should now be foreseeable how a principled response to these awkward arguments may run. The truth is that each Deduction is actually fine, as far as it goes—(to the stated line 8 in the case of the Simple Deduction, and line 10 in the case of the EC-Deduction.) The problem, rather, consists in a non sequitur in the way their conclusions were interpreted. Take the EC-Deduction. (The response to the Simple Deduction is exactly parallel.) What is actually put up for reductio is the claim that a certain dispute involves no cognitive shortcoming. That is a negative existential claim, so the reductio is in the first instance a proof of its negation—that is, a doubly negated claim: that it is not true that A’s and B’s conflicting opinions involve no cognitive shortcoming. This is indeed established a priori (if EC is locally a priori). However, to achieve the alleged ­demonstration of cognitive command—that it is a priori that cognitive shortcoming is involved—we have first to eliminate the double negation. And the needed DNE step, like that involved in the classical ‘proof ’ of LEM and the Sorites-based proof of the unpalatable existential, involves a violation of AG+. As the reader may verify, the reasoning deployed in the EC-Deduction up to its conclusion at line 10 draws on no resources additional to those involved in the proof of the double negation of LEM save modus ponens and the suggestion that one who holds a mistaken view of a knowable matter is per se guilty of cognitive shortcoming. Neither of those additions seems contestable, so the EC-Deduction should be acknowledged as absolutely solid. However, the transition from its actually doubly negated conclusion to the advertised, ­double negation-eliminated result—that cognitive command holds wherever conflict of opinion is possible—demands, in the presence of EC, that there be an identifiable shortcoming in A’s and B’s conflicting opinions—for the shortcoming precisely consists in holding the wrong view about a knowable matter. If the example is one of quandary, the DNE step is thus a commitment to the view that an error may be identified in a case where we do not know the right opinion, do not know how we might know, and have no general reason

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250  The Riddle of Vagueness to suppose that there is, or could be, a way of knowing nonetheless. Once again, the logical and other resources involved in the simple proof (up to line 10) seem manifestly inadequate to sustain a conclusion with that significance. So, although indeed in position to rule out the suggestion that any dis­agree­ ment is cognitively blameless, just as the two Deductions show, we remain— in the light of the enhanced principle of ignorance AG+—unentitled to the claim that there will be cognitive shortcoming in any difference of opinion within a minimally truth-apt discourse. We remain so unentitled precisely because that would be a commitment to a locatability claim for which the proof of the double negation provides no sufficient ground and for which we have, indeed, no sufficient ground. The immediate lesson is that it is an error (albeit a natural one) to characterize failures of cognitive command—or indeed what is involved in True Relativism generally—in terms of the possibility of blameless differences of opinion.47 Indeed, it is the same root error as the characterization of failures of Bivalence in terms of third possibilities, truth-value gaps, and so on. Failures of cognitive command, like failures of Bivalence, must be viewed as situations where we have no warrant for a certain claim, not ones where—for all we know—its negation may be true. We do know—the two Deductions precisely teach—that the negation will not be true. But that is not sufficient for cognitive command. The distinction once again turns on the intuitionistic insight that one may, in contexts of evidential constraint and potential quandary, fall short of knowledge of a claim whose negation one is nevertheless in position to exclude. The point does not depend on the sources of any potential quandary. But my implicit proposal in Truth and Objectivity—the reason why the cognitive command constraint was formulated so as to exempt disagreements owing to vagueness—was that it is a feature of discourse concerning the comic, the attractive, and the merely minimally truth-apt generally, that differences of opinion in such regions may present quandaries for reasons other than vagueness. It is not (just) because ‘funny’ and ‘delicious’ are vague in the way ‘red’ is that the kind of differences of opinion about humour and gastronomy are possible which we do not know how to resolve, do not know how we might get to know, and do not know that there is, or could be, any getting to know. Merely minimally truth-apt discourses, in contrast with discourse exerting cognitive command, provide examples of indeterminacy in re. But we need to 47  Regrettably, the error is encouraged by the wording of some passages in Truth and Objectivity. See, e.g., Wright (1992b, pp. 94, 145).

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On Being in a Quandary  251 correct the usual understanding of this, epitomized by the rhetoric of phrases like ‘no fact of the matter’. That rhetoric, it should now be superfluous to say, is simply inconsistent with the most basic constitutive principles concerning truth and negation. The indeterminacy consists rather in the fact that provision exists for quandaries which, because they arise in contexts governed by evidential constraint, enforce agnosticism about principles—like Bivalence— which, if they could be assumed to hold, would ensure that there was a ‘fact of the matter’, about which we would merely be ignorant. It is a matter, if you like, of lack of warrant to believe in a fact of the matter, rather than a reason to deny one—a subtle but crucial distinction whose intelligibility depends on a perception of the inadequacy of AG and the basic intuitionistic insight.

VII Epistemic Indeterminacy VII.1. Let me return to the issue of the relation between the epistemic conception of indeterminacy I have been proposing and the rampantly realist Epistemic Conception. Writing in criticism of Williamson’s and Sorensen’s respective defences of the latter, I once observed: Perhaps the most basic problem for the indeterminist—the orthodox op­pon­ent of the Epistemic Conception—is to characterize what vagueness consists in—to say what a borderline case is. It is also one of the least investigated. The epistemic conception should not be allowed to draw strength from this neglect. There is no cause to despair that the situation can be remedied. (Wright 1995, p. 146, this volume, Chapter 6)

Well, how close do the foregoing considerations come to remedying the situ­ ation? My proposal in that earlier paper was that borderline cases of F should be characterized in the natural way, using an operator of definiteness, as cases which are neither definitely F nor definitely not-F but—prefiguring what I have been suggesting here—that the definiteness operator should be construed epistemically, with genuine borderline cases marked off from de­ter­min­ate matters lying beyond our ken—including borderline cases as conceived by the Epistemicist—by examples of the latter sort being characterized by the Principle of Bivalence, there characterized as the hypothesis of ‘universal determinacy in truth-value’ (Wright (1995, p. 145; this volume, Chapter 6).

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252  The Riddle of Vagueness Williamson later responded: So far the parties do not disagree; the epistemicist has merely said more than the indeterminist. But that is not the only difference between them. The indeterminist regards the epistemicist’s account of borderline cases as positively incorrect [my emphasis]. The epistemicist is supposed to regard borderline claims as determinate in truth-value, while the indeterminist regards them as not determinate in truth-value. (Williamson 1996a, p. 44)

This gloss on the differences between the protagonists enabled Williamson to advance the following line of criticism. Part of the indeterminist characterization of borderline claims is that they are not determinate in truth value. What does ‘determinate’ mean? If not being determinate in truth value involves lacking a truth value, then we are back with third-possibility indeterminacy. But ‘not determinate in truth value’ cannot just mean ‘not definitely true and not definitely false’ since that claim—with ‘definitely’ understood epi­stem­ic­ al­ly, as now by both sides in the dispute—is one the Epistemicist is prepared to make; whereas the denial of determinacy was supposed to crystallize a point of disagreement between the indeterminist and the Epistemicist. So, Williamson concluded, the indeterminist bugbear—of giving some nonepi­stem­ic account of borderline cases—recurs. This was a curious criticism, given that the notion of determinacy in truth value was involved in the first place only as a paraphrase of the Principle of Bivalence. For in that case, Williamson’s supposition that my indeterminist was someone who regarded borderline claims as not determinate in truth value would be equivalent to attributing to them the thesis that Bivalence failed for such claims. And then, given that I explicitly did not want any traffic with third possibilities, Williamson would have had a much more forceful criticism to make than merely that the implicated notion of determinacy had still not been properly explained. In fact, however—the important point for our present concerns—Williamson mischaracterized the opposition in the first place. It was a misunderstanding to suppose that the ‘indeterminist’—my theorist in the earlier paper— regarded borderline claims as ‘not determinate in truth value’. Rather, the difference between that theorist and the Epistemicist was precisely that the former draws back from, rather than denies, a view which the Epistemicist takes: the negation belongs with the attitude, not the content. The ‘indeterminist’ regarded the Epistemicist’s bivalent view of borderline cases (the view of them as determinate in truth value) not as positively wrong—where that is

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On Being in a Quandary  253 taken to mean: something they are prepared to deny—but as positively unjustified: something which they know of no sufficient grounds to accept. In fact, the involvement of an (unexplicated) notion of determinacy was inessential to the view that was being proposed. The claim of determinacy in truth value just is the claim that Bivalence holds in the cases in question. So the heart of the ‘indeterminist’ thesis was just that borderline cases are these: cases where—in an appropriate epistemic sense of the definiteness operator— a target predication is not definitely true and not definitely false and where there is no extant warrant for the assertion of Bivalence. Williamson’s short response contained nothing to threaten the stability of this view. That said, it merits acknowledgement that ‘indeterminist’ was not the happiest label for the type of position I was trying to outline, and that it may have misled Williamson. For it is hard to hear it without gathering a suggestion of a semantic or an ontological thesis: of vagueness conceived as involving matters left unresolved not (merely) in an epistemic sense, but, in fact, by the very rules of language, or by the World itself. For someone who wants one of those directions made good—and who read my remark quoted above as calling for just that—the direction taken in my earlier discussion, and in this one, will puzzle and disappoint. In any case—save in one crucial detail—it is still no part of the view I have been developing in this chapter to regard the Epistemicist’s account of borderline cases as ‘positively incorrect’. There is agreement that the root manifestations of vagueness are captured by epi­stem­ic categories: bafflement, ignorance, difference of opinion, and uncertainty— and that to conceive of the phenomenon in semantic or ontological terms is to take a proto-theoretical step which, without any coherent further development, there is cause to suspect may be a mistake. The ‘crucial detail’ of dis­ agree­ment—prescinding, of course, from the major conflict over warrant for the Principle of Bivalence—is merely over the thesis that borderline cases are known to defy all possibility of knowledgeable opinion. While the coherence of the Quandary view depends on its rejection, Williamson perceives it as a theoretical obligation of his own view to defend it.48 But, setting that apart, it 48  I am not myself certain that the Epistemicist does have any obligation to defend anything so strong. Someone who believes that vague expressions have sharp extensions ought to explain, sure, why we don’t actually know what they are nor have any clear conception of how we might find out. But there would seem to be no clear obligation to conceive of them as unknowable—(though that might be a consequence of the theorist’s best shot at meeting the less extreme explanatory demand). I suspect that matters proceed differently in Williamson’s thinking: that he regards the impossibility of  knowledgeable (positive or negative) opinion about borderline cases as a datum, which would straightforwardly be explained by semantic and ontological conceptions of indeterminacy (could we but explain them) and of which he therefore conceives that his own, Bivalence-accepting conception must provide an alternative explanation. I do not think it is a datum.

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254  The Riddle of Vagueness deserves emphasis that the view of vagueness here defended is consistent with the correctness of the Epistemic Conception (and on the other hand, pari passu, with agnosticism about whether it even could be correct.) The Quandary view is consistent with the correctness of the Epistemic Conception in just the sense in which the Intuitionist philosophy of math­em­at­ics is consistent with the actual correctness of the Principle of Bivalence and classical mathematical practice. The basic complaint is not of mistake—though the Epistemic Conception may well prove to be committed to collateral mistakes (for instance, about the conditions on possible semantic reference: on what it takes for a predicate to stand for a property)—but of lack of evidence. VII.2. One—albeit perhaps insufficient—reason to retain the term ‘indeterminist’ for the conception of vagueness defended in my earlier paper was the retention of a definiteness operator and the characterization of borderline cases as ‘not definitely . . . and not definitely not . . .’. But I now think that was a mistake—and the operator itself at best an idle wheel. My earlier proposal was that P is definitely true just if any (what I called) primary opinion—any opinion based neither on testimony nor on inference, nor held groundlessly—that not-P would be ‘cognitively misbegotten’—that is, some factor would contribute to its formation of a kind which, once known about, would call its reliability into question in any case and could aptly be used to explain the formation of a mistaken opinion. No doubt this proposal could be pressured in detail, but—with the notion of cognitive command recently before us—the guiding idea is plain: the definite truths were to be those disagreements about which would have to involve cognitive shortcoming tout court, with no provision for excuses to do with vagueness. So a claim which is not definitely true and not definitely false ought to be one—I seem to have wanted to suggest—about which ‘neither of a pair of conflicting opinions need be cognitively misbegotten’ (Wright  1995, p. 145, this volume, Chapter 6). This proposal was intended to capture the idea that the phenomenon of permissible disagreement at the margins is of the very essence of vagueness . . . the basic phenomenon of vagueness is one of the possibility of faultlessly generated—cognitively un-misbegotten—conflict. (Wright 1995, p. 145, this volume, Chapter 6)

However we have in effect seen that this will not do. What we learned from the EC-Deduction was that, wherever we have evidential constraint, hence each of the conditionals

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On Being in a Quandary  255 P → it is feasible to know that P, and ¬P → it is feasible to know that ¬P, the idea of a ‘faultlessly generated’ disagreement rapidly destabilizes. For, if the disagreement were faultless, it could not be that it was feasible to know either of the protagonists’ opinions to be correct, or there would have to be fault in the generation of the other. And, in that case, contraposing on both conditionals, contradiction ensues. But we know the remedy now: retreat to the double negation and invoke the enhanced principle of ignorance, AG+. My proposal should have been not that faultlessly generated disagreements are possible where vague claims are concerned, but that we are in no position to claim that any disagreement about such a claim involves fault. Thus the root phenomenon of vagueness cannot after all, when cautiously characterized, be that of permissible dis­ agree­ment at the margins; rather, it is the possibility of disagreements of which we are in no position to say that they are impermissible, in the sense of involving specific shortcomings of epistemic pedigree. We are in no position to say that because, notwithstanding the incoherence of the idea that such a dis­ agree­ment is actually fault-free, the claim that there are specific shortcomings involved must, in the presence of EC, involve a commitment to their identifiability, at least to the extent of pointing the finger at one disputant or the other. And that is exactly what we have no reason to think we can generally do. The upshot is that, even when ‘definitely’ is interpreted along the epistemic lines I proposed, we should not acquiesce in the characterization of borderline claims as ones which are neither definitely true nor definitely false.49 Rather, they will be claims for which there is no justification for the thesis that they are definitely true or definitely false—again, with ‘definitely’ epistemic— nor any justification for the application of Bivalence to them. But now the former point is swallowed by the latter. For, in the presence of EC, justification for Bivalence just is justification for the thesis that any statement in the relevant range is knowably—so definitely—true or false. So the definiteness operator is (harmless but) de trop. One more very important qualification. None of this is to suggest that we may give a complete characterization of vagueness along these Spartan lines: that vague statements are just those which give rise to quandary and for which Bivalence is unjustified. That’s too Spartan, of course. The view proposed has indeed, after all, no need for the expressive resource of an operator of 49  I leave it as an exercise for the reader to adapt the EC-Deduction to a proof of this claim.

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256  The Riddle of Vagueness in­de­ter­min­acy. But some quandaries—Goldbach’s conjecture, for instance— feature nothing recognizable as vagueness; and others—that infidelity is all right provided nobody gets hurt, perhaps—may present quandaries for reasons other than any ingredient vagueness. So the task of a more refined taxonomy remains—the notion of quandary is just a first step.50 But if the general ­tendency of this discussion is right, it is a crucial step.

VIII Summary Reflections VIII.1. To recapitulate the gist of all this. A proposition P presents a quandary for a thinker T just when the following conditions are met: (i) T does not know whether or not P (ii) T does not know any way of knowing whether or not P (iii) T does not know that there is any way of knowing whether or not P (iv) T does not know that it is (metaphysically) possible to know whether or not P. The satisfaction of each of these conditions would be entailed by (v) T knows that it is impossible to know whether or not P, but that condition is excluded by Quandary as we intend it—a quandary is uncertain through and through. Note that, so characterized, quandaries are relative to thinkers—one person’s quandary may be part of another’s (presumed) information—and to states of information—a proposition may present a quandary at one time and not at another. There are important classes of example which are ac­know­ ledged to present quandaries for all thinkers who take an interest in the matter. Goldbach’s conjecture is currently one such case. But, for the protagonists in an (intransigent) dispute of inclination, it will naturally not seem that the 50  Relevant initial thoughts, already bruited, are these: it is known—in our present state of information, in the absence of proof—that nobody’s opinion about Goldbach is knowledgeable; whereas, on the view proposed, we precisely do not know that a positive or negative verdict about a borderline case of ‘x is red’ is unknowledgeable. And, unlike ‘red’, predications of ‘funny’ have no definite cases—they are always contestable.

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On Being in a Quandary  257 target claim presents a quandary; likewise when conflicting verdicts are returned about a borderline case of some vague expression. Yet, to a third party, the contested claim in such cases—and hence the question who is right about it—may always reasonably be taken to present a quandary nonetheless. It should seem relatively uncontroversial to propose that unresolved mathematical conjectures, borderline cases of vague expressions, and the foci of disputes of inclination meet the four defining conditions of quandary. To say that much is simply to report on our epistemic situation in relation to the claims in question. It is to say nothing about their metaphysical or semantical status. What is not uncontroversial, of course, is the contention that clause (v) fails—that we do not know that there is no knowing the truth of either of two conflicting verdicts about a borderline case, or either of the two conflicting views in a dispute of inclination. As I have acknowledged, this modesty may go against the grain. But it is imposed if we accept that the disputed statement is subject to EC.51 And it is imposed in any case if we are inclined to think that we should be permissive about such disputes—for otherwise we ought to convict both disputants of over-reaching, of unwarranted conviction about an undecidable matter, and they should therefore withdraw. The thought that they are, rather, entitled to their respective views has to be the thought that we do not know that they are wrong to take them—do not know that neither of their views is knowledgeable. I do not expect many immediate converts—at least not from among those who start out convinced that clause (v) should be part of the account of vagueness. But maybe I have done a little to erode that conviction—or at least to bring out other intuitions and theses that it holds hostage. In any case, Epistemicists will abjure the role played by Evidential Constraint in the foregoing discussion. And indeterminists proper will equally abjure the suggestion that the proponents of the Epistemic Conception of vagueness have the matter half right: that indeterminacy is an epistemic matter, that borderline cases should be characterized as cases of (a complicated kind of) ignorance. According to the present view, the Epistemic Conception takes us in the right general direction. It goes overboard in its additional (gratuitous and unmotivated) assumption that the Principle of Bivalence holds for all statements, including quandary-presenting ones, so that we are constrained to think, for example, of predicate expressions which are prone to give rise to such statements as denoting—by mechanisms of which no one has the slightest inkling how to give an account—sharply bounded properties of which we may lack 51  See also n. 42.

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258  The Riddle of Vagueness any clear conception.52 But the general conception of vagueness it involves is otherwise—at least in the round—quite consistent with the present proposal. I have suggested that the Intuitionists’ revisionism is best reconstructed as driven by a mixture of quandary and evidential constraint: the belief that truth in mathematics cannot outrun proof, together with a recognition that unresolved mathematical conjectures can present quandaries in the sense characterized. If this is right, then, my point has been, the revisionary argument will generalize, and classical logic—especially the Law of Excluded Middle and, correlatively, the principle of double negation elimination— should not be accepted (since it has not been recognized to be valid) for any area of discourse exhibiting these two features. The result, I have argued, is that we have the resources for a principled, broadly intuitionistic response to the Sorites paradox. And we can stabilize the contrast between minimal truth aptitude and cognitive command against the Deductions that threatened to subvert it, and which do indeed show that it is unstable in the setting of clas­ sic­al logic. To be sure, we do not thereby quite recover the materials for a coherent true relativism as earlier characterized—which involved essential play with the possibility of fault-free disagreement. But an anti-relativistic rubric in terms of cognitive command: that it hold a priori of the discourse in question that disagreements within it (save when vagueness is implicated) involve cognitive shortcoming, may once again represent a condition which there is no guarantee that any minimally truth-apt discourse will satisfy. The relativistic thesis, for its part, should accordingly be the denial that there is—for a targeted discourse—any such a priori guarantee (or merely the claim that it is unwarranted to suppose that there is). Thus the ancient doctrine of relativism, too, now goes epistemic. I do not know if Protagoras would have approved. VIII.2. It merits emphasis, finally, that—for all I have argued here—these proposals can be extended no further than to discourses which exhibit the requis­ite combination of characteristics: quandary-propensity and evidential constraint. Without that combination, no motive has been disclosed for suspension of classical logic53—but classical logic would serve to reinstate the intended conclusions of the two Deductions and to obliterate the distinction between the proper conclusion of the Sorites paradox—the denial of its major premise—and the unpalatable existential. One question I defer for further work is whether the two characteristics co-occur sufficiently extensively to allow the mooted solutions to have the requisite generality. 52  See n. 6.

53  But see, however, n. 39.

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On Being in a Quandary  259 Two initially encouraging thoughts are these. First, as noted earlier, people’s ordinary willingness to think in terms of ‘no fact of the matter’ in cases of intransigent disputes of inclination is in effect the manifestation of an acceptance of evidential constraint for the relevant discourse. (For, if they were comfortable with the idea that such a dispute could in principle concern an undecidable fact, why would they take its intransigence as an indicator that there wasn’t one?) I therefore conjecture that whatever exactly it is that we are responding to when we engage in the kind of taxonomy I illustrated right at the beginning with the two ‘crude but intuitive’ lists, the contents which we are inclined to put in the first list will indeed be cases where we will not want to claim any conception of how the facts could elude appreciation by the most fortunately generated human assessment. Second, if classical logic is inappropriate, for broadly intuitionistic reasons, for a range of atomic statements, it could hardly be reliable for compounds of them, even if the operations involved in their compounding—quantifiers, tenses, and so on—were such as to enable the construction of statements which are not subject to EC. Thus, what the intuitionistic response to the Sorites requires is not that all vague sentences be both potentially quandarypresenting and evidentially constrained but only that all atomic vague sentences be so. The standard examples of the Sorites in the literature—‘red’, ‘bald’, ‘heap’, ‘tall’, ‘child’—do all work with atomic predicates, and all are, plausibly, evidentially constrained. But that is merely suggestive. If a finally satisfactory intuitionistic philosophy of vagueness is to be possible, we need an insight to connect basic vague expressions and evidential constraint. The notions of observationality, and of response dependence, would provide two obvious foci for the search. For now, however—in a contemporary context in which a few theorists of vagueness have argued against its prospects but most have simply paid no serious heed to the idea at all—it will be enough to have conveyed (if I have) something of the general shape which a stable intuitionistic philosophy of vagueness might assume.54 54  Versions of the material on revisionism were presented at colloquia at the University of Bologna, the City University of New York Graduate Center, and at Rutgers University in Autumn 1998. I was fortunate enough to have the opportunity to present a discussion of all three problems at two seminars at Ohio State University in December of that year, and to have a precursor of the present draft discussed at the Mind and Language seminar at NYU in April 1999, where Stephen Schiffer’s commentary resulted in a number of improvements. The NYU draft also provided the basis for three helpful informal seminars at Glasgow University in May 1999. More recently, I took the opportunity to present the material on the Sorites at an Arché Workshop on Vagueness which, with the sponsorship of the British Academy, has held at St Andrews in June 2000. I am extremely grateful to the discussants on all these occasions, and in addition to John Broome, Patrick Greenough, Riki Heck, Fraser MacBride, Sven Rosenkranz, Mark Sainsbury, Joe Salerno, Tim Williamson, and a referee for Mind for valuable comments and discussion. Almost all the research for the chapter has been conducted during my tenure of a Leverhulme Research Professorship; I gratefully acknowledge the support of the Leverhulme Trust.

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8 Rosenkranz on Quandary, Vagueness, and Intuitionism Sven Rosenkranz provides a searching critique of a line of argument in my 2001 paper ‘On Being in a Quandary’ (henceforward ‘Quandary’) towards a broadly intuitionistic conception of the nature and logic of vagueness (Wright 2001b: 78 ff). I think his principal criticism scores some hits. In the three sections to follow I shall respectively outline what I take to be the essence of his criticism, propose a modification of my argument that I think can resist it, and then suggest a different, more economical route to the proposed intuitionistic distinctions for vague statements.

I Our concern is with the usual kind of Sorites series for a vague predicate F, with x′ the immediate successor of any of its elements x. The argument on which Rosenkranz is focusing occurs at pp. 78 ff of ‘Quandary’ (this volume, Chapter 7). There it is suggested that, whenever Evidential Constraint (EC) holds for atomic predications of F and their negations (I will term both kinds of statement ‘F-predications’), then any truth of the form Fa & ¬Fa′ will be knowable and hence that the conditional (1)  (∃x )(Fx & ¬Fx′) → (∃x )FeasK(Fx & ¬Fx′)1 will hold good (and be known if EC is). My proposal was then that, since any sharp cut-­off would lie in the region of quandary for F—the region of cases where we do not know that knowledge of the truth of an F-predication is so much as metaphysically possible—the consequent of (1) will also present as a quandary. We accordingly do not know that it holds nor therefore that (1)’s 1  As in ‘Quandary’ I write ‘FeasK [. . .]’ to express the feasibility of knowledge that [. . .]. I (and Rosenkranz) assume that the relevant range of F-predications may be so understood as to give rise to no good concern about ‘quantifying in’.

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0009

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262  The Riddle of Vagueness antecedent does. However, since the Sorites reasoning itself disproves the negation of the antecedent, we do know that the double negation of (1)’s antecedent holds. So we have a solid motive to refuse double negation elimination in this context. And once we do so, we are empowered to treat the Sorites as a harmless reductio of its major premise, without the ensuing uncomfortable commitment, imposed by classical logic, to claiming that there is a sharp cut-­ off on the series in question. It is, of course, essential to this strategy that the consequent of (1) is no worse than a quandary. For reason to deny the consequent would contrapose into reason to deny its antecedent—and so would motivate the major premise for the paradox. That is the potential Achilles heel at which Rosenkranz is probing. Let ‘KX(A)’ express that A is known by methods X and consider (2)  ¬(∃x )(FeasKM (Fx & ¬Fx′)), where M denotes our actual normal range of means for assessing F-predications. Rosenkranz claims that we have undefeated empirical evidence (I will call it the Rosenkranz evidence) for (2). Now suppose (3) Methods M are comprehensive —that any predication of F which can be known at all can be known by application of methods M. (2) and (3) together presumably entail the neg­ation of the consequent of (1), and hence the negation of its antecedent, (4)  ¬(∃x )(Fx & ¬Fx′). But (4) of course engenders a Sorites paradox. So the friend of EC who acknowledges the Rosenkranz evidence must regard (3) as false. That is certainly worrying if one reads it as a commitment to the usual (classical) negation of (3)—namely, (5)  There are methods besides M for competently assessing F-predications. If that conclusion is allowed, then it appears that it is the defender of EC who is reduced to making a claim for which there is no evidence (assuming his evidence for EC did not already involve a defence of (5)). However, this reading is questionable in the present dialectical context. The original argument was that the combination of classical logic, EC, and quandary leads to claims for which there is no warrant and that, if we want a logic which is conservative with respect to warrant, then intuitionistic distinctions are required, in particular the distinction between a proposition and its

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Rosenkranz on Quandary, Vagueness, and Intuitionism  263 ­ ouble negation. The argument above, for its part, seems to show that EC, d classical logic, and the Rosenkranz evidence likewise lead to an unwarranted claim. So there is once again the option of rejection of classical logic and the enforcement of intuitionistic distinctions. If we take that option, we can rest on the intuitionistic negation of (3) (6) It is not the case that methods M are the only competent means for assessing F-predications without commitment to the dubious existential claim made by its classical counterpart. There are, however, concerns about a commitment, issuing in this way, even to (6). The position is that, in the presence of the Rosenkranz evidence, there are grounds for EC only if there are grounds for denying the comprehensiveness of existing methods for assessing F-predications. But it does not seem that, for the general run of vague predicates, we have any such grounds. What reason can anyone produce for supposing that the knowable heaps, bald men, and adolescents are not exactly those which can be recognized as such on the basis of our standard grounds for applying such concepts? However, if EC really does enforce the conditional (1), then—in the presence of grounds for (2)—we must have some such reason, or lack any sufficient reason for EC in the first place. Rosenkranz is suggesting that the situation is the latter. The situation is actually yet more awkward, once account is taken of the kind of special motivation for EC envisaged in ‘Quandary’. Consider again the case of colour and take ‘F’ as ‘red’. In support of regarding EC as good for ­colour I wrote as follows: It is a feature of the ordinary concept of colour that colours are transparent under suitable conditions of observation: that if a surface is red, it . . . will appear as such when observed under suitable conditions; mutatis mutandis if it is not red. Colour properties have essentially to do with how things visually appear and their instantiations, when they are instantiated, may always in principle be detected by our finding that they do indeed present appropriate visual appearances. So, according to our ordinary thinking about ­colour . . . EC is inescapable . . . (Wright 2001b, p. 72, this volume, Chapter 7, p. 236)

But, if that is the motive, then the case for EC for simple colour predications is  exactly a case for regarding all true such predications as recognizable by

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264  The Riddle of Vagueness methods M—a case for the relevant version of (3). So the derivation of (1) from EC—however exactly it is envisaged as going (we will come to that in a minute)—will presumably go over into one of (1)M (∃x )(Fx & ¬Fx′) → (∃x )(FeasKM (Fx & ¬Fx′)). But now the Rosenkranz evidence will sustain a contraposition, and thereby reinstate a Sorites. Therefore the following appears enforced: if we really do have the Rosenkranz evidence, then the (a priori) motivation for EC, in so far as it is typified by the thought outlined for colour, must be regarded as de­fect­ ive. And in that case—since that was the motivation I offered—the ‘Quandary’ argument for intuitionistic revisions collapses. Rosenkranz himself does not elaborate on the nature of the Rosenkranz evidence. But it would not be a good response to question whether we really do have any such evidence. It does seem clear that no one with a normal training in and understanding of the general run of vague predicates ever feels confident about drawing a line in a Sorites series—so much is indeed a datum of the problem, a basic manifestation of vagueness itself. If belief is a necessary condition of knowledge, then evidence that it is not possible to move someone to hold an opinion of the form, Fa and ¬Fa′, on the basis of reliance on our ordinary methods of assessment of F-predications is evidence that it is not possible to acquire an item of knowledge of that form on the basis of just those methods either.

II A better reaction is rather to look again at the pedigree of the conditional (1). Rosenkranz himself observes that, in order to derive (1) from the two ingredient conditionals of the relevant case of EC: Fa → FeasK(Fa) ¬Fa → FeasK(¬Fa) we need to presuppose that feasible knowledge is additive in the relevant cases, that the feasibility of knowledge that Fa and that ¬Fa′ individually implies the feasibility of knowledge of their conjunction. He points out that additivity is not exceptionless: that it fails, for example, with the conjuncts of Fitch sentences and will fail wherever feasible knowledge is disciplined by margins of error in the manner argued by Williamson and a and a′ are sufficiently

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Rosenkranz on Quandary, Vagueness, and Intuitionism  265 similar.2 He might have added that it also may fail in cases where the realization of knowledge involves significant opportunity costs. To be sure, none of those considerations seems germane to the type of case at hand. The conjunctions in question display nothing analogous to the knowledge-­limitative character of the Fitch conjunctions; the relevant margin of error principle FeasK(Fa) → Fa′ is not available in the presence of EC,3 and there do not seem to be relevant opportunity costs associated with the application of standard (casual observational) methods for assessing predications of the kind that concern us. But what Rosenkranz should also have observed is that the Rosenkranz evidence itself presents a powerful and relevant objection to additivity. If the evidence is that reliance on our actual methods for assessment of predications of F is impotent to generate conviction about the truth of a conjunction of the form Fa & ¬Fa′, then one who believes in EC for F-predications ought to suppose both that, if some such conjunction is true, then each of its conjuncts is individually knowable and that (there is no reason to think that) any such conjunction is knowable on the basis of actual methods. So there is no basis for additivity in the relevant context. Rosenkranz’s discussion leads him to the general conclusion that there is a tension, not between EC itself and the admission of quandary, but between our possession of compelling a priori warrant for EC while we simultaneously recognize that some of the statements within its scope present quandaries. However, there is no reason to accept that general conclusion unless EC does indeed impose the conditional (1)—and that we have just seen reason to doubt. The required additivity step is suspect. Yet, without conditional (1), does not my case for a distinction between the acceptability conditions of (∃x)(Fx & ¬Fx′) and those of its double negation collapse anyway? Well, not for that reason at any rate. There was never any essential need for conditional (1). Any conditional of the form (∃x )(Fx & ¬Fx′) → A for which EC does provide compelling reason and whose consequent presents a quandary will subserve the ‘Quandary’ argument. Consider for instance (1)* (∃x )(Fx & ¬Fx′) → (∃x )(FeasK(Fx ) & FeasK(¬Fx′)). 2  The basic reference is Williamson 1992a. 3  Since, in conjunction, the two principles will obviously introduce Sorites-­form paradoxes.

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266  The Riddle of Vagueness No question but that the two schematic EC conditionals: Fa → FeasK(Fa) ¬Fa → FeasK (¬Fa) entail (1)*. But, since any verifying instance, x, of the consequent of (1)* must fall in the borderline region for F, the conception of that region as consisting in cases where F-predications make for quandary implies that we are in no position to claim that the two items of knowledge demanded by the consequent—knowledge of some such instance, a, that it is F, and knowledge of its immediate successor that it is not F—are feasible. Accordingly we are in no position to claim knowledge of the consequent of (1)* itself, nor therefore, if knowledge of EC, and hence of (1)* is assumed, knowledge of its antecedent. From there, the argument can proceed as before. The reliance on additivity, involved in the derivation of (1) from EC, was an unnecessary hostage to fortune. Once again, this is no progress unless the consequent of (1)* is indeed a quandary and not something which, like the consequent of (1), we have reason to deny given additional plausible assumptions—or anyway assumptions which we cannot reasonably claim grounds to reject. Do analogues of the problems that attended (1) in this regard also afflict (1)*? The assumption that posed the problems for (1) was assumption (3), that all true predications of F may be known by reliance on our existing methods M. It seemed clear—the Rosenkranz evidence is that—no conjunction of the form Fa & ¬Fa′ may be so known. So, on the assumption of EC, an arguably unwarranted denial of (3) seemed to be enforced. But the Rosenkranz evidence makes no similar trouble for (1)*, since it concerns our inability to know conjunctions, and the consequent of (1)* precisely makes no claim about that. Evidence that would make trouble for (1)* in a similar way would, of course, be evidence that, for any x and x′, there is no possibility that Fx is known and that ¬Fx′ is known. Do we have any such evidence? According to the view of borderline cases proposed by ‘Quandary’, no one presented with a borderline case of F can know that someone who takes a suitably qualified but nevertheless committal—positive or negative—view of it does so incorrectly. Borderline cases are cases where one’s springs of opinion dry up—cases where one fails to come to a view despite enjoying every normal cognitive advantage relevant to a judgement of the kind concerned. But they are not cases where

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Rosenkranz on Quandary, Vagueness, and Intuitionism  267 one recognizes that no knowledgeable verdict is possible. 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 both polar verdicts, but to find oneself unable to come to a polar verdict. Such an inability is consistent with recognizing that other, competent judges may, tentatively, feel able to do so without betraying their competence. For any borderline pair a and a′, about which I myself am moved to no view, it may be, for all I can rule out, that to judge that Fa, or ¬Fa′, is to judge truly. And if such judgements are arrived at by careful exercise of an appropriate competence in what we think of as relevantly unimprovable circumstances, that is surely to grant that, if correct, they are knowledgeable. So viewing a case, a, as borderline, on this conception, does indeed fall short of e­ xclusion of the feasibility of know­ledge that Fa and of the feasibility of knowledge that ¬Fa′. That conclusion still leaves a gap. Suppose a and a′ are cases about which we have a ‘drying of the springs of opinion’. What has just been suggested is that we cannot, in that case, discount the idea that the verdict Fa is true nor, if it is arrived at by someone competent in the right kind of way, that it is know­ ledge­able. Likewise for the verdict ¬Fa′. But it is a further step to conclude that we cannot rule out that both those verdicts are knowledgeable—which is what regarding the consequent of (1)* as a mere quandary requires. For all that has been argued, it might yet be that a necessary condition for the know­ledge­abil­ity of the verdict Fa is precisely that there can be no knowledge of ¬Fa′. This gap, however, was implicitly closed by the penultimate point in the line of thought outlined: the claim that if someone arrives at a true judgement about an F-predication by careful exercise of the normal competences in rele­ vant­ly unimprovable circumstances, then his judgement is perforce know­ ledge­able. If this is accepted, then to allow that, while we cannot exclude the possibility that the judgement ¬Fa′ is knowledgeable, we can exclude its being so in any case where there is also a knowledgeable judgement of Fa, is implicitly to grant that, whenever ¬Fa′ holds for some a, it cannot be that Fa is true. For, if it were, someone might judge it to be so in the circumstances just granted to make that judgement knowledgeable. But then we have implicitly granted that, whenever ¬Fa′, ¬Fa holds also, and so set the Sorites running again.

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268  The Riddle of Vagueness I propose, then, that the line of thought in ‘Quandary’ which attracts Rosenkranz’s objection may be so modified, by replacing (1) by (1)*, as to draw its sting.

III My principal concern in the parts of ‘Quandary’ which deal with vagueness was to explore the possibilities for an intuitionistic treatment whose mo­tiv­ation would be as close as possible to (what I regard as) the most powerful case for Intuitionism in mathematics. Let me therefore conclude by observing— which I did not do in ‘Quandary’4—that actually the most basic case for ­suspension of classical logic where vague statements are concerned is independent of their subjection to Evidential Constraint. First, note that any sufficient case, EC-­driven or not, for agnosticism about Bivalence over the predications in a Sorites series is a reason for agnosticism about the existence of a sharp cut-­off. The point is simply that F−ness monotonically decreases in such a series. So reason to believe in the existence of an {Fx; ¬Fx′} pairing would be reason to believe that all of x’s predecessors were F and all the successors of x′ were not, and thereby to accept that Bivalence held over all the F-predications involved. Any sufficient case for agnosticism about Bivalence is therefore a sufficient case for agnosticism about the existence of cut-­offs—about the truth of the relevant instance of ‘(∃x)(Fx & ¬Fx′)’—notwithstanding the elementary and unexceptionable reductio provided by the Sorites itself of such statements’ negations. Rosenkranz does not question that the recognition that a range of statements is both subject to EC and includes quandaries must rationally undercut an adherence to Bivalence as a known principle concerning them. The thrust of his intended criticisms is against the possibility of any such recognition— against the existence of any motivation for EC in the presence of quandary. However, while—pace Rosenkranz—I stick to it that (enough, basic) vague predications may be recognized to meet these twin conditions, the most basic cause for concern about Bivalence as applied to vague statements is just a direct corollary of what it commits us to, even without EC. Critics of standard Epistemicism (for instance, Wright 1995; this volume, Chapter 6; Schiffer 1999) have generally fastened onto its perceived hostages 4 But see, however, my ‘Vagueness; a Fifth Column Approach’ (Wright 2003c, this volume, Chapter 9).

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Rosenkranz on Quandary, Vagueness, and Intuitionism  269 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, for example, is associated (in any given context of use) with an absolutely definite extension. Where is the theory that tells us what constitutes these associations? What makes it the case, for example, that my use of ‘red’ has the particular sharply bounded extension that it does? Epistemicists often go pragmatist at this point and invoke the (alleged) advantages of clas­ sic­al logic as a ground to accept that there is indeed such an extension. But semantic facts must somehow be constituted in linguistic practice and the (alleged) pragmatic advantages of classical logic provide no clue as to how the association might, even in principle, have been set up by our practice. How can we reasonably believe in the existence of something for which we have no direct evidence and no adequate conception of how its existence is possible? Timothy Williamson has tended to reply to this concern that the connections between language and the world are something of which we lack an  adequate philosophical account in general, vagueness apart—that Epistemicism ‘has not been shown to be inconsistent with anything taught by the theory of reference’ (Williamson 1996a, p. 43). That may be. But the one reasonably clear model (or type of model) we have of how the determination of the extension of 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, and every reason to doubt that we shall ever have a generally satisfactory semantics of natural language—especially a semantics of predication—which discharges Epistemicism’s debts. But let that opinion pass. The question is: can anyone at all, in our present state of information, 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 their hand on their heart and say that they know 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 in this context too. Such agnosticism is little more than proper intellectual modesty.

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9 Vagueness A Fifth Column Approach

I 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-­and-­ ready 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, 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 1  The 1975 Synthese special number on the topic was a crucial spur (On the Logic and Semantics of Vagueness, Synthese, 30/3–4).

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0010

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272  The Riddle of Vagueness 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’, ‘It neither is nor isn’t’, and so on. 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 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

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Vagueness: A Fifth Column Approach  273 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 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

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274  The Riddle of Vagueness but—so I should contend— block it in a way that acknowledges it as a reductio of the major premise and accordingly incorporates a diagnosis both of that premise’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 premises, insisting that there are sharp but unidentifiable thresholds to, for example, colours, heaps, and the territory of Morocco. What, though, is Epistemicism’s story about the plausibility of those premises? Why, according to the epistemic conception, are we tempted to suppose that, for example, if a man with merely n hairs of normal thickness, and so on, is bald, so is one with n+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 does not 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, for example, 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/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 do not apply. Thus ‘Jones is tall’ is 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.

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Vagueness: A Fifth Column Approach  275 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 (cf. Williamson 1994, p. 162). 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 in­tu­ itions 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 def­ in­ition 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 a semantics and proof theory for vague claims which treats 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 supervaluations 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 general2 is much more difficult to make sense of 2  As opposed to some special cases.

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276  The Riddle of Vagueness 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, for example, ‘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 non-­heap 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 manifestly unintuitive for the general case. Nothing in the imagery of blurred shadows helps us understand the vagueness of quantifiers, such as ‘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 ob­ject­ ive­ly 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

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Vagueness: A Fifth Column Approach  277 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 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.

II What are Borderline Cases? Let us 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.

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278  The Riddle of Vagueness 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 Foster3 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 us 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.4 Of course what is 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 φ 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 us 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 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. 3  In graduate classes in Oxford long ago.

4  Cf. Williamson (1990, p. 107) on ‘dommal’.

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Vagueness: A Fifth Column Approach  279 Third Possibility entails but is not entailed by Verdict Exclusion. Verdict Exclusion says that no verdict about such a proposition is knowledgeable5— 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  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 5  Or, if that is different, that no such verdict is warrantable.

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280  The Riddle of Vagueness 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 my having that impression a commitment to regarding any verdict as non-­knowledgeable. If it were, then, in regarding the case as borderline, I would 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 do not 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 is 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

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Vagueness: A Fifth Column Approach  281 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 their competence that they be in it. But that it is understandable and expectable that they and others get into such a predicament is not something which empowers them 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 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.

III 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

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282  The Riddle of Vagueness 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

II

III

Indeterminism Exclusive Pessimism Epistemicism Third accept Possibility Verdict accept Exclusion Bivalence not accept

IV

V

Non- exclusive Agnosticism Epistemicism (Intuitionism)

not accept not accept not accept not accept accept

accept

not accept 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 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—both of Third Possibility and of 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

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Vagueness: A Fifth Column Approach  283 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 Exclude Middle but refusal of the law itself, is exactly the Mathematical Intuitionist trademark.6 But what motivation is there for it in the present setting? In the concluding section I will outline three possible lines. But before that, let us review how the Column V position may address at least two of the challenges that confront any satisfactory treatment of vagueness.

IV The Misconceived Conditional and the Sorites Any satisfactory treatment of vagueness must, at a minimum, (i) say what is wrong with the following conditional (∃x )(Fx & ¬Fx′) → F is not vague (the misconceived conditional7) and (ii) solve the Sorites paradox. The tasks are of course interrelated. The classic formulation of the Sorites presents an inconsistent triad {F 0, ¬Fn, (∀x )(Fx → Fx ′)} ⇒ Λ 6  The possible utility of intuitionistic distinctions in a philosophical treatment of vagueness was first proposed by Putnam (1983a, pp. 271–86). I myself was among the original critics of his proposal (Read and Wright 1985, this volume, Chapter 3). It was only much later that I realized how the principal objections of that note might be answered. 7  Prominent in Timothy Chambers’ objections to Putnam (Chambers 1998).

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284  The Riddle of Vagueness over a suitably ordered finite series. Naturally this is only a paradox because all the premises seem well motivated. But there is a tendency for the motivations for the major premise   (∀x )(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 = ‘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 premise 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 premise. And how that may be done is exactly what the misconceived conditional 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:    {F 0, ¬Fn, ¬(∃x )(Fx & ¬Fx ′)} ⇒ Λ —that the inconsistency remains even after the major premise 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.8 Both standard Epistemicism and its Column IV relaxation are in no difficulty in doing so. The misconceived conditional will fail because vagueness 8  Or at least with its contrapositive.

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Vagueness: A Fifth Column Approach  285 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.9 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 their position precludes them 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, they have 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, 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 would 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 they 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— open-­ minded—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 they know is false but an antecedent about which they 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 9  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 ¬(∃x)(Fx & ¬Fx′) 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 neg­ ation 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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286  The Riddle of Vagueness treating the Sorites reasoning as just what it appears to be, a demonstration that the major premise is false. That is an unsatisfactory proposal only if there is strong independent motivation to regard that premise as true. But that motivation was the thought that the very vagueness of F should suffice for the truth of the major premise, taken in the form: ¬(∃x)(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 (∃x) (Fx & ¬Fx′). The Sorites reasoning itself enforces the denial of the major premise—that is really only common sense, shared by virtually all responses to the problem. A solution has to consist in explaining why that premise 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, and so on—which broadly are about us and which entail nothing about the actual distribution of instances and 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 could not 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 premise for the No Sharp Boundaries paradox is poorly motivated by the phenomenon, and that the para­ dox may be taken as a simple reductio of that premise. 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 ¬¬(∃x )(Fx & ¬Fx ′)

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Vagueness: A Fifth Column Approach  287 —established by the reductio—to (∃x )(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, once again, (∃x)(Fx & ¬Fx′), taken as the statement of the existence of a sharp cut-­off,10 cannot be regarded as known to hold in the series in question unless Bivalence is. What is that case?

V 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: (EC) P → it is feasible to know that P.11 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

10  — rather than, e.g., read supervaluationally. 11  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.

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288  The Riddle of Vagueness 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 (see Wright 2001b, this volume, Chapter  7). A statement P presents a quandary for a thinker T just when the following conditions are met: (i) T does not know whether or not P (ii) T does not know any way of knowing whether or not P (iii) T does not know that there is any way of knowing whether or not P (iv) 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.

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, and so on, 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

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Vagueness: A Fifth Column Approach  289 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 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 not-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

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290  The Riddle of Vagueness 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 Epistemicism (for instance, Wright 1995, this volume, Chapter 6; Schiffer 1999) 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, for example, 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?) 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’ (Williamson 1996a, p. 43). 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, and so on, 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

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Vagueness: A Fifth Column Approach  291 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 their hand on their heart and say that they know 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.

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.

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10 Vagueness-­Related Partial Belief and the Constitution of Borderline Cases I For all post-­1970s effort expended on the topic, the most central and im­port­ ant question about vagueness—what it is: what, specifically, something’s being a borderline case of a vague expression consists in—has seldom been tackled with the theoretical explicitness necessary if issues expectably downstream of it, like the nature of valid inference among vague statements, or the Sorites paradox, are to receive a properly motivated treatment.1 The great interest of Chapter V of The Things We Mean (Schiffer 2003) (together with Schiffer 1998; 2000a,b) is that it points the way towards a new kind of approach, according to which vagueness is constitutively a psychological phenomenon, grounded in the characteristic propositional attitudes of practitioners of vague discourse. It is uncontroversial that a vague expression is one whose presence in a sentence contributes towards there being, at least in principle, situations which present borderline cases of its truth; for Schiffer, it is the status of such a situation as a borderline case that is grounded in the characteristic psychology of thinkers who appraise the sentence in that situation. Sorites-­prone concepts—red, tall, bald, child—are typically associated with a proper comparative: redder than, taller than, balder than, less mature than. Presented with a Sorites series initiated by paradigms of these concepts and well ordered under the associated comparative, competent practitioners will normally experience gradually decreasing confidence in the predications of the concept concerned, culminating perhaps in a case or cases where they are as drawn to a verdict as to its negation, and then a gradual reinstatement of confidence in the opposite verdicts, culminating in a resolute willingness to deny predications of the concept concerned. This claim may involve certain simplifications of the actual sociology of judgement in borderline cases, but the basic phenomenon: that borderline cases of the kinds of vague concepts 1  Epistemicism is not open to this complaint.

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0011

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294  The Riddle of Vagueness listed are essentially manifested not in consensual silence about them nor in agreement that there is ‘no fact of the matter’, but in weakening of opinion and then strengthening of contrary opinion, seems solid. This observation— that vagueness is characteristically associated with a kind of partial belief—is the springboard of Schiffer’s whole approach.

II Partial belief—believing in a way that involves less than full conviction—is, of course, a pervasive phenomenon, and nothing especially to do with vagueness. So, initially at least, the prospects of its yielding a constitutive account of borderline cases might seem remote. What improves them, in Schiffer’s view, is that—with a qualification I shall come on to—the kind of partial belief associated with vagueness—what Schiffer calls vagueness-­related partial belief (VPB, or v-­belief )—is sui generis. Schiffer suggests several respects in which v-­belief contrasts with the traditional, well-­studied kind of partial belief. The latter—what he calls standard partial belief (SPB, or s-­belief )—is the kind of attitude of which examples are ‘your believing to some degree or other—that is, more or less firmly—that you left your glasses in your office, that it will rain tonight, or that the Atlantic City Flounders will win the Super bowl’ (Schiffer 2003, p. 200). Some of the features advanced by Schiffer as characteristic of s-­belief and lacking in the case of v-­belief are these: (i) That it would be open to us to s-­believe, even if our language were perfectly precise. (ii) That s-­belief is a measure of one’s uncertainty about the truth of some matter, and hence an expression of a degree of ignorance. (iii) That standard partial beliefs generate likelihood beliefs: thus if I s-­believe to some quite high degree that I have a spare pair of glasses in my office, then I will believe (absolutely) that it is pretty likely that I have a spare pair of glasses in my office. (iv) That in standard cases where one s-­believes some proposition to some degree intermediate between certainty and certainty-­that-­not, one would typically regard oneself as not being in the best possible pos­ ition to take a view on the truth value of the proposition in question, even if one has no doubts about the quality of the (limited) evidence that one possesses.

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Vagueness-Related Partial Belief  295 Standard partial beliefs are, of course, characteristically treated as sub­ject­ ive numerical probabilities, and are supposed to be subject, where rational, to the laws of the classical probability calculus. But, even if one is, for whatever reason, dubious about their assignment of precise quantitative measure, the considerations marshalled by Schiffer do seem to open up a whole plethora of disanalogies with the kind of partial belief that seems to be operative over vague judgements. When I find myself less confident than I was before, as a march-­past of successively smaller soldiers goes by, that the men now before me are tall, it is not that I think there is some further fact about their tallness, or lack of it, for which I am in context restricted to imperfect evidence; and the idea that the men at this stage are less likely to be tall than their predecessors seems like some kind of conceptual solecism,—in stark contrast, for example, to the idea that they are less likely to be more than 5' 10''. Relatedly, I may regard myself as being in a relevantly unimprovable position for judging whether they are tall or not, and my uncertainty seems to have nothing at all to do with the notion that my evidence, although unimprovable, is less than would be needed to settle the matter. It is clear that Schiffer is on to something. However, the foregoing considerations do not provide the kind of functional characterization of v-­belief that its load-­bearing role in Schiffer’s treatment demands. But he offers a crucial additional consideration. Consider the example of the plucking of the hairs of Tom Cruise (Schiffer 2003, pp. 202 ff.). Suppose that, each time a single hair is plucked, Tom also gains a gram of body fat—or whatever quantity is necessary in order for his passage from slimness to corpulence roughly to match his rate of progress from a full head of hair to baldness. Let the process have reached a borderline stage for each transition and consider the propositions (1)  Tom is fat (2)  Tom is bald (3)  Tom weighs more than 220 lbs (4)  Tom has fewer than 7,000 hairs on his scalp Imagine you have no means of weighing Tom and are not in a position to count his hairs. You are accordingly uncertain about all four propositions. What will be your attitude to the two conjunctions: (5)  Tom is fat and Tom is bald (6) Tom weighs more than 220 lbs and Tom has fewer than 7,000 hairs on his scalp?

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296  The Riddle of Vagueness Well, suppose you reckon your chances of correctly accepting either of the precise claims—the conjuncts of (6)—no better than 0.6. Then, assuming their independence, you ought to reckon that your chances of being right about the conjunction are no better than 0.36—at least from the standpoint of the classical probability calculus. So (if pressed) you should be inclined to accept either (3) or (4) individually but to reject (6). And even prescinding from precise probabilities, the intuitive thought is compelling—or should be—that, since your merely visual evidence for (3) and (4) individually provides only a moderate, readily improved-­upon grounding for either claim, you will be compounding your risk in any view you take about their conjunction. You may incline to the view that Tom now weighs more than 220 lbs, and incline to the view that Tom has fewer than 7,000 hairs; but, if you are rational, and these opinions are each marginal, you will be much more squeamish when invited to endorse (6). However, no analogue of this point would appear to apply to the vague conjunction (5). Here, it seems, there is no rational requirement that the credences placed in the individual conjuncts should ‘multiply down’. A thinker who regards each of the conjuncts (1) and (2) as borderline, but is inclined, in a suitably qualified and hesitant way, to accept them both, can be expected to take the same view of the conjunction—that it is, on balance, just about acceptable—and, intuitively, is open to no complaint of irrationality in doing so. So the kind of partial belief characteristic of vagueness would seem to be distinguished from standard partial belief—classical credence—by failing to conform to the laws of classical probability. Schiffer’s own suggestion is that vagueness-­related partial belief is structured in accordance with the patterns exhibited by the Łukasiewicz matrices for (continuum) many-­valued logics, with conjunction and disjunction, for example, taking respectively the min­imum and maximum values of their components, rather than being determined as product and sum.

III Note that, if Schiffer’s proposals about the attitudinal psychology of vague judgement are correct, even in outline, a new and serious objection to Epis­ temicism emerges. If vagueness is as classical Epistemicism conceives it, then the difference between the two precise claims, (3) and (4), and the two vague claims, (1) and (2), in the double Sorites considered above is only that in the case of the latter we are ignorant of the principles that determine the sharply

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Vagueness-Related Partial Belief  297 bounded extensions in which the epistemicist believes—ignorant of the nature of the sharply extended properties that ‘bald’ and ‘fat’ respectively denote. In all four cases, therefore, we are making judgements in a state of uncertainty, induced by insufficiency of evidence, about determinate matters. So the kind of partial belief induced as we enter the range of hard cases should correspondingly be the same: it should be standard partial belief throughout. Since, if Schiffer is right, it is not, Epistemicism would appear confounded by the attitudinal psychological facts.

IV All this, of course, presupposes that the notion of VPB is in good standing, and that it is indeed a sufficiently natural and pervasive aspect of vague judgement to promise a constitutive explanation of the phenomenon. How does Schiffer foresee such an explanation as proceeding? When a vagueness-­related partial belief in a proposition P is formed under conditions which are epistemically ideal for the purpose of appraising P, Schiffer terms the v-­belief in question a VPB*. His strategy is then to seek a biconditional with x is a borderline case of being F on one side and some appropriate clause embedding the notion of VPB* on the other.2 Apparently he would have liked to propose something like this: x is a borderline case of being F if and only if someone could v*-believe that x is F, but he considers that although the ‘only if ’ direction is acceptable, the ­converse—right to left—conditional fails. That someone could v*-believe that x is F is not sufficient, in his view, for x’s being a borderline case of F—not if borderline status is in turn understood as something peculiar to the kind of vague concepts with which he is concerned (Schiffer 2003, pp. 208–9). What, he thinks, the possibility of v*-belief that P does signal is merely that P is afflicted

2  Supposing a true such biconditional can be found, the task will still remain of justifying the claim that it can be seen as constitutive—as giving an account of that wherein borderline-­case status consists. But the first job is to find a formulation that is true.

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298  The Riddle of Vagueness with some kind of indeterminacy. The problem is that being a borderline case of a vague concept of the intended kind is only one kind of in­de­ter­min­acy. So, he concludes, the condition needs strengthening. That Schiffer believes that it is a more general notion of indeterminacy, rather than vagueness proper, that is associated with the possibility of v*-belief would seem to be clear from his endorsement of formulation (C) (Schiffer 2003, p. 209): (C)  P is indeterminate if and only if someone could v*-believe that P. But the matter is clouded by his remarks about ‘schmadulthood’ (Schiffer 2003, p. 212), where ‘the concept of a schmadult is exhausted by these two conditions: anyone who has reached his or her 2nd birthday is a schmadult, and anyone who has not yet reached his or her 17th birthday is not a schmadult’. Here, Schiffer suggests, if considering someone we know to be 19 years of age, ‘we might well not v-­believe that she is a schmadult but rather s-­believe that it is not true that she is a schmadult and not true that she isn’t a schmadult’. Such a case of mere incomplete definition, and the indeterminacy it generates, thus apparently contrasts with the target case of vagueness proper precisely because it may induce a response of s-belief in the existence of a gap, rather than v-­belief in the appropriate proposition about schmadulthood and its negation. This is puzzling: if the kind of indeterminacy spawned by incomplete definition is not characteristically associated with v*-belief, what is the motive for thinking that v*-belief is associated with other forms of in­de­ter­ min­acy besides vagueness proper? Schiffer, in my opinion, is right to insist on a contrast between vagueness proper and semantic indecision generated by incomplete definition. But I don’t think he is right, as far as his own explanations go, to dissociate the latter from VPB. To see this, take another concept of the same structure—say ‘schmall’, where it is given that an individual is not schmall if they are more than six feet tall, that they are schmall if they stand less than five foot six— and suppose that Tim, our 19 year old, stands five feet ten. If one has, in these circumstances, any kind of partial belief in the propositions: ‘Tim is a schmadult’ and ‘Tim is schmall’, it seems clear that it is a kind of partial belief that does not ‘multiply down’. After all, one’s level of confidence in the conjunction ‘Tim is a schmadult and Tim is schmall’ will presumably be no less than in each conjunct. This, then, seems to be an example where indeterminacy of a kind contrasting with vagueness proper is associated with VPB. That somebody might be inclined simultaneously to s-­believe that there is ‘no fact of the

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Vagueness-Related Partial Belief  299 matter’ in such cases—or, as Schiffer expresses it, to s-­believe that it is not true that P and not true that not-P—does not seem to be in any tension with this. Indeed there seems no reason why the same should not happen with vagueness proper: confronted with a ‘plumb’ borderline case of red and orange, there seems no clear reason why a subject might not v-­believe ‘x is red’ and ‘x is orange’ to equal degrees, while at the same time s-­believing that there is no ‘fact of the matter’ about which colour it is. It may well be that the latter belief would be philosophically unfortunate, or confused; but a propensity to v-­belief is not, presumably, an inoculation against philosophical confusion. It seems to me therefore that, Schiffer’s remarks about schmadult notwithstanding, VPB and VPB* are characteristic of a wider class of forms of in­de­ ter­min­acy. Whether or not (C) is correct as it stands, a true biconditional with ‘x is a borderline case of being F’ on one side and some clause suitably exploiting the notion of VPB* on the other, can be nothing so simple.

V Schiffer’s proposal is that vagueness proper is marked by v-­belief which is F-­concept driven, where a v-­belief—more accurately, VPB*—is, he defines, F-concept driven if one is in ideal circumstances for judging x to be F and one’s concept of being F precludes one from s-­believing to any positive degree either that x is F or that x is not F and determines one to v-­believe to some positive degree that x is F. On this basis, Schiffer proposes: (E)  x is a borderline case of being F iff someone could have an F-concept driven VPB* that x is F. One immediate concern about this is that it is not clear why v-­belief in circumstances of indeterminacy generated by partial definition would not count as F-concept driven by the terms of Schiffer’s formulation—for it does not seem wrong, for all that has so far been explained, to say that my concept of being a schmadult precludes me from s-­believing to any positive degree that a 19 year old is a schmadult or that they are not a schmadult, and does determine me to v-­believe to some positive degree that they are a schmadult. But there seems to be a more fundamental problem about the direction which Schiffer has taken. Part of the content of (E) is that a borderline case of a property F is something for which it is possible to have a VPB* in the prop­ os­ition that it is F. It is instructive to consider a little further why such an

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300  The Riddle of Vagueness account would be insufficient—why there is the need for an additional condition which Schiffer’s introduction of F-concept drivenness is meant to address. One reason, of course, is whatever cause there is to think that other kinds of indeterminacy besides vagueness are associated with v-­belief. But there is a consideration that would apply even if there were no such kinds. It is that v-­belief, if it is characterized purely by points of contrast with s-­belief that pertain just to its out-­rules, so to say—to its characteristic mani­fest­ ations—may always, as a matter of metaphysical possibility, occur under any kind of circumstance whatever—so under circumstances quite unrelated to vagueness. Someone might, for example, be physically so constituted as to form VPBs in response to successive propositions of the form ‘Tom now has fewer than fifteen thousand hairs on his scalp’ in the plucking example. Or I might form a VPB in relation to the proposition that it will snow tomorrow, purely as a result of a neurological disorder. So any constitutive account needs to say something more. And the more that needs to be said must somehow place controls on the provenance—the in-­rules—of v-­belief. That is exactly what F-concept drivenness seems to be intended to do. Witness Schiffer’s talk of the concept ‘precluding’ some and ‘determining’ others among possible attitudinal responses. However, if that is right, then the approach confronts a dilemma: (Horn 1) If one’s concept of being F—the way one understands the associated predicate—really can constrain one’s responses to putative instances in the way involved in the idea of F-concept drivenness, precluding s-­belief and determining v-­belief, then it becomes difficult to see how the leading aspect of Schiffer’s constitutive proposal, that vagueness is a psychological phenomenon and is somehow constituted in our propensities to v-­belief, can be upheld. For now it seems that those propensities in turn are driven by ulterior aspects of our concepts, that is, by ulterior aspects of the way we understand the expressions in question. If the very same psychological state of partial belief may be generated in different ways, and is a phenomenon of vagueness only in some cases, when appropriately ‘driven’ by the concept concerned, then—or so an opponent of the psychological account may insist—we should be looking to what it is about the concept in question that precludes s-­belief and determines v-­belief if we want to say something constitutive about what borderline case vagueness is. V-­belief will be an effect of vagueness, and so unsuited to provide the basis for a constitutive account. (Horn 2) If, on the other hand, one’s concept of being F cannot really enforce one’s attitudinal response in the way suggested by the idea of F-concept

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Vagueness-Related Partial Belief  301 drivenness, then v-­belief—even when it amounts to VPB*, that is, when it is formed under conditions of ideal understanding and information—still cannot play the constitutive role that Schiffer wants. It cannot do so not so much because it may naturally and normally also occur in cases of different kinds of indeterminacy, but because it may, as a sui generis psychological phenomenon, in principle occur in any kind of circumstances, and in connection with any kind of predication.

The dilemma, in brief, is that, if the vagueness of the relevant concept is conceived as leading the response of v-­belief, as the terminology of ‘F-concept drivenness’ suggests, then it cannot be constituted by it. But, if v-­belief, even when ‘F-concept driven’, is in no real sense led by the character of the predicated concept, then it may in principle occur—as a mere metaphysical possibility— in association with any kind of judgement whatever, and so is unfitted to underwrite a necessarily true biconditional of the kind needed for the sort of account Schiffer seeks. This dilemma should certainly not be fatal to all prospect of a constitutive account of vagueness in terms of attitudinal states. But it needs a clear response.

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11 ‘Wang’s Paradox’ I Introduction There is now a widespread accord among philosophers that the vagueness of natural language gives rise to some particularly deep and perplexing problems and paradoxes. It was not always so. For most of the first century of analytical philosophy, vagueness was generally regarded as a marginal, slightly irritating phenomenon—receiving some attention, to be sure, in parts of the Philosophical Investigations (Wittgenstein 1968) and in the amateur linguistics enjoyed by philosophers in Oxford in the 1950s, but best idealized away in any serious theoretical treatment of meaning, understanding, and valid inference. Frege, as is well known, had come to be thoroughly mistrustful of vagueness, supposing that a language fit for the purpose of articulating scientific and mathematical knowledge would have to be purified of it. Later trends in philosophical logic and semantics followed his lead, not indeed in setting about the (futile) task of expurgating vagueness from natural language but by  largely restricting theoretical attention to artificial languages in whose workings vagueness was assigned no role. During the 1970s this broadly Fregean disdain for vagueness was completely turned about. The thirty years since have seen a huge upsurge in interest in the topic and publications about it, most of them by philosophers with not one iota of sympathy with the approach of ‘Ordinary Language Philosophy’ or the seemingly haphazard and anti-­theoretical remarks of Wittgenstein. The reasons for the sea change are no doubt complex, but my own belief is that a crucial im­petus was provided by a single publication: the 1975 special number of Synthese1 in which a number of subsequently influential papers were published for the first time but in which the single most important contribution—the paper one would recommend to a philosopher who was only ever going to read one essay on the topic—was Michael Dummett’s ‘Wang’s Paradox’ (Dummett 1975). 1  On the Logic and Semantics of Vagueness, Synthese, 30/3–4.

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0012

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304  The Riddle of Vagueness Dummett’s paper was actually written some five years earlier and had already had a significant degree of circulation.2 It focuses only partially on vagueness, giving as much or more attention to strict finitism as a rival to the intuitionist philosophy of mathematics—arguably, indeed, as the proper lo­gic­al conclusion of the intuitionistic tendency—and to the nature of observational language. The impact of the paper was in no small measure due to these simultaneous concerns: to the connections it made among topics— vagueness, observationality, and finitism in the philosophy of mathematics— which had received no systematic concerted discussion before. But it also contained a strikingly clear display, unmatched in any previous discussion of which I am aware, of the simple essential architecture of one form of the Sorites paradox. The form of the paradox in question involves an observational3 predicate, F, and a finite series of items connecting a first F-element with a final non-F element, but with each element relevantly indistinguishable (at least by unaided observation) from those immediately adjacent to it. Such a series is possible, of course, because and only because observational indistinguishability is not a transitive relation. Since the thought is, at first blush, utterly compelling that an observational predicate cannot discriminate between observationally indistinguishable items, it seems we have to accept that anything in the series that is F is adjacent only to things which are also F (call this principle the major premise). And that, on the stated assumptions, is enough for paradox. I think some foggy notion had prevailed earlier that the anti­ nomy was somehow due to applying to vague expressions principles of ­reasoning appropriate only to precise ones. Dummett seems to have been the first explicitly to register simultaneously both the utterly plausible character of the major premises in a very wide range of examples and the utterly basic character of the logic required: if vague expressions are not fit for reasoning involving merely iterated applications of modus ponens and universal instantiation, it is hard to see how they can be fit for reasoning at all. I do not think that the depth of the crisis for common sense which this paradox involves had really been properly appreciated before Dummett’s discussion. Dummett’s own reaction was dramatic and is worth quoting in some detail. What the paradox should teach us, he writes, is that ‘the use of vague 2  It was the single most important influence on my own first study of the topic, ‘On the Coherence of Vague Predicates’ (Wright 1975, this volume, Chapter 1), and also I believe on Kit Fine’s proto­typ­ ic­al supervaluational treatment of the semantics of vagueness, ‘Vagueness, Truth and Logic’ (Fine 1975), both of which were first published in the same number of Synthese. 3  Dummett (1975, p. 320) characterizes an observational predicate as one ‘whose application can be decided merely by the employment of our sense organs’.

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‘Wang’s Paradox’  305 predicates—at least when the source of the vagueness is the non-­transitivity of a relation of non-­ discriminable difference—is intrinsically incoherent’ (Dummett 1975, p. 319; emphasis added). He then elaborates this conclusion in four subsidiary claims: (1)  Where non-­ discriminable difference is non-­ transitive, observational predicates are necessarily vague (2)  Moreover, in this case, the use of such predicates is intrinsically inconsistent (3) Wang’s paradox merely reflects this inconsistency. What is in error is not the principles of reasoning involved, nor, as on our earlier diagnosis, the [major premise]. The [major premise] is correct, according to the rules of use governing vague predicates such as ‘small’; but these rules are themselves inconsistent, and hence the paradox. Our earlier— [proto-­supervaluationist]—model for the logic of vague expressions thus becomes useless: there can be no coherent such logic. (Dummett 1975, pp. 319–20)

The fourth subsidiary conclusion, in keeping with (2), then dismisses as correspondingly incoherent the conception of mathematical totalities as the extensions of vague predicates advanced by strict finitism. On first encounter, Dummett’s principal conclusion, that vagueness infects natural language with inconsistency, seems desperate. And there would be, to be sure, more than a suspicion of non sequitur in the transition from an argument that observational expressions have to be both vague and governed by inconsistent rules to the conclusion that vague expressions per se are governed by inconsistent rules. Observationality, understood in a way that suffices for the major premise, entails both vagueness—an observational predicate will fail to draw a line between indistinguishables—and inconsistency, at least whenever Fs can be ancestrally linked with non-Fs via a chain of indistinguishable pairs. That suffices for the conclusion that vague predicates are one and all governed by inconsistent rules only if all vague predicates are observational. And that is not true—recall the strict finitist’s predicates of practical intellectual possibility—‘intelligible’ as applied to numerals, ‘surveyable’ as applied to proofs, and so on—and indeed ‘small’ as applied to numbers. Moreover, even in observational cases, it is apparently the observationality itself that directly generates the problem, rather than the lack of sharp boundaries that it enjoins. Dummett, that is to say, has

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306  The Riddle of Vagueness described a paradox of observationality and—it is tempting to say—its solution must consist in an improved understanding of what an expression’s possessing observational content involves. But he has made no case for saying that vagueness is in­trin­sic­al­ly paradoxical—which is just as well when one considers that, whether or not there are any purely observational expressions in the sense the paradox exploits, vagueness is the norm for expressions of natural language. However, a little reflection shows that the concern does indeed ramify across vague expressions as a class. It is constitutive of an expression’s being vague, surely—or so one might think—that it should fail to draw a sharp boundary in a suitable Sorites series. Consider a column of soldiers marching past their commanding officer, ranging from five feet six to six feet six inches in height and so lined up that each soldier is marginally shorter than the soldier who immediately succeeds them. The precision of the predicate ‘is more than six feet tall’ consists in the fact that, no matter how small the differences in height between one soldier and the next, there is certain to be a first soldier to which it applies. Correspondingly, since vagueness is the complement of precision, the vagueness of ‘short’ should consist in the fact that, on the contrary, if the differences in height are sufficiently marginal, there need be no sharp bound on the short soldiers in the march-­past: no last soldier who is short followed by a first soldier who is not. But, if there is no such boundary, then whenever a soldier is not short, they cannot be immediately preceded by a short soldier. Since all the soldiers who are not short come relatively late in the march-­past and hence do have predecessors, the latter likewise cannot be short. Lack of sharp boundaries as such thus does seem to imply paradox. To  say that F lacks sharp boundaries in a series of the germane kind is to say,  it seems, that there is no element, x, which is F but whose immediate ­successor, x′, is not. That is a claim of the form Ø($x )(Fx & ØFx ¢) and is accordingly classically equivalent to the major premise ("x )(Fx ® Fx ¢) for exactly the kind of Sorites that Dummett focused on. In brief, in the presence of classical logic, vagueness apparently consists in the holding of the major premise for the Dummettian Sorites. For a predicate to lack sharp boundaries does indeed imply, in that setting, that it is subject to inconsistent rules of application.

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‘Wang’s Paradox’  307 Worse, this form of the Sorites paradox—the No Sharp Boundaries paradox— survives even if we drop the classical logical setting. Where k ancestrally succeeds 0, the logic required for eliciting paradox from the trio {F0, Fk, ¬(∃x) (Fx & ¬Fx′)} needs nothing to allow conversion between the quan­ti­fiers but consists merely in the standard introduction rules for conjunction, for neg­ ation (that is, reductio ad absurdum), and for ‵∃′. These principles are intuitively no less constitutive of the content of the constants they govern than are universal instantiation and modus ponens. So there is a powerful-­seeming line of argument that appears to drive us towards Dummett’s—as it appeared initially, overstated—conclusion. Vagueness per se does appear to infect natural language with inconsistency. David Pears once remarked that the characteristic effect of a Dummett intervention in a philosophical conversation was as if to turn on a light which the others had overlooked in a gloomy room. Sometimes, of course, what better lighting shows up is not the solution to a problem but its real contours. That was the kind of illumination shed by ‘Wang’s Paradox’. My own thinking about vagueness and finitism and the associated cluster of issues whose connections Michael’s paper displayed had benefited from discussions with him going back almost a decade before its publication to when I was a Ph.D. student. My hope is that, in this distinguished volume to debate and celebrate his wonderful contributions to modern philosophy, he will enjoy the spectacle of my still wrestling, almost forty years on, with the same conundrum.

II Vagueness as Semantic Incompleteness Frege’s disparaging view of vagueness is seldom explicitly argued for in his writings, but the little he says does indeed suggest he thought that the phenomenon threatens the stability of basic logic. However, he does not cite the Sorites paradox in support of this complaint. Frege takes it that, if a term is vague, that will be tantamount to its being only partially defined—so that it will only be of things of a certain kind that it will make sense to say that it either applies to them or that it does not. But that will give rise to failures not just of Excluded Middle, but of other laws too—contraposition, for example. Everything to which the predicate applies will be a thing of the presupposed kind; but it will not be correct to affirm, conversely, that everything not of that kind is something to which the predicate does not apply—since the range of

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308  The Riddle of Vagueness cases of which the predicate may significantly be denied is likewise restricted to the kind of thing in question.4 The most interesting thing about this line of thought is not its conclusion, but its premise: the equation of vagueness with partial definition. To fix ideas, consider an artificial example of Tim Williamson’s: the predicate ‘dommal’, whose satisfaction conditions are stipulated as met by a creature which is a dog, and failed by a creature which is not a mammal (Williamson 1990, p. 107). Nothing else is said by way of determination of its meaning, so the effect is that it is undefined whether ‘dommal’ applies to mammals other than canines. The conception of vagueness in ordinary language to which Frege gives one of the earliest expressions is in effect exactly that it is in general the naturally occurring counterpart of the artificially generated indeterminacy of ‘dommal’. Our training gives us rules for the application of ‘bald’—roughly, looking sufficiently like certain paradigms—rules for applying ‘not bald’— again, looking sufficiently like certain paradigms—and these rules once again fail between them to cater for all cases. The exceptions—the borderline cases—are those for which we lack any sufficient instruction. They stand to ‘bald’ essentially as non-­canine mammals stand to ‘dommal’. This idea conditioned virtually all mainstream work on vagueness until quite recently. It has been so widely accepted as to be either unnoticed or received as a datum of the problem. So conceived, vagueness is a matter of semantic incompleteness. A vague expression is one for which we have mastered rules for assenting to its application and rules for denying it which between them leave space for a gap—a range of cases where the rules simply do not give us any instruction what to do. I do not know whether this way of thinking about vagueness implicitly originated with Frege, or whether it is much older. In any case it is, un­deni­ ably, extremely natural.5 To be sure, not all writers about vagueness nowadays think of it as a semantic phenomenon at all, but, among the majority who still do, the Fregean conception is entrenched. It is, for example, a presupposition of the whole idea that vague expressions allow of a variety of alternative but 4  I take this to be the argument of Grundgesetze (Frege 1893, vol. I, §65). Frege does mention ‘the Heap’ itself at Begriffschrift (Frege 1879, §27), in the context of his definition of what it is for a property to be hereditary in a series, but remarks on no threat to logic, suggesting merely that, since ‘heap’ is not everywhere sharply defined, we may regard the major premise as indeterminate. (That is no option of course if we take it that the minor premise, that 0 grains of sand cannot constitute a heap, and the conclusion, that—say—200,000 grains of sand cannot constitute a heap, are, respectively, true and false. An indeterminate statement cannot be inconsistent with the facts! Even Frege, it seems, underestimated the Sorites.) 5  Thus Kit Fine (1975, p. 265): ‘I take [vagueness] to be a semantic notion. Very roughly, vagueness is deficiency in meaning.’

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‘Wang’s Paradox’  309 admissible sharpenings, whose effect will be to include or exclude certain items from their range of application whose status was previously in­de­ter­ min­ate. This idea in turn, of course, is presupposed by the widely accepted supervaluational approaches to the semantics and logic of vague discourse. Natural as it may be, however, the Fregean conception now seems to me to be almost certainly wrong. At the least, it is open to serious objection on two major counts. First, it is a very poor predictor of our actual linguistic practices. It gives the wrong prediction about our responses to—and responses to  responses to—borderline cases of standard Sorites-­prone predicates. Someone who has mastered ‘dommal’ will know better than to apply it, or its contrary, to non-­canine mammals. Asked if a cat is a ‘dommal’, they will, or should, say that they are not empowered to judge—for all there is to go on is a sufficient condition for being a ‘dommal’ and a necessary condition for being a ‘dommal’, and cats neither pass the first nor fail the second. By contrast, what we find in borderline cases of the distinction between, say, people who are bald and people who are not, it is exactly not a general recognition that there is no competent verdict to return but rather a phenomenon—spreading both among the opinions of normally competent judges and, across time, among the opinions of a single competent judge—of weak but conflicting opinions, unstable opinions, and—between different judges—agreement to differ. It is true that sometimes a competent judge may simply be unable to come to a view, but it is not a necessary characteristic of the borderline-­case region that it comprises just—or even any—cases where competent judges agree in failing to come to a view; and any case about which a competent judge fails to come to a view may, without compromising their competence, provoke a (weak) positive or negative response from them on another occasion. Moreover, failure to come to a view is not the same as the judgement that there is no competent view to take—and it is the latter that is appropriately made of a cat by someone competent with the use of ‘dommal’. These considerations are radically at odds with what the Fregean conception would lead one to expect. If someone takes the view that some particular cat is a dommal, then, ceteris paribus, they show that they have misunderstood the explanation of the word. If someone understands the explanation properly, they will not return a verdict about a cat. In contrast, our responses to those who do return verdicts in the borderline area of ‘bald’ is that, provided those verdicts are suitably sensitive and qualified, it is permissible so to do. We are liberal about judgements in borderline cases. Our thought is not that they are cases about which one ought to have no view but rather merely that they are cases about which it is probably pointless to try to work through

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310  The Riddle of Vagueness differences of opinion. The psychologist testing the responses of a variety of, by normal criteria, competent subjects down a Sorites series would expect divergences in the borderline area—indeed, that is what the borderline area is: an area of expectable and admissible divergence. If the ‘dommal’ model were correct, the expectation would be of consensual silence. The second major area of difficulty for the Fregean conception concerns one of the most arresting and disconcerting features associated with the kind of vagueness that interests us, that of so-­called higher-­order vagueness. There are various ways of eliciting the impression that the phenomenon is real and needs to be reckoned with. One line of thought is outlined by Dummett (1959, repr. 1978, p. 182): if I was to introduce a new word—say ‘sparsey’—to apply just to borderline bald people, we would find that the boundary between the bald people and the sparsey people was itself vague. Another line of thought is to observe that, in a typical Sorites series, there will be no de­ter­ min­ate first case of which we are content to judge that doing something other than returning the initial positive verdict is appropriate. How is this to be explained under the aegis of Fregean conception? According to the Fregean conception, borderline cases are cases in which we have not provided for a verdict—cases that we failed to cover by the relevant semantic rules; so the remaining cases are ones for which we have so provided and a negative or positive version is appropriate. How can this distinction in turn be one for the drawing of which we could somehow have made insufficient provision? No doubt it is up to us what provision we have made—Williamson might, for  example, have provided even less for ‘dommal’, restricting the sufficient condition to Corgis. But, whatever provision is made, it should not then need a further provision to settle which cases it does and does not cover re­spect­ ive­ly. Giving just the provision that he did, Williamson thereby settled that the borderline cases of ‘dommal’ comprise all non-­canine mammals. He didn’t merely settle that some kinds of case are to be borderline for ‘dommal’, leaving it open whether others are borderline or are cases in which a determinate verdict is mandated. That matter was determined for him by the nature of his omission. It did not need further determination by him—indeed, it was not for him to stipulate at all—how far the omission extends. Of course, a defender of the Fregean conception has possible responses to this. It is intelligible, for instance, how an expression with a semantic architecture like that of ‘dommal’ might nevertheless allow vagueness on the boundary between the things which satisfied its sufficient condition and the things that neither satisfied its sufficient condition nor failed its necessary one. These—second-­order—borderline cases may precisely be things for which it

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‘Wang’s Paradox’  311 was indeterminate whether or not they satisfied the relevant sufficient condition. Such indeterminacy could arise because the concept giving that sufficient condition—dog in the Williamson example—might itself have the same kind of semantic architecture. Similar possibilities would apply to the ­concept—mammal—that gave the original necessary condition. Since there seems to be no limit to the extent to which the pattern might be iterated, it looks as though the Fregean conception might after all be able to recover some of our preconceptions about higher-­order vagueness, to whatever extent they run, provided the relevant concepts that would be successively invoked are all of the illustrated semantic-­architectural kind. That is a proposal. But it seems to have very little mileage in it. When we reflect on the prototypical Sorites-­prone predicates—predicates such as ‘red’, ‘short’, ‘bald’, ‘heap’, and so on—the most salient feature about them is their immediacy: it simply is not credible that our conception of their conditions of  proper application is informed by an indefinitely extending structure of  partial definitions, each one deploying novel concepts distinct from those  employed in the sufficient, or necessary, conditions articulated by its predecessor. These are serious difficulties. But higher-­order vagueness poses a further and I think decisive problem for the Fregean conception. Simply: it allows of no coherent description in terms of the Fregean template for what a borderline case is (see Wright 2003c, p. 89, this volume, Chapter 9). The borderline cases of ‘dommal’ are cases in which there is mandate neither to apply ‘dommal’ nor to apply its contrary. Borderline cases of borderline cases, on this model, will thus be cases in which there is neither mandate to apply a predicate or to apply its contrary, nor mandate to regard them as borderline—to regard them as cases in which neither the predicate nor its contrary is mandated to apply. So they are cases where there is no mandate to apply the ori­ gin­al predicate (otherwise they would not be borderline cases at all), no mandate to apply its contrary (for the same reason), but also no mandate to characterize as cases in which there is neither mandate to apply the predicate nor mandate to apply its contrary! That is absurd. Whenever—as the first two conjuncts say—a case is such that there is no mandate to apply the predicate to it and no mandate to apply its contrary, then it will be true—and hence mandated—to say just that; but that is exactly what the third ­conjunct denies. In sum: the Fregean conception—the conception of vagueness as semantic incompleteness—is in tension with our actual sometimes positive or negative reactions to borderline cases, at odds with the liberality of our reactions to

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312  The Riddle of Vagueness others’ reactions to them, and, on two counts, can provide no room for the—at least apparent—phenomenon of higher-­order vagueness. It may be rejoined in mitigation that we should by now have learned to expect that any possible broad conception of vagueness will have problems in accommodating one or another of our preconceptions about the matter, or in respecting certain aspects of the phenomenology of vagueness or our linguistic practice with vague expressions. That, indeed, is what makes the whole issue so hard. Even if resolute prosecution of the Fregean conception is going to require a theory which is indeed in tension with certain aspects of the apparent data, that—it may be said—should be a decisive consideration against it only if some other theory can match the available advantages of such a theory, including those made possible by the apparatus of supervaluation, while causing fewer, or less serious casualties. But I disagree. I think the problems I have just outlined are sufficiently serious to justify persisting in the assumption that the Fregean conception, however natural, is mistaken: borderline cases of vague expressions typified by the ‘usual suspects’—‘red’, ‘bald’, ‘heap’, ‘short’, and so on—simply are not to be conceived in terms of the idea of, as it were, one’s semantic instructions giving out, of the rules of language failing to provide guidance. But, if they are not to be so conceived, what are the alternatives?

III Vagueness as Unknown Precision If the borderline cases of a vague expression are not to be viewed as cases for which its governing rules fail to prescribe any verdict, then there seem to be just two remaining possibilities: either the rules are inconsistent—Dummett’s idea—and thus prescribe conflicting verdicts in every case, or we should swallow hard and accept that consistent verdicts are indeed prescribed in every case. In any Sorites series of the normal ‘monotonic’ kind, the latter is tantamount to an acceptance that the rules governing the affected predicate mandate a sharp cut-­off. Bivalence is therefore assured, and with it classical logic. What is missing, though, is any account of what vagueness is or why it arises. As we know from the work of Williamson and others,6 this can be made to be a much more resilient proposal than at first appears. Its proponent—the 6  The leading systematic account of the epistemicist view is Williamson (1994). See also Sorensen (1988, ch. 6).

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‘Wang’s Paradox’  313 epistemicist—is thinking of our use of vague expressions as broadly com­par­ able to the sorts of judgements we should make about the application of precise predicates like ‘is more than six feet tall’ if excluded from reliance on canonical means of measurement and forced to estimate by eye. There would be cases in which it would be clear that a subject was more than six feet tall, cases in which it was clear that they were not, and cases in which observation left one unable to form a view, or where our views were weak and unstable. This model thus predicts something very like our actual patterns of application of vague expressions in borderline cases. Presumably these patterns of reaction would be picked up by someone who was trained in the use of ‘is more than six feet tall’ purely by immersion in the practice of those restricted to visual estimation and to whom no communication was made of the real meaning of the predicate, in virtue of which its extension is actually sharp. We can construct a fantasy around that idea. Call those privy to the real meaning the Priests and those kept in the dark the Acolytes. We might imagine that, as time goes by, the Priests die out but the practice of the Acolytes survives in more or less stable form, though uninformed by any adequate conception of what determines the extension of ‘morensicksfittle’ as the phrase begins to be written in their dialect. I do not know if Williamson, or any other defender of the epistemic conception, would take any comfort in this parable. Certainly, it is no part of their view that we should think of ourselves as having, over generations, lost touch with earlier fully explicit conceptions of the meanings of vague expressions. What cannot be avoided, however, is the admission that, if the epistemicist view is right, we do not as a matter of fact have satisfactory such conceptions and have no idea how to go about recovering them. This is something that Epistemicism needs to explain, and—so far as I am aware—no explanation has ever been offered. It is true that Williamson, for one, has worked hard to explain why we cannot know where the (postulated) sharp cut-­offs come in Sorites series (see especially Williamson 1994, ch. 8). But it is not an explanation of that that is being asked for. Estimations of the application of ‘is more than six feet tall’ based on unaided observation would indeed be subject to margins of error, and for my present purposes we can grant that Williamson explains why, accordingly, we cannot know by such means where, in a march-­past of fifty men beginning at five foot six inches tall and increasing by small variable margins less than one quarter of an inch each time, the first man more than six feet tall is to be found. But that does not explain why I cannot know the alleged principle or principles that stand to my use of ‘bald’, as the condition expressed by the predicate ‘is more than six foot tall’ stands

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314  The Riddle of Vagueness to the practice—at least according to the suggestion of the parable—of the Acolytes. The real difficulty, though, is to make anything of the suggestion that the Acolytes’ practice is indeed actually so informed. It is only because the Priests understood what it is to be more than six feet tall that they had any inkling about when it is safe to apply the predicate, or to deny its application, under circumstances of unaided observation. The practice so informed may be im­it­able in a relatively stable and transmissible way, but, insofar as it is, there is no longer any motivation for saying that it is driven by the original precise principle. What drives it, rather, is a sense of the patterns of use—in which, for the Priests, the precise principle was once instrumental—purely as patterns in their own right. There is no sense in the idea of a continuing add­ ition­al undertow exerted by an originating principle which nobody any longer grasps. The most fundamental difficulty with the epistemicist proposal is not  merely—to put it unkindly—the element of superstition involved in the  suggestion that there are indeed, in the case of all vague expressions which  contribute to a successful linguistic practice, underlying principles which determine sharp extensions. The charge of superstition might be thought to be addressed, at least to a degree, by the reflection that what at  this  point may appear to be the only possible alternatives—Semantic Incompleteness, as in Frege, and Incoherentism, as in Dummett—also appear radically unsatisfactory. But the more acute problem is that, so long as the epistemicist has to concede that we have no inkling of what the relevant principles are or how they might be ascertained, it is merely bad philosophy of mind to suppose that our linguistic practice consists, in some sense, in their implementation—that it is, in any meaningful sense, regulated by them at all.

IV Vagueness as Incoherence If both Incompleteness and Epistemicism are unacceptable, then, rather than merely dismiss it as ‘radically unsatisfactory’, we should reconsider Dummett’s incoherentist response before going any further. The leading thought is that the rules governing a vague expression do indeed provide guidance right through Sorites series, rather as a powerful river current guides items of flotsam over the waterfall! The idea that, in our mastery of natural language, we  are governed by inconsistent rules is counterintuitive to be sure. But is

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‘Wang’s Paradox’  315 there some more fundamental objection? A broadly successful practice can be informed by incoherent rules: think of simple division taught to children without any explicit proscription of division by zero, or the game of croquet which, I am told, was for years codified by official rules which contained an inconsistency about the permissibility of iterated roquets. Why should it not be so with linguistic competence? Well, one crucial question is what explanation the incoherentist can offer of our characteristic patterns of response to borderline cases. Why in a Sorites series do we start out with strong positive opinions, then gradually lapse into weak, defeasible opinions, conflicting opinions, unstable opinions, and failures to come to an opinion, before finally gradually reverting to strong negative opinions? Why does confusion not reign throughout the range? The incoherentist may try to suggest that the deterioration of our linguistic practice from broad consensus into paralysis and conflict is a function of our realization that it is towards disaster that the rules are taking us. But that does not explain the subsequent stabilization of confidence in negative verdicts. And it is in any case a poor explanation of the distinctive patterns of reaction in the borderline area. It is a poor explanation of those patterns for the simple reason that it does not need the context of a Sorites paradox to elicit the characteristic responses involved: they are elicited anyway by confrontation with borderline cases, from thinkers who have no inkling of the Sorites or even— in the case of, say, young children—any capacity to follow the reasoning of the paradox and see its point. There is more. The suggestion that in our linguistic practice with vague expressions we follow incoherent rules—rules that do indeed actually mandate the application both of an affected predicate and of its contrary to the very same object—is powerless to explain the basic point that, when we do confront a Sorites paradox, we have not the slightest inclination to weigh the two limbs of the contradiction equally. There is absolutely no inclination to regard the verdict reached by the Sorites chain as correct. It is utterly dom­ in­ated by the verdict with which it conflicts, and the paradox initially impresses us as the merest trick. By contrast, where we really do have conflict in the rules governing a concept—for example, the concept of course of values introduced by Basic Law V of Frege’s Grundgesetze—the two components of the paradox are balanced in our esteem, and we have not the slightest sense that one is to be preferred to the other. This is a basic datum which any satisfactory account of vagueness should accommodate and explain, but  which—at least on the face of it—the incoherentist view is powerless to explain.

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316  The Riddle of Vagueness As observed, it is not to be denied that sense can be made of a successful (in certain respects) activity being informed by inconsistent rules. The society that teaches rules of division which are correct except for the detail that they allow division by zero may get away with it because it never occurs to anyone to attempt a calculation involving a ratio with zero as its de­nom­in­ ator. Or it may be that practitioners are aware of the contradiction, but do not exploit it and also manage to avoid situations in which it matters. In any such case, though, the warrant to identify contradiction in the rules is wholly dependent on the way they are officially explicitly codified: the character of the informed practice, considered just in its own right, furnishes— and, in so far as it is ­stable, successful, and communicable, can furnish—no grounds to propose a theory of it which represents its rules as inconsistent. The point is surely equally good for our pervasive and remarkably successful linguistic commerce involving vague expressions. To represent it as the  product of inconsistent rules purely on the basis of a paradox which nobody actually accepts provides not merely for poor explanations of ­linguistic practice, in the respects just noticed, but is empirically entirely unmotivated.

V Vague Discourse as Unprincipled Now, however, we appear to have hit a complete impasse. What are we to think of the rules which govern the use of a vague expression and of what they instruct us to do as we advance down a Sorites series into the borderline area? It seems that there are just three possibilities: either there is no prescription in the borderline area, or the prescription remains the same as it was, or it changes abruptly—Semantic Incompleteness, Incoherentism, and (in effect) Epistemicism. Yet we have reviewed serious causes for discontent with each of  these three proposals. Each, indeed, comes short in the most basic way, by  failing to offer a satisfactory explanation of one or another aspect of competent speakers’ linguistic practice with vague expressions. Semantic Incompleteness fails to explain, for instance, our tolerance of conflicting verdicts in the borderline region; Incoherentism fails to explain, among other things, the disparity in our reaction to the two components of the contradiction; and Epistemicism, insofar as it is content to appeal to underlying semantic features transcending any sense that competent speakers have of proper

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‘Wang’s Paradox’  317 linguistic performance, seems to disclaim any ambition of giving an ex­plan­ ation of our linguistic practice with vague expressions at all. The puzzle is intense. But we get a pointer to what I have long believed to be the direction in which to find the correct response to it if we consider the so-­called Forced March variation on the Sorites paradox. The Forced March involves no reasoning to a contradiction from premises, plausible or otherwise. Rather, we simply take a hapless subject case-­by-­case down a Sorites series, ranging from things that are clearly F at one end to things that are clearly not-F at the other, and demand a verdict at every point. The subject starts out, naturally, with the verdict, F. And there are just two possibilities for what happens afterwards. Either they go on returning that verdict all the way down, or at some point they do something different—if only refusing to issue a verdict at all. If they do the former, then eventually they will say something false, and hence will betray incompetence. And, if they do the latter, then they will draw a distinction by their responses which (i) will have no force of pre­ ce­dent for verdicts in other contexts, and (ii) will correspond to no relevant distinction that they can call attention to between the last case in which they gave the original verdict and the first case in which they change. So their verdicts will be unprincipled. Conclusion: anyone who uses a Sorites-­prone expression can be forced to use it in ways that are either incompetent or unprincipled. If we add the plausible-­seeming supposition that competent linguistic practice is always essentially principled—always consists in the proper observance of semantic and grammatical rules—then the conclusion, on either horn, is that the use of vague language is bound to be incompetent. The solution must be to break the tie between ‘unprincipled’ and ‘incompetent’. But that is to say there has to be sense in which competent classification utilizing vague expressions does not consist in the implementation of the requirements of semantic rules. When the question is, what do the semantic rules, subservience to which constitutes competence for a vague expression, require when it comes to borderline cases, the answer we should give is not any of the three canvassed. The position is neither that the requirements give out, nor that they remain in force driving us on towards paradox, nor that they mandate a sharp cut-­off (of some kind). Rather, it is that competence with basic vague expressions is not a matter of subservience to the requirements of rule at all. This is, indeed, the conclusion to which I came in my paper on the Sorites published in the same volume as ‘Wang’s Paradox’.7 The great difficulty now, 7  Cf. Wright (1975, this volume, Chapter 1; 1976, this volume, Chapter 2).

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318  The Riddle of Vagueness as then, is to understand what we are committing to if we accept it—and the difficulty is all the more acute when one reflects that, on at least one way of understanding the issue, to claim that basic classifications effected using Sorites-­prone expressions are not rule governed is simply preposterous. It is preposterous because such classifications are, of course, genuine classifications, apt to be correct or incorrect. And it seems the merest platitude that correctness, or incorrectness, has to be a matter of fit, or failure of fit, between an actually delivered verdict and what ought to be said. There is therefore no alternative but to construe vague classification as in some sense subject to norm—and now, if the correctness of our verdicts in general is to be a matter of competence, rather than accident, there seems no option but to concede that we are in some way masters of these norms and follow them in our linguistic practice. I grant there is indeed no option but to concede that. The point, however, is that such a concession may not amount to very much—in particular, it may not amount to anything which sets up the trilemma we confronted above. In order to illustrate how this may be so, we can invoke a comparison with some thoughts about truth which I have canvassed in other work.8 To speak the truth is to ‘tell it like it is’, represent things as they are, state what cor­res­ ponds to the facts. Understood in one way, these phrases are platitudes and  in­corp­or­ate no substantial metaphysics of truth, at variance with, say, coherentist or pragmatist conceptions. The patter of correspondence, platitudinously understood, should motivate no questions about the nature of facts—what kind of entities they might be (as it were, sentence-­ shaped objects?) or how they might somehow be fitted to correspond in an appropriate way to beliefs and thoughts. I want to suggest that, in a similar way, the conception of basic classificatory linguistic practice as consisting in learning rules and following them is likewise open to a minimalist, or platitudinous construal, but that the trilemma: , arises only on a richer, non-­deflated construal. The proper understanding of the idea that such classificatory competence is unprincipled is exactly that the relevant kind of richer construal is inappropriate. So: that is the shape of a proposal. The question is how to fill it out. 8 See for instance Wright (1998, pp.  31–74; in a special issue on Pragmatism, guest edited by Cheryl Misak).

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‘Wang’s Paradox’  319

VI The Modus Ponens Model of Rule-­Following Here is one possible direction—one that takes us to the heart of the agenda of the Philosophical Investigations. Take as a simple, uncontroversial example of a rule-­governed practice the case of castling in chess. The rule states (something like): If the squares between the king and one of its rooks are unoccupied, and if neither the king nor the rook has previously moved in the course of the game, and if the king is not in check, nor would move through or into check, then it may be moved two squares towards the rook and the latter then placed on the adjacent square behind it.

Following this rule involves assuring oneself that the antecedent of the conditional articulating the requirement of the rule is satisfied in the circumstances of a particular game, and then—if one chooses—availing oneself of the permission incorporated in the consequent. Generalizing, what is suggested is a model of rule-­following which involves (implicit) reasoning, of the form of a modus ponens, from a conditional statement whose antecedent formulates the initial conditions of the operation of the rule, and whose consequent then articulates the mandate, permission, or prohibition that the rule involves. We can call this the modus ponens model of rule-­following.9 The qualification, ‘implicit’, is suggested because the appropriateness of the  modus ponens model is not restricted to cases where rule-­following is informed by self-­conscious inference. Following a familiar rule may be, very often, phenomenologically immediate and unreflective. But the important consideration, as far as the appropriateness of the modus ponens model goes, concerns what the rule-­follower would acknowledge as justifying their performance. It is enough, in order for the modus ponens model to be appropriate, that the explicitly inferential structure of reasons it calls for should surface in that context. A practised chess-­player may decide to castle without any conscious thought but that of protecting their king from an attack down the left flank. But, if the legality of the move were questioned, they would be ­prepared to advert explicitly to the relevant pattern of reasons of the modus 9 This idea, and its limitations, are anticipated in Wright (1989, sect. V; repr. Wright 2001c, pp. 170–213).

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320  The Riddle of Vagueness ponens type, whose conditional premise embodied a formulation of the rule for castling. Plausibly, the modus ponens model thus extends a long way into the class of phenomenologically immediate decisions and judgements. How far does it extend? In particular, can the competent application of an expression always be conceived as rule- following in accordance with this simple prototype? More specifically still, can we regard the classifications we effect using basic vague expressions, of the kind typified by the usual suspects, as an example of rule-­governed activity in accordance with the modus ponens model? The answer, I believe, is no, and the reasons why not are instructive. Suppose, to the contrary, that competent classification using ‘red’, for ex­ample, is a rule-­governed practice in the sense articulated by the modus ponens model. Then there should be a conditional which specifies the conditions under which predication of ‘red’ to an item should be assented to, and such that each competent classification can be seen as (implicitly) inferentially grounded in the recognition, of a presented item, that it fulfils the antecedent. The picture, in other words, is that each informed, competent classification of an item, x, as red is underwritten by a pair of reasons of the form: If something is X, then it should be classified as ‘red’ and x is X So OK: what is X? What is the property whose instantiation underwrites the proper application of ‘red’ as fulfilment of the antecedent for the rule for cast­ ling underwrites the judgement that the particular situation in the game is one in which one is permitted to castle? It seems that we do not know of any plausible candidate answer, apt for all cases, except to identify X with red. The same goes for the general run of predicates that make up the usual suspects: basic, vague predicates used to record the results of casual observation. In such cases, the correct answer to the question, what is the condition common to the minor premise and the antecedent of the conditional for the modus ponens?, seems to be irreducibly homophonic. This observation has a striking consequence. It is that the price of continuing adherence to the modus ponens model in these cases is that we are forced to think of grasping the concept of what is to be, say, red as underlying and informing competent practice with the predicate ‘red’. If the explanation of

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‘Wang’s Paradox’  321 competent practice with ‘red’ adverts to the intention to follow a rule the grasp of which demands a prior understanding of what it is for something to be red, then the latter grasp has to stand independent of the linguistic competence. So, in effect, we commit ourselves to the picture of language-­mastery displayed in the quotation from Augustine with which Wittgenstein begins the Philosophical Investigations—not, to be sure, in the aspect of that picture which involves thinking of the semantics of expressions generally on the model that of a name and its bearer—but rather the aspect which we find made explicit only later, at §32: And now, I think, we can say: Augustine describes a learning of human ­language as if a child came to a strange country and did not understand the  language of the country; that is, as if he already had a language, only not this one. Or again: it is as if the child could already think, only not yet speak. And “think” would here mean something like “talk to himself ”. (Wittgenstein 1968)

Wittgenstein, of course, intends his reader to take on board the thought that Augustine prototypically committed a major philosophical mistake. If we agree with him and simply repudiate the Augustinian picture, then we will have to conclude that, at the level of the basic, casual–observational classifications expressed by the usual suspects, the modus ponens model is inappropriate. The classifications effected by means of vague expressions of this basic kind are not supported by reasons, for the only possible candidates to constitute their reasons would demand, when so conceived, that we think of grasp of the concepts expressed by the vague predicates in question as something prior to and independent of the mastery of these predicates. In his ‘later’ work Wittgenstein gives expression to an epistemology of understanding wherein language-­ mastery is systematically conceived not merely as a means for the expression of concepts but as the medium in which our possession of them has its very being. As a reaction against the dia­met­ric­ al­ly opposed view, that thought—at whatever degree of sophistication—may always intelligibly be conceived as constitutively independent of the thinker’s possessing means for its expression, Wittgenstein’s stance is surely compelling. But neither polar view—that language is merely the means of expression of thought, and that nothing worth regarding as thinking is possible without language, respectively—is correct. The correct view of the matter overall is presumably something more nuanced. There is much possible intelligent activity which, though wordless, would naturally call for explanation by

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322  The Riddle of Vagueness ascription to the agent involved of conceptual—content-­bearing—states of varying degrees of sophistication. Imagine, for instance, a chimpanzee who, off their own bat and after many days of manipulations in the corner of their cage, is suddenly able repeatedly to solve Rubik’s cube; or the wonderful behaviour, even in an unfamiliar city, of a well-­trained guide dog. On the other hand, there are plenty of concepts—take, for instance, that of a parametric variable in a step of existential elimination—of which it strains cred­ ibil­ity beyond breaking point to suppose that they could be fully grasped by someone with no linguistic competence at all. So, if its implication of the Augustinian picture when applied to ‘red’, ‘bald’, ‘heap’, and so on, is to be sufficient reason to reject the modus ponens model of competence in those cases, we need a special consideration as to why. Is there one? We would need further detailed discussion fully to vindicate the assumption, which I will from hereon make, that Wittgenstein is right at least about the concepts to which we give expression by basic vague expressions—those for which retention of the modus ponens model would demand siding with Augustine. Here let me just gesture at one line of thought. We can certainly imagine a creature without language—the chimpanzee again—exhibiting some sort of concepts of colour: they may, for example, manifest a preference, among variously coloured but identically smelling and tasting boiled sweets, for green ones. More generally, there are any number of kinds of colour-­ sorting behaviours that could be—and in many cases, are—exhibited by prelinguistic children. But grasping the colour concepts we actually have is not merely a matter of dispositions of appropriate response to paradigms. To grasp any classificatory concept, one needs not just to learn to respond appropriately to central cases but also to acquire a sense of its limits. With the usual suspects, however, it is the very pattern of our linguistic practices that sets the limits, imprecise though they may be. We learn by immersion in the language how far one can stretch from paradigm cases of red, or blue, before classification starts to become acceptably controversial or difficult. Prelinguistic children no doubt have some sort of grasp of colour saliencies. But the raw concepts they have, or could have, do not match the vagueness of colour concepts as linguistically captured. The reason is that it is precisely the onset of hesitancy, disorder, and weak conflict in the linguistic verdicts of the competent that constitutes the gradual intrusion into the borderline area and sets the limits of the concepts in question. I do not deny that the chimpanzee might behave in ways which went some way towards giving sense to the idea that the concepts they were working with were vague—they might hesitate over a turquoise sweet, for example, while vigorously discarding a blue one.

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‘Wang’s Paradox’  323 What seems to make no sense, however, is that their concept of green so manifested might be identical, vagueness and all, to the concept which, miraculously endowed with the power of speech and perfect basic English, they went on to display in their uses of ‘green’. Competence with vague concepts is an essentially linguistic competence because the extent of their vagueness is an essentially linguistically manifest, socially constrained phenomenon. If the broad direction of these remarks is correct, we must abandon the modus ponens model if we are to understand the sense in which the exercise of vague classifications is a form of rule-­governed activity. One reaction would be to deny that it is, properly speaking, rule governed at all— that it is sensible to think of an activity as governed by a rule only if one could in principle articulate the content of what would be the appropriate rules in such a way that a suitable thinker could generate a competence by observing them, and justify their performance by their lights. If I am right, that condition is indeed unattainable where competence with vague classifications is concerned. That competence is, precisely, not to be viewed as a product of any possible anterior body of information which, in principle, could be used explicitly to inform the moves that competence requires. But, actually, I do not think it matters whether we say that there is, properly speaking, no rule-­ following in such cases or whether we say merely—as I originally announced—that we need a more minimalistic conception of what in the rele­ vant cases rule-­ following involves. Certainly, even with judgements involving vague expressions, there are still all of correctness and incorrectness, criticizability, proper responsibility, the intention to get things right, and a wide range of contexts in which it is important to succeed in that intention. But what there is not is a body of information which underlies competence, in a way in which knowledge of the rule for castling together with knowledge of the history of the particular game and the present configuration of pieces en­ables (what may well be an unreflective) awareness by an expert player that castling is an option here if they want.10 If this is right, then we can see our way around the trilemma. The trilemma arises if but only if it is a good question concerning the rules which govern our competence with ‘red’: what do they really have to say when our colour classifications fall into the complex patterns which show that we are in the borderline area?—what is their message, what would we do if we were to do exactly what they require and nothing else? That the three answers canvassed seem, among them, to exhaust all of the alternatives, yet each to be 10  Additional difficulties for the modus ponens model are raised in Boghossian (2005).

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324  The Riddle of Vagueness objectionable, might have made one suspect that there is something wrong with the question. Now we see how that suspicion might be substantiated. There is no requirement imposed by the rules—not if we understand such a requirement as something whose character may be belied by our practice in the borderline area and into which there is scope for independent inquiry. There is no such requirement because to suppose otherwise is implicitly to  commit oneself to the modus ponens model of rule-­following for the classifications in question, and hence, as we have seen, to open an—at least locally—unsustainable gap between conceptual competence and the linguistic capacities that manifest it.

VII The Sorites Paradox of Observationality To abandon the modus ponens model of competence with vague expressions is to open the way for the thought that the characteristic manifestations of vagueness in linguistic practice are, in a sense, the whole story. Vagueness is constituted at the level of use, rather than at the level of explanation of use. Here are some of the kinds of fact that are salient at the level of use. First, competence with a vague expression is, in the usual run of cases, mastery of a practice which is communicated not by verbal explanations and characterizations but by immersion in that very practice. A large part of the acquisition of competence with a vague expression will typically consist in learning to use it to make judgements which we think of as immediately responsive just to how things strike us. Such judgements impress as, in a sense, undiscussable; we have to hand no repertoire of reasons with which to negotiate about them. Second, there is only such a thing as competence with such expressions because the relevant kind of training does generate—fortunately—a high degree of intersubjective—and cross-­temporal intra-­subjective—constancy in  judgements, both positive and negative. Third, however, the training we receive also results in ranges of cases in which constancy breaks down, wherein otherwise perfectly competent subjects differ, views are char­ac­ter­is­ tic­al­ly unstable, and we often find it difficult to be moved to a view at all. My suggestion is that a philosophical account of the nature of vagueness is to be sought in the, by all means properly refined and nuanced, elaboration of facts such as these. There is no scope for an, as it were, information-­theoretic standpoint whose goal is to outline rules which we implicitly follow in our use

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‘Wang’s Paradox’  325 of vague expressions, and thereby to explain what it is about those rules that gives rise to the characteristic manifestations. There is no such legitimate explanatory project. Competence with a vague expression is not the in­tern­al­ iza­tion of a set of rules that prescribe the agreed verdicts in cases in which agreement occurs, and whose other features explain the cases in which it breaks down—in any of the ways that Incompleteness, Epistemicism, and Incoherentism respectively offer. The breaking-­down of constancy has to do, not with incompleteness, or inconsistency in the rules we tacitly follow, nor with the invisibility to us of the lines they draw, but is a matter of how we naturally respond, without reasons, to cases of significant distance from paradigms. This shift in perspective provides for a major change in the impression given by Dummett’s observational Sorites. Take what is surely the most daunting type of example: the Sorites associated with predicates like ‘looks red’ over a series of pairwise indistinguishable square colour patches, running from, say, crimson to orange. Here adjacent patches look just the same. And ‘looks red’ is surely an observational predicate in Dummett’s sense if anything is. But observationality must require— must it not?— that the meaning of ‘looks red’ enjoins that it is properly applied to both, if to either, of any pair of things which look just the same. So the paradox seems iron-­cast. Yet this appearance precisely depends on a tacit construal of the observationality of ‘looks red’ and its kin which rests on the modus ponens model of competence in their use. In effect, we are seduced into thinking of competence as involving the internalization of a set of rules which connect appearances with the proper application of the predicate: rules of the form, roughly, If x appears thus-­and-­so, then it should be classified as ‘looks red’. Once into this way of thinking of the matter, it is indeed impossible to understand how things which appear the same could possibly deserve any but the same classification. But the allure of the major premise dies away once the idea is taken on board that the classifications effected by these predicates are, in basic cases, essentially unsupported by any articulatable structure of ­reasons—more specifically, that they can be and normally are competently made without the mandate of any internalized conditional rule of the schematized kind. Since they need no such mandate, they are not properly regarded as unprincipled—at least not just on that account—when they collectively take a shape over a Sorites series—as they must if they are to be ­consistent—which would violate such a rule. Competent classification over a

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326  The Riddle of Vagueness Sorites series for ‘looks red’ will—indeed must—at some point involve differential responses to presentations that look exactly alike. The (crazy) idea that competence somehow accordingly involves disrespecting the rules is an artefact of a misplaced adherence to the modus ponens model at a level where it involves an incoherent over-­rationalization of our practices. As Wittgenstein (near enough) says (1968, §289), not everything judged rationally is judged for reasons. I said earlier that the natural reaction to the Dummettian Sorites is to seek ‘an improved understanding of what an expression’s possessing observational content involves’. I meant that reaction to be in contrast with one of mistrust of the very notion of an observational predicate. To be sure, if what I have been saying has any truth in it, then we should certainly mistrust the idea of an expression’s being governed by rules which prescribe like verdicts about like appearances. But that is no cause for suspicion of Dummett’s own working characterization of observational expressions as those ‘whose application can be decided merely by the employment of our sense organs’ (cf. footnote 3). What is wrong is the little piece of theory which, under the aegis of the modus ponens model, links the Dummettian notion with governance by rules prescribing like verdicts about things our sense organs cannot distinguish. But the right reaction is not to retain the modus ponens model and look to somehow complicate or refine the rules which are characteristic of the competent employment of an observational predicate. The right response is to realize the implications of the point that competence with an observational predicate is not, even tacitly or ‘in principle’, an inferentially controlled competence.11 Dummett put the spotlight on observational predicates and I too have ­spoken of the usual suspects as predicates of ‘casual observation’. But if the general drift of the foregoing is correct, vagueness will be an expectable characteristic of a more inclusive class of predicates (or other kinds of expression): all those which give rise to judgements for which competence is not, in basic cases, to be explained in terms of the modus ponens model—that is, judgements competence with which is not a matter of sensitivity to inferentially organized reasons but which are characteristically both rationally and 11  The point I am making has, I believe, close connections with what Wittgenstein is driving at when he speaks in the famous passage at Investigations (1968, §201) of ‘a way of grasping a rule which is not an interpretation, but which is exhibited in what we call “obeying the rule” and “going against” in actual cases’. It would not stray very far from the intent of that passage, in my opinion, if we gloss it as: there is, and has to be, a kind of rule-­following in which the ingredient steps are performed without the possibility of an articulated justification in the light of a statement of the rule—and in that sense are performed blindly. (Cf. Wittgenstein 1968, §219.)

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‘Wang’s Paradox’  327 phenomenologically immediate. Observational judgements are a species, but only one species, within this wider genus. My conjecture is that all vagueness of the kind which interests us here—vagueness of the kind which seems to be associated with higher-­order vagueness and seems to give rise to Sorites-­ susceptibility—originates at the level of non-­inferential judgement.

VIII Denouément Unfortunately these considerations do not yet take us out of the wood. They provide the means to address one motivation for the major premises in Sorites paradoxes for observational predicates. If they are compelling, and if that (illicitly rationalistic) motive exhausts the field, then we can indeed explain away the plausibility of those premises—an essential part of any solution to a paradox, properly so regarded—and are now free, presumably, to regard each such Sorites simply as a reductio of its major premise. But now a new concern surfaces. To deny the major premise in the ‘looks red’ Sorites, for example, is to conclude that it is not true that each patch that looks red in the series we imagined is succeeded by a patch that looks red . . . But how is that not true? After all, we precisely so conceived the example that no patch that looks red is succeeded by one that does not. Surely no implicit over-­rationalization of competence with observational predicates is involved in the thought that no pair of indistinguishable patches can be such that one looks red while the other does not . . .? It would be relevant to observe that treating the reasoning of the paradox as a reductio of a major premise of the form ("x )(Fx ® Fx ¢) is a commitment to affirming the corresponding claim of the form ($x )(Fx & ØFx ¢) only on the assumption that classical logic is good for vague statements. And, for what it is worth, I do maintain that something like an intuitionistic logic— at the least, a logic in which double negation elimination does not hold unrestrictedly—will be required in any fully satisfactory treatment (and will say

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328  The Riddle of Vagueness more on this shortly). But someone could agree about that and still have the misgiving just aired. Even if we have disposed of the motivation for the universally quantified form of major premise in the observational Sorites, there still appears to be every motivation for the negative existential major premise for a corresponding observational instance of the No Sharp Boundaries paradox. Again: surely there just nowhere occurs in the relevant kind of series any patch that looks red while its immediate successor does not. Is not the negative existential major premise just flat out true in this case? The problem is, indeed, more general. Nothing has so far been said to address the overarching motivation, noted at the conclusion of Section I, for the major premises in No Sharp Boundaries paradoxes as a class—namely, the simple thought that vagueness per se consists in absence of a sharp cut-­off in the relevant series, hence in the truth of a statement of the form Ø($x )(Fx & ØFx ¢). What is vagueness if it is not a truth-­conferrer on statements of that form? These are very awkward-­looking questions. But I think the work we have done has taken us closer to being able to answer them. Let me conclude by sketching how. In dropping the modus ponens model, we drop the idea that truth for vague statements is a matter of fit with the requirements of semantic rules which competence internalizes. Rather, truth must now be viewed as grounded in the patterns of linguistic practice which competence, uninformed by any such rules and reconceived purely as the ability to participate successfully in that very practice, exhibits. There is a temptation to try to say something detailed and specific about the nature of this grounding—perhaps in the form of a developed account of some sort of response-­dependence. I do not know if any such attempt can be fully successful. But we can get some mileage out of a theoretically more modest standpoint. Vagueness, of the kind that interests us, gives rise to definite cases as well as borderline ones. There are things that definitely look red, and things that definitely do not. These definite cases are cases which elicit a firm and stable consensus in judgement from those considered competent in conditions considered as relevantly suitable. And, in discarding the modus ponens model, we now treat that sociological fact as, in a sense, the whole truth about the cases in question: there is nothing deeper, in particular no tacitly understood set of guiding requirements that explain and underwrite the convergence in verdicts about them, and no sense accordingly in the idea that the convergence might somehow be defeasible by

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‘Wang’s Paradox’  329 further standards, implicit in the meanings of the expressions in question. The way is therefore open to us to regard such convergence as unimprovable evidence of the truth of the statements in question, and—pari passu—inconsistency with verdicts that enjoy such convergence should be regarded as unimprovable evidence of falsity. The immediate effect of that consideration is that there is no option but to regard the verdicts about the end points of a Sorites series—schematically, F0 and ¬Fk—as straightforwardly true. For there is now nothing on which their truth might depend which is left out of the picture after competent convergence on them in good enough conditions is factored in. But that entails that there really is no alternative but to regard the negative existential major premise Ø($x )(Fx & ØFx ¢) as false. It must be regarded as false because it is inconsistent with a consistent pair of statements for whose truth there is unimprovable evidence. Corres­ pondingly, we have therefore no option but to accept its negation ØØ($x )(Fx & ØFx ¢). This realization sets us two problems. The first is to understand how we can deny the negative existential major premise without controverting the fact of the vagueness of F within the relevant series. The second, closely related, is to explain how, after that denial, we can now avoid an apparently preposterous— pace Williamson—endorsement of the unpalatable existential ($x )(Fx & ØFx ¢), which seems tantamount to an endorsement of F’s precision. First, then, on denying the negative existential. Does the vagueness of F in an appropriate series not just consist in the absence of a sharp cut-­off and is that not just what the negative existential directly states? Well, no—or rather ‘yes’ to the second question and ‘no’ to the first. Vagueness, we have to learn, does not consist in the absence of sharp cut-­offs. What the negative existential states—even in intuitionist logic—is the same as what is stated by ("x )(ØFx ¢ ® ØFx ¢),

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330  The Riddle of Vagueness which, in the presence of ¬Fk, is tantamount to the false statement that there are no Fs in the series. So that cannot be the correct way to characterize the vagueness of the boundary between the Fs and the non-Fs in a series that contains both. What is a fact is the absence of any pattern of responses that would provide evidence for the existence of a sharp cut-­off. The relevant pattern of responses would involve convergence on the initial elements of the series up to some last positive verdict, Fi, followed by convergence on the negative verdict, ¬Fi′, and then sustained convergence on negative verdicts up to the end. What we have instead is just the pattern of judgements dis­tinct­ ive of vagueness: polar constancy flanking the gradual breakdown of convergence in verdicts of any kind in the borderline region. That indeed, on the present perspective, is what vagueness consists in. So it does not consist in anything which makes the negative existential true. Rather, if the existence of a sharp cut-­off is identified with the truth of the unpalatable existential, then we should say this: that what is distinctive of a predicate’s vagueness is not that there is no sharp cut-­off, but rather that there is nothing in our practice with the predicate that grounds the claim that there is a sharp cut-­off (and that—again, pace Williamson—we have no satisfactory conception of how that claim might be grounded in an independent theoretical way). So, as far as the first problem—that of explaining how we can deny the negative existential without controverting F’s vagueness—is concerned, then, I am saying the following: that, so far from the negative existential’s being a satisfactory characteristic statement of a predicate’s vagueness, our practice with vague expressions provides unimprovable evidence that the negative existential is false, and hence that its negation—the double negation of the statement of the existence of a sharp cut-­off—is true. The pattern of practice constitutive of vagueness, so far from providing evidence for the truth of the negative existential, is inconsistent with it and needs to be described quite differently. As to the second problem, that of avoiding endorsement of the unpalatable existential, ($x )(Fx & ØFx ¢), we have so far observed merely that nothing in our patterns of judgement about each Fi provides any evidence for it, and that—pace Williamson—we have no other theoretical reason for thinking it true. These, on the present, ‘post-modus ponens model’ perspective, are constitutive facts: vague expressions

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‘Wang’s Paradox’  331 are ones for which sharp boundaries are neither drawn by actual competent practice nor posited by any known well-­motivated theory. So there is no reason whatever to assert that there are such boundaries.12 The temptation to think that this adds up to enough to justify denying the unpalatable existential—and so asserting the negative existential major premise—is very strong. It is of a species with a general tendency to deny things for which we have found absolutely no evidence (—the spirit in which we would, many of us, deny that there are leprechauns.) Usually, these somewhat arrogant denials do no serious harm. In this context, however, their nemesis is the No Sharp Boundaries paradox. Let me relate these proposals to the specific case of the No Sharp Boundaries paradox for ‘looks red’. We left discussion stalled on the thought that it seems merely to be the simple truth that no patch in the relevant series that looks red is followed by the one that does not. But that is not the simple truth. What is simply true is that nobody competent will willingly make a judgement which identifies such a pair of patches. And, since none of us will do that singly, there is no question of a convergence on a cut-­off. So nothing in our practice provides a ground for the claim that there is a cut-­off, nor do we have any defensible conception of how there might be one nonetheless which our practice fails to reflect. On the other hand, the equivalent of the ‘simple truth’— namely, that no patch in the series that does not look red is immediately preceded by one that does, entails, when some of the later elements do in fact not look red, that none of them does. So the ‘simple truth that no patch in the series that looks red is followed by one that does not’ is actually inconsistent with the way the elements collectively look and is therefore not a truth of any kind. As before, what has happened is a confusion of the complete absence of grounds for the existence of a cut-­off with the possession of grounds for denying one. But in this case I think there is an additional seductive fallacy. It is that we confuse its not looking as if there is a sharp cut-­off in the series—which is true: if it did look as if there is a cut-­off, we would converge on it—with its looking as if there is none. The latter is false. It does not look as if there is no cut-­off— at least not if that is taken to mean that the series collectively looks as it would if there was indeed no cut-­off. For, if there were no cut-­off, the series would

12  This claim presupposes, of course, that classical logic does not provide well-­motivated the­or­­etic­al reason to affirm the unpalatable existential on the ground of our acceptance, just argued for, of its double negation. That is exactly my position. I shall say more about it in a moment.

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332  The Riddle of Vagueness have to look to be uniformly composed of squares that all looked red, or all did not look red. Which by hypothesis it does not.13 We are having to draw some quite subtle distinctions. It may help to consolidate a sense of what I am proposing in general if the reader reflects that, assuming ‘monotonicity’ for the series in question, the unpalatable existential is equivalent to the Principle of Bivalence when restricted in range to each ingredient statement, Fi.14 The temptation to deny the unpalatable existential is thus on all fours with the temptation to think that vagueness should prompt denial of Bivalence. That denial, however—at least when truth is ­disquotational—likewise familiarly leads to contradiction. I am suggesting an attitude to the unpalatable existential which I take to be comparable in all essentials to the Intuitionists’ attitude to Bivalence in those areas of math­em­ at­ics where they regard it as unacceptable. It is there regarded as unacceptable because, when the truth of statements to which Bivalence is to be applied is conceived as constrained in ways that are taken to be independently mo­tiv­ ated, it goes beyond all evidence to suppose that each instance of Bivalence is true. In the case of mathematical statements, the Intuitionists take it that there is independent motive to require truths to be constructively provable. I do not know whether the historical motivation for that thought is wholly independent of what I have proposed about vague statements: that their truth and falsity have to be thought of as determined by our very practice, rather than by principles which notionally underlie it. But the effect is similar. The denial of any instance of Bivalence leads by unexceptionable basic logic (and the usual truth rules) to contradiction. So we must deny any such denial. But, when Fi is borderline, the breakdown in convergence of verdicts leaves us without uncontroverted evidence for its truth or for the truth of its negation. Since we lack convincing theoretical grounds to think that one or the other must be true nonetheless—because such grounds would have to regard the de­ter­min­ ants of truth and falsity as constituted elsewhere than in our linguistic ­practice—we are left with no compelling reason to regard either Fi or its neg­ ation as true. We should therefore abstain from unrestricted use of the law of

13  The conflation is comparable to one at work in the idea that one can get an inkling of what experience of disembodied survival of death would be like by suffering total sensory and proprioceptive anaesthesia. The latter might indeed generate experience in which one’s body was in no way presented to one. But its not appearing as if one has a body is not the same thing as a pattern of experience which represents one as having none. 14  For, if (∃x)(Fx & ¬Fx′) is true, then monotonicity ensures that each y < x will be F and each z > x′ will be non-F. So every element will be determinately either F or non-F. Conversely, if Bivalence holds, and there are both Fs and non-Fs in the series, then there will have to be an adjacent pair, the earlier F and the other not.

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‘Wang’s Paradox’  333 ­ouble negation elimination. We do so on the grounds that it is non-­ d conservative of knowledge—exactly the intuitionistic reservation. The thought that vagueness might require a non-­classical logic is, of course, hardly original. But it has usually been clothed in broadly semantic proposals about third truth values, truth-­value gaps, and degrees of truth. All these proposals see the principle of Bivalence as breaking down where vague statements are concerned. I am suggesting that the correct basic complaint to have about Bivalence where vague statements are concerned is not that it breaks down but that it goes beyond the evidence. I have supported this contention by argument that vagueness is a phenomenon of judgement unsupported by reasons, and that this makes for an especially intimate connection between patterns in competent linguistic practice and truth. It is this intimacy of connection that allows us to rest content with our stable verdicts about the poles of a Sorites series and so to enforce denial of the major premises involved. It is the same intimacy of connection that allows us to say that there is no reason — not merely: no reason provided by our actual patterns of judgement, but no reason whatever—to regard the unpalatable existential as true. So double negation elimination is, in this instance, epistemically non-­conservative. There is a lot more that would need to be discussed and developed in a fully explicit treatment, but I must draw this particular essay to a close. It is, of course, conspicuous among Michael Dummett’s many philosophical achievements to have done more than anyone else to rehabilitate the intuitionistic outlook and to transform it into a significant contemporary force not just in the philosophy of mathematics and logic but in metaphysics and the theory of meaning generally. If there is anything in what I have been saying, he may, ironically, have overlooked one especially apt and helpful application of it, to the riddle of vagueness.

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12 The Illusion of Higher-Order Vagueness It is common among philosophers who take an interest in the phenomenon of vagueness in natural language not merely to acknowledge higher-order vagueness but to take its existence as a basic datum—so that views that lack the resources to account for it, or that put obstacles in the way, are regarded as deficient just on that score. My main purpose in what follows is to loosen the hold of this deeply misconceived idea. Higher-order vagueness is no basic datum but an illusion, fostered by misunderstandings of the nature of or­din­ ary (if you will, ‘first-order’) vagueness itself. To see through the illusion is to take a step that is prerequisite for a correct understanding of vagueness, and for any satisfying dissolution of its attendant paradoxes.

I The Ineradicability Intuition One standard motive for acknowledging higher-order vagueness is given proto­typ­ic­al expression by Michael Dummett (1959, repr. 1978, p. 182): Now the vagueness of a vague predicate is ineradicable. Thus ‘hill’ is a vague predicate, in that there is no definite line between hills and mountains. But we could not eliminate this vagueness by introducing a new predicate, say ‘eminence’, to apply to those things which are neither definitely hills nor definitely mountains, since there would still remain things which were neither definitely hills nor definitely eminences, and so ad infinitum [sic].

This thought—the ineradicability intuition—may be generalized like this. Take any pair of concepts, F and G, with a vague mutual border. If you attempt to eradicate the vagueness by introducing a new term, H, to cover the shared borderline cases of F and G, your nemesis will be that the F–H and G–H

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0013

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336  The Riddle of Vagueness borders will be vague in their turn. It follows, seemingly,1 that the distinction between the Fs and the F–G borderline cases is itself already vague. Likewise for the Gs. So, iterating, we have a hierarchy of levels of borderline cases of F, and another hierarchy of levels of borderline cases of G, each continuing indefinitely. Notice how Dummett, like so many others, equates the lack of a sharp boundary between the Fs and the Gs with the (potential) existence of borderline cases, viewed as a kind of thing: things that are neither definitely F nor definitely G. I will henceforward term this characterization the Basic Formula. Moreover, Dummett does not, plausibly interpreted,2 intend to allow that things which are neither definitely F nor definitely G might yet be F or G all the same—only just not definitely so. He is thinking of the kind in question as cases that in some way come short of being either F or G: if x is an ‘eminence’, then it fails to qualify either as a hill or as a mountain. So for there to be no definite line between hills and mountains is for there to be (potential) things ‘in between’ that are, in some way, of a third sort. Thus the mutual vagueness of F and G, on this understanding, consists in the existence of a certain kind of buffer zone between their respective (potential) extensions. Yet this buffer zone had better be blurry on both edges in turn, or F and G will turn out to be not mutually vague but sharply separated by a mutual neighbour. And now it seems we have no option but haplessly to allow the blurred buffer-zone model to reiterate indefinitely. Dummett’s thought is closely related to, though distinct in detail from, that at work in these remarks of Russell (1923): The fact is that all words are attributable without doubt over a certain area, but become questionable within a penumbra, outside of which they are again certainly not attributable. Someone might seek to obtain precision in the use of words by saying that no word is to be applied in the penumbra, but unfortunately the penumbra itself is not accurately definable, and all the vaguenesses which apply to the primary uses of words apply also when we

1  It does follow, provided we assume that the introduction of the new term effects no alteration in the respective extensions of the original concepts; I will come back to this point later. 2  In ‘Wang’s Paradox’, he writes: ‘For, in connection with vague statements, the only possible meaning we could give to the word “true” is that of “definitely true” ’ (Dummett 1978, p. 256) No doubt here are no borderline cases of ‘definitely P’ which are clear cases of P. The question is whether we should allow, as part of the intended meaning of the definiteness operator, that it consists with something’s being a borderline case of ‘Definitely P’ that it yet be a case of P. Dummett is here saying no to that. We can call Dummett’s Principle the thesis that there are no truthful instances of the conjunctive form: P but not definitely P. As will emerge later, there is actually considerable pressure against the principle.

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The Illusion of Higher-Order Vagueness  337 try to fix a limit to their indubitable applicability. (From Russell, “Vagueness”, at pp. 63–4 in Keefe and Smith 1996)

Here Russell envisages not the introduction of a new term but rather a moratorium on applying any term. If it is not certain that F is properly applied, then it is not to be applied—the penumbra is to be an exclusion zone. Still Russell’s idea, like Dummett’s, involves the notion of a kind of case separating those where the applications of F and not-F are respectively mandated, or ‘indubitable’. And it is clear that he confidently expects judgements about membership in this kind to involve no less ‘vaguenesses’ than we started out with. The ineradicability intuition impresses as highly plausible. The linguistic stipulations respectively envisaged by Dummett and Russell would indeed— surely—not have the effect of introducing precision. But can that really be enough to enforce the vertiginous hierarchy of borderline kinds?

II The Seamlessness Intuition The ineradicability intuition provides one motive for postulating higher-order vagueness. A prima facie distinct motivation emerges from the idea that vagueness consists in the possession of borderline cases, together with one natural notion about how borderline cases, as characterized by the Basic Formula, come about and the apparent phenomenological fact of seamless transition. Consider a case where, as many would allow, something akin to vagueness is induced by deliberate definitional insufficiency. Suppose we characterize the notion of a pearl as follows.3 (i) It is to be a sufficient condition for being a pearl that a candidate have a certain specified chemical constitution and appearance and be nat­ur­ al­ly produced within an oyster.

3  The example is John Foster’s from classes in Oxford in the early 1970s. Compare Kit Fine’s ‘nice1’ (1975, p. 266), Timothy Williamson’s ‘dommal’ (1990, p. 107; 1994, pp. 213–14), and Mark Sainsbury’s ‘child*’ (1991, p. 173).

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338  The Riddle of Vagueness (ii) It is to be a necessary condition for being a pearl that a candidate have that same specified chemical constitution and appearance. What about artificial pearls? They satisfy the specified necessary condition but not the specified sufficient one. One thing we might say is this: since there is no sufficient basis for classifying them either as pearls (for they do not satisfy the only specified sufficient condition) or as non-pearls (for they do satisfy the only specified necessary condition), it is so far indeterminate whether artificial pearls are pearls.4 There is no fact of the matter. Now (this is the natural notion mentioned) suppose we think of borderline cases of naturally occurring vague predicates—‘bald’, ‘heap’, ‘red’, and the other usual suspects—as relevantly like artificial pearls: cases which are left in classificatory limbo by a broadly analogous but naturally occurring kind of semantic incompleteness. Thus they are cases that do not meet any practiceestablished sufficient condition for satisfying the relevant predicate but do satisfy all practice-established necessary ones. This is, seemingly, a very intuitive way of thinking of the Basic Formula as being underwritten. The (def­in­ ite) truths, and falsehoods, are what are determined as true, or false, by the facts and the semantic rules for the language in question. Borderline cases arise when the facts and semantic rules somehow fail to deliver.5 Next contrast the following two cases. Case 1: You have a collection of 2-inch-square colour patches, each of a uniform shade, collectively ranging in hue from red to orange, and numerous and varied enough to allow that every patch is matched by something that matches something in the collection that it does not match.6 You have to arrange them in a ‘monotonic’ series; specifically, one such that the first patch is red and each subsequent patch is immediately preceded by something that is at least as red as it is. So your selection will consist in an initial batch of red patches followed by some which hover around the red–orange border 4  One who, like Timothy Williamson, believes that Bivalence, like the Articles of the United States Constitution, is a self-evident truth, has of course to move differently: to deny that ‘pearl’ has so far been endowed with a meaning, or—as proposed by Williamson himself—to regard artificial pearls as non-pearls purely by dint of their failure to satisfy any established sufficient condition for being pearls. See Williamson (1994, p. 213; 1997a, sect. 3). The availability of this proposal to Williamson is queried in Heck (2004, p. 112). 5  This type of view goes back to Frege and was for a long time regarded as datum, rather than theory. For modern exponents, see McGee and McLaughlin (1995, pp. 209 ff.) and Soames (1999, ch. 7, passim). For criticism, see Wright (2007, pp. 419–23, this volume, Chapter  11). Some of the criticisms there lodged are presented as depending on higher-order vagueness. I postpone to a future discussion the question whether they can survive in a qualified form if the conclusions of the present study are accepted. 6  At least one commentator (Fara 2001) has argued that this is impossible. I beg to differ—but the example could easily be reworked so as to finesse the issue.

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The Illusion of Higher-Order Vagueness  339 followed by some orange ones, the whole giving the impression of a perfectly seamless movement, without regression, from red to orange. Case 2: You have a collection of pearls, artificial pearls and costume (plastic) pearls, and, again, have to arrange them in a monotonic series; spe­cif­ic­ al­ly, a series such that the first selection is a pearl and each subsequent selection is immediately preceded by something whose case to be a pearl is at least as strong. Then your selection will consist in a string of pearls, followed by a string of artificial pearls, followed by the fakes. The thought suggestive of higher-order vagueness is then simply this. Both series—we are currently supposing7—contain indeterminate cases, conceived as generated by semantic incompleteness. However, in the pearl series, the transitions from the pearls to the indeterminate cases, and from the latter to the non-pearls, occur sharply, at specific places. And, associatedly, there is no second-order indeterminacy—no indeterminacy in turn in the pearl-in­de­ter­ min­ate and indeterminate-fake pearl distinctions. So, the thought occurs, how to explain the manifest difference in the phenomenology of the changes occurring within the two series if not by postulating second and, indeed, indefinitely higher-orders of indeterminacy in the red-to-orange series? How else to accommodate the fact that we are absolutely at a loss to identify specific first and last borderline cases of the red–orange distinction in that series, or indeed abrupt changes of any kind? The key thoughts again: the vagueness of pearl and red is held to consist in the existence of borderline cases of these concepts, conceived as items that are not definitely classifiable as ‘pearls’, or as ‘red’, and not definitely classifiable as something else, on account of the semantic incompleteness of the relevant expressions. The sharpness of the distinction between the pearls and the borderline pearls shows in the abruptness of the transition between them in the relevant monotonic series. By contrast, the smoothness of the transitions between the reds and the borderline cases, and between the borderline cases and the oranges, enforces the idea that these distinctions are vague in turn. So it follows that they too admit of borderline cases. And so on ad infinitum. We can call the driving intuition here the seamlessness intuition.8 In general: unless we have an indefinite hierarchy of kinds of borderline case, it seems there will have to be sharp boundaries in any process of transition

7  In case it is not obvious, I do not think that this is the right way to conceive of the vagueness of the ‘usual suspects’. 8  I prefer ‘seamlessness’ to ‘continuity’. The relevant notion is pre-mathematical and intuitive. Cf. Fara (2003).

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340  The Riddle of Vagueness between instances of one vague concept and instances of another. Or so it anyway appears. But we will return to explore this thought in some detail. The resulting broad conception of full-blown higher-order vagueness: the conception of an infinite hierarchy of kinds, each potentially serving to provide an exclusion zone and thereby prevent a sharp transition, in a suitable series, between instances of distinctions exemplified at the immediately preceding stage of the hierarchy, may be termed the Buffering view. I shall argue for each of the following claims: (i) That the Buffering view is not well motivated by either the ineradicability or the seamlessness intuitions. (ii) That there is serious cause to question whether the Buffering view is fit for purpose. (iii) That for the kinds of vague concepts—the ‘usual suspects’—in which we are interested, the view that they exhibit higher-order vagueness on the model of the Buffering view is at odds with the broadly correct conception of their (‘first-order’) vagueness.

III Potential Confusions about Higher-Order Vagueness—Three Distinct Notions Within limits disrespected by Humpty Dumpty, philosophers are free to mean by the phrase ‘higher-order vagueness’ whatever they choose. But the fact is that at least three distinct putative phenomena have been earmarked by it in the literature, without—perhaps—all of those who have so earmarked them being clear that their discussions concerned potentially different things. One is (a) That the distinction between the things to which a vague expression applies and its first-order borderline cases—the cases where it is in­de­ter­ min­ate whether it or its complement applies—does itself, in the cases that characteristically interest us, admit of borderline cases; that the distinction between the things to which a vague expression applies and this secondorder of borderline cases also admits of borderline cases; that the distinction between the things to which a vague expression applies and this third-order of borderline cases also admits of borderline cases; and so on indefinitely.

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The Illusion of Higher-Order Vagueness  341 When, in the fashion noted, borderline cases are thought of as an intermediate kind, distinguished from the kinds of which they are borderline cases, this idea becomes the Buffering view.

Standing apparently unrelated to that is (b) The vagueness of Vague: there are concepts which are borderline cases of the vague–precise distinction itself—concepts which are neither definitely vague nor definitely precise—and, further, there are borderline cases of membership of this range of concepts in turn, and borderline cases of those in turn. . . and so on.9

Then finally there is the thought (c) That the usual kind of definiteness operator—that is: one introduced for the purpose of allowing us to characterize the borderline cases of F in accordance with the Basic Formula—ineluctably gives rise to a hierarchy of new, pairwise inequivalent vague expressions, ‘definitely F ’, ‘definitely def­in­ ite­ly F’, and the like (see, for example, Williamson 1999). (Definitization modifies truth conditions but does not eliminate vagueness.)

It seems obvious enough that there is little connection between (b) and the other two. It seems quite consistent with holding to the Buffering view, or with thinking of ‘definitely P’ as vagueness-inheriting though precisionincreasing when applied to a vague claim P, that the notion of vagueness itself should divide all expressions into two sharply bounded kinds—that there is never any vagueness about the question whether an expression is vague or not. Conversely, one might think of the distinction between vague expressions and others as admitting of borderline cases but hold to a view of the nature of vagueness according to which there are no higher-order borderline cases; and one might simultaneously just repudiate any operator of def­in­ite­ness, or take the view that any legitimate such operator generates only precise claims. At any rate, these are all prima facie compatibilities. If there are deeper tensions, that would be interesting—but they remain to be brought out. I will say nothing further here about thesis (b). Of potentially more im­port­ ance for our purposes is the apparent distinctness of thesis (a) and thesis 9  This discussion seems to originate in Sorensen (1985). See Hyde (1994; 2003) and Varzi (2003)].

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342  The Riddle of Vagueness (c), the thesis that applications of the definiteness operator, while they shift truth conditions (since they take any originally indefinite claim to a false one), are never­ the­less impotent to eliminate vagueness: if P is vague, so is ‘definitely P ’. Thesis (a) takes the distinction between F and (any order of) its borderline cases to be vague. F ’s higher-order vagueness consists, at each nth order, n >1, in the (potential) existence of borderline cases of the distinction between F and its borderline cases of the immediately preceding order. The thought embodied by thesis (c), by contrast, changes the terms of the relation of mutual vagueness. At second order, for example, it is not F but ‘definitely F ’ that is assigned a vague borderline. More specifically, letting ‘Def ’ be the def­in­ite­ness operator, the ‘second order’ of borderline cases countenanced by thesis (c) may be schematised thus:    Ø Def (Def (F ))& ØDef (ØDef (F )& ØDef (ØF )). And in general each successive nth order of vagueness, n > 1, is conceived as consisting in the vagueness of the boundary between the Defn−1(F)s—the things that are definitely . . ..definitely (n−1 times) F—and the definite borderline cases of order n−1, that is, as consisting in the (potential) existence of cases satisfying the condition:    Ø Def n (F ) & ØDef (Borderlinen -1F ) Now, as a construal of the notion of higher-order vagueness as suggested by the ineradicability and seamlessness intuitions, thesis (c) initially just seems wayward. Those intuitions motivate a thesis about the existence of a hierarchy of orders of vagueness of a single originally targeted concept. Thesis (c) by contrast goes in for a hierarchy of kinds of first-order vagueness which successively concern different concepts: definitely F, definitely definitely F, . . . and so on,—a hierarchy produced as an artefact of the introduction of the def­in­ite­ness operator. The preoccupation of much of the discussion with thesis (c) might therefore seem to offer one more example of philosophers taking their collective eye off the ball. It is hardly intuitively evident that natural language contains any operator that behaves like this. And, even if it does, what can that have to do with the proper understanding of the nature of vagueness, which presumably comes fully formed, as it were—and therefore fully ‘higher-orderized’, if the phenomenon is indeed real—even in languages lacking any definiteness operator? Aspects of the behaviour of such an operator cannot constitute higher-order vagueness as originally motivated. What does thesis (c) have to do with anything?

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The Illusion of Higher-Order Vagueness  343 Here is one arguable connection. When the first-order borderline cases of the distinction between F and its negation are characterized by the Basic Formula, they will be, one and all, things that are not definitely F. So they will fall under the negation of ‘definitely F ’ and will thus, none of them, be borderline-cases of ‘definitely F ’.10 Now thesis (a) requires that there are borderline cases of the distinction between F and its first-order borderline cases. These will all, presumably, be clear cases of ‘not definitely not F ’. So, if they are borderline cases of the Basic Formula’s characteristic conjunction, they must be borderline cases of ‘not definitely F’. But, if they were definite cases of ‘def­ in­ite­ly F ’, they would not be borderline cases of its negation. So they must be borderline cases of ‘definitely F’ too, which is therefore vague if thesis (a) is true of F and borderline cases are characterized by the Basic Formula. Very well. However, thesis (c) involves two components: that definitization does not eliminate vagueness, just argued for, and that it generates statements which are not, in general, equivalent to those definitized. Since it is, intuitively understood, a factive operation, the second component is tantamount to the claim that a definitized statement is in general logically stronger than its prejacent. This too is, as will emerge, plausibly taken to be a consequence of thesis (a) and the characterization of borderline cases given by the Basic Formula. What about the converse direction? Is thesis (a) a consequence of thesis (c), assuming the Basic Formula? Again, arguably so. Let G be any predicate such that the F–G distinction is vague. Then F has borderline cases, characterized as cases which are not definitely F and not definitely G. But by thesis (c), ‘def­ in­ite­ly F’ is vague if F is. And, since by hypothesis G is vague, so likewise is ‘definitely G’. Since vagueness is, presumably, preserved under negation, ‘not definitely F’ and ‘not definitely G’ are likewise vague. Since vagueness is presumably preserved under (consistent) conjunction, so is ‘not definitely F and not definitely G’—so the notion of a borderline case of F is itself vague, and hence has borderline cases. These cannot be definite cases of F or they would fail the first conjunct and hence not be borderline cases of the conjunction. So they must be borderline cases of F and of the notion: borderline case of F and G. The latter notion is then available for choice in place of ‘G’, and the reasoning can be iterated indefinitely. So, given that the vagueness of a predicate consists in its susceptibility to borderline cases and the thesis that these are one and all to be characterized as per the Basic Formula, there is a case—we can put it no stronger than that—that thesis (a) and thesis (c) are equivalent. If that is right, it offsets the 10  This step, nota bene, applies Dummett’s Principle. See n. 2.

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344  The Riddle of Vagueness charge of irrelevance against intended investigations of higher-order vagueness that have taken thesis (c) to be a constitutive matter. On the other hand, if thesis (a) depicts an illusion, the equivalence will mean that the illusion persists in thesis (c) as well. Work on the semantics and proof theory of the definiteness operator directed towards the elucidation and stabilization of thesis (c) will then be so much misdirected effort.

IV The Basic Formula and Lack of Sharp Boundaries So let us assume for the sake of argument that borderline cases are felicitously described by the Basic Formula, and—thesis (a)—that certain concepts sustain an infinitely ascending hierarchy of orders of borderline case, each characterizable by a suitable application of the Basic Formula. What reason is there, in this setting, to think that the definiteness operator should comply with the proof-theoretic part of thesis (c): the claim that definitization increases logical strength? In fact there is quite powerful pressure towards that thought. It comes from reflection on that form of the Sorites paradox—what I once called the No Sharp Boundaries paradox—which seems to connect most directly with the very nature of vagueness (Wright 1987a, this volume, Chapter 4). I will make the point in some detail over this and the succeeding section. The standard form of major premise for the Sorites is a universally quantified conditional, usually motivated by tolerance intuitions. But the major premise for the No Sharp Boundaries paradox takes the form of a negative existential, (i)  Ø($x )(Fx & ØFx ¢), seemingly tantamount merely to the affirmation that F is indeed vague in the series in question. For vagueness is just the complement of precision, and precision (relative to the relevant kind of series) is, it seems, perfectly captured by    ($x )(Fx & ØFx ¢). But, whereas it may be doubted that vague predicates really are tolerant, it hardly seems doubtful that they really are vague! In affirming (i), accordingly,

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The Illusion of Higher-Order Vagueness  345 we seem merely to have affirmed that F is vague.11 So vagueness appears paradoxical per se. Enter the definiteness operator. What, it may be suggested, really constitutes precision is a sharp boundary between definite cases. Hence what is really tantamount to an expression of F ’s vagueness in the relevant series is not the negative existential statement (i) above but rather: (ii)  Ø($x )(Def (Fx )& Def (ØFx ¢)) —the thesis that there is no last definite case of F in the series immediately followed by a first definite non-F. But (ii), unlike (i), gives rise to no immediate paradox. We can show, of course, by appeal to it that any n such that Def(¬Fn′), must be such that ¬Def(Fn). But then—absent further proof-theoretic resources for the definiteness operator—we seem to have no means to commute the occurrences of ‘¬’ and ‘Def ’ to generate something soritical. What, though—other than the reflection that we can apparently finesse the paradox thereby—is available to justify the claim that it is indeed (ii), rather than (i), that gives proper expression to F’s vagueness in the kind of series in question? There is a very good argument for that claim if we can legitimately have full recourse to classical logic. Take it that what F’s vagueness in the series consists in is the presence there of (first order) borderline cases of F, and that these are suitably characterized by the Basic Formula. Specifically, suppose that there is such a borderline case of F: 11  We obtain a Sorites paradox from the negative existential major premise without reliance on any distinctively classical moves, by running right-to-left, as it were—by beginning with a minor premise of the form ¬Fa, and reasoning through successive steps via the rules for conjunction, existential introduction, and the (intuitionistically acceptable) negation-introduction half of reductio. It merits emphasis that the intuitive motivation for the major premises for Sorites paradoxes varies quite dramatically across forms that are classically equivalent. Consider, for instance, the three genres of premise: (i)   ("x )(ØFx Ú Fx ¢) (ii)  ("x )(Fx ® Fx ¢) (iii)  Ø($x )(Fx & ØFx ¢) The last, as noted, is naturally motivated just by the thought that it is constitutive of the vagueness of a predicate that its extension in a suitably constructed series of objects not run right up against that of its negation. This thought involves no intuitive dependence on Bivalence. The second is driven, more specifically, by tolerance intuitions, of the kind discussed in Wright (1975, this volume, Chapter 1), that in turn draw on folk-semantical ideas about observational and phenomenal predicates which have little explicit connection with vagueness. These ideas, again, involve no intuitive dependence on Bivalence but are stronger than the thought that motivates (iii) since someone who embraced a ‘Third Possibility’ view of borderline cases could accept (iii) while rejecting (ii): vagueness might be conceived as, in typical cases, intolerant of the distinction between some Fs and some borderline cases of F, even though sustaining No Sharp Boundaries principles in the form of (iii). (i), finally, is entailed by either of the other two if, but only if, Bivalence is assumed for predications of F. It is thus natural to conceive of (i) through (iii) as of decreasing strength. It is a significant weakness of the classical outlook that it stifles these intuitive differences.

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346  The Riddle of Vagueness (iii)  ($x )( ØDef (Fx )& ØDef (ØFx )) but also, for reductio, that there is a last definite case of F in the series immediately followed by a first definite non-F: (iv)  ($x )(Def (Fx )& Def (ØFx ¢)). Contradiction follows on the assumption of the monotonicity of the series (intuitively, that all the F-relevant changes manifested in it are one directional), which we may capture by the pair of principles:    ("x )(Def (Fx ¢) ® Def (Fx )) —the immediate predecessor of anything definitely F is definitely F—and    ("x )(Def (ØFx ) ® Def (ØFx ¢)) —the immediate successor of anything that is definitely not-F is likewise def­ in­ite­ly not-F. For suppose m is a witness of (iv); that is,    Def (Fm )& Def (ØFm¢). Then the monotonicity principles will ensure that every element preceding m in the series is definitely F and every element succeeding m′ is definitely notF; and hence that none satisfies the rubric for borderline cases given by the Basic Formula, contrary to (iii). We supposed that the vagueness of F in the series in question consists in the presence of borderline cases of F, as characterized by the Basic Formula. The reasoning we just ran through establishes that one who accepts that supposition thereby commits themselves to (ii). So, in order to show that it is (ii), not the soritical (i), that is tantamount to an acceptance that F is vague in the series in question, we now require the converse direction: that someone who accepts that there is no last definite F element immediately succeeded by a first definite non-F element is thereby committed to the existence of borderline cases of F in the series concerned, as characterized by the Basic Formula. Straightforward—though classical—reasoning establishes the point. The series, we can take it, is such that    (1) Def(F0) and (2) Def(¬Fn)

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The Illusion of Higher-Order Vagueness  347 Suppose (ii) above and for reductio the negation of (iii):      (3) ¬(∃x)( ¬Def(Fx) & ¬Def(¬Fx)), —there are no borderline cases of F in the series. Then   (4) Def(¬Fx´) → ¬Def(Fx), — from (ii). So    (5) ¬Def(Fn−1), — from (2) and (4). Suppose  (6) ¬Def¬(Fn−1). Then    (7) (∃x)( ¬Def(Fx) & ¬Def(¬Fx)), —contrary to 3. So     (8) ¬¬Def(¬Fn−1). This routine may be repeated eventually culminating in contradiction of 1. At that point (3) may be discharged by reductio, on (1), (2) and (ii) as remaining assumptions, a final step of double negation elimination then yielding (iii). Our result, then, is that—granted classical logic—F ’s vagueness, identified with its possession of borderline cases as characterized by the Basic Formula, is equivalent not to the soritical (i)  Ø($x )(Fx & ØFx ¢), but to the apparently harmless (ii)  Ø($x )(Def (Fx )& Def (ØFx ¢)). It is the latter, then, which, we may accordingly be encouraged to think, is the canonical expression of F’s lack of sharp boundaries in the relevant kind of series. This result is the first point towards uncovering the advertised impetus towards the proof-theoretic component of thesis (c). I will pursue that further in the next section. It may also seem (as it once did to me) to be the first step towards a dissolution of the No Sharp Boundaries paradox. Obviously, however, it is at most a first step. For one thing, the reliance on classical logic is, of course, of some significance in this context. The question under review is whether, and if so, how a correct understanding of the nature of vagueness escapes a commitment to a soritical version, such as (i), of the No Sharp Boundaries intuition. In exploring the matter, we therefore must resort only to principles of inference which are sound for vague languages. Those who share the doubts of the present author whether classical logic is in that case should therefore regard the reasoning just run through with at most qualified enthusiasm.

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348  The Riddle of Vagueness Even were we satisfied that classical logic is fit for duty in this setting, however, there is a further issue. For, unless we are prepared to allow that the boundary between the definite Fs and the borderline cases of F is sharp, there is the same intuitive motivation as previously to affirm (i)*  Ø($x )(Def (Fx )& ØDef (Fx ¢)), and this, if allowed, will in turn subserve a Sorites paradox (this time subverting the distinction between the borderline cases and the definite cases of F). To be sure, the reply can be that the proper way to do justice to the vagueness of the second-order borderline is to affirm not (i)* but (ii)*  Ø($x )[Def (Def (Fx ))& Def (ØDef (Fx ¢))] —there is no sharp cut-off separating the definite cases of ‘definitely F’ from the definite borderline cases of F. And in general, for an arbitrary pair of mutually vague, contrary concepts φ and ψ, exemplified in the series in question, it may be proposed, generalizing the reasoning above, that the proper way to give expression to a lack of sharp boundaries between them is to affirm the negative existential, (*)  ¬(∃x)(Def (φx) & Def (ψx′)) So we need never, apparently, be committed at any level to a soritical claim. But where is this leading? If the seamlessness intuition is to be upheld, then it seems that it must be possible, in principle, so to describe a Sorites series that no abrupt transitions of any relevant kind take place between adjacent elements within it. So every pair of contrary concepts φ and ψ mani­fest­ed in the series must sustain the truth in it of the relevant instance of (*). More specifically: if the mutual vagueness of any pair of concepts Def(…x…) and Def¬(…x…) is viewed as consisting in the existence of borderline cases as characterized by the Basic Formula, and, if the seamlessness intuition is accepted, then we are committed to each of the following principles:    Ø($x )(Def (Fx )& Def (ØFx ¢))    Ø($x )[Def (Def (Fx ))& Def (ØDef (Fx ¢))]    Ø($x )(Def [Def (Def (Fx ))]& Def [ØDef (Def (Fx ¢))])   . . . and so on.

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The Illusion of Higher-Order Vagueness  349 Given the reliance on classical logic of the reasoning worked through above, it would be tendentious to proclaim these Gap principles12 to be respectively characteristic of the putative successively higher-orders of borderline case of the predicate F. But they are at least, it may seem, among our commitments if we accept that a series is possible in which a seamless, monotonic transition is effected from instances of F to instances of not-F, and in which any borderline cases of any distinction exemplified within it are characterised by the Basic Formula as applied to that distinction. Let us take stock. It is hard to reject the idea that the seamlessness intuition is sound in some form: the transition from Fs to non-Fs in a Sorites series can be effected without abrupt, noticeable change of status at any point. The thought that leads from seamlessness to the postulation of higher-order vagueness can be refined as follows. Define a monadic predicate (open sentence) as F-relevant if it is formulated using just F, the truth functional connectives, and the definiteness operator. Conceive of seamless transition as the circumstance that the ranges of each pair of incompatible F-relevant predicates exemplified in a Sorites series running from instances of F to instances of its negation are buffered: between the instances of any such pair of predicates intervenes at least one element to which neither definitely applies: an element which is a borderline case of the distinction they express, according to the characterization of borderline cases given by the Basic Formula. As we saw, this conception, assuming monotonicity in the transition concerned, ensures that a Gap principle—an instance of (*)—holds for any such pair of predicates. On classical assumptions, the holding of such a Gap principle is equivalent to the presence in the series of a borderline case, characterized as per the Basic Formula, of the original distinction. So the train of thought is this: • Seamlessness requires buffering of all F-relevant distinctions exemplified in the series. • Such buffering requires the presence, in the series, of borderline cases (characterized as per the Basic Formula) of each such distinction. • The presence of such borderline cases requires (indeed, classically, is tantamount to) the holding of appropriate Gap principles. That said, though, note that a plausible connection between seamlessness and the Gap principles can, of course, be made out more directly. If any of the 12  Delia Fara’s nice term in Fara (2003). Each such principle (Fara actually formulates them slightly differently) classically ensures that the instances in a suitable series of a pair of contrary concepts of the form Defn(φx) and Def(¬Defn−1(φx)) are separated by a gap—in our terminology above, a buffer zone.

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350  The Riddle of Vagueness existential statements which the Gap principles respectively directly contradict is true in a Sorites series, then there is an abrupt, non-seamless change of status between the element that witnesses that statement’s truth and its immediate successor. So seamlessness, it appears, requires the Gap principles to hold anyway, whether or not we take that to be equivalent, as classically it is, to the presence of borderline cases of each appropriate higher order.13

V Thesis (c) and the Paradox of Higher-Order Vagueness Let us now connect the foregoing with the proof-theoretic component of thesis (c). I once argued that, so far from resolving the No Sharp Boundaries paradox, to corral our no-sharp-boundaries intuitions into an endorsement of principles of the (*)-form merely generates new soritical problems.14 The argument utilized a proof theory incorporating the rule:    ( DEF )       

 

{A1 ¼ An } Þ P {A1 ¼ An } Þ Def (P ),

13  Note that anyone content with classical logic in this region who accepts the idea that seamless transition is possible and that it is correctly construed as requiring the Gap principles to hold en masse should worry about this: that no finite Sorites series can exemplify borderline cases of every higher order unless some borderline cases instantiate multiple, indeed infinitely many orders. (This is noted at Fara 2003, p. 205.) Given the ways, reviewed earlier, in which acceptance of higher-order vagueness is standardly motivated, this—egregious violation of Dummett’s Principle—is an idea for which we are wholly unprepared, indeed an idea of questionable intelligibility. 14  Wright (1992a, this volume, Chapter 5). The argument was there presented as a reductio of the very idea of higher-order vagueness. In fact, what it puts under pressure is any set of assumptions entailing an nth-order Gap principle, n > 1. The picture of higher order vagueness captured by the Buffering view incorporates one such set of assumptions, as we have seen. But we have also noted that the very idea of seamless transition appears to enforce the Gap principles as well. Focused on the case second-order Gap principle, presumed itself to be a Definite truth, the argument was this: (1) Def¬(∃x)[Def(Def(Fx)) & Def(¬Def(Fx´))] 2 (2) Def(¬Def(Fk´)) 3 (3) Def(Fk) 3 (4) Def(Def(Fk)) 2,3 (5) (∃x)[Def(Def(Fx)) & Def(¬Def(Fx´))] 1 (6) ¬(∃x)[Def(Def(Fx)) & Def(¬Def(Fx´))] 1,2 (7) ¬Def(Fk) 1,2 (8) Def(¬Def(Fk)) 1 (9) Def(¬Def(Fk´)) → (Def(¬Def(Fk))

1

Assumption Assumption Assumption 3, DEF. 2, 4, ∃-intro. 1, Def-elim. 3, 5, 6, Reductio 7, DEF 2, 8 Conditional Proof

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The Illusion of Higher-Order Vagueness  351 where {A1 . . . An} contains only ‘fully definitized’ propositions (that is, pro­ posi­tions prefixed by ‘Def ’). Once Def ’s proof theory incorporates this rule,15 each of the Gap principles corresponding to the successive higher orders of vagueness becomes soritical.16 But the Gap principles, as we have seen, are seemingly imposed by the possibility of seamless transition across a sorites series. Moreover, classically, each is tantamount to—and each is anyway a consequence of—an affirmation of the existence of a corresponding order of borderline cases, when characterized in accordance with the Basic Formula. So the postulation of any higher order of borderline cases is soritical unless the DEF rule fails. And, if seamless transition does indeed entail the Gap principles, then—even without classical logic—we must likewise accept that the DEF rule fails provided we believe that seamless transition is possible.17 To reject the DEF rule is to allow that Def(P) can be a consequence of a set of (fully definitized) premises, even though Def(Def(P)) is not. Since the entailment from Def(Def(P)) to Def(P) is unquestioned, to reject the DEF rule is thus to regard the definitization of a sentence as potentially increasing its logical strength. That is the proof-theoretic component of thesis (c).

VI A Revenge Problem for the Buffering View Let us review the dialectic to this point. In the cases that interest us (the ‘usual suspects’), it is not, claimed Dummett and Russell, possible to eliminate vagueness by annexing a new expression to the borderline cases of a 15  In effect, just an S4 rule for ‘Def ’. 16 See the proof schema illustrated in n. 14. Note that the general applicability of the schema assumes, in addition, that the Gap principles are definite truths, and that there are definite borderline cases of the relevant order. These points would need defence in a fully rigorous presentation of the line of thought currently under development. 17  This précis ignores a number of subtleties. As Richard Kimberly Heck (1993) pointed out, the reasoning of my original ‘paradox’ of higher-order vagueness involved, besides the DEF rule, free recourse to standard rules allowing for the discharge of assumptions, specifically reductio ad absurdum and conditional proof. The DEF rule is under pressure from the paradox only if its combination with the standard introduction rules for the conditional and negation is acceptable. But one might independently doubt that. There are a variety of conceptions of the meaning of ‘Def ’ which will have the effect that the deduction theorem fails: for instance, any broadly many-valued set-up will underwrite a failure of the deduction theorem which (i) construes entailment as preservation of a designated value, (ii) regards DefP as designated if P is, but as taking a lower undesignated value than P when P is undesignated, and (iii) regards the conditional as undesignated just when its consequent takes a lower value than its antecedent. One of the interesting points about Fara’s (2003) reconstruction of the paradox is that it obviates the need for conditional-introduction steps.

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352  The Riddle of Vagueness distinction, since the distinctions between items to which the new expression applies and those that fall under either of the original concepts will both remain vague. However, it is typically possible so to arrange the elements of a soritical series for a concept φ that an apparently seamless transition is effected from instances of it to instances of some contrary concept, where seamlessness involves that no salient, relevant changes occur between any element of the series and its successor. Higher-order vagueness is meant to provide a natural and plausible explanation of both these putative items of data. Annexure of a new expression to the borderline cases of a distinction never results in precision because the concept to which the term is thereby annexed is itself a vague concept in its own right. Seamless transition is possible because it is possible so to en­gin­eer a soritical series that every pair of contrary concepts manifested within it are buffered by borderline cases of their contrast. This in turn requires the failure of the DEF rule, if Sorites paradoxes are not to recur. Where P is vague, Def(Def(P)) must in general be logically stronger than Def(P), although still vague.18 There are a number of issues on which a fully satisfactory development of the Buffering view would have to elaborate. Three in particular are especially salient. First, it will not do, obviously, just to reject the DEF rule on the grounds that paradox will otherwise be reinstated. Rather, an explanatory semantics is wanted for the definiteness operator to underwrite the failure of the rule and explain more generally what form an appropriate proof theory for the operator should assume. Second, any genuinely explanatory such semantics had better be grounded in further insight into the nature of borderline cases—an insight somehow serving to explain why the borderline cases of any vague distinction are themselves a vaguely demarcated kind. Third, it needs to be explained how exactly a finite Sorites series can indeed provide for a seamless transition between incompatible descriptions. It is not enough to gesture at the idea of buffering by borderline cases: we need to be told in detail how a seamless transition may be fully adequately described, according to the Buffering view.19 18  But see n. 17. 19  This problem—what Mark Sainsbury (1992) christened the Transition Question—for any adequate account of vagueness has not drawn the attention in the literature meted out to other problems of vagueness. It is in effect the issue raised by the Forced March sorites: the problem of explaining how a competent subject who is charged to give nothing but correct, maximally informative verdicts may respond, case by case, to the successive members of a soritical series without at any point committing themselves to some kind of abrupt (and incredible) threshold. If the Buffering view can genuinely provide an account of seamless transition, it will provide the descriptive resources that the hapless subject of the Forced March needs. I shall pour cold water on the prospects—and, in a sense, on the problem—later.

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The Illusion of Higher-Order Vagueness  353 I do not believe that the Buffering view can deliver on these obligations. I shall not here, however, further consider what might be done to address the first.20 For the second, the notion that the borderline cases of a vague distinction constitute a further vague kind taking a place, so to speak, in the same broad space of possibilities as the poles of that distinction—this notion is exactly the illusion that I aim to expose. The third issue—the Transition Problem—will occupy us in the next section. The task for this section is to table an argument that, even before any further development is attempted, the Buffering view is susceptible to a new paradox. The paradox is a kind of ‘revenge’ problem, consequent on the possibility— as it appears—of defining a distinct operator of absoluteness in terms of that of definiteness as follows:

Abs(P ) is true if and only if each Def n ( P ) is true for arbitrary finite n.



There seems no reason to contest that such an operator is well defined if Def is, nor that, intuitively, it should have some actual cases of application. Consider, for instance, Kojak, a man microscopic examination of whose scalp—under whatever degree of magnification—reveals no distinction, in point of the presence of hair fibres, from the surface of a billiard ball. Does it make any sense to suppose that any of

  

Def ( Kojak is bald ) , Def 2 ( Kojak is bald ) , Def 3 ( Kojak is bald )¼ Def n ( Kojak is bald ) ,¼



fails of truth or is somehow less acceptable than a predecessor in the series? By its definition, Abs(P) entails Def(P); so in particular any statement of the form Abs(At) entails Def(At), and therefore any statement of the form (∃x)Abs(Ax) entails the corresponding (∃x)Def(Ax). Contraposing, any statement of the form ¬(∃x)Def(Ax) entails the corresponding ¬(∃x)Abs(Ax). Since any Gap principle for definiteness is—assuming that Def distributes across conjunction and collects conjuncts in the obvious way—equivalent to something of the former form, acceptance of any Gap principle for def­in­ite­ ness is a commitment to acceptance of the corresponding Gap prin­ciple for absoluteness. 20  For development of some misgivings about the ability of supervaluational approaches, at least, to deliver on this aspect, see Fara (2003).

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354  The Riddle of Vagueness That is all as intuitively it should be. But now observe that, whatever the position with Def, the absoluteness operator, so defined, has to be iterative across the conditional.21 So the effect, just provided that the relevant Gap principle is itself absolute, and that the relevant polar verdicts are assumed absolute, is to reintroduce a version of the No Sharp Boundaries paradox. The proof is just the obvious adaptation: (1) Abs(¬(∃x)[Abs(Abs(Fx)) & Abs(¬Abs(Fx´))] 2 (2) Abs(¬Abs(Fk´)) 3 (3) Abs(Fk) 3 (4) Abs(Abs(Fk)) 2,3 (5) (∃x)[Abs(Abs(Fx))) & Abs(¬Abs(Fx´))] 1 (6) ¬(∃x)[Abs(Abs(Fx)) & Abs(¬Abs(Fx´))] 1,2 (7) ¬Abs(Fk) 1,2 (8) Abs(¬Abs(Fk)) 1 (9) Abs(¬Abs(Fk´)) → Abs(¬Abs(Fk)) 1

Assumption —absoluteness of 2nd order Gap principle for Abs Assumption of polar absoluteness Assumption for reductio (3), iterativity of Abs (2),(4), ∃-intro. (1), Abs-elim. 3,5,6, RAA. 7, iterativity and closure for Abs 2,8 CP.

In sum: the Gap principles may or may not be directly soritical when augmented by whatever may prove to be the appropriate proof theory for Def. But, even if they are not, there seems no objection to introducing the Abs operator as defined, if there is no objection to Def in the first place. If, as argued, Abs is iterative, and if it is an absolute truth that a (first-order) borderline case of F is not an absolute case of F, and if the Gap principles for Def are absolute truths

21  This excellent observation is due to Elia Zardini. Here is a sketch of one plausible demonstration of it: 1 (i) Abs(A) Assumption 1 (ii) Def(A) & Def(Def(A)) & . . .…. (i) Definition of Abs 1 (ii) Def(Def(A)) & Def(Def(Def(A))) & . . ... (ii) &E 1 (iv) Def[Def(A) & Def(Def(A)) & . . .….] (iii) collection for Def over conjunction 1 (v) Def(Abs(A)) (iv) Definition of Abs   (vi) Abs(A) → Def(Abs(A)) (i), (v) Conditional Proof   (vii) Def[Abs(A) → Def(Abs(A))] (vi) Def Intro — see below*   (viii) Def(Abs(A)) → Def(Def(Abs(A))) (vii) Closure of Def over entailment 1 (ix) Def(Def(Abs(A))) (v), (viii), MPP   (x) Abs(A) → Def(Def(Abs(A))) (i), (ix) Conditional Proof and so on. Thus each Defn(Abs(A)) can be established on Abs(A) as assumption. Abs(Abs(A)) is accordingly a semantic consequence of Abs(A). * The principle appealed to is that if ⊨A, then ⊨Def(A). This should be uncontroversial—presumably all necessary truths are definite.

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The Illusion of Higher-Order Vagueness  355 (whence those for Abs are also), then the Gap principles for Def do ultimately spawn a Sorites paradox in any case, even if they are innocent of paradox when worked on merely via the appropriate proof theory for Def.

VII The Transition Problem No doubt, there are lines of resistance for a defender of Gap principles to explore.22 But we must delay no further in attending to a more basic difficulty which has been shadowing the discussion all along and is in the end, I suggest, decisive that the attempt to capture the seamlessness intuition by means of an apparatus of ascending Gap principles, a fortiori by means of limitless buffering,23 is fundamentally misconceived. Let us step back. The seamlessness intuition, as interpreted by the Buffering view, has it that, in any Sorites series for a concept F, no pair of adjacent elem­ ents are characterized by incompatible F-relevant predicates.24 Somehow a seamless transition is effected from (Definiten) Fs at one end to (Definiten) non-Fs at the other. The move to an apparatus of Gap principles is a response to this thought, which interprets it as requiring that every incompatible pair of predicates, Φ and Ψ, formulable using just F, Def and negation, which are exemplified in the series must be buffered—there have to be intermediate elem­ents whose strongest F-relevant characterization is compatible with both Φ and Ψ. These are the borderline cases of the Φ–Ψ distinction. One direct corollary of this way of handling seamlessness which it is time— rather belatedly—to take proper note of is that, if the Basic Formula is to offer a viable characterization of borderline cases, we have to think of ‘¬Def(Φx) & ¬Def(Ψx)’ as compatible with both Φx and Ψx. So ‘Φx & ¬Def(Φx)’ has to be a consistent description; and hence, it appears, we have after all to take ser­ious­ly the possibility that there are items which satisfy it—things which, while being a certain way, are not definitely that way. Dummett’s Principle has to be repudiated if

22  One is to query the status of the minor premises. To treat the reasoning outlined as a Sorites paradox, properly so termed, requires that its conclusion—Abs(¬Abs(F0))—confounds an acceptable such premise. Indeed it does if F(0) is absolutely true. But, if F(0) were, say, merely definitely true (!), might that not be consistent with its also being an absolute truth that it is not absolutely true? For considerations in this direction, see Williamson (1997a) and Dorr (2010). 23  Which, recall, is classically the same thing. 24  Recall that a predicate is F-relevant if it is formulated using just F, negation, conjunction, and the definiteness operator.

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356  The Riddle of Vagueness the Buffering view is to have any chance of delivering seamlessness. And with it goes any Third Possibility interpretation of borderline status. The rejection of Dummett’s Principle can easily seem like nonsense. We might try to set aside that impression as owing to the intrusion of inappropriate ­resonances associated with the English word ‘definitely’. We are, after all, it may be said, introducing a term of art for certain theoretical purposes. But that would be a pretty brass-necked response, given that it was exactly the resonances of the natural language word that made the Basic Formula seem apt in the first place. Be that as it may, the basic problem remains that, even after Dummett’s Principle is surrendered, the idea of limitless buffering in accordance with the Basic Formula, rather than providing for a lucid understanding of the possibility of seamless transition, seems, when pressed, merely to plunge into ­aporia. The difficulty is best elicited in the context of a version of the Forced March. Suppose you are the subject and that you have returned a correct verdict—Φ—concerning ­element m. If Φ and ¬Def(Φ) are compatible, then you now have the option of describing m´ as an instance of the latter without explicit concession of a change in Φ-relevant status. Well and good. Nevertheless, since Def is factive, some ­elements correctly describable as ¬Def(Φ) will be so because they are Ψ. And m´ had better not be one of those, or the transition from m to m´ will mark a sharp boundary in the series after all. On the other hand, if m´ is also Φ—as compatibly with its correct description as ¬Def(Φ) it may, after the jettison of Dummett’s Principle, now be—then the buffer zone is merely narrowed by one element and we can push on to m″ and raise the same possibilities again: is m″ an instance of ¬Def(Φ) because it is Ψ?—in which case there is a sharp boundary—or is it also an instance of Φ?—in which case the buffer zone narrows again. Obviously, the buffer zone must not narrow too far, or there will be a sharp cut-off between Φ and Ψ in any case. So it appears that we have to think in terms of there being cases which are correctly describable as ¬Def(Φ) but not because they are Ψ, and which also—if narrowing of the buffer zone is to be halted—do not exploit the compatibility of Φ and ¬Def(Φ) by being Φ. These cases will constitute a distinctive kind of borderline case between Φ and Ψ: cases that qualify for characterization in terms of the Basic Formula without exploiting the compatibility, after the surrender of Dummett’s Principle, of Φ with ¬Def(Φ) and of Ψ with ¬Def(Ψ). It is essential that such cases occur if a seamless transition is to be effected. For, if they do not, each case within the region characterized as ¬Def(Φ) and ¬Def(Ψ), will be either Φ and/or Ψ. So, to solve the Transition Problem, you—the subject—need to be provided with the means in principle, whatever epistemological difficulties you might encounter in practice, to mark the occurrence of such cases. But how can that be done?

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The Illusion of Higher-Order Vagueness  357 This is already a fatal objection to the prospects for solving the Transition Problem using the resources at hand, since we now appear to be committed to recognizing a kind of indeterminacy for which the apparatus of Φ-relevant and Ψ-relevant predicates and the Basic Formula provides no adequate means of expression—cases whose description in accordance with the Basic Formula masks their distinction from others which it also characterizes but which are, so to say, tacitly polar. There is, therefore, no prospect of your doing justice to seamless transition using just the notion of buffering by borderline cases, conceived in accordance with the Basic Formula, since we have given you no resources adequately to characterize the masked cases. But, even if we had, a second lethal consequence looms large. In order to preserve seamlessness, we now need to avoid the postulation of a sharp boundary between a last Φ and a first exemplar of this new genre of indeterminate cases, the non-tacitly polar instances of the Basic Formula applied to Φ and Ψ (let us call these the Δ’s.) So, on the Buffering view, we now need in turn to buffer the contrast between Φ and Δ, however exactly the instances of the latter are to be described. But, strategically, the means at our disposal are just the same as—and hence no better than—those just deployed for the Φ–Ψ distinction—except that now, of course, there are fewer elements to subserve the buffering of the distinction, since the Φ–Δ series is shorter than the Φ–Ψ one. Since exactly the same form of problem is going to recur at every stage and the series is finite overall, the strategy cannot succeed. The root of the trouble is that there is, simply, no satisfactory conception of what a borderline case is that is serviceable for the explanation of seamlessness. Obviously no ‘Third Possibility’ conception is to the purpose: if one is trying to explain seamless transition between contrasting situations, it does not help to interpose a third category of situation contrasting with both. But if, recoiling from that, we essay to think of the interposed category as compatible with each of the originally contrasted statuses (so dropping Dummett’s Principle), then in assigning an object to that category we fall silent concerning what if any shift from polar status it instantiates. To fall silent is not to explain anything. Moreover, when pressed, as we saw, it seems we are forced to postulate a ‘Third Possibility’ type of case—Δ-cases—after all. At which point, the game is effectively lost. We should conclude that there is no prospect of a stable elucidation of seamless transition by means of the conception of an endless hierarchy of orders of borderline cases. So, far from being well motivated by the possibility of seamless transition between instances of incompatible vague predicates, the Buffering view winds up in compromise and confusion.

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358  The Riddle of Vagueness Where does that leave the Transition Problem? Well, it is striking that the kind of difficulty just outlined will afflict any attempt to do justice to the nature of the changes, stage by stage, involved in a process of seamless transition across a finite series of stages between contrary poles. It has nothing especially to do with vagueness or our having recourse to the notion of a borderline case. For, suppose we have somehow turned the trick: we have somehow succeeded in fully correctly describing, stage by stage, a process of seamless transition. We will have had to say incompatible things about some of the stages. Let m and n be a pair where we did that and which are as close together as any pair where we did that. They will not have been adjacent. Let F be the description given of m, and G that given of n. So m´ will have received a verdict, F´, compatible with both F and G. Is F true of m´? If it is, then G is not. So, since compatible with G, F´ does not do full justice, in relevant respects, to m´, even if true of it. So, if we did somehow do full justice to all the stages, F cannot be true of m´. But then the series was not seamless after all: there is a sharp boundary at m. Conclusion: the Transition Problem is insoluble in any vocabulary if the ‘full justice’ requirement is enforced. So, far from demanding recourse to a baroque apparatus of borderline cases of arbitrarily high orders, the requirement that seamless transition somehow allow of a fully adequate description, stage by stage, was unsustainable all along. When the task is to explain how seamless transition is possible in a way that involves doing full justice, in all relevant respects, to the elements in a finite series that manifests as effecting such a transition, it is about as helpful to believe in higher-order vagueness as to believe in fairies. Dissatisfaction may persist. Forget about doing full justice to seamless transition. Do we not at least have invoked concepts of higher-order vagueness and buffering if we are to describe the relevant kind of series in a fashion consistent with seamless transition, even if the description does not do full justice to it? Well, no. Once the ‘full justice’ requirement is relaxed, and we need merely to avoid adjacent incompatibilities, we can perfectly well describe the stages of a seamless transition, without misrepresentation, using only precise vocabulary. Suppose Cameron grows seamlessly from 5 feet tall to 6 feet tall between their fourteenth and eighteenth birthdays and consider a series of appropriately dated true descriptions: Cameron is now exactly 5 feet tall Cameron is now exactly 5 feet tall, give or take an inch Cameron is now exactly 5 feet 1 inch tall Cameron is now exactly 5 feet 1 inch tall, give or take an inch

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The Illusion of Higher-Order Vagueness  359 . . . and so on. If the ‘full justice’ requirement is in force, the spandrel-plagued apparatus of the Buffering view is to no avail; if the requirement is not in force, and we are allowed to give less than all relevant information, it is easy to turn the trick without involving anything of the kind. One last try. Notice that, when the admissible substitutions for ‘F’ are restricted to predicates in the range used in the example in describing Cameron’s changing height, the result is not, of course, to provide a model of the original no-sharp-boundaries principle, (i)  Ø($x )(Fx & ØFx ¢) —since, for any choice of F in the range of predicates concerned, there will be a last case of which it is true. By contrast, is it not forced on us that each of the hierarchy of Fara’s Gap principles is true in a finite series exemplifying seamless transition between instances of contrary vague concepts? If so, then at least from a classical point of view, that enforces acceptance of the hierarchy of borderline kinds, even if we are thereby no better placed when it comes to doing justice to the phenomenon of seamless transition. But this has to be a bad thought. If, after we introduce the definiteness operator, seamlessness enforces the Fara Gap principles, then before we introduced the definiteness operator, it had already enforced the major premise of the No Sharp Boundaries paradox. What we considered earlier was an argument, impressive in the context of classical logic, that (i) is not an adequate capture of F’s vagueness, which is rather canonically expressed by (ii)  Ø($x )(Def (Fx )& Def (ØFx ¢)). Let that conclusion stand. Then the vagueness of F, qua canonically expressed by (ii), does not impose (i). But nothing has been done to disarm the impression that the seamlessness of the relevant transition does. That is another matter. If seamlessness enforces the higher-order Gap principles, it enforces (i) too, and the No Sharp Boundaries paradox rearises as a paradox of seamlessness. There are two directions on which to look for a response to the situation. One, proposed recently by Fine,25 is to restrict the underlying logic of 25  In his monograph, Fine (2017) rejects the rule of ‘Conjunctive Syllogism’:

A, Ø ( A & B ) ØB



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360  The Riddle of Vagueness ­ egation in such a way as to block the ‘right-to-left’ reasoning of the No Sharp n Boundaries paradox. In that case, (i) and the members of the hierarchy of Gap principles will all be acceptable as mandated by seamlessness, however inchoately understood. But the needed weakening of the logic of negation is apt to impress as hugely counterintuitive, indeed as a betrayal of principles that are constitutive of the notion of negation. My own preference, accordingly, is to explore the thought that relevant instances of ‘unpalatable existential’ claims of the form    ($x )(Fx & ØFx ¢) are rendered ungrounded, rather than false, by the phenomenon of seamless transition, which is therefore in urgent need of a less inchoate understanding, and that F’s vagueness in the relevant series likewise renders the unpalatable existential ungrounded. I have no space here to pursue these suggestions.26 In any case, enough has been done, I trust, to discredit the seamlessness in­tu­ ition as a motive for the Buffering view.

VIII The Ineradicability Intuition Once More It remains to re-scrutinize the ineradicability intuition, expressed in rather different ways by Dummett and Russell. Both implicitly started from the idea of the vagueness of the borderline between Φ and Ψ as consisting in a region of uncertainty—a ‘penumbra’ in Russell’s seminal image—and envisaged an additional stipulation to try to bring this region under linguistic control: a new predicate in Dummett’s case, a moratorium on description in Russell’s and therefore the intuitionistically acceptable half of classical reductio:

G, A Þ^ G Þ ØA

26 Wright (2001b, this volume, Chapter  7; 2003c, this volume, Chapter  9; 2007, this volume, Chapter 11) offer argument in some detail that acceptance of a predicate’s vagueness need not involve denial of a relevant unpalatable existential—i.e., endorsement of an instance of (i). Those arguments, if effective, equally militate against acceptance of higher-order Gap principles as a response to the vagueness of the predicates concerned. I have not elsewhere attempted to explain why seamlessness, properly understood, should not motivate acceptance of Gap principles. But the basic point that I believe that a proper treatment should develop is that seamlessness is an epiphenomenon of our discriminative limitations. It is merely a projective error to read it back into the characterization of the elements in a seamless series.

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The Illusion of Higher-Order Vagueness  361 case.27 Both then simply asserted—plausibly but, notably, without any argument whatever—that the proper application of the new stipulation would itself be vague: that there would be cases where it would be uncertain how to apply the new term, or whether they fell within the scope of the moratorium. The assertion is plausible. But it should, on reflection, seem puzzling why it is plausible. The claim that there are borderline cases of a certain concept is, after all, partly an empirical sociological claim: to make it is to predict that possessors of the concept will not react with verdicts about its application that collectively converge on a sharp distinction between positive and negative cases. How do Russell and Dummett know this in advance, sitting in their armchairs? Who is to say that, after ‘eminence’, for instance, was introduced in the manner Dummett envisages, we would not in fact respond with a stable, consensual practice converging on an agreed range of applications for all three concepts—hill, eminence, and mountain—and responding in no case with the characteristic manifestations of vagueness? So why is our reaction to the ineradicability claim not, ‘How do you know? What’s the evidence?’ Why do we not feel it necessary to leave the armchair and try it out and see? The answer, presumably, is that we think we know already what the outcome of an experiment would be. But why do we think that? It is not, after all, as if we have often made stipulations of the Dummett–Russell sort and experience has taught that they do not work. I suggest that the explanation of the armchair plausibility has to do with a sense of the limited guidance that the envisaged kind of stipulation would be able to give us. In going along with the prediction of uneliminated vagueness, we are reporting something about our own sense of limitation in response to the kind of stipulation hypothetically envisaged; the phenomenon is broadly—not exactly—of a piece with the ability to predict uncertainty in your application of rules which you know you have only partially understood; or the ability knowledgeably to say ‘No’ to the question ‘Do you understand?’, when what is at issue is competence for some form of subsequent task. Our sense is that, in contrast to the corresponding Dummettian, or Russellian, stipulation for cases like ‘dommal’ or ‘pearl’, we are not clear enough about which the borderline cases are—which are the cases to trigger the stipulation—to be confident in general how to apply it. The key is to see that this uncertainty does not demand explanation in terms of the idea of higher-order vagueness.

27  A third move in the same spirit would be to extend Ψ, if it is the complement of Φ—or, in any case, to extend the sphere of application of one of the concepts concerned.

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362  The Riddle of Vagueness I will enlarge on that diagnosis in a moment. First, we need to consider an objection to the alleged connection between the ineradicability intuition and higher-order vagueness that was prefigured at the beginning.28 The objection is that an additional presupposition is required before any connection with higher-order vagueness is even apparent. That presupposition is that the introduction of a linguistic stipulation of the kind envisaged by Russell and Dummett will have no impact on the identity of the concept—Φ—whose borderline cases it aims to provide means of denoting or otherwise differentially treating. This presupposition is actually quite implausible. Consider a small child tidying up his play-bricks, so far without any colour words save ‘red’, ‘blue’, ‘green’, and ‘yellow’, who is told to put the reds into one bin and the blues into another, although the bricks include many shades of red, blue, mauve, purple, pink, orange, and so on. It seems quite expectable that he will place many reddish purples and bluish purples, for instance, in the red and blue bins respectively which, if we were to single out a few royal purple bricks and others of similar shades, and give him the word and a new bin with the instruction to tidy the purples into it, he would then prefer to house there. In general, it is to be expected that provision of the resources to mark an intermediate category will have the effect of disturbing—narrowing—the accepted extensions of the concepts which flank it to include fewer uncomfortable cases, and thereby of modifying the original concepts themselves. But, if the effect of regulating the response to the borderline cases would be to modify the concepts concerned, then the ineradicability intuition provides no argument for thinking of them as being even second-order vague—rather we have a situation where the introduction of the new resources afforded by a Dummett/Russell stipulation merely generates three new concepts which then exhibit ordinary—first-order—vagueness in relation to each other. This is an important point. But I do not think that, on its own, it takes us to the heart of the issue. There is a second questionable assumption at work in Dummett’s and Russell’s line of thought—an assumption which indeed is still unchallenged even in the point just registered. It is the assumption that the invitation to annex a new word to the borderline cases of a distinction, or to respond to them with a moratorium on classification, or some other kind of new, distinctive treatment, is in general one that can so much as be taken up. In order to respond to such an invitation, one must first be able to corral the borderline cases—those, after all, are the only cases to which the new practice, whatever it involves, is to be applied. The question this goes past is 28  See n. 1.

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The Illusion of Higher-Order Vagueness  363 whether the reactions that characteristically manifest the borderline status of a case involve the exercise of a concept somehow contrasting with the polar concepts; or whether what they betray is, rather, a subject’s difficulty in bringing it under one of the polar concepts—a ‘drying of the springs of opinion’, a slide into quandary.29 If it is the latter, then the reason why the invitation will not have the effect of generating precision—a new, sharply tripartite practice of some kind—is not because the separation between the cases to which the new convention is to apply and the rest is itself vague on both borders, but because we have no settled concept of those cases in the first place. We need to go carefully here. I am not, of course, denying that there is such a thing as the judgement that a case is borderline—denying that we have any concept of what it is for a colour, for instance, to be a borderline case of red and orange. The question is: what is the content of such a judgement? Does regarding a case as borderline red–orange involve bringing it under a concept that competes, so to speak, within the same determinable space as the relevant polar concepts, red and orange? If so, its force, like theirs, will be normative and exclusive. The judgement will imply, for example: ‘Here you should not take either polar view—the case is too far removed from the clear cases of red and orange.’ Or is the judgement, rather, something that does not involve the application of a competitor concept in that way? It might, for example, be best interpreted as a projection of the characteristic phenomenology of attempted judgement in the particular case, so that its force is broadly sociological: say: ‘Here competent people in excellent epistemic position still have weak and unstable views, struggle to come to a view, and so on.’ The difference is crit­ic­al. The roots of the Buffering view of higher-order vagueness, when mo­tiv­ated by ineradicability, lie entirely in the former way of thinking. That may be fine for some cases—typified by the example of purple and the child’s toy bricks. But it cannot be the way to think about the general run of mutually vague concepts. Borderline cases of a vague distinction, Φ — Ψ, are not in general things that form a kind unified under a concept that stands to the poles, Φ and Ψ, as purple stands to blue and red. In all cases, the borderline region is indeed, as Russell stresses, one of uncertainty where we struggle to bring elements under either polar concept—but, where basic vagueness is concerned, this is for r­ easons that have nothing to do with there being a third concept of the same broad kind, a competitor with the originals in the same determinable space, which seems preferable to both. When there is such a third concept, the invitation

29  I mean this notion only in an intuitive sense here, though the remark just made will bear interpretation in terms of the more specialized sense of ‘quandary’ developed in Wright (2001b; this volume, Chapter 7).

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364  The Riddle of Vagueness to annex a new word to it, or some other practice, will be intelligible enough. But the range of cases on the borders of this concept and the two originals will, again, be likely to defeat our powers of conceptualization—or, if they do not, iteration of the process will anyway bring us eventually to mutual distinctions for which the model of purple, the model of an intervening kind, gives out. At that point, the reason why we will not be able to eradicate vagueness by proposing a differential form of classification, or treatment, of the borderline cases will not be because the concept—borderline case of Φ and Ψ—that would control the new practice will itself be vague, but because we have no concept of such borderline cases that we can exercise in contradistinction to both Φ and Ψ, as we can exercise purple in contradistinction to both red and blue. When borderline cases are exactly things that defeat our ability to apply any of the relevant concepts, borderline case of the Φ — Ψ distinction is nothing we can regulate a new practice by. This is the point of connection, suggested above, with the phenomenon of avowably imperfect understanding. The reason why it may be confidently predicted that a Russell/Dummett stipulation will not have the effect of introducing precision is indeed broadly comparable to the reason why I can be confident that I will not be able to give the right answers when applying a rule I realize I have imperfectly understood. (Of course, in both cases there is the bare possibility that I will surprise myself.) Simply: I do not know how to apply such a stipulation because I lack any stable concept of the kind of cases which are meant to trigger it. My characteristic reaction to such cases is one of a failure to bring them confidently under either polar concept, but not because I am clear that I should bring them under neither. I do not, precisely, grasp them as a third kind. But that is exactly what I would need to do in order to be able to work the stipulation in a stable, discriminating way. Since I am not able to form a settled view about whether they are cases of the sort for which the new stipulation is not called for: that is, cases of Φ or of Ψ, I cannot be confident about when to invoke the new stipulation. Again: if one’s characteristic reaction in the borderline area is a ‘drying of the springs of opinion’—an inability to bring a case under either polar concept that is not associated with a better alternative—then of course the invitation to introduce a new predicate, covering cases whose status is to contrast with polar cases, will not result in clear guidance, let alone precision; that is, in confident and complete classifications across the range. The content of the quandary was precisely whether to apply a polar concept and if so which. So the invited new predicate, or new policy, the application of which will pre-empt either original polar judgement, will be bound to inherit that quandary.

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The Illusion of Higher-Order Vagueness  365 There is, as we noted, what we might term the sociological option: to annex a sociological conception of the borderline cases of a distinction to a stipulation of the Russell/Dummett kind. (In the case of a single judge, ‘borderline case’ will then become a concept grounded in his own characteristic psychological reactions.) But the obvious point to make in that case is that no such conception of the borderline cases of Φ gives any literal sense to the idea of the boundary between the Φs and the borderline Φs being vague. As a first approximation: if the content of a judgement that a case is borderline is broadly sociological, or psychological, then, whereas in judging that a case is Φ, we are making a judgement about the case, in judging that a case is borderline Φ, we are recording a judgement about us; so the idea that this distinction might itself be vague is incoherent—mutual vagueness requires a common domain of predication. I have been suggesting that it is a fundamental error to think of the borderline cases of a vague distinction as if they were shades of purple and the given distinction were like that between red and blue. Entrenched though the error is, it takes only a little reflection to see that this cannot be the nature of the general run of cases. In particular, it cannot be the nature of the distinction between the Φs and the non-Φs. Even setting that case to one side, there is an intuitive notion of adjacency for vague concepts that compete in a single space—in the way that red and orange, for example, or blue and purple are adjacent in colour space, or moderately uncomfortable and painful, perhaps, are adjacent in the space of sensations. Intuitively, when you move from red in the direction of yellow, the next thing you come to is orange. Where concepts are adjacent in this intuitive sense, we will have no third competitor concept to characterize a buffer zone between them, in the way in which purple buffers the blues and reds. We may indeed be able to master a narrower concept that applies in the borderline area (for example, blood orange), but this will not compete with the originals (red and orange) as they compete with each other. It will be open whether it is a determinate of either. And, if we make it clear that it is not to be so viewed, and annex a word to it, the result will be the narrowing phenomenon we noted above. The root error in the Buffering view is to think of borderline cases as instances of what I have elsewhere called Third Possibility. I have given other arguments against that broad conception and will not rehearse them here.30 The ineradicability intuition is indeed a commitment to the Buffering view when taken under the aegis of Third Possibility. And the lesson to learn is that the inference of buffering from ineradicability goes wrong by—draws the wrong conclusion as a 30  For elaboration, see Wright (2001b, this volume, Chapter 7; 2003c, this volume, Chapter 9; 2016, this volume, Chapter 13).

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366  The Riddle of Vagueness result of—passing over a conception of mutually vague concepts not as demarcated from their neighbours by a borderline area conceived on Third Possibility lines but as, though adjacent—there is nothing of any other kind that separates them—characterized by the inability of those who have ­mastered the concepts concerned to run them right up against each other in stable judgement. The conflation of these two ideas—the failure to see that the second (the inability to run the extensions up against each other) does not require the first (a sensitivity to an intervening kind)—is the car­dinal source of the illusion of second-order blurred boundaries. The second is the idea that  Mark Sainsbury (1990) gestures at when he speaks of boundaryless concepts. But I do not think the point of that perceptive piece of ter­min­ology has been generally understood.31

31  I am grateful to the members of Arché’s AHRC-funded project on Vagueness: its Nature and Logic (2003–6) for helpful discussion and critical comments during the seminars that saw the gestation of this paper. My special thanks to Elia Zardini, who gave me detailed written comments on the draft I prepared for the 2007 Arché conference, and to Mark Sainsbury, my commentator on that occasion. A proper response to all their observations and suggestions would have demanded a much more extended and doubtless much improved treatment.

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13 On the Characterization of Borderline Cases It is a great pleasure to have the opportunity to contribute to this volume1 dedicated to the critical celebration of Stephen Schiffer’s very considerable philosophical achievements. My focus will be on his recent work on vagueness.2 The broad direction of Schiffer’s researches in this area has been to give priority to what we may call the characterization problem: the problem of say­ ing what the vagueness of expressions of natural language consists in or, more specifically—since Schiffer takes it as a given that the vagueness he is target­ ing consists in a propensity of vague expressions to give rise to borderline cases—the problem of saying what being a borderline case of the concept expressed by a vague expression consists in. This has not been a main pre­ occu­pa­tion of most of the work in the field since the vagueness ‘boom’ started in the mid-­1970s. There has been a tendency to jump straight into devising semantic theories for vague languages, usually aimed at the twin desiderata of saving classical logic and dissolving the various paradoxes of vagueness, with a principal focus on the standard sorites, and occasional glances at the Forced March, and others.3 Of course, such work has inevitably implicated commit­ ment to broad conceptions of vagueness, and of borderline cases, of various kinds. The classical epistemicist approach, for example, conceives of border­ line cases as instances whose correct classification in terms of the relevant concept is, for reasons it attempts to explain, unknowable. Semantic indeter­ minist approaches, by contrast, tend (often implicitly) to conceive of border­ line cases as items to which the concept in question neither applies nor fails to 1 This chapter was originally written for Gary Ostertag’s edition of the festschrift for Stephen Schiffer, Meanings and Other Things (Oxford: Oxford University Press, 2016). 2  See Schiffer (1998; 2000a; 2000b; 2003; 2006; 2020). Schiffer’s focus, of course, is on soritical vagueness—the kind of vagueness that is characteristic of expressions that are prone to give rise to a Sorites paradox. Other linguistic phenomena that are sometimes described as involving vagueness include generality (lack of specificity), partial definition, family resemblances, and criterial conflicts. I am concerned only with soritical vagueness in what follows. 3  e.g., the so-­called Problem of the Many (for a definitive overview, see Weatherson 2016).

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0014

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368  The Riddle of Vagueness apply and as coming about because our practice with the concept leaves it, in effect, merely partially defined and so ‘gappy’. A variation on this, still seman­ tic indeterminist, regards vagueness as consisting in a phenomenon akin to divided reference, whereby a predicate, for example, may be associated with a range of extensionally distinct best candidates to be the property it refers to; borderline cases are then items which exemplify some but not all of these properties. Finally, some have attempted to see vagueness as constituted in rebus—in the world, rather than in meaning or in our ignorance: being a bor­ derline case, so viewed, is a matter of being situated within a penumbra, as it were, like the position of a point between the light and the dark in the image cast by an intense but blurred shadow.4 Generally speaking, however, pro­pon­ ents of these various kinds of view have not devoted the same degree of atten­ tion to elucidating and defending their (implicit) commitments concerning the nature of vagueness and borderline cases as they have devoted to the development of formal semantical theories, and to criticizing opposing views and attempting to address the paradoxes. Yet one would naturally suppose that the characterization problem should be a locus of developed discussion rather than one of presupposition. For, until we have a properly argued account of what vagueness is, how can one possibly expect to know what kind of semantic theory for vague expressions might be best motivated, let alone how the most appropriate kind of semantic theory might assist with the dis­ arming of the Sorites paradox and other problems? I find much to agree with in both the general approach and the details of Schiffer’s work on the problems of vagueness. We concur, first and foremost, in prioritizing the question of the nature of vagueness and the char­ac­ter­iza­ tion of borderline cases—though I confess to being a little less confident than Schiffer that it is possible, or necessary, to accomplish this in an exceptionless, biconditional formulation. We are also agreed in rejecting standard semantic indeterminist, epistemicist, and in rebus views, proposing instead that the characterization of vagueness will best proceed, broadly, in terms of aspects of the distinctive attitudinal psychology involved in the exercise of judgement involving vague concepts. And we are at one in repudiating ‘third-­possibility’ conceptions of borderline cases: conceptions according to which borderline case propositions—propositions ascribing a vague concept, or its contrary, to items in the borderline area of the concept in question—enjoy a status incon­ sistent with that of simple truth, or simple falsity. Examples of Third Possibility status include lack of truth value, the possession of a third truth value, the 4  As far as I am aware, no one has attempted a fully general view of vagueness along these lines.

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On the Characterization of Borderline Cases  369 possession of any of a range of intermediate truth values, and the (dialetheist) idea that they possess both (polar) truth values. Significant differences, though, remain. Some of the more major concern the epistemic status of borderline cases—in particular Schiffer subscribes to the widespread view of Verdict Exclusion (VE)—namely, that (it is known that) no polar verdict about a borderline case proposition is knowledgeable. We also disagree about the kind of attitudinal psychological story to try to tell about borderline cases: Schiffer holds that it should give a central place to his notion of vagueness-­related partial belief (VPB), although he finds some merit in my notion of quandary;5 whereas I would prefer to centralize the notion of quandary, although I think there is insight contained in Schiffer’s notion of VPB. Third, although neither of us is prepared straightforwardly to endorse the use of classical logic in reasoning with vague concepts, Schiffer proposes no alternative, holding merely that it is indeterminate whether certain prin­ ciples of classical logic—including, strikingly, modus ponens—hold good for reasoning among vague judgements, whereas I have argued that the strongest logic justified for such reasoning would involve qualification of the law of excluded middle for atomic statements and of double negation elimination for compound ones (and so approximate an intuitionistic logic). The considerations to follow will focus on these points of disagreement. They are offered in a spirit of collaboration, in the hope of furthering progress towards the best possible version of the broad genre of account to which Stephen and I are both drawn.6

I Vagueness-­Related Partial Belief It is plausible enough that in the passage along the elements of a typical Sorites series for a predicate F, we pass from cases where we are completely confident in the correctness of the verdict, ‘F,’ through a region where our satisfaction with any particular verdict is qualified and much diminished, and then on to a region where confidence builds again in the contrary of the original verdict 5  More accurately, he finds merit in a notion of quandary in whose characterization his notion of VPB plays a central role, and which is accordingly rather different from mine. 6 My own views are principally developed (and developing) in Wright (2001b, this volume, Chapter  7; 2003a, this volume, Chapter  8; 2003c, this volume, Chapter  9; 2007, this volume, Chapter 11).

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370  The Riddle of Vagueness culminating, as the series progresses, in complete satisfaction once more. I think the phenomenology of whatever it is that seemingly decreases and then increases again in this way needs subtle description, and that it may not be most felicitous to regard it as a form of partial belief. But I do not think Schiffer fundamentally disagrees about that. What is striking is that the atti­ tude involved, although allowing of degree, seems to differ from standard partial belief (SPB) or credence in a number of respects. As Schiffer observes, credences greater than 0 but less than 1 are typically based on evidence which is conceived of as falling short of the best possible evidence, and so are often attended by a conception of how the relevant evidence might be improved. They also tend to be associated with beliefs about likelihood: to believe to some quite high degree that it will rain tomorrow will tend to be to believe (absolutely) that it is quite likely that it will rain tomorrow. Neither of these features is replicated by the kind of partial satisfaction, or confidence, one may have in a verdict that applies a vague concept to a case that lies outside its polar regions but is, say, rather closer to one than the other. But the defining and single most striking feature of VPB is its apparent departure from the laws of classical probability. Classically, my confidence in the conjunction of a pair of (independent) propositions in each of which I place merely partial confidence should approximate the product of the degrees of belief thereby reposed in the conjuncts—and so, when both those degrees are quite small, should be very small. But it does not seem that we would be inclined to regard a conjunction whose conjuncts were a pair of independent propositions, each ascribing a vague property to a more or less ‘central’ borderline case of it, as very much less credible than its conjuncts. Indeed, there seems no clear sense in which it would be rational to be any less confident about the conjunction that about the conjuncts. Imagine a cube of a metal which gradually changes colour as it changes temperature—though neither as a result of the other— and suppose it comes to sit simultaneously on the borderlines between red and orange and between warm and hot. Imagine we find we have no prefer­ ence for the verdict ‘Cube is warm’ over ‘Cube is hot’ and vice versa; and simi­ larly with respects to the verdicts ‘Cube is orange’ and ‘Cube is red’. It does not seem likely that, asked whether we would assent to ‘Cube is orange and Cube is warm’, we would feel any more negative about doing so than about assent to the individual conjuncts.7 And no reason is apparent why we should.

7  As Elia Zardini has emphasized to me in conversation, however, Schiffer’s point is served merely if any lowering of satisfaction in the conjunction is (appreciably) less than would correspond to the ‘multiplication’.

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On the Characterization of Borderline Cases  371 The nature of this kind of partial confidence, or satisfaction—for in truth I am not sure how best to represent it—could certainly stand further clarifica­ tion. What exactly is it that varies in degree, and how may the variation be measured? Betting behaviour, a classic recourse for the explanation of the functional role of standard partial belief, is obviously inappropriate here, since there is no question of an ‘outcome’ where a borderline case is con­ cerned—so nothing to bet on. But, although it is arguable that Schiffer has (so far) left issues to do with the functional role and explanatory potential of VPB less clear than is desirable, I think it is hard to dismiss the phenomenon and that the prima facie case for its reality is strong. The question is whether it can take the weight that Schiffer wants to place upon it in his account of vagueness. About that I continue to have misgivings. Schiffer’s strategy is to try to characterize borderline cases directly in terms of the notion of VPB. More accurately, it is to characterize them in terms of the notion of VPB*— vagueness-­related partial belief formed under epistemically ideal circum­ stances. I have already noted elsewhere (Wright 2006, this volume, Chapter 10) certain difficulties with the proposal developed in Schiffer (2003), to which Schiffer (2006) has since responded. I will not pursue the detail of that particular discussion here, but will outline a more general, though related consideration. Even if we take the notion of VPB, and its characteristically non-­classical behaviour, to be adequately clear, and attested for Schiffer’s purposes, any claim that it is of the essence of grasping vague concepts to be prone to VPB with respect to appropriate judgements that involve them looks to be too strong on at least two counts.8 Consider first the case of Tim. Tim passes, by any reasonable tests, as a master of a wide range of vague concepts expressible in his natural language. But he has persuaded himself, by a variety of more or less philosophically questionable moves, of the correctness of the classical epistemicist conception of vagueness. Accordingly, he conceives of each of the vague expressions in his language as expressing a property (or other appropri­ ate form of semantic value) that actually has a completely sharply bounded extension. The effect is that although his atomic vague judgements are per­ fectly orthodox, he is very insistent on the use of classical logic in reasoning with vague judgements and very confident about the Principle of Bivalence as  applied to them. Moreover, and crucially, although his confidence in the 8  I take Schiffer to be suggesting such a claim in Schiffer (2006), though it is another question whether he needs to.

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372  The Riddle of Vagueness vague (atomic) judgements that he makes varies in degree after the broad manner of VPB, he does—as he should—regard the conjunction of hard cases as increasing the risk of error, and so is prone to ‘multiply down’—to reject conjunctions in cases where he is not prepared to flat out reject either of the (independent) conjuncts. It would be, I think, strained to insist that Tim’s superstitions about the semantics of vague expressions are inconsistent with his perfectly well understanding them—with his fully possessing the concepts that they express. Rather he is making a philosophical mistake—the history of the philosophy of language is littered with such mistakes—about the kind of meanings possessed by expressions which he perfectly well understands. But if that is the right description of him, it is not of the essence of a grasp of vague concepts to enter into attitudes of VPB towards certain judgements involving them. Alternatively, consider Hugh, an individual who is maximally opinionated.9 Hugh’s opinions know no half measures. If he takes a view about anything, he takes it with complete conviction. Yet the pattern and spread of his judgements involving vague concepts are otherwise normal. Thus, in the borderline region of some concept, he sometimes has no view, or returns a verdict inconsistent with one he has given before—in which case he takes the line that what he said before was ‘completely wrong.’ He may even dis­ play signs of hesitancy in judging borderline cases—but if he finally over­ comes it and is moved to judgement, that judgement is once again completely confident. In short, while the extensional profile of Hugh’s judgements involving vague concepts is normal, there is none of the phe­ nomenology of partial belief. And again, it would seem an overreaction to the case to view this psychological quirk as calling into question Hugh’s grasp of the concepts concerned. I think these cases show that, while Schiffer is correct to have emphasized the role of something akin to partial belief in the normal attitudinal psych­ ology of vague judgement—and deserves credit for the insight that this kind of partial belief may, perfectly rationally, display the non-­classical features he emphasizes—it is wrong to see it as belonging to the nature of the under­ standing of vague concepts to involve a disposition to partial belief of this kind, or to write it into the possession conditions for vague concepts that one should be so inclined.

9  I am indebted here to Elia Zardini.

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On the Characterization of Borderline Cases  373

II Verdict Exclusion ‘Verdict’ is here a term of art. A verdict about a statement involving a vague concept is a judgement that it takes one in particular of the two polar values, true and false. The thesis of VE is that, for borderline cases of the concept concerned, no such verdict can be an expression of knowledge. Concerning VE itself, three possible views are relevant: (i) that it is known to be false—so known that knowledgeable verdicts, of truth or falsity respectively, are pos­ sible for, say, predications of a vague concept of its borderline cases; (ii) that it is known to be true—so known that borderline cases are cases where know­ ledge of the relevant kind of verdict is impossible; and (iii) agnosticism—that neither view about the status of VE is mandated. Schiffer’s view is (ii), a con­ tention that he finds intuitive and regards as enforced in any case by the role of VPB in the individuation of borderline cases (see Schiffer 2020, sects 6, 7). I, by contrast, believe that the correct stance overall is (iii): we should be agnostic about the possibility of knowledgeable verdicts in borderline cases. The disagreement is crucial. Verdict Exclusion is no less a nodal issue for the proper characterization of vagueness than Third Possibility, in whose rejection Schiffer and I, as noted, concur. In my view, a repudiation of both Third Possibility and Verdict Exclusion—that is, a rejection of both theses as unjustified—belongs with a more general liberal stance concerning the epi­ stem­ic nature of borderline cases. In this section, I will outline some mo­tiv­ ations for Liberalism and defend it against certain objections, including some of Schiffer’s. However, what I take to be Schiffer’s principal reason for endors­ ing VE—and hence for rejecting Liberalism—has to do with his conception of  VPB as independently excluding potentially knowledgeable belief.10 The issues around that impress me as very difficult, and I must reserve discussion of them for another occasion. Schiffer and I are both impressed by the datum that ordinary speakers do not treat the borderline area of a vague distinction as one where competence mandates silence or suppression of any inclination to offer a verdict, however qualified. If something is on the borderline between red and orange, say, it will not call your competence with those concepts—or your eyes—into 10  I take his view to be that the sort of qualified acceptance involved in VPB-­ing P to some signifi­ cantly greater degree than not-P is categorically unsuited to serve as the doxastic ingredient in knowledge.

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374  The Riddle of Vagueness question if you are inclined to describe it as red, or for that matter as orange. One is entitled, if one is so moved, to a verdict in the borderline area, so en­titled—presumably—to the opinion which that verdict expresses. A perhaps more forceful expression of this entitlement intuition—since free of any demand on one’s preferred understanding of ‘borderline cases’—is this. Consider any typical soritical series running from clear instances of F to clear instances of non-F by steps small enough to service a Sorites paradox as plausible as any. Then—this is the intuition—there will be no element of this series about which it is mandatory to return neither the verdict ‘F’ nor the verdict ‘non-F’. A polar verdict is always permissible provided it is sincere; there are no cases that mandate a Third Possibility type of response. If some­ one prefers to understand ‘borderline case’ in such a way that the knowledge that something is a borderline case ought to inhibit any verdict, then a way of expressing the entitlement intuition is that there are no clear—definite—bor­ derline cases in a typical Sorites series. The clear cases, on the other hand, are those where only one polar verdict is permissible. Liberalism is the simplest theoretical accommodation of the entitlement intuition. It is the view that it is always permissible to return a verdict about a borderline case simply because it is—in a sense we need to clarify—open what to think about such cases and open, indeed, whether, in thinking one thing in particular, you are knowledgeable. This openness is, at a minimum, what goes with agnosticism about Third Possibility and Verdict Exclusion re­spect­ ive­ly: if neither thesis is justified, there is no call to stifle aspects of our natural practice and inclinations to judgement that would be inappropriate if they were true.11 Let me quickly summarize some considerations which I take to support Liberalism and then—in the next section—respond to some objections to it. First, to emphasize the implausibility of the idea of a datum of Third Possibility. If Third Possibility were known to obtain, even if in just one case, we would know that any polar opinion about that case was mistaken. But we do not behave as though we think we know any such thing: someone who returns a (perhaps suitably qualified) polar verdict about a borderline case is never thereby automatically treated as revealing a mistake, or incompetence— rather we feel, to repeat, that they are, ceteris paribus, precisely entitled to their opinion. The manifestations of judgement of borderline cases include 11  In refusing to affirm that no verdict about a borderline case can be knowledgeable, the liberal does not, of course, intend to exempt such verdicts from other normal controls on knowledge ascrip­ tion—the point is only: we have no justification for thinking that knowledgeability is excluded just by a verdict’s concerning a borderline case.

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On the Characterization of Borderline Cases  375 hesitation, inability to form an opinion, weakness of opinion, instability of opinion, and conflict among judges whose competence is not in question. But convergence in patterns of hesitation, or non-­opinion, still less any sense that ‘no opinion’ is sometimes the uniquely appropriate response, is at best contin­ gently (and doubtfully) involved. Mastery of a vague concept seems to involve no essential exercise of a concept of any kind of Third Possibility. Now to Verdict Exclusion. If VE were known to obtain, even if in just one case, we would know that any polar opinion about that case was, whatever else, not the product of a successful feat of cognition. It would be something caused, no doubt, by relevant features of the case but not a fitting cognitive response to them. And, if VE were known to obtain generally in the border­ line area, then our propensity to verdicts—albeit weak and unstable verdicts— about cases lying within it would seem to amount to no more than a kind of cognitive incontinence. It seems that to know such a thing about opinions one is inclined to form should have the effect of undermining them. So knowledge of Verdict Exclusion is also in tension with the entitlement in­tu­ ition. Our sense is that, no matter what case in a Sorites series we consider, it is consistent with full perceptual and conceptual competence if someone takes a (perhaps suitably qualified but) polar view of it. There are no cases, even towards the ‘middle’, where it is eo ipso incompetent to have a (suitably qualified) polar view. However, a presumption of knowledgeability—or at least warrant—is a condition of rational opinion: for a rational judge, judging that P is judging that P is what one ought to think. But one who thinks that VE is known says in effect that we know that there is no mandate for opinion in some cases in a Sorites series—there is, in borderline cases, nothing that ought to be thought. In such cases, thinking that P is what one ought to think is therefore mistaken. VE thus has the effect that opinions which, according to the entitlement intuition, are consistent with competence are ungrounded and hence not competent. If VE is part of the best theory of vagueness, then the best theory is one according to which our actual practice in all but definite cases is irrational/incompetent. It is hard to envisage that a best theory should have that feature. Finally, there are difficulties for VE involved in its interaction with prin­ ciples of Evidential Constraint (EC). The best formulation of the latter for any particular type of judgement is doubtless going to be controversial. But EC is undeniably intuitive for secondary qualities generally and for a wide class of predicates of casual observation—‘heap’, ‘bald’, and so on—that foster Sorites paradoxes. We are not, intuitively, up for the idea of baldness that cannot be recognized as such in principle, or heaps whose heaphood would elude the

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376  The Riddle of Vagueness detection of even the most fortunately situated judge. Yet, if Verdict Exclusion were known to obtain, even if in just one case, we would know that EC—in the form of the two conditionals:  If P, then it is feasible to know P   If not-P, then it is feasible to know not-P —would have to fail in that case (on pain of the obvious contradiction). This, it seems to me, is simply too strong a result to swallow. Maybe EC is controversial,12 and perhaps it is implausible for certain vague expressions.13 But it is already a problem if it is not implausible for all—since our knowledge of VE is being supposed to be characteristic of borderline cases without exception: if VE and EC are (known to be) inconsistent, the plausibility of taking VE to be known for an arbitrary vague predicate, say, is hostage to that of taking it that we actually know that EC fails for it. The proponent of VE owes an explanation of what they take the content of this knowledge to be. And if it is, as naturally construed, that colours, baldness, heaps, and so on can all be undetectable, even under unimprovable conditions of observation, then I do not think we do know anything of the sort, for a very wide class of vague concepts.14 12  It is, of course, in tension everywhere with Williamson’s thesis (in 2000a) that knowledge is sub­ ject to margins of error in a sense that requires that knowing that P holds of circumstances C entails that any case within some fixed margin of difference of C is also a case where P holds. I will come back to this in Section III. 13  Schiffer (2020) suggests ‘brave.’ 14  Schiffer (2020) suggests that the motivation for EC confuses the above conditionals with the (in the relevant, plausible cases) acceptable weakenings: If definitely P, then it is feasible to know P If definitely not-P, then it is feasible to know not-P This is no help, however, without a developed account of the difference in the truth conditions of the original and weakened formulations, and a story about how exactly the original versions are sup­ posed to fail. Recall that, as noted in the main text, even the supposition that we do not know that the original conditionals fail is inconsistent with the claim that VE is known. Here is a related point, due to Zardini. Consider the claim—with which the epistemicist is comfort­ able, though not, I would imagine, Schiffer—that some nth case in the series is such that either it is F (say, looks red) and we cannot feasibly know that, or it is not-F and we cannot feasibly know that. Formally, ($n)[(Fn & Ø it is feasible to know(Fn)) Ú (Ø Fn & Ø it is feasible to know(ØFn))]. That is something which, if we do not believe that there is any such n, we may very well want to deny. Intuitionistically and classically, the denial is equivalent to: (*) ("n)[Ø(Fn & Ø it is feasible to know(Fn)) & Ø(ØFn & Ø it is feasible to know(ØFn))]. And (*) is already enough to yield aporia when conjoined with VE. VE is inconsistent with the denial that there are any elements in a Sorites series for looks red which look red but cannot be known to do so, or which do not look red and cannot be known to do that. If (pace the epistemicist) one can rationally doubt there are any such elements, one cannot coherently endorse VE.

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On the Characterization of Borderline Cases  377

III Stabilizing Liberalism On the other side, there are a number of more or less intuitive objections to Liberalism, some amounting to direct arguments for VE. While the matter requires a much fuller treatment than I have space for here, I will try to address at least some of the anti-­liberal considerations known to me and point up some of the issues on which further clarity is needed. Schiffer himself offers two intuitive thoughts on behalf of VE. The first, very simply, is that it’s as much of a platitude to say that If someone knows that S, then it’s determinately true that S as it is to say that If someone knows that S, then S. (Schiffer 2020, p. 167) Assuming the standard kind of characterization of S’s being borderline— namely, that it is neither determinately true that S nor determinately true that not-S, it follows that no one ever knows a borderline S. But I reply that Schiffer’s platitude is clearly that only if ‘determinately’ carries the sense of a kind of particle of emphasis, like ‘actually’ in one common kind of use, or ‘indeed’. When it is so understood, to characterize a borderline case for S as a situation where it is neither determinately true that S nor determinately true that not-S is, in effect, to endorse Third Possibility—which Schiffer does not. Schiffer’s second intuitive thought invites us to reflect on one’s epistemic position when confronted with what one knows to be a borderline proposition. Suppose that you are holding a ball in your hands in circumstances that are as good as you can conceive of them getting for judg­ ing the ball’s color. You are certain you know what color the ball is, whether or not you have a word for that color in your vocabulary; you know that you have mastery of the concept of red; and you know with certainty that the color you know the ball to have is not one you can now justifiably say either is or is not red. Furthermore, you cannot even conceive of how, given all you know, you could come to have warrant for judging either that the ball is red, or that it isn’t red. You rightly take yourself to be in the best possible position to verify whether or not the ball is red, and you can’t imagine what you could conceivably find out that would give you knowledge that the ball was, or

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378  The Riddle of Vagueness wasn’t, red. Given all that, I would think that you would be entirely justified in thinking that it’s impossible, given the obtaining facts, for you, or anyone else, to know whether or not the ball is red. Examples involving any other sorites-­prone concept can be used to make the same point. (Schiffer 2020, pp. 167–8)

I think the portion of this up to but excluding the last two sentences nicely characterizes some of the phenomenology of quandary.15 It is indeed in such a situation hard to conceive of any improvement in one’s epistemic position which one could foresee would sway the balance. But to find oneself unmoved to an opinion by evidence of which one has no clear conception of a possible improvement justifies the claim that there is no knowing in the case in ques­ tion only if one knows, first, that one’s lack of a clear concept of a possible improvement is an indicator that there is no possible improvement—which raises interesting issues which I will not here go into—and, more crucially, second, that one’s present evidence is not enough for knowledge: that had one been moved to a verdict on the basis of the very same experience and collat­ eral beliefs, one’s present quandary discloses that being so moved would have been inappropriate, rather than the other way round. But you do not know that. Had you in the same circumstances been moved—marginally—to the opinion that the ball is red, it would go with that opinion to think that coming to no view in the same circumstances would—marginally—underplay the evidence. Do you know now that that would be an inappropriate reaction? Here next is an argument (from Rosenkranz 2005) that purports to commit Liberalism to an actual contradiction. Suppose that Aye and Nay come to ­different (but suitably qualified) polar verdicts about a borderline case of F. Liberalism—specifically, agnosticism about VE—seems committed to say­ ing that it is not known that Aye is not knowledgeable, and it is not known that Nay is not knowledgeable. But this pair of claims, given only uncontro­ versial proof-­theoretic properties of knowledge, is unstable. For the latter claim seems to imply that ØK (ØK (ØFk )), and the former that ØK (ØK (Fk )) 15  Although the bit about knowing ‘with certainty that the color you know the ball to have is not one you can now justifiably say either is or is not red’ is too strong.

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On the Characterization of Borderline Cases  379 And from these, given factivity and closure for ‘K,’ we get, by elementary moves, that ØK (Fk ) & ØK (ØFk ). So: on the—apparently—liberal supposition that it is not known that either polar verdict is not knowledgeable, we appear to have shown that neither polar verdict is knowledgeable! This conclusion will go for all polar disagree­ ments about borderline cases. So, in no such disagreement is any knowledge­ able opinion involved, just as VE requires, and contrary to Liberalism. I think the right reaction to this objection is that it points up the need for a distinction in the interpretation of the various occurrences of the operator ‘K’. Liberalism indeed cannot be coherently expressed otherwise. But what dis­ tinction? The general idea is that we try to be open to the possibility that any particular opinion about a borderline case may be knowledgeable, including both one’s own, if one has one, and that, possibly conflicting, of others. But obviously, if Aye’s opinion is that P, then she ought to think that anyone who thinks otherwise is wrong, so not knowledgeable. What Liberalism requires is that, consistently with thinking that he does not know, Aye should somehow nevertheless allow that Nay could be knowledgeable, even thinking what he actually does in the world as it actually is. What is the modality there? It is the same as that whereby, when you hold any opinion of which you are not entirely sure, you may concede, consistently with retaining that opinion, that you could be wrong. I carefully count the marbles in a bag and get a lar­ gish number. If I am right about the number, then, given the way I arrived at my belief—by a careful count—it seems reasonable to say that I know what it is. But perhaps you nag me that even careful counts can involve error when largish numbers are involved, and I am thereby moved to concede that I could be wrong. What is the content of that concession? It had better not be tanta­ mount to the admission that I do not know—after all, my belief may be true and formed by careful execution of a reliable, indeed canonical method. So, the concession needs to be weaker. But nor does it seem entirely happy to view me as admitting that I do not know that I know; for, if we are under­ standing knowledge in enough of an externalist way to allow that I may still actually know the number of marbles in the bag, the thought invites itself that I may still (so to say, externally) actually know that I know that I do. A better suggestion is that what I concede is the right to claim to know. I opine that P but I do not claim to know that P, though nor do I admit that I do not know it. So, let us try that distinction in the context of Aye and Nay: Aye’s opinion that P commits her to holding that Nay does not know not-P; but Nay’s

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380  The Riddle of Vagueness knowing not-P is nevertheless consistent with everything that Aye regards herself as in position to claim to know—since she does not regard herself as in pos­ition to claim to know P. Nay’s knowing not-P is a possibility for Aye in that somewhat qualified sense of epistemic possibility. Again, this is not the same as saying that it is consistent with everything Aye in fact knows—not if one thing she may in fact know is P. The crucial thing is thus that, when moved to a verdict in the borderline area, one is in no position to claim to know (although still in position to ­consider that one may know). This requires, to stress, that the things which one is in position to claim to know are a potentially narrower class than those which in fact one does know. Indeed, it requires, I readily acknowledge, a more general account of the notion of one’s ‘being in position to claim’ and an ex­plan­ation of its potential to carry a narrower extension than one’s actual knowledge. I do believe in the good standing of and need for such a notion, but I cannot argue for that here (it is argued for in Wright 2008). To review the original argument in the light of this distinction. Let ‘R(P)’ express that one is in position to claim to know P. Then the ingredients in the liberal supposition become ØR (ØK (ØP )), and ØR (ØK (P )) Given that R is closed, (and K factive) we still get ØR (P ) and ØR (ØP ) But that is now no problem—just the (agnostic) result that no one is in pos­ ition to claim knowledge of a verdict in a borderline case. To summarize, both the following liberal-­seeming claims— 1. No one is in position to know that any polar verdict about a borderline case is, just in virtue of its subject matter and specific polarity, not knowledgeable. 2 No one should commit themselves to thinking that any particular polar verdict about a borderline case is, just in virtue of its subject matter and specific polarity, not knowledgeable. —are incoherent and are no part of the commitments of Liberalism. The first commits one who endorses it to the premises of the Aye–Nay reductio. The second is inconsistent with the entitlement intuition, since anyone who takes

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On the Characterization of Borderline Cases  381 a polar view of a borderline case naturally commits themselves to thinking that the opposing view is false, and therefore not knowledgeable. What, I am suggesting, is a commitment of Liberalism is this: 3. No one is in position to claim to know that any polar verdict about a borderline case is, just in virtue of its subject matter and specific polar­ ity, not knowledgeable. We are entitled, when so moved, to have polar opinions about borderline cases. When we do, we are committed to thinking that such contrary opinions are not true, and so not knowledgeable. But we are in no position to claim to know that a given such opinion is not true, or not knowledgeable. And we are also in no position to claim to know that a given such opinion is true, or is knowledgeable. There is one final powerful-­looking extant argument in favour of VE, and hence antithetical to Liberalism. It is Timothy Williamson’s argument that inexact knowledge requires a margin for error, with the latter notion under­ stood in such a way as to entail that cases within a fixed margin of difference from a case known to be F are likewise F. This enforces the negation of the claim that all the EC-­conditionals of the form, If Fk, then it is feasible to know that Fk, for k an element in a given sorites series, are true.16 It does not, except clas­sic­ al­ly, enforce the idea that some in particular are false; it does not, except clas­ sic­al­ly, even preclude thinking that there is none that is false; nor does it force us to conceive of the falsity of such a conditional as consisting in the F-ness of the relevant k but unfeasibility of knowledge that Fk. There is therefore a pro­ ject, for those, like the present author, for whom intuitionistic distinctions will be respected in the logic of choice for vague statements, of exploring whether or the extent to which Williamson’s argument might somehow be stopped short of full-­out contradiction with Liberalism. But my own ex­pect­ ation is that something will have to give here. Williamson’s proposal is driven by two thoughts: (i) that knowledge requires reliability of the relevant method of belief formation, and (ii) that, in creatures of limited powers of dis­crim­in­ ation, reliable detection of a characteristic will be compromised if changes too slight for them to discriminate can make the difference between its applying 16  We assume that each successive element in the Sorites lies within the relevant margin of differ­ ence of its predecessor.

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382  The Riddle of Vagueness and its failing to apply. There is evident merit in both thoughts. Where there may be room for manoeuvre is over whether, as best understood, they prop­ erly combine to enforce the general truth of Williamson’s putative corollary, that cases within a fixed margin of difference from a case known to be F are likewise F. But the matter needs a careful separate treatment—something I cannot embark on here.17 Specific arguments apart, it is clear that there is a strong intuitive pull, felt by many philosophers of very different theoretical predispositions about vagueness, to accept VE. What is the root of it? One motivation, undoubtedly, is the intuitive predilection for Third Possibility conceptions of borderline cases. Another is the questionable idea that knowledge requires subjective certainty, coupled with the thought that the borderline region is one of often tentative, uncertain opinion. A third would regard the characteristic instabil­ ity of opinions in the borderline region as disqualifying them as knowledge— but that would not justify the modal component in VE: those drawn to it are unlikely to feel any better about a verdict about a borderline case which just happened to be stable. My guess is that the single most powerful pre-­theoretic motive for endorsement of VE is the idea that the opposed contention, that knowledge in borderline cases is possible, owes a concrete conception of a number of matters that present as moot, to say the least—indeed as impon­ derable. What kind of fact could it be that a case on the borderline of ‘red’ is actually red? Will one not need recourse to something approaching the epi­ stem­icist’s idea of inscrutable semantic mechanisms linking the predicate to a property of which we may have no adequate conception to make sense of such a fact? And, if one goes in that direction, what would make an opinion that happened to coincide with the fact knowledgeable? In short, it looks as though a worked-­out denial of VE will involve all the problems that discourage most from endorsing Epistemicism, together with the additional baggage of explaining how the kind of sublimated facts in which Epistemicism believes are in principle open to knowledge. But, if this is the primary, most basic motive for endorsing VE, it should come with a sense of liberation to realize that a justifiable recoil from this cluster of issues and problems should actually motivate no such thing. We are not, in denying that VE is justified, affirming its negation. We incur none of the distinctive obliga­ tions of that affirmation. The liberal view is that we do not know VE; but nor do we claim to know that knowledge is possible in the borderline area. We 17  Useful discussion bearing on the issue may be found in Mott (1998), Williamson (2000b), and Égré (2006).

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On the Characterization of Borderline Cases  383 have, accordingly, no obligation to provide a further account of the facts of which it would be knowledge or of how knowledge of them might be reached.

IV Quandary and the Characterization Problem I turn to the question of whether the notion of quandary might be usefully deployed to address the characterization problem. An early decision needs to be taken about the form in which to try to address the problem, since there are several interrelated questions here. They include: (1)   In what does the vagueness of a vague expression consist? (2)   What is a borderline case object/item? (3)   What is a borderline proposition? One natural proposal is to try to answer (3) first, after which the answers to questions (1) and (2) can respectively consist in elaboration of these two basic thoughts: (1)  That the vagueness of E consists in the fact that its presence in token utterances results, in certain circumstances, in their expression of bor­ derline propositions; (2)  That a borderline case item is the subject of a borderline proposition. The following remarks are offered under the aegis of this natural proposal. In earlier work (Wright 2001b, this volume, Chapter 7), I characterized the notion of a quandary as follows: Proposition P presents a thinker T with a quandary in circumstances (of evaluation) C18 if and only if, in C, (i) T does not know whether or not P (ii) T does not know any way of knowing whether or not P

18  The relativity to circumstances of evaluation is for the obvious reason: the very same proposition can be a quandary in one set of circumstances and clearly true in another. But the relativity might be compounded for vagueness-­unrelated reasons if one’s conception of proposition allows propositional truth to be relative to time, or standards, etc.

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384  The Riddle of Vagueness (iii) T does not know that there is any way of knowing whether or not P (iv)  T does not know that it is (metaphysically) possible to know whether or not P I also suggested that it would be wrong to add (v)  T knows that it is not (metaphysically) possible to know whether or not P for reasons included in and elaborated upon by the discussion of the preced­ ing sections. It is a consequence of the proposal sans clause (v) that, the less cognitively adept a thinker, the easier it will be to confront them with ­quandary—since they only have not to know various things. Indeed, they do not even have to be able to wonder whether or not P. Clearly there is therefore no prospect of capturing the notion of a borderline proposition by reference to the propensities of actual thinkers to be put in quandary by it. We will need to consider thinkers of some degree of conceptual and cognitive sophistication. A first shot at a characterization in terms of quandary of a borderline propos­ition might run as follows: P is borderline in circumstances C iff a conceptually and perceptually fully competent thinker T could be put in quandary by P in C The intuitive thought is that, while quandary is never a mandated response to a borderline situation, it is always a possible one quite consistently with full and proper cognitive functioning and grasp of all relevant concepts. In a clear case, by contrast, to fall into quandary is not consistent with full and proper cognitive functioning in appropriate respects together with grasp of all rele­ vant concepts, since the clear cases mandate a verdict. Notice that this strat­ egy of characterization grants that quandary is not a characteristic mental state associated with the appraisal of a borderline proposition—not if ‘charac­ teristic’ means widespread and typical. It is the possibility of quandary, con­ sistently with full competence in all relevant respects, that is crucial, not the actual prevalence of it. The first shot characterization, however, is still manifestly insufficient. The problem is that there are non-­vagueness-­related quandaries. A thinker of arbitrary conceptual and perceptual—indeed, mathematical—competence may very well meet all four clauses with respect to Goldbach’s Conjecture, for example: a proposition in which there is no relevant form of vagueness, and which accordingly has no claim to be a borderline proposition. And moral

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On the Characterization of Borderline Cases  385 dilemmas hold out the prospect of another region of quandary: finding myself in a situation in which all possible courses of action conflict with one or another of my values, I may consider myself apprised of all relevant non-­ moral facts and quite reasonably be perfectly agnostic about the prospects of adjudicating between the competing values concerned. In that case P—that is, some proposition of the form ‘A is the best thing to do in the circumstances’— may very well present me with a quandary. True, there are almost certain to be vague concepts involved in P; but it seems likely to be a misdiagnosis of the source of the quandary posed by a moral dilemma to locate it there. Goldbach’s conjecture and the wider class of propositions that it repre­ sents—propositions, whether or not mathematical, for which we have no assurance of decidability—are, I think, easy to exclude by a well-­motivated modification of the first shot proposal. What is striking about these cases is that quandary is mandated—a fully competent thinker ought to regard Goldbach as presenting a quandary. That is in crucial contrast to the situation of borderline cases, at least as construed by Liberalism. So it looks as though the first shot should be followed by a second shot beginning like this: P is borderline in circumstances C iff (i) a conceptually and perceptually fully competent thinker T could, con­ sistently with those competences, be put in quandary by P in C; and (ii)  Such a T is not required to be put in quandary by P in C; and . . . —though only so beginning, since something now needs to be added to address the quandary of moral dilemma. There are other possible quandaries too: if one is persuaded of the possibility of faultless disagreement in matters of taste, for example, there looks to be potential for quandary in the situation of a bystander considering which of the protagonists in such a disagreement may be right. There is now a tactical issue. We could proceed by trying to track through all possible varieties of quandary, hoping to find a clause for each to distin­ guish it from the vagueness-­related cases. But it is not clear what degree of insight might be expected along that path. Better, if we can, to say something which captures the nature of vagueness-­related quandary in one cast, as it were, thereby systematically contrasting it with all the other kinds. How might that be accomplished? Here is a tempting thought: perhaps this is the place to reinvoke VPB. It is not implausible that either of the conflicting claims in a moral dilemma might—by one for whom it is not a dilemma—be endorsed as strongly as you

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386  The Riddle of Vagueness like without any necessary implication of some form of incompetence. Likewise for the ingredient claims in a putatively faultless disagreement on a matter of taste. But can any thinker who is fully competent in relevant respects endorse a verdict about a vague judgement strongly—as strongly as you like— which, for another relevantly fully competent thinker, presents a quandary? In a borderline case, where quandary is consistent with competence, must not that fact surface in at least some, perhaps quite significant degree of qualifica­ tion in the confidence of one who, as according to Liberalism they are en­titled, endorse a verdict? If so, we can add to the second shot along the follow­ ing lines: (iii) T is required to repose at most a relatively low degree of VPB in P (or its negation) in C, and hope thereby to have at least the overall shape of a successful char­ac­ter­ iza­tion (even if further refinement will certainly be needed). However, there is, if an earlier point is correct, a problem for this—the same problem caused by our friend Hugh and the possibility of ‘maximal opinionation’ for the general idea that VPB is a (non-­contingently) character­ istic mental state of vague judgement. If maximal opinionation is consistent with mastery of vague concepts, VPB is at most contingently so characteristic— and the requirement proposed in clause (iii) as formulated is not a require­ ment. I see no way round this objection at present. What we should like to propose would be a clause that captured the idea that, in borderline cases, a thinker who is put into quandary will be so because of the nature of the prevailing circumstances of evaluation and the vagueness of the judgement concerned: the quandary is an upshot of the vagueness. But, of course, to say that would be to surrender the project of accounting for the nature of vagueness in terms of the propensity to induce quandary in fully competent thinkers—rather we will have slipped into thinking of vagueness as an underlying cause of the attitude of quandary, whose nature will there­ fore have to be explained independently.19 Surely, though, the basic thought has to be right that it goes with the nature of the kind of vagueness we are concerned with that quandary may be induced without demanding explanation by defective cognitive or conceptual 19  Schiffer confronts an exactly analogous problem when he speaks of VPB* as ‘F-concept driven’ in borderline cases—for development of the challenge, see Wright (2006, this volume, Chapter  10). Schiffer responds in Schiffer (2006).

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On the Characterization of Borderline Cases  387 competence. And, when quandary is so induced, it is presumably induced by something. The project of attempting a broadly attitudinal–psychological response to the characterization problem means that we cannot rest content with an account of the cause that involves an unreconstructed appeal to the vagueness of the proposition concerned. But it may quite properly appeal to a certain kind of characteristic property of the relevant circumstances of evalu­ ation—if we can find one. And surely we can. The very form of the Sorites paradox itself provides the means. All the vague concepts with which we are concerned are, after all, Sorites prone—that defines the kind of vagueness with which we are concerned. And every Sorites paradox involves gradual change in the values of some parameter, Π, with the following characteristic: that there is a finite strict ordering of such values whose early instances pro­ vide for clear cases of F and whose later ones for clear cases of its contrary. The kind of quandary distinctive of vagueness is one induced by this process of gradual change—or, more accurately, since quandary is not restricted to soritical contexts, one induced, in the circumstances in question, by the value for the rele­vant parameter taken by the object of the judgement that presents a quandary. Call such a parameter a parameter of supervenience for F. For bald, a parameter of supervenience will be number and distribution of hairs on the scalp; for heap of sand, a parameter of supervenience will be number and arrangement of grains of sand; for red, a parameter of supervenience will be colour. I take it these ideas are sufficiently clear for the present purpose. Perhaps, then, the following clauses point in a potentially profitable direction: P is borderline in circumstances C iff P configures some concept F for which Π is a parameter of supervenience such that (i)  A conceptually and perceptually fully competent thinker T could be put in quandary for P in C by the value taken by Π in C; and (ii) A conceptually and perceptually fully competent thinker T is not required to be put in quandary for P in C. However, it is important to keep in mind the general pressures from which this proposal springs. In general there is no hope, I think, of a successful ­char­ac­ter­iza­tion of vagueness in terms of the attitudinal states of those who make vague judgements unless we include some form of causal constraint on the provenance of these states. Any successful such account will therefore be bipartite: it will proffer some putatively distinctive attitudinal feature(s) of

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388  The Riddle of Vagueness vague judgement and it will tie significant instantiations of it/them to causes that are somehow an essential feature of the broader landscape of vague judgement. What I have suggested in this section is merely a rather simple-­ minded illustration of this model: the selected distinctive feature is the ­association of vagueness with judgemental paralysis in certain circumstances— glossed as quandary—and the relevant feature of the broader landscape is the value taken by the parameter, gradually shifting in a suitable Sorites paradox, on which instantiation and non-­instantiation of the relevant vague concept supervenes. It remains to observe that, if VPB is a well-­conceived phenomenon at all— which I do not doubt—then there has, of course, to be scope for other ver­ sions of this general form of bipartite proposal which seek to centralize it, rather than quandary as characterized. Such versions will presumably be more congenial to Schiffer and may well have advantages. I regard the area as very open.

V The Logic of Vagueness There is no immediate connection between the proposal that borderline cases should somehow be characterized in terms of a certain distinctive kind of partial belief and a treatment of the Sorites paradox. Schiffer effects one by proposing a simple set of characterizations of the degrees of VPB—degrees of V-­credibility—that a rational subject will assign to compounds of vague state­ ments on the basis of the degree of V-­credibility that they assign to their con­ stituents. The matrices he proposes follow the pattern of Łukasiewicz’s tables for infinitely many valued logics. Thus the V-­credibility of a negation is 1 minus the V-­credibility of the negated statement; the V-­credibility of a con­ junction is the minimum of the V-­credibilities of its conjuncts; and the V-­credibility of a disjunction is the maximum of those of the disjuncts. The universal and existential quantifiers respectively take the greatest lower bound and least upper bound of the V-­credibilities possessed by their instances. Now consider a Sorites paradox, with a major premise taken in the form ¬(∃x)(Fx & ¬Fx′). Classically (and intuitionistically) this forms an inconsist­ ent triad with the two minor premises F0 and ¬Fk, with k selected so as to ensure that the latter is effectively incontrovertible, while F0 is beyond dis­ pute. So it is natural to think that first base for any solution must be to

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On the Characterization of Borderline Cases  389 acknowledge that the Sorites reasoning reduces the major premise to absurd­ ity and so demonstrates ¬¬(∃x)(Fx & ¬Fx′). The question is then how to block the classical entailment of the unwelcome ‘unpalatable existential’ (∃x)(Fx & ¬Fx′), or somehow make out that it is not really unpalatable. The landscape changes, however, with Schiffer’s proposals. This becomes clear if we ask what degree of V-­credibility attaches to the unpalatable existential. It will be the maximum of the V-­credibilities of the k instances: conjunctions of the form Fx & ¬Fx′, whose V-­credibility in turn will be the minimum of those of their conjuncts. In the polar regions this will be very low, since one or the other conjunct will be roundly disbelieved. But the figure will climb as one enters the borderline region, culminating, at least in principle, with a V-­credibility of, or very close to, 0.5 (though never higher.) So the unpalatable existential, and hence also its negation—the major premise for the paradox— will also have V-­credibility very close to 0.5. The upshot is that, according to the Łukasiewicz-­style matrices, both are paradigms of indeterminacy. Is that a good result? It may seem not—after all, the major premise is very plausible; that is why we had a paradox. So do we not want it to turn out to have quite a high VPB? Well, yes—inasmuch as we want to explain the plausi­ bility of the major premise (in particular, its plausibility over its negation) but—you might suppose—also no, since we have to fault it somehow. Suppose—as I take it Schiffer intends—that the computation we have just sketched is meant to be normative: to deliver a measure of VPB that it is rational to have. Had the computation delivered the result that the major premise did have quite a high VPB, then the account would be saying that we ought to accept it over its negation—in effect, confirming the rational accept­ ability of the premises of the paradox! So it is good that we do not get that result. But, as it is, the result is that rationally there is absolutely nothing to choose between the major premise of the Sorites and its negation. That leaves the pull that the premise exerts on us—which is the whole source of the para­ dox—as unexplained, indeed as irrational. And surely, one may protest, it is not irrational—after all, the negation is tantamount to assertion of the exist­ ence of a sharp cut-­off—that is, a denial of the vagueness of F in the series in question, which is a datum of the problem. Why on Schiffer’s account does the negative existential form of the major premise for the Sorites attract us? Rationally, if his matrices for VPB are on the right lines, we should find nothing to favour it over its negation. I am not sure how Schiffer sees the options here. One thing he can say is that we are under no pressure to choose—since the Law of Excluded Middle, conceived as a general schema, is foursquare indeterminate too (an exercise for the

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390  The Riddle of Vagueness reader). That suggests the following diagnostic: the major premise attracts us because acceptance of its negation repels. But the truth is that there is no rational pressure to accept either if the Law of Excluded Middle is in­de­ter­ min­ate. However, this leaves unexplained why the unpalatable existential is unpalatable. For, if it really is tantamount to a denial of the vagueness of F in  the series in question, it is false, not indeterminate. And if it is not, the repulsiveness of that denial does not explain its unpalatability. How exactly, in any case, does getting in position to regard the major premise as indeterminate dispose of the paradox? It may seem obvious: the paradox was that we seemed driven to a contradiction from acceptable ­premises—the contradiction, for example, that a person with exactly 37 cents both is not—of course—and is—by the Sorites—rich. Now, since one of the Sorites premises turns out to be indeterminate, we are no longer under any pressure to accept one of the components in the contradiction. What is salient, though—and of course Schiffer is absolutely aware of and explicit about this—is that that cannot be his whole story. There is a commit­ ment in his account to a kind of overkill. Not merely have we disposed of the acceptability of the major premise. We have also got into a position where we have to regard it as indeterminate whether the Sorites reasoning is valid. For, since the minor premise is true, and the conclusion is false, and since—at least for Schiffer—the indeterminacy of P is to be consistent with the truth of P, the situation is one where it is consistent with everything we know that the Sorites premises are both true and its conclusion false—so it is consistent with everything we know that it is an invalid argument. A corollary, as Schiffer observes, is that—since we may run the Sorites just using hundreds of special conditional premises instead of the usual major premise—we have to regard the validity of modus ponens as an indeterminate issue too! Is this something that can be lived with? There may be some temptation to reply that Schiffer’s account treats the validity of modus ponens, or other classical rules, as indeterminate only when we are involved with indeterminate premises and/or conclusions. Elsewhere there is no problem. But I do not think such an attempt to ring-­fence the sin­ gularity makes sense. Validity—at least, classical validity—turns on compos­ sibilities of truth-­value. If certain argument patterns have instances involving indeterminate premises and false conclusions—or, more graphically, true premises and indeterminate conclusions20—then they are not known to be 20  Consider the inference from the minor premise and negation of the conclusion of the Sorites to the negation of the major premise.

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On the Characterization of Borderline Cases  391 truth-­preserving. So why should we trust them anywhere? And, in any case, do we not need reliable rules in terms of which to bring out the commitments of a thinker who takes a view—however qualified—on an indeterminate issue? I have little space to pursue this fundamental matter further. To be sure, Schiffer’s predicament—if that is what it is—is a function of his choice of the Łukasciewicz-­style matrices, and they have independent discomforts in this setting; for example, they mispredict the level of partial belief in which nor­ mal thinkers will repose in conjunctions of vague but incompatible conjuncts, and they predict variations in the V-­credibility of conjunctions of the form Fk and ¬Fk′, k and k′ adjacent in a Sorites series, which are surely not em­pir­ic­al­ly confirmed—one would expect a uniformly low valuation, irrespective of the place in the series of k.21 But the point I would like to emphasize in closing is that the position for which Schiffer proposes to settle is one in which he has, given other things he accepts, to doubt that certain classically valid inferences are knowledge-­preserving. The inference, for example, from F0 and ¬Fk, for suitably ‘distant’ k, to the relevant unpalatable existential is classically valid and has—one would suppose—known premises. But the conclusion, on Schiffer’s calculation, is indeterminate—so cannot be claimed to be known. Moreover, if, with Schiffer, we accept VE, we will have to say that it cannot be known. In that case—whatever the situation with truth—classical validity fails to pre­ serve knowledge. Strikingly, that is, in effect, exactly the intuitionists’ com­ plaint about it: that it permits the derivation from warranted premises of conclusions for which there is no warrant—in particular, none elicitable from the warrant for the premises and the derivation. My own treatment of this agenda is precisely fashioned around argument for and consequences of the thesis that classical logic is epistemically non-­conservative where vagueness is concerned. Schiffer’s views commit him to the same. What he has yet to pro­ vide is his own account of the shape a logic should assume to remedy the defect.

Conclusion The foregoing has concentrated on points of disagreement and difference. It would be entirely inappropriate to end on anything but a note of admiration

21  Elia Zardini has suggested in discussion that a restriction of the Łukasiewicz clauses to inde­ pendent propositions would be a natural and attractive way of trying to get around these awkward­ nesses (assuming a case can be made that ¬Fk′ is relevantly negatively dependent on Fk).

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392  The Riddle of Vagueness for the work I have been commenting on. For the philosopher who wishes better to understand the nature of linguistic competence and linguistic representation, vagueness presents challenges of exceptional importance and the greatest intellectual difficulty. Stephen Schiffer has responded to these challenges with a rare mix of breadth of philosophical vision, resourcefulness, and dialectical and technical expertise. I look forward to the products of his continuing engagement.22

22  I am most grateful to Elia Zardini for detailed comments on the penultimate draft, and to him and other members of the Arché Vagueness project for feedback on the ideas herein canvassed at vari­ ous project seminars over the last few years.

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14 Intuitionism and the Sorites Paradox I Introduction: The Basic Analogy Mathematical Platonism may be characterized as the conviction that in pure mathematics we explore an objective, abstract realm that confers determinate truth values on the statements of mathematical theory irrespective of human (finite) capacities of proof or refutation. This conviction crystallizes in the belief that classical logic, based on the semantic Principle of Bivalence, is the appropriate logical medium for pure mathematical inference even when, as of course obtains in all areas of significant mathematical interest from numbertheory upwards, we have no guarantee of the decidability by proof of every problem. In this respect—the conviction that the truth-values, true and false, are distributed exhaustively and exclusively across a targeted range of statements irrespective of our cognitive limitations—an epistemicist conception of vagueness1 bears an analogy to the Platonist philosophy of mathematics. Let us characterise a vague predicate as basic just if it is semantically unstructured and is characteristically applied and denied non-inferentially, on the basis of (casual) observation. The usual suspects in the Sorites literature—bald, yellow, tall, heap and so on—are all of this character and will provide our implicit focus in what follows. Epistemicism postulates a realm of distinctions drawn by such basic vague concepts that underwrite absolutely sharp ‘cut-offs’ in suitable soritical series,2 irrespective of our capacity to locate them. For the epistemicist, the Principle of Bivalence remains good for vague languages— or, if it does not, it is not vagueness that compromises it—and classical logic remains the appropriate medium of inference among vague statements. An indiscernible difference between two colour patches in a soritical series for 1 As supported by, among others, Cargile (1969), Sorensen (1988),  Horwich (1998), and, in its most thoroughgoing development, Williamson (1994). See also Magidor (2019). 2  We understand a ‘suitable’ Sorites series for a predicate F to be monotonic—i.e., one in which any F-relevant changes involved in the move from one element to its immediate successor are never such that the latter has a stronger claim to be F than the former.

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0015

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394  The Riddle of Vagueness yellow may thus mark an abrupt transition from yellow to orange; the impression of the indeterminacy of that distinction is merely a reflection of our misunderstanding of our ignorance of where the cut-off falls. For the epistemicist, the Sorites paradox is accordingly easily resolved. It is scotched by the simple reflection that its major premise will always be subject to counterexample in any particular soritical series. If the initial element is yellow and the final elem­ent is orange, then there must be an adjacent pair of elements one of which is yellow while the next is orange. It is just that we, in our ignorance, are in no position to identify the critical pair. On one understanding of it, the ur-thought of Intuitionism as a philosophy of mathematics is a rejection of the idea of a potentially proof-transcendent mathematical reality as a superstition: something that there is, simply, no good reason to believe in. For the intuitionist, the mathematical facts are justifiably regarded as determinate only insofar as they are determinable by proof, and the relevant notion of proof needs accordingly to be disciplined in such a way as to avoid any implicit reliance on the Platonist metaphysics. So, in any area of mathematics where we lack any guarantee of decidability, the logic deployed in proof construction cannot rely on the Principle of Bivalence and hence—according to the intuitionist—cannot justifiably be classical. In particular, the validity of the Law of Excluded Middle, which Intuitionism understands as depending on the soundness of Bivalence, can no longer be taken for granted. There is evident scope for a similar reaction to the epistemicist conception of vagueness. The latter is a commitment to a transcendent semantics for vague expressions which construes them as somehow glomming onto semantic values—properties in the case of vague predicates—that are possessed of absolutely sharp extensions, potentially beyond our ken. The conception that vague expressions work like that may likewise impress as the merest superstition. Perhaps a little more kindly, it may impress as merely ad hoc, for there is not the slightest reason that speaks in favour of it except its convenience in the context of addressing the Sorites.3 Say that an object, o, is F-surveyable just in case o is available and open to as careful an inspection as is necessary to justify the application of F to it whenever it can be justified. For the epistemicist, reality is such that the application of any meaningful basic vague predicate, F, to an F-surveyable object must result in a statement 3  I have sometimes encountered in discussion the impression that the motivational shortfall here is addressed by Williamson’s argument that knowledge everywhere requires a margin of error. Not so. What that argument establishes, if anything, is that, if there is a sharp cut-off in a Sorites series, we will not be able to locate it. The argument provides no reason to suppose that the antecedent of that conditional is true. I will say a little more about the Williamsonian argument below.

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Intuitionism and the Sorites Paradox  395 of determinate truth value, true or false. For an intuitionistic conception of vagueness—one conceived on the model of mathematical Intuitionism—a satisfactory semantics and logic for basic vague predicates must eschew commitment to any such claim. The avoidance of such a commitment is, of course, common ground with any instance of the long tradition of theories about vagueness that construe borderline cases as examples of semantic indeterminacy: as cases where the rules of the language leave us in the lurch, so to speak, by issuing no instruction for any particular verdict. Here, though, the intuitionist credits the epi­ stem­icist with a crucial insight: that vagueness is indeed a cognitive rather than a semantic phenomenon; that our inability to apply the concepts on either side of a vague distinction with consistent mutual precision is not a consequence of some kind of indeterminacy or incompleteness in the semantics of vague expressions but is constitutive of the phenomenon. Consider this example. Suppose we are to review a line of 100 soldiers arranged in order of decreasing height and to judge of each whether they are at least 5' 10'' tall—but to judge by eye rather than by using any means of exact measurement. Let the soldiers’ heights range from 6' 6'' to 5' 6''. This provides a toy model of a Sorites series as conceived by the epistemicist. For, while there is indeed a sharp cut-off—there must be a first soldier in the line who is less than 5' 10'' tall—our judgements about the individual cases will ex­pect­ ably divide between an initial range of confident positive verdicts and a later range of confident negative verdicts between which there will be a region of uncertainty, where we return hesitant, sometimes mutually conflicting, verdicts and sometimes struggle to return a verdict at all. Here, of course, we have a conception of canonical grounds determining whether a hard case has the property expressed by the predicate at issue, so that is a point of contrast with our situation when we face a Sorites series for a vague predicate as conceived by Epistemicism. But, putting that disanalogy to one side, it remains that, in the soldiers’ scenario, our patterns of judgement will have exactly the physiognomy that the epistemicist regards as the hallmarks of vagueness. Hence, in their view, there is nothing in our practice with vague concepts that distinguishes it from judgements concerning sharply bounded properties about whose specific nature we are ignorant. The intuitionist agrees with Epistemicism that such a physiognomy of practical judgement is characteristic of vagueness. But Intuitionism drops both the assumption that the judgements concerned are answerable to the extension of a sharply bounded property and the notion that a different kind of explanation of these characteristic judgemental patterns is called for, in terms,

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396  The Riddle of Vagueness roughly, of shortcomings in—our lack of guidance by—the semantic rules that fix the meanings of the expressions concerned. For the intuitionist, the vagueness of a predicate consists in these distinctive patterns in our use of it. They are the whole story. The intuitionist conception of vagueness is thus a deflationary conception: it holds that there is no more to the phenomenon than meets the eye, so to speak—that it is unnecessary, is indeed a mistake, to look to some underlying feature of the semantics of vague expressions to explain our characteristic patterns of judgement in the borderline area. (That is not to say that one should not look for an explanation of a different kind.) It is the view of the intuitionist that both Epistemicism and Indeterminism commit versions of this mistake. The justification for this charge when the canvassed alternative is semantic indeterminism rests on a complex variety of considerations whose details, for reasons of space, I cannot rehearse here.4 However, there is one such consideration—one relevant aspect of our practice with vague concepts—which is particularly important for the grounding of the most distinctive aspect of the intuitionist approach. Semantic Indeterminism interprets borderline cases of a distinction as cases where there is no mandate to apply either of the expressions concerned—where the rules for their use prescribe no verdict. That suggestion does a poor job of predicting one salient aspect of our judgemental practices with vague concepts—namely, our uncritical attitude to polar—posi­ tive or negative—judgements concerning items in their borderline regions. Provided a verdict is suitably qualified and evinces an awareness that the case is a marginal one, it is not treated as a mark of incompetence, or mistake, to have a view, positive or negative, about any single borderline case.5 Suppose X struggles to have an opinion whether some shade from the mid region of the yellow–orange Sorites is yellow enough to count as yellow but Y is of the opinion that it is—just about—yellow. Our sense is that such divergences are just what is to be expected, and that each reaction can be as good as the other. X need not be regarded as coming short; Y need not be regarded as overreaching. Each reaction is quite consistent with full mastery of yellow and due attention to the hue concerned. According to semantic indeterminism, this laissez-faire attitude should be regarded as cavalier, for X and Y cannot both be operating as the relevant semantic rules require; the rules cannot both be silent on the relevant hue and 4  For some elaboration, see Wright (2003c, this volume, Chapter 9; 2007, this volume, Chapter 11; 2010, this volume, Chapter 12). Also Williamson (1994). 5  For the purposes of this claim, we may take a borderline case to be any that tends to elicit the judgemental physiognomy characterized earlier among a significant number of competent judges.

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Intuitionism and the Sorites Paradox  397 mandate Y’s qualified verdict of ‘yellow’.6 Yet our ordinary practice reflects no sense of that. We are characteristically open to—as I have elsewhere expressed it (Wright 2003c, this volume, Chapter 9; 2007, this Volume, Chapter 11), liberal about—polar verdicts about borderline cases. To be sure, the indeterminist might be tempted to interpret this liberality as reflecting a sense of respect for our ignorance about in just which cases the rules do in fact fall silent— which are the true borderline cases. But since, if so, there is no evident means of remedying that ignorance, that again would be a step in the direction of objectionably transcendentalising the semantics of vagueness. For the in­tu­ ition­ist, in contrast, there really need be no sense in which one who returns a (qualified) polar verdict about a borderline case does worse than one who fails to reach a verdict. The point may seem slight, but it is crucial. For respecting this aspect of our practice as in good standing requires that, in contrast to the view of Semantic Indeterminism, we should not regard borderline cases as presenting truthvalue gaps.7 If borderline cases are truth-value gaps, then someone who returns a polar verdict about such a case actually makes a mistake. And that is just what, according to Liberalism, we have no right to think. It follows that we have no right to regard borderline cases as counterexamples to the prin­ ciple of Bivalence, and hence that vagueness, as now understood, provides no motive to reject Bivalence. Since, by rejecting epistemicism, and recognizing that we cannot in general settle questions in the borderline region either, we have also undercut all motive to endorse the principle, the resulting position is exactly analogous to the attitude of the mathematical intuitionist to Bivalence in mathematics: that it is a principle towards which we should take an agnostic stance. With these preliminaries in place, let us turn to review how an in­tu­ition­istic treatment of the Sorites may be developed in more detail. 6  To be sure, there is another possibility: we might try to think of the rules as, in the borderline area, issuing permissions. Then both a tentative verdict and a failure, or unwillingness, to reach a verdict, may be viewed as rule compliant. But it is very doubtful that any satisfactory proposal lies in this direction. Presumably among the clear cases the rules must mandate specific verdicts rather than merely permit them. So we need to ask about the character of the transition from cases where a posi­ tive verdict about F is mandated to cases where it is merely permitted. If this is a sharp boundary, then, since there is again no possibility of knowing where it falls, the proposal will have ‘transcendentalized’ the semantics of F in a manner different from but no less inherently objectionable than Epistemicism. But, if the transition is accomplished by a spread of further borderline cases—cases that are borderline for the distinction between ‘mandatorily judged as F’ and ‘permissibly judged as F’—then the question arises how, in point of mandate or permission, cases in this category are to be described. For argument that contradiction ensues, see Wright (2003c, this volume, Chapter 9). 7  Or indeed as having any kind of ‘Third Possibility’ status inconsistent with each of truth and falsity simpliciter. For further discussion, see Wright (2001b, this volume, Chapter 7).

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398  The Riddle of Vagueness

II The Tolerance and ‘No Sharp Boundaries’ Paradoxes The classic deductive8 Sorites paradoxes vary in two respects: first in the formal character of the major premise involved, and second—where the major premise is shared—in the manner in which that premise is made to seem plausible. And, of course, different forms of major premise will call for cor­res­ pond­ing­ly different deductive sub-routines in the derivation of the paradox. Perhaps the most familiar form of the deductive Sorites is what we may call the Tolerance paradox. As normally formulated, it uses a universally quantified conditional major premise:    TP :

("x )(Fx ® Fx ¢)

and proceeds on the assumption of one polar premise, F1, and n-1 successive steps of universal instantiation and modus ponens to contradict the other polar premise, ¬Fn. As for motivation, the key thought is, as the title I have given to the paradox suggests, that, such is its meaning, the application of F, and/or the justification for applying it, tolerates whatever small changes may be involved in the transition from one element of the series to the next: for instance, that, if a colour patch is (justifiably described as) red, a pairwise indiscriminable (or even just barely noticeable) change in shade must leave it (justifiably described as) red; that, if a person is bald, the addition of a single hair will not relevantly change matters, and so on. For the examples with which we are concerned, claims of this ilk can seem thoroughly intuitive; and they can be supported by a variety of serious-seeming theoretical con­sid­er­ ations.9 In some cases, indeed, the claim of tolerance may seem absolutely unassailable: how could ‘looks red’, for example, fail to apply to both, if to either, of any pair of items that look exactly the same? Unfortunate, then, that ‘looks exactly the same’ is not a transitive relation. The Tolerance paradox, however, impressive as it may be in particular cases, is not, or at least not obviously, a paradox of vagueness per se. Vagueness is not, or at least not obviously, the same thing as tolerance. Precision must imply non-tolerance, of course, but the converse is intuitively less clear. Ought 8  As distinguished from the so-called Forced March Sorites. For reasons of space, I must forgo discussion of that here. See Oms and Zardini (2019, pp. 14–16) 9  For elaboration of some such, see Wright (1975, this volume, Chapter 1).

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Intuitionism and the Sorites Paradox  399 there not somehow to be some distance between a predicate’s possession of borderline cases and its being tolerant of some degree of marginal change? While the claim may indeed seem intuitive, it requires—in the presence of paradox—argument to suggest that yellow, heap, bald, and so on, are tolerant. But no argument is required to suggest that they are vague. That these predicates are vague is a datum. The No Sharp Boundaries paradox, by contrast, impresses as a paradox of vagueness par excellence. It works with a negative existential major premise,    NSB:

Ø($x )(Fx & ØFx ¢ )

that may very plausibly seem simply to give expression to what it is for F to be vague in the series of objects in question. For vagueness, surely, is just the complement of precision, and the sentence of which that negative existential is the negation—namely, what I have elsewhere (Wright  2007, this volume, Chapter 11) called the unpalatable existential    UE:

($x )(Fx & ØFx ¢)

surely just states that F is precise in the series in question: that there is a sharp boundary between the Fs and the non-Fs, and so no borderline cases. If, then, F is in fact vague, the negative existential seems imposed just by that fact, indeed to be a statement of exactly that fact. And now contradiction follows by iteration of a different but no less basic and cogent-seeming deductive subroutine, involving conjunction introduction, existential generalization, and reductio ad absurdum as a negation introduction rule.10 With both paradoxes, there is the option of letting the reasoning stand as a reductio of the major premise. If we take that option with the Tolerance paradox, we treat it as a schematic proof that none of the usual suspects is genuinely tolerant of the marginal differences characteristic of the transitions in a soritical series for it. Tolerance, in that case, is simply an illusion. And that is a conclusion we might very well essay to live with, provided we can provide a satisfactory explanation of why and how the illusion tends to take us in, and of what is wrong with the ‘serious-seeming theoretical considerations’ apparently enforcing tolerance that I have already alluded to.

10  That is, the intuitionistically valid half of classical reductio, where the latter also allows reductio ad absurdum inferences that serve to eliminate negations.

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400  The Riddle of Vagueness But not so fast: even if those obligations can be discharged, the proposed response, in the presence of classical logic, is not yet stable. For (allowing its ingredient conditional to be material) the negation of TP, now regarded as established by the paradoxical reasoning, is a classical equivalent of the unpalatable existential. So, if our logic is classical, non-tolerance does after all collapse into precision, and, to the extent that one feels there should, as remarked above, be daylight between them, that should impress as a black mark against classical logic in this context. Moreover, that impression is only reinforced when one considers the option of letting the No Sharp Boundaries paradox stand as a refutation of NSB. For then all that stands between that result and affirmation of the unpalatable existential is a double negation elimination step. And now, once constrained by classical logic to allow the inference to UE, we seem to be on the verge of admitting that vagueness itself is an illusion. That, surely, is not anything we can live with. Intuitionism, by contrast, aims at winning through to a position where we can accept each of the Tolerance and No Sharp Boundaries paradoxes as a reductio of its major premise but refuse in a principled way the inference onwards to the unpalatable existential. We also aim to retain the ordinary conception of an existential statement as requiring a determinate witness for its truth, and thus to avoid any form of the implausible semantic story that construes the statement ‘There is in this series a last F element followed immediately by a first non-F one’ as neutral on the question of the existence of a sharp cut-off as intuitively understood. Our path will be to explore, in the light of the general, deflationary conception of the nature of vagueness outlined earlier, what motivation it may be possible to give for broadly in­tu­ition­ ist restrictions on the logic of inferences among vagueness-involving statements. In this we follow a suggestion first briefly floated at the end of Hilary Putnam (1983).11 If, in particular, we can justify a rejection of double negation elimination for molecular vague statements in general, then it may be possible comfortably to acknowledge that both the Tolerance and the No Sharp Boundaries paradox do indeed disprove their respective major premises without any consequent commitment to the unpalatable existential, nor consequent obligation either (with the epistemicist) to believe it or (with the supervaluationist) somehow to reinterpret it in such a way that it does not mean what it seemingly says. 11  Early discussions of Putnam’s proposal, besides my own work from Wright (2001, this volume, Chapter  7) onwards, include Read and Wright (1985, this volume, Chapter  3), Putnam (1985), Schwartz (1987), Rea (1989), Putnam (1991), Schwartz and Throop (1991), Mott (1994), Williamson (1996b), and Chambers (1998).

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Intuitionism and the Sorites Paradox  401

III Constraints on an Intuitionistic Solution I propose that we set the following three constraints on the project. First (Constraint 1), and most obvious, we need to motivate the required restrictions on classical logic in general and, in particular, to explain how a valid reductio of TP, or NSB, can fail to justify the unpalatable existential. Second (Constraint 2), as with all attempts to solve, rather than merely to block a paradox, we must offer a convincing explanation of why the premises that spawn aporia impress us as plausible in the first place, of what mistaken assumptions we have implicitly fallen into that give them their spurious credibility. So, in the present instances, we must contrive to explain away the continuing powerful temptation to regard the major premises for the Tolerance and No Sharp Boundaries paradoxes as true. I have said much elsewhere to attempt to defuse the attractions of tolerance premises.12 Here we will focus on the challenge to explain why NSB is not a satisfactory characterization of F’s vagueness in the series in question. (We have already implicitly shown our hand on this.) Finally (Constraint 3), I think it reasonable to require, although I grant it is not wholly clear in advance exactly what the requirement comes to, that Constraints 1 and 2 should, so far as we can manage it, be satisfied in a way that draws on an overarching account of what the relevant kind of vagueness consists in (that is, of the nature of the relevant kind of borderline cases). We are proposing restrictions on what, from a classical point of view, are entrenched, tried, and tested patterns of inference. If such restrictions are justified, it may be, to be sure, that that justification is global, applying within discourses of every kind. That is the character, for example, of the metasemantic considerations about acquisition and manifestation of understanding originally offered by Michael Dummett half a century ago in support of a global repudiation of the Principle of Bivalence except in areas where decidability is guaranteed. Whatever one’s estimate of such arguments, what Constraint 3 is seeking is a justification for relevant restrictions on classical logic that is specifically driven by aspects of the nature of vague discourse. In

12  Such an attempt must perforce be somewhat ramified, in order to match the diverse sources of such attraction. My own diagnostic forays run from Wright (1975, this volume, Chapter 1) through Wright (1987, this volume, Chapter 4) to Wright (2007, this volume, Chapter 11).

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402  The Riddle of Vagueness the present context, that will require putting to work the deflationary conception of vagueness sketched in Section 1.

IV Addressing Constraint 1: The Basic Revisionary Argument In the mathematical case, as remarked, the intuitionistic attitude flows from a rejection of the Principle of Bivalence, based on a repudiation of Platonist metaphysics and insistence that truth in pure mathematics can only consist in the availability of proof.13 In the case of vague statements, many would be pre-theoretically willing to grant that Bivalence is generally unacceptable anyway. Certainly, the metaphysics of meaning implicit in epistemicism has none of the intuitive appeal of, say, arithmetical Platonism. But, even if it is granted that Bivalence is principium non gratum where vagueness is concerned, re­pudi­at­ing the principle is one thing and motivating revision of classical logic a further thing. Classical logic need not necessarily fail if Bivalence is dropped. How should the intuitionist argue specifically that the logic of vague discourse should not be classical? What I once called the ‘basic revisionary argument’ is designed to accomplish that result. It runs for any range of statements that are not guaranteed to be decidable but are subject to a pair of principles of evidential constraint (EC). That is, for each such statement P, each of these conditionals is to hold:

 EC: P → it is feasible to know P  Not P → it is feasible to know not-P

Now, it is plausible—but with caveats, to be considered in a moment—that each of the usual suspects (yellow, bald, tall, heap, and so on) generates atomic predications that exhibit this form of evidential constraint; that is, intuitively, if something is, in the sense characterized earlier, surveyable for yellow (that is, it is available for inspection in decent conditions, and so on), and it is yellow, then we will be able to tell that it is; and if it is not yellow, we will be able to tell that. Intuitively, what colour something is cannot hide if and when 13  This argument is central in Wright (1992b; 2001b, this volume, Chapter  7; 2007, this volume Chapter 11).

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Intuitionism and the Sorites Paradox  403 conditions present themselves in which it is possible to have a proper look at it.14 And analogously for baldness, tallness, and ‘heaphood’. The basic revisionary argument is then the observation that, if the Law of Excluded Middle is retained for all such predications, P, then reasoning by cases across the EC-conditionals will disclose a commitment to the disjunction:   D:

It is feasible to know P V It is feasible to know not-P.

In effect, the thesis that P is decidable. But of that, if the relevant predicate is associated with borderline cases, we have no guarantee. Accordingly, we have no guarantee of the validity of the Law of Excluded Middle in application to such statements and therefore have no business reckoning it among the lo­gic­al laws. Simply expressed, the thrust of the argument is that a range of statements may be such as both to lack any general guarantee of decidability in an arbitrary instance and to have a guarantee that, if any of them is true, it will be recognizably true and, if false, recognizably false. Imposition of the Law of Excluded Middle onto such statements will then enforce the conclusion that each of them is decidably true or false—contrary to hypothesis. It will amount to the pretence of a guarantee that we do not actually have. Arguably a very large class of statements are in this position, including not merely vague predications but, for instance, evaluations of a wide variety of kinds, including expressions of personal taste, humour, and perhaps (some aspects of) morality. And of course the argument will run for any region of discourse where we reject the idea that truth can outrun all possibility of recognition but have to acknowledge that we lack the means to decide an arbitrary question—exactly the combination credited by the intuitionists for number theory and analysis. Suppose then that we disdain the Law of Excluded Middle on this (or some or other) basis. The soritical series we are considering involve a monotonic direction of change: that is, any F element is preceded only by F elements, and any non-F element is succeeded only by non-F elements. The reader will observe, accordingly, that, once the Law of Excluded Middle is rejected, the sought-for distinction between the unpalatable existential and its double neg­ ation is enforced. For the latter is surely established by the inconsistency of NSB with the truths expressed by the polar assumptions. But, given monotonicity, the unpalatable existential is equivalent to the Law of Excluded Middle over the range of atomic predications of F on the series of elements in question. 14  The claim that EC holds good for these cases is thus not subject to ‘killer yellow’ issues.

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404  The Riddle of Vagueness

V One Objection to the Basic Revisionary Argument So far, so good. But now for the caveats. The EC-conditionals are chal­lenge­able on a number of serious-looking counts. First, they are in direct tension with the upshot of Timothy Williamson’s recently influential ‘anti-luminosity’ argument (2000a, ch. 4). Familiarly, Williamson makes a case that, if know­ledge generally is to be subject to a certain form of (putatively) plausible safety constraint, then it must be controlled by a margin of error: in particular, if a subject knows that F holds of an object a, it cannot be that F fails to hold of any object that they could not easily distinguish, using the same methods, from a. The effect is thus that, for elements, x, in a soritical series for F, the following conditional is good:   (It is feasible to know that Fx) → Fx′,

which, paired with the first of the EC-conditionals, immediately provides the means to show that F applies throughout the soritical series. Here is not the place for a detailed engagement with Williamson’s thesis. But there are a couple of fairly immediate misgivings about it that deserve notice. One is whether the notion of safety that it utilizes is indeed a wellmotivated constraint on knowledge everywhere, whatever the subject matter and methods employed. Williamson’s intuitive thought, if I may venture a précis (cf. Williamson 2000a, p. 97), is that, if a subject comes to the judgement that Fa, and a' is pairwise indistinguishable from a, then the subject must be significantly likely also to judge that Fa'—and now, if the latter is false, they are therefore very likely to make a false judgement using the very methods they used in judging Fa. So those methods are not generally reliable, in which case the judgement that Fa, based upon them, ought not to count as knowledgeable in the first place. Yet, if that is the intuitive thought, one salient question is why we should require that, in order to count as a reliable means for settling a question about one item, a method must also be reliable about others that, however similar, differ from it. Why could not a machine— a speedometer, for example—that issues a varying digital signal in response to a varying stimulus have an absolutely sharp threshold of reliability, so that its responsive signals are reliable up to and including some specific value, k, in its inputs but then go haywire for inputs of any greater value. In that case, its signals may be regarded as ‘knowledgeable’ for any input value i, less than but as close to k as you like. If it is not a priori ruled out that our judgements, for

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Intuitionism and the Sorites Paradox  405 some particular pairings of subject matter and methods, are like that, then it is not a priori guaranteed that Williamsonian safety is everywhere a necessary condition of knowledge. One can envisage the likely rejoinder that as a matter of anthropological fact we are not in any area of our cognitive activities comparable to such a machine. Still, even if that is so, it seems incredible that such a contingency could somehow entail that there are yellows, and instances of baldness, and so on, that lie beyond our powers of recognition even in the best of circumstances. But now grant that the general requirement of safety proposed—again: the proposal that, in order to know that P in circumstances C, my methods must be such that they could not easily lead me astray in circumstances sufficiently similar to C—grant that this is well-motivated everywhere. A second misgiving is that, in the way that Williamson puts the proposal to work, no account is taken of the possibility of response dependence: the idea that some kinds of judgement—and here the critic is likely to be thinking of exactly the kinds of judgement, about sensations and other aspects of one’s occurrent mental state, that Williamson means to target in directing his argument against the traditional idea of our ‘cognitive home’—are not purely discriminatory of matters constituted independently but are such, rather, that the subject’s own judgemental dispositions are somehow themselves implicated in the facts being judged. For any area of judgement where this idea has traction, the supposition that in perfectly good conditions of judgement we might easily respond to what is in fact a non-F case in the way we do to an F case that is very similar to it is in jeopardy of incoherence. Simply, if F-ness and non-Fness are response-dependent matters, then it cannot legitimately be assumed that, purely on the basis of their similarity in a particular case, we will be at risk of responding to a non-F case in the way we do to an F case. To be sure, the heyday of the recent discussion of response dependence has passed, and rigorous but still dialectically useful formulations of it proved hard to come by when the debates were at their height. Still, many may feel that there is an elusive truth in it, with qualities instantiated in one’s phenomenal mental life and Lockean secondary qualities of external objects generally providing two examples of domains to which philosophical justice can be done only by keeping a place for the idea of response dependence on our philosophical agenda.15 15  Concerns of this character, although he does not mention the notion response- (or judgement-) dependence by name nor relate his discussion to the literature about it, are nicely elaborated in Berker (2008).

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406  The Riddle of Vagueness I do not think, accordingly, that Williamson’s argument, in our present state of understanding, comes anywhere near to establishing that the basic revisionary argument is hobbled by its reliance on the EC-conditionals. Rather, the argument sets up yet another philosophical paradox: prima facie plausible thoughts about knowledge in general and a putative requirement of the safety/reliability of methods whereby beliefs are formed prove to conflict with prima facie plaus­ ible thoughts about the luminosity of a range of concepts for which, we would probably otherwise be inclined to think, the EC-conditionals look good. Something has to give. But here cannot be the place further to investigate what.

VI Two Further Objections There are, however, two less theoretically loaded reservations about the role of the EC-conditionals in the basic revisionary argument that should be tabled when what is envisaged is its application specifically to vague expressions. First, no connection has actually been explained to link the EC-conditionals with vagueness as such. All that has been offered is the suggestion that the conditionals are plausible for some examples of vague predicates—for the usual suspects. A general theoretical connection is wanted before there can be any firm prospect of a solution by this route to the Sorites paradox in general. One senses that a development may be possible of a general connection between vague judgement and response-dependent judgement, grounded in the thought that the status of something as a borderline case is a response-dependent matter. That suggestion, though, once again in the present state of our understanding, is merely speculative. Second, and perhaps more threatening to this particular strategy for underwriting an intuitionistic treatment of the Sorites, is the conflict between the EC-conditionals and a principle I have elsewhere called Verdict Exclusion (VE): VE: Knowledge is not feasible about borderline cases. EC and VE are pairwise inconsistent (since, as the reader will speedily see, they combine to enforce contradictory descriptions of borderline cases). So someone who accepts EC must deny VE. But VE may well impress—indeed has impressed a number of expert commentators (Williamson  1996b; Rosenkranz 2003; Schiffer 2016)—as a datum. In any case, the principle may

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Intuitionism and the Sorites Paradox  407 seem to have powerful intuitive support from the very deflationary conception of vagueness which, I have proposed, should be seen as the mainspring of an intuitionistic treatment. On that conception, borderline cases are constitutively cases whereby subjects characteristically fall into weak, inconstant, and mutually conflicting opinions. Any opinion a subject holds about such a case is one that they might very easily, using just the same belief-forming methods, not have held. Surely on any reasonable interpretation of a safety, or reliability, constraint on knowledge, that must count as inconsistent with such an opinion’s being knowledgeable. Elsewhere (Wright 2003c, this volume, Chapter 9), I have suggested that an endorsement of VE proves, on closer inspection, to be in tension with Liberalism. Let me here make a different point. Once it is given that something is a borderline case, I think the line of argument just outlined for VE is likely to prove compelling. But the crucial consideration is that, of any particular element in a Sorites series, it is not a given—except as a contingent point about the sociology of a particular group of judges—that it is a borderline case. Being a borderline case is judge relative: x may be such as to induce the characteristic judgemental difficulties and variability in some but not other competent judges. Let the proposition that x is yellow elicit those characteristic responses in some of us but suppose that Steady Freddy consistently judges x yellow (though acknowledges that it is near the borderline). Must we deny that Freddy’s verdict is knowledgeable? After all, it is, we may suppose, the verdict of someone who gives every indication otherwise of a normal competence in the concept, has normal vision, and is judging in good conditions—and judging in a way consistent with their judgement of the same shade on other occasions. It is harsh to say they do not know.16 And, if it is at least indeterminate whether Freddy knows, then we do not know VE. On the other hand, if we take it that the EC-conditionals are known to hold good for surveyable predications of F, must we not also accept the strange claim that VE is known to be false for such cases and hence that each element in a soritical series for F allows in principle of a knowledgeable verdict about its F-ness? It is not clear. There is a double negation elimination step in the drawing of that conclusion whose legitimacy might be viewed as sub judice in the present dialectical context. Rather than take a stand on the matter, it may

16  Some will no doubt say that Freddy has a different concept. But that seems merely ad hoc. What does the difference consist in? Why not say instead that they are steadier than we are in their judgements involving a concept we share?

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408  The Riddle of Vagueness seem that prudence dictates, pro tem., that we reserve judgement on both VE and EC, committing to neither. Prudence, though, comes at a cost. Unfortunately for the would-be in­tu­ ition­ist, that agnostic attitude requires that we must also be agnostic about the basic revisionary argument. If, in our current state of philosophical information, the strongest relevant claim we can justifiably make about the EC-conditionals is that it is epistemically possible that they hold good for surveyable predications of ‘yellow’, ‘bald’, and so on, then, supposing we accept the validity of the Law of Excluded Middle, we can validly reason only to the epistemic possibility that D above holds good—that is, that it is epi­stem­ ic­al­ly possible that, for each P in the relevant class of statements, it is feasible to know P or it is feasible to know not-P. But that double-modalized conclusion does not look uncomfortable—or, anyway, not uncomfortable enough to put pressure on the acceptance of Excluded Middle. In particular, if it is epi­stem­ ic­al­ly possible that Steady Freddy indeed knows, then for each P in the relevant range, there epistemically possibly could be a steady subject who knowledgeably judges that it is true (or that it is false). The revisionary import of the basic revisionary argument requires more than that the EC-conditionals are epistemically possibly correct.

VII Addressing Constraint 1: A Different Tack—Knowledge-Theoretic Semantics So what now? Well, a suspension, perhaps temporary, of confidence in the basic revisionary argument in this context need not surrender all prospect of a strong motivation for an intuitionistic approach to the logic of vague discourse. The basic revisionary argument attempts to garner the desired result without any particular assumptions about semantics. Let us therefore now instead consider directly what might be the most desirable shape for a semantics to take that is to be adequate for a language—a minimally sufficient sor­iti­ cal language for F—that has just enough resources to run instances for a particular vague predicate F of both the Tolerance and No Sharp Boundaries paradoxes. Such a language thus contains the predicate F, a finite repertoire of names, one for each member of a suitable soritical series, brackets subject to the normal conventions, and the standard connectives and quantifiers of firstorder logic.

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Intuitionism and the Sorites Paradox  409 Let L be such a language. Since we wish to avoid any commitment to the idea that, when F is applied to an object that is surveyable for it, the result can take a truth value beyond our ken, we have no interest in any semantic theory for L that works with an evidentially unconstrained notion of truth. But nor, since we are now (even if temporarily) agnostic about EC (and therefore also about VE), should such a semantics work instead with a verificationist notion of truth. It follows that we should not choose a truth-theoretic semantics at all.17 But then what? Well, what any competent practitioner of L has to master are the conditions under which its statements may be regarded as known or not. We may therefore pursue a semantic theory that targets such conditions directly, in a spirit of aiming at a correct description of what we are in pos­ ition to regard as knowledgeable linguistic practice. It will be for the critic to make the case, if there is a case to make, that we thereby misdescribe the practice we actually have. How to make a start? We do not have much to go on. What is solid to begin with is only that there is a range of polar cases where there is no doubt that Fx may be known, a range of polar case where there is no doubt that ¬Fx may be known, and a range of cases that manifest the uncertainty and variability of judgement that our governing deflationism regards as constitutive of vagueness. But consider the following controversial principle (CP): All the knowable statements in L are knowable by means of knowing the truth values of atomic predications—(which we are assured of being able to do only in polar cases.)

According to CP, any of the molecular statements of L can be known, if it can be known at all, by knowing some of L’s atomic statements. So the semantic clauses for the connectives and quantifiers by means of which any particular molecular statement is constructed ought—if that statement is to be reckoned knowable—to reflect an upwards path, as it were, whereby the acquisition of such knowledge might proceed. If we accept CP, we will be looking therefore

17  I am not here assuming that merely to give a truth-theoretic semantics for some region of discourse must involve explicit commitment to one horn or the other of this alternative. But the question may legitimately be pressed, and the point I am making in the text is that we cannot answer unless at the cost of surrender of agnosticism about EC. Better, therefore, not to invite the question. However, there is more to say about the motivation for the style of semantics about to be proposed. I will return to the matter at the end.

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410  The Riddle of Vagueness for a semantics that recursively explains conditions of knowledge for the molecular statements of L in terms of those of their constituents.18 Presumably, we are not going to want to accept CP. ‘Controversial’ somewhat flatters the principle. We will surely want to admit a range of exceptions, cases where a molecular statement plausibly holds good even when its constituents are borderline. Some, for instance, will be general statements that are arguably analytic of the specific vague predicate concerned, such as ‘Everything red is coloured’; others may be nomologically grounded in the property concerned, such as maybe ‘All heaps are broadest in the base’. A more interesting class of exceptions are what Kit Fine once characterized in terms of the notion of penumbral connection (Fine 1975). They will concern vague predicates in general. Epistemicists will regard some instances of the Law of Excluded Middle as coming into this category. We will not follow them in that, but we should want to allow, for example, that, no matter what F may be, all instances the Law of NonContradiction are knowable as, with respect to the kind of series we are concerned with, are all monotonicity conditionals—that is, statements of the form Fx′ → Fx,

notwithstanding whether x is borderline for F. The same will hold for the corresponding generalizations:    ("x ) Ø(Fx & ØFx ), ("x )(Fx ¢ ® Fx ), ( "x )(ØFx ® ØFx ¢), ¼ To be sure, that such claims are knowable is not uncontroversial. It is a familiar feature of many-valued treatments of vagueness that such principles as these are sometimes parsed as indeterminate—when, for instance, in­de­ter­ min­acy in a conjunct is treated as depriving a conjunction of determinate truth, or a conditional with an indeterminate antecedent and consequent is regarded as thereby indeterminate. We are not here taking a stand on the question whether such treatments are appropriate when one accepts their governing assumption—namely, that being a borderline case is a kind of alethic status, contrasted with both truth and falsity. But we are rejecting the governing assumption. And when instead borderline-case status is viewed as a cognitive status, as on our governing assumption, there is no evident reason to demur at the suggestion that principles of penumbral connection can be known. We can know of structural constraints that knowledge, were it but 18 For ease of formulation, I here count the instances of a quantified statement as among its ‘constituents’.

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Intuitionism and the Sorites Paradox  411 attainable, of the truth values of a range of statements would have to satisfy without having any guarantee that we can get to know those truth values. These considerations suggest we pursue a theory of knowledge for L that has CP as a motivating base but includes a range of permitted exceptions to it. The theory will incorporate a knowledge-conditional semantics for L and a logic based upon it, but may also contain additional, primitive axioms of penumbral connection and perhaps other axioms analytic of or otherwise somehow guaranteed for a particular choice for F. The semantics will comprise recursive clauses that determine, for each of the quantifiers and con­ nect­ives of L, the conditions that are necessary and sufficient for knowledge of L-statements in which that operator is the principal operator on the basis of the knowledge conditions of its constituents. The natural approach will be something in the spirit of the Brouwer– Heyting–Kolmogorov (BHK) interpretation of intuitionist logic (see, for ex­ample, Troelstra 2011, sect. 5.2), which, as is familiar, proceeds in proof-theoretic rather than truth-theoretic terms. There is, however, an important point about the BHK interpretation that we need to flag before moving to propose specific clauses for the theory for L that we seek. In logic and math­em­at­ics, or so one might plausibly hold, all knowledge (other than of axioms) is conferred by, and only by, proof. So it can look as though BHK-style semantics is already nothing other than a local version of knowledge-conditional semantics. So it is, but expressing matters that way may encourage an oversight. While proofs in logic and mathematics confer knowledge of what they prove, that is not all they do. They also vouchsafe knowledge of what is proved as knowledge. Someone who comprehendingly works through a math­em­at­ic­al proof that P learns not merely that P is true but also—assuming their grasp of the concept of know­ ledge, and so on—that P may now be taken to be part of their knowledge. They establish a right to include P as part of what they may legitimately claim to know. Say that knowledge is certified when accomplished in a fashion that legitimizes that claim: accomplished in such a way that a fully epistemically responsible, sufficiently conceptually savvy epi­stem­ic agent will be aware that they have added to their knowledge. The clauses to follow are to be understood in terms of knowledge that is cer­ti­fi­able—c-knowledge—in this sense.19 Adapting BHK-style clauses in a natural way, we may accordingly propose: 19  It is a consequence of some kinds of knowledge externalism that not all knowledge need be c-knowledge. But the externalist will presumably grant that knowledge often is c-knowledge, since it is not supposed to be a consequence of externalism that our claims to knowledge are mostly imponderable without further investigation. The crucial assumption I am making in what follows is that know­ ledge achieved by canonical means—typically, casual observation—of clear cases of the ‘usual suspects’ will be c-knowledge.

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412  The Riddle of Vagueness ‘A & B’ is knowable just if it is knowable that ‘A’ is knowable and that ‘B’ is knowable. ‘A ∨ B’ is knowable just if it is knowable that either ‘A’ is knowable or ‘B’ is knowable. ‘(∀x)Ax’ is knowable just if it is knowable that, for any object in the sor­ iti­cal domain and term ‘a’, known to denote that object, ‘Aa’ is knowable.20 ‘(∃x)Ax’ is knowable just if it is knowable that, for some object in the soritical domain and term ‘a’, known to denote that object, ‘Aa’ is knowable. (What about the conditional? Actually, we do not strictly need a treatment of the conditional for the present purposes.21 And this is fortunate, since the natural proposal

‘A → B’ is knowable just if it is knowable that, if ‘A’ is knowable, ‘B’ is knowable,

raises an awkwardness which I will explain below.22) We can now assert the following Thesis (verification is left to the reader23): Where validity is taken as c-knowability-preservation, and c-knowledge is taken to be factive and closed over c-knowable logical consequence, the clauses above justify rules of deduction for the listed operators coinciding with the common ground for those operators—the standard rules for conjunction introduction and elimination, disjunction introduction and elim­in­ation, universal generalization and instantiation, and existential generalization and instantiation—recognized by both classical and intuitionist first-order logic. What about negation? An adaptation of the BHK-style clause along the above lines would run: 20  Recall that L will contain a known name for every element of the soritical domain. 21  When the major premise for the Tolerance paradox is formulated, as standardly, as a universally quantified conditional (rather than, e.g., as involving a binary universal quantifier), then the paradox does of course depend on the unrestricted use of modus ponens. But the intuitionist resolution of the paradox to be proposed will pick no quarrel with that and is thus neutral on the semantics of the conditional to that extent. 22  See n. 25. 23 ‘And how’, the dear reader may ask, ‘am I supposed to do that when you have nominated no specific logic for the metalanguage—here English!—in which I am supposed to run through the relevant reflections?’ Touché. But the meta-reasoning concerned will require, besides the noted properties of c-knowledge, no more than the rules of inference for ‘and’, ‘or’, ‘any’, ‘some’, and ‘if ’ which constitute common ground between classical and intuitionist first-order logic.

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Intuitionism and the Sorites Paradox  413 ‘¬A’ is knowable just if it is knowable that ‘A’ is not knowable. But that, obviously, will introduce calamity into any account that accepts VE. Our official stance at this point is one of agnosticism towards VE, but it would be good to have the resource of a treatment of the paradoxes that would be robust under the finding that VE was after all philosophically mandated. In any case, and perhaps more telling, BHK-style negation has always been open to the intuitive complaint that it provides a licence to convert grounds for thinking we are doomed to ignorance on some matter into grounds for denial and thus distorts negation as intuitively understood. There is, however, a natural and much more intuitive replacement: ‘¬A’ is knowable just if some ‘B’ is knowable that is knowably incompatible with ‘A’.24 Or, more generally, ‘¬A’ is knowable just if some one or more propositions are knowable that are conjointly knowably incompatible with ‘A’. There is no space here to undertake a proper exploration of the philosophical credentials of this proposal. Still, the reader may find it intuitively plausible that mastery of negation, at least at the level of atomic statements, is preceded in the order of understanding by mastery of which of them exclude which 24  The reader should note that there is a question, drawn to my attention by Timothy Williamson, whether we may stably combine this proposed knowledge-theoretic clause for negation with the knowledge-theoretic clause for the conditional flagged earlier: ‘A → B’ is knowable just if it is knowable that if ‘A’ is knowable, ‘B’ is knowable. For suppose VE is accepted and A is such that (i)  It is knowable that A is borderline. (ii)  Then it is knowable that A is not knowable (by VE). (iii)  So it is knowable that if A is knowable, then B is knowable (by substitution in ¬A ⇒ (A → B) (ex falso quodlibet) and closure of knowledge across knowable entailment). (iv)  So it is knowable that, if A, then [take some arbitrary contradiction for B] (from iii, by the knowledge-theoretic clause for the conditional). (v)   So it is knowable that ¬A (by the proposed clause for negation, letting B be: if A, then [contra­ dic­tion], and presuming that to be knowably incompatible with A). So the proposed clause degrades after all into the BHK-style knowledge-theoretic clause for negation: ‘¬A’ is knowable just if it is knowable that ‘A’ is not knowable, which is what we were trying to improve on. True, the argument as presented depends on VE, which we have not endorsed. But it will run for any ‘A’ that is knowably unknowable. Maybe there are no such statements formulable in a minimally sufficient soritical language. Maybe one should look askance at ex falso quodlibet. Still a concern is raised that will need disinfection in a fully satisfactory general treatment. I will not pursue the matter here.

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414  The Riddle of Vagueness others: being not yellow, for example, is, among coloured things, initially understood as the having of some colour that rules out being yellow. The above proposal reflects the thought that we may take incompatibility among atomic statements as epistemically primitive. Matters change, of course, once molecular statements enter the mix. For molecular statements, incompatibility will, conversely, sometimes be recognizable only by recognizing that one or more of them entail the negation of something entailed by the other. That is, A pair of (sets of) propositions are knowably mutually incompatible if there is some proposition ‘A’ such that the one knowably entails ‘A’ and the other knowably entails ‘¬A’. If we now, for convenience, avail ourselves of a dedicated constant, ‘⊥’, to express the situation when a set of propositions, X, incorporates both of some pair of knowably incompatible propositions, thus:    X Þ ^, then the first of the displayed clauses above mandates the following negation introduction rule

  

( ØIntro )

X È { A} Þ ^ X Þ ØA

while the second displayed clause mandates the following negation elim­in­ ation rule:

  

( ØElim )

X Þ ØA Y Þ A X ÈY Þ ^

That is, intuitively, if a set of propositions entails the negation of some prop­ os­ition, then adding to it any set of propositions that entail that proposition will result in incompatibility. Whatever deep justification these proposals may be open to, it will be enough for present purposes if they seem plausibly knowability-preservative in the light of the reader’s intuitive understanding of negation. Their most immediately significant consequence is that they allow us to justify intuitionist

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Intuitionism and the Sorites Paradox  415 reductio as a derived rule.25 Given the Thesis flagged above, we thus have all the rules (&I, ∃I, reductio) needed to run both the No Sharp Boundaries paradox and—assuming no question is raised about modus ponens—the Tolerance paradox as well. The upshot, in the presence of assumed knowledge of the polar assumptions, is the following important corollary: Corollary: When the quantifiers and connectives are understood as above, there is no option but to regard the negations of the major premises of the No Sharp Boundaries Sorites as known.

VIII Addressing Constraint 1 (Cont.): The Payoff Constraint 1 requires that we explain how and why the reductio of the major premises accomplished by the paradoxical reasoning fails to justify the unpalatable existential. This is now straightforward. By the clause for ‘∃’, the knowability of the unpalatable existential requires that for some object in the soritical domain and term, ‘a’, known to denote that object ‘Fa & ¬Fa' ’ is knowable—requires, in short, the knowability of a witness to a sharp cut-off. We neither have nor have any reason to think we can obtain that knowledge: no ‘a’ denoting any clear case furnishes such a knowable witness. And we have absolutely no reason, either given by the Sorites reasoning itself or other­wise, to think that such a witness may be knowledgeably identified in the borderline area. Since, by the Corollary emphasized at the conclusion of the preceding section, we do know the negations of each of NSB and TP, the respective 25  At least they do so if we may assume the Cut rule. Intuitionist reductio may be represented as the pattern:

X È A Þ B Y Þ ØA X È Y Þ ØA

Suppose we have an instance of right-hand premise. From that and B ⇒ B we may obtain by ¬Elim: Y ∪ {B} ⇒ ⊥

From that and the left-hand premise we have, by Cut: X ∪ Y ∪ {A} ⇒ ⊥ So by ¬Intro, we have

X ∪ Y ⇒ ¬A. 

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416  The Riddle of Vagueness classical inferences from the negations of NSB and TP to the unpalatable ex­ist­en­tial fail to guarantee knowability and are thus are invalid in the present knowledge-theoretic setting. So there is the needed daylight. Constraint 1 is met and the discomfort involved in regarding the soritical reasoning simply as a reductio of its major premise is thus relieved.

IX Addressing Constraint 2 At least, it is relieved if, as required by Constraint 2, we can neutralize the persistent temptation to regard the major premises for the paradoxes as true. The epistemicist—and indeed almost all theorists of this topic26—also share this obligation, of course, so here we, most of us, can march in step. There are a number of sources for the temptation. I will touch on four. Projective Error The core attraction of NSB is, naturally, simply the other face of the un­pal­at­ abil­ity of the unpalatable existential. And that in turn springs from our in­clin­ ation to accept that NSB is simply a statement of what it is for F to be vague in the series in question. On our overarching conception of what vagueness is, this is a tragic mistake. It is, indeed, the pivotal mistake, ‘the decisive step in the conjuring trick’ that our intuitive thinking plays on us here. For F to be vague is for it to have borderline cases, but its possession of borderline cases is, according to the overarching deflationary conception of vagueness here proposed, a matter of our propensity to certain dysfunctional patterns of classification outside the polar regions.27 F’s being vague is thus a fact about us, not about the patterns that may or may not be exhibited by the Fs and the non-Fs in a Sorites series. Nothing follows from its vagueness about thresholds, or the lack of them, in a Sorites series, or indeed about the details of its extension at all.

26  The exceptions are those theorists who prefer to look askance at the underlying logic of the paradox; for instance, at the assumption of the transitivity of logical consequence in this setting (see Zardini 2019) or at intuitionist reductio (Fine). 27  This much is common ground with Epistemicism as I understand it. The difference is that we reject the further step of postulating a sharply bounded property our inability to keep track of whose extension explains the dysfunctionality.

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Intuitionism and the Sorites Paradox  417 The diagnosis of projective error chimes nicely with Constraint 3: the overarching conception of borderline case vagueness we are working under is invoked to undergird the proposed means of satisfying Constraint 2. Inflated Normativity However, there are other kinds of seductive untruth at work in conjuring the attraction that the major premises exert. One such involves an implicit inflation of the legitimate sense in which competent practice with the usual suspects is constrained by rule and is a crucial factor in the allure of tolerance premises. An example is the general thought that the rules for the use of any of the usual suspects that can be justifiably applied or denied purely on the basis of (varying degrees of casual) observation must mandate that elements in the soritical domain between which there is no relevant (casually) observable difference should be described alike. (For how otherwise could mere observation enable us to follow the rules?) In fact, none of the expressions with which we are concerned is governed by rules that mandate any such thing. But the illusion that they are—indeed must be—so governed has deep sources. As announced earlier, I must forbear to go further into these matters here.28 An Operator Shift? Both the foregoing, though ultimately misguided, are nevertheless respectable reasons for our inclination to accept the major premises, involving subtle philo­ soph­ic­al mistakes. I am not completely confident that some of us, over the last four decades of debate of these paradoxes, may not have fallen prey to a less respectable reason. (I am sure no present reader would be guilty of this.) There is a fallacious transition available in this context of a kind that we know it is easy to slip into: an operator shift fallacy. The transition concerned is that from Nothing in the meaning of F (the way we understand it) mandates a dis­ crim­in­ation between adjacent elements in the soritical series, to The meaning of F (the way we understand it) mandates that there is to be no discrimination between adjacent elements in the soritical series, so that ‘Fx & ¬Fx'’ is everywhere false. 28  Wright (1975, this volume, Chapter 1) rehearses what is still the best case known to me for thinking otherwise and points an accusatory finger at the implicit inflation of normative constraint; the inflation is further explained and debunked in Wright (2007, this volume, Chapter 11).

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418  The Riddle of Vagueness Irrelevant Truths Finally there are a number of truths in the vicinity that may tempt one to accept NSB but that, on the present deflationary conception of vagueness, adjoined with Liberalism, simply have no bearing on it. It is true, for example, that no clear cases bear witness to the unpalatable existential, that nobody could justify claiming to have identified a witness in the borderline area, and that we (normal speakers) have no conception of what it would be like even to have the impression that we had identified a witness. But these all merely reinforce the impression that UE, the unpalatable existential, is nothing we can justify. The mere possibility of a coherent Epistemicism should teach us, if nothing else, at least that such considerations do not parlay into good reasons for its denial. The temptation though dies hard. ‘Granted’, it may be said, ‘that, if we are epistemicists, considerations like the above provide no good reason for denial. But what if we are not epistemicists? What if our attitude is, as the governing conception of vagueness that you are proposing itself involves, that there are here no facts behind the scenes: that our best practice exhausts the relevant facts—that ‘nothing is hidden’? If there are no truth-makers for predications of F and not-F save aspects of our best practice, then—given that our best practice determines no sharp cut-off for F in a suitable series—must we not conclude that there is none? And then is not some version—employing some suitably ‘wide’ notion of negation—of NSB going to be forced on us?’ The question is in effect: how can we avoid treating a fact about us and our judgemental limitations as a fact about the properties of the elements of the series unless we are prepared, with the epistemicist, to invoke some form of transcendent fact? I reply that, while our judgemental reactions no doubt are caused by and reflect properties of the elements of the series, it is a further, unwarranted step to draw conclusions from them about the extension of F. The transition from Our collective best practice does not converge on a sharp boundary for F in the series to There is no sharp boundary for F in the series is still a non sequitur even if we do not admit any practice-transcendent truthmakers for the statements in question. What does follow is at most a conclusion about indeterminacy—that something has not been settled by a convergence in

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Intuitionism and the Sorites Paradox  419 our collective practice. But indeterminacy, conceived as proposed by the in­tu­ ition­ist, is not an alethic status, so not something that excludes truth. Is this too much to swallow? Let the critic have another go: ‘Suppose it is agreed’, they may say, ‘that there are just two kinds of ways in which an instance of ‘Fx & ¬Fx′’ can hold true in the series. One is the epistemicist way. The other is as grounded in the linguistic practice of competent judges in good conditions. Suppose we reject epistemicism. Then surely we are forced to accept the conditional: If, for no element in the series, is there sufficient agreement among competent judges in good conditions on the ruth of the relevant instance of ‘Fx & ¬Fx′’, then no such element is true? So then, since there is no foreseeable such agreement, each such statement is untrue; and now it must be possible to run an NSB Sorites in terms of a suitably wide negation.’ What obstructs this train of thought, as the reader may anticipate, is Liberalism. If we accept Liberalism about verdicts in the borderline area, we must reject the displayed conditional anyway, even without epistemicism as a background assumption. Liberalism requires that we not insist on available convergence about an atomic statement among competent judges as a necessary condition for its truth. Why should it be any different for molecular statements in general and an instance of ‘Fx & ¬Fx′’ in particular? ‘Not so fast’, the critic may continue. ‘If we are to leave open the possibility of an instance of ‘Fx & ¬Fx′’ holding true, and if this possibility is not to be understood as the epistemicist understands it—as a matter of a cut-off in the extension of a property that is the semantic value of F but of whose nature we are not fully aware—and if, moreover, the possibility is not to be understood, either, as realized by a convergence in our best practice on each conjunct, then what is it a possibility of? What other kind of state of affairs, if it obtained, could conceivably be a truth-maker for an instance of ‘Fx & ¬Fx′’?’ The critic is assuming that there can be no truth without a truth-maker. But let us not question that and consider how an intuitionist might answer her question directly. Let ‘Fa’ be a non-polar statement. Suppose that a is the last element in the Sorites series for which a competent judge in good conditions—Freddy again—returns a steady positive, if suitably nuanced, verdict. Liberalism requires that we not discount Freddy’s verdict about a. But suppose also that a′ is the first element in the series for which Teddy, Freddy’s epistemic peer, returns a steady negative, if suitably nuanced, verdict.

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420  The Riddle of Vagueness Liberalism requires that we not discount Teddy’s verdict either. Should we not then be liberal towards their conjunction? The displayed conditional, however, would force us to dismiss the conjunction of Freddy’s and Teddy’s respective verdicts as untrue, for that conjunction elicits nobody’s assent, however competent, however good the conditions. The critic may be unpersuaded. ‘One can perfectly reasonably be liberal about a pair of verdicts individually but illiberal about their conjunction? Change the example. What if Teddy and Freddy were steadily to disagree about whether Fa? Now there is no option of regarding both as right—yet that does not preclude our taking a liberal view of each verdict on its own. So why should liberalism about Freddy’s and Teddy’s respective verdicts either side of the putative cut-off provide any leverage towards liberalism about them taken together?’ I reply that such leverage is the default: that liberalism about any pair of judgements individually should extend to their conjunction except in cases where there is antecedent reason to recognize tension—for example, flat contra­dic­tion!—between the judgements concerned. Unless, therefore, one is independently inclined to see the truth of ‘Fa’ as in tension with the truth of ‘¬Fa′’, there is no reason to look askance at the conjunction of Freddy’s and Teddy’s respective steady verdicts. If, however, you consider that you do have good reason to be independently so inclined, you will presumably be independently inclined to accept NSB. The resurgent paradox will then be a deserved nemesis.

X Addressing Constraint 3 The third constraint we imposed on an intuitionistic treatment of the Sorites was the requirement that the first two constraints—explaining how and why there can be a deductive gap between the negation of the major premise and the unpalatable existential, and explaining the spurious plausibility of the different forms of major premises—be met in a way that is informed by an overarching conception of what vagueness consists in. Have we done this? It is arguable that the second constraint is not really motivated in the case of the Tolerance paradox. To be sure, the vagueness of a predicate, deflationarily conceived, is nothing that should suggest that it be tolerant. However, the principal motivations to regard, for example, the usual suspects as tolerant

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Intuitionism and the Sorites Paradox  421 have, as remarked, little to do with their vagueness per se and need a separate treatment, not embarked on here. On the other hand, the diagnosis of pro­ject­ ive error as responsible for the thought that an NSB premise just states what it is for F to be vague in a relevant soritical series draws heavily and specifically on the deflationary conception of vagueness that I have represented as the heartbeat of an intuitionistic approach. But what about the first constraint—explaining the deductive gap? Assume that the knowledge-theoretic semantics offered performs as advertised. We have to acknowledge that a semantic theory of this kind might be proposed, for certain purposes, for almost any factual discourse. So the question becomes: what, if anything, is it about vagueness as deflationarily conceived that makes such a semantics appropriate for vague discourse specifically? Recall that it is essential to our deflationism not merely to regard certain judgemental patterns among competent judges as constitutive of an expression’s vagueness but to reject the demand for explanation of these patterns in terms of underlying semantic phenomena—for instance, sharply bounded but imperfectly understood semantic values, incomplete (or conflicting) semantic rules, or the worldly side of things being such as to confer truth statuses other than truth and falsity. That precludes any semantic theory that works with a bivalent notion of truth, truth-value gaps, or postulates any kind of third truth status. Admittedly, the possibility is left open of working with a verificationist truth-conditional theory, as would be mandated by EC. However, no reason is evident why the knowledge-theoretic style of semantics proposed could not amount to one way of implementing the semantic import of EC, nor hence why all the crucial parts of the treatment of the paradoxes proposed could not survive were we to quash any reservations about EC. So we have not closed that particular road by going about things the way I have here. But nor have we committed to travelling it.29

29  Versions of this material were presented at the Staff Research Seminar at Stirling, the Philosophy of Maths seminar at Oxford, and an Arché seminar at St. Andrews in the autumn of 2016, and at colloquia at Brown University and the University of Connecticut in the spring of 2017. I am grateful to all who participated in these meetings for useful feedback, and to Ian Rumfitt and Josh Schechter for helpful additional discussion. Special thanks to Elia Zardini and Sergi Oms, the editors of the volume in which this was originally published, for very searching critical comments that have led to many improvements.

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APPENDIX TO CHAPTER 14

The ‘Forced March’ Paradox ‘The’ Sorites paradox is a misnoma. Sorites paradoxes come in various stripes. All purport to elicit some form of aporia from the assumptions—surely incontestable—concerning any predicate F among the ‘usual suspects’ that it may, without qualification, truly be applied at one pole of a suitable soritical series for it and that its application at the opposite pole is, without qualification, false. So in every case there are these polar assumptions. But what else the paradox merchant adds to the brew in order to foment aporia varies. In the kind of case on which we have been focused, they add a plausible-­seeming extra premise—the major premise—and deduce a contradiction by seemingly unexceptionable deductive moves from its combination with the polar assumptions. But what of the quite different routine involved in the so-­called Forced March Sorites? Here there is no deductive path to a formal contradiction. Rather, we are invited to envisage a hapless subject who is, as it were, marched through the successive elements in a suitable soritical series and required to return a verdict in point of F-ness about each element. If the subject is competent, they must return the appropriate verdicts concerning the clear cases at the poles. Hence—the paradox merchant continues—since the verdicts required at the poles are different, they must at some point give some kind of discriminatory responses to an adjacent pair of elements between which—for so the series is constructed—the subject can discern no relevant difference. Since it is a form of incompetence to purport to discriminate cases between which one discerns no relevant difference, it appears to follow that, when a sufficient range of cases has to be considered, consistently competent use of any of the usual suspects is metaphysically impossible. That is an uncomfortable conclusion rather than a contradictory one. We might just accept it. Indeed, it may seem that that the epistemicist, for one, is committed to some such conclusion in any case, since it goes with the view, at least as developed by Williamson, that no set of verdicts about F’s application throughout a Sorites series can be both comprehensive and everywhere knowledgeable. Still, that is not quite the same point. What the Forced March Sorites seems to foist on us is the conclusion that no such comprehensive set of verdicts can both respect the clear cases of F and not-F and be everywhere

The Riddle of Vagueness. Crispin Wright, Oxford University Press (2021). © Crispin Wright. DOI: 10.1093/oso/9780199277339.003.0016

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424  The Riddle of Vagueness principled—that is, reflective only of distinctions that the subject can actually draw. Whether that conclusion is something we could live with or not, there is a natural response to block it. It is true, of course, that, as the march progresses, the hapless subject must respond differently to some adjacent pairs of elem­ ents if they are not to misdescribe polar cases. At some stage, they must stop returning the verdict ‘F’; and at some stage, they must start returning the verdict ‘not-F. But whenever they do something different, this does not have to amount to the returning of a, perhaps subtly,1 differing verdict. They may, for instance, simply fail to come to a view. And if they do, they are not forced to, as it were, project this change in their response onto the items concerned and pretend to have discerned a relevant difference in them. The reasoning of the Forced March paradox misses the distinction between differential responses that purportedly mark a relevant distinction between adjacent pairs of elem­ ents in the series and differential responses that are merely different. It is only the occurrence of the latter that is imposed by the requirement of differential verdicts at the poles. If what is troublesome about the Forced March routine is as suggested, then this point, it seems to me, defuses any attendant paradox. And the point carries no commitment to any particular theory of vagueness but may be made by all hands. Still, there is a respect in which the deflationary conception of vagueness with which we have been working may here claim an add­ ition­ al advantage. Both epistemicism and semantic indeterminism are committed to countenancing further facts of the matter to which the subject’s differential responses, when they occur, may or may not correspond. Suppose a is the last case to which they return a clear verdict, ‘F’. Confronted with its successor, aʹ, they do something different. For the epistemicist, their latter response may actually serendipitously (albeit unknowledgeably and no doubt improbably) have aligned with the actual cut-­off of F’s extension in the series. Likewise for the indeterminist, the subject’s response to aʹ may mark the transition to a point at which the semantic rules for ‘F’ have ceased to mandate any determinate verdict. Even without any problematical suggestion that full competence should involve sensitivity to such differences, both views must still admit that there is a question of alignment between the subject’s response and a further fact about how matters really stand with a and aʹ in point of

1  ‘Perhaps subtly’ differing because the most challenging development of the paradox will impose no limitations on the repertoire of qualifiers—including intonation, facial expression, linguistic hedges (‘sort of F’, ‘F-ish’, etc.)—with which the subject may be permitted to soften or qualify a verdict.

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Appendix to Chapter 14  425 F-ness; hence they must allow that there is a question of the extent of the fe­li­ city in general of the Forced March subject’s responses, understood in terms of the extent to which, outside the polar regions, they actually align with whatever F-relevant changes are taking place as they progress along the series. I propose that it is desirable for a would-­be theorist of vagueness to avoid any such admission. The intuitionistic approach, with its integral ­repudiation of any idea of vagueness as constituted in semantic facts that somehow underlie and explain the distinctive patterns of use of vague ­expressions, and its consequent commitment to liberalism about verdicts in the borderline region, does exactly that.

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432 References Williamson, Timothy (1994). Vagueness. London: Routledge. Williamson, Timothy (1996a). ‘Wright on the Epistemic Conception of Vagueness’, Analysis 56 (1): 39–45. Williamson, Timothy (1996b). ‘Putnam on the Sorites Paradox’, Philosophical Papers 25 (1): 47–56. Williamson, Timothy (1996c). ‘Cognitive Homelessness’, Journal of Philosophy 93: 554–73. Williamson, Timothy (1997a). ‘Imagination, Stipulation and Vagueness’, Philosophical Issues 8: 215–28. Williamson, Timothy (1997b). ‘Reply to Commentators: [Horgan, Gomez-Torrente, Tye]’, Philosophical Issues 8: 255–65. Williamson, Timothy (1999). ‘On the Structure of Higher-Order Vagueness’, Mind 108 (429): 127–43. Williamson, Timothy (2000a). Knowledge and Its Limits. Oxford: Oxford University Press. Williamson, Timothy (2000b). ‘Margins for Error: A Reply’, The Philosophical Quarterly 50 (198): 76–81. Wittgenstein, Ludwig (1968). Philosophical Investigations. Translated by G. E. M. Anscombe, third edition. Englewood Cliffs, N.J: Pearson. Wittgenstein, Ludwig (1978). Remarks on the Foundations of Mathematics. Oxford: Blackwell. Wright, Crispin (1975). ‘On the Coherence of Vague Predicates’, Synthese 30 (3/4): 325–65. Wright, Crispin (1976). Language Mastery and the Sorites Paradox. In: Gareth Evans and John McDowell (eds.) Truth and Meaning: Essays in Semantic, pp. 223–47. Oxford: Oxford University Press (Clarendon Press). Wright, Crispin (1982). ‘Strict Finitism’, Synthese 51 (2): 203–82. Wright, Crispin (1983). Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University Press. Wright, Crispin (1986a). Does Philosophical Investigations  I.  258-60 Suggest a Cogent Argument Against Private Language. In: John McDowell and Philip Pettit (ed.) Subject, Thought, and Context. Oxford: Oxford University Press (Clarendon Press). Wright, Crispin (1986b). Rule-Following, Meaning and Constructivism. In: Charles Travis (ed.) Meaning and Interpretation, pp. 271–97. Oxford: Blackwell. Wright, Crispin (1986c). Theories of Meaning and Speakers’ Knowledge. In: Realism, Meaning and Truth, 204–38. Oxford: Blackwell. Wright, Crispin (1986d). Theories of Meaning and Speakers Knowledge. In: Stuart Shanker and Croom Helm (eds.) Philosophy in Britain Today, pp. 267–307. Beckenham: Croom Helm. Wright, Crispin (1987a). ‘Further Reflections on the Sorites Paradox’, Philosophical Topics 15 (1): 227–90. Wright, Crispin (1987b). ‘On Making Up One’s Mind: Wittgenstein on Intention’, Proceedings of the 11th International Wittgenstein Symposium. Wright, Crispin (1988). ‘Realism, Antirealism, Irrealism, Quasi-Realism. Gareth Evans Memorial Lecture, Delivered in Oxford on June 2, 1987’, Midwest Studies in Philosophy 12 (1): 25–49. Wright, Crispin (1989). Wittgenstein’s Rule-Following Considerations and the Central Project of Theoretical Linguistics. In: Alexander L. George (ed.) Reflections on Chomsky, pp. 233–64. Oxford: Basil Blackwell. Wright, Crispin (1992a). ‘Is Higher Order Vagueness Coherent?’ Analysis 52 (3): 129–39. Wright, Crispin (1992b). Truth and Objectivity. Cambridge MA: Harvard University Press. Wright, Crispin (1993). Realism, Meaning and Truth, second edition. Cambridge: Wiley-Blackwell.

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References  433 Wright, Crispin (1995). ‘The Epistemic Conception of Vagueness’, The Southern Journal of Philosophy: Spindel Supplement 33 (S1): 133–60. Wright, Crispin (1996). ‘Human Nature?’ European Journal of Philosophy 4 (2): 235–54. Wright, Crispin (1998). ‘Truth: A Traditional Debate Reviewed’, Canadian Journal of Philosophy 28 (sup1): 31–74. Wright, Crispin (2001a). Minimalism, Deflationism, Pragmatism, Pluralism.. In: Michael  P.  Lynch (ed.) The Nature of Truth: Classic and Contemporary Perspectives. Cambridge MA: MIT Press. Wright, Crispin (2001b). ‘On Being in a Quandary: Relativism Vagueness Logical Revisionism’, Mind 110 (437): 45–98. Wright, Crispin (2001c). Rails to Infinity. Cambridge MA: Harvard University Press. Wright, Crispin (2003a). ‘Rosenkranz on Quandary, Vagueness and Intuitionism’, Mind 112 (447): 465–74. Wright, Crispin (2003b). Saving the Differences: Essays on Themes from Truth and Objectivity. Cambridge MA: Harvard University Press. Wright, Crispin (2003c). Vagueness: A Fifth Column Approach. In: JC Beall (ed.) Liars and Heaps: New Essays on Paradox, pp. 84–105. Oxford: Oxford University Press. Wright, Crispin (2004). ‘Intuition, Entitlement and the Epistemology of Logical Laws’, Dialectica 58 (1): 155–75. Wright, Crispin (2006). ‘Vagueness-Related Partial Belief and the Constitution of Borderline Cases’, Philosophy and Phenomenological Research 73 (1): 225–32. Wright, Crispin (2007). Wang’s Paradox. In: Randall E. Auxier and Lewis E. Hahn (eds.) The Philosophy of Michael Dummett, pp. 415–45. Chicago and La Salle: Open Court Publishing Company. Wright, Crispin (2008). ‘Internal-External: Doxastic Norms and the Defusing of Sceptical Paradox’, Journal of Philosophy, 105: 501–17 (special number on Epistemic Norms, edited by J. Collins and C. Peacocke). Wright, Crispin (2010). The Illusion of Higher Order Vagueness. In: Richard Dietz and Sebastiano Moruzzi (eds.) Cuts and Clouds, pp. 523–49. Oxford: Oxford University Press. Wright, Crispin (2016). On the Characterization of Borderline Cases. In: G. Ostertag (ed.) Meanings and Other Things, pp. 190–210. Oxford: Oxford University Press. Wright, Crispin (2020). Intuitionism and the Sorites. Replies to Rumfitt and Schiffer. In: Alexander Miller (ed.) Logic, Language and Mathematics. Themes from the Philosophy of Crispin Wright, pp. 354–83. Oxford: Oxford University Press. Zardini, Elia (2008). ‘A Model of Tolerance’, Studia Logica: An International Journal for Symbolic Logic 90 (3): 337–68. Zardini, Elia (2019). Non-Transitivism and the Sorites Paradox. In: Sergi Oms and Elia Zardini (eds.) The Sorites Paradox, pp. 168–86. Cambridge: Cambridge University Press.

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General Index ‘n.’ after a page number indicates the footnote number. A absoluteness  37, 218 n.10, 353–4 agnosticism  21 n.39, 34, 231–4, 373, 374, 378, 408, 413 Bivalence  268, 269, 287–91 borderline cases  7, 32, 245 n.42, 279, 380 anti-luminosity/luminosity arguments  14 n.28, 24, 404, 406 anti-realism  6, 209, 229, 230–4, 240, 242 n.39 B Basic Formula  336, 337, 338, 344–50, 355–7 Basic Revisionary Argument  229–33, 236, 238, 240, 402–8 see also logical revisionism behaviouristic approach  2, 23 n.42, 43, 77, 101, 108, 112, 157, 161 beliefs see partial belief BHK interpretation of intuitionistic logic (Brouwer–Heyting–Kolmogorov)  34, 411–13 biconditional formulation  18, 127–8, 146, 158–61, 166, 185, 187, 193, 368 partial belief  297, 299, 301 Bivalence  4, 7, 26–7, 38, 195, 251–3, 273, 275, 282–3, 287–91, 393, 394, 397, 401, 402 agnosticism  268, 269, 287–91 distinctions of degree and  63–4, 93–4 epistemicism  195–6, 221 n.12, 257–8, 393 intuitionism and  4 n.9, 332 rejection of  27, 332, 394, 397, 401, 402 vague predicates  27, 38, 269, 333 Williamson on  27, 38, 183–7, 338 n.4 borderline cases  4, 5–6, 7, 29, 36, 38–40, 45, 82, 127, 184–5, 187, 279–81, 289, 343, 368, 373–5, 406 1st-order borderline cases  146, 169–70, 245 n.42, 340, 343, 345, 354

2nd-order borderline cases  310–11, 340, 348, 366 3rd-order borderline cases  340 agnosticism  7, 32, 245 n.42, 279, 380 characterization problem  xiii, 272, 280, 291, 367 concept of  29, 38–40, 272, 277–81, 367–8, 374 definite borderline case  146, 148–9, 154, 170, 171, 177, 187, 195, 251, 280, 348, 351 n.16 epistemicism  5–6, 28, 38–9, 181–2, 195–6, 221 n.12, 251–3, 273, 281–2, 367, 369 higher-order vagueness  145, 146–8, 168–70 indeterminacy  187–8, 192, 234, 273 indeterminist approach  184–5, 187, 195–6, 197, 252–3, 257, 367–8 partial belief  xiii, 294, 297, 299–300, 371 permissibility  280–1, 397 n.6 polar verdicts  267, 279–81, 369, 374, 378, 380, 397 as quandary cases  29–30, 33, 238–9, 241, 246, 288, 384–6 rules and rationality on the borderline 15–24 semantics  25, 338, 339, 367–8, 395, 396–7, 424 Third Possibility  279–81, 345 n.g11, 365, 368–9, 374–5, 382 truth-value gap and  184, 187–8, 397 vague predicates  6–7, 82, 181, 280, 328 borderline cases: authors on Frege, Gottlob  45, 82, 310, 338 n.5 Rosenkranz, Sven  29, 266–7 Sainsbury, Mark  177–9, 366 Schiffer, Stephen  xiii, 367 Williamson, Timothy  11, 29, 192, 310 Wright  11, 28–30, 36, 38–9

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436  General Index borderline region  11, 12, 20, 24, 35, 266, 316, 330, 363, 382, 389, 397, 425 boundaryless predicate/concept (M. Sainsbury)  177–9, 366 Buffering view  340–4, 350 n.14, 351–5, 357, 359, 360, 363, 365 C C-paradox  119–21, 123, 138–41, 143 see also colour characterization problem  xiii, 174–5, 272, 280, 291, 367–8, 383–8 Chisholm’s paradox  xi n.7 classical logic  xi, xii, 63, 181, 182, 199, 269, 345, 347, 348, 350, 351, 367, 369, 391, 393, 401, 402, 412 suspension of  230, 241–2, 248, 258, 268 unpalatable existential  227, 389, 400 see also logical revisionism classical probability  295, 296, 370 cognition  5–6, 151, 152, 159, 194–6, 254, 393, 395 cognitive command  217, 219–22, 220, 223 n.16, 226, 249–50, 254, 258 cognitive shortcoming  220–4, 249, 250, 254, 258 quandary  384, 386–7 see also epistemicism colour  18, 21–2, 31–2, 49, 51–2, 86–7, 134–5 ascriptions of colour  17, 19–21, 75, 99, 236 colour chart  51, 67–8, 72–3, 86, 96–7, 98 colour predicates  49, 51, 53–4, 84, 86, 88, 90 colour vocabulary  51, 75, 86, 99, 115–16, 123 Goodman Chart/Shade  68–72 indiscriminability  10, 12, 13, 54, 58–62, 66, 88, 91–2, 95 observationality of colour predicates  53–4, 58–9, 62, 88–91 see also C-paradox colour spectrum  48–53, 54, 67–73, 84, 86–7, 97, 98, 178 colour shade as non-observational notion  54–62, 75–7, 88–92, 96–7, 98, 99–101 contextualism  2 n.4, 16 n.30, 20 n.35

D definiteness  35–6, 37, 150, 191–7, 251, 269, 341–4, 351, 352, 353 cognitively misbegotten  194–6, 254 definite borderline case  146, 148–9, 154, 170, 171, 177, 187, 195, 251, 280, 349, 351 n.16 definiteness operator  140 n.19, 144–6, 149–50, 153–4, 168–9, 176, 192–8, 251, 253, 254–6, 336 n.2, 341–2, 344–5, 349, 352–3, 359 higher-order vagueness  147–88, 192 n.10, 196–7 operator of absoluteness in terms of definiteness  37, 353 Williamson on  192–3 degree of human maturity  48, 50, 52–3, 84, 87, 151–2, 155 n.26 degree theoretic approach  2 n.4, 26, 36 criticism of  3, 132–43, 162 degree of change  13, 19, 49, 84–5, 189 degrees of truth  63, 93, 129 n.8, 131, 132, 141, 333 distinctions of degree  63–5, 92–4 modus ponens  136, 138 Peacocke, Christopher  3, 131–43 disjunction elimination  230–1 n.25 Disjunctive Syllogism  105 n.1 Disquotational Scheme  183–6, 234 n.32, 235 ‘dommal’  184 n.4, 308–11, 361 double negation  x, 167 n.2, 174, 231, 283 DNE (double-negation elimination)  xi, 4, 106, 139–40, 167 n.2, 174, 228, 249, 258, 262, 327, 332–3, 400 see also negation Dummett’s Principle  336 n.2, 350 n.13, 356–7 E EC (Evidential Constraint)  xii, 24, 32–3, 34, 230, 232, 236–43, 246, 261–6, 268, 282, 289–90 Basic Revisionary Argument  229–30, 232, 236, 238, 240, 402–3, 406–8, 421 EC-conditionals  381, 403, 404, 406–8 EC-Deduction  223–6, 249–50, 254–5 logical revisionism  236–43, 245 n.42, 246, 247 n.45 Verdict Exclusion  375–6, 406–8

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General Index  437 Epistemic Conception of vagueness  212–14, 220–1, 251, 254, 257–8 Sorensen, Roy  209–10, 212 n.5, 393 n.1 Williamson, Timothy  209–10, 212 n.5, 393 n.1 epistemicism  viii, 5–6, 12, 26, 195–6, 204–7, 220, 248, 269, 273, 284–5, 312–14, 393–5, 402 Bivalence  195–6, 221 n.12, 393 borderline cases  5–6, 28, 38–9, 181–2, 195–6, 221 n.12, 251–3, 273, 281–2, 367, 369 criticism of  5, 6, 12, 27, 182–3, 192, 202–3, 204, 207, 213 n.6, 268–9, 274, 277, 290–1, 296–7, 316–17, 369, 394, 396 Forced March Sorites paradox  423, 424 indeterminacy  181–2, 187, 204–6, 209, 394 margin of error  11–12, 200–3, 394 n.3 reference/referential relations  204, 207, 213 n.6, 269, 277, 290–1 sharp boundaries  5, 182–3, 188–91, 197–8, 199, 207, 213 n.6, 253 n.48, 268–9, 290, 312–14, 393–4 Sorensen on  182, 183, 188–91, 197–8 Sorites paradox solution  274, 394 vagueness as matter of ignorance  182, 199, 274, 277 Williamson on  viii, 5, 6 n.14, 182, 183–7, 199–203, 252, 269, 273, 279, 290–1, 423 epistemology  22, 23, 77, 117, 159 Equivalence Scheme  184, 186–7 Excluded Middle (LEM—Law of Excluded Middle)  4, 6, 41, 79 n.1, 139 n.17, 174, 283, 389–90, 394, 403 Basic Revisionary Argument  229–31, 232, 236, 240, 403 logical revisionism and  237, 239–41, 242 n.39, 249, 258 F finitism  vii, 162, 304–5, 307 see also philosophy of mathematics Fitch’s paradox  33 n.64, 242 n.39, 247 n.43, 264–5 Forced March Sorites paradox  viii, 40, 317, 352 n.19, 356, 367, 398 n.8, 423–5

G Gap principles (D.G. Fara)  348–51, 353–5, 359, 360 n.26 Goldbach’s Conjecture  30, 32, 256, 287–8, 384–5 governing view  viii, 2, 7–8, 10, 15–17, 43, 77, 157, 166 1st thesis  43, 44–5, 46, 76, 81, 82, 83, 100, 107–8, 113 1st thesis, rejection of  114, 117–23, 124 2nd thesis  8, 43, 44–5, 46, 53, 54, 62, 81–2, 83, 86, 87–8, 107, 113, 115 ascriptions of colour and  19–21 Dummett on  3, 107, 114–16, 304–305 Frege–Russell view of vagueness and  46 implicit knowledge of rules  113, 117, 161 inconsistent rules  115, 116, 305, 306, 314, 316 language mastery  107–110, 314 language use  2–3, 7, 43–4, 81 observational predicates  90–1, 124, 127, 326 Peacocke on  3 semantic incoherence  45, 46, 53, 65–74, 75, 77, 81–2, 83, 95–8, 99–101, 113, 315–16 Sorites paradox and  3, 107, 108–114 Sorites paradox major premises and  124, 151–62 use of language from within  7–8, 10, 23 n.42, 43, 81 Wittgenstein on  44 Wright on  2, 9–10, 15, 19–20, 44 H heap/heap paradox  10–11, 49–51, 53, 56, 83–4, 85, 86, 97, 98, 276 Megarian paradox  47–8 higher-order vagueness  xiii, 179, 276 borderline cases  145, 146–8, 168–70 characterization problem  174–5 definitization/definiteness operator  147–8, 192 n.10, 196–7 Dummett on  335 Edgington on  34 Fara on  35, 338 n.6, 351 n.17 Fine on  35 Frege on  310–12

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438  General Index higher-order vagueness  (cont.) Heck on  34 as illusion  40, 170, 335, 344, 358, 365–6 No Sharp Boundaries paradox and  143–50, 163, 168–76 Sainsbury on  34, 170–8 Wright on  5, 34–40 I immediacy  20, 21, 311 incoherentism  190–1, 314–16, 318, 325 indeterminacy  44, 150, 173, 181–2, 188, 204–6, 233–4, 272–3, 275–7, 339, 389–90, 394 borderline cases  187–8, 192, 234, 273 epistemic indeterminacy  209, 251–6 epistemicism  181–2, 187, 204–6, 209, 394 logical revisionism  233–5, 243 partial belief  298, 299, 300, 389 Quandary view of indeterminacy  235–6 semantic indeterminacy  182, 205, 275–6, 395, 396–7, 424 Sorensen on  209 Williamson on  209 indeterminist approach  181, 183, 196, 285 borderline cases  184–5, 187, 195–6, 197, 252–3, 257, 367–8 criticism of  183, 187, 190–1, 197, 252–3, 396 definiteness 195–6 vagueness  197, 272–3, 277, 281–2, 367–8 Williamson on  252–3 indiscriminability  4 n.8, 65–6 change in time  60–1 colour  10, 12, 13, 54, 58–62, 66, 88, 91–2, 95 non-transitivity of indiscriminability  54, 58–62, 66, 88, 91–2, 95, 107, 114, 121, 122, 128, 305 infinity  48 n.4, 164 n.34 instantiation  103, 304, 307, 320, 388, 398, 412 intuitionism  viii–x, 6, 29, 228, 242, 327–8, 394, 397, 401, 408, 412 BHK interpretation of intuitionistic logic  34, 411–13 Bivalence  4 n.9, 332, 394, 397, 402 Dummett on  333 Forced March Sorites paradox  425

logical revisionism  xii, 209, 241, 258 philosophy of mathematics/ Mathematical Intuitionists  ix, 26, 33, 241, 254, 268, 283, 287, 304, 332, 333, 394–5, 397, 402 Putnam on  4, 104, 143–4, 144 n.21, 167, 227, 283 n.6, 400 solution to the Sorites  242–8, 258, 259, 401–21 Sorites paradoxes  104–6, 143–4, 227 Third Possibility  234, 382 vagueness  24–34, 40, 105–6, 259, 261, 272, 282, 395–6, 400, 402, 407, 409, 416, 418, 420–1, 424, 425 Wright on  viii–x, 24–34, 40 K knowledge 411–12 feasible knowledge  223, 237, 239, 255, 264, 287–8, 404, 408 implicit knowledge  3 n.7, 109, 110, 112, 113–14, 117–19, 161 knowledgeable/not knowledgeable polar verdicts 379–81 margin of error  11–12, 200–3, 264, 265, 376 n.12, 381, 394 n.3, 404 L language  vii, 2, 17–18, 42, 43, 77, 80, 157 comparison with a game  43, 76–7, 100–101, 108–12, 115, 319–20, 323 language and the world  65, 95, 269, 271 vagueness of ordinary language  41, 79, 303, 308 see also natural language language mastery  vii, 42, 79, 81, 111, 271, 314 comparison with a game  108–12 implicit knowledge  110–12, 113–14 philosophy of language  109, 111, 271 self-conscious masters  7, 43, 81 Wittgenstein on  321 language use  15, 42, 45, 80 governing view  2–3, 7, 43–4, 81 use of language from within  7–8, 10, 23 n.42, 43, 81 Law of Excluded Middle see Excluded Middle

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General Index  439 liberalism  373–4, 377–83, 385, 386, 396, 397, 407, 419–20, 425 logical revisionism  xii, 209, 229–48, 258–9 anti-realism  209, 229, 230–3, 240, 242 n.39 Basic Revisionary Argument  229–33, 236, 238, 240 Dummett on  209, 229 LEM (Law of Excluded Middle)  237, 239–41, 242 n.39, 249, 258 revisionism saved  233–42 suspension of classical logic  230, 241–2, 248, 258, 268 see also Basic Revisionary Argument M meaning, theory of  vii, 101, 111, 161, 165 objectivity of meaning  163–4, 165–6 philosophical theory of meaning  42, 80 Wittgenstein on  163–6 memory/memorability  51, 53, 62, 86–7, 89, 112, 154, 155 metaphysics  viii, 271, 318, 333, 394, 402 modus ponens (rule-following)  15–16, 17–18, 27, 48, 84, 304, 319–24, 326 degree theoretic approach  136, 138 rejection of  17, 328 Schiffer on  390 Sorites paradoxes  131 Tolerance paradox  412 n.22 Wittgenstein on  321–2, 326 n.11 see also governing view; semantic rules modus tollens  27, 105 n.1, 133, 222 monotonicity  332, 338, 339, 346, 349, 393 n.2, 403, 410 Moorean paradox  245 n.42, 276 N natural language  vii, 167, 269 vagueness  8, 79, 167, 271, 272, 303, 305 negation  171, 173, 174, 185, 234, 414 see also double negation negative existential  249, 329–31, 348 negative existential major premise  144, 328, 329, 331, 344–5, 345 n.11, 389, 399 No Sharp Boundaries paradox  144, 328, 331, 344–5, 399 Schiffer on  389 ‘no fact of the matter’  223–5, 251, 259, 272, 294, 338

No Sharp Boundaries paradox  3, 4, 25, 26–7, 143–50, 154–5, 163, 167–77, 344, 347, 354, 359, 415–16, 420–1 dissolution of  347 Dummett on  306–7 higher-order vagueness and  143–50, 163, 168–76 indefinite hierarchy of orders of vagueness 147–9 lack of sharp boundaries  149, 152, 328, 331–2 logical formulation  399 negative existential major premise  144, 328, 331, 344–5, 399 reductio of its major premise  399–400, 401, 415 Sainsbury on  170–6 see also sharp boundaries, lack of non-transitive matching  67–8, 69, 72, 95–6, 135 O objectivity  31–2, 58–9, 163–4, 166 observational predicates/observationality  2, 8, 16, 23, 55, 58–9, 123–31, 326 appearance  56, 59, 65, 129–30, 157–8, 161–2, 325 colour predicates  53–4, 58–9, 62, 88–92 connection between observationality and ostensive definition  57–8, 65, 90, 126–7 Dummett on  2, 8, 126, 128, 129, 155, 161, 304–5, 325–6 governing view  90–1, 124, 127, 326 indistinguishability/indiscriminability 55, 57–62, 88, 128, 131, 141, 157, 161 lack of sharp boundaries  83, 140 non-transitivity indiscriminability  54, 58–62, 66, 88, 91–2, 95, 121, 128, 304–5 ostensively definable  56, 57, 90, 156 paradigms and  72–4, 96–7, 98 Peacocke, Christopher  121, 123, 126, 127, 128–9 n.8, 129, 130 n.10 rules for the use of  143 Sorites paradox of observationality  324–7 Sorites paradox premises  126, 128, 284, 327 Tachometer paradox  125, 127, 128, 143, 163

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440  General Index observational predicates/ observationality (cont.) tolerance  55, 62, 73, 95, 98, 123–31, 161 see also degree theoretic approach; criticism; phenomenal predicates operator  174, 379, 411, 417 absoluteness operator  37, 354 definiteness operator  140 n.19, 144–6, 149–50, 153–4, 168–9, 176, 192–8, 251, 253, 254–6, 336 n.2, 341–2, 344–5, 349, 352–3, 359 opinions  195–6, 309, 315 conflicting opinions  195, 217, 245, 249, 345 n.42, 407 difference of opinion  211, 220, 223 n.16, 238, 239, 246, 250, 253 disputes of inclination  211–20, 222, 223, 226, 256–7, 259 ‘drying of the springs of opinion’  266–7, 281, 286, 288, 363, 364 Oxford University  vii, xi, 2, 107, 278 n.3, 303, 337 n.3 P paradigms  67, 72–4, 96–7, 98, 137, 214, 293, 308, 322, 325 partial belief  xiii, 294, 297–300, 371 Łukasiewicz matrices  296, 388, 389, 391 Schiffer on  xiii, 294–301, 369–72, 373, 388 SPB (s-belief/standard partial belief)  294, 297, 298–9, 370 SPB/VPB distinction  294–5, 296 V-credibility  388–9, 391 VPB (v-belief/vagueness-related partial belief)  294, 299, 300, 301, 369–72, 385–91 VPB*  297–8, 299–301, 371, 386 n.19 pearl example (John Foster)  278, 337–9, 361 perception  21, 22 n.41, 23, 125, 251 permissibility  28, 31, 187–8, 195, 280–1, 315 pessimism  32, 41 n.2, 282, 290 phenomenal predicates  21, 23, 32, 33, 58, 141–3, 163 see also observational predicates phenomenal Sorites  12–14, 23–4, 33, 40 philosophy of language  vii, 3 n.7, 9, 101, 207, 271, 372

language mastery  109, 111, 271 objectivity of meaning  164, 166 philosophy of mathematics  vii, 62 finitism  vii, 162, 304–5, 307 Mathematical Intuitionists  ix, 26, 33, 241, 254, 268, 283, 287, 304, 332, 333, 394–5, 397, 402 Platonism  165, 199 n.13, 393, 402 philosophy of science  124, 130 n.10 polar verdicts  xi, 188, 234, 279, 354, 374, 379–81 borderline cases  267, 279–81, 369, 378, 380, 397 Q quale/qualia (N. Goodman)  68, 71, 131, 134–5 quandary  237–8, 241–2, 246, 247, 363 borderline cases as cases of quandary  29–30, 33, 238–9, 241, 246, 288, 384–6 characterization problem and  383–8 conditions for  256–7, 288, 383–4 Schiffer on  369 quantified conditional  152–3 see also universally quantified conditional R realism  6, 199 n.13, 210, 212–14, 222, 251 Dummett on  224 n.18 see also anti-realism relativism  32, 209, 210–26, 249–51, 258 Relevant Logics  105 revenge problem  36, 351–5 revisionism see logical revisionism S S4M modal logic  34 semantic incoherence  44, 48, 49, 84 Dummett on  128, 314–16 governing view  45, 46, 53, 65–74, 75, 77, 81–2, 83, 95–8, 99–101, 113, 315–16 vagueness as semantic incoherence  144, 168, 314–16 semantic rules  9 n.20, 42, 44, 77, 107, 117–18, 159 correct use of language and  43, 45, 81 schematic rule  42–3, 44, 80 see also governing view

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General Index  441 semantics  x, xi–xii, 40, 45–6, 48, 74, 98, 229, 230, 268, 269, 290, 393, 409, 421 borderline cases  25, 338, 339, 367–8, 395, 396–7, 424 semantic indeterminacy  182, 205, 275–6, 395, 396–7, 424 vagueness as semantic incompleteness  307–12, 314, 316 vagueness as semantic phenomenon  viii, 6, 53, 88, 272, 367–8 Williamson on  199 sensitivity  14, 120–1, 130, 160, 161, 163, 424 insensitivity  189, 190 n.8 Sorensen on limited sensitivity  188–91, 198 sharp boundaries (sharp cut-offs)  viii–ix, xii, 45, 82, 286, 356, 358 epistemicism  5, 182–3, 188–91, 197–8, 199, 207, 213 n.6, 253 n.48, 268–9, 290, 312–14, 393–4 existence of  183–7, 188–91, 197–8 sharp threshold  154, 276, 286, 404 sharpening of a vague predicate  139 n.17, 140 n.19 Sorensen on limited sensitivity  188–91 Sorensen’s clones  197–8 unpalatable existential  330–1 why can’t we know where the sharp cut-offs lie?  199–203 Williamson on Bivalence  183–7 sharp boundaries, lack of  45–6, 48, 49–50, 52–3, 82–3, 85–6, 336, 344–50, 358–9 Dummett on  336 heap paradox  49–50, 276 No Sharp Boundaries paradox  149, 152, 328, 331–2 vagueness as absence of sharp boundaries  103, 328, 329 see also No Sharp Boundaries paradox Simple Deduction  220–2, 224–6, 249–50 sociological issues  38, 40, 328, 361, 363, 365 sociology of judgement  293, 407 Sorites paradoxes  vii, xi, 1, 26–7, 103–6, 131, 181 Dummett on  vii, 2, 107, 128 epistemicism  274, 394 governing view and  3, 107, 108–114

intractable forms of  12–13, 15, 23, 25 intuitionism  104–6, 144, 227 misconceived conditional and the Sorites 283–7 phenomenal Sorites  12–14, 23–4, 33, 40 restricted class of cases  108, 117–18 Sorites paradox of observationality  324–7 Sorites-prone predicates  152, 191, 288, 293, 309, 311, 317, 318, 378 Sorites-susceptibility  8, 139 n.17, 143, 152, 167, 327 vagueness and  103–4, 167, 273–4 variety of  viii, 162–3, 398, 423 Sorites paradox premises  4, 12, 25, 26–7, 39, 85, 108, 113, 226–7 classical logic  227, 327 denial of  248, 258, 274, 286, 333 Dummett on  304, 306 false premises  xi, 25, 123, 124, 126, 136, 149, 150 governing view and  124, 151–62 major premise of  3–4, 132–3, 150, 227, 398, 423 minor premises  354 n.22, 388, 390 negative existential major premise  328, 329, 331, 345 n.11, 389–90, 399 observational predicates  126, 128, 284, 327 Sorites reasoning as reductio of its major premise  103–4, 149, 150, 167 n.2, 228, 242, 262, 274, 327, 399–400, 416 see also universally quantified conditional Sorites paradoxes solution  xii, xiii, 25, 46, 63–5, 92–4, 167–8 n.2, 228 epistemicism  274, 395 governing view, jettison of first claim of  114, 117–23 intuitionistic solution  242–8, 258, 259, 401–21 phenomenal Sorites  23–4, 33, 40 supervaluationism  2 n.4, 34 n.69, 36, 37, 140 n.19, 182 n.1, 228 n.22, 274–5, 285, 400 borderline cases  38–9 proto-supervaluationist model  305 Williamson on  275 Wright on  26 Synthese  271 n.1, 303–4, 304 n.2

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442  General Index T Tachometer paradox  3, 12–15, 19, 21, 40, 108, 123, 125, 127, 128, 141, 143, 162–3 Peacocke, Christopher  119–21, 123 Tarski biconditionals/T-sentences  7, 24, 27 Third Possibility  26, 36, 234, 278–83, 355, 357, 382 borderline cases  279–81, 345 n.11, 365, 368–9, 374–5, 382 definition  28, 36, 278 rejection of  36, 368, 373 Schiffer on  373 third possibility indeterminacy  233–4, 237 third truth value  278, 333, 368 tolerance  10, 49, 50, 52–3, 54, 62, 63, 65–6, 73–4, 76, 84–5, 86–7, 89, 92, 95, 96–8, 100, 103, 123–31, 279 observational predicates/observationality and  55, 62, 73, 95, 98, 123–31, 161 tolerance as putatively a priori  7–12, 39 vagueness/tolerance distinction  191–2 Tolerance paradox  398–400, 401, 414–15, 420–1 negation of major premises  414–16, 420–1 universally quantified conditional  398, 412 n.21 truth  24, 229 degrees of truth  63, 93, 129 n.8, 131, 132, 141, 333 truth-apt discourses  184–5, 217, 219–20, 222, 249, 250, 258 truth status  149–50, 185–6, 193, 221 n.12, 234, 235, 421 truth value  82, 165, 176, 195, 220, 251–3, 273, 393 classical truth values  193–4 indeterminacy as failure of truth value  188, 233, 252 Third Possibility/third truth value  278, 333, 368 truth-value gap  27 n.52, 171, 184, 185, 187–8, 193–4, 250, 333, 397, 421 U universally quantified conditional  3–4, 107, 112, 132, 144, 167, 328, 344, 398, 412 n.22

unpalatable existential  242–4, 247–9, 329–32, 359–60, 390–1, 399, 400, 401, 403, 415, 416, 420 classical logic  227, 389, 400 denial of  227–8, 243, 258, 331, 332, 360 n.26, 389, 399, 418 endorsement of  329–30, 400 No Sharp Boundaries paradox  399, 400, 415 as a quandary  246, 247 sharp boundaries  330–1 V vague predicates  2, 6, 8, 25, 40, 188–9, 203, 269, 272–3, 365 Bivalence  27, 38, 269, 333 borderline cases  6–7, 82, 181, 280, 328 Dummett on  8, 191 Fine on  139 n.17 Frege on  41 Wright on  7–12, 15, 19–20 vagueness  vii, 4, 5–6, 103–6, 177, 178–9, 182, 248, 273–4, 328, 329, 367, 395 1st order vagueness  335, 340, 342–3, 362 2nd order vagueness  41, 171, 362 as boundarylessness  177–9 characterization of vagueness  176, 248, 255, 368, 373 deflationary conception of  396, 400, 402, 407, 409, 416, 418, 420–1, 424 Dummett on  144 n.22, 304–5, 307, 314–16 Frege on  82, 181, 303, 307–12, 315 in rebus conception of  272–3, 275, 276–7, 281, 282, 368 indeterminist approach  197, 272–3, 277, 281–2, 367–8 intuitionism  24–34, 40, 105–6, 259, 261, 272, 282, 395–6, 400, 402, 407, 409, 416, 418, 420–1, 424, 425 natural language  8, 79, 167, 271, 272, 303, 305 nature of  46, 83, 178, 261, 271–2, 324, 341–2, 344, 347, 368, 385, 386, 401 non-classical/special logic for  41, 53, 63, 79, 93, 333 ordinary language  41, 46, 79, 303, 308 as psychological phenomenon  xiii, 293, 296, 300–1

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General Index  443 Putnam on  104–6, 167 Sainsbury on  177–9 as semantic incoherence  144, 168, 314–16 as semantic incompleteness  307–12, 314, 316 as semantic phenomenon  viii, 6, 53, 88, 272, 367–8 tolerance/vagueness distinction  191–2 vagueness trilemma  271–7, 282, 318, 323

Verdict Exclusion  32, 34, 279, 369, 373–6, 377, 378–83, 391, 406–8, 413 borderline cases  32, 245 n.42, 279–81, 289, 373, 375, 406 liberalism  378–80, 382–3, 407 Schiffer on  373, 376 n.14, 377, 391 Third Possibility  279, 282, 382 W ‘Wang’s Paradox’ (M. Dummett)  vii, xiv, 2, 3, 24 n.44, 114–16, 303–7, 336 n.2

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OUP CORRECTED AUTOPAGE PROOFS – FINAL, 15/06/21, SPi

Index of Names ‘n.’ after a page number indicates the footnote number. A Augustine  321, 322 B Barnes, Elizabeth  xiv Beall, J. C.  ix Berkeley, George  66 Black, Max  vii n.2 Blackburn, Simon  216 Bobzien, Susanne  xiv, 34 Brouwer, Luitzen Egbertus Jan  31 n.59 BHK interpretation of intuitionistic logic  34, 411, 413 C Campbell, Richmond  182 n.2, 210 n.4 Cappelen, Herman  20 n.35 Cargile, James  182 n.2, 210 n.4, 393 n.1 Chambers, Timothy  25, 246–7, 283 n.7 Christensen, David  31 n.59 Clark, Peter  106 n.2 D Davidson, Donald  viii, 21 n.37 Dietz, Richard  xiv Diogenes Laertius  47 n.3 Dummett, Michael  ix, xiii, 144, 168, 191, 351 anti-realism  6, 229 behaviourism  23 n.42 Bivalence 401 Dummett’s Principle  336 n.2, 350 n.13, 355–7 governing view  3, 107, 114–16, 305 higher-order vagueness  335 ineradicability intuition  335–6, 337, 360–2, 365 intuitionism 333 lack of sharp boundaries  336

logical revisionism  209, 229 No Sharp Boundaries paradox  307 observational predicates  2, 8, 126, 128, 129, 155, 161, 304–5, 325–6 Reply to Wright  ix n.5 ‘slowly moving pointer’ example  121–2, 126 Sorites paradox  vii, 2, 107, 128 Sorites paradox premises  304, 307 vague predicates  8, 191 vagueness  145 n.22, 304–5, 307, 314–16 ‘Wang’s Paradox’  vii, xiv, 2, 3, 24 n.44, 114–16, 303–7, 336 n.2 ‘What Do I Know when I Know a Language?’  24 n.42 E Edgington, Dorothy  34 Eklund, Matti  vii n.3 Esenin-Volpin, Alexander  48, 84 Eubulides of Megara  vii, 47 n.3 Evans, Gareth  viii n.4, 277 F Fara, Delia Graff  20 n.35 Bivalence 38 Gap principles  349–51, 354–5, 359, 360 n.26 higher-order vagueness  35, 338 n.6, 351 n.17 phenomenal Sorites  13 n.27 Field, Hartry  xiv, 37 n.78 Fine, Kit  viii, xiv, 26 n.48, 139, 140 n.19, 416 n.26 ‘Conjunctive Syllogism’  359 n.25 higher-order vagueness  35 penumbral connection  410–11 vague predicates  139 n.17 ‘Vagueness, Truth and Logic’  304 n.2

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446  Index of Names Fine, Kit  (cont.) vagueness as semantic phenomenon  308 n.5 Fitch, Frederic B.  264–5 Fitch’s paradox  33 n.64, 242 n.39, 247 n.43 Foster, John: pearl example  278, 337–9, 361 Frege, Gottlob  2, 9, 27, 56, 66, 178 borderline-cases  45, 82, 310, 338 n.5 criticism of  309–12 Grundgesetze  308 n.4, 315 higher-order vagueness  310–12 vague predicates  41 vagueness  82, 181, 303, 307–12, 315 vagueness of ordinary language  41, 46, 79, 308 G Gibbard, Allan  216 Glanzberg, Michael  ix, 28 n.53 Goldbach, Christian: Goldbach’s Conjecture  30, 32, 256, 287–8, 384–5 Goodman, Nelson  46, 133 Goodman Chart  68–70 Goodman-Shade 68–72 quale/qualia  68, 71, 131, 134–5 Structure of Appearance, The 67–8 Greenough, Patrick  xiii–xiv, 24 n.44, 29 n.54, 224 n.19, 259 n.54 Unknowability Problem  30 n.57 H Harman, Gilbert  215 Heck, Richard  xiii, xiv, 1–40, 351 n.17 higher-order vagueness  34 ‘Non-Conceptual Content and the “Space of Reasons” ’  18 n.33 ‘Semantic Accounts of Vagueness’  5 n.12, 21 n.39 Hegel, Georg W. F.  xii Horwich, Paul  184 n.3, 210 n.4, 393 n.1 Hyde, Dominic  25 n.45 J Johnston, Mark  160 n.30 K Kindermann, Dirk  xiv

Körner, Stephan: Philosophy of Mathematics 62 Kripke, Saul A.  34 n.69, 269, 290 L Lepore, Ernest  20 n.35 Locke, John  66, 124, 405 Łukasiewicz, Jan  296, 388, 389, 391 M McDowell, John  viii n.4, 21 n.37, 23 n.42 Moline  vii n.1 Momtchiloff, Peter  xiv Morruzzi, Sebastiano  xiv O Oms, Sergi  x P Peacocke, Christopher  3, 14 ‘Are Vague Predicates Incoherent?’  108 C-paradox  119–21, 123 degree theoretic approach  3, 132–43 observational predicates  121, 123, 126, 127, 128–9 n.8, 129, 130 n.10 Pears, David  307 Priest, Graham  25 n.46 Putnam, Hilary  viii, 4, 104–6, 269, 290 criticism of  246, 283 n.6 intuitionism  4, 104, 143–4, 144 n.21, 167, 227, 283 n.6, 400 ‘Vagueness and Alternative Logic’  104 R Raffman, Diana  16 n.30 Rayo, Agustin  xiv Read, Stephen: ‘Hairier than Putnam Thought’  4, 103–106, 144 n.21 Rosenkranz, Sven  xii, xiii–xiv, 261–9, 378 borderline cases  29 liberalism 378 ‘Wright on Vagueness and Agnosticism’ 29 Rumfitt, Ian  ix, x, xiv, 34 Russell, Bertrand  45, 72, 351 ineradicability intuition  336–7, 360–4 Inquiry into Meaning and Truth 68 Tractatus  41, 79 vagueness 82

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Index of Names  447 vagueness of ordinary language  41, 46, 79 S Sainsbury, Mark  ix, 34 boundaryless predicate/concept  177–8, 179, 366 higher-order vagueness  34, 170–9 No Sharp Boundaries paradox  170–6 Simple Deduction  221 n.12 Transition Question  352 n.19 vagueness 177–8 Salerno, Joe  209 n.3 Schechter, Josh  9 n.19, 35 n.75 Schiffer, Stephen  x, xii–xiii, xiv borderline cases  xiii, 367 characterization problem  367 modus ponens 390 negative existential  389 partial belief  xiii, 294–301, 369–72, 373, 388 quandary 369 soritical vagueness  367 n.2 Things We Mean, The 293 Third Possibility  373 vagueness as psychological phenomenon  xiii, 293, 300–1 Verdict Exclusion  373, 376 n.14, 377, 391 Shapiro, Stewart  xiv, 222 n.13 Sider, Theodore  8 n.18 Slaney, John  106 n.2 Soames, Scott  16 n.30 Sorensen, Roy  182, 183 Epistemic Conception of vagueness  209–10, 212 n.5, 393 n.1 epistemicism  182, 183, 188–91, 197–8 indeterminacy 209 on limited sensitivity  188–91, 198 Sorensen’s clones  197–8 Stanley, Jason  16 n.30 Sudbury, Aidan  vii, xiii T Tappenden, Jamie  2 n.5 Thomson, Judith  215 n.7 U Unger, Peter  149 n.24, 167, 191

W Weir, Alan  25 n.46 Wiggins, David  194 Williamson, Timothy  viii, 12, 24, 312–13, 413 n.24 absoluteness 37 anti-luminosity/luminosity arguments  14 n.28, 24, 404, 406 Basic Revisionary Argument  404–6 Bivalence  27, 38, 183–7, 338 n.4 borderline cases  11, 29, 192, 310 borderline region  11, 12 definiteness 192–3 disjunction elimination  230–1 n.25 ‘dommal’  184 n.4, 308–11 Epistemic Conception of vagueness  209–10, 212 n.5, 393 n.1 epistemicism  viii, 5, 6 n.14, 182, 183–7, 199–203, 252, 269, 273, 279, 290–1, 423 indeterminacy 209 indeterminism 252–3 indiscriminability  4 n.8 margin for error  11–12, 200–3, 264, 376 n.12, 381, 394 n.3, 404 semantics 199 supervaluationism 275 Wittgenstein, Ludwig  22, 326 language mastery  321 modus ponens model  321–2, 326 n.11 Philosophical Investigations 44, 163–4 n.34, 303, 319, 321, 326 n.11 Remarks on the Foundations of Mathematics 163 theory of meaning  163–6 Wright, Crispin  1–7, 40 ‘Anti-Realism, Timeless Truth, and Nineteen Eighty-Four’  33 n.64 Bivalence 27 borderline cases  11, 28–30, 36, 38–9 epistemicism  5–6, 12 governing view  2, 9–10, 15, 19–20, 44 higher-order vagueness  5, 34–40 intuitionism  viii–x, 24–34, 40 rules and rationality on the borderline  15–24 Saving the Differences: Essays on Themes from Truth and Objectivity  6 n.15 Sorites paradox  xi, 1

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448  Index of Names ‘Strict Finitism’  vii, 24 n.44 supervaluationism 26 Tachometer Paradox  3, 12–15 tolerance as putatively a priori  7–12 Truth and Objectivity  6, 24 n.44, 217, 219, 220, 224, 250 vague predicates  7–12, 15, 19–20

Y Yu Guo  xiv Z Zardini, Elia  x, xiv, 8 n.17, 25 n.45, 26 n.48, 370 n.7, 376 n.14, 416 n.26 on Lukasiewicz clauses  391 n.21