Cuts and clouds: vagueness, its nature, and its logic

C U TS A N D C LO U D S

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CUTS AND CLOUDS Vagueness, Its Nature, and Its Logic Edited by

R I C H A R D D I E T Z AND S E B A S T I A N O M O RU Z Z I

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Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © the several contributors 2009 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2009 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloguing in Publication Data Cuts and Clouds: Vagueness, its Nature, and its Logic / edited by Richard Dietz and Sebastiano Moruzzi. p. cm. Includes bibliographical references and index. ISBN 978–0–19–957038–6 1. Vagueness (Philosophy) I. Dietz, Richard. II. Moruzzi, Sebastiano. B105.V33C88 2009 110—dc22 2009027993 Typeset by Laserwords Private Ltd, Chennai, India Printed in Great Britain on acid-free paper by Antony Rowe Ltd, Chippenham ISBN 978–0–19–957038–6 1 3 5 7 9 10 8 6 4 2

Acknowledgements Except for Hartry Field’s paper, all contributions to this collection emerged from a series of workshops and a conference hosted by the Arch´e Philosophical Research Centre for Logic, Language, Metaphysics, and Epistemology at the University of St Andrews in 2003–7. These events were part of the Arch´e research project Vagueness: Its Nature and Its Logic, which was directed by Crispin Wright and funded by the AHRC in 2004–6. This event series would not have been possible without the further support of the Leverhulme Trust, the Analysis Trust, the Mind Association, and the Scots Philosophical Club, to which also our thanks are due. For helpful correspondence, informal discusssion, and comments, we are grateful to four anonymous referees, Andrea Bottani, Francesco Berto, Richard Davies, Raf DeClercq, Dag Westerst˚ahl, and, Crispin Wright. We thank Susan Beer (copy-editor), Tessa Eaton (production editor), and Angela Anstey-Holroyd (proofreader) for their excellent work. Peter Momtchiloff was a great help to us in bringing this volume to publication. Finally, and most especially, we would like to thank our families for all sorts of support, at every step. RD and SM

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Contents List of Contributors

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Introduction

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RICHARD DIETZ AND SEBASTIANO MORUZZI

PART I The Nature of Vagueness

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I What is Vagueness?

21

1 A Metasemantic Account of Vagueness

23

2 The Possibility of Partial Definition

46

AGUST´IN RAYO

SCOTT SOAMES

3 Vagueness and Second-Level Indeterminacy

63

MATTI EKLUND

4 Vagueness as Indeterminacy

77

BRIAN WEATHERSON

5 Sorensen on Vagueness and Contradiction

91

DOROTHY EDGINGTON

II Vagueness in Reality

107

6 Vague Properties

109

STEPHEN SCHIFFER

7 Vagaries about Vagueness

131

NATHAN SALMON

8 Vagueness, Metaphysics, and Objectivity

149

STEWART SHAPIRO

III Tolerance and Paradox

163

9 Agnosticism and Vagueness

165

SVEN ROSENKRANZ

10 Vague Intensions: A Modest Marriage Proposal

187

JC BEALL

11 This Magic Moment: Horwich on the Boundaries of Vague Terms HARTRY FIELD

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Contents

12 Perceptual Indiscriminability and the Concept of a Color Shade

209

LEON HORSTEN

13 The Sorites, Linguistic Preconceptions, and the Dual Picture of Vagueness 228 ´ MARIO GOMEZ-TORRENTE

14 Vagueness and Central Gaps

254

PETER PAGIN

IV Vagueness in Context

273

15 Hold the Context Fixed—Vagueness Still Remains

275

˚ JONAS AKERMAN AND PATRICK GREENOUGH

16 Saying More (or Less) Than One Thing

289

ANDREA IACONA

17 Vagueness as Semantic

304

18 How to Respond to Borderline Cases

327

PART II The Logic of Vagueness

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V Supervaluationism

343

19 Supervaluationism and the Report of Vague Contents

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20 Supervaluationism, Indirect Speech Reports, and Demonstratives

360

¨ MAX KOLBEL

´ DAN LOPEZ DE SA

MANUEL GARC´IA-CARPINTERO ROSANNA KEEFE

21 Scope Confusions and Unsatisfiable Disjuncts: Two Problems for Supervaluationism

373

DELIA GRAFF FARA

VI Paraconsistent Logics

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22 The Prospects of a Paraconsistent Response to Vagueness

385

DOMINIC HYDE

23 Non-Transitive Identity

406

GRAHAM PRIEST

VII Many-Valued Logics

417

24 Identity and the Facts of the Matter

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GRAEME FORBES

Contents 25 Fuzzy Epistemicism

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JOHN MACFARLANE

26 Indeterminacy and Truth Value Gaps

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MARK RICHARD

27 Supernumeration: Vagueness and Numbers

482

PETER SIMONS

28 Degree of Belief is Expected Truth Value

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NICHOLAS J. J. SMITH

VIII Higher-Order Vagueness

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29 Demoting Higher-Order Vagueness

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DIANA RAFFMAN

30 The Illusion of Higher-Order Vagueness

523

CRISPIN WRIGHT

31 Iterating Definiteness

550

CIAN DORR

Index

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List of Contributors ˚ Jonas Akerman Jc Beall University of Connecticut Richard Dietz Cian Dorr

University of Leuven and Arch´e, University of St Andrews

University of Oxford

Dorothy Edgington University of Oxford and Birkbeck College Matti Eklund

Cornell University

Delia Graff Fara

Princeton University

Hartry Field New York University Graeme Forbes University of Colorado, Boulder Manuel Garc´ıa-Carpintero

University of Barcelona

Mario G´omez-Torrente Institute of Philosophical Research, UNAM Patrick Greenough

University of St Andrews and Arch´e, University of Sydney

Leon Horsten University of Bristol and University of Leuven Dominic Hyde University of Queensland Andrea Iacona

L’Aquila University

Rosanna Keefe University of Sheffield Max K¨olbel University of Birmingham, ICREA and University of Barcelona Dan L´opez de Sa St Andrews John MacFarlane

ICREA, University of Barcelona and Arch´e, and University of University of California, Berkeley

Sebastiano Moruzzi University of Bologna and Arch´e, and University of St Andrews Peter Pagin Graham Priest CUNY, University of Melbourne, and Arch´e University of St Andrews Diana Raffman

University of Toronto

Agust´ın Rayo Massachusetts Institute of Technology and Arch´e, and University of St Andrews Mark Richard

Tufts University

Sven Rosenkranz

ICREA and University of Barcelona

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List of Contributors Nathan Salmon University of California, Santa Barbara Stephen Schiffer

New York University

Stewart Shapiro

Ohio State Arch´e, University and University of St Andrews

Peter Simons Trinity College Dublin Nick J. J. Smith University of Sydney Scott Soames University of Southern California Brian Weatherson Rutgers University and University of St Andrews Crispin Wright

New York University

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Introduction Richard Dietz and Sebastiano Moruzzi

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P U Z Z L E S O F VAG U E N E S S

Vagueness is a feature that we know from many expressions in natural languages. While being a familiar phenomenon, on closer inspection, it gives rise to various puzzling questions. As some ancient philosophers already noticed, vague terms are susceptible to a particular type of paradoxes, also known as sorites paradoxes, which challenge the very idea that we can coherently represent reality in vague terms. Common sense may be a good guide inasmuch we want to use vague expressions correctly. But it is not helpful inasmuch as we try to get a clearer understanding of what vagueness is and how to deal with the paradoxes it gives rise to. Ancient philosophy made a start. With the revival of the philosophical discussion in the last three decades, the questions and answers have become much more differentiate and substantive. Vagueness has been one of the driving motivations for the development of numerous non-classical logics. It is one of the core issues in contemporary philosophy of language and epistemology. And some authors have suggested that a theory of vagueness is not only pivotal to our understanding of how we represent reality, but also of what reality is like, irrespective of our ways of representing it. According to this view, vagueness is a key subject of metaphysics. We focus here on two types of puzzles that philosophical discussion has been centred on and that also play a cenral role in this collection. One feature that is characteristic of vague general terms is that they admit of borderline cases of application. We only need to reflect a bit on our linguistic behaviour in order to grasp the concept of being a borderline case. Unlike Borel set, neutrino or Turing machine, no technical terminology is required for explaining the concept. An everyday example does. Here is one: just take this book. If you have read the first 500 pages, it is clear that you have read the most of this book. On the other hand, ten pages clearly are not the most of this book, for less than a half of something does not suffice for being the most of it. Both for 500 and ten, there is a precise answer to the question of whether the number is the most of the number of pages of this book. But there are numbers in between, for which it seems fair to say that it is vague whether the respective number of pages is the most of this book. Being asked for any such number n whether n pages is the most of this book, it would be appropriate to give a hedging response such as ‘I would not say that n pages it is the most, nor that it is not’, or ‘it depends on how you look at it’, or ‘one can go either way’ or ‘it is mostish’, or ‘it is kind of in between’. In other words, we could also say ‘n pages is a borderline case of being the most of this book’. To contrast borderline cases with cases where 1

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we lack relevant information, consider the case where I pick out randomly a particular number of pages and you only know that I have picked out a number that is greater than i and smaller than i + 3. Being asked whether n is the number I have picked out, it would seem inappropriate to respond in a hedging way. Rather it would be fair to grant that there is only one correct answer to the question and that we just don’t know it. In borderline cases of ‘most’, by contrast, there seems to be no such uniquely correct answer, even if there is no more relevant information we could acquire—if n is a borderline case of being a number of pages that makes the most of this book, then even if you know that n is the number I just picked out, your evidence seems neither to rule out the number of pages just picked out to be the most of this book, nor to be not the most. Borderline predications are a familiar feature of vague general terms. But what are they exactly? Do they have truth values? Are we in a position to have knowledge whether n pages is the most of this book, if n is a borderline case of being a number of pages which is the most of this book? And why are we prompted to give hedging assessments with respect to cases we regard as borderline predications? The nature of borderline predications is just one puzzle. Let us turn to sorites paradoxes, another, even more notorious puzzle about vagueness. There are a number of variants of sorites. For our purposes, it suffices to introduce the standard version. Consider a strictly decreasing series of natural numbers, beginning with 500 and ending with 10, where for each pair of adjacent numbers, the first member is by one larger than the second member. For any particular pair of adjacent numbers, it seems odd to say that it marks a cut-off between being the most of the number of pages of this book or and being not the most of the number of pages of this book. For if any number n makes the most of the number of pages of this book, then whatever the number is, then one page less still should make the most of this book. Starting from 500, this reasoning leads us to the conclusion that also 499 pages is the most of this book. For 500 pages is clearly the most of this book, and since at no point in the series, we can accept a transition from being the most to being not the most to take place, every number of pages in the series is the most of this book, and thus also 10—which is absurd. In effect, we have apparently a valid argument with true premises and a false conclusion. This is the paradox. Let us be clearer about its logical structure. We have an instance of the sorites paradox for a predicate F  if sentences of the following are all individually acceptable:¹ (A) (B) (C) (D)

Fa ∼Fz (∀x)(∀y)((Fx & Rxy) → Fy) (∃b1 . . . bn )(Rab1 & Rb1 b2 & . . . & Rbn−1 bn & Rbn z)

a is to be a clear case of F -ness, and z is a clear case of F -ness. R is to be any relation for which (C) and (D) are acceptable. Sentence (C) says then that F  is tolerant with respect to the relation R. Sentence (D) says that there is a sorites series for F , with a being the first member and z being the last one. With these premises in place, contradiction is derivable. We only need universal instantiation (i.e. the inference ¹ We follow here Fara (2000:49–50).

Introduction

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rule that allows us to infer from any universally quantified sentence to the truth of each instance), modus ponens (i.e. the inference rule according to which given P and P → Q, we can derive Q) and transitivity of deduction (i.e. the requirement on deduction that if A is derivable from a set of premises  and B is derivable from the union of  and A, then B be also derivable from  alone): By universal instantiation, from (C), all sentences of the form (C ) (Fi & Rij) → Fj are derivable. As by (D), R applies to all adjacent pairs in a sorites series, we can derive from (A) and the relevant instance of the form (C ), by modus ponens, that the successor of a is an F. From this, by parity of reasoning, we infer the same for the successor of the successor of a, and so on, ending with the conclusion that the last object in the series (i.e. z) is an F . By transitivity of deduction, from (A), (C) and (D) alone, it follows that z is an F —which contradicts premise (B). As a result, we have an argument for a conclusion that contradicts our expectation that vague terms provide consistent means of classifying reality. That is, either sorites arguments are flawed, or our expectations regarding vague terms are flawed. One group of authors, who may be labelled ‘incoherentists’, argue that we do have to evaluate the paradox in the latter way—in contexts where there is a sorites series for a vague term, the term is inconsistent. The received approach is to offer an escape route from this radical conclusion and to locate the flaw in sorites arguments instead. There are various options of taking this line. For one, one may cast doubt on the validity of some logical proviso. Transitivity of deduction may be abandoned. Alternatively, one may reject modus ponens as invalid. A closely related line suggests that modus ponens is valid, but does not ensure conclusions to be as acceptable or as close to truth as the premises. The prevailing group of resolutions to sorites paradoxes, by contrast, diagnose the tolerance principle (C) to be the actual source of the paradox. Adherents to classical logic contend that the tolerance principle is false. In a classical framework, this is equivalent to the existential assumption (∃x)(∃y)(Fx & Rxy & ∼Fy), that is to say, there is a counterinstance to the tolerance principle. Typically, plausible candidates of R also satisfy a monotonicity constraint: For every x and y, if Rxy and Fy, then Fx. Given this, our existential assumption says furthermore that there is a sharp cut-off between Fs and non-Fs for the vague predicate F . Other authors have argued for non-classical frameworks that allow us to reject the tolerance principle without being committed to the existence of a counterinstance to tolerance (or even the existence of a sharp cut-off ). Another option of blocking sorites arguments is to abandon instead the existential assumption of a sorites series (D)—according to this, whatever relation may validate tolerance, its transitive closure will not apply to any pairs of a clear case of F -ness and a clear case of non-F -ness.² ² It is not the aim of this introduction to give a state of the arts report. For a systematic introduction into the philosophical discussion of vagueness, see Williamson (1994) and Keefe (2000). For collections of seminal papers, see Keefe & Smith (1996) and Fara and Williamson (2000).

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The contributions of this collection explore new directions in approaching puzzles about borderline vagueness and the sorites paradoxes. Part one focuses on the nature of vagueness, with contributions on what sort of facts vagueness is grounded in (chapters 1–5), on whether there is vagueness in reality (chapters 6–8), on whether tolerance principles are defensible, on what sort of relation is in play in tolerance principles (chapters 9–14), and lastly on the things we do with vague words in communication (chapters 15–18). Part two focuses on the logic for vague languages, with contributions on supervaluationist logic (chapters 19–21), on paraconsistent logics (chapters 22–23) and many-valued logics and degree theories (chapters 24–28). A special problem in this context is the logic of expressions in terms of which we express vagueness (e.g. ‘it is vague whether’), which is at issue in the last three contributions (chapters 29–31). 2

W H AT I S VAG U E N E S S ?

Take a vague predicate, say ‘bald’. It is a fact that this predicate has borderline cases. It is also a fact that it apparently lacks a sharp boundary. But what kind of facts do we have to do with here? Some authors argue that vagueness is grounded in epistemic facts about us, in the mouth of whom it is true to say that there is vagueness. On this view (also known as epistemicism), ‘bald’ does in fact have a sharp boundary, it only apparently lacks a sharp boundary, since we—that is, ordinary speakers of English—are (for certain reasons characteristic of vagueness) not in a position to know where to locate exactly the boundary. According to this, that Alfred is a borderline case of ‘bald’ amounts to the fact that we are (for certain reasons characteristic of vagueness) not in a position to know that ‘bald’ applies to Alfred, nor that it does not. The most popular view holds that vagueness is grounded in facts about languages, or more generally (in so far as also non-linguistic representations may engender vagueness), in facts about the way reality is represented. On this view, ‘bald’ does lack a sharp boundary, in the sense that there is no point in a sorites series where the linguistic meaning fixes a ‘bald’-predication of one object to be true and the same predication of the adjacent object to be false. According to this, that Alfred is a borderline case of ‘bald’ amounts to the fact that the linguistic meaning of ‘bald’ does not determine a ‘bald’-predication of Alfred to be true, nor to be false. Still other authors suggest that vagueness is grounded in facts about reality itself—irrespective of how it is represented, and irrespective of the epistemic capacities subjects may have. On this view, ‘bald’ apparently lacks a sharp boundary, because it denotes a property with no sharp boundary—there is no pair of objects in a sorites series where there is a fact of the matter that the one object bears the property, and that the other one fails to bear it. According to this, the fact that Alfred is a borderline case of ‘bald’ amounts to the fact that he is a borderline case of the property expressed by ‘bald’. Scott Soames’s, Matti Eklund’s, and Agust´ın Rayo’s discussions all start from the basic idea that vagueness is a fact about the way reality is represented. Soames offers a defence of the view that vague predicates are partially defined, in the sense of being

Introduction

5

governed by rules of application which for some objects do not fix a truth value. He aims in particular at an argument advanced (in different variants) by Michael Dummett and Michael Glanzberg, according to which the norm of assertion rules out the existence of partially defined predicates. Soames gives reasons for combining partiality with context-sensitivity in modelling vague terms (on contextualism about vagueness, see § 5). For one, (he suggests that) a context-sensitive approach supplies means of explaining why some violations of ‘laws’ of classical logic are deemed more acceptable than other ones. For another, (it is suggested that) if we model vague predicates as partially defined, we do not need to assume the existence of a sharp cut-off of which we are ignorant. A number of authors (e.g. Soames) hold the view that linguistic vagueness is to be characterized in terms of special semantic values. Eklund calls this position ‘first-level indeterminacy view’ and offers arguments for an alternative approach, which he calls ‘second-level indeterminacy view’. According to this, borderline vagueness is in fact not a feature of semantic values, but of semantic valuations—the linguistic meaning of vague sentences is indeterminate in the sense that for some possible circumstances of utterance, it does not fix any classical valuation as the uniquely correct one. Eklund argues that considerations both on the meta-semantics and the pragmatics of vague expressions tell in favour of the second-level view and against the first-level view. Rayo’s ‘metasemantic’ theory of vagueness provides a more specific model of vagueness as ‘second-level indeterminacy’. On this account, what renders an expression meaningful is the prevalence of suitable linguistic conventions—where prevalence of convention may come in degrees. According to this, an individual forms a borderline case of application of a predicate just in case conventions governing the use of the predicate are only prevalent to a low degree. Brian Weatherson’s and Dorothy Edgington’s discussions shed a critical light on the way the agenda for any theory of vagueness is used to be conceived of. The received view holds that tolerance is in some way or other fundamental to vagueness (on tolerance, see § 3). Weatherson argues to the contrary: there may be vagueness without tolerance. Scrutinizing various recent attempts of characterizing vagueness in terms of tolerance, he suggests that these accounts all fail to provide an adequate way of systematizing our intuitions while respecting theoretically important commonalities and distinctions. On his account, we need to redefine the goal for any theory of vagueness. The central issue is not to give an account of tolerance (which is rather an epiphenomenon of vagueness), but an account of indeterminacy in denotation. Edgington’s considerations cast doubt on the very idea that the question of what vagueness consists in is pivotal for any theory of vagueness. In her criticism of Roy Sorensen’s theory of vagueness, she compares the philosophical enterprise of giving an account of vagueness with other attempts made in philosophy to reduce fundamental concepts such as knowledge, or causation, or possibility—all of which so far did not succeed. She suggests treating vagueness as a primitive concept, which may be elucidated in different terms after all. For example, she argues that standard epistemic or objective interpretations of probability are of no avail for an adequate modelling

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of reasoning in vague languages, and that we need to reinterpret probability in a way characteristic of vagueness. 3

VAG U E N E S S I N R E A L I T Y

For our purposes, it is helpful to distinguish here between two kinds of issues that are typically associated with the label ‘vagueness in reality’ and often conflated. For one, the issue as to whether there is vagueness in reality may pertain to the question of what the bearers of vagueness are. The received view says that only states of affairs regarding matters of representation may be vague—all vague states of affairs accordingly concern representations and their semantic relations to things in reality. Some authors have pronounced the idea that the scope of vague states of affairs may go beyond that. For example, a proponent of the idea that it is vague of Alfred whether he bears the property of baldness—and not only vague of him whether ‘bald’ is true of him—would be a proponent of vagueness in reality in the above sense. The question of whether there may be vague, non-representational states of affairs has been discussed especially with regard to states of affairs of compositionality (may it be vague of some molecule whether it is part of Mount Everest?), and states of affairs of identity (may it be vague of this and that object as to whether the former is identical with the latter?). In association with the idea of vague states of affairs, some authors have claimed that even objects or properties themselves may be bearers of vagueness. For another, the issue as to whether there is vagueness in reality may pertain to the question of what sort of facts vagueness is grounded in. Some authors have pronounced the idea that there are cases of vagueness that are grounded in reality, that is, irrespective of the ways in which we may represent it and irrespective of our epistemic capacities—according to this, we could not even possibly give a comprehensive account of all states of affairs in a fully precise language, nor would reality be more precise if we were omniscient. The second variant of the idea of vagueness in reality is scrutinized in Stewart Shapiro’s discussion. He contends that the previous attempts to impart some more specific content to this thesis fail to be coherent or to be intelligible, or they fail to capture the intended intuitive idea. In particular, he argues that even if there is no possible way of representing reality in fully precise ways, from this, it does not follow that there is vagueness irrespective of the ways in which we may represent reality. Stephen Schiffer submits that there are non-representational properties (such as being bald) that have borderline cases—or to speak in his terminology, that there are vague properties. On his account, vagueness of general terms (adjectives, predicates, etc.) is grounded in the vagueness of the properties they denote. This does not mean though that Schiffer subscribes to the radical thesis that there is vagueness in reality that is not grounded in any facts about the way reality is represented or facts about the epistemic capacities of subjects who mean to represent reality by using languages. On his view, vague properties essentially differ from precise properties in their ontological and in their epistemic status. For instance, the vague property of baldness is mind-dependent in that its individuation principle is defined by the way we

Introduction

7

use language (specifically, the way we use general terms such as ‘is bald’). As a consequence, we cannot acquire individuating knowledge of that property before learning to use general terms that denote it. This alone does not rule out the possibility that there is vagueness in reality irrespective of facts about linguistic representations and the epistemic capacities of their users. But it says at least that the particular vague properties we may refer to are self-made. Nathan Salmon, by contrast, makes a case for the more radical thesis that there are vague states of affairs that are not grounded in facts about our language or our epistemic situation. According to his argument, resentments against this very idea are not only ill-motivated but self-undermining. Giving an appraisal of various ways of putting the idea of vagueness in language more specifically, he concludes that there is no way of explaining this sort of vagueness but by imputing vagueness regarding objects and their linguistic attributes that is grounded in these objects and attributes themselves—irrespective of how we represent them, or how good our epistemic capacities are. E.g., he contends that there is no other way of explaining the vagueness of ‘Princeton, New Jersey is the fusion of Princeton Borough and Princeton Township’ but by imputing vagueness to some spatial areas and the property of being designated by ‘Princeton, New Jersey’: there is the municipality-fusion and the property of being designated by ‘Princeton, New Jersey’, but there is no fact as to whether the first has the second—where the ‘no fact of the matter’ is to be understood in a brute ontological, and not in a semantic or epistemic sense. Once we grant that there is vagueness affecting objects and their linguistic properties (such as the property of being denoted by ‘is bald’) in this sense, there is just a small—and, according to Salmon, no problematic—step to acknowledging also vagueness regarding nonlinguistic properties of objects. 4

TO L E R A N C E A N D PA R A D OX

Sorites arguments suggest that once we assume for any given vague predicate, (i) the existence of a sorites series, (ii) the existence of clearly true and clearly false application cases in the series and (iii) an associated tolerance principle, we are logically committed to the conclusion that all objects in the series are true application cases—thus, the conjunction of (i), (ii) and (iii) is provably false, given that sorites arguments are valid. One escape route from paradox is to sustain the assumption of a sorites series with clear cases of truth and clear cases of falsity, in which case tolerance is false. One may argue that this price is too high, in so far as tolerance is the only way of cashing out more exactly the familiar (if obscure) idea that vague predicates lack a sharp boundary—or in so far this form of constraint at least is to be implied by any plausible account of what it is to lack a sharp boundary. For another, there is still another familiar (and less obscure) idea that needs to be accommodated, namely that vague predicates are useful tools for making classifications about reality. For instance, applications of the term ‘is the most of this book’ to numbers of pages of this book are not idle in the sense there is no distinction to be made between numbers of pages with respect to being the most of this book—some applications are clearly true, whereas

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others are clearly false. Mario G´omez-Torrente’s, Peter Pagin’s and Jc Beall’s contributions explore different ways of defending tolerance without committing us to the radical conclusion that vague predicates fail to provide us classifications in terms of which we can distinguish some objects from other ones. Leon Horsten’s discussion of the structural features of the similarity relation involved in the tolerance principle bears on a recent debate about sorites paradoxes for phenomenal vague predicates. Sven Rosenkranz and Hartry Field explore ways of accommodating the idea of apparent boundarylessness for vague predicates without the tolerance principle. Jc Beall sides with the incoherentist view that vague predicates are fully tolerant and thus without sharp cut-offs. On this view, they are not useful means of representing reality in that they fail to supply means of distinguishing some objects from others. On the other hand, he suggests, they are useful in that in our language, for every vague predicate, there is a class of homonyms (or ‘precisifications’) with sharp cut-offs, which are intended for cutting distinctions. On this proposal, incoherentists are right about vague predicates. Coherentists on the other hand, inasmuch as their accounts yield sharp cut-offs for vague predicates, are in effect giving an adequate account of the ‘sharp off-spring’ of vague predicates, not an account of vague predicates themselves. Mario G´omez-Torrente presents a novel type of incoherentism (for incoherentism see §1). According to him, in situations where there is an instance of the sorites paradox, it is not the linguistic meaning which supports the categorical premises or the tolerance premise. Rather these premises represent linguistic pre-conceptions of ordinary speakers, that are, sentences we are disposed to accept firmly. Preconceptions regarding vague general terms come in two types, which he refers to as ‘paradigmatic’ (expressing the application of the predicate to its clear cases) and ‘generic’ (the tolerance principle) respectively. In regular contexts of use, that is contexts where there is no instance of a sorites paradox, these two types of preconceptions supply sufficient means of fixing classical satisfaction-conditions, which partition a relevant domain of objects into an extension and an anti-extension of the term. In other contexts, however, these preconceptions conflict with each other and fail to fix any truth-conditions. The author suggests that regular contexts represent the everyday case and that we can therefore rely on vague terms typically as consistent means of classification. Similarly to Gom´ez-Torrente’s proposal, Peter Pagin’s account rests essentially on the idea that the relevant domain over which we make quantificational statements may vary with utterance contexts—plausibly depending on what we pay attention to or what is salient. Pagin develops a formal semantic framework that covers both paradoxical and non-paradoxical utterance contexts. In contexts of the latter type, some objects are excluded from the domain of quantification to the effect that there is no pair of a true application case and a false application case such that any relation that satisfies tolerance applies to them. As a result, the joint assumption of tolerance and the existence of clear and of false predications are jointly consistent. In effect, what is given up in this kind of case is the existential assumption of a sorites series. Consider a sorites series for any phenomenal vague predicate, say of coloured patches, for ‘looking red’. Each pair of adjacent patches in the series looks the same. But the patch at one end of the sorites does not look the same as the patch of the other

Introduction

9

end. This reasoning suggests that indiscriminability is not transitive (where transitivity means: if a looks the same as b and b looks the same as c, a looks the same as c). Some authors have argued in favour of the contrary conclusion. Indiscriminability is transitive, and the associated tolerance principle holds. But the sorites paradox is blocked, for absence of a sorites series. Or so they argue. In his thorough reconstruction of this recent debate, Leon Horsten urges considerations which cast doubt on the thesis that indiscriminability is transitive. He argues that there is no good evidence for the transitivity thesis about indiscriminability, in so far as we want to give an account of phenomenal intuitions that maximizes their coherence. According to Sven Rosenkranz, we should rather take an agnostic position regarding the existence of sharp boundaries for vague terms, that is neither commit ourselves to its existence, nor to its non-existence. He argues that proponents of a classical semantics for vagueness are hostage to commitments in the theory of reference which we do not know how to redeem. Rosenkranz follows Crispin Wright in endorsing an intuitionist logic for vague languages. In contrast to Wright, whose position is mainly primarily motivated by an antirealism about truth, Rosenkranz’s agnostic position foregoes commitment to any theoretical view regarding the nature of truth though. Consider the following argument: (1) 400 pages are the most of this book. (2) Zero pages are not the most of this book. (3) Thus, there is a number n such that n pages are the most of this book, and for any k < n, k pages are not the most of this book. Undeniably, if i pages are the most of this book, so are i + 1 pages. Given this, (3) says that there is a sharp boundary between numbers of pages which are not the most of this book and numbers of pages which are. Our argument is valid, given the least number principle is valid. Hartry Field contends that for vague predicates, the argument can be blocked. According to him, classical logic fails for borderline case predications, with the consequence that only a weakening of the least number principle is valid. 5

VAG U E N E S S I N C O N T E X T

What does vagueness in language more specifically come to? Can we give a comprehensive account of this phenomenon only in terms of facts about the meanings of expressions encoded in language? Or do we need to get also facts about the contexts where we use expressions into the picture? The contributions of this section suggest that the answer to the latter question is to be given in the positive. It is suggested that in leaving out contexts of use, we thereby miss out also certain factors (such as the psychological state or the interests of the speaker, the conversational background, or other kinds of facts) that are relevant to a theory of linguistic vagueness—or to an important part thereof. Contextualism about vagueness is the generic view that reference to contexts is indispensable for a theory of the semantics of languages involving vague expressions. For example, consider Alice. Is she tall? It seems fair to say that whether Alice is tall depends on her height—if she is tall, then anybody else with the same height is tall as well, and if she is not, then nobody else with the same height is either. This said, one may argue that whether her actual height makes for a true predication case of ‘is tall’

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Richard Dietz and Sebastiano Moruzzi

in turn depends on standards that may vary with contexts. According to this, ‘Alice is tall’ may vary in truth value also with contexts, where her height does not vary—in which case, relative to one context, ‘is tall’ is true of anybody with Alice’s height, and relative to a different context, ‘is tall’ is false of anybody with Alice’s height. Contextualist accounts of sorites paradoxes standardly describe them as involving some type of context-shifting fallacy. For example, in a sorites series for ‘is tall’, when assessing any particular relevant instance of tolerance, ‘If anybody with the height n × 0.001 inches is tall, then anybody with the slightly lower height of (n − 1) × 0.001 inches is tall’, we thereby bring forward a context at which it is true. But even though for every particular instance, there is a context at which it is true, there is no context at which all relevant instances of tolerance are true. Or this would be a line typically taken in contextualist accounts (leaving aside differences in the way semantic valuations relative to contexts and entailment are modelled). A standard objection to contextualism is this: keep the context fixed, and even then predicates will typically show all features characteristic of vagueness—in particular, they will be also susceptible to sorites paradoxes. ˚ J onas Akerman and Patrick Greenough agree with this point, but they contend that it only highlights the explanatory limits of a semantic theory of vagueness. The authors make two alternative proposals of giving an account of vagueness within a context: the one proposal is epistemological in kind and invokes the idea that vagueness may be characterized as a species of ignorance of speakers. The other one is a metasemantic account invoking the idea that any available metalinguistic means of referring to contexts are themselves affected by vagueness. Contextual factors that are relevant to the truth value of sentences may play two different roles: (i) to determine what is said, and (ii) to determine the truth value of what is said. If a contextual factor plays the role (i) with respect to the truth values of certain truth values, then these sentences say something about that factor—e.g. on standard indexical semantics for the first person (‘I’), sentences like ‘I am Alice’ as uttered say some thing about the speaker at the respective context. If a contextual factor plays the role (ii) with respect to the sentences of certain sentences, then what is said by these sentences is true with respect to that factor—metaphysically contingent sentences (e.g. ‘Alice’s height is 5 feet and 1 inch’) are standardly modelled as sentences with a content that is true with respect to some possible worlds, and false with respect to others. Max K¨olbel argues for a contextualism about vagueness that models vagueness in a way analogous to metaphysical contingency, and not as an instance of indexicality. On his account, sentences that are vague in truth value have a content that is true with respect to some ‘standards of precisification’, and false with respect to others. In a second step, this contextualist semantics is applied to an account of assertability. The basic idea is to model assertability relative to a context as truth relative to all standards of precisifications admissible at that context—where what counts as an admissible standard may vary with contexts. On this account, for any instance of tolerance, there is a context at which it is assertable, but no context where all instances are jointly assertable. Some authors have suggested that borderline cases admit of faultless disagreement, in the sense that as assessed from one context, it is correct to assent to the predication,

Introduction

11

whereas as assessed from another context, it is correct to dissent from the predication. Dan L´opez de Sa makes a case against the very thesis that there may be faultless disagreement over borderline cases. Whether vagueness is a species of ignorance or a species of semantic indeterminacy, for either case, plausible normative constraints on assertion rule out that there may be any room for faultless disagreement regarding borderline cases. Or so his point may be summarized. L´opez de Sa suggests that in the default case, the only rational attitude towards borderline cases is to abstain both from assertion and denial. What speakers mean to communicate in uttering a sentence frequently diverges from what the sentence literally says—e.g. consider what you would typically mean to say when uttering sentences such ‘Is there a doctor in the house?’, or ‘Alice is Alice’. In so far as what a sentence says may be affected by vagueness, plausibly what a speaker means in uttering the sentence may be vague as well. This suggests another type of vagueness, which emerges in contexts, in association with the communicative intentions of speakers. Andrea Iacona develops a semantic framework for modelling vagueness at the level of speakers’ meanings. It is suggested that the same sort of classical semantic models that may be used for an account of vagueness of what is said, may be applied as well to an account of vagueness of what is meant by speakers.

6

S U PE RVA LUAT I O N I S M

Supervaluationist logic combines elements of classical logic with the semantic view that vague sentences are truth value gaps. Whereas on many-valued semantics, classical constraints of truth-functionality are generalized for a wider range of semantic values, supervaluationist semantics are generated by quantifying over a given set of ‘admissible’ classical interpretations of the language—where admissibility is subject to constraints that ensure that precisifications respect semantic connections not only between logical constants (e.g. an interpretation on which ‘Alfred is tiny’ and ‘Alfred is tall’ are both true should not count as admissible).³ A sentence is true (false) simpliciter just in case it is true (false) relative to all admissible precisifications. Here are some results of supervaluationism which have been found objectionable by a number of authors: (i) If a sentence P is true, it is ‘ultratrue’, in the sense that if the language contains a sentence operator ‘D’ for truth, we may attach however many D-operators to P, and the result will be in turn true (i.e. DP is true, DDP is true, and so on). As sentences which express the truth value status of other sentences are either truth or false, it follows that questions of vagueness or precision have always a precise answer—which clashes with a widely held view that there may be higher orders of vagueness (on higher-order vagueness, see § 9). (ii) As entailment is defined as preservation of truth simpliciter, and as admissible precisifications are classical valuations, ³ In standard supervaluationist terminology, truth and falsity simpliciter are labelled ‘supertruth’ and ‘superfalsity’ respectively. For brevity, we stick to the terms ‘truth’ and ‘falsity’ here.

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Richard Dietz and Sebastiano Moruzzi

all classical tautologies are valid. However, classical inference rules that involve subarguments such as conditional proof (if B follows from  ∪ {A}, then A ⊃ B follows from ), reductio ad absurdum (if contradiction follows from  ∪ {A}, then ∼A follows from ), argument from cases (if C follows both from  ∪ {A} and from  ∪ {B}, then it also follows from  ∪ {A ∨ B}), and contraposition (if B follows from  ∪ {A}, then ∼A follows from  ∪ {∼B}) are invalid. (iii) Standard logical connectives other than negation fail to be truth-functions: a disjunction can be true without any disjunct being true (e.g. ‘Alfred is bald or Alfred is not bald’ is in any case true, even if ‘Alfred is bald’ and its negation are gappy); conversely, a conjunction can be false without any conjunct being false (e.g. consider ‘Alfred is bald and Alfred is not bald’, where both conjuncts are gappy). Correspondingly, existential quantificational sentences can be true without having any true instance, and universal quantificational sentences can be false without having any false instance. This feature is exploited in the supervaluationist solution to sorites, according to which the tolerance principle is false, without there being any false instance of the tolerance schema. Specifically, all instances involving either a vague antecedent or a vague consequent are untrue without being false. To illustrate: ‘For every pair of persons, x and y, if x has only one more hair on his head, then y is not bald, if x is not bald’ is false—even though no particular instance of the form ‘a has only one hair more on his head than b, but whereas a is not bald, b is bald’ is true. Failure of truth-functionality for standard logical connectives other than negation is widely (outside the supervaluationist camp at least) held to be a disadvantage of supervaluationism. Delia Graff Fara’s discussion reveals that connectives fail to meet even certain basic modal truth-functional intuitions. For in so far as supervaluationism allows for sentences that are vague, it is even committed to true disjunctions where either disjunct is impossibly true—even though (as Fara argues) one should expect disjunctions in such a case fail to be true. Another objection concerns the issue of how to explain the prima facie plausibility of the tolerance principle, which is, according to the supervaluationist account of sorites arguments, false. Kit Fine and Rosanna Keefe have attempted to give such an account. Fara highlights several problems that tell against the success of these proposals. Stephen Schiffer has raised various objections that are meant to show that supervaluationism is doomed to fail to give an adequate account of indirect speech reports. Supervaluationism suggests that ‘Carla said that Bob is tall’ is true just in case for every proposition p that precisifies the content of ‘Bob is tall’, Carla said p. In defence of supervaluationism, Rosanna Keefe contends (pace Schiffer) that this result is far from absurd, considering that on supervaluationist first-order logic, a universal statement can be true without there being any true instance. Furthermore, confronting particular problems with indirect speech reports involving demonstratives, she contends that vagueness raises no new problems that did not already face a theory of demonstratives. Manuel Garc´ıa-Carpintero’s defence of supervaluationism against Schiffer’s objections is essentially different in that it allows not only for imprecise denotations but also for imprecise entities (particulars, properties or kinds) denoted in language. On Keefe’s account, ‘Al said that Ben was there’ is true just in case it is true to say that

Introduction

13

there is a precisely delimited region of space such that Al said of it that Ben was there. On Garc´ıa-Carpintero’s account, by contrast, ‘Al said that Ben was there’ is true just in case there is an imprecise region of which it is true to say that Al said of it that Ben was there.

7

PA R AC O N S I S T E N T LO G I C S

The explosion principle (also known as ex contradictione quodlibet principle) says that anything follows from any contradictory premise set. It is part of classical logic and also of most non-classical logics that have been proposed for vague languages. On paraconsistent logics, this principle fails. E.g., starting from explosion, on the joint assumption of Alice is tall and of Alice is not tall, anything follows. But one may argue that not any sentence whatsoever (e.g. The moon is made out of cheese) follows from this premise set as well. Or this is a way of illustrating the intuitive case against explosion. In this section, two particular paraconsistent logics are at issue that have been proposed for vagueness. Both logics are motivated by the underlying intuitive idea that instances of vagueness in truth value are both true and false—in other words, they are truth value gluts, and not gaps. Dominic Hyde’s subvaluationism can be easily obtained from supervaluationism. Frames are constructed the same way as in supervaluationist semantics (see § 6). We just need to make the following important modifications: (i) instead of modelling truth (falsity) simpliciter as truth (falsity) relative to all admissible precisifications, we model truth (falsity) simpliciter as truth (falsity) relative to some admissible precisifications.⁴ (ii) Entailment is defined as preservation of truth simpliciter in the subvaluationist sense. As on supervaluationism, the semantics for logical connectives departs from classical compositionality constraints, albeit not exactly in the same way—e.g. on subvaluationism, a disjunction may be true, even though both disjuncts are false (consider instances of the law of excluded middle A ∨ ∼A where A is a truth value glut). Like supervaluationist logic, subvaluationist logic departs from classical logic, again not exactly in the same way. Importantly, modus ponens fails for any instance where the antecedent A in the relevant conditional A → B is true and false, and the consequent B is only false (for on the standard material account of the conditional, A → B is equivalent to ∼A ∨ B, and this disjunction is true if A is false). As a result, the standard sorites argument can be blocked at the first instance of tolerance where the antecedent expresses a borderline case (i.e. a truth value glut), and where the consequent expresses a predication which is clearly false (a sentence that is only false). Is there any good reason for preferring subvaluationism to its kin, supervaluationism? In his contribution to this volume, Hyde confronts supervaluationist arguments to the effect that subvaluationism involves a more radical departure from classical logic (we have failure of adjunction, that is a principle of single-conclusion ⁴ In analogy to the supervaluationist terms for truth (falsity) simpliciter, Hyde chooses the label ‘subtruth’ (‘subfalsity’) for the subvaluationist notion of truth (falsity) simpliciter respectively. For brevity, we stick here to the term ‘truth’ (‘false’).

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logic that says that assuming a pair of sentences, the associated conjunction must be true). As Hyde points out, once we widen the scope of our appraisal to multipleconclusion logic, supervaluationist turns out to be as radical in its logical revisionism (e.g., we have failure of subjunction, that is a principle of multi-conclusion logic that says that assuming a disjunction, at least one of the disjuncts must be true). In Graham Priest’s discussion, a second-order version of a famous paraconsistent logic, labelled LP, is applied to a particular type of sorites paradoxes that concern the identity relation. To illustrate, consider a ship that is going to be completely refurbished, plank by plank. Every minute, an old plank is replaced with a new one. After a couple of hours, no old plank has been left. Intuitively, for any time t in the refurbishment process, one may (following Priest) plausibly accept each tolerance instance of the form The ship at stage t is identical with the ship one minute after. By standard logic of identity however, according to which identity is transitive, the acceptance of every instance of the relevant tolerance principle commits us to the conclusion that the old completely unrefurbished ship is identical to the completely new ship, after all parts of the old ship have been replaced—which (according to Priest) may be deemed as absurd. On Priest’s logic, this reasoning can be blocked, since identity fails to be transitive—e.g. it may be true to say a = b, true to say b = c, but only false to say a = c. Priest suggests that this point bears not only on the solution to identityrelated sorites paradoxes. It supplies (he suggests) also means of giving an account of what makes us prone to go into the trap of sorites arguments in general. Consider any sorites series for any vague predicate F , where we are prone to make no difference as to F -ness between any pair of adjacent objects. Speaking in terms of the truth values of F -predications in the series, this amounts to the case where for each pair of adjacent objects ai and ai+1 , the associated metalinguistic identity statement of the form the truth value of F (ai ) = the truth value of F (ai+1 ) is acceptable. That is, we have a corresponding sorites series for identity with respect to truth values of F -predications, with a corresponding metalinguistic sorites paradox. And in so far as we have a solution to the latter, we have a metalinguistic account of what gives rise to standard sorites paradoxes for a given language. 8

D E G R E E - T H E O R I E S A N D M A N Y- VA LU E D LO G I C S

In degree-theoretic semantics, the classical range of two values, {0, 1}, is replaced by infinitely many values, standardly the unit interval of reals, [0, 1]. Instead of the classical dichotomy of truth and falsity, there is hence a continuum of values that can be assigned to predications in a sorites series, which admits of a smooth transition of semantic valuations from clear cases of truth (i.e. value 1) to clear cases of falsity (i.e. value 0). Some authors have suggested adopting probabilistic constraints on the structure of degrees for logical compounds. More popular is a group of degree-theories where classical truth-functionality principles for logical connectives and quantifiers are generalized—common labels for this type of degree-theories are ‘fuzzy semantics’ and ‘fuzzy logic’. Most prominent, at least amongst philosophers with a sympathy for fuzzy semantics, is Łukasiewicz’s infinitely-many-valued system, which is a

Introduction

15

generalization of his three-valued system.⁵ The following valuation rules for standard logical connectives are the same in both systems: (i) v(∼P) = 1 − v(P); (ii) v(P ∨ Q) = max[v(P), v(Q)] (i.e. disjunctions take the maximum of the values of their components); (iii) v(P & Q) = min[v(P), v(Q)] (i.e. conjunctions take the minimum of the values of their components); (iv) v(P → Q) = 1, if v(P) ≤ v(Q), otherwise v(P → Q) = 1 + v(Q) − v(P). The treatment of disjunction and conjunction can be generalized to the existential and the universal quantifier respectively: (v) For finitely many values, (∃x)() takes the maximum of all values of  relative to xassignments, for infinitely many values, it takes the lowest upper bound of all values. (vi) Correspondingly, universal quantifications take the minimum (for finitely many values) or the greatest lower bound (for infinitely many values). With this type of semantics in place, sorites arguments are rejectable as unsound—irrespective of the definition of entailment (whether it is defined as preservation of clear truth (value one), or as preservation of degree of truth in general). Take a sorites series for baldness. As predications take lower values progressively when we go down the series, starting from a clear case of baldness, all instances of tolerance where not both objects are clearly bald, or clearly lack baldness, will receive a lower value than one. That is, assuming that truth is identifiable by value one, some instances of tolerance are untrue. And since the greatest lower bound of all instances of tolerance is lower than one, so is the value of the tolerance principle—that is, it is untrue as well. Degree theories and epistemic theories of vagueness are usually presented as a dichotomy of approaches for the same type of explanatory problems in theory of vagueness, which it would be pointless to opt for in combination. John MacFarlane’s novel argument in favour of degree theories suggests that this common picture of the dialectics between degree theories and epistemic theories is misleading. On his account, we need degree theories for representing partial states of belief adequately. On the other hand, on his account, we also need an epistemic account of ordinary speakers’ ambivalence towards vague propositions. According to the classical conception, subjective probabilities can be modelled as expected truth values, with truth and falsity being the mutually exclusive and exhaustive possibilities of semantic valuation. In non-classical semantic frameworks, this conception of subjective probability requires reconsideration. Nick J. Smith’s suggests generalizing the classical conception for Łukasiewicz’s infinitely-many logic. As a consequence, as far as languages are concerned that involve vague sentences, his account suggests normative formal constraints on degree of belief that deviate from classical probability. Smith’s proposal to give a unified account of subjective probability, both for fully precise and vague languages, turns against recent suggestions that there are two kinds of degrees of belief, one type of degree of belief arising from uncertainty, and another one arising from vagueness. Some authors have found fuzzy semantics objectionable on the ground that tautologies need not receive the value one, nor—even worse—do contradictions need to receive the value zero. Peter Simons’s so-called ‘supernumeration’ approach may be ⁵ See Priest (2008: 224–27 and 457–61).

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seen as an attempt to have a fuzzy-like semantics where tautologies and contradictions are evaluated by the maximum and minimum value respectively. It combines elements of fuzzy semantics with elements of supervaluationism. From the former, it takes the idea of assigning numbers in the unit interval and of generalizing standard compositional constraints for connectives and quantifiers for a wider range of semantic values. From the latter, it takes the idea of a range of different valuations that respect penumbral connections. Graeme Forbes develops a fuzzy semantic framework for handling various sorts of puzzling cases each of which (he argues) implies that certain de re claims of identity are to be vague—that is, the thesis is that there is vagueness about certain pairs of objects and the concept of identity as to whether the latter applies to the former. On the suggested account, identity statements of pairs of objects may be nearly true but not totally true, and transitivity of identity is only a safe inference principle for the limiting case where only totally true identity statements are involved. Lastly, Mark Richard presents a way of modelling the apparent lack of sharp boundaries for vague predicates and the idea of higher-order vagueness in terms of a three-valued logic (specifically, Strong Kleene logic). He gives a thorough appraisal of some famous arguments against truth value gaps, such as the Dummett-Glanzberg argument from the norm of assertion, and Timothy Williamson’s argument from disquotational truth. According to Richard, denial is a sui generis speech act that is distinct from assertion, and logical connectives may contribute not only to truthconditional content but also to force.

9

H I G H E R - O R D E R VAG U E N E S S

Depending on how ‘first-order vagueness’ is defined, one may distinguish between different uses of the technical term ‘higher-order vagueness’. One way of defining first-order vagueness for a given predicate F  in a language is to say that it is firstorder vague just in case there is an object of which it is borderline whether it is an F . F  is then defined to be second-order vague just in case there are objects of which it is borderline whether it is borderline whether they are vague—and so on, for higher orders. Some authors have argued that this definition of higher-order vagueness is problematic in so far as it depends on the expressive means in the object-language we investigate. A different metalinguistic definition of first-order vagueness circumvenes this problem: a predicate F  in a given language is first-order vague just in case for some object it is borderline whether F  is true of it. Accordingly, a predicate F  is second-order vague just in case for some object, it is borderline whether ‘it is borderline whether F ’ is true of it—and so on, for higher orders. Diana Raffman makes a case against the idea of higher-order vagueness in both mentioned senses of the word. The idea of higher-order borderline cases is motivated by the idea that in making statements of first-order vagueness with regard to a given predicate F , such as Alice is a borderline case of tallness, we use more

Introduction

17

fine-grained classifications in addition to F . Raffman argues that this very underlying idea misdescribes our linguistic behaviour in sorites series. Against the alternative, metalinguistic notion of higher-order vagueness, she objects that it is in fact empty in extension, and that apparent cases of higher-order vagueness in this sense can be explained away. In effect, it is suggested that however we may plausibly put the hypothesis of higher-order vagueness, it is of no theoretical use, and also of no need. Crispin Wright suggests that the case for higher-order vagueness stands or falls insasmuch as higher-order vagueness provides a proper explanation for two basic intuitions about vagueness: for one, the ineradicability intuition, that is, the intuition that if you try to cover the borderline area between two polar opposite vague concepts by introducing a new concept you will in turn end up with a vague concept; for another, the seamless transition intuition, that is, the intuition that in considering a sorites series, we have the impression of a seamless transition from the beginning to the end. Wright shows that on weak provisos, the assumption of higherorder vagueness is not only hostage to paradox. He furthermore argues that even if the assumption of higher-order vagueness is coherent, it lacks any explanatory value regarding the seamless transition intuition. On Wright’s account, the idea of higherorder vagueness is wrongheaded in that it draws on the idea of borderline cases as involving a third status between two polar statuses. He submits that borderline cases are best understood as cases where our use of vague predicate lacks clear guidance. The notion of borderline vagueness is closely connected with the notion of definite truth. To say Alice is a borderline case of tallness comes to saying She is neither definitely tall, nor definitely not tall. The logic of definite truth thus bears on the logic of being a borderline vague, and thus also on the question of whether there are any higherorder borderline cases (in the sense of the first definition of higher-order vagueness, see above). Cian Dorr’s investigation focuses on the question of whether definite truth iterates: If it is definitely true that Alice is tall, does from that follow that it is definitely true that it is definitely true that she is tall? Suppose you attach another ‘definitely’ to definitely, Alice is true, and the same for the resulting sentence you obtain from this, and so on, so that you have countably many iterations of ‘definitely’ (that is, for every natural number n, the sentence begins with at least n iterations of ‘definitely’). If this sentence is true, we say it is ultratrue. Dorr puts forward an argument for the conclusion that there is no ultratrue sentence. Definite truth is modelled as applying to sentences relative to possible circumstances and communities of speakers. Dorr’s constraints on definite truth suggest that definitely fails iterating at a rather low level—probably just after five iterations. Re f e re n c e s Fara, D. G. (2000), ‘Shifting sands: an interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. Published under the name ‘Delia Graff ’. Fara, D. G. and Williamson, T., eds. (2002) Vagueness (International Research Library of Philosophy), Aldershot, Ashgate. Published under the name ‘Delia Graff ’.

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Keefe, R. (2000), Theories of Vagueness, Cambridge, Cambridge University Press. Keefe, R. and Smith, P., eds. (1996), Vagueness: A Reader, Cambridge (MA), MIT Press. Priest, G. (2008), An Introduction to Non-Classical Logic: From if to is, 2nd edn., Cambridge, Cambridge University Press. Williamson, T. (1994), Vagueness, London, Routledge.

PA RT I T H E N AT U R E O F VAG U E N E S S ?

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I What is Vagueness?

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1 A Metasemantic Account of Vagueness Agust´ın Rayo

1.0

THE PLAN

In this chapter I will defend what might be called a metasemantic account of vagueness.¹ The proposal divides naturally into three parts. The first will be developed in section 1.1, and is based on the following three theses: Conventionality The meaningfulness of our sentences depends on the prevalence of suitable linguistic conventions.² I wrote this chapter in 2003, and have since grown dissatisfied with its content. I am unhappy with a number of issues, but the main misgiving is this: conventionality-based conceptions of linguistic representation are not very plausible as accounts of linguistic competence (Laurence (1996)), and I have come to believe that an adequate account of vagueness must be closely linked to an explanation of what our ability to use language consists in. At the same time, I remain convinced of the basic insight of the Metasemantic Account: that the root of vagueness lies not in the type of semantic-value that is best associated with an expression, but in the type of linguistic practice that renders the expression meaningful. I have decided to publish this chapter in spite of its many deficiencies because it seems to me that the Metasemantic Account is underrepresented in the literature, and that the chapter presents a version of the account that is both intuitive and detailed enough to allow for serious assessment. I have deleted a section on mental content in order to satisfy a request from the editors for a shorter chapter, but refrained from making any other modifications to the original version of the paper. (For my current position on vagueness, see Rayo (2008).) I am grateful to audiences at the University of Bologna, the University of California at Irvine, Glasgow University, the Jagiellonian University, MIT, the Ohio State University at Columbus, Princeton University, the University of St Andrews, Warsaw University, and the University of Wisconsin at Madison. Special thanks are due to Jonathan Cohen, Juan Comesa˜na, Roy Cook, Richard Dietz, Patrick Greenough, Øystein Linnebo, Sebastiano Moruzzi, Graham Priest, Elliott Sober, Robbie Williams, Crispin Wright, and an anonymous referee. I gratefully acknowledge that most of this chapter was written during the tenure of a research fellowship at Arch´e, the AHRC Research Centre for the Philosophy of Logic, Language, Mathematics, and Mind. ¹ Related proposals are set forth in Black (1949b) and Dorr (2003). For criticism of Black, see Williamson (1994) §3.2. ² In claiming that language is conventional I do not wish to claim, for example, that speakers generally mean by their words whatever they intend to mean. Conventionality amounts to little more than the truism that words lack ‘natural’ meanings.

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Agust´ın Rayo

High degree of prevalence

Low degree of prevalence M1

M100,000

Figure 1.1 Prevalence of conventions about how to use the Mi (1 ≤ i ≤ 100, 000), where Mn is the sentence An event involving exactly n persons is multitudinous.

Degree Whether or not a convention prevails in a given population needn’t be an all-ornothing matter; it can come in degrees. Borderline Cases Whereas conventions about how to use sentences involving only attributions of vague predicates to clear cases are often prevalent to a very high degree, conventions about how to use sentences involving attributions of vague predicates to borderline cases are prevalent, if at all, to lesser degrees. (See Figure 1.1.) These theses leave open the question of what a semantics for vague terms should look like. The second part of the metasemantic account addresses this issue, and will be the focus of section 1.2. Once the semantic proposal is in place, a number of issues in pragmatics will need to be taken up. Doing so constitutes the third part of the metasemantic account, and will be the task of section 1.3.

1.1

M E TA S E M A N T I C S

1.1.1 Convention What is it for a convention to prevail among members of a given population? The metasemantic account of vagueness is to a large extent independent of the details of one’s answer to this question, but it will be useful to fix ideas by considering the account of convention set forth in Lewis (1969) and Lewis (1973).³ According to Lewis, a convention is a regularity of a certain kind. Specifically, a regularity R is a convention in a population P just in case the following conditions obtain within P: 1. There is general (or almost general) conformance to R. ³ Lewis’s work on convention is criticized in Schiffer (1972), Burge (1975), Jamieson (1975), Grandy (1977), Schiffer (1987) and Gilbert (1989). Rival account of convention—on the basis of which the metasemantic account of convention could also be defended—are set forth in Schiffer (1972) and Gilbert (1989).

A Metasemantic Account of Vagueness

25

2. Everyone (or almost everyone) believes that there is general (or almost general) conformance to R. 3. The belief that there is general (or almost general) conformance to R gives everyone (or almost everyone) good and decisive reason to conform to R herself. 4. There is a general (or almost general) preference for general conformity to R rather than slightly-less-than-general conformity. 5. R is not the only possible regularity meeting conditions 3 and 4. 6. It is a matter of common knowledge (or almost a matter of common knowledge) that conditions 1–5 obtain. Consider an example. The regularity of driving on the left-hand side of the road counts as a convention among drivers on the British Isles. This is because they meet the following conditions: (1) almost everyone drives on the left-hand-side of the road most of the time; (2) almost everyone believes that almost everyone drives on the left-hand side of the road most of the time; (3) the belief that almost everyone drives on the left-hand side of the road most of the time gives almost everyone good and decisive reason to drive herself on the left-hand-side of the road (since driving on the right-hand-side of the road is extremely dangerous when others are driving on the left-hand-side, and almost everyone is adverse to dangerous driving); (4) there is a near general preference for everyone’s driving on the left-hand-side of the road rather than almost everyone’s doing so (since even a slight non-conformance to the regularity of driving on the left-hand-side of the road makes driving in Britain much more dangerous, and almost everyone is adverse to dangerous driving); (5) driving on the left-hand-side of the road is not the only regularity satisfying (3) and (4) (since the regularity of driving on the right-hand-side of the road would have done just as well); and (6) it is a matter of common knowledge (or almost a matter of common knowledge) that (1)–(5) obtain.

1.1.2 Convention as a matter of degree On one of the hills surrounding the football stadium there is a small village. Whether a great many people attend the daily football match is important to villagers. It is important to village drivers because traffic becomes unbearable. It is important to village bar-owners because, after well-attended matches, football fans are drawn in great numbers to the village for drinks. It is important to villagers normally uninterested in football because they might be interested in running down to the stadium to share the excitement. It is important to villagers who like football but dislike crowds because they might consider watching the match on television rather than going to the stadium. And so forth. As a result, an interesting practice has taken hold in the village. A single villager—a different one each day—phones the stadium box-office early in the morning. After learning how many of the available 10,000 tickets have been sold for the day’s match, he or she decides whether to walk to the center of town and raise the village flag.

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Frequency of flag-raising

0% 0

tickets sold

10,000

Figure 1.2 Correlation between flag-raising and box-office sales.

The practice emerged gradually and informally, and no explicit rule about when to raise the flag was ever set forth. Nonetheless, there is a clear correlation between flagraising and box-office sales. When box-office sales are high, the flag is nearly always raised; when box-office sales are low, the flag is nearly never raised; when box-office sales are middling, the flag is sometimes raised and sometimes not. More specifically, the correlation is as represented by Figure 1.2. Say that box-office sales are ‘high’ if between 9,000 and 10,000 tickets were sold. Then regularity of raising the flag when box-office sales are high constitutes a convention among villagers, in Lewis’s sense. For: (1) almost every villager raises the flag almost every time box-office sales are high (villagers are sometimes absent minded, and sometimes engage in pranks); (2) almost every villager believes that almost every villager raises the flag almost every time box-office sales are high; (3) the belief that almost every villager raises the flag almost every time box-office sales are high gives nearly every villager good and decisive reason to herself raise the flag when box-office sales are high (unless they are engaged in pranks, villagers are eager to avoid misleading others); (4) there is a near general preference for every villager’s raising the flag when box-office sales are high rather than nearly everyone’s doing so (since nonconformance to the regularity is likely to mislead villagers); (5) raising the flag when box-office sales are high is not the only regularity satisfying (3) and (4) (ringing the village bell would have done just as well); and (6) it is a matter of common knowledge that (1)–(5) obtain. (Similarly, the regularity of refraining from raising the flag when box-office sales are low constitutes a convention among villagers.) Now consider the following question: Is it a convention among villagers to raise the flag when box-office sales are between 8,000 and 9,000? What about sales between 7,000 and 8,000, or between 6,000 and 7,000, or between 5,000 and 6,000? As we consider lower and lower ranges of box-office sales, there is less and less of a flagraising regularity in place, and less of a widespread belief in the presence of a flagraising regularity. When does it stop being a convention to raise the flag? It is hard to give a principled answer to these questions. For, on the definition above, ‘convention’ inherits the vagueness of occurrences of ‘almost’ in clauses 1–6. Fortunately, as Lewis himself observes, it is possible to substitute a parameter for each (extensional) occurrence of ‘almost’ in his definition: 1 . Conformance to R takes place x1 percent of the time.

A Metasemantic Account of Vagueness

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2 . x2 percent of the population believes that there is general (or almost general) conformance to R.  3 . The belief that there is general (or almost general) conformity to R gives x3 percent of the population good and decisive reason to conform to R herself.  4 . x4 percent of the population prefers general conformity to R rather than slightlyless-than-general conformity.  5 . R is not the only possible regularity meeting conditions 3 and 4 . 6 . It is a matter of common knowledge (to degree x6 ) that conditions 1 –5 obtain.⁴ One could also speak of degrees of belief and preference, but I will avoid doing so here for the sake of brevity. We can now say that a regularity R is a convention to degree x1 , x2 , x3 , x4 , x6 just in case conditions 1 –6 obtain within P. This puts us in a position to give principled answers to questions like ‘To what degree is there a convention among villagers to raise the flag when box-office sales are between 6,000 and 7,000?’ For, on the assumption that we have enough information about the village (and, in particular, enough information about the beliefs and desires of villagers), we will be in a position to say that a convention prevails to degree x , for some particular vector x . Of course, establishing that a convention prevails to degree x won’t always answer the question of whether the convention prevails simpliciter, in the original sense of ‘convention’. But it is important to note the the two senses of ‘convention’ are not in competition. Each of them can fruitfully be used to address a different range of issues. Here we will be concerned with the notion of conventionality to a degree. It should now be clear why I think that the second of the three theses I introduced in section 1.0 is true: Degree Whether or not a convention prevails in a given population needn’t be an all-ornothing matter; it can come in degrees. In the case of the village and the stadium, for example, raising the flag when box-office sales are between 9,000 and 10,000 is a convention to a very high degree, raising the flag when box-office sales are between 6,000 and 7,000 is a convention to a lesser degree, and raising the flag when box-office sales are between 0 and 1,000 is a convention to a very low degree.⁵

1.1.3 Meaning-fixation Let us now turn to the first of the three theses introduced in section 1.0: ⁴ There are different ways in which one could assign a degree of common knowledge to a population, but the details matter little for present purposes. ⁵ Since degrees of conventionality are n-tuples of real numbers rather than real numbers, something needs to be said about what makes one degree greater than another. The details matter little  for present purposes, but  to fix ideas one might say that x1 , . . . , xn ≤ y1 , . . . , yn iff (α1 x1 )2 + . . . + (αn xn )2 ≤ (α1 y1 )2 + . . . + (αn yn )2 (for suitable weightings α1 , . . . , αn ).

28

Agust´ın Rayo Conventionality The meaningfulness of our sentences depends on the prevalence of suitable linguistic conventions.

What is a ‘suitable linguistic convention’? As before, the metasemantic account of vagueness is to a large extent independent of one’s answer to this question. But it will be useful to fix ideas by considering a slight modification of the account of meaningfixation set forth in Lewis (1973). Assume, as Lewis does, that propositions are sets of possible worlds. The suggestion is that a sentence s is used by population P to mean∗ a proposition p just in case it is a convention in P to act (or believe) in accordance with the following principles:⁶ 1. Truthfulness Assent to an utterance of s only if you believe p; dissent⁷ from an utterance of s only if you believe p. 2. Trust Acquire the belief that p when others assent to an utterance of s; acquire the belief that p when others dissent from an utterance of s. (where p is the set-theoretic complement of p; I use ‘mean∗ ’ rather than ‘mean’ so it is clear that I do not wish to claim that a sentence s expresses a proposition p, as used by P, just in case s is used by P to mean∗ p). Thus, ‘Snow is white’ is used to mean∗ that snow is white if (a) there prevails a convention to assent to utterances of ‘Snow is white’ only if one believes that snow is white and dissent from utterances of ‘Snow is white’ only if one believes it is not the case that snow is white, and (b) to acquire the belief that snow is white when others assent to utterances of ‘Snow is white’ and acquire the belief that it is not the case that snow is white when others dissent from utterances of ‘Snow is white’. And, of course, one can say that s is used to degree x by P to mean∗ p just in case the relevant convention prevails in P to degree x .⁸ ⁶ I assume for the sake of simplicity that s is devoid of context-sensitivity and ambiguity. ⁷ I take dissent into consideration in order to avoid the conclusion that, if s is used by P to mean∗ p, then s is used by P to mean q whenever belief in p necessitates belief in q. A more complete characterization of meaning∗ would presumably take into account speech-acts such as supposition in addition to assent and dissent. ⁸ It is worth emphasizing that the present account of meaning-fixation is compatible with semantic externalism. Let speakers on Earth and Twin-Earth display identical linguistic behavior. There will still be a difference in what ‘water is wet’ is used to mean∗ on Earth and Twin-Earth, provided there are suitable differences in the contents of the beliefs of Earthlings and TwinEarthlings. Similarly, there will be a difference between what sentences involving the term ‘mass’ are actually used to mean∗ by English speakers, and what they would have been used to mean∗ by English speakers had Newtonian physics been correct, provided there would have been suitable differences in the contents of our beliefs had Newtonian physics been correct. Could there be meaning∗ in a population consisting of a single member? There could, provided the speaker’s practice gives way to regularities over time which constitute appropriate conventions of truthfulness and trust.

A Metasemantic Account of Vagueness

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The Lewisian account of meaning-fixation can be expected to deliver the following result: for a wide range of cases, s is used by English speakers to mean∗ p, where p captures the truth-conditions which are intuitively associated with s.⁹

1.1.4 Vagueness For each n between 1 and 100,000, let Mn be the sentence An event involving exactly n persons is multitudinous, and assume that the space of possible worlds under consideration is restricted to worlds in which the actual linguistic conventions prevail. Then, to a very high degree, M1 is used among English speakers to mean∗ the empty proposition.¹⁰ We get the same result for M2 and M3 : when attention is restricted to worlds in which the actual linguistic conventions prevail, M2 and M3 are, to a very high degree, used by English speakers to mean∗ the empty proposition. As k gets larger, however, Mk is used among English speakers to mean∗ the empty proposition to a lesser and lesser degree. For, e.g. the regularity to refrain from assenting to utterances of Mk prevails to a lesser and lesser degree, and the belief that there is general (or almost general) conformance to this regularity prevails to a lesser and lesser degree. ⁹ See, however, footnote 16. ¹⁰ This is because the following conditions obtain: 1. Truthfulness (assent) English speakers nearly always conform to the regularity of refraining from assenting to utterances of M1 . And their conformity is independent of what they believe, as long as they believe that actual linguistic conventions prevail. In addition, English speakers nearly always believe that there is general (or almost general) conformance to this regularity. [And so forth.] 2. Truthfulness (dissent) It is trivially true that English speakers conform to the regularity of dissenting from utterances of M1 only if they believe the set-theoretic complement of the empty proposition. In addition, English speakers nearly always believe that there is general conformance to this regularity [And so forth.] 3. Trust (assent) If someone assents to an utterance of M1 , it is to be expected that others will be led to revise their beliefs about whether the assent was serious, or about the utterer’s linguistic competence, or about their own linguistic competence. In other words, English speakers nearly always conform to the regularity of opting out when someone assents to an utterance of M1 : they opt out of the assumption that it is a case of genuine assent, or opt out of the assumption that the space of possible worlds under consideration is restricted to those in which the actual linguistic conventions prevail. In addition, English speakers nearly always believe that there is general (or almost general) conformance to this regularity [And so forth.] 4. Trust (dissent) It is trivially true that English speakers nearly always conform to the following regularity: if someone dissents from an utterance of M1 , then others will acquire the belief that the the set-theoretic complement of the empty proposition obtains (or, in this case, retain that belief ). In addition, English speakers nearly always believe that there is general conformance to this regularity [And so forth.]

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High degree

Low degree M1

Figure 1.3 Degree to which English speakers use Mk to

M100,000

mean∗

the empty proposition.

High degree

Low degree M1

M100,000

Figure 1.4 Degree to which English speakers use Mk to mean∗ the universal proposition.

By the time k is close to 100,000, Mk is used among English speakers to mean∗ the empty proposition to an extremely low degree, if at all. The situation is represented by Figure 1.3. Similarly, when attention is restricted to worlds in which the actual linguistic conventions prevail, M100,000 is, to a very high degree, used among English speakers to mean∗ the universal proposition. The same is true for M99,999 and M99,998 . As k gets smaller, however, Mk is used among English speakers to mean∗ the universal proposition to a lesser and lesser degree. By the time k is close to 1, Mk is used among English speakers to mean∗ the universal proposition to an extremely low degree, if at all. The situation is represented by Figure 1.4. It should now be clear why I think that the third of the three theses introduced in section 1.0 is true: Borderline Cases Whereas conventions about how to use sentences involving only attributions of vague predicates to clear cases are often prevalent to a very high degree, conventions about how to use sentences involving attributions of vague predicates to borderline cases are prevalent, if at all, to lesser degrees. For values of k such that events involving exactly k persons are clear cases of multitudinousness or non-multitudinousness, Mk is to a very high degree used by English speakers to mean∗ a proposition. But for values of k such that events involving exactly k persons are borderline cases of multitudinousness, Mk is used to lesser degrees, if at

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31

High degree

Low degree M100,000

M1

Figure 1.5 Degree to which English speakers use Mk to mean∗ a proposition.

all, by English speakers to mean∗ a proposition. The situation is as represented by Figure 1.5.¹¹ This concludes my defense of the three theses introduced in section 1.0. The next task is to say something about what form a compositional semantics for vague terms might take.

1.2

SEMANTICS

1.2.1 Degrees of truth? One might be tempted to think that the preceding section leads to a degrees-of-truth account of vagueness. In particular, one might think that the ‘degree of truth’ of a sentence should be identified with the degree to which the sentence is used to mean∗ a true proposition in the relevant linguistic community. That would be a mistake. To begin with, the choice of terminology is unfortunate. For talk of a sentence’s ‘degree of truth’ fails to distinguish between the claim that it is a matter of degree whether a sentence is used to mean∗ a given proposition (true or otherwise) and the claim that it is a matter of degree whether a proposition is true. The present proposal holds the former as a basic tenet, but has nothing to do with the latter. More importantly, it is no part of the present proposal that the degree to which a logically complex sentence is used to mean∗ a true proposition is a function of the degrees to which its components are used to mean∗ a true proposition. (Conspicuously, it is not the case that a sentence is used to mean∗ a true proposition to degree x just in case its negation is used to mean∗ a true proposition to degree 1 − x .¹² Quite the contrary: it is crucial for Borderline Cases that sentences involving attributions of vague predicates to borderline cases not have this feature.) But the view that the ¹¹ As before, I assume that attention is restricted to worlds in which the actual linguistic conventions prevail. I also assume uses among English speakers of one of the Mk to mean∗ anything other than the empty proposition or the universal proposition are of low enough degree to be negligible. ¹² Let 1 be the maximum degree, 0 be the minimum degree, and x1 , . . . , xn − y1 , . . . , yn be x1 − y1 , . . . , xn − yn .

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‘degree of truth’ of a logically complex sentence is a function of the ‘degrees of truth’ of its components is a central assumption of standard degrees-of-truth accounts of vagueness. Actually, the very idea of directly reading-off the truth-status of sentences from the degrees to which they are used to mean∗ true propositions is hopeless. Among other things,¹³ no sentence too complex to be parsed by humans can be expected to be used to mean∗ a proposition to any significant degree. In order to properly assess the truthstatus of such sentences, what one needs to do is ascribe a compositional semantics to the relevant linguistic community on the basis of information pertaining to sentences which are used to mean∗ propositions to significant degrees.

1.2.2 The absolute-semantics picture Let me fix some terminology. To assign truth-conditions to a sentence is to specify what the world must be like in order for the sentence to be true.¹⁴ A (compositional) semantics is a (compositional) assignment of truth-conditions to sentences. A classical semantics is a compositional semantics in the style of Lewis (1970).¹⁵ If C is a classical semantics, C(s) is the set of possible worlds in which the truth-conditions which C assigns to s are satisfied; C is vindicated in P with respect to s (to at least degree x ) just in case s is used by P to mean∗ C(s) (to at least degree x ). According to the Absolute-Semantics Picture —as I shall call it—the ascription of a classical semantics C to a given population is subject to the following constraint: C should only be ascribed to P if it is vindicated by P to a sufficiently large degree with respect to an appropriate range of sentences.¹⁶ ¹³ See also footnote 16. ¹⁴ I do not presuppose that speakers must somehow have cognitive access to such specification. ¹⁵ For sentences of the form P(a) the idea is roughly as follows. The semantic value of the name a is a function taking each centered-world w to an individual in w; the semantic value of the predicate P is a function taking each centered-world w to a set of individuals. P(a) is true relative to a centered-world w just in case the individual assigned to w by the semantic value of a is a member of the set assigned to w by the semantic value of P. An utterance u of P(a) is true simpliciter just in case P(a) is true relative to the actual world centered on u. Use of centered-worlds will be omitted in what follows for the sake of simplicity. ¹⁶ I have ignored an important qualification. As Stalnaker (1979) has emphasized, the proposition communicated by an assertion is not always the proposition expressed by the asserted sentence. In general, the proposition communicated by an assertion of s is the diagonal —the set of (contextually relevant) worlds w such that ‘s is true’ is true-in-w. One should therefore expect meaning∗ to track diagonals, rather than tracking propositions expressed (as the notion of vindication defined above presupposes). One way of addressing the problem is by enriching the notion of meaning∗ : instead of taking meaning∗ to relate a sentence to a proposition, one could take it to relate a sentence to a function f from possible worlds to propositions. At a first approximation, meaning∗ might then be characterized as follows: sentence s is used by population P to mean∗ function f just in case (1) it is a convention in P to act (or believe) in accordance with the following two principles: (a) assent to an utterance of s only if you believe the proposition consisting of worlds w such that w ∈ f (w) (and correspondingly for dissent), and (b) acquire a belief in the proposition consisting of worlds w such that w ∈ f (w) when others assent to an utterance of s (and correspondingly for dissent); and (2) if s contains no counterfactual conditionals, c is a counterfactual conditional of the form s → s and s is used to mean∗ f  , then c is used to mean∗ g, where g(w) is the set of worlds w

A Metasemantic Account of Vagueness

33

I will argue in the following section that, in spite of its initial plausibility, the Absolute-Semantics Picture must be given up.

1.2.3 A semantics for vague terms For i a natural number, let the Ci be a family of classical semantics. They all agree about the semantic lexicon from which sentences are to be built and about the particular semantic structure that is to be assigned to each sentence. They also agree about which semantic values should be assigned to each item in the semantic lexicon, with one important exception: for each n, Cn assigns to ‘multitudinous’ the function taking each world w to the set of events in w involving at least n persons. (Assume, in addition, that the meaning∗ of the Mn in P is as in section 1.1.4, and that relevant lexical items are assigned ‘homophonic’ semantic values by the Ci .)¹⁷ As long as ascriptions of semantic theories are understood in accordance with the Absolute-Semantics Picture, no individual Ci can provide us with a fully accurate description of P’s linguistic practice. Suppose, for example that C70,000 is ascribed to P. Then, by the lights of the Absolute-Semantics Picture, one should expect M0 through M69,999 to be used by P to mean∗ the universal proposition to a very low degree, and M70,000 through M100,000 to be used by P to mean∗ the universal proposition to a very high degree; and this is only a rough approximation of the truth (see Figure 1.6). Ascribing a more complex semantics to P does not, by itself, constitute significant progress. Little is gained, for example, by ascribing a range of Ci to P, rather than an individual Ci . Suppose, for example, that C40,000 through C80,000 are ascribed to P. Then, by the lights of the most natural way of extending the Absolute-Semantics Picture to range-ascriptions, one should expect M0 through M39,999 to be used by P to mean∗ the universal proposition to a very low degree, M40,000 through M79,999 to such that some world in the intersection of f  (w) and f (w) is ‘closer’ (in some appropriate sense) to w than any world in the intersection of f  (w) and the complement of f (w). (Here I assume that the language in question is rich enough to contain counterfactual conditionals, and that there is some satisfactory way of picking them out.) The notion of vindication can then be redefined as follows: C is vindicated in P with respect to s (to at least degree x ) just in case s is used by P to mean∗ f (to at least degree x ), where C(s) is the result of applying f to the actual world. ¹⁷ More specifically, we shall assume that the following is true for every i ∈ N: (a) Ci assigns to each of the Mn the semantic structure ∀x[(Event(x) ∧ ∃!n y(Person(y) ∧ Involved(x, y))) → Multitudinous(x)] (with standard definitional equivalences); (b) the semantic value of ‘Event’ according to Ci is the function taking each world w to the set of events in w; (c) the semantic value of ‘Person’ according to Ci is the function taking each world w to the set of persons in w; (d ) the semantic value of ‘Involved’ according to Ci is the function taking each world w to the set of ordered pairs x, y for x an event in w and y involved in x in w; (e) the semantic value of ‘=’ according to Ci is the function taking each world w to the set of ordered pairs x, x ; (f ) the semantic value of ‘∃x’ according to Ci is the function taking each world w to the set of non-empty sets of objects in w; (g) the semantic value of ‘¬’ according to Ci is the function taking each set of worlds to its complement; and (h) the semantic value of ‘∧’ according to Ci is the function taking each pair of sets of possible worlds to their intersection.

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Agust´ın Rayo Actual Practice

C70,000’s Prediction

High degree

Low degree M 70,000

M1

M100,000

mean∗

Figure 1.6 Degree to which Mk is actually used to the universal proposition in P vs. degree to which Mk should be used to mean∗ the universal proposition in P according to C70,000 . Range Prediction

Actual Practice High degree

Low degree M1

M40,000

M80,000

M100,000

∗

Figure 1.7 Degree to which Mk is actually used to mean the universal proposition in P vs. degree to which Mk should be used to mean∗ the universal proposition in P according to a range ascription consisting of C40,000 through C80,000 .

be used by P to mean∗ the universal proposition to a middling degree, and M80,000 through M100,000 to be used by P to mean∗ the universal proposition to a very high degree. And, again, this is only a rough approximation of the truth (see Figure 1.7). In order for genuine progress to be made, the Absolute-Semantics Picture must be given up. Rather than attempting to describe P’s linguistic practice by ascribing a particular Ci to P and insisting that the ascription satisfies the constraint imposed by the Absolute-Semantics Picture, one should set forth a method M for gauging how apt a description of meaning∗ in P each of the Ci is, and describe P’s linguistic practice by saying of one or more of the Ci how apt it is by the lights of M. It is important that M be chosen in such a way that it is clear what predictions about meaning∗ one is entitled to make on the basis of an assignment of a particular degree of aptness to a classical semantics. But as long as that constraint is met, the choice of M can be made on the basis of the particular needs of semanticists. A method M might measure aptness in two different ways. On the one hand, M might measure the aptness of a semantics relative to a given expression. To fix ideas,

A Metasemantic Account of Vagueness Comparative aptness

35 Degree of meaning*

Good approx. / High degree

Bad approx. / Low degree C1 / M1

C100,000 / M100,000

Figure 1.8 Comparative aptness of the Ci (relative to ‘multitudinous’) / Degree to which Mk is used to mean∗ the universal proposition in P.

think of the degrees of aptness of the Ci relative to ‘multitudinous’ as given by the first derivative of the curve in Figure 1.4 (see Figure 1.8).¹⁸ On the other hand, M might deliver an overall degree of aptness for a given semantics. To fix ideas, think of the overall degree of aptness of Ci as a weighted average of Ci ’s degrees of aptness relative to a suitable class of expressions and the degrees to which Ci is vindicated (in the sense of section 1.2.2) with respect to a suitable range of sentences. Here is one example of a semantics for vague terms which is in accordance with the new approach. Say that a weighted semantic space is a pair S, W such that: (a) S is a set of classical semantics, all of which agree about a semantic lexicon from which sentences are to be built, and about the particular semantic structure that is to be assigned to each sentence; and (b) W is a function assigning a degree δα to each Cα and a degree δαE to each pair Cα , E (where Cα ∈ S and E is an expression). Assume, moreover, that M is an agreed method for gauging the degrees of aptness of the Cα . The proposal is then that a semantics for vague terms is a weighted semantic space, and that one is justified in ascribing a space S, W to a population P only if the degrees assigned by W are the degrees to which the members of S approximate the truth about meaning∗ in P, by the lights of M. Here is another example of a semantics for vague terms which is in accordance with the new approach. Assume, as before, that M is an agreed method for gauging the degrees of aptness of the Cα . Let a semantics for vague terms be a classical semantics, and say that one is justified in ascribing a classical semantics Cα to the relevant population just in case Cα is an ‘optimal’ approximation of the truth about meaning∗ in that population, by the lights of M. (One could say, for example, that the semantics ¹⁸ More precisely, assume that x has m components, and let xj be the jth component of x . For j ≤ m, let fj : N → [0, 1] be such that fj (k) = x j , where x is the degree to which Mk is used by P to mean∗ the universal proposition, and let gj : R → [0, 1] be the error function best approximating fj . Say that the degree of approximation of Cn is g1 (n), . . . , gm (n) . If degrees of approximation need to be ordered, one can use the method described in footnote 5. Relative to expressions such as ‘intelligent’, which have multiple dimensions of vagueness, one would need the degree of aptness of Ci to be an n-tuple with different components corresponding to different dimensions.

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which is optimal among the Ci with respect to ‘multitudinous’ is whichever comes closest to the point in which the second derivative of the curve in Figure 1.4 is zero.) From the present perspective, the ascription of Cα to P should lead one to expect that Cα is an optimal description of meaning∗ by the lights of M, but not that Cα is a correct description of P’s linguistic practice by the lights of the Absolute-Semantics Picture. Different types of semantics for vague terms will capture different amounts of information about meaning∗ , and will be more or less succinct. The first of the two proposals suggested above is capable of delivering more information about meaning∗ than the second, but is less succinct. Fortunately, the proposals are not in competition. When properly understood, they both deliver accurate descriptions of the linguistic practice of the relevant population. One can therefore make use of whichever is best suited to one’s theoretical needs.

1.2.4 Supervenience Any account of vagueness should allow for an explanation of how it is that the semantic facts supervene on the non-semantic facts. From our present perspective, the semantic facts pertaining to a population P are facts about meaning∗ in P and facts about which semantics should be ascribed to P. The non-semantic facts include facts about the beliefs and desires of members of P, facts about the regularities that members of P conform to, facts about the environmental conditions under which members of P operate, and so forth. According to the account of meaning-fixation set forth in section 1.1.3, a sentence s is used by P to mean∗ p (to degree x ) just in case an appropriate convention of truthfulness and trust prevails in P (to degree x ). So, thanks to Lewis’s account of convention, one has an explanation of how it is that the facts about meaning∗ in P supervene on facts about the beliefs and desires of members of P, facts about the regularities that members of P conform to, and so forth. On the present proposal, one should provide a semantics for vague terms by setting forth a method M for gauging how apt a description of meaning∗ in P a given classical semantics is, and go on to say of one or more classical semantics how apt it is by the lights of M. So, for any choice of M which makes clear how the degree of aptness of a classical semantics is to be determined on the basis of facts about meaning∗ , one will have an explanation of how it is that facts about which semantics should be ascribed to P supervene on facts about meaning∗ in P.

1.2.5 Characterizing vagueness An answer to the question of what it is about vague terms that makes them vague flows naturally from the metasemantic account. Let S, W be a weighted semantic space describing the linguistic practice of English speakers, and assume that S consists of a large and diverse family of classical semantics. One can then say that an expression E is vague to the extent that there is a range of semantics in S which disagree about the semantic value of E along some appropriate parameter and to which W assigns a

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significantly diverse but smoothly distributed range of degrees of aptness (relative to E). In particular, one can say that ‘multitudinous’ is vague because there is a range of semantics in S which disagree about the minimum number of persons an event must involve if it is to fall under ‘multitudinous’, and to which W assigns a significantly diverse but smoothly distributed range of degrees of aptness (relative to ‘multitudinous’). I would like to mention a virtue of the proposed characterization of vagueness. Because no appeal has been made to the notion of definiteness or its variants, there is no need to supplement the proposal with a characterization of higher-order vagueness.¹⁹ The full depth of the notion of vagueness is captured from the start.

1.2.6 Linguistic competence Suppose that S, W is a weighted semantic space describing the linguistic practice of English speakers. Should one conclude that the linguistic competence of English speakers consists of (implicit or explicit) knowledge of S, W ? I would like to suggest not. It is sufficient for basic linguistic competence to know of some classical semantics that it is a rough approximation of the truth. For no more is required to be in a position to exercise reasonable conformance to conventions of truthfulness and trust prevailing in the relevant linguistic community.²⁰ To be minimally competent in the use of ‘multitudinous’, for example, it is sufficient to know of, e.g. C70,000 that it is at least a rough approximation of the truth. If one knows, in addition, that none of the Ci is a perfect approximation (and that the different degrees of aptness vary along the lines described in section 1.2.5), then one also knows that ‘multitudinous’ is vague. So one has more than minimal competence in the use of the term.

1.2.7

Truth

Suppose M is an agreed method for gauging the aptness of classical semantics, and imagine a perfect linguistic community: for any sentence s there is a proposition p such that speakers use s to mean∗ p to degree 1 (and use s to mean∗ any other proposition to degree 0 ). If all goes well, one can expect to find a classical semantics which is a perfectly apt description of the relevant linguistic practice by the lights of M. For better or worse, English speakers do not constitute a perfect linguistic community: single sentences with simple enough syntactic structures are used to mean∗ different propositions to different degrees. If the description in section 1.1.4 is accurate, for example, then M80,000 is used both to mean∗ the universal proposition to a high degree and to mean∗ the empty proposition to a low degree. It is therefore unreasonable to expect to find a classical semantics which is perfectly apt by the lights of M. Instead, one can expect to find a family of classical semantics to which M assigns ¹⁹ For a detailed characterization of higher-order vagueness, see Williamson (1999). ²⁰ Provided, of course, that one is competent with the workings of pragmatics.

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varying degrees of non-perfect aptness (and, perhaps, a classical semantics to which M assigns ‘optimal’, though non-perfect, aptness). When it comes to a perfect linguistic community, one can speak of the truthstatus of any given sentence with no further ado: if C ∗ is the classical semantics which describes the linguistic practice with perfect aptness (by the lights of M), one can say that s is true, as used by members of the perfect linguistic community, just in case C ∗ (s) contains the actual world as a member. Not so for the imperfect community of English speakers. Even if C ∗ turned out to be an ‘optimally apt’ description of the practice of English speakers (by the lights of M), it would be misleading to say that a sentence s is true, as used by English speakers, just in case C ∗ (s) contains the actual world as a member. For by doing so one would ignore the differences between the community of English speakers and a perfect linguistic community. It is better to say that a sentence s is true with respect to a classical semantics which is optimally apt just in case C ∗ (s) contains the actual world as a member. More generally, if S, W is a weighted semantic space describing the practice of English speakers, one can say that a sentence s is true with respect to a classical semantics which is apt to degree δ just in case some Cα ∈ S is such that W(Cα ) = δ and Cα (s) contains the actual world as a member. (Similarly, one can say that s expresses proposition p with respect to a classical semantics which is apt to degree δ just in case some Cα ∈ S is such that W(Cα ) = δ.) I am not, however, suggesting that the notion of truth simpliciter be forgone. I will argue in section 1.3.2 that, even if an English sentence cannot be said to be true or false simpliciter, it can be used to make assertions which are true or false simpliciter. Before bringing this section to a close, I would like to address some potential sources of concern: 1. Couldn’t one give a supervaluational characterization of truth simpliciter? In particular, if S, W is a weighted semantic space describing the practice of English speakers, couldn’t one say that a sentence s is true simpliciter just in case Cα (s) contains the actual world as a member for every Cα ∈ S? It is unclear that much would be gained by setting forth such a definition. For different sentences would be counted as true simpliciter on different choices of S. But the question of which semantics to include in S is to be answered on the basis of the theoretical needs of semanticists, not on the basis of information about the linguistic phenomenon under study. This is because by removing a semantics from S, or adding a semantics to S and extending W in accordance with M, one does not alter the accuracy with which meaning∗ is described by S, W ; what changes is the amount of information delivered. (And, of course, when S contains every classical semantics which is apt to some positive degree, one should expect to get the unwelcome result that a sentence like M100,000 is not counted as true simpliciter, since M100,000 is likely to be false relative to some classical semantics which is apt to a positive—albeit very low—degree.) 2. Don’t we get a degrees-of-truth account of vagueness after all? For suppose S, W is a weighted semantic space describing the practice of English speakers, and assume that the Cα ∈ S are normalized, in the sense that the sum of the degrees

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of aptness assigned by W to the Cα ∈ S relative to any item in the shared semantic lexicon is 1 . Couldn’t one then say that the degree of truth of a sentence s is the sum of the degrees of aptness (relative to some lexical item occurring in s) of the Cα ∈ S such that Cα (s) contains the actual world? One could certainly speak of such ‘degrees of truth’ if one wished, but it would be misleading to call the resulting proposal a degrees-of-truth account of vagueness. For, to return to a theme from section 1.2.1, one will not generally get the result that the ‘degree of truth’ of a logically complex sentence is a function of the degrees of truth of its components, which is a central assumption of standard degrees-oftruth accounts of vagueness.

1.3

1.3.1

P R AG M AT I C S

The Stalnaker–Lewis picture

Here is a rough outline of the Stalnaker–Lewis picture of conversation.²¹ To each stage of a non-defective conversation corresponds a context-set: a set of possible worlds which are ‘open-possibilities’ as far as the presuppositions of participants at that stage in the conversation are concerned.²² As the conversation evolves, the context-set changes. In particular, the context-set changes when an assertion is made. One way in which it changes is by registering the fact that an assertion has been made, i.e. by shifting so as to contain only possible worlds in which the relevant assertion takes place. But there is another, more important, way in which an assertion can alter the contextset: if participants accept the assertion, then the context-set shifts so as to include only possible worlds which are compatible with the content of the assertion. This is the assertion’s essential effect. On this picture, one may take the content of an assertion to be a subset of the context-set on which the assertion has its essential effect: the subset consisting of those worlds which are left as ‘open possibilities’ in light of what has been asserted. In this section I will try to explain, from the perspective of the metasemantic account of vagueness, how it is that the content of a given assertion may be determined. The most straightforward proposal would be this: the content of an assertion of sentence s is the intersection of the context-set on which the assertion has its essential effect and the set of possible worlds in which the proposition expressed by s is true. Unfortunately, this suggestion is not available to proponents of the metasemantic account of vagueness. For, on the metasemantic account, it is not fully accurate to say of a sentence that it expresses a proposition simpliciter. One must say that it expresses a proposition relative to a given semantics (which is apt to a certain degree). ²¹ See Stalnaker (1999) (especially chapters 1–5) and Lewis (1979). ²² For some purposes, it is best to think of ‘open possibilities’ not as (maximally specific) possible worlds, but as the elements of a partition defined on a set of (maximally specific) possible worlds. I shall ignore this complication for expositional purposes.

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Fortunately, there is a different proposal available. Say that a semantics Cα selects A as the content of an assertion just in case A is the intersection of the context-set on which the assertion has its essential effect and the set of possible worlds in which the proposition expressed by the sentence asserted relative to Cα is true. The proposal is this: if among the sets of possible worlds selected as the content of an assertion by a sufficiently large range of semantics there is one which is salient to participants in the conversation, then that proposition is the content of the assertion; otherwise, the assertion is defective, and has no content. Here are a couple of examples: 1. An assertion of ‘John is bald’ is to have its essential effect on a context-set consisting of worlds of the following two sorts: α-worlds, in which John is a clear case of baldness, and β-worlds, in which John is a clear case of non-baldness. One can expect that any semantics which is apt to anything but the lowest of degrees (relative to ‘bald’) will select the set of α-worlds as the content of the assertion. This should make the set of α-worlds salient to participants in the conversation. Accordingly, the content of the assertion is the set of α-worlds. 2. An assertion of ‘John is bald’ is to have its essential effect on a context-set consisting of worlds of following two sorts: α-worlds, in which John has no hairs on his head, and β-worlds, in which John has n hairs on his head (appropriately distributed) for some n such that men with n hairs on their heads (appropriately distributed) are borderline cases of baldness. One can expect that any semantics which is apt to anything but the lowest of degrees (relative to ‘bald’) will select either of the following two propositions as the content of the assertion: (a) the set of α-worlds, or (b) the set of all worlds in the context-set. This should make the set of α-worlds salient to participants in the conversation, since accepting the set of all worlds in the context-set as the content of the assertion would make the assertion pointless. Accordingly, the content of the assertion is the set of β-worlds. So far so good. But now consider a conversation in which nothing special is presupposed about the abundance of John’s hair. The context-set might start out containing a world in which John has n hairs on his head (appropriately distributed), for each n between 0 and 150,000. If that is the context-set on which an assertion of ‘John is bald’ is to have its essential effect, then the assertion is likely to be defective. For one can expect that there will be a large number of rival propositions, each selected by a similarly broad range of semantics which are apt to similarly substantial degrees (relative to ‘bald’); so no one proposition will be salient to participants in the conversation. But how can this be right? Doesn’t everyday conversation supply examples of cases in which it is appropriate to assert that someone is bald even if one’s audience presupposes nothing about how much hair that person has? The way out of the problem is to note that, regardless of what the original contextset consists in, an assertion of ‘John is bald’ will not usually have its essential effect on a context set containing too many worlds. For everyday conversation is governed by what Lewis (1979) calls ‘rules of accommodation’, according to which the contextset should evolve so as to make it the case that no assertion is defective. Thus, one can expect the assertion in our example to have its essential effect on a modified

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context-set: the result of excluding as many worlds as necessary from the original context-set for participants in the conversation to regard a selected proposition as sufficiently salient, and thereby forestall defectiveness. This will normally mean excluding worlds in which John is a borderline case of baldness, so that we are left with a situation such as the one described in example 1 above.²³ ( Just which worlds are presupposed-away from the context-set is something it is up to each participant in the conversation to decide. If the modifications of different participants are close enough not to affect common goals, then the conversation may carry on without incident. Otherwise, there is a risk of defectiveness, and further adjustments may be necessary. See Stalnaker (1979).) It is worth emphasizing that defectiveness-averting modifications of the context-set may not always be possible (as when a discussion of the phenomenon of vagueness is salient). Our discussion of assertoric content has been developed on the basis of the Stalnaker–Lewis picture of conversation. But I hope that the plausibility of the following conclusions is relatively independent of some of the details of that picture: 1. Even if a sentence doesn’t express propositions simpliciter, but only relative to some semantics or other, an assertion of that sentence can enjoy a determinate content because the situations which are regarded as ‘open possibilities’ for the purposes of the assertion can be fairly limited. 2. An assertion of a sentence involving vague terms may fail to enjoy determinate content if—despite the presence of rules of accommodation—enough scenarios are regarded as ‘open possibilities’. If so, the assertion is pragmatically defective, and should be avoided.

1.3.2 Truth simpliciter I am now in a position to say why I think that a sentence can be used to make an assertion which is true or false simpliciter, even if the sentence itself cannot be said to be true or false simpliciter, but only true or false relative to a semantics (which is apt to a certain degree). To wit: when the assertion enjoys a determinate content, its truth value simpliciter may be identified with the truth value of its content. Since an argument’s logical status does not depend on the meaning of its nonlogical vocabulary, there is no need to give a pragmatic characterization of logical consequence: the standard model-theoretic characterization will do.²⁴ In particular, one gets the following welcome result. Let aφ and aψ be (non-defective) assertions of ²³ It is worth noting that this sort of story predicts that, in contexts where nothing special is presupposed about the abundance of John’s hair, one will only assert ‘John is bald’ if one takes John to be a clear case of baldness. For, if John is a borderline case of baldness, the context-set will exclude the actual world when it contracts to avert defectiveness, thereby falsifying the content of the assertion. ²⁴ Whether or not any English sentences turn out to be logically valid will depend on whether English particles such as ‘and’ and ‘not’ turn out to be precise terms expressing the logical connectives.

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φ and ψ, respectively, and assume that there is no difference in the context-sets on which they have their essential effects. If aφ is true simpliciter and φ  ψ, then aψ must also be true simpliciter.

1.3.3 The Sorites Paradox Here is a version of the Sorites Paradox as it arises with respect to the Mn : Each of the following appears to be true: 1. M100,000 2. ¬M1 3. ∀k(Mk+1 → Mk ) But, by iterated applications of universal instantiation and modus ponens, one can derive a contradiction from 1–3.²⁵ In order to address the paradox, I will proceed in two stages. First I will argue that the metasemantic account can explain why 1–3 appear to be true. Then I will show that a contradiction is nonetheless averted from the perspective of the metasemantic account. It is straightforward for a proponent of the metasemantic account to explain why M100,000 and ¬M1 appear to be true. For, according to the metasemantic account, typical assertions of M100,000 and ¬M1 are true (provided the conversation carries no false presuppositions). Indeed, in the absence of false presuppositions, the truth of such assertions is guaranteed by the relevant linguistic conventions, together with the norms governing the way in which an assertion acquires a content in a given context.²⁶ So all we need to explain the appearance of truth of M100,000 and ¬M1 is the assumption that speakers’ intuitions are suitably informed by their linguistic competence. The real challenge is explaining the appearance of truth of ∀k(Mk+1 → Mk ). I will proceed by offering an explanation of why it seems so natural to think that there could be no counterexample to ∀k(Mk+1 → Mk ). Here is the explanation. It is natural to suppose that, in order to produce a counterexample to ∀k(Mk+1 → Mk ), one must identify a true instance of the following: (∗) Event e is multitudinous, but event e is not; ²⁵ I trust the reader will forgive use/mention of infelicities in 2 and 3. ²⁶ This can easily be verified when propositions are assumed to be sets of possible worlds. For relative to all but a few compositional semantics (none of which is apt to anything but the lowest of degrees relative to ‘multitudinous’), M100,000 expresses the universal proposition. So, in all but the most devious of contexts, an assertion of M100,000 must have a true content—provided the context-set on which it is to have its essential effect is not marred by false presuppositions. (As Stalnaker has taught us, the information conveyed by such an assertion is unlikely to coincide with its content. The content of the assertion is utterly uninformative because it consists of every world in the context-set. So the information conveyed must be given by the diagonal, which excludes worlds in the context-set where ‘multitudinous’ is used non-standardly. See Stalnaker (1979).)

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(where it is presupposed that e involves precisely one more person than e). But it is a consequence of the metasemantic account that no such identification is possible. For, in any reasonable context, instances of (∗) will be unassertable as a matter of principle. The reason is this. If the discussion in section 1.3.1 is broadly along the right lines, an assertion can only escape defectiveness if the situations which are regarded as ‘open possibilities’ for the purposes of the assertion can be successfully classified by conversational participants into two distinct groups. But, when it comes to assertions involving ‘multitudinous’, our linguistic practice provides no more than rough guidance as to how such a classification is to be carried out (since ‘multitudinous’ is assigned no unique extension). This means that defectiveness can only be averted if some classification of the ‘open possibilities’ is salient enough to conversational participants that it can be singled out merely on the basis of the rough guidance that linguistic practice supplies. The standard way for a classification to be rendered salient in such cases is for the range of ‘open possibilities’ to be fairly limited, as illustrated in section 1.3.1.²⁷ But it becomes impossible to exclude the right sorts of ‘open possibilities’ when it comes to an assertion of (∗), for its conjuncts impose incompatible demands on the ‘open possibilities’ which are to be left out. When propositions are taken to be sets of possible worlds, the argument can be spelled-out in further detail. Let Ck be a semantics that places the cut-off point of ‘multitudinous’ at k. Then the content selected by Ck for (∗) can only contain worlds in which e involves precisely k persons and e involves precisely k + 1 persons, since these are the only worlds in which (∗) can be true, according to Ck . But this means that no two semantics which differ in the extension they assign to ‘multitudinous’ can select the same non-empty content for (∗). So no non-empty content can be rendered sufficiently salient to participants in the conversation. The assertion must therefore be defective.²⁸ So much for my explanation of why the Sorites premises appear to be true. How is it that a contradiction is nonetheless averted? From the perspective of the metasemantic account, the basic answer is straightforward: assertions of ∀k(Mk+1 → Mk ) are false in any reasonable context because, for 1 ≤ k ≤ 100, 000, ∀k(Mk+1 → Mk ) expresses a necessarily false proposition according to all but a few compositional semantics (none of which is apt to anything but the lowest of degrees relative to ‘multitudinous’). I would like to suggest, however, that the fact that typical assertions of ∀k(Mk+1 → Mk ) are false is not an especially illuminating piece of information about the semantic status of ∀k(Mk+1 → Mk ). It is more a reflection of the theoretical machinery we chose to describe the practice of speakers—i.e. classical semantics—than of the practice itself.²⁹ In order to say something illuminating about the semantic status of ²⁷ Given a sufficiently unusual context, there might be other ways. For instance, conversational participants might somehow presuppose that it is safe for 59,792 persons or less to get together, but unsafe for 59,793 persons to get together. ²⁸ Here I assume that the context-set on which (∗) is to have its essential effect contains no worlds that disagree about what our linguistic conventions are like. Otherwise, the assertion might be non-defective by having a determinate and non-empty diagonal content. ²⁹ It is worth emphasizing that, as far as the metasemantic account is concerned, one’s theoretical machinery needn’t be based on classical semantics. An approach based on non-classical semantics

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∀k(Mk+1 → Mk ) we need a finer instrument than truth simpliciter; we need to say something such as the following: For any choice of Ci meeting the constraints of section 1.2.3, the Ci display a smooth and monotone gradation of degrees of aptness relative to ‘multitudinous’, which go from very low to very high as i increases.

1.3.4

Contextualism

A virtue of the present proposal is that it allows us to explain why it is that the sorts of contextual phenomena that motivate contextualist accounts of vagueness take place.³⁰ Here are a couple of examples: 1. My only copy of War and Peace is colored a very light grayish-blue. In context A it is sitting among bright red books. When I assert ‘I would like you to bring me the blue book’, you bring me War and Peace. In context B my copy of War and Peace is sitting among bright cobalt-blue books. When I assert ‘I would like you to bring me the gray book’, you bring me War and Peace. Explanation: In context A the context-set contains only open-possibilities according to which I would like you to bring me a book which is in fact bright red and open-possibilities according to which I would like you to bring me a book which is in fact War and Peace. In context B the context-set contains only open-possibilities according to which I would like you to bring me a book which is in fact bright cobalt-blue and open-possibilities according I would like you to bring me a book which is in fact War and Peace. The explanation can then proceed as in example 2 from section 1.3.1. 2. When it is contextually salient that two men have similar amounts and distributions of hair on their head, speakers will refrain from asserting that one of them is bald and the other is not. Explanation: Parallel to the argument in section 1.3.3 for the conclusion that (∗) is unassertable.

1.4

C O N C LU S I O N S

I have argued for an account of vagueness according to which the root of vagueness lies not in the type of semantic-value that is best associated with an expression, but in the type of linguistic practice that renders the expression meaningful. I suggested, in particular, that conventions about how to use sentences involving attributions of might do, provided one is able to set forth a suitable method for gauging how apt a description of meaning∗ a given semantics of the relevant kind is, and supply a suitable pragmatics. ³⁰ See Kamp (1981), Soames (1999) chapter 7, Raffman (1994), Raffman (1996) and Fara (2000). My examples follow Fara (2000).

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vague predicates to borderline cases prevail to lesser degrees than conventions about how to use sentences involving attributions of vague predicates to clear cases. This approach led to the conclusion that a sentence shouldn’t generally be said to express a proposition simpliciter, but only relative to a given semantics (which is apt to a certain degree). I noted, however, that typical assertions will nonetheless turn out to be true or false simpliciter. In arguing for the metasemantic account I sometimes presupposed particular theories of conventionality, meaning-fixation and conversational pragmatics. This was necessary to provide a detailed defense of the proposal. But, as far as I can tell, the proposal does not depend on the details of these presuppositions. Re f e re n c e s Black, M. (1949a), Language and Philosophy, Cornell University Press, Ithaca, NY. (1949b), ‘Vagueness: An exercise in logical analysis’, Philosophy of Science 4, 427–55. Reprinted in Black (1949a) and Keefe & Smith (1996). Br¨auerle, R. et al., eds (1979), Semantics from Different Points of View, Springer-Verlag. Burge, T. (1975), ‘On knowledge and convention’, The Philosophical Review 84, 249–55. Dorr, C. (2003), ‘Vagueness without ignorance’, Philosophical Perspectives 17, 83–113. Fara, D. G. (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28, 45–81. Originally published under the name ‘Delia Graff’. Gilbert, M. (1989), On Social Facts, Princeton University Press, Princeton, NJ. Grandy, R. (1977), ‘Review of Lewis’s Convention: A Philosophical Study’, The Journal of Philosophy 74, 129–39. Jamieson, D. (1975), ‘David Lewis on convention’, Canadian Journal of Philosophy 5, 73–81. Kamp, H. (1981), ‘The paradox of the heap’. in M¨onnich (ed.) Aspects of Philosophical Logic. Keefe, R. and Smith, P., eds. (1996), Vagueness: A Reader, MIT Press, Cambridge, MA. Laurence, S. (1996), ‘A chomskian alternative to convention based semantics’, Mind 105, 269–301. Lewis, D. (1969), Convention: A Philosophical Study, Harvard University Press, Cambridge, MA. (1970), ‘General semantics’, Synthese 22, 18–67. Reprinted in Lewis (1983). (1973), ‘Languages and language’, Versus 4, 2–21. Appears in Italian translation as ‘Lingue e lingua’. Reprinted in English in Lewis (1983). (1979), ‘Score-keeping in a language game’, The Journal of Philosophical Logic 8, 339–59. Reprinted in Br¨auerle et al. (1979); reprinted in Lewis (1983). (1983), Philosophical Papers, Volume I, Oxford. Raffman, D. (1994), ‘Vagueness without paradox’, Philosophical Review 103(1), 83–111. (1996), ‘Vagueness and context-relativity’, Philosophical Studies 81, 175–92. Rayo, A. (2008), ‘Vague representation’, Mind 117, 329–73. Schiffer, S. R. (1972), Meaning, Clarendon Press, Oxford. (1987), Remnants of Meaning, MIT Press, Cambridge, MA. Soames, S. (1999), Understanding Truth, Oxford University Press, Oxford and New York. Stalnaker, R. C. (1979), ‘Assertion’, Syntax and Semantics 9, 315–22. Reprinted in Stalnaker (1999), 78–95. (1999), Context and Content, Oxford University Press, Oxford. Williamson, T. (1994), Vagueness, Routledge, London and New York. (1999), ‘On the structure of higher-order vagueness’, Mind 108, 127–44.

2 The Possibility of Partial Definition Scott Soames

The view of vagueness I favor is one according to which vague predicates are partially defined, in the sense of being governed by rules that provide sufficient conditions for them to apply, and sufficient conditions for them not to apply, but no conditions that are both individually sufficient and disjunctively necessary for them to apply, or not to apply, to an object. Objects for which such a predicate P is undefined are those for which neither the claim that P applies to them, nor the claim that it doesn’t, is sanctioned. For any name n, which we know to refer to o, we accept the claim that P applies to o just in case we accept Pn, which we accept just in case we accept the claim that Pn is true. When P is undefined for o, these sentences and claims are also undefined. Since even complete knowledge of all linguistic and non-linguistic facts wouldn’t justify accepting or believing them, such acceptance or belief is always mistaken. The extension of P is the collection of things to which P applies; the antiextension is the collection to which P doesn’t apply. The determinate extension of P is the set of objects o, such that claim that P applies to o is a necessary consequence of the rules of the language plus all relevant underlying non-linguistic facts. For some objects o the claim that o is not in the determinate-extension of P is true, whereas the claim that o is not in the extension of P is undefined. Similar remarks apply to the antiextension and determinate-antiextension of P. Corresponding to these distinctions, there is also a distinction between truth and determinate truth. In addition to being partially defined, vague predicates are context sensitive. Given such a predicate P, one begins with its (default) determinate-extension and (default) determinate-antiextension. P is undefined for o just in case o is in neither of these sets. Since the sets don’t exhaust all cases, speakers have the discretion of adjusting the extension and antiextension to include initially undefined cases. When one does this by predicating P of o, or by denying such a predication, and one’s hearers go along, the extension (or antiextension) of P is contextually adjusted to include o, plus all objects that bear a certain relation of similarity to it. Observation predicates like ‘is blue’—which we learn by example rather than definition—are good illustrations. When learning the word, we note that certain objects Thanks to Jeff King, John MacFarlane, Sebastiano Moruzzi, and Nathan Salmon for their helpful comments.

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are uniformly called ‘blue’, while certain others are uniformly called ‘not blue’. People say of o—which we note to be of a certain shade BE1—‘That’s blue,’ while saying of o∗ —which we observe to be of shade BA1—‘That’s not blue.’ On this basis, we come to accept the rule Blue 1. Blue 1 If o is BE1, then ‘is blue’ applies to o If o is BA1, then ‘is blue’ does not apply to o Further experience leads us to accept additional rules involving different shades, until, at some point, we are counted as understanding the predicate. At this point, our rules provide a rich set of sufficient conditions for application, plus a similar set for nonapplication. However, the requirement that the rules be adhered to by the great majority of speakers ensures that these conditions won’t be jointly exhaustive. Since there are shades of color, and objects having them, about which the rules say nothing, the predicate is partially defined. Context sensitivity means that we are free to adjust the extension or antiextension of the predicate to include objects for which it is undefined by the rules of the language. Suppose I call such an object o ‘blue’, and my hearers go along. Then, the extension of ‘is blue’ is contextually expanded to include o, plus others discriminately bluer than, or perceptually indiscriminable in color from, o. Let BEc be a shade that applies to precisely this class. The rule—If an object is BEc, then ‘is blue’ applies to it— is thereby implicitly adopted in the conversation. Although not a rule of the language governing the predicate, it is one that speakers may adopt in particular contexts. The basic rule of the language governing the predicate (by providing its default determinate extension and antiextension) is Blue-English, where BE and BA are families of shades uniformly characterized as blue, and not blue, respectively (leaving a gap). Blue-English If o exemplifies one of the shades in BE, then ‘is blue’ applies to o If o exemplifies one of the shades in BA, then ‘is blue’ does not apply to o What are these shades, and how do they become associated with the word ‘blue’? Colors are natural kinds, and color shades are surface reflectance properties of objects. Their association with ‘blue’ is illustrated by a simplifying idealization. Imagine a small, homogenous community introducing the word into their language. They notice a set BEo of perceptually similar objects (of varying shades within BE) which are easily discriminable from another set BAo of objects (of varying shades within BA). They introduce the word ‘blue’ with a reference-fixing stipulation. Intro. ‘Blue’ is to designate the property of object surfaces causally responsible for the fact that (nearly) all members of BEo appear similar to one another, and different from BAo . Hence, ‘is blue’ will apply (at any world-state) to all and only those objects the surfaces of which have the property which (in the actual

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Scott Soames world-state) causally explains why members of BEo look similar to us, and different from members of BAo .

This stipulation is, of course, a fantasy. The term ‘blue’ could have been introduced in this way, and it behaves pretty much as if it had been so introduced. However, no such stipulation need ever have occurred. It is enough if speakers simply started calling things ‘blue’, with the intention that the predicate was to apply, not only to certain objects they had encountered, but also to those sharing the property of surfaces that explained their appearance. Finally, it must be remembered that in discussing the idealized stipulation, as well as the more realistic process of introduction it summarizes, we are not talking about a semantic rule of the language, mastered by speakers, stating the meaning or reference of a term. Although the stipulation mentions particular objects involved in the introduction of ‘blue’, it is not a semantic rule of English that this, that, or the other object is blue. Instead, the stipulation summarizes a crucial element in the explanation of how the word ‘blue’ acquired the semantic properties it has—among them, the property of being partially defined.¹

2.1

T H E A L L E G E D I M P O S S I B I L I T Y O F PA RT I A L DEFINITION

That, in brief, is the account of vague predicates I favor. I now turn to an objection that seeks to establish, not just that the account is wrong, but that it is incoherent. According to the objection, made by Michael Glanzberg, there aren’t, and couldn’t have been, partially defined predicates in any language.² His main argument, which is presented as an elaboration of one given earlier by Michael Dummett, is based on global claims about assertion.³ It is supposed to show that there can be no truth value gaps—all propositions must be either true or false. Glanzberg states the argument as follows: (i) Speech acts, including assertions, are moves within a practice of using language which is (partially) rule-governed . . . As such, speech acts have intrinsic purposes [norms]. (ii) The intrinsic purpose [norm] of assertion is to convey the information that something is the case, i.e. to assert s is to convey the information that s.⁴ (iii) Combining (ii) with the idea that propositional contents encapsulate truth conditions implies a form of the ‘truth-assertion platitude’, for the intrinsic purpose ¹ This sketch of vague predicates, including color terms, summarizes more detailed discussions in Soames (1999, chapter 7), Soames (2003), and Soames (2004). ² ‘Against Truth Value Gaps,’ in Liars and Heaps. ³ Michael Dummett, ‘Truth,’ Proceedings of the Aristotelian Society, 59, 1959, 141–62. ⁴ Glanzberg’s use of metalinguistic variables and corner quotes requires correction. The final clause of (ii) should be understood: i.e. to assert (the proposition expressed by) s is to convey the information (proposition) denoted by that s.

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[norm] of assertion: the intrinsic purpose [norm] of assertion is to assert that truth conditions obtain. (iv) The truth of a claim is thus fundamentally a matter of a purposive act achieving its intrinsic purpose [conforming to its intrinsic norm].⁵ Elaborating on this conclusion, Glanzberg says: Assessing for truth is a matter of assessing a purposive act for success. We may thus think of truth itself as having a point or purpose, in so far as it is correctly applied exactly when a purposive act achieves its purpose. The same may be said for truth values . . . Any assignment of truth value amounts to an assessment of whether a purposive act has achieved its purpose. [my emphasis] [T]he value true corresponds to the intrinsic purpose of an assertion being achieved, and false corresponds to it failing to have been achieved. It appears evident that these are the only ways that an assertion can be assessed for whether it has achieved its intrinsic purpose. It either has or has not done so.⁶ [my emphasis]

This points to the following conception of the intrinsic purpose, or norm, of assertion. The Glanzberg–Dummett Account of the Norm of Assertion (GD1) For any proposition p, an assertion of p is correct (satisfies the intrinsic norm of assertion) just in case p is true. (GD2) An assertion is incorrect (fails to satisfy the intrinsic norm governing assertion) just in case p is false. The import of this conclusion for theories of truth value gaps, and/or partial definition, is easy to see. Any theory that maintains both that some propositions are neither true nor false, and that the assertion of such a proposition is incorrect because it violates the norm of assertion, is incompatible with (GD2). Thus, establishing (GD2) would be sufficient to refute any such theory. It would also be sufficient to refute theories that embrace partial definition, in my sense. On the one hand, these theories insist that asserting an undefined proposition violates the norm of assertion, and so is incorrect. On the other hand, in calling the proposition undefined, the proponent of partial definition is committed to rejecting the claim that it is untrue—thereby violating (GD2). Thus, accepting (GD2) requires rejecting partial definition. Moreover, the friend of partial definition—who doesn’t assert the existence of propositions that are neither true nor false, doesn’t object to identifying falsity with untruth, and is happy with contraposition (in the sense of accepting If ∼B, then ∼A whenever he accepts If A, then B)—recognizes that (GD1) entails (GD2). Thus, (GD1) is incompatible with partial definition. But how, exactly, is (GD1) supposed to follow from the premises of Glanzberg’s argument? Premise (ii) tells us that the aim of asserting the proposition p expressed by a sentence S is to convey that which S expresses, namely p. But the claim that conveying p is the aim of asserting p doesn’t advance the argument. Nor does premise (iii), which says, in effect, that the aim of asserting p is to assert that the truth conditions ⁵ Glanzberg (2003, 159).

⁶ Glanzberg (2003, 159, 165–6).

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of p ‘‘obtain.’’ Since for conditions to ‘‘obtain’’ is just for them to be satisfied, this amounts to the claim that the aim of asserting p is to assert that p is true—which is, at best, parasitic on the triviality that the aim of asserting p is to assert p. The problem reappears in a further remark Glanzberg makes. The intrinsic purpose of assertion is to say that the truth conditions expressed obtain. This purpose is achieved just when the proposition expressed is true.⁷ [my emphasis]

But this is a non-sequitur. If my purpose is simply to say that the truth conditions of p obtain, and hence to commit myself to the claim that p is true, I can easily achieve that purpose even if p is false, or undefined. After all, it is perfectly possible to say of any proposition that it is true. Thus, Glanzberg has no argument for (iv), which is supposed to abbreviate (GD1). However, this needn’t be fatal, since (GD1) and (GD2), which can be broken into pairs of quantified conditionals, have some impendent plausibility.⁸ (GD1a) For any proposition p, and assertion A(p) of p, if A(p) is correct (satisfies the intrinsic norm of assertion), then p is true. (GD1b) For any proposition p, and assertion A(p) of p, if p is true, then A(p) is correct (satisfies the norm). (GD2a) For any proposition p, and assertion A(p) of p, if p isn’t true, then A(p) is incorrect (doesn’t satisfy the norm). (GD2b) For any proposition p, and assertion A(p) of p, if A(p) is incorrect (doesn’t satisfy the norm), then p isn’t true. (GD1a) and its contraposed version, (GD2a), are unproblematic. Since the assertion of an undefined proposition p violates the norm of assertion, instances of these principles corresponding to p will be true—by falsity of antecedent in the case of (GD1a), and by truth of the consequent, in the case of (GD2a). Thus, it is only (GD1b) and (GD2b) that are potentially problematic for partial definition. However, these principles are incorrect. In the presence of the (a) principles, what the (b) principles tell us is that all there is to the intrinsic norm of assertion is the directive to assert truths. But, as Timothy Williamson has argued, this is implausible.⁹ Assertion isn’t the only speech act that aims at truth. That other truth-directed acts—like conjecturing or predicting—put less stringent demands on the agent than does assertion suggests that there is more to assertion than aiming at truth. This is born out by cases—e.g. those involving lotteries—in which we aren’t warranted in asserting certain truths, even though they are highly probable on our evidence. In these cases one believes, but fails to know, some true proposition p, even though the odds in favor of p are very heavy. The fact that one isn’t warranted in asserting p, despite reasonably believing p to be true, suggests that assertion requires what is missing in these cases—knowledge. As Williamson ⁷ Page 164. ⁸ In discussing these issues I take ‘correct’ and ‘incorrect’ to be jointly exhaustive (when applied to assertion). Although this is an idealization, it doesn’t affect the issues at hand. ⁹ Williamson (1996).

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notes, this explains why the question ‘How do you know?’ is a standard way of challenging an assertion. The question presupposes that an agent who has asserted p should know p—which is just what one would expect if knowledge, rather than truth, was the norm of assertion. These and related considerations support replacing the Glanzberg–Dummett truth-based norm with the Williamsonian knowledge-based norm. Williamson’s Account of the Norm of Assertion (W1) For any proposition p, an assertion of p is correct (satisfies the norm of assertion) just in case the agent knows p. (W2) An assertion of p is incorrect (fails to satisfy the norm) just in case the agent doesn’t know p. As before, we can divide each of these claims into a pair of claims. (W1a) For any proposition p, and assertion A(p) of p, if A(p) is correct (satisfies the norm of assertion), then the agent knows p. (W1b) For any proposition p, and assertion A(p) of p, if the agent knows p, then A(p) is correct (satisfies the norm). (W2a) For any proposition p, and assertion A(p) of p, if the agent doesn’t know p, then A(p) is incorrect (doesn’t satisfy the norm). (W2b) For any proposition p, and assertion A(p) of p, if A(p) is incorrect (satisfies the norm), then the agent doesn’t know p. Glanzberg’s principles (GD1a)/(GD2a)—which, as we have seen, are unproblematic for theories of partial definition and undefined propositions—are entailed by (W1a)/(W2a), and so have the status of derived norms of assertion. Since (GD1b)/(GD2b) conflict with (W1a)/(W2a), they must be rejected. What remains of his argument against partial definition is, therefore, reducible to the question of whether accepting partial definition is compatible with accepting (W1)/ (W2). Since these two principles are interderivable, we may concentrate on (W2). Can I admit that the assertion of an undefined proposition is incorrect because it can’t be known, without attributing its unknowability to it’s not being true? I should think so. From the beginning, I have said that it is a mistake to assert an undefined proposition p because even complete knowledge of linguistic and nonlinguistic facts wouldn’t justify accepting p, as opposed to its negation. If this point can be extended to an explanation of why one can’t know p, then partial definition and undefined propositions will have been rendered compatible with the correct account of the norm of assertion—and the Glanzberg–Dummett argument will have been rebutted.

2.2

T H E U N K N OWA B I L I T Y O F T H E U N D E F I N E D

Why, then, can’t one know the undefined? Since the Glanzberg–Dummett argument purports to rule out the very possibility of a language containing partially defined predicates, I will frame my rebuttal around a simple, artificial example, which

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parallels, for color words, an example I have used in other contexts.¹⁰ Imagine members of a small linguistic community living on a desert island, cut off from the outside world. Sharing no antecedent common language, they set about to create one. Color words are introduced by authoritative stipulation. One of these, ‘bluege’, is introduced by applying it to examples. As it happens, the island contains objects of various shades of blue each of which is stipulated to be bluege, and various shades of green and other colors, which are stipulated not to be bluege.¹¹ However, a few shades remain unclassified, because they aren’t exemplified on the island. Among them are shades intermediate between the least blue of those stipulated to be bluege and the most blue-like of the greens stipulated not to be bluege. Since there is no pressing need to decide the status of these shades, the gap goes unremarked. At this point, speakers agree that their language contains a meaningful term ‘bluege’, governed by the authoritative, meaning-giving stipulations summarized in Bluege-Island. (BIE is the family of exemplified blue shades on the island; BIA is a family of exemplified nonblue shades.) Bluege-Island If o exemplifies one of the shades in BIE, then ‘bluege’ applies to o If o exemplifies one of the shades in BIA, then ‘bluege’ does not apply to o Is ‘bluege’ partially defined? Consider a shade, INT, intermediate between blue and green. A sequence of four barely discriminable shades separates INT from the least blue shade in BIE, and a similar sequence separates it from the most blue-like shade of green in BIA. Does ‘bluege’ apply to objects exemplifying INT? The stipulations don’t tell us. Objects exemplifying INT haven’t been stipulated to be bluege, or not to be. Since they are as perceptually similar to those stipulated to be bluege as they are to those stipulated not to be bluege, the case for classifying them one way is no better than the case for classifying them the other. The issue isn’t how to extend the meaning of ‘bluege’. That’s a future matter for the Islanders to consider. Rather, the issue is whether ‘bluege’ already applies, or doesn’t apply, to objects exemplifying INT. Since there is no more support for one of these alternatives than for the other, the facts don’t determine either one. Hence ‘bluege’ is undefined for objects exemplifying INT, and complete knowledge of all relevant facts would neither justify taking ‘bluege’ to apply to them, nor justify taking it not to apply to them. This explains why someone omniscient about all the relevant facts wouldn’t know any proposition p predicating ‘bluege’ of an object for which it was undefined.¹² That explanation doesn’t say that p isn’t true. It says that knowing p isn’t possible because the agent lacks the justification needed for knowledge. In general, an agent whose justification for a proposition is no better than his justification for its negation won’t ¹⁰ I refer to the smidget example (the idea for which was originally suggested to me by Nathan Salmon) discussed in Soames (1999, chapter 6). ¹¹ We may imagine that the comparative, ‘blueger than’, is similarly introduced. ¹² Here and throughout I adopt the simplifying assumption that, for all the cases under discussion, an agent who understands a sentence S that expresses a proposition p (in a context C) will accept/believe-true (be justified in accepting/believing-true) S (in C) iff the agent believes (is justified in believing) p. Although Kripkean Pierre-type cases show that this principle needs modification, the complications don’t affect the issues raised here.

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know that proposition, even if he believes it. But if agents who know all the ‘bluege’relevant facts lack the justification required for knowledge of p, the same will be true of ordinary, non-omniscient agents, who know less. To be sure, the acquisition of additional information bearing on a hypothesis q sometimes puts an agent with partial information about q in a worse position to know it—as when coming to know a true, but misleading, defeater undermines one’s initial knowledge of q. In all such cases, however, further knowledge, defeating the defeater, reinstates one’s original knowledge. Although selective bits of additional knowledge sometimes put one in a worse position to know something, knowing all the relevant facts never does. Thus, if the non-omniscient could know p, the relevantly omniscient could too. But since the omniscient can’t, no one else can either. Appreciating this point requires distinguishing justification, in the sense that it is needed for knowledge, from mere reasonableness of belief. Undefined propositions can be highly probable on one’s evidence, making it perfectly reasonable to believe them. However, this doesn’t provide the justification needed for knowledge. As lottery examples have taught us, even when the probability of a proposition on one’s evidence is arbitrarily high, one may fail to know it—because one’s evidence isn’t of the right sort. This point, which holds for true propositions, doesn’t cease to hold for the undefined. When justification is understood as the cognitive requirement needed for knowledge, believing the undefined is unjustified, no matter how reasonable it is in certain cases. This completes my rebuttal of the Glanzberg–Dummett argument that the intrinsic norm of assertion rules out partial definition. The first step was to replace their inadequate conception of this norm with a better one—according to which what assertion requires is knowledge of the proposition asserted. The second step was to explain why undefined propositions can’t be known in a way that doesn’t commit one to their not being true. Combining both, we have an account of why asserting an undefined proposition violates the norm of assertion, and so is a mistake. 2.3

PA RT I A L D E F I N I T I O N A N D T H E E XC LU D E D M I D D L E

Rebutting the objection doesn’t, of course, establish that partial definition really is possible. It does, however, justify giving that possibility some weight. Absent compelling arguments to the contrary, we are, I think, prima facie justified in taking the possible term ‘bluege’ to be partially defined. What about the ‘law’ of the excluded middle? It is often thought that acceptance of partial definition brings with it rejection of some instances of the ‘law’—on the grounds that a disjunction is undefined when both disjuncts are. While plausible, this point is less obvious than it first seems. When S is undefined, the rules governing S, plus the totality of facts relevant to evaluating S, don’t determine that S is true, or that it isn’t. As a result, one can’t know that S is true, and asserting that it is is a mistake. Similar points hold for the proposition expressed by S. But when S is a disjunction  or , how do we show that S is undefined in this sense if  and are? It does not, in general, follow from the fact that asserting each of two propositions is a mistake that asserting their disjunction is.

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Nor does it follow that one who fails to know each of two propositions also fails to know their disjunction. What about determination of truth by the totality of linguistic and non-linguistic facts? Does the claim that the truth of a disjunction is not a necessary consequence of those facts follow from the claim that the truth of neither disjunct is? That depends on what counts as a necessary consequence of what—which in turn depends in part on whether  or ∼ is itself necessary. Since this is an instance of the very question we are trying to decide, we must be careful not to presuppose the answer we are trying to justify. How, then, might one combine partial definition with unqualified acceptance of excluded middle? On scheme for doing so is supervaluationism. One starts with an intended model M that assigns interpretations in which some sentences are true, some are false, and some are neither. A sentence S is counted as true simpliciter iff S is true in every admissible bivalent extension of M. S is false simpliciter iff S is false in all such extensions. Otherwise S is neither true nor false. Since S or ∼S is true in all bivalent extensions, the ‘law’ of the excluded middle is preserved, even when both disjuncts are neither true nor false. Nevertheless, classical supervaluationism doesn’t reconcile acceptance of excluded middle with the kind of partial definition given here. For the classical supervaluationist, the claim that S isn’t true follows from the claim that S is undefined. For me, it doesn’t; rather, the claim that S isn’t true is undefined when S is. This mismatch could, in principle, be repaired. Instead of holding that S is true iff S is true in all admissible bivalent extensions of the initial partial model M, and false iff S is false in all such extensions, one might stipulate that S is true, if S is true in all admissible bivalent extensions of M, and that S is not true, if S is false in all such extensions. Since these stipulations give sufficient conditions for being true, and sufficient conditions for not being true, while saying nothing about sentences with different truth values in different classical extensions, such sentences will be undefined in my sense. The resulting system preserves excluded middle, while allowing a form of partial definition that allows one to assimilate sentences and propositions that are not true to those that are false. However, it is still not what we want. In either the classical or the revised form, supervaluationism violates the truism that  or  is true just in case  is true or is true, and hence that a disjunction can’t be true unless one of its disjuncts is. Since this truism is essential to our ordinary understanding of ‘or’ and ‘true’, supervaluationism doesn’t give the right account of the truth conditions of complex sentences of natural language. It is also explanatorily baroque. In order to determine whether S is true or not, supervaluationism requires one to first determine whether S is true in all admissible bivalent models, false in them all, or true in some and false in others. This presupposes a notion of truth in a model antecedent to the official supervaluationist notion of truth, plus an antecedent logic used to calculate which sentences are true in which models. The idea that there is both a hidden truth and a hidden logic—conceptually prior to the ordinary notion of truth we apply in language, and the logic we employ when using it—is implausible, as well explanatorily tendentious. Since the (classical) laws of the hidden logic are simply taken for granted, it is hard to see how supervaluationism can be used to explain or justify them.

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The ineffectiveness of supervaluationism as a semantics doesn’t preclude limited supervaluationist reasoning from resolving certain kinds of indeterminacy. Suppose it is determinate that an agent asserts some proposition, but indeterminate which of p1 to pn is asserted.¹³ Although what is asserted is indeterminate, supervaluationist reasoning might still be used to explain how it is determinate that the speaker said something true, or something untrue—provided p1 to pn are all true, or all untrue. However, this isn’t the kind of indeterminacy to which partially defined predicates give rise. Suppose a speaker says of o ‘That’s bluege,’ when in fact o exemplifies the shade, INT, indeterminate between blue and green. There is no indeterminacy about what is asserted in this case. It is not as if there is a family of totally defined properties B1 . . .Bn such that it is determinate that the agent asserted the proposition that o has one of these properties, but indeterminate which. Rather, the asserted proposition predicates the partially defined property being bluege of o. The reason it is indeterminate whether what is said is true, is that the truth value of this proposition—which was determinately asserted—is indeterminate. Supervaluationism doesn’t fit this kind of case. This leaves us back where we started—trying to decide whether partial definition requires rejecting some instances of excluded middle. We have seen that supervaluationism does not show that partial definition can be combined with unqualified acceptance of the law. But, I haven’t yet argued that the two can’t be combined. Can a disjunction be determinately true, even if its disjuncts are undefined? Can the claim that  or  is true be a necessary consequence of the rules of the language, plus the underlying nonlinguistic facts, even though neither the claim that  is true, nor the claim that is true, is? When  and are unrelated, it’s hard to see how the rules of the language, which are silent about both, could be definitive about the disjunction. How about when one disjunct is the negation of the other? Is there something about this case that gives it a special status, rendering the attribution of truth to (1) a necessary consequence of the rules of the language, plus the underlying facts, even though the same can’t be said for the disjuncts, (2a) and (2b)? 1. N is bluege or N is not bluege. 2. a. N is bluege. b. N is not bluege. It is difficult to see what it might be. Suppose we did take (1) to be true. Surely, we would have to say the same about (3) and (4), where ‘N∗ ’ names the same object as ‘N’, and ‘M’ names a qualitative duplicate of that object (both exemplifying INT). 3. N is bluege or N∗ is not bluege 4. N is bluege or M is not bluege Intuitively (1), (3), and (4) should be treated similarly—all true, or all undefined. However, we have no explanation of how to get the result that the truth of the latter is a necessary consequence of the rules of the language plus the underlying facts. ¹³ See (Soames 2002a, 81–3, 337–8) for potential examples.

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5. It is a necessary consequence of the rules of the language, plus the underlying facts, that N is bluege, N∗ is bluege and M is bluege are all true, all untrue, or all undefined. To guarantee that (3) and (4) are (determinately) true, if (1) is, we need something like (6). 6. It is a necessary consequence of the rules of the language, plus the underlying facts, that substitution of M is bluege or N∗ is bluege for N is bluege in any true disjunction always preserves truth. Though (6) is quite reasonable, the rationale for it makes it difficult to assign truth to (1). The reason we find (6) plausible is, I think, that we find two ideas compelling: R1. The status of a disjunction is entirely dependent on the status of its disjuncts. R2. We have as much reason for taking N∗ is bluege and M is bluege to be true as we have for taking N is bluege to be true. But if R2 is compelling, so is R3—which, in the presence of R1, leads to (7). R3. We have as much reason for taking N is not bluege to be true as we have for taking N is bluege to be true. 7. It is a necessary consequence of the rules of the language, plus the underlying facts, that substitution of N is not bluege for N is bluege (or vice versa) in any true disjunction always preserves truth. Although (7) is, I think, as well motivated as (6), it clearly precludes taking (1) to be true. The lesson to take from this is that accepting partial definition and undefined propositions, while trying to retain a completely unrestricted version of excluded middle, is a dubious business. Although it is formally possible to combine the two, the resulting systems seem ill motivated. I therefore conclude that our prima facie justification for taking partial definition to be possible provides prima facie justification for rejecting the unrestricted excluded middle. We aren’t, of course, justified in accepting the negations of any of its instances. That would be incoherent. Rather, we have reason to believe that some of those instances are undefined—in the sense that asserting them would be a mistake, that knowing them is impossible, and that their truth is not determined by the totality of linguistic and nonlinguistic facts. I haven’t argued that partial definition or undefined sentences exist in English. However, I have tried to make such analyses more plausible by addressing the familiar objection that they impose ‘the high price of giving up classical logic.’ Of course, this talk of price is metaphorical. The issue is descriptive, not volitional. We aren’t deciding how to reason, and looking for the most economical way of doing so. The issue is whether unrestricted versions of all classical ‘laws’ are true. The idea that a theory pays a high price for refusing to agree that they are is just the idea that the ‘laws’ seem, initially, to be so. Thus, a theory that doesn’t embrace them in full generality has some

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explaining to do. I have tried to provide the beginning of such an explanation. Later, I will say a word about what the explanation has to say about seemingly more obvious ‘laws’ like ∼( and ∼)  2.4

PA RT I A L D E F I N I T I O N , C O N T E X T S E N S I T I V I T Y, AND IGNORANCE

The view I favor is one in which vague predicates in natural language are both partially defined and context sensitive. In order to more closely approximate the natural language case, let’s add context sensitivity to the rules governing ‘bluege’. The imagined situation is as before, with ‘bluege’ being introduced by authoritative stipulation, accepted by everyone in the community. The result is a partially defined predicate the meaning of which is stable, due in part to the fact that there are few, if any, objects for which it is undefined among those standardly talked about. At some point, the introduction of such objects changes the situation. Some speakers notice the new shades, and when speaking about them use expressions like ‘bluegish’, ‘kind of bluege’, and ‘more bluege than greenge’. Both the shades previously called ‘bluege,’ and those called ‘not bluege,’ continue to be uniformly so characterized. But, there is contextual variation in how shades in the intermediate range are described, and sometimes speakers find themselves at loss for words. Responding to this need, speakers start allowing themselves the freedom to apply ‘bluege’ to objects for which it had initially been undefined. There is, unsurprisingly, variation in how this is done. In some contexts—depending on the audience, subject, and time—speakers are more expansive in what they are willing to call ‘bluege’ than they are in others. However, no one worries about this. No one thinks that there is just one right way to apply the word to objects within its initially undefined range. Instead, it is recognized to be a matter of decision—with different reasons yielding different results in different cases. Nor is there any sense that the standards adopted in a given context must settle, for each possible shade of color, whether objects of that shade are to be in the extension of ‘bluege’ or not. It is enough if the adopted standard allows determinate evaluation of all conversationally relevant propositions. If the predicate does remain undefined for some objects, agents presented with such an object, and pressed to classify it, will often be indifferent—and say things like ‘Its sort of bluege and sort of not,’ or ‘You really can’t say that its one or the other,’ or ‘It doesn’t matter, call it what you like.’ At this stage, ‘bluege’ is both partially defined and context sensitive. The determinate extension and antiextension of the original term have become the default determinate-extension and default determinate-antiextension of the new contextsensitive term. It’s meaning is a function from contexts of utterance to members of a restricted family of properties (many of which are partially defined). One member of the family—the default semantic content of ‘bluege’—is a partially defined property that applies to all members of the default determinate-extension, fails to apply to all members of the default determinate-antiextension, and is undefined for everything else. This property plays two roles. First, it is the semantic content of the

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term unless something about the context selects a different property. Second, it fixes the boundaries of allowable contextual variation in the use of ‘bluege’. Each object it (determinately) applies to is one that every contextually possible semantic content of the term (determinately) applies to, and each object it (determinately) doesn’t apply to is one to which no such content (determinately) applies. Properties in the family of possible semantic contents of ‘bluege’ differ only in how they divide up the objects for which the default content of ‘bluege’ is undefined. Of course, not every way of dividing up those objects corresponds to a property in the family. For every such property P and object o, if P (determinately) applies to o, then P (determinately) applies to all objects as blue as, or bluer than, o. When an object onew for which the term had previously been undefined is called ‘bluege’ in a context Cnew , and conversational standards are thereby adjusted, the property selected as semantic content in Cnew (determinately) applies to objects perceptually indistinguishable in color from onew , plus those discriminately bluer. The resulting line between items to which ‘bluege’ applies and those for which it remains undefined will be as sharp as the notion of objects being perceptually indiscriminable in color.¹⁴ Although it is easy to be deceived about precisely where this line is, its location can, in principle, be identified.¹⁵ Let oI be perceptually indistinguishable from onew , and o2 be perceptually indiscriminable from o1 —each located just one indiscriminable step beyond its predecessor in a sorites sequence. Looking at the pair and correctly judging them to be indiscriminable, one can come to know of o1 that it falls within the extension of ‘bluege’ in Cnew . A similar point holds for o2 —which, though indiscriminable from o1 , is discriminately less blue than onew . Correctly judging this to be so by looking at onew and o2 together, one can come to know that ‘bluege’ is undefined for o2 in Cnew . So far, we haven’t found any facts about ‘bluege’ that are in principle unknowable. The best place to look for such facts is the line between the default determinateextension of ‘bluege’ and the objects for which it is initially undefined. Up to now, I have treated this line as if it were sharp. However, that’s an artifact of the example. In my fable, ‘bluege’ is originally introduced as a partially defined term, in which the boundary between the objects to which it applies and those for which it is undefined is determined by a universally agreed upon, authoritative stipulation. When context sensitivity was introduced, I imagined this boundary simply remaining in place, with the initial semantic content becoming the default content of the new term. The ¹⁴ This statement of the rule for adjusting the extension of the predicate is a rough approximation. A more accurate statement stipulates that ‘bluege’ comes to apply to all objects with the surface spectral reflectance property causally responsible for the perceptual indiscriminability from onew of all, or nearly all, of those objects the color of which is indiscriminable from onew . This opens up the possibility of isolated instances of ‘fools bluege,’ in which something may be indiscriminable from onew for nonstandard reasons, without having the surface spectral reflectance property required to count as bluege, as well as the possibility that some objects with the property might—for extraneous reasons—not be perceptually indiscriminable from onew . (Similar remarks apply to clause concerning objects discriminately bluer than onew .) Although these standard natural-kind complications are mostly irrelevant to our concerns, they do introduce another source of potential vagueness (‘casually responsible . . . . for all or nearly all . . .’). ¹⁵ For discussion, see Soames (1999, 212–17).

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resulting semantic model—though realizable in principle—isn’t realistic in practice. In more realistic cases, the default semantic content of a context-sensitive term will depend on an overall pattern of agreement among uses by different speakers. In such cases, the boundary between what’s in its default extension and what isn’t will be blurred. Since this boundary is indeterminate, no single partially defined property can, strictly speaking, be said to be the default semantic content of the term.¹⁶ Instead, there will be a family of barely differing properties, each of which is a candidate for being that content. Since it is indeterminate which of these properties is the default semantic content of the term, there will be a limited range of cases in which it is indeterminate whether the claim that ‘bluege’ applies to o is true because the rules of the language, plus facts about o, determine that it’s true, or because the speaker has exercised minimal discretion in adjusting the extension of ‘bluege’ to include o. What precisely is the range of this indeterminacy? Although speakers typically won’t care, the question is both troubling and theoretically puzzling. I don’t see any good alternative to the Williamsonian view that—somehow—the question has a definite answer, even though we can’t find it. If that’s right, then context sensitivity and partial definition don’t tell the whole story about vague language.

2.5

A M O D E S T C A S E F O R PA RT I A L D E F I N I T I O N AND CONTEXT SENSITIVITY

I conclude with a pair of modest arguments—first, that partial definition is part of the story, if context sensitivity is, and second, that context sensitivity is part of the story, if partial definition is. In other words, accounts of vague language that include both partiality and context sensitivity have advantages over those that employ only one. The first argument rests on an assumption that at least some vague terms are context sensitive, in the sense of having different semantic contents in different contexts of utterance. The meaning of such a term is a function from contexts to members of a family of related properties. Our question is whether all of these properties have to be totally defined. The case that they don’t is based on the idea that the context sensitivity of a vague term is limited. For example, there are limits on what can truly and literally be called ‘green’, and limits on what can truly and literally be called ‘not green’. If ‘green’ is context sensitive, these limits will be reflected in the family of contextually possible semantic contents of ‘green’. No member of the family will apply to scarlet red roses, or fail to apply to healthy, well-watered patches of grass. If, in addition, ‘green’ is partially defined, then the set of things to which every member of the family (determinately) applies will be its default determinate-extension, while the set of things to which no member (determinately) applies will be its default determinateantiextension. If, on the other hand, the term is required to be totally defined, there will be a corresponding pair of sets. Call them the minimal non-negotiable extension ¹⁶ For a detailed explanation of the reasons for, and consequences of this, see Soames (2003).

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and antiextension of the term. Either way, both theoretical models must specify the range of linguistically possible contextual variation in literal uses of the term. In both cases, this range will be subject to seeming indeterminacy and/or impenetrable Williamsonian ignorance. The two models differ in that, in the totally defined model, this ignorance comes on top of a different, and prior, form of impenetrable ignorance. For each contextually legitimate use of the totally defined term, there must be a precise, but unknowable boundary separating things to which it applies, from those to which it doesn’t. How this boundary is determined is, according to the model, shrouded in mystery. There is no corresponding mystery with the partially defined model. Once the range of contextual variation is determined, the extension and antiextension of the term, when used with its default semantic content, is completely given. Since subsequent contextual adjustments are essentially matters of stipulation, plus a similarity relation associated with the term, any further indeterminacy or ignorance is limited to vagueness or indeterminacy about the stipulation, or the similarity relation—unclarity about whether the speaker has really referred to this object o and called it ‘green’, or about whether, if he has, some other object o∗ is discriminably greener than, or perceptually indiscriminable in color from, o.¹⁷ Although there may be cases in which there are no knowable, or even determinate, answers to these questions, this area of indeterminacy or mysterious ignorance is sharply limited, and highly circumscribed. One loses this, if one insists on bivalence and total definition in every case. Since it is better to have less extensive theoretical mystery than more, there is something to be said for including partial definition in one’s account of vague terms—if one decides that they are also context sensitive.¹⁸ The second argument—for the desirability of context sensitivity given partial definition—involves the explanation of our differential reactions to apparent violations of different logical ‘laws.’ I have already argued that the case for partial definition provides prima facie justification for rejecting certain instances of the ‘law’ of the excluded middle. When F is undefined for o in the sense I have explained, and n names o, it is natural and correct to reject both the claim that Fn is true and the claim that ∼Fn is true. But then, the truism that a disjunction can be true only if one of its disjuncts is, leads one to reject both the claim that Fn or ∼Fn is true and Fn or ∼Fn itself. Whatever one’s ultimate verdict on this result, the force of the reasoning is clear, and the attractiveness of the conclusion is easily recognizable. The same cannot be said for other consequences of the position. Corresponding to the truism about disjunction is a similar truism about conjunction—namely that a conjunction ¹⁷ The similarity relation in question is the one that determines which objects are included in the contextually adjusted extension of the predicate as a result of the decision to include the new object o. On the partially defined model this is a relation born to o by all objects discriminately greener than o, plus those perceptually indiscriminable from o. On the totally defined picture, the relation will be much less restricted—applying to all objects discriminately greener than o, plus (perhaps) those objects that are more similar in color to o than to the least green object in the antecedently determined antiextension. Typically, the latter will allow much greater room for indeterminacy or mysterious ignorance. ¹⁸ An earlier statement of this argument can be found on in Soames (2002b, 447–9).

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can be untrue only if one of its conjuncts is.¹⁹ Given both this truism and the undefinedness of Fn and ∼Fn, one has no choice but to reject the claim that Fn and ∼Fn is untrue —in which case one must reject both the claim that its negation, ∼(Fn and ∼Fn), is true, and ∼(Fn and ∼Fn) itself. But rejecting the law of noncontradiction seems far less defensible than rejecting excluded middle. Proponents of partial definition need some explanation of this. In my opinion, there is no plausible way to retain the undefinedness of Fn or ∼Fn, and its negation, without recognizing the similar undefinedness of (Fn and ∼Fn), and its negation—when F is undefined for o. Thus, rejection of some instances of excluded middle should bring rejection of corresponding instances of non-contradiction in its wake. The reason that the rejected instances of noncontradiction seem correct, in a way that the corresponding instances of excluded middle don’t, is that the former are prey to a certain metalinguistic confusion that the latter aren’t. When, in addition to being partially defined, F is also context sensitive, it is easy to confuse the obvious truth 8. No matter what contextual standards one adopts governing F, it will not be the case that those standards (plus the nonlinguistic facts) dictate that Fn and ∼Fn are both true—and hence that Fn and ∼Fn) is too. with the undefined (and hence properly rejected) claims (9a–c). 9a. (Fn and ∼Fn) is never true. b. ∼(Fn and ∼Fn) always true. c. ∼ (Fn and ∼Fn) It is because of this confusion that we are reluctant to reject instances of the ‘law’ of noncontradiction, even when they are undefined. A similar confusion of the metalinguistic triviality (10a) with the undefined claim (10b) explains our (misguided) reluctance to reject undefined instances of the ‘law’ (10c). 10a. If a contextual standard dictates that Fn is true, then it will dictate that Fn is true. b. If Fn, then Fn is always true. c. If Fn, then Fn The proper explanation of these cases parallels the proper explanation of familiar so-called penumbral truths involving vague predicates. For example, although (11a) is undefined, it is easily confused with a metalinguistic truth (11b) that says something important about the partially defined, context sensitive predicate ‘is bald’.²⁰ 8a. If a man is bald, then he would be bald if he had one less hair. b. No matter what standards governing ‘is bald’ we adopt in a context, if according to those standards he is bald applies to a man, then according to those same standards it would apply to him if he had one less hair. ¹⁹ Thanks to Nathan Salmon for reminding me of the parallel. ²⁰ See Soames (2002b, 440–1).

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By contrast, there is no metalinguistic truth corresponding to the ‘law’ of the excluded middle that makes us reluctant to reject undefined instances of them. The reason they seem easier to reject than some other classical ‘laws’ is that rejecting them isn’t subject to the same pragmatic interference we encounter with the other laws. Logically, the different laws are in the same boat. Pragmatically, they differ in what they suggest about the context-sensitive semantic effects of context change. In this way, the proponent of partial definition can explain our differential reaction to violations of different classical ‘laws’—provided that the predicates in question are context sensitive, as well as partial. Re f e re n c e s Dummett, Michael (1978), ‘Truth’ in Truth and Other Enigmas, Harvard University Press: Cambridge, 1–24. Originally published in Proceedings of the Aristotelian Society, 59 (1969), 141–62. Glanzberg, Michael (2003), ‘Against truth value gaps’, In Jc Beall, ed., Liars and Heaps: New Essays on Paradox, Clarendon Press, Oxford, 151–94. Soames, Scott (1999), Understanding Truth, Oxford University Press, New York. (2002a), Beyond Rigidity: The Unfinished Semantic Agenda of ‘Naming and Necessity’, Oxford University Press, New York. (2002b), ‘Replies,’ Symposium on Understanding Truth in Philosophy and Phenomenological Research 65, 429–52. (2003), ‘Higher-order vagueness for partially defined predicates’, in Jc Beall, ed., Liars and Heaps, Clarendon Press, Oxford, 128–150. (2004), ‘Knowledge of manifest natural kinds’, Facta Philosophica 6, 2004, 159–81. Williamson, Timothy (1996), ‘Knowing and Asserting’, Philosophical Review 105: 489–523.

3 Vagueness and Second-Level Indeterminacy Matti Eklund

My theme here will be vagueness. But first recall Quine’s arguments for the indeterminacy of translation and the inscrutability of reference. (I will presume these arguments to be familiar.) If Quine is right, then there are radically different acceptable assignments of semantic values to the expressions of any language: different assignments of semantic values that for all that is determined by whatever it is that determines semantic value are all acceptable, and all equally good. Quine even argued that the indeterminacy is so radical that some sentences are true under some acceptable assignments but false under others.¹ Still, Quine does not allow intermediate truth values or truth value gaps. (As I will put it, avoiding the disjunctive formulation: does not allow that there are sentences which are neuter.) Quine holds on to classical logic and bivalence and requires each acceptable assignment to be classical and bivalent.² A general point is this. There are, in principle, two fundamentally different ways in which a sentence can be, somehow, indeterminate in truth value. First, there can be a determinately best way of assigning a semantic value to the sentence, but the semantic value in question is neuter.³ Second, there can fail to be a determinately best way of assigning a semantic value to the sentence. There can be several acceptable assignments, and the sentence can have different semantic values under the different acceptable assignments. Many thanks to David Liebesman, Agust´ın Rayo, Roy Sorensen, Robbie Williams, and the audience at a St Andrews vagueness seminar (in 2005) for helpful comments and criticism. ¹ Throughout I will set phenomena such as context sensitivity to the side, and simply speak of sentences as true and false. I will also restrict my focus to declarative sentences used for making assertions. ² See Quine (1960) and, primarily, (1968). My brief presentation of Quine is unabashedly unhistorical. It is in (1960), where he discusses translation manuals, that he stresses that such manuals in ‘countless places’ will ‘diverge in giving, as their respective translations of a sentence of the one language, sentences of the other language which stand to each other in no plausible sort of equivalence however loose’ (27). It is in (1968) that Quine discusses word–world relations and not only translation manuals. ³ Consider two phenomena quite distinct from indeterminacy: reference failure and presupposition failure. If these phenomena give rise to there being sentences that are neuter, they present examples of this.

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Introducing some terminology, let us say that the distinction just drawn is between first-level and second-level indeterminacy. A sentence is second-level indeterminate just in case it has different truth values under different acceptable assignments of semantic values. A sentence is first-level indeterminate just in case there are acceptable assignments under which the sentence lacks a classical, determinate truth value. Analogously, we can define first-level and second-level indeterminacy for subsentential expressions. Of course, not all cases of lack of classical determinate truth values are strictly cases of indeterminacy. But let us stick with these labels, for here the focus will be on varieties of indeterminacy. The point is then that Quine’s thesis is that there is second-level indeterminacy, but Quine is by no means committed to there being first-level indeterminacy.⁴ Turn now to vagueness. It is widely assumed that vagueness is associated with some sort of (semantic or ontological) indeterminacy. Among those who make this assumption, it is also in effect widely assumed that the indeterminacy in question is first-level. That is, it is assumed that if vagueness indeed is associated with some sort of indeterminacy, what this means is that the semantic values best ascribed to vague sentences are such that these sentences lack classical, determinate truth values: witness the different non-classical semantics that have been proposed as the right semantics of vagueness. Now, epistemicists often argue as against those who take the proper semantics of vagueness to be non-classical/non-bivalent that a strategy of the kind just described cannot possibly provide part of the solution to the philosophical problem of vagueness.⁵ Consider first someone who presents a three-valued logic and a three-valued semantics. The problem here that epistemicists point to is that it is as implausible that there should be a sharp boundary between the true and the neuter sentences as that there should be a sharp boundary between the true and the false sentences. A similar point holds with respect to appeal to fuzzy semantics: there will still be a sharp boundary between the last sentence with truth value 1 and the first sentence with truth value <1.⁶ ( This is the problem of higher-order vagueness. Note that talk of ‘higherorder vagueness’ is quite different from the talk of higher-level indeterminacy here introduced.) On the strength of negative arguments like this, epistemicists hold that opting for a deviant semantics is unwarranted, and they go on to identify the apparent indeterminacy with some kind of principled unknowability of truth value. But a major seeming problem with the epistemicist’s own preferred view is that it seems very improbable that whatever determines the meanings and semantic values of vague expressions should determine sharp cutoffs between truth and falsity—e.g. that for a predicate like ‘looks roughly like a robin’ and every object o, ‘o looks roughly like a ⁴ I have earlier brought up, and made important use of, the distinction between first-level and second-level indeterminacy. See my (2002), 271, and my (2005), 44. But in these other articles this distinction is bound up with other theoretical commitments. The distinction, and its application to problem of vagueness, is worth stressing in its own right. ⁵ See e.g. Williamson (1994), chapters 4–5. ⁶ For discussion, see e.g. Williamson, op.cit., and also Sainsbury and Williamson (1997) and Keefe and Smith (1997).

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robin’ is determined to be true or determined to be false.⁷ (Call this the determination problem.)

3.1

VAG U E N E S S A N D S E C O N D - L EV E L I N D E T E R M I N AC Y

I am going through some standard moves in the philosophy of vagueness very quickly here. Needless to say, the proponents of the various positions discussed have responses—and responses worth taking seriously—to the problems brought up.⁸ But rather than discussing the actual and possible responses and counterresponses, let me present a positive proposal that promises to get around the whole cluster of problems we are getting into. It is that we should take vagueness primarily to be associated with second-level indeterminacy, rather than first-level indeterminacy. On the view that vagueness is associated with second-level indeterminacy, one can endorse the epistemicist criticism of common proposals to the effect that the proper semantics for vagueness is non-classical/non-bivalent, and hold that every acceptable assignment of semantic values to vague expressions is classical and bivalent. (Naturally there may be reasons independent of vagueness for abandoning classical logic and bivalence: but let us not here get into them.) At the same time, the determination problem is avoided. For if vagueness is associated with second-level indeterminacy, the fact that whatever factors determine the meanings of our expressions have not determined exact boundaries for vague predicates is reflected in there being different acceptable assignments of semantic values to these expressions. Not that it is in any way strange or unreasonable initially to assume vagueness rather to be associated with first-level indeterminacy. The intuitive picture associated with the semantics of vague predicates is that their extensions sort of fade off, in a way that may reasonably be thought best captured by, e.g., fuzzy logic. But intuitive though this view may be, the philosophical discussion of vagueness has arguably taught us that no specific assignment of semantic values to vague predicates, not even a fuzzy one, can fully satisfy our conception of what the extensions of vague predicates are like. Having set out the view that the indeterminacy with which vagueness is associated is second-level indeterminacy, let me straight away turn to objections. The discussion of the objections is also aimed at making the view clearer. First, it may be objected that also the view that vagueness is associated with second-level indeterminacy is committed to the kind of tripartite division that the epistemicist objection is directed against. In this case, the tripartite classification is that between sentences that are true under all acceptable assignments, sentences ⁷ Being a standard objection to epistemicism, this objection is presented in many places; but two nice presentations are found in Schiffer (2003), 182f and Sainsbury and Williamson (1997), 480f. ⁸ For an epistemicist response to the determination problem, see Williamson (1994). For defense of appeal to non-standard logic against the objection pressed, see e.g. Tye (1994).

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that are false under all acceptable assignments, and sentences that are second-level indeterminate. This point, however, mistakes the nature of the epistemicist objection. The objection is not directed against a tripartite division per se: a bipartite division is not in itself better. The objection can only be meant to show that the problem of vagueness is not solved by substituting a bipartite classification with a tripartite or fuzzy one. The problem that the epistemicist stresses is that of higher-order vagueness. If a bipartite division between what ‘tall’ is true of and what it is false of is intuitively unacceptable, a tripartite division is likewise intuitively unacceptable, regardless of whether we conceive of the tripartite division as one between true, neuter and false, or we conceive of it as one between truth under all assignments, falsity under all assignments, and shiftiness. But the lesson of the epistemicists’ objection to accounting for the phenomena of vagueness by appeal to some deviant semantics of this kind is precisely that whatever the correct story is it will involve positing an intuitively unacceptable cutoff. Saying that the individual acceptable assignments are all classical certainly goes no way toward avoiding this counterintuitiveness. Saying that there is a multitude of acceptable assignments does not do so either. But this is not a strike against the account I am proposing. The idea that vagueness is associated with second-level indeterminacy is not meant to rebut the epistemicists’ argument. Rather, it is supposed to provide a way of accepting the conclusion of the argument while avoiding the determination problem. In his (2003), Richard Heck argues for a tripartite classification of vague sentences. He does not bring up anything like the distinction between first-level and secondlevel indeterminacy, but it appears that his arguments rely on the distinction. His argument for a tripartite division is not that a tripartite division of sentences into true/neuter/false is more intuitively acceptable than other divisions. The prospects of a convincing argument along those lines would be slim. Rather, Heck’s point is that whatever it is that determines meaning is unlikely to determine every sentence as either true or false. Heck does not draw the distinction between first-level and second-level indeterminacy. However, having drawn this distinction we can distinguish between two theses along the lines suggested by Heck’s considerations. One is that there is a unique best assignment of semantic values, and under that assignment some sentences are indeterminate in truth value. The other is that some sentences have different truth values under different acceptable assignments. What Heck’s argument in the first instance suggests is the latter conclusion. Below I will discuss Heck’s view further. Having defended the claim that the view I am proposing is acceptable even if it entails that there is a sharp tripartite division of the kind described, I should note a complication. It can be vague or otherwise indeterminate what the acceptable assignments are. But even if this is a complication, and it forces further distinctions, it does not affect the theoretical point I am concerned to make, that vagueness is (primarily) associated with second-level indeterminacy. A second objection to the suggestion that vagueness is associated with second-level indeterminacy is that obviously vagueness is not the same phenomenon as the kind of indeterminacy Quine sought to establish the existence of. But the proposal is not that

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vagueness is the very same kind of phenomenon as Quinean indeterminacy: all that is suggested is that these are both instances of second-level indeterminacy. No one would argue against the suggestion that vagueness is associated with first-level indeterminacy on the ground that vagueness is a different phenomenon from reference failure, or presupposition failure. A third objection—if it really should be called an objection—to my suggestion that vagueness is second-level indeterminacy is that contrary to how I have advertised it, what I have presented is not really a novel view on vagueness at all. Saying that vagueness is bound up with there being a multitude of acceptable assignments rings bells: is this not what supervaluationists have been urging for some time?⁹ However, what supervaluationists are typically talking about when they are talking about different assignments of semantic values to a vague expression is a multitude of precisifications, a multitude of ways of making precise vague expressions, and that is quite different from acceptable assignments in the present sense. From the supervaluationist point of view, there is a unique best semantic value to ascribe to borderline sentences: such sentences are neuter.¹⁰ The appeal to precisifications comes in only when the semantic values of complex sentences of which this sentence is part are determined. (Supervaluationists do often use the slogan that vagueness is ‘semantic indecision’, and this slogan can in principle be taken as an affirmation of the idea that vagueness is associated with second-level indeterminacy.¹¹ But the slogan is open to several interpretations, and insofar as it is taken to justify the standard form of supervaluationism it is incompatible with the second-level indeterminacy interpretation.) However, that said, I should note that I am not the first to associate vagueness with second-level indeterminacy. David Lewis (1969) indicates that vagueness is a matter of it not being determinate which of all possible languages there are that we speak; see too Linda Burns (1991), who in this respect follows Lewis, defending what she calls a pragmatic view on vagueness. This is compatible with it being the case that all the languages we are indeterminate between are classical.¹² Also, it seems that what Richmond Campbell (1974) means by ‘semantic uncertainty’ is something like second-level indeterminacy.¹³ In more recent years, Cian Dorr (2003), Nicholas ⁹ The classic defense of supervaluationism about vagueness is Fine (1975). See also McGee and McLaughlin (1995) and Keefe (2000). ¹⁰ Some supervaluationists prefer the classification determinately true/indeterminate/determinately false to the classification true/neither/false. See e.g. McGee and McLaughlin (1995). It may sound as if a supervaluationist who operates with this classification should be more hospitable to the idea of second-level indeterminacy. But is there really more than a terminological difference between these two forms of supervaluationism? ¹¹ The slogan is from Lewis (1986), 212. ¹² Lewis holds that languages—abstractly conceived—generally are classical, and for a variety of reasons a community can fail to speak a determinate one among these languages. He does not apply the idea specifically to the puzzles of vagueness. (Where Lewis does focus more on vagueness—see (1970), 228–30 and (1979), 244–6—he adopts a supervaluationist approach.) Although Burns holds the view of Lewis (1969) and focuses specifically on vagueness, her resolutions of the puzzles of vagueness centrally invoke a number of other considerations, e.g. an appeal to context. ¹³ Campbell (1974). In the course of defending classical logic and semantics in the face of vagueness, Campbell says, ‘. . . a borderline case of a short man is one where (a) it is uncertain

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Smith (2008), and Agust´ın Rayo (2008, this volume) have also put forward similar views, but in the contexts of broader discussions. For example, in Rayo’s work the distinction is bound up with bigger issues in the theory of meaning. As I will seek to illustrate in the next section, I think that very different views on meaning allow for the distinction to be drawn.¹⁴ A fourth objection—related to the third—is that second-level indeterminacy in a way simply collapses into first-level indeterminacy. One might suggest that a sentence is true (false), simpliciter, just in case the sentence is true (false) under all acceptable assignments, in the present sense of ‘acceptable assignment’. A number of different remarks on this suggestion are in order. First, it deserves stressing that the suggestion by no means represents the only possible way to go with respect to talk about truth. Call a sentence determinately true just in case it is true under all acceptable assignments. One hypothesis is then that ‘true’ expresses determinate truth. A different hypothesis about the ordinary language predicate ‘true’ is that it expresses disquotational truth: that for every sentence S and for every acceptable assignment, ‘S is true’ is true under that assignment just in case S is true under that assignment (and correspondingly for ‘false’).¹⁵ A third, more complicated hypothesis is that ‘true’ is—in some way or other—indeterminate as between determinate truth and disquotational truth.¹⁶ The objection gets off the ground only if we should opt for the first of these three hypotheses. Second, even if ‘true’ does express determinate truth, what has been described here is a route to there being sentences which are neither determinately true nor determinately false which is quite distinct from those routes others, like supervaluationists, have tended to stress. So even if there is a collapse of sorts, the interest of the appeal to second-level indeterminacy remains. Third, and most importantly, suppose we start out with a class of classical assignments of an extension to a vague predicate. Then, by the collapse suggestion, we can define, whether or not it is a case of a short man, (b) the uncertainty is not due to lack of information about the man’s height or about the distribution of heights among men in general, and (c) the uncertainty is not due to linguistic incompetence . . . For convenience I shall speak of this uncertainty as semantic uncertainty, since the uncertainty is apparently due to the fact that the meaning of ‘‘short man’’ is vague or inexact’ (180). ¹⁴ See Dorr (2003), section 2; Smith (2008); Rayo (2008), esp. 365; and Rayo (this volume), esp. section 1.2.1. ¹⁵ The distinction between determinate truth and disquotational truth is from McGee and McLaughlin (1995). They do not however draw the present distinction between first-level indeterminacy and second-level indeterminacy. McGee and McLaughlin’s discussion suggests that to opt for disquotational truth here is to decide in favor of deflationism. But deflationism typically requires more, for example that the equivalence between S and ‘S is true’ somehow exhausts the notion of truth. ¹⁶ Compare perhaps the discussion in McGee and McLaughlin (1995). McGee and McLaughlin suggest that ‘true’ is indeterminate as between determinate truth and disquotational truth. (But McGee and McLaughlin do not discuss what I here call second-level indeterminacy.) A complication, impressed upon me by Roy Sorensen, is that there may be no best assignment or assignments but only a sequence of better and better assignments. Here is a fix. Instead of talking about a sentence being true on some or all best assignments, one might talk about a sentence being true on some (sufficiently adequate) assignment while there is no better (or: equally good or better) assignment on which it is not true.

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on the basis of these classical assignments, a three-valued one: and this is the correct assignment of an extension. But what we have obtained is merely yet another candidate for being the right assignment, and if we already had reason to think that the acceptable assignments are all classical, then this three-valued one has a strike against it. The third point deserves stressing. Return to Heck’s (2003) discussion. At one point, Heck brings up a possible objection to his view: Consider some vague predicate, say, ‘chair’, and suppose what I’ve said so far is correct: there are many classical extensions this predicate might have consistent with its use. Then the thought is that we should simply take the extension of ‘chair’ to be the intersection of these various possible extensions, that is, the set of objects of which it is determinately true, so that it is false of any object of which it is not determinately true.¹⁷

Heck describes the construction suggested by the objection as the ‘closing-off construction’, and responds by saying, appropriately enough, that ‘. . . the closing-off construction begins with some set of extensions that our use of the predicate does not rule out as possible extensions. But if so, however, then, by hypothesis, there are extensions the predicate could have, consistent with our use of it, other than the one delivered by the closing-off construction’.¹⁸ This corresponds to my response to the present objection. It is then odd that Heck goes on to defend what appears to be an orthodox form of supervaluationism. He says, e.g., If vague predicates lack determinate classical extensions, then there will be no classical semantics for a language containing vague predicates. There are, to be sure, plenty of ways to proceed. But it is natural at this point . . . to suggest that we try to provide such a semantics in terms of what is already available to us, namely, a set of extensions that ‘red’ could have, consistently with our use of it. So consider an utterance of ‘That is red’. When shall we regard it as true? Well, surely, if it would have been true (false) no matter which of the possible extensions for ‘red’ it might have had, it is hard to call it false (true). Conversely, if there are extensions ‘red’ could have, consistently with our use of it, according to which this utterance would not have been true (false), it is hard to call it true (false).¹⁹

Why is the response to the closing-off suggestion not equally available here? That is: by hypothesis, the classical extensions of the predicate are extensions our use does not rule out as possible extensions. But then there are extensions the predicate could have, consistently with our use of it, other than the supervaluational (or generally, three-valued) ones. Worse still: if the classical extensions are supposed to be all the candidate-extensions, the supervaluational semantics is not even among the candidates. The defender of the supervaluational semantics has a comeback. She can insist that even if we start out holding the view that there is a multitude of acceptable ¹⁷ Heck (2003), 112. ¹⁸ Heck (2003), 113 f. The label ‘closing-off’ is taken from a particular move concerning the liar paradox that Kripke discusses in (1975). ¹⁹ Heck (2003), 121f.

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assignments, but they are all classical and bivalent, we should anyway upon reflection want to identify truth with truth under all assignments: the new assignment which results—call it a superassignment —will be an assignment better than the bivalent ones. But then the bivalent assignments are not acceptable after all. In order to assess this properly, one would have to carefully compare this superassignment with the bivalent assignments. Let me here just emphasize the following. What is rendered problematic, if the superassignment is indeed better than the classical ones, is in the first instance just the idea that vagueness is not associated with first-level indeterminacy. It can still be that there are several different supervaluational assignments that each are ‘best’. If so, vagueness is still associated with second-level indeterminacy. The possibility of combining the view that the acceptable assignments are supervaluational with the view that vagueness is associated with second-level indeterminacy further illustrates the point that supervaluationism and appeal to secondlevel indeterminacy are different.²⁰

3.2

VAG U E N E S S A N D M E A N I N G

The thesis that vagueness is associated with second-level indeterminacy is neutral between different accounts of what, specifically, vagueness is, and between different views on meaning. But in part to further illustrate the issues here and in part to further illustrate the claim that vagueness is associated with second-level indeterminacy, let me present some different types of views on vagueness and on meaning given which the thesis is likely to be true. Let me begin with my own preferred view.²¹ I defend an account of vagueness given which vague expressions on account of their vagueness give rise to second-level indeterminacy. Very briefly, my view is an inconsistency view on vagueness, according to which the vagueness of an expression consists in it being necessary for full competence with the expression to be disposed to take it to be tolerant—despite the fact that tolerance is an incoherent feature of expressions, and no expression can actually be tolerant.²² On an inconsistency view like this, the semantic values of vague ²⁰ In general, the idea that vagueness is associated with second-level indeterminacy is obviously compatible with all sorts of claims regarding what first-level indeterminacy it is associated with. The acceptable assignments could—at least in principle—be fuzzy, or paraconsistent. Or some acceptable assignments could be classical and bivalent and others would be fuzzy, and still others supervaluational. ²¹ See my (2002) and (2005). ²² Where, to a first approximation, a predicate F is tolerant just in case it satisfies a tolerance principle of the following form: Whereas large enough differences in F’s parameter of application sometimes matter to the justice with which it is applied, some small enough difference never thus matters. The notion is from Wright 1975, 334. There are worries regarding the exact formulation of tolerance principles. For example, as characterized, tolerance can only be a feature of predicates. But also expressions of other linguistic categories can be vague. See Weatherson (this volume) for relevant discussion.

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expressions cannot actually be such as to make true the principles that are meaningconstitutive for the expression. Rather, they are such as to come as close as possible to making these meaning-constitutive principles true. It is plausible that there will be ties for closeness, and hence several different acceptable assignments of semantic values to vague expressions. My own preferred view involves much that is controversial—primarily, of course, the idea that meanings can somehow be inconsistent —and irrelevantly so for present purposes. There are other views too which naturally yield that vagueness is associated with second-level indeterminacy. On one influential view on reference-determination (defended or relied on, in different ways, by theorists like David Lewis, Mark Johnston, and Frank Jackson) the semantic values of many of our expressions are determined by associated ‘folk theories’, firmly entrenched and generally shared conceptions of the subject at hand.²³ The semantic value of the expression is what satisfies the claims of the associated folk theory, if there is such a satisfier, and is what comes ‘near enough’ otherwise (and if nothing comes near enough the term lacks a semantic value). This view too is hospitable to a distinction between first- and second-level indeterminacy. To see this, simply note that elements in folk theories have the same role in determining reference on this view as meaning-constitutive principles do on the inconsistency view. Given the apparatus of the folk-theoretic account of referencedetermination, one might say that tolerance principles, despite being inconsistent, are part of the folk theories for vague expressions, and hence are reference-determining for vague expressions. Both my view and folk-theoretic semantics are broadly descriptivist, broadly Fregean views on reference. It should however be clear that the distinction between first-level and second-level indeterminacy is relevant also given other views on reference. Whatever determines the semantic values of expressions, it would appear that it is possible to think of the semantic values of expressions as the semantic values the expressions have under the assignment that best satisfies some given constraints. Descriptivist theories focus on descriptive constraints; other theories focus on causal constraints. And under any view on what the constraints are, there would appear to be a distinction between (a) there being ties between different assignments all satisfying the constraints equally well, and (b) the semantic values of expressions being indeterminate under some or all of the assignments that best satisfy the constraints. Under (a) we have second-level indeterminacy. Under (b) we have first-level indeterminacy. Having noted that under many, perhaps all, reasonable accounts of how semantic values are determined, second-level indeterminacy is an option, let me note what may be regarded as a possible problem with the story I am telling. Although on each of the accounts I have presented it is likely that vagueness should introduce second-level indeterminacy, none of them comes near entailing that vagueness does. It may be thought that the link between vagueness and indeterminacy is tighter than I should also note that the talk about speaker dispositions here is quite rough. See my (2007) for further discussion. ²³ See Lewis (1970a, 1972, 1997), Johnston (1992), and Jackson (1998).

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this—specifically, that vagueness surely must entail indeterminacy—and that there thus is something important which is missing from the present story about the relation between vagueness and indeterminacy. By way of response, here are a few remarks on the relation between vagueness and indeterminacy. Return to the debate between epistemicists and those theorists—supervaluationists, fuzzy logic theorists, . . .—who adopt some deviant logic or semantics in an effort to solve the puzzles associated with vagueness. On one side there are the theorists who deny the existence of genuine indeterminacy.²⁴ On the other side there are the theorists who link vagueness so closely with indeterminacy that they think the key to the puzzles associated with vagueness is to be found through an investigation of the indeterminacy associated with vagueness. Let me give two examples to illustrate this latter stance. First, proponents of fuzzy logic and semantics think that the indeterminacy associated with vagueness is such that vague sentences take fuzzy truth values. They then apply this to the puzzles like the sorites paradox. Compare Tom with his near duplicate Tom-minus, just like Tom except that one hair has been plucked from his head. The sentences ‘Tom is bald’ and ‘Tom-minus is bald’ are then almost identical in truth value. The sorites paradox arises because we mistake near-identity for identity. Second, proponents of supervaluationism think that the indeterminacy associated with vagueness is such that sometimes a sentence of the form ‘there is an x such that x is bald and x is not bald’ is true without there being any object a such that ‘a is bald and a is not bald’ is true (where ‘ ’ denotes a kind of successor function, taking us to the next element in a sorites series). The sorites paradox arises because we naively miss the difference between these two claims. Clearly there is in principle room for a third kind of view. According to this view vagueness typically—or perhaps always—gives rise to an associated kind of indeterminacy, but the indeterminacy does not hold the key to the puzzles of vagueness in the way typically envisaged by ‘fuzzy’ theorists and supervaluationists. Taking vagueness to be associated with second-level rather than first-level indeterminately naturally, although not inevitably, suggests this third view on the relation between vagueness and indeterminacy. 3.3

W R I G H T O N PE R M I S S I B L E D I S AG R E E M E N T

Let me close by remarking on another consequence of taking seriously the possibility that vagueness is associated with second-level indeterminacy. Crispin Wright has urged that vague sentences can permissibly be called true and also permissibly be called false, even in situations where the speakers have perfect information, and this is something that must be respected by any adequate account of vagueness.²⁵ Vagueness gives rise to permissible disagreement. Wright holds that both epistemicism and ordinary accounts of vagueness as semantic indeterminacy fail to respect this. On the ²⁴ Or who insist that the indeterminacy with which vagueness is associated is only ‘epistemic’; that the so-called indeterminacy amounts to a principled unknowability. ²⁵ Wright (1995), 138. See too Wright (2001).

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strength of this type of consideration he proposes his own ‘quandary view’ on vagueness, which I will describe shortly. It may seem that all sorts of accounts of vagueness have the means to account for permissible disagreement. The epistemicist, who holds that the truth value of a borderline sentence is for principled reasons unknowable, can say that when the truth value of a sentence is in that way unknowable, both verdicts on its truth value are permissible. The supervaluationist can say that when a sentence is true under some precisifications and false under others, both verdicts are permissible. (Generally, that the judgment that a sentence is true [false] is permissible just in case the sentence is true [false] under some precisification.) The ‘fuzzy’ theorist can say that when a sentence has a truth value between 0 and 1 (or perhaps in some interval right around 0.5), both verdicts are permissible. However, these suggestions trade on taking ‘permissible disagreement’ in a way that is not intended. Let me try to clarify Wright’s idea. What Wright says when introducing the idea of permissible disagreement is, 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 truth-conferrer 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 indeterminate, one takes a view with any direct bearing on the question of the truth value of P. I know of no way of making that idea intelligible except by construing indeterminacy as some sort of epistemic status.²⁶

The reason why orthodox accounts of vagueness fail to accommodate permissible disagreement is in Wright’s view that they assign to borderline sentences a status incompatible with truth and falsity. Perhaps the idea can be presented as follows. To assert is to present as true. But then if a sentence assertively uttered by a speaker has a status incompatible with the truth, the speaker has done something impermissible, provided she has perfect information. Ordinary views on vagueness then have the consequence that a borderline sentence cannot permissibly be assertively uttered when the speaker has perfect information. Wright’s idea of permissible disagreement is that borderline sentences (and hence also their negations) can be permissibly assertively uttered in such situations. Once the idea of permissible disagreement is specified in this way, one may well come to think that permissible disagreement is impossible. But Wright’s thought is that there is a way of construing indeterminacy that allows for permissible disagreement. And whatever in the end the fate of this idea, it has a certain intuitive attraction. One might well think that if ‘Harry is bald’ really is a borderline sentence, then assertively uttering the sentence and assertively uttering its negation are both permissible even in situations of perfect information. This is connected to the sentence being genuinely indeterminate, as opposed to simply having a third status. ²⁶ Wright (2001), 476 f.

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I will not here attempt to evaluate Wright’s claim that vagueness gives rise to permissible disagreement in the sense outlined.²⁷ I just want to stress that if Wright is correct about permissible disagreement, that is something the secondlevel indeterminacy view can help explain. The account of vagueness as secondlevel indeterminacy can respect this point of Wright’s, on the (fairly plausible but not undeniable) assumption that a sentence is permissibly called ‘true’ (‘false’) if there is at least one acceptable assignment of semantic values under which it is true (false). Taking vagueness to be a matter of second-level indeterminacy does not commit one to Wright’s permissible disagreement view: but it allows for this view.²⁸ As the above quotation makes clear, Wright thinks that the only way to accommodate what for him is the datum of permissible disagreement is by identifying indeterminacy as some sort of epistemic status. He concludes that we should characterize borderline cases as quandaries, as 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’.²⁹ But what we have just seen is that the considerations adduced by Wright can be accounted for without indeterminacy being construed as some sort of epistemic status. Suppose borderline sentences are second-level indeterminate, and suppose further that ‘true’ does not express determinate truth. Then the indeterminacy of a borderline sentence S is not conceived of as a status incompatible with truth or falsity.³⁰

3.4

C O N C LU D I N G R E M A R K S

I have here distinguished between first-level and second-level indeterminacy, and raised the question: what sort of indeterminacy is vagueness primarily associated with. My suggestion has been that vagueness is associated with second-level indeterminacy. This avoids problems faced by other accounts of vagueness. ²⁷ It is not obvious that Wright’s view is correct. Here is one immediate objection. If one can permissibly assertively utter ‘Harry is bald’, that is something that is known in situations of perfect information. Now suppose I know that the sentence is permissibly assertively uttered. Is that not sufficient to make it the case that I ought not to assertively utter its negation? (See Sorensen (1995) for criticism of Wright along these lines.) ²⁸ What I am urging is just that appeal to second-level indeterminacy helps with the points of Wright’s that I mention. Wright’s views on vagueness are very rich and the suggestion is not that I can accommodate everything that Wright wants to say. ²⁹ Wright (2001), 477. ³⁰ For Wright, borderline cases due to vagueness are sources of essential bafflement: we can come to no determinate description of their status. On my view by contrast—or rather on my view supplemented so as to accommodate Wright’s purported data concerning permissible disagreement—it holds that although we cannot say anything determinate about the truth values of borderline sentences, we can, at least in principle, say something determinate about their truth values under the different acceptable assignments.

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Re f e re n c e s Burns, L. (1991), Vagueness: An Investigation into Natural Languages and the Sorites Paradox, Kluwer Academic Publishers, Dordrecht. Campbell, R. (1974), ‘The Sorites Paradox’, Philosophical Studies 26, 175–91. Dorr, C. (2003), ‘Vagueness without ignorance’, Philosophical Perspectives 17, Language and Philosophical Linguistics, 83–114. Eklund, M. (2002), ‘Inconsistent languages’, Philosophy and Phenomenological Research 64, 251–75. (2005), ‘What vagueness consists in’, Philosophical Studies 125, 27–60. (2007), ‘Meaning-constitutivity’, Inquiry 50, 559–74. Fine, K. (1975), ‘Vagueness, truth and logic’, Synth`ese 30, 265–300. Heck, R. (2003), ‘Semantic accounts of vagueness’, in Jc Beall, ed., Liars and Heaps: New Essays on Paradox, Clarendon Press, Oxford. Jackson F. (1998), From Metaphysics to Ethics: A Defense of Conceptual Analysis, Oxford University Press, Oxford. Johnston, M. (1992), ‘How to speak of the colors’, Philosophical Studies 68, 221–63. Keefe, R. (2000), Theories of Vagueness, Cambridge University Press, Cambridge. Keefe, R. and Smith, P. (1997), ‘Introduction: Theories of Vagueness’, in Keefe and Smith, eds., Vagueness: A Reader, MIT Press, Cambridge, Massachusetts. Kripke, S. (1975), ‘Outline of a Theory of Truth’, Journal of Philosophy 72, 690–716. Lewis, D. (1969), Convention: A Philosophical Study, Harvard University Press, Cambridge, Massachusetts. (1970), ‘General semantics’, Synth`ese 22, 18–67. Reprinted in Lewis (1983), 189–232. (1970a), ‘How to define theoretical terms’, Journal of Philosophy 67, 427–46. (1972), ‘Psychophysical and theoretical identifications’, Australasian Journal of Philosophy 50, 249–58. (1979), ‘Scorekeeping in a language game’, Journal of Philosophical Logic 8, 339–59. Reprinted in Lewis (1983), 233–49. (1983), Philosophical Papers Volume I, Oxford University Press, Oxford. (1986), On the Plurality of Worlds, Blackwell, Oxford. (1997), ‘Naming the colours’, Australasian Journal of Philosophy 75, 325–42. McGee, V. and McLaughlin, B. (1995), ‘Distinctions without a difference’, Southern Journal of Philosophy, suppl. vol. 33, 203–51. Quine, W.V.O. 1960, Word and Object, MIT Press, Cambridge, Massachusetts. 1968, ‘Ontological relativity’, Journal of Philosophy 65, 185–212. Reprinted in Ontological Relativity and Other Essays, Columbia University Press, New York (1969), 26–68. Rayo, A. (2008), ‘Vague representation’, Mind 117, 329–73. , ‘A metasemantic account of vagueness’, this volume. Sainsbury, M. and Williamson, T. (1997), ‘Sorites’, in B. Hale and C. Wright, eds., A Companion to the Philosophy of Language, Blackwell, Oxford. Schiffer, S. (2003), The Things We Mean, Oxford University Press, Oxford. Smith, N. J. J. (2008), Vagueness and Degrees of Truth, Oxford University Press, Oxford. Sorensen, R. (1995), ‘The epistemic conception of vagueness: Comments on Wright’, Southern Journal of Philosophy, suppl. vol. 33, 161–70. Tye, M. (1994), ‘Sorites paradoxes and the semantics of vagueness’, in J. Tomberlin, ed., Philosophical Perspectives 8: Logic and Language, Ridgeview, Atascadero. Weatherson, B. ‘Vagueness as indeterminacy’, this volume.

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Williamson, T. (1994), Vagueness, Routledge, London. Wright, C. (1975), ‘On the coherence of vague predicates’, Synth`ese, 30, 325–65. (1995), ‘The epistemic conception of vagueness’, Southern Journal of Philosophy, suppl. vol. 33, 133–59. (2001), ‘On being in a quandary’, Mind 110, 45–98. Reprinted in Saving the Differences, Harvard University Press, Cambridge, Massachusetts (2003), 443–509.

4 Vagueness as Indeterminacy Brian Weatherson

Recently there has been a flurry of proposals on how to ‘define’ vagueness. These proposals are not meant to amount to theories of vagueness as, for instance, epistemic or supervaluational theories of vagueness are. That is, they are not meant to provide solutions to the raft of puzzles and paradoxes traditionally associated with vagueness. Rather, they are meant to give us a sense of which terms in the language are vague, and to use Matti Eklund’s phrase, in what their vagueness consists. Doing this might be a prelude to a successful theory of vagueness, or it might just be an interesting classificatory question in its own right. When this activity started, most notably with Patrick Greenough’s ‘A Minimal Theory of Vagueness’, I suspected that it would be a hopeless project. Imagine, I thought, trying to give a definition of what causation is that didn’t amount to a theory of causation. That project seems hopeless, and I didn’t think the prospects for a definition of vagueness were much better. I now think I was wrong, and we can learn a lot from thinking about which terms are vague, independent of our theory of vagueness. (As we’ll get to below, Greenough’s theory isn’t marketed as a definition of vagueness, but rather a ‘minimal theory’ to which all parties can agree. But it has been taken, e.g. by Eklund and Nicholas Smith, to be providing a rival to genuine definitions of vagueness, and I’ll follow Eklund and Smith in this respect.) The point of this exercise is not to give an analysis of how the man on the Clapham omnibus uses ‘vague’ and its cognates. As is widely recognized, ‘vague’ is often used in ordinary language as a predicate that applies to claims like The Grand Canyon is between 2 and 2 trillion years old, i.e. claims that are consistent with a wide range of possible worlds. That’s not the sense of ‘vague’ which philosophers use, nor the sense we are trying to define. But nor should we think we are just trying to analyse philosophical use of ‘vague’. The philosophers’ usage may be our starting point, but if we find philosophers have traditionally being ignoring theoretically important commonalities, or blurring theoretically important distinctions, our best definition may well amount to a revision of philosophical usage. The game, I think, is one of setting goals for what a theory of vagueness should do. It is a legitimate objection to a theory of vagueness that it isn’t comprehensive, Thanks to David Chalmers, Matti Eklund, Delia Graff Fara, Patrick Greenough, Mark Johnston, Daniel Nolan, Nicholas J. J. Smith, Andrew Strauss, and audiences at ANU, Princeton and Arch´e.

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that it doesn’t cover the field. If supervaluationism was only a theory of how vague words that started with consonants behaved, for example, that would be a problem for supervaluationism. But to press objections of this form we must have an antecedent answer to the question of which words are in the field, and hence should be covered. That’s the good question which these definitions of vagueness address. Because I take this to be the important issue, I’m going to start this paper with a bunch of examples of apparently vague, and apparently non-vague, terms. We’ll then look at which theories do the best job at systematizing intuitions about these cases. I’ll then argue that the best way to systematize our intuitions about these cases while respecting theoretically important commonalities and distinctions is to take vagueness to be indeterminacy, while staying silent for now on whether the indeterminacy is semantic or epistemic. In doing so I’m returning to a traditional view of vagueness, one that is discussed in such classic works as Kit Fine’s statement of supervaluationism (Fine 1975). So I make no claim to originality in my conclusions here, though I hope at least some of the arguments are original. 4.1

EXAMPLES

I’m going to introduce five classes of examples, which will serve as our data in what follows. I’ll give a fairly tendentious description of each class to orient us before starting. Our five classes are (a) words that are indeterminate but not vague, (b) vague words that are not predicates, especially predicate modifiers, (c) vague predicates whose conditions of application are contentious, (d) vague predicates whose application depends on discrete states of the world, and (e) vague predicates that do not determine boundaries.

4.1.1 Indeterminacy without vagueness Many philosophers, if asked, would say that vague words are those that have borderline cases. As noted above, Fine (1975) takes exactly this view. My preferred view, that vagueness is indeterminacy, is a simple generalization of this view to nonpredicates. But it is a commonplace of the literature on definitions of vagueness that this won’t do because of examples of indeterminacy without vagueness. Two examples are commonly used. One of these is Sainsbury’s example child ∗ (Sainsbury 1991). By definition, the extension of child ∗ is the set of persons under sixteen years old, and its anti-extension is the set of persons eighteen years old or older. Sixteen and seventeen year-olds are borderline cases. The intuition is that even though child ∗ has borderline cases it is not vague, because there are sharp boundaries to its borderline. A similar case arrives with mass as it is used by a Newtonian physicist. (I’m grateful to Delia Graff Fara for pointing out the connection here.) As Field (1973) showed, mass is indeterminate between two meanings, rest mass and proper mass. But it is intuitively not vague, because it is determinate that it means either rest mass or proper mass. These cases are well discussed in the existing literature, and I won’t say much

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more about them here, save to note that one of the examples that is usually taken to be very problematic for the vagueness as indeterminacy view, child ∗ , is not synonymous with any term in any natural language. This is not a reason that it could not serve as a counterexample, because a definition should cover all terms actual and possible.

4.1.2 Non-predicate vagueness Not only predicates are vague. There is an extensive literature on vague singular terms. Arguably many determiners are vague. And, as I’ll stress here, many predicate modifiers are vague. We can make an intuitive distinction between vague and precise predicate modifiers. Compare the following two (obviously artificial) predicate modifiers. (I owe these examples to David Chalmers.) Where F is a predicate such that Fa is true iff for some variable v, v(a) > x and v has a natural zero value (e.g. like height and unlike utility) then we can define doubly F and bigly F. It is true that a is doubly F iff v(a) > 2x and a is bigly F iff v(a)/x is big. Now there’s a good sense in which doubly is a precise modifier, for the modification it makes to its attached predicate can be precisely defined, while bigly is a vague modifier. That’s the sense in which I mean some modifiers are vague and others are precise. Note that even though doubly is precise it can be a constituent of a vague predicate, such as doubly tall. That makes sense; just as a vague sentence need only contain one vague word, so need a vague complex predicate need only contain one vague word. Now we might well ask whether natural language modifiers like very are vague or precise. I’m sad to say that I really don’t have an answer to that question, but I think it’s an excellent question. To get a sense of how hard it is, note one awkward feature of very —it is most comfortable attaching to words that are themselves vague. For instance (1a) is a sentence of English while (1b) is not. (1) a. Jack is very old. b. ∗ Jack is very forty-seven years old. I don’t know whether this is a universal feature of very. My best guess is that it is though in conversation some people have proposed interesting putative counterexamples. (I’m grateful here to Daniel Nolan.) But to avoid that complication, I’ll introduce a new word very∗ . This modifier is defined such that if F is vague then very∗ F means the same thing as very F, and if F is not vague then very∗ F is meaningless, like very forty-seven years old. It’s an excellent question whether very∗ is vague, and I think it’s a requirement on a definition of vagueness that it allow this question to be asked. As we’ll see, this is sadly not true of most proposed definitions of vagueness on the market.

4.1.3 Philosophically interesting vague terms It’s morally obligatory that someone with my standard of living donate 1% of their income to charity. It’s not morally obligatory that someone with my standard of living

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donate 100% of their income to charity. What is the largest x such that it’s morally obligatory that someone with my standard of living donate x% of their income to charity? (As a moralistic cheapskate I’d rather like to know.) Arguably this is vague. But perhaps only arguably. On some divine command theories it is precise, because there’s a fact about what God wants me to do, however hard this is to figure out. (It’s even a knowable fact, since God knows it.) But on more standard secular moral theories this may indeed be vague. There are two lessons to draw from this case. First, if two philosophers can debate what the correct theory of morality is while one thinks it is vague and the other thinks is precise, as I think could happen in a dispute between a divine command theorist and a virtue ethicist, then knowing that a vague term is vague is not required for understanding the term. (I assume here the divine command theorist is not so confused that she’s not really talking about goodness.) Second, it is important to remember that for some vague terms competent users of the term need not know in virtue of what they apply. Much of the literature on vagueness focuses on words like tall, thin and bald where all competent users know which kinds of underlying facts are relevant to their application. But not all vague terms are like that, as good illustrates. And this phenomena extends beyond the normative, at least narrowly conceived. If you believe Tom Wolfe (2000) then among the youth of America going out with is vague and many do not know exactly in virtue of what it applies. It’s a familiar point in philosophy of mind that competent users can disagree about what kinds of features a thing must have to satisfy is thinking. And we can multiply instances of this by considering any area of philosophy we like.

4.1.4 Discrete vague terms An academic with one child has few children for an academic. An academic with five children does not have few children for an academic. (I’ll omit the comparison class ‘for an academic’ from now on.) Where is the borderline between those with few children and those not with few children? (I don’t ask out of personal interest this time.) This question, like the question of how much giving is morally obligatory, feels vague. But note that we cannot generate a compelling Sorites paradox using has few children. Let’s see how badly this Sorites argument fails. (2) a. An academic with one child has few children. b. If an academic with one child has few children, then an academic with two children has few children. c. If an academic with two children has few children, then an academic with three children has few children. d. If an academic with three children has few children, then an academic with four children has few children. e. If an academic with four children has few children, then an academic with five children has few children. f. So an academic with five children has few children.

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Arguably premise e is plausible because as a material conditional it can be seen to be true via the falsity of the antecedent. And at a pinch I can see d as compellingly true for the same reason. But neither b nor c strike me as at all compelling. If someone presents this argument as a Sorites paradox, I simply deny that the paradox-mongerer can know these premises to be true, or that I have a reason to believe they are true. To be sure, I don’t know which premise is false. (If you think you know b to be false replace academics in the example with a more fertile professional group.) But just because I don’t know where the argument fails doesn’t mean it presents any kind of paradox. When I have no reason to accept two, maybe three, of the premises, the argument falls well short of being paradoxical. A small note on terminology. Contemporary scientific theories imply that many familiar vague predicates apply in virtue of facts about the world that are, at some level, discrete. What I’m interested in under this heading are predicates where the differences between salient adjacent cases are easily observable, such as the difference between having two and three children.

4.1.5 Vagueness without boundaries The letter of Patrick Greenough’s proposal (to be discussed in section 4.3 below) suggests that every vague term has only vague boundaries. This is not true. The predicate in one’s early thirties has a sharp boundary at the lower end and a precise boundary at the upper end. But it isn’t too hard to amend his theory to allow for such cases, by saying (in effect) that a vague term is a term with at least one vague boundary. Nicholas Smith makes basically that move in his paper. But such a move won’t work, because some vague predicates don’t have boundaries. Indeed, some predicates can be vague even though they are satisfied by every object in the domain. The examples here are a little more complicated than in the rest of the paper, but I think they are important enough to warrant the complexity. For the next several paragraphs the domain will be adult Australian women, and when I use tall I’ll mean tall for an adult Australian woman. I don’t know enough facts to know where the boundaries are for tall in this context, but I’ll stipulate that a woman shorter than 170 cm is determinately not tall, and a woman taller than 180 cm is determinately tall. I claim here neither that I know where these boundaries are nor that I could know where they are. But I assume there are boundaries. I’m making these stipulations because it is easier to follow the examples if I use 170 and 180 rather than variables like y and z. It will become obvious that the particular numbers won’t matter, as long as there’s separation between them. It also doesn’t matter whether we use a semantic or epistemic account of determinacy here. It will matter that we use classical logic at various points (e.g. in assuming there are boundaries), but I think that’s perfectly reasonable in this context. (Here I follow the arguments in section 2 of Greenough 2003.) Consider the class of predicates defined by the following schema. tall x =df tall or shorter than x cm

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For x < 170, tall x has all the same borderline cases as tall, and is presumably vague in anyone’s book. For x > 180, tall x determinately applies to everyone in the domain, and for now we’ll say that makes it not vague. ( Though note it need not determinately determinately apply to everyone in the domain, and we’ll see below that might be a reason to group it with the vague predicates.) When x is between 170 and 180, tall x has some very odd properties. The borderline cases are those women whose height is between x and 180 cm. When x is close to 180, this might be a very small border. While we’re assuming classical logic, we can assume that there is a value y such that women taller than y cm are tall and those shorter than y cm are not tall. We need not here assume the value of y is either epistemically or semantically determinate. Consider a value of x, say 179, such that x > y. (Again it’s not a necessary assumption that 179 > y, but it makes the example easier to understand if I use a particular number.) Now tall 179 has some interesting properties. It has borderline cases, those women between 179 and 180 cm tall. But it is satisfied by every woman, since every woman is either tall or shorter than 179 cm. I think the existence of the borderline cases is sufficient to make tall 179 vague. Note that these cases are quite different to child ∗ , because at the upper boundary there is no sharp jump from borderline cases to clear cases—the two blur together in just the way borderline cases and clear cases of tall blur together, so whatever reasons we had to worry about child ∗ being vague are not applicable here. Still the ‘borderline cases’ are mislabelled here for there is no border they fall on. Every woman satisfies the predicate. So no definition of vagueness in terms of having a vague boundary, indeed of having a boundary at all, can work. One might object here that a definition of vagueness is only meant to apply to words not phrases. But just as we can worry about a possible word child ∗ , we can worry about a possible atomic word gish that means the same thing as tall 179 , so that move won’t help here. We now have enough data on the table. In the next section I argue that taking vagueness to be indeterminacy provides a satisfactory treatment of the data. In the third section I argue that none of the live alternatives is so satisfactory. So I conclude, somewhat tentatively, that we should define vagueness as indeterminacy. 4.2

VAG U E N E S S A S I N D E T E R M I N AC Y

Back when I was a supervaluationist, I thought that what it was for a term to be vague was for it to refer to different things on different precisifications. That won’t do as a theory-neutral definition, for it presupposes supervaluationism, which is not only a theory but a false theory. But we can capture the essential idea in slightly less loaded language. I will have to make three possibly controversial assumptions. First, I assume a broadly Montagovian perspective, on which we can talk about the referent of an arbitrary term. (See Montague 1970, 1973 for more details.) That referent might be an object, or a truth value, or a function from objects to truth values, or a more complicated function built out of these. Second, I assume we can sensibly use an expanded

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Lagadonian language where objects can be names for themselves, truth values can be names for themselves, functions from objects to truth values can be names for themselves, and so on. (See Lewis 1986 for more on Lagadonian languages.) Third, I assume there is no metaphysical vagueness, so each of these Lagadonian names is not vague. Those assumptions let us make a first pass at a definition of vagueness, as follows. A term t is vague iff there is some object, truth value or function l which can serve as its own name such that the following sentence is neither determinately true nor determinately false. (3) t denotes l. That delivers the intuitively correct account in four of the five cases we discuss above, all except the cases like child ∗ . I’ll say much more about that case below. But it is in one respect slightly too liberal, and we need to make a small adjustment or two to fix this. Consider a predicate F that is defined over a vague domain, but which is determinately satisfied by every object in the domain. Intuitively it is a partial function, which maps every member of its domain to true. And assume for sake of argument that it is determinate that it maps every member of the domain to true. (Say, for example, it means is self-identical when applied to a member of the domain.) Such a predicate is not, I think, vague. But since it is indeterminate which partial function it denotes, the above theory suggests it is vague. We need to make a small adjustment. To state the corrected theory, we will stipulate that every term denotes a function. What were previously thought of as terms denoting constants will be treated as terms denoting constant functions. So instead of a name like Scott Soames denoting Scott Soames, we’ll take it to denote the function that takes anything whatsoever as input, and returning Scott Soames as output. Given that, our second take at a definition of vagueness is as follows. t is vague iff ∃x, y1 , y2 such that y1 = y2 and it is indeterminate whether ∃l such that t denotes l and l(x) = y1 , and it is indeterminate whether ∃l such that t denotes l and l(x) = y2 . To get a sense of the definition, it helps to translate it back into supervaluational talk, and look at the special case where t is a predicate. Then the definition comes to the claim that there is some object that is in the extension of t on one precisification, and in the anti-extension of t on another, which seems like what was intended. Arguably even that is not enough of a correction. (I’m indebted in the following three paragraphs to Mark Johnston.) Frequently there are debates in semantics over the appropriate type of various terms.¹ For instance, a straightforward account would say that in She ran yesterday, yesterday modifies the intransitive verb run, so it denotes a function of type <<e,t>, <e,t>>. But on a Davidsonian semantics, ¹ In what follows I’ll refer to functions of type <X, Y>. These are functions from things of type X to things of type Y, where the basic types are entities, represented by e, and truth values, represented by t. So a function of type <e, t> is a function from objects to truth values, or, equivalently, the characteristic function of a set. A function of type <<e, t>, <e, t>> is a function from characteristic functions of sets to characteristic functions of sets. This is plausibly the semantic value of a predicate modifier like very.

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yesterday denotes a property of the running event being discussed, so its type is simply <e,t>. Now it is at least a philosophical possibility that there should be no fact of the matter which of these theories is correct. There are two things we might say about such a possibility. On the one hand, it doesn’t at all seem right that a word should count as vague because it is indeterminate what its type should be. That suggests the above definition needs modification. On the other hand, the above definition doesn’t imply t is vague whenever there are two distinct functions that could be the denotation of t; it must also be the case that these functions have overlapping domains. The most natural cases of syntactic indeterminacy are cases where the two possible denotations are functions of quite different types. That suggests the above definition needs no modification. I think the case for modification is a little stronger. That’s partially because the possibility of type-shifting suggests there’s a possibility, perhaps a distant one but a possibility, that the second suggestion could fail. And partially because even if there are no uncontroversial cases of syntactic indeterminacy that will mistakenly be treated as cases of vagueness by this theory, the mere possibility of classifying a case of syntactic indeterminacy as a case of vagueness should be enough to warrant concern. And there is a way to modify the definition that does not look like it will lead to mistakenly ruling out any cases of vagueness that should be ruled in, as follows. t is vague iff ∃x, y1 , y2 such that y1 = y2 and y1 is of the same type as y2 , and it is indeterminate whether ∃l such that t denotes l and l(x) = y1 , and it is indeterminate whether ∃l such that t denotes l and l(x) = y2 . That implies that if yesterday is indeterminate merely because it is indeterminate what type of function it denotes, it won’t count as vague, and that’s all to the good. So this is our final definition of vagueness. Still there’s a problem with child ∗ . Many people have thought that it should not be considered vague for one reason or another. Sometimes this is just asserted as a raw intuition, as in Smith and Eklund. There’s no arguing with an intuition, so I won’t try arguing with it. Rather I’ll just repeat a point I made at the start. We aren’t here in the business simply of summarizing ordinary or philosophical intuitions. Rather we are looking for a definition that captures all the cases that fall into the most theoretically important categories. And intuitions about theoretical importance are less impressive than demonstrations of theoretical importance. Patrick Greenough (2003) suggests that the problem with terms like child ∗ is that they aren’t vague, but rather that they are simply undefined for the alleged borderline cases. If that’s true, and perhaps for some of the examples people had in mind in this area it is, then our definition agrees that they are not vague. For a term that carves a precise division out of part of the domain, and then stays silent, is precise not vague on my account. Greenough also suggests that the problem with child ∗ is that it is not higher-order vague.² But as he says this can hardly be the entirety of the problem. For it does not ² When I say a term is higher-order vague, I mean that it is subject to higher-order vagueness, not that it is vague whether the term is vague.

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seem to be definitional that the vague terms are also higher-order vague. True, there is a theoretically important category of terms that are vague and higher-order vague. But it is not a category that we cannot represent. A term t is in this category just in case t is vague, and definitely t is vague, and definitely definitely t is vague, and so on. So we can capture that category, even if we don’t call only members of that category the vague terms. And this doesn’t seem to diminish the theoretical importance of the category of terms I called vague. It might be thought that what is wrong with child ∗ is that it cannot be used to generate a Sorites argument. If you think that’s what is centrally important to vague terms, then there’s a theoretical reason to separate child ∗ from the genuinely vague. But we should have seen enough by now to show that that can’t be right. It’s hard to know what it is for a predicate modifier to be Sorites-susceptible, and our last two examples of vague predicates, has few children and tall 179 cannot be used to set up Sorites arguments. So that child ∗ does not generate a Sorites paradox is no reason to classify it outside the vague. So I take it there is no compelling reason to classify child ∗ and similar terms as precise rather than vague. Admittedly there is an intuition that they are not vague, and perhaps that should be respected. But if the cost of respecting that intuition is that we misclassify several other terms, we should reject the intuition. That’s what I’ll argue in the next section.

4.3

R I VA L D E F I N I T I O N S

I just mentioned the idea that a vague predicate could be defined as one that is susceptible to a Sorites argument. This account is sometimes attributed to Delia Graff Fara (2000), but it seems quite a widespread view. For instance, Terence Horgan (1995) says that it is distinctive of vague predicates that they can be used to generate inconsistency because the Sorites premises attaching to them are true. As I mentioned, such views are vulnerable to a wide variety of counterexamples. Many of these counterexamples also apply to rival definitions of vagueness. Matti Eklund (2005) develops a similar kind of definition. He starts with Crispin Wright’s (1975) famous definition of what it is for a predicate F to be tolerant. Whereas large enough differences in F ’s parameter of application sometimes matter to the justice with which it is applied, some small enough difference never thus matters. Eklund’s position then is that F is vague iff it is part of semantic competence with respect to F to be disposed to accept that F is tolerant. Eklund agrees that it is inconsistent to assert that F is indeed tolerant. But as he has argued extensively elsewhere, the falsity of the tolerance principle is compatible with it being part of competence that one is disposed to accept it. (A view in the same family is put forward in Sorensen 2001.) I have no wish to dispute this part of Eklund’s theory. Indeed that meaning principles can be false, even inconsistent, it seems to have been a fairly fruitful idea

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in a variety of areas of Eklund’s philosophy. But I don’t think it helps with vague terms. Three of the problems with this have already been given. It is not clear what a parameter of application for a non-predicate like very even is, so it isn’t clear what it means to say that very is tolerant. It surely is not required of competent users of few children that they are disposed to accept the premises in our earlier Sorites argument. And for some vague predicates, like tall 179 , the tolerance principle is not plausible to a competent speaker because it is not plausible that a ‘large enough’ difference in the parameter of application (presumably height) matters. These problems all seem to carry over from the problems associated with Sorites based definitions. I suspect, though I’m less certain here, that the philosophically interesting cases also pose a problem for Eklund’s view. When we look at philosophically interesting cases, like being good, there are two distinct ways to read Eklund’s claim that competent speakers are disposed to accept the tolerance principle. These are the wide scope and the narrow scope reading. To see the ambiguity, let’s write out Eklund’s principle in full. Competent speakers are disposed to accept that whereas large enough differences in F ’s parameter of application sometimes matter to the justice with which it is applied, some small enough difference never thus matters.

Here’s the wide scope reading of this. F ’s parameter of application is such that whereas competent speakers are disposed to accept that large enough differences in it sometimes matter to the justice with which F is applied, some small enough difference never thus matters.

And here is the narrow scope reading, with a phrase added for emphasis. Competent speakers are disposed to accept that whereas large enough differences in F ’s parameter of application, whatever it is, sometimes matter to the justice with which it is applied, some small enough difference never thus matters.

To see the difference between the two cases, assume for the sake of argument that a competent speaker thinks that to be good is to do actions whose consequences have a high enough utility, whereas in reality to be good is to obey enough of God’s commands. In each case being good is vague, because we are using satisficing versions of consequentialism and divine command theory. So the parameter of application for being good is the number of God’s commands you obey. The competent speaker will not accept the wide scope version of tolerance with respect to being good, because they don’t think that large differences with respect to how many of God’s commands you obey matter to the justice with which being good is applied. Such cases can be multiplied endlessly to show that the wide scope version of Eklund’s principle cannot generally be true, because it makes it the case that competent speakers have correct views on contentious philosophical matters the resolution of which goes beyond semantic competence. For these reasons Eklund has said (personal communication) that he intends the narrow scope version.

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But the narrow scope version also faces some difficulties. The most direct problem is that one can be a competent user of a term like food or dangerous or beautiful without having any thoughts about parameters of application. I suspect I was a competent user of these terms before I even had the concept of a parameter of application. Even bracketing this concern, there is a worry that competence requires knowing of a term whether it is vague or not. But this seems to be a mistake. It is not a requirement of competence with moral terms like good that one know whether they are maximizing or satisficing terms. Tom Wolfe and the students he observed while writing I Am Charlotte Simmons seemed to disagree about whether going out with is vague, but they were both competent users, they simply disagreed on something like a normative question. (See Wolfe 2000 for more on his take on matters.) And it seems that two users of language could disagree over whether is thinking is vague without disagreeing over whether either is a competent semanticist. They may well disagree over whether either is a competent philosopher of mind, but such disagreements are neither here nor there with respect to our present purposes. So I don’t think that either disambiguation of Eklund’s principle can properly account for vagueness in philosophically interesting terms. Nicholas Smith argues for a definition of vagueness that uses some heavier duty assumptions about the foundations of semantics. In particular, he sets out the following definition, Closeness If a and b are very similar in F -relevant respects, then ‘Fa’ and ‘Fb’ are very similar in respect of truth. and goes on to say that vague predicates are those that satisfy non-vacuously satisfy Closeness over some part of their domain. For this to work there must be, as Smith acknowledges, both degrees of truth and something like a distance metric defined on them. ( These are separate assumptions; in the theory of Weatherson 2005 the first is true but not the second.) I won’t question those assumptions, but rather focus on the problems the definition has even granting the assumptions. As with the two definitions considered so far, it is hard to see how this could possibly be generalized to cover vagueness in non-predicates. It’s true (given our assumptions) that if a and b are similar in very tall-relevant respects, then ‘a is very tall’ and ‘b is very tall’ will be similar in respect of truth. But that doesn’t show very is vague, for the same condition is satisfied when we replace very with the precise modifier doubly. This isn’t an argument that Smith’s definition couldn’t be extended to cover modifiers, but a claim that it is hard to see how this will work. The definition also has trouble with tall 179 for this satisfies Closeness vacuously. Though, to be fair, given the logical assumptions Smith makes, it is possible that no predicate with the properties I’ve associated with tall 179 can be defined. More seriously, there is a problem with predicates like has few children. It just isn’t true that ‘An academic with two children has few children’ is close in truth value to ‘An academic with one child has few children’. In general Smith’s theory has trouble with, i.e., rules out by definition, vague terms where the underlying ‘relevant respects’ are highly discrete. Note that the problem here extends to some predicates where the underlying facts are continuous. Consider the predicate is very late for the meeting. At

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least where I come from, a person who is roughly ten minutes late is a borderline case of this predicate. But which side of ten minutes late they are matters. (In what follows I make some wild guesses about how numerical degrees of truth, which aren’t part of my preferred theory, should operate. But I think the guesses are defensible given the empirical data.) If Alice is nine and three-quarters minutes late, and Bob is ten and a quarter minutes late, then the degree of truth of ‘Alice is very late’ will be much smaller than the degree of truth of ‘Bob is very late’. The later you are the truer ‘you are very late’ gets, but crossing conventionally salient barriers, like the ten minutes barrier, matter much more to the degree of truth than crossing other barriers like the nine minutes thirty-three seconds barrier. Smith (in conversation) has suggested that he’s prepared to accept that is very late for the meeting is only partially vague if the truth values ‘jump’ at the ten minute mark as I’m suggesting. But this seems improper, for this is as clear a case of a vague predicate as we have. Still, it’s worth remembering as always that every definition has its costs, and this may be a cost one chooses to live with. Personally I think it is excessive. Patrick Greenough did not put forward his theory as a definition of vagueness, but rather as a minimal theory to which all partisans could agree. Like Eklund, Greenough plays off Crispin Wright’s idea of tolerance. Roughly, a vague predicate is one that is epistemically tolerant—it’s one where you can’t know that a small difference makes a difference. Here’s a less rough statement of it, though note this is heavily paraphrased. Let τ be a variable that ranges over truth states (e.g. true, determinately true, not determinately determinately not determinately true, etc.) v a function from objects to real numbers such that whether x is F depends only on the value of v(x) (i.e. v is F ’s parameter of application) and c a suitably small number. Then F is vague iff the following claim non-vacuously holds. ∀τ ∀α∀β∀a∀b, if |v(α) − v(β)| < c and a names α and b names β and it is knowable that Fa is τ then it is not knowable that Fb is not τ . Less formally, we can’t know where any boundary at any order of definiteness for F lies. (It isn’t clear in Greenough’s presentation exactly what the non-vacuous condition comes to. He only explicitly says that for the special case where τ is is true there must be an a and a b such it is knowable that Fa is τ and Fb is not τ , but maybe that should be extended to all τ .) Because of cases like in one’s early thirties this cannot do as a general definition, but it is easy enough to repair it by restricting the quantifier attaching to a and b to a range over which F has only vague boundaries. Doing this amounts to weakening Greenough’s claim from the view that vague terms have only vague boundaries to the view that they have some vague boundaries, which seems plausible. But still there are problems. Most obviously, tall 179 does not non-vacuously satisfy the tolerance requirement. And like all the tolerance-based theories it is far from clear how it should be extended to vagueness in non-predicates. On the other hand, Greenough’s theory might well handle the discrete cases like has few children. I say might rather than does because it is rather hard to work out how the higher-orders of vagueness behave for such terms. I’ll simply note that there are some plausible enough epistemic models on which has few children satisfies his requirement.

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There is a problem which is distinctive to Greenough’s view of his theory as a minimal theory. As Smith notes, Greenough makes it a requirement that vague boundaries are unknown. But this is controverted in some mainstream theories, for example the version of supervaluationism in Dorr (2003). Since Dorr’s theory should not be ruled out by a minimal theory or a definition, this is a weakness in Greenough’s theory. The more philosophically interesting problems concern, appropriately enough, the philosophically interesting terms. Greenough has a proof that his definition is equivalent to a definition in terms of borderline cases. The proof has several assumptions, one of which being that we know what the parameter of application of a vague term is. More precisely, he assumes that we know everyone older than an old person is old, which is unproblematic, but he also assumes that the proof generalizes to all vague cases, and this amounts to the assumption that we know parameters of application. As we’ve seen, this isn’t true of philosophically interesting vague terms. This leaves open the possibility that Greenough’s theory, unlike Smith’s and Eklund’s theories, overgenerates. The following is probably not a live possibility in any interesting sense, but it isn’t I think the kind of thing a definition (or minimal theory) should rule out by definition. It is possible that a kind of mysterianism about ethics is true, and we cannot know whether good is vague or precise. For a concrete example, let’s assume it is knowable that some kind of divine command theory is true, but it is unknowable whether to be good one must obey all of God’s commands or merely enough of them, where it is vague what counts as enough of them. In fact morality requires obeying all God’s commands, but this is not knowable—for all we know the satisficing version is the true moral theory. If this is the case then good will be epistemically tolerant, for we cannot know that a small difference in how many of God’s commands you obey makes a difference to whether you are good, or determinately good etc. But in fact good is precise, for it precisely means obeying all of God’s commands. Earlier I objected to Eklund’s theory because semantic competence does not require knowing parameters of application, especially as such. This is the converse objection—I claim that a term’s being precise does not imply that we know, or even could know, that it applies in virtue of a precise condition. All that matters is that it does apply in virtue of a precise condition. It’s a constant danger in philosophy that one infer from the falsity of all extant rivals that one’s preferred theory is correct. I certainly don’t want to argue that because Eklund’s, Smith’s and Greenough’s definitions are incorrect that the traditionalist definition I have offered must be right. But we can make that conclusion more plausible by noting how widely the arguments levelled here generalize. The philosophically interesting cases seem to tell against any definition of vagueness in terms of semantic competence, for they show that competent users can have exactly the same attitude towards vague terms as they have towards precise terms. And our moral example suggests that any definition in terms of epistemic properties will be in trouble for it might not be knowable whether a particular term is vague or precise. Finally, the cases of vague predicate modifiers raise difficulties for any attempt to define the vagueness of a term in terms of properties of sentences in which it is used rather than mentioned.

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For it seems that as long as very∗ attaches only to vague predicates, then whether very∗ is vague or precise will make no salient differences to the sentences in which it appears. So we have to look at sentences in which the allegedly vague term is mentioned. And while I don’t have a definitive argument here, I think looking at the range of cases we want to cover, and in particular at the range of cases where tolerance-type principles fail to be non-vacuously satisfied, our best option for completing these sentences is to look whether the term has a determinate or indeterminate denotation. We can then pass the questions of what determinacy consists in, and in particular the question of whether it is an epistemic or semantic feature, to the theorist of vagueness. Re f e re n c e s Dorr, Cian (2003), ‘Vagueness without ignorance’, Philosophical Perspectives 17, 83–113. Eklund, Matti (2005), ‘What vagueness consists in’, Philosophical Studies 125, (1), 27–60. Field, Hartry (1973), ‘Theory change and the indeterminacy of Reference’, Journal of Philosophy 70, (14), 462–81. Fine, Kit (1975), ‘Vagueness, truth and logic’, Synthese 30, (3–4), 265–300. Fara, Delia Graff (2000), ‘Shifting sands: An interest-relative theory of vagueness’, Philosophical Topics 28, (1), 45–81. [This paper was published under the name ‘Delia Graff ’.] Greenough, Patrick (2003), ‘Vagueness: A minimal theory’, Mind 112, (446), 235–81. Horgan, Terrence (1995), ‘Transvaluationism: A Dionysian approach to vagueness’, Southern Journal of Philosophy 33, Spindel Conference Supplement, 97–125. Lewis, David (1986), On the Plurality of Worlds, Blackwell, Oxford. Montague, Richard (1970), ‘Universal grammar’, Theoria 36, (3), 373–98. (1973), ‘The proper treatment of quantification in ordinary English’, in K. J. J. Hintikka, J. M. E. Moravcsik, and P. Suppes, eds., Approaches to Natural Language Reidel, Dordrecht 221–42. (1974), Formal Philosophy. Selected Papers of Richard Montague, edited and with an introduction by Richmond H. Thomason. New Haven/London: Yale University Press. Sainsbury, Mark (1991), ‘Is there higher-order vagueness?’, Philosophical Quarterly 41, (163), 167–82. Smith, Nicholas J. J. (2005), ‘Vagueness as closeness’, Australasian Journal of Philosophy 83, (2), 157–83. Sorensen, Roy (2001), Vagueness and Contradiction, Oxford University Press, Oxford. Weatherson, Brian (2005), ‘True, truer, truest’, Philosophical Studies 123, (1–2), 47–70. Wolfe, Tom (2000), Hooking Up, Farrar, Strauss and Giroux, New York. Wright, Crispin (1975), ‘On the coherence of vague predicates’, Synthese 30, (3–4), 325–65.

5 Sorensen on Vagueness and Contradiction Dorothy Edgington

This chapter explores some themes from Roy Sorensen’s fascinating book, Vagueness and Contradiction (2001). It is an expanded version of my contribution to a symposium on the book (Edgington 2005). I discuss some of the further elucidation and replies to critics in Sorensen’s contribution to that symposium (Sorensen 2005a, 2005b), and take up some further questions raised by his work. 5.1 One second after noon is noonish. 10,000 seconds after noon is not noonish (it is after 2.30 pm). So the generalization ‘For all n, if n seconds after noon is noonish, n + 1 seconds after noon is noonish’ must be false. So its negation must be true: for some n, n seconds after noon is noonish, and n + 1 seconds after noon is not. That is, there is a last noonish second; though we don’t—can’t—know which the last noonish second is. Sorensen takes this to be the only legitimate response to a sorites argument: the induction step is false. There is a sharp boundary to the concept ‘noonish’, but it is epistemically inaccessible to us (see 1; and 2: ‘Therefore, amazingly, all vague predicates have sharp boundaries’).¹ The real problem, says Sorensen, is to solve the meta-paradox: why so many intelligent, philosophically and logically acute people do not accept this simple and unassailable response—why even he himself, after seventeen years of propounding it, continues to suffer cognitive dissonance about it. I agree with much of the opening argument. I agree that the induction step is false, that is, its negation is true. There is just one word I would challenge, and that is the word ‘sharp’.² Not all boundaries are sharp. ‘The boundary between A and B is not sharp’ (where A and B are adjacent lands, say) is not an oxymoron. Consider a location l somewhere in this vague boundary, such that it is not clear (definite, determinate) whether it is in A or in B. l is clearly in A or B (there are no other alternatives), ¹ Page references are to Sorensen (2001) when not otherwise attributed. ² Sorensen reiterates his ‘commitment to sharp boundaries for vague terms’ in his pr´ecis of the book (2005a, 681).

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but it is not clear which. So l is in some land, but it’s not clear which. Similarly, the boundary is not sharp between the noonish, and the later-than-noonish. There are times which are not clearly on one side or the other. There is—there must be—a last noonish second, but it is not clear which that is. There is no second which is clearly the last. Of course, it may turn out on further reflection that an epistemic reading of ‘clearly’ is the only viable one, or the most viable one. But that is, at this stage, an open question. There are theories of a variety of kinds which accept the above reasoning: supervaluationism (Kit Fine 1975; Rosanna Keefe 2000); the closely related theory developed by Vann McGee and Brian McLaughlin (1995); the sort of degree theory I have proposed (Edgington 1996); and a theory I dub ‘primitivist’ (David Barnett, 2009, which I discuss in §6). There are, of course, approaches to vagueness which reject the above reasoning; and I do not claim that it is obviously correct. Given the amount of philosophical effort devoted to vagueness in recent years, it seems unlikely that any theory is obviously correct, rather than, on balance, best. My point here is that rejecting the induction step as false does not immediately commit one to sharp boundaries of which we are ignorant. 5.2 As mentioned, Sorensen takes one of his tasks to be to defuse the meta-paradox: why is the simple solution to the sorites—that the induction principle is false—so hard to accept? Why is it so tempting to take it to be true? He points to other instances of this kind: generalizations which are false, but almost true in the sense that they have very few exceptions (perhaps only one), for example, what he calls ‘anti-superlatives’ like ‘no matter how rich you are, there’s always someone richer’ (mutatis mutandis for smarter, taller, stronger, . . . ). He had acquired a fondness for wise sayings of this form, until a friend refuted him by giving him a copy of the Guinness Book of Records. Except in an infinite domain, there is at any time someone such that there is no one richer (taller, etc.). The anti-superlatives are false; but pragmatically, we do better to treat these falsehoods as truths, thereby capturing the vast predominance of their true instances over their false instance, and without having to enter into the messy business of assigning probabilities to an arbitrary instance’s being true. And the induction step in a sorites—for instance ‘For all n, if n seconds after noon is noonish, n + 1 seconds after noon is noonish’—he takes to be an especially deep-seated and hard-wired instance of this phenomenon, helped by the fact that we cannot produce the counterexample. (Unfortunately the Guinness Book of Records doesn’t inform us about the richest poor man, the tallest short man, the reddest orange patch, etc.). He also draws analogies with perceptual illusions to which we cannot but succumb, even when we know that they are illusory, and with ‘impossible figures’ as seen in Escher drawings. Similarly, we have ‘illusions’ of boundarylessness to which we cannot but succumb, and which usefully prevent us from wasting time looking for boundaries which are inaccessible to us.

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This thesis of the naturalness, indeed inevitability of the error is developed in a very strong form: There is a clear sense in which our beliefs are massively contradictory. The contradictions are inescapable because they issue from linguistic competence itself. . . . All speakers are inconsistent by virtue of their mastery of a natural language. . . . When predicates are vague, we fall under the command of a simplifying epistemology. . . . We are forced to accept some contradictions as analytic truths. . . . We are forced to accept a wide plume of contradictions. (57)

Consider the ‘noonish’ sorites again, this time in its long form, with 9,999 conditional premises, ‘If 10 seconds after noon is noonish, 11 seconds after noon is noonish’, etc. Premise 0, ‘One second after noon is noonish’ is an analytic truth, he tells us, and the conclusion, ‘10,000 seconds after noon is noonish’, is an analytic falsehood. ‘Therefore’, says Sorensen, ‘there must be a conditional in the chain that has an analytically true antecedent and an analytically false consequent’ (58). This is a non-sequitur. Any chain-argument of this form with an analytically true intitial premise and an analytically false conclusion must have a false premise, i.e. a premise whose antecedent is true and conclusion false. But it may have only a synthetic, contingently false premise. Note that the conditionals in question are material conditionals—Sorensen is explicit about this, and that is all they need be to generate the sorites. Let ‘1 = 1’ and ‘1 = 0’ be the analytic truth and falsehood. Consider: (0) (1) (2) (3)

1 = 1; 1 = 1 ⊃ it’s raining; It’s raining ⊃ 1 = 0; 1 = 0.

(0) and (3) are, respectively, true and false, and (by assumption) analytically so. The argument is valid. So one of the premises must be false. But which premise is false depends on whether or not it is raining. If it is, (1) is true and (2) is false. If it is not, (1) is false and (2) is true. And concerning the sorites: anyone with any time for the notion of analyticity will agree that ‘One second after noon is noonish’ is analytically true and ‘10,000 seconds after noon is noonish’ analytically false: anyone with the concept ‘noonish’ (and a little arithmetic) is capable of recognizing these statements as true and false respectively.³ But there is little intuitive appeal to the idea that a borderline statement, say, ‘12.23 pm is noonish’ is either analytically true or analytically false. Many will hold that our concept of noonish neither clearly dictates that it is true, nor clearly dictates that it is false. It is certainly not analytic in the epistemological sense of that notion, according to which a statement is analytic if understanding it is sufficient for recognizing its truth—for we understand it yet cannot recognize whether it is ³ At a stretch, there may be some reason to doubt even this: consider an experiment in astronomy requiring extremely accurate measurement of time. One second after noon may be too late to be noonish in this context. And conversely, extremely lax standards might be in operation. Nevertheless, anyone with the concept ‘noonish’ is capable of recognizing, relative to a context, the truth values of these statements when they are not borderline.

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true or false. I shall return to the question of the status of borderline statements in Section 5.3. For the moment, let us accept Sorensen’s assumption. Call the conditional with the analytically true antecedent and analytically false consequent X. The argument consists entirely of analytical truths and X, which is an analytic falsehood. All of the premises seem analytically true, he tells us, because they manifest our understanding that ‘noonish’ is tolerant.⁴ This is not just a psychological fact. It is normative. To be inducted into the language is to submit to conditionals like these. We have an obligation to believe them. If you wish to use ‘noonish’, you must cooperate by ignoring insignificant differences. But we are doomed to violate our obligation because our obligations are not jointly satisfiable (58). The rules of language are like an inconsistent moral code: to be a competent language user, you need to be inconsistent! Tolerance conditionals are worthy of assent, on a priori grounds. You could avoid believing X by not believing any such conditional. But this would mean sacrificing a huge number of true beliefs to prevent a single false belief. Conditional X appropriately resembles a bunch of analytically true statements. Such appropriate resemblance is a source of entitlement. X is believed on a priori grounds although it is analytically false. (One should be fallibilist, he argues, about a priori warrant: that one warrantedly believes on a priori grounds that p, does not entail that p. So we can have, for a given domain of propositions: (∀p) it is a priori that p; yet it is a priori that not for all p, p.) Each speaker has a right and a duty to ignore insignificant differences. This makes each step of the sorites individually a priori. Yet the negation of the conclusion is also a priori. Hence our a priori beliefs are jointly inconsistent. The rules of language command assent to propositions that cannot be true in conjunction (62–5). Now, as a psychological story of the appeal of tolerance principles, this rings true; and I am willing to concede some low-level normative force: for the sake of quick and simple thinking, ignore sufficiently small differences when, as is almost always the case, doing so leads to no trouble—no danger of being led down slippery slopes. But I think Sorensen exaggerates, suggesting we can do no better. We can and should do better when careful thinking is required if we are not to be led astray. The structure of the phenomenon is well known: that of the lottery paradox and the paradox of the preface. This structure is made perspicuous in terms of a framework which allows one’s epistemic attitudes to come in varying degrees of closeness to certainty. If belief entails certainty, one cannot consistently believe each of p1 , p2 , . . . pn yet disbelieve their conjunction. If belief does not entail certainty, but only a high degree of confidence which may be less than certainty, then there is, by one’s own lights, more risk of being wrong about a conjunction than there is about either conjunct (provided neither conjunct entails the other). If you are very close to certain, and ⁴ A predicate is said to be tolerant if small enough differences along the relevant dimension of variation do not affect whether it applies. The induction step is sometimes called the tolerance principle, and the related conditionals tolerance conditionals. Hence the claim that vague predicates are tolerant amounts to acceptance of the induction step. This use of ‘tolerance’ is due to Crispin Wright (1975).

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there are only a few conjuncts, say two, this additional risk can harmlessly be ignored. Idealizing with numbers to make the structure clear: if you are at least 99% certain that p and at least 99% certain that q, this entitles you to be at least 98% certain that p & q.⁵ In that sense it is approximately true to say that if you believe that p and believe that q, you should believe p & q. But each added conjunct that is less than certain (and not entailed by the others) increases the risk of error. If belief does not entail certainty, then one can, without cognitive dissonance, believe each of p1 , . . . pn but disbelieve their conjunction. And, from an epistemicist standpoint, that should be your attitude to the tolerance conditionals. One of them is false. You don’t—can’t—know which. Each individually deserves a high degree of confidence, a conjunction of any two, a slightly lower degree of confidence; and so on. Sorensen never considers the fact that epistemic attitudes come in different degrees of closeness to certainty, and that probability theory provides a good, idealized account of the structure of these epistemic attitudes. About the preface paradox he says ‘Theoretically, authors can escape by not believing anything they write. They can just assign probabilities’ (67). But on one way of understanding the matter, this is not ‘not believing anything they write’! They can be extremely confident of everything they write. But if they are not absolutely certain of each thing they write, they can consistently be pretty certain that an error will have slipped in somewhere. The same applies to his remark, ‘People can make a special effort not to believe the tolerance conditionals. . . . They can merely assign a high probability to each tolerance conditional’ (69). Sorensen seems to assume that assigning probabilities is changing the subject-matter of what is believed. But a venerable tradition takes one application of the concept of probability to be a measure of the strength of one’s epistemic attitude, rather than part of the content of what is believed. For most purposes, we can happily ignore the fact that the statements in question are not quite certain. They are close enough, and it won’t get us into trouble to classify them simply as beliefs (which they are). But in the special pathological contexts of the preface, or the sorites construed epistemically, it is this structural fact about uncertain beliefs which saves us from any pernicious inconsistency, and gives us a perspicuous account of rational epistemic attitudes.

5.3 Sorensen’s epistemicism differs from Timothy Williamson’s (1994). Williamson maintains that the threshold of a vague predicate is determined by its usage in the linguistic community and the environment: no difference in threshold without a ⁵ The probability that ‘p & q’ is false is the probability that ‘¬p ∨ ¬q’ is true. This cannot be greater than the sum of the probabilities of ¬p and of ¬q, i.e. the sum of the probabilities of falsity of p and of q. (Of course, it also cannot be greater than 1: the upper limit for ‘¬p ∨ ¬q’ is either 1 or the sum of the values of ¬p and ¬q, whichever is lowest. In the interesting case where the probabilities of ¬p and ¬q are both small, it is the sum of their probabilities which is the operative upper limit.)

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difference in use; but the matter is so complex that it is far beyond our cognitive capacities to discover the threshold. Beings of vastly superior cognitive power could perhaps do so. An omniscient being would of course know the threshold. Sorensen does not accept the supervenience of the threshold on use. He cites with approval an instance of the common complaint that it is incredible: We cannot imagine why it should be the case that any particular candidate for precise boundaries for the term ‘heap’, as opposed to numerous other candidates, would be the unique boundaries that actually supervene on the total use-pattern governing this term. . . . Even given some specific overall use-pattern as the total supervenience base, it seems there would still be numerous equally good candidates for precise boundaries, and there would be no reason whatever why one of these candidates should win out over the others. ( Terence Horgan, 1997, 231–2, cited by Sorensen, 2001, 178)

For Williamson the thresholds are unknowable for beings like us. For Sorensen they are absolutely unknowable by any being whatsoever. ‘I do not make an exception of God’, he says. ‘If God exists, God is all knowing. But then God would know whether predicates yield truths when applied to their absolute borderline cases. That is impossible!’ (2005a, 680).⁶ The metaphysical notion Sorensen introduces to explain absolute borderline cases is that of a truthmaker gap. To motivate the idea, he considers what he calls the ‘no–no paradox’. On each side of a piece of paper is written ‘The statement on the other side of this paper is false’. This is not a liar paradox (it would be if we changed one of the occurrences of ‘false’ to ‘true’). We can consistently assign truth values to these statements. Indeed they are jointly consistent provided they are given opposite truth values. It is nevertheless embarrassing that there are two distinct consistent assignments, one sentence true and the other false, with nothing to choose between them (in this respect this case resembles the truth-teller paradox, ‘This sentence is true’). It is further embarrassing that two tokens of the same sentence-type, symmetrically referring to each other, must differ in truth value. No relief is gained by declaring them neither true nor false. For we could raise the same problem by considering the pair ‘The statement on the other side is not true’. If they both lack truth values, they are both true. So they can’t both lack truth values. But if one of them has a truth value, the other has the opposite truth value, as before. According to Sorensen, we have with the no–no paradox an under-specific truthmaker. The two no–no sentences make true ‘Exactly one of the sentences is true’; but neither one of these sentences has a truthmaker. He gives a larger-scale version of the same phenomenon, a list of 100 sentences, each of which reads ‘Some other sentence in this list is false’. To these sentences we may consistently assign 99 Ts and one F in a hundred different ways. (Suppose the first sentence is false. Then all the others are true. Suppose the first sentence is true. Then there must be a false sentence somewhere on the list, in which case all those apart from it are true.) ⁶ Anyone who wished to, could make an exception of God, it seems to me: God just knows, without there being any story to be told of how.

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( There is something rather charming about this list: it reminds me of NIMBY— Not in My Back Yard. Yes, we must have another reform school, or rubbish dump, but it mustn’t be HERE. The hundred regions express this opinion, and one of them must be wrong. And one can see echoes of this structure in the sorites. Whenever you focus on a particular adjacent pair—of colour patches say—you think ‘There’s got to be a cut-off point somewhere, but it certainly isn’t HERE’.) A further example is added for fun: he bought a package containing two toothbrushes, enticed by the offer printed on the package: ‘Buy one, get one free’. Which toothbrush did he buy, and which did he get free? Had the offer been two for the price of one, it would have been financially equivalent. ‘But’, he says, ‘[the two styles of offer] differ metaphysically, not just psychologically. When I purchase two for the price of one, I do not get anything for free. Those who believe that each contingent truth has a truthmaker are forced to conclude that the typical ‘‘Buy one, get one free’’ offer is false. The believer in truthmaker gaps has the option of concluding that one was bought and one was free even though there is no telling which was which’ (183).⁷ The no–no paradoxes are his model for the sorites.⁸ Nothing makes some particular second the last noonish second, any more than anything makes some particular sentence on our list the unique false one. There is no truthmaker. A threshold just groundlessly exists. It is absolutely unknowable. There are ‘epistemic islands’ to which we have no access. (Not to ignore higher-order vagueness, he adds ‘Unlike the previously mentioned islands, the borders of this epistemic island are vague. The epistemic island is surrounded by accessible propositions and there is no way of telling where they stop and the island begins’ (177).) Sorensen does not give us a fully worked-out theory of truthmakers. He cites David Lewis’s formulation of what Lewis took to be the minimal insight behind the demand for truthmakers: